onQM

Ontology & Quantum Mechanics

2099-07-01T21:31:00.011-04:00

This blog is an experimental way to discuss topics in philosophy of physics, especially interpretation of quantum mechanics (QM), and some philosophy of mind.

I tend to support the many worlds interpretation (MWI) of QM, and computationalist philosophy of mind. However, I try to be objective and lay out the difficulties clearly.

I welcome comments and criticism. However, if you think that mainstream physics is nonsense and that you are a lone genius, please go elsewhere until you learn some physics.

Jacques Mallah, Ph.D. (jackmallah@yahoo.com)

Table of Posts:
Ontology & Quantum Mechanics

Chapter 1: Basics of Quantum Mechanics

1.1. Simple proof of Bell's Theorem
1.2. Why MWI?
1.3. Top 12 things to know about physics
1.4. on external links
1.5. Studying Quantum Mechanics: the Delayed Choice example
1.6. Key definitions for QM: Part 1
1.7. Key definitions for QM: Part 2
1.8. Key definitions for QM: Part 3
1.9. Studying Quantum Mechanics: Measurement and Conservation Laws
1.10 Studying Quantum Mechanics: Decoherence and Macroscopic Superpositions
1.11. Further Study

Chapter 2: Probability in Many Worlds Interpretations

I. Interlude: Anticipating the 2007 Many Worlds conference
II. Interlude: The 2007 Perimeter Institute conference Many Worlds @ 50

2.1. Meaning of Probability in an MWI
2.2. Measure of Consciousness versus Probability
2.3. Why 'Quantum Immortality' is false
2.4. Early attempts to derive the Born Rule in the MWI
2.5. Decision Theory & other approaches to the MWI Born Rule problem, 1999-2009
2.6. MWI proposals that include modifications of physics
2.7. The Computationalist approach to Measure
2.8. Dualism and hidden variables

2.9. The Everything Hypothesis: Its Predictions and Problems
III. Interlude: The Everything-List

Chapter 3: Making Computationalism Precise
3.1. Basic idea of an implementation
3.2. The Putnam-Searle-Chalmers theorem
3.3. Restrictions on mappings 1: Independence and Inheritance
3.4. Restrictions on mappings 2: Transference
3.5. Watch only the Road: Activity vs. Computation

IV. Interlude: The Partial Brain thought experiment

V. Interlude: Indiscernibles are Not Identical

3.6. Counting implementations: The Problem of Size
3.7. Linear dynamics, independence, & noise
3.8. Counting implementations: proposal 1: Substate-style criterion
3.9. Counting implementations: proposal 2: Just allow any starting states
3.10. Counting implementations: proposal 3: Use different physical ranges
3.11. Implications for quantum mechanics
3.12. Possible changes to the model of physics
3.13. The problem of Boltzmann Brains

VI. Interlude: Ideas on quantum gravity

3.14. Implications for artificial intelligence

Interlude: The Partial Brain thought experiment

2011-12-29T14:58:00.003-05:00

In the previous post, the case was made that it is not plausible that unused components of a system would affect consciousness, even though they can affect the counterfactual behavior of the system. This problem was dealt with by generalizing computationalism such that consciousness only depends on causal chains instead of on whole computations. Causal chains involve counterfactuals, but only "local" ones that affect the behavior of "active" components that affect the formal states that the system's state is mapped to during the actual run.

Mark Bishop's 2002 paper "Counterfactuals Cannot Count" attacked the use of counterfactuals in computationalism using a different kind of argument: a neural replacement scenario, in which the components (e.g. neurons) of a brain (which may already be an artificial neural net) are replaced one at a time with components that merely pass through a predetermined sequence of states that happens to be the same sequence that the computational components would have passed through. Eventually, the whole brain is replaced with the equivalent of a player piano, unable to perform nontrivial computations.

There is a long history in philosophy of mind of using the neural replacement thought experiment to argue that various factors (such as what the components are made of) can't affect conscious experiences. For example, David Chalmers used it as an argument for computationalism. It works like this: the rest of the brain can't have any reaction to the replacement process, since by definition the replaced components provide the same signals to the rest of the brain. It's been argued that it doesn't seem plausible that the brain could be very much mistaken about its own experiences, so a gradual change in or vanishing of consciousness is taken to be implausible. A sudden change isn't plausible either, since there's no reason why a particular threshold of how far the replacement has gone should be singled out.

Bishop's argument is really no different from other neural replacement thought experiments, except in the radical (to a computationalist) nature of its conclusions. So, if neural replacement thought experiments do establish that consciousness must be invariant in these scenarios, then computationalism must be rejected.

My Partial Brain thought experiment shows that neural replacement thought experiments are completely worthless. It works like this: Instead of replacing the components of the brain with (whatever), just remove them, but provide the same inputs to the remainder of the brain as the missing components would have provided.

What would it be like to be such a partial brain? Some important features seem obvious: it is not plausible that as we let the partial brain decrease in size, consciousness would vanish suddenly. But now it's not even possible (unlike in neural replacement scenarios) that consciousness will remain unchanged; it must vanish when the removal of the brain is complete.

Therefore, progressively less and less of its consciousness will remain. In a sense it can't notice this - it beliefs will disappear as certain parts of the brain vanish, but they won't otherwise change - but that just means its beliefs will become more wrong until they vanish. For example, if the higher order belief center remains intact but the visual system is gone, the partial brain will believe it is experiencing vision but will in fact not be.

The same things would happen - by definition - in any neural replacement scenario in which the new components don't support consciousness; the remaining brain would have partial consciousness. So neural replacement scenarios can't show us anything about what sorts of components would support consciousness.

The partial brain thought experiment also shows that consciousness isn't a unified whole. It also illustrates that the brain can indeed be wrong about its own conscious experiences; for example, just because a brain is sure that it has qualitative experiences of color, that is not strong evidence in favor of the idea that it actually does, since a partial brain with higher-order thoughts about color but no visual system would be just as sure that it does.

Restrictions on mappings 2: Transference

2011-12-14T14:10:00.006-05:00

In the previous post, Restrictions on mappings 1: Independence and Inheritance, the "inheritance" of structured state labels was explained; it allows the same group of underlying variables to be mapped to more than one independent formal variable. In the example a function on a 2-d grid was mapped to a pair of variables.

Transference is something like the reverse process: It allows a set of simpler variables to be mapped to a structured state function on a grid.

This allows ordinary digital computers to implement wave dynamics on a 3-d spaces, which could matter for the question of whether the universe could be ultimately digital. The AdS/CFT correspondence in some models of string theory would need something similar if the bulk model is to be implemented on the boundary in the computational sense.

Transference can be Direct or Indirect. It works like this:

Direct Transference could be used in a mapping by taking the value from a given variable and turning it into a label for structuring a set of new variables.

For example, if there is a single integer variable I(t), we can transfer its value to label to a set of bits B(j) which each only depend on whether I(t) equals the value of its label, e.g.

B(j) = 1 if I = j
B(j) = 0 if I does not = j

These bits can be considered an ordered series of "occupation tests" of the different regions that the underlying variable's value could be in.

Of course, only one of these bits at a time will be nonzero. But they are to be considered independent variables. At this point you might object: If you know the value of the nonzero one, don't you know the other bit values must be zero? But just as Inheritance carved out an exception to the rule for independence, so would Direct Transference carve out an exception to it.

Going the other direction is no problem: If we restrict a mapping such that only one bit in an ordered set B(i) is nonzero, then a new variable I can be constructed such that I has a value equal to the index i of the nonzero bit. Here we are doing the reverse.

We can't double count, though; if we make the new set of variables b(i), we can't make a second independent new set of variables c(i) which gets its label transferred from the same underlying variable I(t) for the same values of I.

If we have two underlying variables I and J, we could similarly use Direct Transference to map them to a 2-d grid of bits, B(i,j), in which only one bit is nonzero.

If we then re-map this grid using inheritance we could arrive back at our original I and J variables. So, basically, what Direct Transference is saying is that these two pictures are really equivalent.

We could also map the two of them to a single 1-d series of variables, e.g. which are the sum of the respective 1-d series of bits. (Since the value of the sum becomes 2 when X=Y, these are trits, not bits.)

Can the variables that were obtained using Direct Transference be used to make a mapping so flexible that it must not be allowed? Something like a clock and dial mapping? The answer to that certainly appears to be, no. And that may be justification enough for allowing it; my philosophy is to be liberal in allowing mappings, as long as those mappings don't allow implementations of arbitrary computations.

Indirect Transference is a little more complicated. Consider a computer simulation of dynamics on a 2-d grid, f(x,y). When the value of f is updated at the pair of parameters x and y, this can be done by setting one variable equal to x, another equal to y, and using them to find the memory location M in the computer at which the corresponding value of f is to be changed. Since updates of f at (x,y) always involve fixed values for each of those parameters, f(x,y) can be labeled by those values. In this way, mapping of the values of f to the actual function of x and y, f(x,y), is considered a valid mapping, even though the computer's memory is not laid out in that matter. This is an example of Indirect Transference. It can be generalized to any case in which a parameter of a function is used.

Watch only the Road: Activity vs. Computation

2011-12-11T11:59:00.005-05:00

One criticism that is sometimes raised (e.g. by Maudlin, by Marchal) against the idea that a valid criterion for implementation of a computation can exist is that what consciousness exists should not depend on "what would happen" in counterfactual situations. Yet, counterfactuals are needed to define the causal relationships that distinguish a run of a computation from a mere sequence of states, and a such mere sequence could be easily found in random or repetitive data and should not be conscious.

This criticism gains intuitive force - and begins to seem plausible - when the counterfactual situation would be a complicated one.

For example, consider the following system:

I1,I2 are inputs
If I1 = I2 then Black Box 1 is activated, making state A = a; otherwise A = 0
If I1 does not = I2 then Black Box 2 is activated, making state B = b (b is not equal to a); otherwise B = 0
Output O1 = A + B
The value of O1 can then be used for further steps in a computation, e.g.
O2 = TRUE if O1 = a, else O2 = FALSE

This system implements the computation (I1, I2, O2) ==> (I1 , I2 , (I1=I2))

Now assume that I1 = I2. Black Box 1 was activated, and in the end O1 = a and O2 = TRUE.

We also know that if I1 had not been = to I2, then Black Box 2 would have been activated, and in the end the outputs would have been O1 = b and O2 = FALSE.

The problem is that Black Box 2 could contain a very complicated apparatus, such as a Rube Goldberg machine. If we didn't already know, it would be very difficult to tell what the effect of activating the box would actually be, without activating it to find out. Besides that, the box might work by sending a signal to a distant planet which is luckily inhabited, waiting many years while the aliens' culture evolves to the point that they can understand the signal, and then receiving back the message "b" from the aliens.

Given that in the actual scenario, Black Box 2 was never even activated, can whether the computation was implemented or not (and thus, if computationalism is assumed, perhaps whether or not an AI was conscious) really depend on such distant and complicated details? What if the aliens would have transmitted the message "a" instead of "b"?

By definition, whether the computation was implemented or not does indeed depend on those global counterfactual details. However, it does not seem plausible that whether an AI was conscious or not (and if so, of what) could have depended on those complicated counterfactual details. Must we then abandon computationalism?

To keep computationalism we need to keep causality, and thus counterfactuals, but avoid complicated chains of counterfactual events. There is a way to do it, and it is by considering equivalence classes of computations with respect to consciousness. If a given run of a computation would have given rise to a conscious observation, then it must be the case that a run of a different computation which had both the same sequence of actual states and the same causal relationships between those states would have given rise to the same consciousness, even if that computation would have behaved differently if the initial state had been different. I will call such runs, together with the causal relationships, a computational "road".

But how do we define those causal relationships, if not by counterfactual relationships?

If we consider the underlying system, there is no real problem if the system is allowed to evolve for a single 'time step'. I call the one-step considerations 'local causality'. The problem arises when complicated chains of events - global causality - would have come into play.

Causality must have the following characteristics:
If A causes B (written as A ==> B), then:

If A occurs, then B _must_ occur.

If A _does not_ occur, then B _might_ occur. In order to find out, we would need to know the laws of the underlying system, which will tell us what independent variables (of the underlying system or of a valid intermediate mapping) B would depend on. (As always, structured states are involved, so independence, inheritance and so on work normally.)

So now we have B(t+1) = function(variables at previous time)

Now, if a single time step of the underlying system is all that's involved, great: we just need to check to see if A is in that list of variables. If it is, then we have causality from A to B.

In other words, if the transition rule is B(t+1) = F(A(t),other(t)) then we have causality from A to B. But if F simplifies to F = f(other(t)) where by the rules of independence other(t) doesn't tell us A(t), then we don't.

If A ==> C, and C ==> B, then A ==> B.

If there is more than one underlying system time step involved (or for a continuous system), then we need to look at a series of time steps (or sequence of moments). Here, starting at the final state, causality can be traced to the previous step. In the end, there must be a chain of local causal relationships which relate the starting formal state A to the final formal state B. I call this a causal chain.

A computational road is also a causal chain, since each step causes the next. At each step along the way, there are "off-ramps" which are counterfactual transitions that are not allowed to occur by the transition rules, and "on-ramps" which are other counterfactual states that would have transitioned to the same state that did occur. The wrong on-ramps (too symmetric with respect to a variable which should be involved in the cause) would spoil the causality. It is OK if an off-ramp (which is not taken) would or would not have led back to the road at a later point by a valid on-ramp.

Back to the example above: Suppose that the aliens _would have_ transmitted back the message "a" instead of "b", but didn't, since I1=I2 in the first place and so they were never contacted.

O2 = TRUE was caused by O1 = a
O1 = a was caused by A = a and B = 0

What if B had been different even though A = a? Then O1 = a would not have happened. But that is OK; since B is an independent variable, we can restrict our mapping to states such that B = 0, which did occur. Within this mapping, O1 = a is caused by A = a.

A = a was caused by Black Box 1's activity
Black Box 1's activity was caused by I1 = I2

So here we have an implementation of a causal chain such that I1 = I2 caused O2 = TRUE.

If I1 had not been equal to I2, it would also have ended up that O2 = TRUE. But at no step along the actual causal chain did we need to explore the path that would have led there.

On the other hand, suppose that the transition rule for O2 had simply been O2(t+1) = f(t). In that case, the sequence of physical states - the activity - could have been the same as in the actual I1 = I2 case, but there would be no causal chain tracing O2 back to I1 and I2.

Formal transition rules of a computation no longer give full information about causal relationships: (I1=I2) ==> O2 = a, together with (I1 not = I2) ==> O2 = a, is equivalent to the simple transition rule O2 = a, and information about causality has been lost.

Causal chains are therefore the more fundamental concept. Like computations, they involve processing of information to cause transitions to specific states. Instead of implementing computations, one should really speak of implementing Structured State Causal Chains (SSCC's), but that doesn't have the same ring to it.

The Everything Hypothesis: Its Predictions and Problems

2011-10-06T11:55:00.008-04:00

There are some basic questions about the world:
1) Why does anything exist, instead of just nothing?
2) What does exist?
3) Why does that stuff exist instead of other stuff?
4) Given that stuff does exist, why is there consciousness instead of just behavior?

These questions are so basic that it would be nice to know the answers to them before worrying about specifics of our own situation.

Regarding these questions we can make highly counter-intuitive observations:

Suppose that somehow you didn't know that anything exists, and you are asked to guess: Does anything exist? My guess would certainly be "No, it would make a lot more sense if nothing exists. That is not only much simpler, it's also a lot easier to understand how it might be that nothing exists."

OK, but things do exist. So why THIS instead of THAT? We don't know. And not only that: It seems impossible to even imagine any reason why one possible thing would be selected over another. You can't say "it's because of this other thing" (whether the other thing is a law of physics, a god, or whatever) because that doesn't explain anything, it just begs the question of "So why does that other thing exist?" and we are back to the start.

There are basically two schools of thought on consciousness: 1] Dualism: Consciousness is a basic thing, which can not be due to something mathematically describable; or 2] Reductionism: Consciousness is an inevitable consequence of certain mathematically describable things. While I tend to fall into the latter camp, both ideas have counter-intuitive implications which I will not address in this post.

At least we do have ideas to debate about consciousness. Are there even any ideas about possible answers to the other questions, controversial or not?

It turns out that there is an idea which might begin to address them to some extent, called the Everything Hypothesis:

What if everything that possibly could exist does exist?

This would seemingly avoid avoid the apparent paradox of some things existing rather than other things despite there being no possible reason why that could be the case. It is also the simplest possible set of things that could have existed (other than just nothing) due to being fully symmetrical.

However, it doesn't really answer question 3). To really put question 3) to rest we would still need to know "Why does everything exist?" which would also cover question 1).

It turns out that there is an idea that could address that; this idea is often called "Platonism" in analogy with certain ideas that Plato had, but it is really a modern idea that was perhaps, although not necessarily inspired by Plato, given a bit of philosophical confidence by his precedent.

The idea is that there is no fundamental physical reality; instead, the fundamental reality is the world of logical and mathematical possibilities (which would thus better be called actualities). Of course, it remains difficult to understand why a logically possible world would automatically be real enough to have real observers inside it; but if that is the way it is, the Everything Hypothesis would have to be true.

While Max Tegmark is credited with the first paper on the Everything Hypothesis, various other people came up with similar ideas on their own, as will be discussed in the post on the Everything List. I was one of them.

There are variations or more limited versions of the Everything Hypothesis based on what is meant by 'Everything'. Tegmark's version of the Everything Hypothesis is explicitly mathematical: namely, that every possible mathematical structure exists. If Dualism is false, then that is equivalent to the full Everything Hypothesis. However, if Dualism is true, then consciousness and laws related to it are other things within the set of Everything.

The next question is: What does the Everything Hypothesis predict?

This can be put into familiar terms for a many-worlds model: What measure distribution on conscious observations does it predict?

At first glance, one might think that a typical observer within the Everything would see a quite random mess. If so, the Everything Hypothesis must be false, since we see reliable laws and a highly ordered universe.

However, taking a computationalist view of mind, only certain mathematical structures would support observers: those that have the equivalent of time evolution with respect to some parameter (or some suitable substitute), and have reliable laws for such dynamics. We should therefore consider the set of such dynamical structures with all possible dynamical laws. The starting state may be completely random, but the state will no longer be so random once time evolution occurs.

While we don't know how deal with the set of all possible dynamical systems, there is one subset that is much easier to handle: Turing machines, which are digital computers. A universal Turing machine can run the equivalent of any digital computer program.

So we can look at a simple universal Turing machine and consider the set of all programs for it. Such a program is just an infinitely long symbol string, which I'll call a bit string for simplicity. The machine has a 'head' that move along the string and changes bits and a few value values internal to the head.

Most programs have only a finite number of bits that actually do anything (the active region), because the head is likely to be instructed either to halt or to enter an infinite loop. The number of programs that are the same within the active region but different in the region of bits that don't do anything decreases exponentially with the number of bits in the active region, because each bit brought into the active region is a bit that can't be randomly varied in the inactive region.

Therefore, shorter and simpler programs have a higher measure (more copies) than longer programs do. The typical laws of worlds simulated by these programs are therefore likely to be as simple as possible, consistent with the requirement that observers exist within them. Perhaps, then, our own world is such a world.

That is an impressive result! It would certainly be interesting - though probably it will always be beyond our capabilities - to know exactly what would be typical for observers in the set of all Turing machine program runs to see.

However, there's a big problem here for the Everything Hypothesis: there are infinitely many possible Turing machines and digital computers in general. We can pick one, but that contradicts the fact that the Everything Hypothesis must have no arbitrary choices - no free parameters - if it is to explain why the world is the way it is. So why not just pick all of them and weight them equally? The problem there is that since there are infinitely many, the order in which we list them makes a difference to the result we get when trying to get the measure distribution. It's a very small difference for 'subjectively reasonable' choices of these parameters, but that's not the point; ANY arbitrariness completely ruins the explanatory power of Platonism and the Everything Hypothesis.

Besides, what about continuous systems? Some people are content to assume that the fact that there appears to be no natural measure for them means we don't need to include them in the Everything even if we are Platonists. However, it seems to me that they are just as much legitimate candidates for existence as digital systems.

Reluctantly, I am forced to conclude that - unless there is some way of overcoming these mathematical problems that we don't know about, which seems unlikely - Platonism does not provide the explanation we were looking for. This is a paradox, because it also seems that Platonism is the only thing that even could have been a real explanation for why things are the way they are.

I still think that the set of all things which exist is probably very simple in some sense and that the physics we see is just a small part of it. We see the part of it that we see due to being fairly typical observers.

Perhaps the right way to derive the Born Rule of quantum mechanics would be to start with something like the set of all possible Turing machine programs and derive from it the measure distribution on conscious observations, but obviously, such a project would be all but impossible to carry out in practice. My work will focus instead on studying the consequences of the standard physics equations (which are based in large part on experimental observations). However, my criteria for implementation of a computation are general and do not depend on the assumption that the underlying system is a physical one, so in principle, they should apply even to underlying systems that are Platonic Turing machines.

See also the post about the Everything List (forthcoming)

Restrictions on mappings 1: Independence and Inheritance

2011-10-04T20:50:00.015-04:00

Previous: The Putnam-Searle-Chalmers Theorem

Chalmers proposed that a computation state be represented by a string of numbers, where each part of the string is "independent"; he proposed that independence can be achieved if each part of the string depends on a different part of the underlying system. He called this a Combinatorial State Automaton (CSA), because the state is given not by a single number but by the combination of the different parts of the string. The CSA avoids the false "clock and dial" implementations discussed in the previous post.

However, as Chalmers noted, there are still false implementations that can be found for the CSA. Suppose that each part of the underlying system which is mapped to part of the string at the current time step contains all of the relevant information about every other relevant part of the system at the previous time step.

In that case, a mapping can be made in which - although each part of the string depends only on its own part of the underlying system - a sequence of values can be assigned to each part of the string in such a way as to mimic any desired computation: enough information is available for the value assigned to that part to depend on any desired combination of the values of the parts of the string at the previous time step.

The amount of information that must be stored would grow exponentially with time, which leads to systems that quickly become physically unrealistic after a few time steps, but the fact that the problem exists in principle is enough to show that a different restriction on mappings is needed.

CSSA:

An implementation criterion will be given for a formal system consisting of a structured set of variables (also called substates), and a transition rule. Such a system will be called a Combinatorial Structured State Automaton (CSSA). It differs from Chalmers' CSA in that the structure of the state is not restricted to that of a string of values. For example, the substates could be arranged into a rectangular matrix, or a higher-dimensional array, and this arrangement and its specification would be considered part of the specification of the CSA.

One reason for including such structure is to allow label inheritance (explained below) from an underlying computation to another computation. While the ultimate underlying system is assumed to be the physical universe (or whatever underlies that!), the definitions must work in general so that the underlying system can be anything that can be described mathematically, because the criteria for implementation of a computation should not depend on experimentally discovered facts of physics but rather on pure mathematics.

It will also be useful in analyzing systems to allow computations to implement other computations. For example, if an object implements a "Windows machine" computation, then we only need to check variables on the level of that computation (rather than looking at fundamental systems variables such as the wavefunction's behavior in terms of electron positions) to see if it also implements the "Firefox" computation. In this case the "Windows machine" is now treated as the underlying system. Many of the simple examples I will use will involve a simple underlying digital system.

The criteria given below are a modification of the ones given in my MCI paper, and will be incorporated into a revised paper.

Independence:

The basic rule for independence is that it must not be possible to determine from knowledge of the physical states that are mapped to the value of the current substate and of the system dynamics

- the values of any of those substates (at the previous time step) that determined what the value of a given substate is at the current time step, except when the current value itself provides enough information or there is only one previous substate that the current value depends on; or

- the values of other substates (at the same time step), except when the current value itself provides enough information.

This rules out clock and dial mappings, because the clock and dial physical state which any substate depends on would reveal all formal values of the other substates and the previous substates. It also rules out the other false implementations discussed above in which the variables record information that would reveal the values of the previous states that determine them.

Inheritance:

However, it is sensible to allow cases in which interacting formal substates share the underlying variables that they depend on. This is especially important with quantum mechanics (see below).

In these cases, independence is instead established with the help of “inheritance” of labels. Underlying label indices are inherited when a substate depends on the values of that index, associating the substate with that index.

Knowledge of the variables that determine the other substates - modulo permutations (or in other words, limited knowledge that would not be affected by such swapping) of the labeling indexes that are inherited by the given substate - must not reveal the value that the given substate has (at the same time step).

The restriction that knowledge of the underlying variables that determine the substate must not reveal the values of the substates at previous time step that determine its value (with exceptions as the basic rule for inheritance above) still applies as well, but modulo permutations of the inherited indexes for the respective substates.

An example helps show what is involved here:

Bits on a 2-d grid plus time --> (mapped to) 2 integers as functions of time

b(i,j,t) --> I(t), J(t)

The computation would then involve the dynamics of I and J, such as
I(t+1) = f(I(t),J(t))
J(t+1) = g(I(t),J(t))

where f and g are appropriate functions.

Each bit takes on the value 0 or 1. If the number of nonzero bits is not equal to 1, this mapping will be undefined (there is no need for the mapping to cover all possible states of the underlying system).

If only one bit is nonzero at a given time, then let
I(t) = the value of i at which b(i,j,t) is 1
J(t) = the value of j at which b(i,j,t) is 1

If we were to swap around the values of the i-index, it would affect the value of I(t) but not of J(t), and similarly for the j-index affecting J(t) but not I(t). Therefore the conditions are met so that I(t) “inherits” the i label and J(t) the j label.

To verify independence - even though both I and J depend on all bits on the grid in the sense that only one nonzero bit is allowed - knowing the full state of the 2-d grid modulo a permutation of the j-index (that is, knowing it only up to a possible swapping around of the j-index values) reveals the value of I(t), but not J(t), and vice versa. Thus they are independent.

In this example, an underlying system which consists of a function ON a 2-dimensional grid was reduced to a system of two variables, which together define a single point IN a 2-d space. This can be summarized as “on --> in”.

In cases where the underlying system has other variables, e.g. the k index in b(i,j,k,t), a mapping might not be symmetric with respect to the other variables. For example, we could have

X(t) = b(1,1,1,t)+ b(1,2,2,t) - b(2,1,3,t) - b(2,2,4,t)
Y(t) = b(1,1,1,t)+ b(2,1,3,t) - b(1,2,2,t) - b(2,2,4,t)

In this case, are X and Y independent, with X inheriting from i and Y from j? If two values of j (say j=1 and j=2) are swapped as indexes of b(), then the value of X could change: b(1,1,1,t) + b(1,2,2,t) is not b(1,2,1,t) + b(1,1,2,t). On the other hand, if we only had a limited knowledge of b() and that knowledge is symmetric with respect to such swaps, we wouldn't know that. This mapping is OK. It is perhaps better to think of it as if the terms in X(t) are labeled by different i-values and j-values (the values the mapping is supposed to inherit from) while ignoring their functional dependence on k (which is not being inherited by anything).

The concept of inheritance can be generalized to cases in which more than one underlying variable is responsible for the distinction being inherited. For example, here X inherits from i, but Y is to inherit from j and k:

X(t) = b(1,1,1,t)+ b(1,2,1,t) - b(2,1,1,t) - b(2,1,2,t)
Y(t) = b(1,1,1,t)+ b(2,1,1,t) - b(1,2,1,t) - b(2,1,2,t)

Inheritance can also occur with multiple levels of functionals. For example, suppose the underlying variables are a functional g of function f(i,j), e.g. if i and j are bits, here g=g(f(0,0),f(0,1),f(1,0),f(1,1)). An intermediate mapping could be made with substates that inherit from f and are functions on (i,j). This can then be mapped to a pair of substates X,Y that inherit from i and j respectively.

In quantum mechanics, the wavefunction is a function ON a high dimensional configuration space. In classical mechanics, the set of particle positions defines a point IN configuration space. It certainly seems that, in the classical limit, the computations that would performed by the related classical system (such as a system of classical switches) would in fact be performed by the actual quantum system. Inheritance of labels - from quantum directions in configuration space to classical position variables - permits that. Quantum field theory, which is even more of a realistic model, involves functional dependency of the wavefunctional on a function of space.

Next: Restrictions on mappings 2: Additional considerations

The Putnam-Searle-Chalmers Theorem

2011-10-03T16:19:00.007-04:00

Previous: Basic idea of an implementation

If a computation is implemented, there must be a mapping from states of the underlying system to formal states of the computation, and the states must have the correct behavior (transition rule) as they change over time.

For example, for an electronic digital computer, we can map a high-voltage state of a transistor lead to a "1" and a low voltage state to a "0". By doing the same for various other circuit elements, we can obtain an ordered string of 1's and 0's. This string will change over time as the voltages change. If, due to the laws of physics and the way the circuit is connected, this string must always change in accordance with the transition rules for computation C, then the system implements C.

We must allow as much flexibility as possible in the choice of mapping, because we are trying to understand the behavior of the system without any reference to things outside the system. Human convenience in recognizing the mapping is not a consideration.

The obvious way to try do this is simply to allow any mapping that is mathematically possible. This leads to what I will call the naive implementation criterion, because while it may sound good at first it is not a viable option for a satisfactory criterion. Chalmers' paper Does a Rock Implement Every Finite-State Automaton? explained this in detail.

The following is my version of what I'll call the "Putnam-Searle-Chalmers (PSC) Theorem" which shows that unrestricted mappings are not a viable option:

Suppose that a system consists of two parts, S and T, each of which has a numerical value, and that the dynamics of our system are as follows:
S(t+1) = S(t)
T(t+1) = T(t) + 1

These dynamics are fairly trivial; S is a dial that maintains a constant setting, while T is a clock.

We will check to see if this system implements a computation, C, which has the transition rule X(t+1) = F(X(t)) where t is a time index.

Here X need not be a single number; it might be a string of bits, for example.

F could be a complicated function, such as F(X) = the Xth prime number (expressed in base 2, where X is an integer expressed in base 2).

Now make a mapping M going from (S,T) to X with the following properties:
X = M(S,T)
M(S,T+1) = F(M(S,T))

We do need to make sure that any possible starting value of X is allowed by the mapping, which we can always do if our system has enough possible dial values.

Now, according to the mapping, X will change as a function of time based on the dynamics of the system as follows:
X(t+1) = M(S(t+1),T(t+1))
= M(S(t),T(t)+1)
= F(M(S(t),T(t)))
= F(X(t))

Therefore, the system would implement the computation if this mapping is allowed. But the system's dynamics are trivial while the computation can be a very complicated function. Obviously, this is not acceptable; the computation does not characterize the behavior of the system at all. This is what I call a "false implementation". All of the complicated dynamics of the computation have been put into the mapping. It is therefore necessary to put restrictions on what mappings are allowed.

One possibility is to require that each part of a string that defines a formal computational state (e.g. each bit in a bit string) takes its value based on a different part of the underlying system. That is basically what Chalmers proposed to overcome the problem.

While it somewhat counter-intuitively rules out distinctions based on the value of a single number (since a single number can still be mapped to any other single number), it goes a long way towards ruling out false implementations, while still allowing important standard examples of implementations that we want to retain, such as mapping switch positions to bit values (assuming classical physics).

However, it is not quite right. There are some systems for which it still allows false implementations, and there are other cases where it rules out what seem to be legitimate implementations - and these become clearly important when quantum mechanics is brought into the picture. In classical mechanics, different particles are different parts of the system; in quantum mechanics, different particles are different directions in the shared configuration space on which the wavefunction evolves.

Next: Restrictions on mappings 1: Independence and Inheritance

Basic idea of an implementation

2011-10-01T16:55:00.011-04:00

In conceptual terms, to say that a system implements a given computation is to say that something about that system - some aspect of its behavior - is described by the computation. It is thus a way of characterizing the system.

For example, if a system can take two numbers as inputs and produces their sum as an output, we could characterize it as an "adder". If instead it can take two numbers as inputs and produces their product as an output, we could characterize it as a "multiplier". "Adding" and "multiplying" are two different _types_ of computations. These behaviors are fairly general, so they can be can be further divided into specific computations if more details are specified, such as what format the inputs are in, internal functioning during intermediate steps, and so on.

Two "multipliers" with different intermediate steps would implement different computations. Thus, while the word "computation" suggests something that is done to get a result, that is misleading in the context of the computationalist view of consciousness, which instead focuses on internal functioning of the system in question.

If we find a third system and discover that it behaves like an "adder", then it has a lot in common with our other "adder": we now know a lot about how it can behave. But there's also a lot that this characterization does NOT tell us. It doesn't tell us what the system is made out of; or what its other behaviors might be; or what internal processes it uses to perform the additions. It also does not tell us whether it performs the addition multiple times, redundantly, perhaps performing the same addition again but sending that (same) result to some other person.

In addition to its capabilities, a system's behavior is further characterized by the exact sequence of computer states that are actually involved in its behavior. For example, an adder which added the numbers 45 and 66 and got 111 is characterized by those numbers, in addition to just being an adder. Such a sequence of numbers (or more precisely the sequence that includes all intermediate states of the computation) can be called the "activity" of the computation, and together with the computation it specifies what is known as a run of the computation.

It is important to note that neither the computation alone nor the activity alone are enough to adequately describe the behavior of the system; both are needed.

Often, I will speak only of computations, but it should be understood by the context that "runs of computations" are often meant as well. For example, if I say that a particular computation gives rise to a particular conscious observation, I mean that a particular run of that computation - corresponding to a particular starting state - gives rise to that observation. A different run of the same computation (such as the same type of brain except in a different starting state) might then give rise to a different kind of observation or to none at all.

In the case of programmable systems, the distinction between computations and runs becomes somewhat arbitrary, but this is not a problem as we can always specify what computations we are interested in and use that to make the choice.

A system could be both an adder and a multiplier; for example, the universe is both if at least one of each exists as a sub-system of it. Yet a hypothetical universe with 100 adders and 1 multiplier would be significantly different from one with 1 adder and 100 multipliers. A more detailed way of characterizing such systems would be to state how many instances of each kind of computation is performed by it: in other words, to give a measure distribution on computations. Such a measure distribution (or more exactly, a measure distribution on runs of computations) is the main tool needed to evaluate the computationalist version of the Many Worlds Interpretation of QM, and will be addressed in later posts.

While artificial computers tend to operate on command, there is nothing about the concept of implementing computations that requires that. For example, a robot that walks around and listens for numbers, then once it has heard two of them, adds them and says the result, then starts again, would still be an "adder". Such a robot would behave according to its internal agenda, ignoring the desires of the people around it even if they beg it to stop.

It's also important to note "outputs" need not be distinguished from internal parts of the system. Any parts of the system that produce the behavior in question are the relevant parts, regardless of how they interact with things outside the system.

For a computation with more than one step, it is useful to define "inputs" as substates that are affected by influences outside the scope of the computation in such a way that their values are not determined by the previous state of the computation. Influences outside the scope of the computation are not necessarily outside of the system (such as the universe) which performs that computation.

Those familiar with computer science might be surprised that the concept of the Turing Machine will play no role in using computations to characterize systems. A Turing Machine is a type of computer that is useful for specifying which functions could be calculated by in principle by digital computers (if unlimited memory - an infinite number of substates - is available, but the transition rule for each substate depends on a finite number of substates) and how easily (setting bounds for how memory needs and processing time scale with the size of a problem).

Because this class of computers is so ubiquitous in computer science, those computer scientists who dare venture into the swamps of philosophy far enough to read a paper or attend a lecture on the implementation problem sometimes completely lose interest in the problem when they realize that no one is mentioning Turing Machines. In fact, it would be quite easy to describe Turing Machines as a special case of the computers that I will describe.

But here we are concerned with characterizing the behavior of actual systems as they exist, not with finding what size of problems they could be programmed to handle. It is also important to note that digital computing is just a special case of the behaviors that could be characterized as computation; analog computing can certainly be considered as well, and most of the definitions I will make are general enough to cover all cases. However, I will focus on digital computation in most of the examples I look at.

The next step is to formalize the idea of implementation by giving a mathematical criterion for whether a computation is implemented by a given mathematically described system. This will be done by requiring a mapping from states of the underlying system to formal states of the computation, and requiring the correct behavior of the states as they change over time.

However, as will be seen in the next post, this approach quickly runs into a problem: without restrictions on allowed mappings, 'almost any' physical system would seem to implement 'almost any' computation. This absurd result would imply that a rock implements the same computations as a brain. The likely solution is to impose restrictions on the allowed mappings, but finding a fully satisfactory set of restrictions has proven to be a difficult task. My proposals for this will be presented in subsequent posts; I think that I have been successful in finding (at least close to) the right set of restrictions.

The Computationalist approach to Measure

2011-09-26T15:14:00.004-04:00

For reference, before reading this post, read the posts on Meaning of Probability in an MWI and Measure of Consciousness versus Probability.

Here, I'll outline the way in which the Computationalist philosophy of mind suggests that it might be possible to calculate the measures of consciousness for various outcomes.

My goal in this is to either derive from the MWI the Born Rule (which allows us to calculate the probabilities in QM), thus providing strong support for the computationalist view of the MWI, or to show that the experimentally discovered Born Rule is inconsistent with these popular views of mind and physics.

Computationalism is basically the idea that a brain gives rise to conscious observations due to its mathematically describable functioning: the motions of its parts together with the laws of physics that restrict and determine those motions.

With the existence of consciousness thus being in principle describable mathematically, we can hypothesize that it might be possible to analyze the mathematical description of a physical system, and based on it determine not only whether consciousness is present, but also to determine that multiple different types of observations are being made in different parts of the system. We can make a reasonable guess as to the nature of those observations.

We can also attempt to determine what the "quantity of consciousness" is for each significantly different type of observation. If we can, then we have a measure distribution, which we can compare to what the Born Rule predicts.

In doing so, it is necessary to work with computations as proxies for the conscious observations which are assumed to accompany them. The nature of the link between computations and conscious observations, while an important topic in its own right, is for the most part not something we need to analyze for this purpose, which is fortunate as understanding it is famously hard.

In comparing the computationalist + MWI prediction to the Born Rule, it would not be a problem if there are very small deviations which would not yet have been detected experimentally. In theory, such a situation could open the door to a straightforward experimental test of the computationalist approach to the MWI, which would be a great thing to have. However, it seems likely that IF small deviations are predicted then any such deviations would be too small to ever measure.

On the other hand, if the measure distribution we get does NOT agree with the Born Rule even approximately, then the computationalist picture of the MWI would be refuted.

Unfortunately, all of this is easier said than done. While the general idea of Computationalism is fairly popular, it has not yet been developed into something precise enough to carry out this program. Needless to say, experiments can be of no help in this regard.

As will be explained in detail in forthcoming posts, I have made proposals as to how to make computationalism precise enough. While it is technically challenging to state necessary and sufficient conditions to be able to say that a given computation is carried out by a given system, I am satisfied that this can be done and that my proposal is at least quite close to the correct way to do so.

Given such conditions, the next thing that we need is a way to determine the accompanying quantity of consciousness. Hopefully, we can avoid dealing with the link between computation and consciousness even here, by determining as a proxy the "quantity of each computation" that is present, and assuming that it is proportional to the quantity of consciousness.

However, the link may not be so easily bypassed, as it turns out that there are different options for determining the "quantity of computation" and naturally the "right" choice would be the one that best suits the application we have in mind, namely consciousness. It is not clear, however, what that one is.

What I can say with certainty given my work so far is that, given the choices that I would tend to pick based upon subjective criteria of what "seems most natural" to me, the application of those choices to the standard physics of the MWI apparently DOES NOT give the Born Rule. This fact is not sufficient for me to declare that the MWI has been disproven; however, it does help motivate a search for MWI proposals that include modifications of physics.

In fact, there is another choice of "quantity of computation" criteria that could be made which apparently DOES give the Born Rule from the standard MWI physics. So why don't I just endorse that choice and say "Look, the MWI works!!!"? The problem is that I pretty much found those criteria not on philosophical grounds but by asking "What criteria would make it work?" With other options to choose from and no independent motivation for picking those criteria, it is not necessarily very meaningful that such criteria exist. While it allows the unmodified MWI to remain a viable option, it doesn't really provide the new philosophical support for the unmodified MWI that I had been hoping for.

However, my intuition on the matter of how to determine "quantities of computations" is not necessarily very good. It's not as if anyone has experience in such matters. That is one reason why I would like other people to learn of and study these issues. Perhaps, in the future, people who have studied these issues deeply and explored the implications and alternatives in great detail will find that the criteria which happen to give the Born Rule from the unmodified MWI are the same that they prefer for quite different reasons.

MWI proposals that include modifications of physics

2009-10-21T13:59:00.013-04:00

Previous: Decision Theory and other approaches to the MWI Born Rule, 1999-2009

The greatest appeal of the Everett-style Many-Worlds Interpretation of QM - that is, the wave equation alone, or standard MWI - is its simplicity in terms of mathematics and physics. While all other interpretations need to add extra things to the wave equation - adding in hidden variables with dynamics of their own, or modifying the wave equation itself to include random wave collapse processes - the Everett MWI states that the standard wave equation alone can explain everything we observe.

Yet, despite numerous attempts and claims to the contrary (and putting aside the possibilities for my own approach for now), the Born Rule probabilities have not been derived from the Everett picture. Thus, it may prove necessary to add new physics to our description of QM after all.

However, some approaches attempt to retain much of the advantage in simplicity of the MWI, as well as its multiple-worlds character, while making such modifications. It's a promising idea, and these MWIs certainly take inspiration from the Everett-style MWI, but they add too much to the pure wave picture to satisfy true Everett-style MWI partisans.

a) Hidden variables introduce complexity not only because of the extra dynamic equations, but because they require some choice of initial conditions. The wavefunction of QM also requires initial conditions, of course, but there is reason to hope that some simple equation could govern those initial conditions; the Hartle-Hawking 'no boundary' condition for the wavefunction of the universe is a well-known example of such a proposal (though it has problems of its own). Particle-like hidden variables do not seem amenable to such simple specification of initial conditions. However, if all possible sets of hidden variable initial conditions are equally real, then the overall simplicity of initial conditions for the multiverse is restored.

'Continuum Bohmian Mechanics' (CBM) is the best-known example of this approach. Like the Pilot Wave Interpretation, it has particle-like hidden variables; but instead of just one set of them, it has a continuous distribution of such sets, which act much like a continuous fluid. In addition to the possibility of simpler initial conditions, CBM may be immune to the fatal flaw of the PWI, which is being 'many-worlds in denial'. In other words, in the 'single-world' PWI, with one set of hidden variables, most of the observers will end up being implemented by the many worlds of the wavefunction, so the hidden variables won't matter. With CBM, the number of hidden variable sets is also infinite, so a typical observer could depend on the hidden variables after all. (This latter claim still needs to be proven compatible with computationalist considerations, but it is plausible.)

The hidden variables in the PWI follow the Born Rule, so CBM should be OK in that regard. But CBM retains the other features of the PWI that many physicists dislike, namely non-locality and a preferred reference frame. It is also not clear how well a relativistic version of the PWI works, and CBM inherits such problems. (Granted, no physics that works for quantum general relativity is known yet.) Also, even CBM is not as simple as the standard MWI.

I regard CBM (and more generally, MWI's with hidden variables) as something useful to keep in mind, as it is one of the few interpretations of nonrelativistic QM that seems to actually work in terms of being compatible with the Born Rule and not having an 'MWI in denial' problem. But other possiblities must be thoroughly explored before I would consider endorsing CBM as being likely to be true.

b) Another approach is to retain a pure wavefunction picture as in Everett's MWI, but to make the wavefunction be discrete instead of continuous. Discrete space is not what is meant here, but rather a discrete nature of the wavefunction itself. Buniy et al advocate such an approach.

The most obvious way to do that might be to assume that the wavefunction is represented by an integer function on configuration space rather than a continuous function. (If configuration space is also discrete, that is one way an approximate discrete numerical representation of a continuous wave function might be done on a digital computer.)

Buniy et al propse a somewhat different different assumption, in which wavefunctions seperated by a term of some mimimum squared amplitude are condidered to be the same.

Because the wave function (or perhaps I should say its 'populated' region) is constantly and rapidly expanding into new areas of configuration space (e.g. as entropy increases), its numerical value is constantly imploding. I will call this the Wavefunction Value Implosion (WVI). If the universe is finite, then this effect will be finite, but exponentially large as a function of the number of particles in the universe. Thus, a discrete wavefunction could not be detected experimentally if its discrete nature is small enough, until such time as the WVI brings the populated part of the wavefunction below that scale, and then presumably time evolution will radically change or effectively stop; I will call this the Crash.

Discrete physics has a certain appeal to some people, independent of any possible role in quantum mechanics. Wolfram's book "A New Kind of Science" discusses such views. Also, the idea that all possible mathematical universes physically exist (the Everything Hypothesis, which will be discussed in a later post) may be somewhat more tractable if it is restricted to digital systems (though, despite its undeniable appeal, it still has problems even then).

If the wavefunction is discrete, would that help explain the Born probabilities? Buniy et al argue that it would, by shoring up the old "frequency operator" attempted derivation by extending it to finite numbers of measurements rather than infinite. This argument notes that, after repeated measurements, terms in the wavefunction which don't have the Born frequencies have much smaller amplitudes than the terms with the 'right' frequencies. With a minimum amplitude cutoff, most of the un-Born terms would be eliminated. This argument does not seem very satisfactory, as we are interested in situations with small numbers of meaurements, and thus small factors of difference in amplitude, while the digital cutoff would have to be very far from a significant fraction of the total amplitude if the Crash is not yet upon us. In practical situations, other factors would affect amplitudes much more. For example, entropy production is not associated with low probability, but it results in numerous sub-branches each of which shares a fraction of the original squared amplitude.

Shoring up the 'Mangled Worlds' argument would seem a more promising approach. There are many sub-branches comprising each macroscopically distinguishable world, and they tend to have a log-normal distribution in squared amplitude. As Hanson showed, a cutoff in the right range of squared amplitudes would lead to Born Rule probabilities. This cutoff must be uniform across branches, which Hanson's 'mangling' mechanism by larger branches actually fails to provide, but a digital cutoff could provide it. I will tentatively say that this is a possible mechanism for the Born Rule, though I need to study it more before I can say for sure that there are no problems that would ruin it. In particular, if the number of worlds changes too much over time or the era in which the Born Rule holds is too short, that would indicate a problem.

c) Another mechanism that improves on 'Mangled Worlds' is my own idea in which random noise in the initial wavefunction means that larger volumes in configuration space per implemented computation are required for low-amplitude sub-branches, which can lead to the Born Rule. This requires new physics in the form of special initial conditions, but hopefully not in terms of time evolution. It is possible that this leads to a Boltzmann Brains problem. I will discuss this, as well as an alternative in which the Born Rule is due to a special way to count computations (which avoids new physics - if it can be justified) in later posts.

d) The Everything Hypothesis (that all possible mathematical stuctures exist) can be used directly in an attempt to predict what a typical observer would observe. Some have argued that this explains what we observe, including the Born Rule. The Everything Hypothesis will be discussed in a post of its own.

e) Other MW schemes for modifying physics have been proposed.

One example is Michael Weismann's idea involving sudden splitting of existing worlds into proposed new degrees of freedom, with a higher rate of such splitting events for higher amplitude worlds. The problem with it is that if new worlds are constantly being produced, then the number of observers would be growing exponentially. The probability of future observations, as far into the future as possible, would be much greater than that of our current observations. Thus, the scheme must be false unless we are highly atypical observers, which is highly unlikely.

David Strayhorn had an idea based on general relativity, in which different topologies correspond to different sub-branches of the wavefunction. This approach is not well-developed as of yet and it has problems that I think will prevent it from working. I discussed it in various post on the OCQM yahoo group.

Decision Theory & other approaches to the MWI Born Rule problem, 1999-2009

2009-09-26T18:13:00.018-04:00

In the previous post, I explained the early attempts to derive the Born Rule for the MWI. These attempts required assumptions for which no justification was given; as a result, critics of the MWI pointed to the lack of justification for the Born Rule as a major weakness of the interpretation.

MWI supporters often had to resort to simply postulating the Born Rule as an additional law of physics. That is not as good as a derivation, which would be a great advantage for the MWI, but it at least puts the MWI on the same footing as most other interpretations. However, it is by no means clear that it is legitimate to do that, either. Many people think that branch-counting (or some form of observer-counting) must be the basis for probabilities in an MWI, as Graham had suggested. Since branch-counting gives the wrong probabilities (as Graham failed to realize), a critic might argue that experiments (which confirm the Born rule) show the MWI must be false.

Thus, MWI supporters were forced to argue that branch-counting did not, in fact, matter. The MWI still had supporters due to its mathematical simplicity and elegance, but when it came to the Born Rule, it was in a weak position.

In the famous Everett FAQ of 1995, Price cited the old 'infinite measurements frequency operator' argument. That was my own first encounter with the problem of deriving the Born Rule for the MWI, and despite being an MWI supporter, the finite-number-of measurements hole in the infinite-measurements argument was immediately obvious to me.

5) The decision-theoretic approach to deriving the Born Rule

In 1999, David Deutsch created a new approach to deriving the Born Rule for the MWI, based on decision theory. He wrote "Previous attempts ... applied only to infinite sets of measurements (which do not occur in nature), and not to the outcomes of individual measurements (which do). My method is to analyse the behaviour of a rational decision maker who is faced with decisions involving the outcomes of future quantum-mechanical measurements. I shall prove that if he does not assume [the Born Rule], or any other probabilistic postulate, but does believe the rest of quantum theory, he necessarily makes decisions as if [the Born Rule] were true."

Deutsch's approach quickly attracted both supporters and critics. David Wallace came out with a series of papers that defended, simplified and built on the decision theory approach, which is now known as the Deutsch-Wallace approach.

Deutch's derivation contained an implicit assumption, which Wallace made explicit, and called 'measurement neutrality'. Basically, it means that the details of how a measurement is made don't matter. For example, if a second measurement is made along with the first, it is assumed that the probabilities for the outcomes of the first won't be affected. This implies that unitary transformations, which preserve the amplitudes, don't matter. That implies 'equivalence', which states that two branches of equal amplitudes have equal probabilities, and which is essentially equivalent to the Born Rule. The Born Rule is then derived from 'equivalence' using simple assumptions cast in the language of decision theory.

Wallace acknowledged that 'measurement neutrality' was controversial, admitting "The reasons why we treat the state/observable description as complete are not independent of the quantum probability rule." Indeed, if probabilities depend on something other than amplitudes, then clearly they can change under unitary transformations.

So he offered a direct defense of the 'equivalence' assumption, which formed the basis of the paper that was for a long time considered the best statement of the DW approach, certainly as of the 2007 conferences. New Scientist magazine proclaimed that his derivation of the Born Rule in the MWI was "rigorous" and was forcing people to take the MWI seriously.

His basic argument was that things that the person making a decision doesn't care about won't matter. This included the number of sub-branches, but he also took care to argue that the number of sub-branches can't matter because it is not well-defined.

Consider Albert's hypothetical fatness rule, in which probabilities are proportional both to the squared amplitudes and to the observer's mass. This obviously violates 'equivalence'. According to Wallace's argument, the decider should ignore his mass unless it comes into play for the decision, so that is impossible. But it is a circular argument; the decider should care about his mass if it in fact affects the probabilities.

My critique of Wallace's approach is presented in more detail here, where I also cover his more recent paper.

In his 2009 paper, Wallace takes a different approach. Perhaps recognizing that assuming 'equivalence' is practically the same as just assuming the Born Rule, he makes some other assumptions instead, couched in the language of decision theory, which allow him to derive 'equivalence'. The crucial new assumption is what he calls 'diachronic consistency'. In addition to consistency of desires over time, it contains the assumption of conservation of measure as a function of time, which there is no justification to assume. Of course, the classical version of diachronic consistency is unproblematic, and only a very careful reading of the paper would reveal the important difference if it were not for the fact that Wallace helpfully notes that Albert's fatness rule violates it.

6) Zurek's envariance

W. Zurek attempted to derive the Born Rule using symmetries that he called 'envariance' or enviroment-assisted invariance. While interesting, his assumptions are not justified. The most important assumption is that all parts of a branch, and all observers in a branch, have the same "probability". Albert's fatness rule provides an obvious counterexample. I also note that a substate with no observers in it can not meaningfully be assigned any effective probability.

He uses this, together with another unjustified assumption that is similar to locality of probabilities, to obtain what Wallace called 'equivalence' and then the Born Rule from that. Because the latter part of Zurek's derivation is similar to the DW approach, the two approaches are sometimes considered similar, although Zurek does not invoke decision theory.

7) Hanson's Mangled Worlds

Robin Hanson came up with a radical new attempt to derive the Born Rule in 2003. It was similar to Graham's old world-counting proposal in that Hanson proposed to count sub-branches of the wavefunction as the basis for the probabilities.

The new element Hanson proposed was that the dynamics of sub-branches of small amplitude would be ruined, or 'mangled', by interference from larger sub-branches of the wavefunction. Thus, rather than simply count sub-branches, he would count only the ones with large enough amplitude to escape the 'mangling'.

Due to microscopic scattering events, a log-normal squared-amplitude distribution of sub-branches arises, as it is a random walk in terms of multiplication of the original squared-amplitude. Interference ('mangling') from large amplitude branches imposes a minimum amplitude cutoff. If the cutoff is in the right numerical range and is uniform for all branches, then due to the mathematical form of the log-normal function, the number of branches above the cutoff is proportional to the square of the original amplitude, yielding the Born Rule.

Unfortunately, this Mangled Worlds picture relies on many highly dubious assumptions; most importantly, the uniformity of the ‘mangling’ cutoff. Branches will not interfere much with other branches unless they are very similar, so there will be no uniformity; small-amplitude main branches will have smaller sub-branches but also smaller interference from large main branches and thus a smaller cutoff.

Even aside from that, while the idea of branch-counting has some appeal, it is clear that observer-counting (with computationalism, implementation-counting) is what is fundamentally of interest. Nonetheless, 'Mangled Worlds' is an interesting proposal, and is the inspiration for a possible approach to attempt to count implementations of computations for the MWI, which will be discussed in more detail in later posts. That does require some new physics though, in the form of random noise in the initial conditions which acts to provide the uniform cutoff scale that is otherwise not present.

In the next post, proposals for MWIs that include modifications of physics will be discussed.

Early attempts to derive the Born Rule in the MWI

2009-09-23T14:57:00.016-04:00

When Everett wrote his thesis in 1957 on the '"Relative State" Formulation of Quantum Mechanics', he certainly needed to address how the Born Rule probabilities fit into his new interpretation of QM. While the MWI remains provocative even today, it was not taken seriously in 1957 except by a few people, to the extent that Everett had to call it "Relative State" rather than "Many Worlds". So it is perhaps fortunate that he did not realize the true challenges of fitting the Born Rule into the MWI, which could have derailed his paper. Instead, he came up with a short derivation of the Born Rule, using assumptions that he did not realize lacked justification.

Of course, the Born Rule issue has long since returned to haunt the MWI. Historically, what has happened several times was that a derivation of the Born Rule that seemed plausible to MWI supporters was produced, but soon it attracted critics. After a few years it became clear to most physicists that the critics were right, and the MWI fell into disrespect until a new justification for the Born Rule was produced. This cycle continues today, with the decision-theoretic Deutsch-Wallace approach being considered the best by many, and now attracting growing (and deserved) criticism.

When considering claimed derivations of the Born Rule in the MWI, it is often useful to keep in mind an 'alternative rule' that is being ruled out, and to question the justification for doing so. Two useful ones are as follows:

a) The unification rule: All observations that exist have the same measure. In this case, branch amplitudes don't matter, as long as they are nonzero (and they always are, in practice).

b) David Albert's fatness rule: The measure of an observer is proportional to the squared amplitude (of the branch he's on) multiplied by his mass. Here, amplitudes matter, but so does something else. This one is especially interesting because it illustrates that not all observers necessarily have the same measure, even if they are on the same branch of the wavefunction. While it is obviously implausible, it's a useful stand-in for other possibilities that may seem better more justifiable, such as using the number of neurons in the observer's brain instead of his mass, or any other detail of the wavefunction.

Another useful thing to keep in mind is the possibility of a modified counterpart to quantum mechanics, in which squared-amplitude would not be a conserved quantity. We would expect that the Born Rule might no longer hold, but some other Rule should, even in the absense of conserved quantities. Presumably, if the modification is small, so would be any departure from the Born Rule. Thus, one should not think that conserved quantities must have any special a priori importance without which no measure distribution is possible.

Let us examine a few of the early attempts to derive the Born Rule within the MWI:

1) Everett's original recipe

In Everett's 1957 paper, he models an observer in a fairly simple way, considering only a set of memory elements. This is a sort of rough approximation of a computational model, but without the dynamics (which are crucial for a well-defined account of computation). Thus, Everett was a visionary pioneer in applying computationalist thinking to quantum mechanics, but he never confronted the complexity of what would be required to do a satisfactory job of it.

He assumed that the measure of a branch would be a function of its amplitude only, and thus would not depend on the specific nature of that branch. This is a very strong assumption, and arguably contains his next assumption as a special case already. [A more general approach would allow other properties to be considered, such as in Albert's fatness rule.]

[Note: Everett's use of the term 'measure' is not stated to refer specifically to the amount of consciousness, but in this context, the role it plays is essentially the same as if it did. Some authors use 'measure of existance' to specifically mean the squared amplitude by definition; obviously Everett did not, since he wanted to prove that his measure was equal to the squared amplitude. I recommend avoiding overly suggestive terms (like 'weight') for the squared amplitude.]

Next, he assumed that measure is 'additive' in the sense that if two orthogonal branches are in superposition, they can be regarded as a single branch, and the same function of amplitude must give the same total measure in either case.

If the definition of a 'branch' is arbritrary in allowing combinations of orthogonal components, the 'additivity' assumption makes sense, since it means that it does not matter how the branches are considered to be divided up into orthogonal components. [An argument similar to that would be presented years later in Wallace's 2005 paper, in which Wallace defended the assumption of 'equivalence' (branches of equal amplitude must have equal measure) against the idea of sub-branch-counting, based on the impossibility of defining the specific number of sub-branches. Everett did not get into such detail.]

With the previous assumption, 'additivity' would only hold if the measure is proportional to the squared amplitude; thus, he concluded that the Born Rule holds.

Everett considered the additivity requirement equivalent to saying that measure is conserved; thus, when a branch splits into two branches, the sum of the new measures is equal to the measure of the original branch. He gave no justification for the conservation of measure, perhaps considering it self-evident.

In classical mechanics, conservation of probability is self-evident because the probability just indicates something about what state the single system is likely to be in. If the probabilities summed to 2, for example, a single system couldn't explain it; perhaps there would have to be 2 copies instead of one. Yet the existance of multiple copies is precisely what the MWI of QM describes, and in this case, there is no a priori reason to believe that the total measure can not change over time.

Everett's attempted derivation of the Born Rule is not considered satisfactory even by other supporters of the MWI, because he did not justify his assumptions. Soon, other attempts to explain the probabilities emerged.

2) Gleason's Theorem

Also discovered in 1957, Gleason's theorem shows that if probabilities are non-contextual, meaning that the probability of a term in the superposition does not depend on what other terms are in the superposition, then the only formula which could give the probabilities is based on squared expansion coefficients. It is straighforward to argue that the correct expansion to use is that for the current wavefunction; thus, these coefficients are the amplitudes, which gives Born's Rule.

Unfortunately, there is no known justification for assuming non-contextuality of the probabilities. If measure is not conserved, the probabilities can not generally be noncontextual. Gleason's theorem is sometimes cited in attempts to show that the MWI yields the Born Rule, but it is not a popular approach since usually those attempts make (unjustified) assumptions which are strong enough to select the Born Rule without having to rely on the more complicated math required to prove Gleason's theorem.

3) The infinite-measurements limit and its frequency operator

The frequency operator is the operator associated with the observable that is the number of cases in a series of experiments that a particular result occurs, divided by the total number of experiments. If is assumed that just the frequency itself is measured, and if the limit of the number of experiments is taken to infinity, the eigenvalue of this frequency operator is unique and equal to the Born Rule probability. The quantum system is then left in the eigenstate with that frequency; all other terms have zero amplitude, as shown by Finkelstein (1963) and Hartle (1968).

This scheme is irrelevant for two reasons. First, an infinite number of experiments can never be performed. As a result, terms of all possible frequencies remain in the superposition. Unless the Born Rule is assumed, there is no reason to discard branches of small amplitude. Assuming that they just disappear is equivalent to assuming collapse of the wavefunction.

Second, in real experiments, individual outcomes are recorded as well as the overall frequency. As a result, there are many branches with the same frequency and the amplitude of any one branch tends towards zero as the number of experiments is increased. If one discards branches that approach zero amplitude in the limit of infinite experiments, then all branches should be discarded. Furthermore, prior to taking the infinite limit, the very largest individual branch is the one where the highest amplitude outcome of each individual experiment occurred, if there is one.

A more detailed critique of the frequency operator approach is given here. The same basic approach of using infinite ensembles of measurements has been taken recently by certain Japanese physicists, Tanaka (who seems unaware of Hartle's work) and (seperately) Wada. Their work contains no significant improvements on the old, failed approach.

4) Graham's branch counting

Neil Graham came out with a paper in 1973 that appears in the book "The Many Worlds Interpretation of Quantum Mechanics" along with Everett's papers and others.

Graham claimed that the actual number of fine-grained branches is proportional to the total squared amplitude of a course-grained macroscopically defined branch. Such sub-branches would be produced by splits due to microscopic scattering events and so on which act as natural analogues of measurements.

If it were true, it could also begin to give some insight into why the Born Rule would be true, beyond just a mathematical proof; that is, each fine-grained branch would presumably support the same number of copies of the observer. (That assumption would still need to be explained, of course.)

Unfortunately, and even aside from the lack of precise definition for fine-grained branches, he failed to justify his statistical claims, which stand in contradiction to straightforward counting of outcomes. He simply assumed that fine-grained branches would on average have equal amplitudes regardless of the amplitude of the macroscopic branch that they split from.

In the next post, the more recent attempts (other than my own) to derive the Born Rule within the MWI will be described.

Why 'Quantum Immortality' is false

2009-09-21T18:09:00.007-04:00

In the previous posts, I explained that effective 'probabilities' in an MWI are proportional to the amount (measure) of consciousness that sees the various outcomes. Because this measure need not be a conserved quantity, this can lead to nonclassical selection effects, with 'probabilities' for a given outcome still changing as a function of time even after the outcomes have been observed and recorded. That can lead to an illusion of nonlocality, which can only be properly understood by thinking in terms of the measures directly, as opposed to thinking only in terms of 'probabilities'.

The most extreme example in which it is crucial to think in terms of the measures, rather than 'probabilities' only, is the so-called 'Quantum Suicide' (QS) experiment. Failure to realize this leads to a literally dangerous misunderstanding. The issue is explained at length in my eprint "Many-Worlds Interpretations Can Not Imply 'Quantum Immortality'".

The idea of QS is as follows: Suppose Bob plays Russian Roulette, but instead of using a classical revolver chamber to determine if he lives or dies, he uses a quantum process. In the MWI, there will be branches in which he lives, and branches in which he dies. The QS fallacy is that, as far as he is concerned, he will simply find himself to survive with no ill effects, and that the experiment is therefore harmless to him.

A common variation is for him to arrange a bet, such that he gets rich in the surviving branches only, which would thus seem to benefit him. Of course in the branches where he does not survive, his friends will be upset, and this is often cited as the main reason for not doing the experiment.

That it is a fallacy can be seen in several ways. Most basically, the removal of copies of Bob in some branches does nothing to benefit the copies in the surviving branches; they would have existed anyway. Their measure is no larger than it would have been without the QS - no extra consciousness magically flows into the surviving branches, while the measure in the dead branches is removed. If our utility function states that more human life is a good thing, then clearly the overall measure reduction is bad, just as killing your twin would be bad in a classical case.

It is true that the effective probability (conditional on Bob making an observation after the QS event) of the surviving branches becomes 1. That is what creates the QS confusion; in fact, it leads to the fallacy of "Quantum Immortality" - the belief that since there are some branches in which you will always survive, then for practical purposes you are immortal.

But such a conditional effective probability being 1 is not at all the same as saying that the probability that Bob will survive is 1. Effective probability is simply a ratio of measures, and while it often plays the role we would expect a probability to play, this is not a case in which such an assumption is justified.

We can get at what does correspond for practical purposes to the concept of 'the probability that Bob will survive' in a few equivalent ways. In a case of causal differentiation, it is simple: the fraction of copies that survive is the probability we want, since the initial copy of Bob is effectively a randomly chosen one.

A more general argument is as follows: Suppose Bob makes an observation at 12:00, has a 50% chance QS at 12:30, and his surviving copies make an observation at 1:00. Given that Bob is observing at either 12:00 or 1:00, what is the effective probability that it is 12:00? (Perhaps he forgets the time, and wants to guess it in advance of looking at a clock, so that the Reflection Argument can be used here.) The answer is the measure ratio of observations at 12:00 to the total at both times, which is therefore 2/3.

That is just what we would expect if Bob had a 50% chance to survive the QS: Since there are twice as many copies at 12:00 compared to 1:00, he is twice as likely to make the observation at 12:00.

Most of your observations will be made in the span of your normal lifetime. Thus QI is a fallacy; for practical purposes, people are just as mortal in the MWI as in classical models.

In fact, there is a general argument to be made against immortality, which applies to immortality of any sort: If we were immortal (or very long-lived), then the effective probability of making an observation before we are older than a normal human lifetime would be zero (or very small). Since we find ourselves within a normal human lifetime, we can rule out immortality in favor of the competing hypothesis which assigns a high probability to such 'normal' observations, namely mortality.

Next up: Early attempts to derive the Born Rule in the MWI

Measure of Consciousness versus Probability

2009-09-16T13:47:00.005-04:00

In the last post, Meaning of Probability in an MWI, it was explained that in a deterministic Many-Worlds model, with known initial conditions, that which plays the role for of a probability for practical purposes is the ratio

(the measure (amount) of consciousness which sees a given outcome)
/ (the total measure summed over outcomes)

I call that the effective probability of the outcome.

Although the effective probability is quite similar to what we normally think of as a probability in terms of its practical uses, there are also important differences, which will be explored here.

The most important differences stem from the fact that measure of consciousness need not be a conserved quantity. By definition, probabilities sum to 1, but that is not all there is to it. In a traditional, single-world model, a transfer of probability indicates causality, while the total measure remains constant over time. This is not necessarily so in a MW model.

For example, suppose there are two branches, A and B. A has 10 observers at all times. B starts off with 5 observers at T0, which increases to 10 observers at T1 and to 20 observers at T2. All observers have the same measure, and observe which branch they are in.

So the effective probability of A starts off at 2/3 at T0, while the effective probability of B is 1/3. At T1, A and B have effective probabilities of 1/2 each. At T2, the effective probability of A is 1/3 and that of B is 2/3.

There are two important effects here. First, the effective probability of B increased with time. In a single-world situation, that would mean that a system which was actually in A was more likely to change over to B as time passes. But in this MW model, there is no transfer of systems, just changes in B itself.

This means that probability changes that would require nonlocality in a single-world model don't necessarily mean nonlocality in a MW model. If A is localized at X1, and B is localized at X2 which is a light-year away, there need not be a year's delay before the effective probability of B suddenly increases.

In a single-world local hidden variable model, probability must be locally conserved, so that the change of probability in a region is equal to the transitions into and out of adjacent regions only. This need not be so in an MW model.

The second important effect of nonconservation of measure in a MW model is that total measure changes as a function of time. Observers can measure, not only what branch they are on, but also what time it is. They will be more likely to observe times with higher measure than with lower measure, just as with any other kind of observation.

A good example of this is a model proposed by Michael Weissman - a modification of physics designed to make world-counting yield the Born Rule. His scheme involved sudden splitting of existing worlds into proposed new degrees of freedom, with a higher rate of such splitting events for higher amplitude worlds. The problem with it is that if new worlds are constantly being produced, then the number of observers would be growing exponentially. The probability of future observations, as far into the future as possible, would be much greater than that of our current observations. Thus, the scheme must be false unless we are highly atypical observers, which is highly unlikely.

It is important to realize that since changes in measure mean changes in the number of observers, decreases in measure are undesirable. This will be discussed further in the next post.

Meaning of Probability in an MWI

2009-09-11T13:43:00.010-04:00

The quantitative problem of whether the Born Rule for quantum probabilities is consistent with the many-worlds interpretation is the key issue for interpretation of QM. Before addressing that, it is important to understand in general what probabilities mean in a many-worlds situation, because ideas from single-world thinking can lead to unjustified assumptions regarding how the probabilities must behave. Many failed attempts to derive the Born Rule make that mistake.

The issue of what probabilities mean in a Many-Worlds model is covered in greatest detail in my eprint "Many-Worlds Interpretations Can Not Imply 'Quantum Immortality'". Certain work by Hilary Greaves is directly relevant.

First, note that for a single-world, deterministic model, such as classical mechanics provides, probabilities are subjective. The classic example is tossing a coin: the outcome will depend deterministically on initial conditions, but since we don't know the details, we have to assign a subjective probability to each outcome. This may be 50%, or it may be different, depending on other information we may have such as the coin's weight distribution or a historical record of outcomes. Bayes' rule is used to update prior probabilities to reflect new information that we have.

In such a model, consciousness comes into play in a fairly trivial way: As long as we register the outcome correctly, our experienced outcome will be whatever the actual outcome was. Thus, if we are crazy and always see a coin as being heads up, then the probability that we see "up" is 100%. Physics must explain this, but the explanation will be grounded in details of our brain defects, not in the physics of coin trajectories.

By contrast, in any normal situation, the probability that we see "up" is simply equal to the probability that the coin lands face up. [Even this is really nontrivial: it means that randomly occuring "Boltzman brains" are not as common as "normal people". As we will see, if we believe in computationalism, it also means that rocks don't compute everything that brains do, which is nontrivial to prove.]

In a many-worlds situation, it may still be the case that we don't know the initial conditions. However, even if we do know the initial conditions, as we do for many simple quantum systems, there would still be more than one outcome and there is some distribution of observers that see those outcomes.

Assume that we do know the initial conditions. The question of interest becomes (roughly speaking): 'What is the probability of being among the observers that see a particular given outcome?'

It is important to note that in a many-worlds situation, the total number of obsevers might vary with time, which can lead to observer selection effects not seen in single-world situations. Because of this the fundamental quantity of interest is not probability as such, but rather the number, or quantity, of observers that sees each outcome. The amount of conscious observers that see a given outcome will be called the measure (of consciousness) for that outcome.

In a deterministic MWI with known initial conditions, it will be seen that what plays the role of the “probability” of a given observation in various situations relates to the commoness of that observation among observers.

Define the 'effective probability' for a given outcome as (the measure of observers that see a given outcome) divided by (the total measure summed over observed outcomes).

1) The Reflection Argument

When a measurement has already been performed, but the result has not yet been revealed to the experimenter, he has subjective uncertainty as to which outcome occurred in the branch of the wavefunction that he is in.

He must assign some subjective probabilities to his expectations of seeing each outcome when the result is revealed. He should set these equal to the effective probabilities. For example, if 2/3 of his copies (or measure) will see outcome A while the other 1/3 see B, he should assign a subjective probability to A of 2/3.

Why? Because that way, the amount of consciousness seeing each outcome will be proportional to its subjective probability, just as one would expect on average for many trials with a regular probability.

2) Theory Confirmation

It may be than an experimental outcome is already known, but the person does not know what situation produced it. For example, suppose a spin is measured and the result is either “up” or “down”. The probability of each outcome depends on the angle that the preparation apparatus is set to. There are two possible preparation angles; angle A gives a 90% effective probability for spin up, while angle B gives 10%. Bob knows that the result is “up”, but he does not know the preparation angle.

In this case, he will probably guess that the preparation angle was A. In general, Bayesian updating should be used to relate his prior subjective probabilities for the preparation angle to take the measured outcome into account. For the conditional probability that he should use for outcome “up” given angle A, he should use the effective probability of seeing “up” given angle A, and so on.

This procedure is justified on the basis that most observers (the greatest amount of conscious measure) who use it will get the right answer. Thus, if the preparation angle really was B, then only 10% of Bob’s measure would experience the guess that A is more likely, and the other 90% will see a “down” result and correctly guess B is more likely.

3) Causal Differentiation

It may be the case that some copies of a person have the ability to affect particular future events such as the fate of particular copies of the future person. The observer does not know which copy he is. Pure Causal Differentiation situations are the most similar to classical single-world situations, since there is genuine ignorance about the future, and normal decision theory applies. Effective probabilities here are equal to subjective probabilities just like in the Reflection Argument.

4) Caring Coefficients

As opposed to Causal Differentiation, which may not apply to the standard MWI, the most standard way to think of what happens to a person when a “split” occurs is that of personal fission. Perhaps this is the most interesting case when an experiment has not yet been performed. Decision theory comes into play here: In a single-world case, one would make a decision so as to maximize the average utility, where the probabilities are used to find the average. What is the Many-Worlds analogue?

If it is a deterministic situation and the decider knows the initial conditions, including his own place in the situation, it is important to note that he should not use some bastardized form of ‘decision theory in the presence of subjective uncertainty’ for this case. It is a case in which the decider would know all of the facts, and only his decision selects what the future will be among the options he has. He must maximize, not a probability-weighted average utility, but simply the actual utility for the decision that is chosen.

Rationality does not constrain utility functions, so at first glance it might seem that the decider’s utility function might have little to do with the effective probabilities. However, as products of Darwinian evolution and members of the human species, many people have common features among their utility functions. The feature that is important here is that of “the most good for the most people”. Typically, the decider will want his future ‘copies’ to be happy, and the more of them are happy the better.

In principle he may care about whether the copies all see the same thing or if they see different things, but in practice, most believers in the MWI would tend to adopt a utility function that is linear in the measures of each branch outcome:

U_total = Σ_i Σ_p m_ip[Choice] q_ip

where i labels the branch, p denotes the different people and other things in each branch, m_ip is the measure of consciousness of person (or animal) p which sees outcome i, and is a function of the Choice that the decider will make, and q_ip is the decider’s utility per unit measure (quality-of-life factor) for that outcome for that person.

The measures here can be called “caring measures” since the decider cares about the quality of life in each branch in proportion to them.

Utility here is linear in the measures. For cases in which measure is conserved over time, this is equivalent to adopting a utility function which is linear in the effective probabilities, which would then differ from the measures by only a constant factor. In such a case, effective probabilities are used to find the average utility in the same way that actual probabilities would have been used in a single-world model in which one outcome occurs randomly.

Next: Measure of Consciousness versus Probability

Interlude: The 2007 Perimeter Institute conference Many Worlds @ 50

2009-09-07T13:14:00.011-04:00

As explained in the previous post, I had long been anticipating a conference on the MWI in 2007, and attended the Perimeter Institute conference Many Worlds at 50, armed with a copy of my then-new eprint on the Many Computations Interpretation.

When I arrived at my hotel the night before the conference, an older couple was checking in at the same time as I was. Someone asked the clerk for directions to the Perimeter Institute. It turned out that this couple was also attending the conference, and they were a couple of the friendliest and most interesting people I met there.

George Pugh had worked with Hugh Everett (founder of the MWI) at a defense contractor, Lambda Corp. (The work Everett did there is not so famous as his MWI but was actually important during the Cold War.) George and his impressive wife Mary had talked about the MWI with Everett himself, and they support it. They asked me which side I was on, as both pro- and con- people were attending the conference. I told them I was in favor of the MWI. They liked to hear that. We ended up having meals together on several occasions over the course of the conference.

The conference itself consisted mostly of lectures in a classroom-like atmosphere, followed by questions from the audience. Appropriately, most of the talks focused on the question of probability in the MWI.

However, and unfortunately, they mainly focused on the attempt to derive the Born Rule from decision-theoretic considerations. That approach was proposed by David Deutsch in 2000, and further developed by Simon Saunders and especially by David Wallace. Saunders and Wallace gave talks that mainly reiterated what is in their papers. There were also talks that (correctly, though of course this was not accepted by Wallace's supporters) pointed out the failures of that approach, such as those by Adrian Kent and David Albert.

The only other approach to the Born Rule that was presented at a talk was that of W. Zurek, who talked about his (equally fallacious) 'envariance' approach. Most people seemed to agree that Zurek's approach was similar to Wallace's. There was little discussion of it beyond that. When Zurek was asked about Wallace's approach during an informal discussion, he basically said that he didn't know if Wallace's approach was correct also, but he didn't seem to think it matters much, because his own approach showed that the Born Rule followed from the MWI. When I tried to point out to him why his approach fails - a task made all the more difficult by his somewhat intimidating large physical presense and lion-like bearded appearance - he didn't understand my point and soon ended the conversation.

Max Tegmark was a speaker, and he briefly discussed his heirarchy of many-worlds types, up to the Everything Hypothesis for which he is known.

Besides that, the only other controversy addressed in the talks was that of the legitimacy and meaning of talking about probability in the deterministic MWI, which is a seperate question than the quantitative problem of deriving the Born Rule. This focused on Hilary Greaves' 'caring measure' approach. She is sometimes lumped in with the decision theoretic approach to the Born Rule, because she uses decision theory in another way, but in fact her ideas are independent of that and are basically correct though not the full story.

The official speakers were basically divided into two camps: Those MWI-supporters who supported Wallace's attempted derivation of the Born Rule or who were considered allies of it (like Zurek and Greaves), versus those who not only rejected it but also were against the MWI in general (like Kent and Albert). Tegmark was neither but his one talk was largely ignored, and he did not address the Born Rule controversy.

Among the attendees, however, the situation was more complicated. I was not the only one who supported some kind of MWI, and considered understanding the Born Rule to be the key issue of interest, but utterly rejected the approaches to the Born Rule that had been presented. The alternatives that we wanted to discuss involved some form of observer-counting as the basis for probabilities in an MWI, even if it required some new physics. This led to a minor rebellion, in which a few of us tried to talk about our ideas during a lunch period in the room set aside for the conference lunch. The only official speaker that we got any help from was Hilary Greaves. We were able to speak in the lunchroom for a little while, but it didn't get much attention.

There was another young woman by the name of Hillary, I think a physicist studying at the Institute, who also helped us set up the lunchtime discussion.

The 'counter' camp included Michael Weissman, who proposed a modification of physics in order for world-counting to yield the Born Rule. His scheme involved sudden splitting of existing worlds into proposed new degrees of freedom, with a higher rate of such splitting events for higher amplitude worlds. This was interesting, but I was skeptical, and after thinking about it for a while I found the fatal flaw in it. If new worlds were constantly being produced, then the number of observers would be growing exponentially. The probability of future observations, as far into the future as possible, would be much greater than that of our current observations. Thus, the scheme must be false unless we are highly atypical observers, which is highly unlikely. While false, Mike's model serves as a good way to discuss the need for approximate conservation of measure for a successful model. In any case, Mike proved to be a good guy to talk to.

Also among the 'counters' was David Strayhorn, who proposed that an indeterminacy in General Relativity could lead to a Many Worlds model in which spacetime topologies were distributed according to, and formed the basis for, the Born Rule. His ideas did not seem fully developed, and I was skeptical of them as well, but we had interesting discussions.

Another guy with us was Allan Randall. He supports Tegmark's Everything Hypothesis, and is also interested in transhumanism and immortality. As I explained to Allen and to Max Tegmark, I wasn't sure about the Everything hypothesis, because of the problem of what would determine a unique measure distribution, but I used to support it and still like it. I think it's important and maybe useful. After all, and like many supporters of the hypothesis, I discovered a version of it on my own long before I ever heard of Tegmark.

Which brings me to a subject that received little official mention at the conference, the 'Quantum Immortality / Quantum Suicide' fallacy which Tegmark had publicized. This is the belief, which many MWI supporters have come to endorse, that the MWI implies that people always survive because some copies of them survive in branches of the wavefunction. I had always regarded this as the worst form of crackpot thinking, and had hoped to discuss it at the conference as something that MWI supporters must crush before it gets out of hand. My brief discussions about it at the conference convinced me that it was not getting the condemnation that it deserves. This ultimately led me to write my own eprint against it, Many-Worlds Interpretations Can Not Imply 'Quantum Immortality', despite my misgivings that even discussing the subject could give the dangerous idea extra publicity.

I also had interesting discussions with Mark Rubin, who had shown an explicit local formulation of the MWI using the Heisenberg picture, which is something I still need to study more. Mark and I had dinner with the Pughs. I liked the Swiss Chalet restaurant and Canadian beer.

I also happened to run into a friend of mine from NYU, where I got my Ph.D. in physics. Andre is a Russian who came to the US to study, and he had a postdoc at the Perimeter Institute. He's not an MWI supporter or really into interpretation of QM, but he knew that I am, so he was not too surprised that I showed up at the conference. I was lucky to run into him, because the next day he was heading to England for a postdoc there, studying quark-gluon plasmas using the methods he learned from models of string theory. He said he might never return to the US.

All in all, it was certainly an interesting experience. Ultimately, though, it was disappointing because I didn't get to discuss my paper much, and I never was able to have a substantive discussion with the well-known figures in the field who were there to present their own work. It was largely a lecture series rather than an egalitarian discussion group. Some discussion took place on the sidelines, such as at meals, but that was limited in who you happened to be next to. Well-known people mainly talked to each other.

One thing that grew out of the discussions on observer-counting was that a group of us decided to continue the discussion on-line. This led to the creation of the OCQM yahoo group, which included David Strayhorn, Michael Weissman, Allan Randall, Robin Hanson, and myself. Robin had not been at the conference, but he was the originator of the Mangled Worlds approach to the Born Rule, and accepted our invitation to join the group. In practice, however, posts to the group largely came from just David and myself. We all supported some form of observer-counting, but our approaches were quite different. We had some very interesting discussions, and it was a good place to 'think out loud', but ultimately even David's posting to the group petered out and it seems dead at this point.

I gave the Pughs my printed copy of the MCI paper. They were compiling a book in which they would quote various people about why Everett's interpretation of QM was important, so I wrote a few lines for them. Ultimately they decided not to use it though. I think they didn't like my criticism of the current status of the Born Rule in the MWI.

Interlude: Anticipating the 2007 Many Worlds conference

2009-08-27T22:02:00.009-04:00

For many years, I knew it was coming. You just had to do the math: Hugh Everett III had published his thesis, which introduced the Many Worlds Interpretation (MWI) of quantum mechanics, in 1957. So, somewhere, there would be a 'Many Worlds at 50' conference in 2007. And I would be there.

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Back in 2000, I attended the conference ‘One Hundred Years of the Quantum: From Max Planck to Entanglement’ at the University of Puget Sound, which commemorated Planck's paper which first introducted the concept of energy quantization, used to explain why the equilibrium density of thermal radiation is not infinite.

I had already started exploring the concepts behind the Many Computations Interpretation (MCI). [I called it the 'Computationalist Wavefunction Interpretation' (CWI) but that just didn't have the same ring to it.] It grew out of David Chalmer's suggestion, in the last chapter of his book The Conscious Mind, that applying computationalism to quantum mechanics was the right way to make sense of the MWI. But I knew that computationalism had to be made more precise before that could be done, and I knew that the Born Rule would be the key issue.

I submitted a short paper about it for the conference book. The paper is still available online at
http://www.finney.org/~hal/mallah1.html

At the conference I met a few well known physicists, the most famous of whom was James Hartle. At the time, the 'Consistent Histories' approach to interpretation of QM was getting a lot of attention, and Hartle and Murray Gell-Mann had written a book about it. As far as I was concerned, that approach was not of much interest, because it pretended that single-world-style probabilities could be assigned to terms in the wavefunction 'once decoherence occurred' despite the fact that decoherence is never truly complete. (Probabilities can not generally be assigned in the sense that, prior to decoherence, interference effects can occur and only be understood as showing the simultaneous existance of multiple terms in the wavefunction.)

It was also maddeningly vague about what exactly was suppposed to really exist, and declared that some questions must not be asked. It was not clear whether it was really just the MWI in drag, deliberately using vague language so as not to scare away those who thought the MWI is too weird, or if it was some new variant of the single world Copenhagen Interpretation. Its advocates publically claimed inspiration from both sources!

I got the chance to ask Hartle a question. I asked him two things:

1) Is Consistent Histories the same as the MWI?

He said it is. That provoked a gasp from the audience! You see, Consistent Histories was looked on quite favorably by many physicists at the time, while the MWI was still largely dismissed as material for science fiction.

2) Is it the same as the Pilot Wave Interpretation?

He said it's not. The second question was necessary because some people, especially those who like the Copenhagen Interpretation, consider experimental predictions to be the only thing that matters - so that they would consider all interpretations which give the same predictions to be the same thing. Now I knew that was not the case with him, so the first answer really did mean something.

Anyway, after that conference I resolved to try to make my interpretation of QM precise in time to discuss it at the inevitable 2007 conference. Seven years should be enough time, right? Of course, it was never my day job, just a hobby of sorts.

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In 2002 I attended ‘Towards a Science of Consciousness’ (TSC), a yearly philosophy conference which was held at the University of Arizona that year and every even year. That was interesting in its own right, as I met interesting people and learned about issues and thought experiments in philosophy of mind which I had not previously been exposed to. (I don't think it would be as interesting to attend another TSC, because many of the issues are the same every year, unless I have published something of my own that will be talked about. But it's not bad so perhaps I will.)

At that 2002 TSC, I participated in the poster session, with a poster called “What Does a Physical System Compute?” which laid out my ideas about an implementation criterion for computations. It got little attention, except that David Chalmers himself was kind enough to stop by and consider it. He made some comments and criticisms. I'd had many false starts at formulating a criterion, and had discussed it by email with him, so he knew what it was about. The criteria I listed weren't good enough, and we both knew it, but I believed it was a step in the right direction.

[Some of the other posters there were interesting, but I remember only one, because it stood out as being the most crackpot idea I'd yet encountered - and I'd encountered many on the usenet newgroups. This guy was combining the kooky notion that humans only became conscious when language was invented, with the crazy idea that only consciousness causes wavefunction collapse, to argue that _the biblical age of the Earth is correct_ (a few thousand years) because that's when the first wavefunction collapse brought the universe into real existence! Quite a combination!]

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So, years passed by and before I knew it the 2007 Perimeter Institute conference Many Worlds @ 50 was approaching. This was it; the conference I'd been looking forward to for so long, in which I hoped to discuss my ideas about the MWI with other supporters of the interpretation. Would I be ready? I'd had some success in refining my implementation ideas, and scrambled to write up what I had.

The Born Rule still eluded me, though. I had hoped that once I found the precise criteria for existence of an implementation, I could apply it to quantum mechanics and the Born Rule might pop out. After all, it's actually fairly easy to get the Born Rule to pop out if you impose certain simple requirements such as conservation of measure. People have been doing it for years without even realizing they'd made unjustified assumptions. All I had to do was find a reason to justify an assumption like that for the counting of implementations.

I didn't find that justification, and time was getting short. I turned to an unusual approach for inspiration - Robin Hanson's 'Mangled Worlds' papers. He had a rather innovative approach to the MWI, in which large terms in the wavefunction 'mangle' small ones, leading to an effective minimum amplitude, and he argued that the Born Rule followed from counting worlds (lumps of wavefunction) in the distribution of survivors. The world-counting appealed to me, as it could easily be translated into implementation-counting, but I did not believe his scheme could work: large worlds would not 'mangle' worlds they had decohered from nearly as much as Hanson had assumed.

To get that kind of thing to work, I had to assume new physics, contrary to Everett. But the new physics was fairly simple: random background noise in the wavefunction (which could be part of the initial conditions rather than new dynamics) could 'mangle small worlds' and if it does the Born Rule pops out (in an interesting new way). There were still some real questions about whether this could work out right, so I explored a more direct approach as well in which I tried to rig the way implementations are to be counted in order for it to come out right. That turned out to be easier said than done, and it remains an open question about whether it can or should be done, though I regard it more favorably now. All of this will be discussed in later posts.

I also discussed other alternatives, such as an MWI with hidden variables, and other ways that a minimum amplitude could be introduced. The basic conclusion was that computationalism strongly favors some kind of MWI over single-world interpretations, even if both have hidden variables, but the details are unknown (and might always remain so).

I wrote all this up and added criticisms of the incorrect attempts to derive the Born Rule in the MWI, including the one based on decision theory, which was widely considered the strongest of the attempted derivations although it had its critics. This became my MCI paper, which I placed on the preprint arxiv: http://arxiv.org/abs/0709.0544

I knew that I was cutting it close, so I emailed some of the people who had written about the MWI and who would attend the conference to tell them about my paper on the arxiv.

It was time to go to Canada and see if the 2007 MWI Perimeter Institute conference would live up to the anticipation.

Futher Study

2009-08-27T21:39:00.003-04:00

I'd like to wait for some comments for this one. What do you want to learn?

I assume you know how to search the web. The Stanford Encyclopedia is good for many topics, as is Wikipedia. Though as always, don't assume that something is true just because you read it there. You must develop an eye for controversial issues.

What I have attempted to do so far here is to provide an easy to understand overview of many issues surrounding interpretation of quantum mechanics. That should be most useful to students who intend to pursue a serious interest in philosophy of physics.

I will add references here on an irregular basis. Traffic on this 'blog' is not high as of yet so there is no typical reader. If that changes, I expect some requests. Unlike a typical blog, I edit these posts as needed to cover a topic, rather than just making new posts all the time.

You can email jackmallah@yahoo.com if you don't want to post a comment.

You can also add your own links in your comments.

From here on out, the focus of the 'blog' will change from review of QM to discussion of contemporary research topics related to the MWI, but still will hopefully be understandable.

Studying Quantum Mechanics: Measurement and Conservation Laws

2009-08-20T14:16:00.006-04:00

When you learned that the results of measurements in quantum mechanics are random, it may have raised a question in your mind: What about conservation laws? Do they only hold on average? For example, if you measure the energy of an atom, you might end up with a different amount of energy than the average, right? If there are random fluctuations in 'conserved' quantities, could the effect be used to violate conservation laws in a systematic way?

For example, consider a spin measurement for spin-1/2 particles. Each particle's spin carries an amount of angular momentum equal to hbar/2 in the direction it points. The particles are prepared so that their spins point in the +Z direction, and then sent into a Stern-Gerlach (SG) device, which we can rotate to measure spin along any direction. If we measure a spin in the X direction, the result is that the spin ends up in either the +X or -X direction. So it looks like we are violating conservation of angular momentum in a systematic way, destroying the +Z direction angular momentum we prepared the particles with. If that were true and the experiment is done in an isolated satellite, we could use it to build up a net angular momentum in the -Z direction.

If conservation laws mean anything, there must be something wrong with the above picture. Perhaps, one might think, there must be some back-action of the particles on the Stern-Gerlach device. That is, the missing angular momentum is being transferred into the SG device, as the particles exert torques on it with their magnetic moments as they come through.

The problem we run into next is that this seems to violate linearity: A +Z spin can be written as a superposition of a +X term and a -X term. After going through the SG device, there is decoherence (or as some people wrongly assume, wavefunction collapse), and what is observed is just a +X result or a -X result. Since QM is linear, the final wavefunction is a linear superposition of the terms that would have resulted if the original spins had been +X or -X. Such terms do not take the original +Z spin into account. So at least as far as an observer within such a term is concerned, there is no residual effect of the original spin direction, such as we would need if the SG device had recieved angular momentum that depended on that direction.

The solution to this puzzle, naturally, is to treat the measuring device as a fully quantum-mechanical system. That means that its angular orientation can not be precisely known, due to its finite uncertainty in angular momentum. (The uncertainty principle applies, limiting how small the product of the uncertainties of angle and angular momentum can get.) As a result, there will be very small 'error' terms in which the wrong spin outcome is measured, i.e. -X instead of +X, or an incoming spin is flipped.

This effect may seem negligable, but it is enough to allow the information about the original direction of the particle spin to be encoded in the final state of the SG device. It works out to be exactly enough of an effect to enforce the conservation law. The uncertainty in the SG device's angular momentum allows a sort of selection effect; in effect, the 'lost' angular momentum does end up in the SG device. The same kind of effect holds for all conservation laws. This is explained in detail in my eprint "There is No Violation of Conservation Laws in Quantum Measurement". It was first studied by Wigner in 1952, and is related to the Wigner-Araki-Yanase theorem (1960).

Key definitions for QM: Part 3

2009-08-12T13:53:00.005-04:00

Previous: Key definitions for QM: Part 2

In this post some additional QM terms will be defined. These often come up in applications of QM and might come into play for interpretation issues.

Hamiltonian: In classical mechanics, the Hamiltonian is a function of the configuration and velocities that equals the energy of the system, giving the energy as a sum of that of the various types of energy in the system. In QM, it is a corresponding linear operator on the wavefunction. The Hamiltonian appears in the Shrodinger equation. Its eigenstates have definite values for energy. A wavefunction that is an energy eigenstate will not undergo change as time passes except as a standing wave, undergoing phase rotations.

commute: Let A and B be operators. They commute if A B psi = B A psi for any function psi. This is written as AB = BA or [A,B]=0.

If two operators don't commute, then measurements associated with one of them will change the probabilities for values of measurable quantities associated with the other, and they can not be measured simultaneously.

Position does not commute with momentum (which is mass times velocity). Spin measurements in different directions also don't commute.

(Heisenberg's) uncertainty principle: There is a minimum uncertainty for the product of measurable quantities that don't commute. This follows from the math (and the Born Rule). Most famously, the product of (spread in position) (spread in momentum) >= hbar/2.

This can be understood roughly as follows: Momentum is related to the wavelength of sinusoidol patterns in the wavefunction - those are its eigenstates (actually there is an imaginary component as well - the wavefunction is complex-number-valued). If the wavefunction is concentrated near a point (small uncertainty in position), then it must be built up out of a superposition of a wide range of sinusoidal functions. If on the other hand it is in a nearly sinusoidal pattern, then it must be spread out over a large range of positions.

(The Born Rule comes into play because we assume the usual relation between probability and the square of the wavefunction.)

There is a similar uncertainty principle that relates uncertainty in energy and time.

Wikipedia's article has a more in-depth explanation.

boson: Particles with "integer spin" have spin component eigenvalues that are integer multiples of hbar. Such particles are bosons, which means that they have a tendency to occupy the same states as identical particles of the same type; technically, their wavefunctions are symmetric with respect to exchanging the particles. Photons (particles of light) are bosons, which lets them reinforce each other and produce the classical-seeming behavior of electromagnetic fields.

fermion: Particles with half-integer spin are fermions, which means they cannot occupy the same state as identical particles of the same type; technically, their wavefunctions are anti-symmetric with repect to exchanging the particles.
The connection between spin values and boson/fermion behavior is a consequence of relativistic quantum field theory (QFT). Actually in QFT there are no particles, just quantized excitations of the fields. It is not surprising that treating excitations of the fields as though they were particles (as is done in the nonrelativistic approximation, used very often in QM) would require some special treatment of the so-called particles with respect to the symmetry of exchanging them.

Pauli exclusion principle: This is the principle that, as mentioned above, no two fermions of the same type can occupy the same state. Electrons are fermions, so this is very important in atomic physics and chemistry. Atoms have various shells of electrons which can be though of as built up by adding one electron at a time. When an inner shell is fully occupied, another electron can't occupy one of those states, so it will end up in the next shell out. (If placed in an even higher shell out, it will fall to the innermost shell it can, emitting a photon to carry away the extra energy.)

Shrodinger Picture: This is the usual formalism in which the wavefunction varies with time while measurable quantities are associated with fixed linear operators. There is a global time.

Heisenberg Picture: This is a formalism in which the wavefunction is static but linear operators vary with time, giving the same Born Rule probabilities for measured outcomes. In relativistic quantum field theory, this picture has fewer problems of being mathematically well defined (with infinite renormalization, or re-scaling of certain quantities) than the Shrodinger picture, and is also the only local formulation of QM. (Local meaning that things defined at points in ordinary space only interact with their neighbors.)

However, this formalism is harder to work with. It is also believed that infinite renormalization will not be necessary for a fundamental model that includes quantum gravity.

The Heisenberg picture has the strange feature that interactions carry labels with them of what has been interacted with, and these proliferate as more and more systems interact. Basically, at each point, a field operator encodes information about the correlations of the field at that point with the set of field configurations over all space. At each point in space the field operator must be capable of carrying an unlimited amount of information about all of space and updating it as time passes. Although there are no shortage of infinities in most models of physics, this would seem surprisingly inefficient for the fundamental working of nature. Of course, nature has surprised us before.

For more detailed and technical information about locality and label proliferation in the Heisenberg picture see
Locality in the Everett Interpretation of Heisenberg-Picture Quantum Mechanics
Locality in the Everett Interpretation of Quantum Field Theory

Next: Further study

Key definitions for QM: Part 2

2009-08-11T14:28:00.016-04:00

In the last post, definitions Part 1, I explained some of the terms that commonly come up in interpretation of QM and described their roles in that context. Here, I will define some other useful terms; these are more technical and less key to understanding most interpretation issues, but still handy in that context (and fun!) You can look up the equations that are involved; my concern here is with what is relevant to interpretations.

spin: This refers to a property of individual particles that behaves like an intrinsic angular momentum. When measured, it has a constant magnitude, and the component of it in the measured direction can only take on a few discrete values.

Spin-1/2 particles, such as an electron, have two possible eigenvalues of their measured spin component: + or - 1/2 hbar. When not measured, they are in a superposition of the eigenstates with the allowed values (or in an entangled state). Such a superposition is always an eigenstate of the spin measurement operator in some other direction, though an entangled state is not.

degenerate: While this term may refer to modern society, in the context of QM it means that there are more than one eigenfunctions of a particular operator with the same eigenvalue. Measurements based on that operator will not cause degenerate eigenfunctions to decohere - for example, if you measure energy and there are two eigenstates with the same energy, those two states will remain in a coherent superposition and the observer will not distinguish out a unique eigenstate.

bra and ket notation: Dirac invented this useful notation in which a function can be represented by a 'ket', or the second half of a bracket, written |label> where "label" is used to describe which function is being referred to. For example, if f(x) = sine(ax)exp(-bx^2), one could write |f_ab>.

A 'bra', or the first half of a bracket, represents the complex conjugate of the same function.

It is written < F_ab|

A 'bracket', such as < f | g >, represents the integral (sum over the configuration space) of the bra function (here, complex conjugate of f) multiplied by the ket function (here, g).

Often (though not always), bras and kets are normalized so that < f|f> = 1.

If you see two kets next to each other, such as |b>=|f>|g>, this means function b is the product of the function f that lives in the configuration space of one system and g which lives in another system: b(x,y) = f(x) g(y).

quantized: Some measurements have discrete possible outcomes, and such quantities are called quantized. For example, the energy levels of an electron in a hydrogen atom are quantized, but the energy of an electron that escapes the atom can take on a continuous set of values so it is not quantized.

It can also refer to obtaining a quantum mechanical model from a classical one.

'Quantum mechanics' originally referred to quantized quantities but is now used to describe the whole branch of physics which deals with related phenomena such as the wavefunction.

Plank's constant, h: This is a constant that appears in the Shrodinger equation. It sets the scale at which quantum phenomena have noticable direct effects. It has units of 'action', units which are those of (mass)(velocity)(position). More commonly encountered is hbar, which is h/2 pi. The plain h is more useful for full oscillating cycles, which are common enough with waves, while hbar is useful for instantaneous rates of change.

h = 6.63 x 10^-34 kg m^2 / s

geometric optics limit: While there are many issues involved in deriving the appearance of a classical world from the wavefunction model, if we grant the validity of the Born Rule for probabilities then an important part of the derivation of classical mechanics is simple after that:

When the wavelength of a wave is much smaller than the size of whatever openings it goes through, the spreading out of the wave become negligable. This is the same reason that light waves can be treated as coming out of a flashlight in straight lines, while sound waves much more noticably bend around corners. There are still small tails where a tiny portion of the wave's squared amplitude will spread off, which is why I invoked the Born Rule, since it lets us neglect that part.

Just as most of a light wave will move in a straight line, most of a quantum matter wave for a non-microscopic (macroscopic) object will follow the trajectory predicted by classical mechanics. The small bit that will not generally has a Born Rule probability so low that it is effectively impossible to ever measure.

Hilbert space: The state of a quantum system is given by the wavefunction, which is a function on configuration space. This can be thought of as representing a 'state vector' in an abstract space.

An ordinary vector in regular 3 dimensional space is a quantity which has both direction and magnitude - for example, velocity. It can be represented by x,y,z components: V = (vx, vy, vz). Thus, in a particular coordinate system, it is written as a function of a discrete index which can only take on 3 values; e.g. V(1) = vx, V(2) = vy, V(3) = vz.

A Hilbert space is a generalization of this to describe any function as a vector in some high-dimensional space.

Philosophically, thinking of the function as a vector implies that the particular coordinate system in which the components are spelled out is not very important. The physical nature of a velocity would be the same no matter what x,y,z coordinate system we are working in, but the components would look different.

quantum mechanical basis: This is analogous to choosing a coordinate system to write the vectors of the Hilbert space in. Changing a basis is analogous to rotating the directions of your coordinate system components.

Examples include 1) position basis, in which the wavefunction is a function of position as expected, 2) momentum basis, in which the wavefunction is a function of particle momentum, which is (mass)(velocity). It may not seem at first that the two pictures are equivalent, because in classical mechanics, knowing the momentum will not tell you the position. But in quantum mechanics, in which the wavefunction is a complex number valued function, knowing the wavefunction at every point in the momentum basis is enough to find it in the position basis and vice versa.

Just as there is rotational symmetry in ordinary space which makes it impossible to know if there is in nature any actual, fundamental coordinate system in which vectors really have three components, it is impossible to know what the "actual basis that nature uses" is in QM. Under the Copenhagen interpretation, the assumption was that there is no such thing - only things we can measure were considered real.

As will be seen, more mechanistic, literal views of mathematical models (such as the MWI is) seem to require some actual basis that nature would use.

Next: Key definitions for QM: Part 3

Key definitions for QM: Part 1

2009-08-07T14:13:00.026-04:00

Before turning to the more advanced issues that I created this blog to discuss, it would be good to give at least a brief summary of a few of the basic terms related to QM that often come up in discussion of interpretations.

Here I will explain some of the terms that seem most likely to cause confusion and which are most directly involved in the interpretation of QM. Rather than technical definitions, here I am concentrating on their role in interpretations. In the next post, definitions Part 2, I'll define some slightly more technical terms which often arise in the context of discussions of QM but which are not as essential for the basic interpretation issues.

wavefunction: The mathematical model of QM, in it most standard formulation, deals with a complex-valued function of the configuration space that undergoes wave-like motions. In a fit of inspired creativity, it was dubbed the wavefunction ;) It is often represented by the Greek letter psi, which looks like a U with a vertical line through the middle and extending below the curve.

measurement: This usually refers to an experiment in which a human observes a macroscopic instrument to determine the outcome. As such, it is an emergent phenomenon and can play no fundamental role in the mathematical model of the physical world. However, it certain plays a fundamental epistemological role in our ability to learn about the world.

When a 'measurement' occurs, the result is one of the allowed results (an eigenvalue of the measurement operator) and subsequently the wavefunction appears to behave as if it had been placed in the corresponding eigenstate (see below).

eigenfunction/eigenvalue: These terms from German, now used in linear algebra, describe mathematical properties of certain functions. Eigen- means characteristic. Each measureable quantity (such as energy) corresponds to a linear operator. Each such operator A give a spectrum of solutions to the equation A psi_i(x,..) = c_i psi_i (x,..) where c_i is a constant. Here i is an index for however many values work for that operator. The different constants are called eigenvalues, and I'll let you guess what the corresponding functions are called ;) A wavefunction that is an eigenfunction is called an eigenstate.

Copenhagen Interpretation: This was "the standard view" of most physicists during much of the 20th century. Essentially, it said that "when a measurement occurs", the wavefunction "collapses" to give one of the allowed outcomes. It was never possible to define the exact circumstances under which a measurement was supposed to occur, or to say what "collapse" was like. This interpretation has become widely recognized as incomplete at best, or less charitably, as ill defined and utterly implausible. In practice, it often meant "shut up and calculate" - it allowed the interpretation question to be swept under the rug so that physicists could work on practical problems instead, like making atomic bombs. It is no longer taken seriously by most philosophers of QM.

Shrodinger equation: This is a deterministic, linear equation that gives the time evolution of the wavefunction. In the MWI, this equation always holds true.

It does not produce any "collapse of the wavefunction" so certain other interpretations must modify it, either explicitly (continuous collapse models) or by hand waving talk about 'measurement' (Copenhagen).

"collapse of the wavefunction": It was long believed that another process, not described by the Shrodinger equation, must occur during 'measurement' in which a single random outcome is chosen - the so-called 'collapse of the wavefunction'. In 1957, Everett argued that no such 'collapse' is needed to explain what we see - he proposed the MWI.

Caution: Even people who believe the MWI sometimes use a sloppy terminology in which they talk about "collapse of the wavefunction" when it is supposed to be understood that they really only mean the illusion of such collapse due to decoherence. I dislike this misleading terminology.

Born Rule: When a 'measurement' occurs, the probability of each outcome is given by the absolute value of the square of the overlap integral of the wavefunction with the corresponding eigenstate. It is an open question as to whether and how the standard (Shrodinger equation only) MWI can explain this, or if not, what does. This is the key issue in interpretation of QM.

linear superposition: One of the most important properties of the Shrodinger equation is that it is linear. This means that if F1 is a solution of the equation, and so is F2, then the sum (superposition) F3 = F1 + F2 is also a solution. (The actual solution that physically occurs depends on the initial conditions - the starting state.)

In a measurement-like situation, and more generally whenever two systems interact, the solution will usually have a branching type of behavior. For example, when a photon hits a half-silvered mirror, there will be part of the wave that is reflected and another part that passes through. The subsequent behavior of these two parts of the wave does not depend on what the other part is doing. If the two parts are brought back together, the resulting wavefunction is a linear superposition of the parts. This can result in an interference pattern.

Linearity forbids collapse because if F1(0) evolves to F1(t), and F2(0) evolves to F2(t), then F1(0) + F2(0) must evolve to F1(t) + F2(t). Collapse, by contrast, would mean replacing the sum by randomly selecting only one, either F1 or F2.

decoherence: This refers to the way in which different branches of the wavefunction stop interfering with each other. Linearity prevents different terms in a superposition from changing each other but it still permits cancellation or reinforement between parts of the functions - interference patterns, e.g. a positive part of F1(x) cancelling a negative region in F2(x).

Decoherence usually means that a system becomes entangled (correlated) with the environment in a robust way. This is generally an irreversible process in the statistical sense, much like an increase in entropy in statistical mechanics.

Once entangled with the environment, interference patterns are no longer seen because the functions F1(x1,x2,...) and F2(x2,x2,...) now have most of their nonnegligable regions in different parts of configuration space: They may still overlap in terms of the x1-dependence (the microscopic system under study), but they occur in different parts of the environment variables' space, e.g. x2.

In principle, an interference pattern could be restored if the x2,... dependence were also brought back into overlap. In practice, there are so many particles in the environment that doing this is not feasible. Thus, decoherence creates the illusion of irreversible 'collapse of the wavefunction".

Also, it is now fairly well understood that in measurement-like situations, in which an interaction exists that tends to seperate out components of the wavefunction that have different eigenvalues for what is being measured, the different eigenstates will tend to decohere. This explains part of the measurement puzzle from the MWI perspective.

entanglement: This means that the wavefunction has correlations between the states of two or more systems. For example, F(x1)G(x2) is a product state and is not entangled, but F1(x1)G1(x2)+F2(x1)G2(x2) is a correlated, entangled state. Entanglement with the environment results in decoherence and the illusion of "collapse of the wavefunction".

Entangled states between small numbers of controlled particles are also important, because they display various non-classical behaviors, such as violations of Bell's inequalities when measured, and are useful in quantum computing and quantum cryptography.

Next: Key definitions for QM: Part 2

Studying Quantum Mechanics: the Delayed Choice example

2009-08-05T15:04:00.011-04:00

Most descriptions of QM are not very good. In particular, the configuration-space-wave-mechanical aspects of QM are usually not fully taken into account; instead, a nearly incomprehensible description is given in more classical terms.

Delayed Choice experiment:

For example, consider a delayed-choice thought experiment in which a photon can take two paths simultaneously. If the experimenter wants, he can "determine which path the photon took" by letting it hit a pair of detectors; it will register in only one detector, randomly chosen, as far as he can tell. The paths are laid out in such a way that in order for it to hit a detector, it must have taken the corresponding path. Taking the other path would cause it to sail past that detector and into the other one.

(image from http://scienceblogs.com/principles/2007/02/thought_experiments_made_real_1.php)

Or, he can insert a 'beamsplitter' (BS2; conceptually, a half-silvered mirror) to recombine the beams, in a way that results in the photon always going to the rightmost detector due to wave interference - in which case it must have taken both paths. He can choose whether to insert the mirror just before the photon reaches the detectors, after most of the paths would have already occurred!

Mysterious stuff, right? It looks like the experimenter reached back in time, changing whether the photon took both paths or chose one randomly!

Sure - if you think about it the wrong way.

In terms of wave mechanics (which is the MWI), the photon took both paths in all cases. If the beam recombiner is not present, the photon becomes entangled with the detectors - that is, becomes correlated with their degrees of freedom in configuration space. In one 'branch' of the wavefunction, one detector clicked; in the other, the other did. There is no 'delayed choice' mystery. [There is only the standard question for the MWI of what explains the appearance of probabilities - the old Born Rule problem.]

Most of the mysterious aspects of QM make a lot more sense when viewed as just wave mechanics in configuation space. But it's hard (impossible?) to find an introductory treatment of QM that even mentions configuration space. An advanced treatment of QM is unlikely to be much better - the equations will be there, but with little explanation.

The next post will be a basic glossary of common QM terms such as 'entangled'. Then I should be getting on to start discussing MWIs in more detail.

on external links

2009-08-05T11:21:00.003-04:00

I wrote in a comment on the "Why MWI?" post:

"External links can be very useful, and thanks for the tips, but there is one problem: There is liable to be something I disagree with at most links. For example, while the article on collapse interpretations that you gave a link for is good, it casts them in a more favorable light than I would. I mentioned collapse in my blog only to say why it is wrong, get it out of the way, and move on to the more interesting stuff :)"

The matter bears some discussion, and I would welcome comments about it, though those remarks might have scared off the guy I was responding to.

There are a few things I want to make clear:

1) I do not want to limit anyone's exploration of ideas or to railroad people into a particular conclusion. This isn't about that at all. What I want is to avoid sending people to read misleading articles until they are ready to detect the ways in which those articles are unintentionally misleading.

2) Now, the best way for you to know if an article omits important information, contains outright untruths, sweeps problems with a claim under the rug, or is otherwise misleading, is for you to read it and decide for yourself! However, in order to decide correctly, you often need considerable background information.

For example, suppose you see an article that says "Bell's theorem, and the experiments that have tested it, prove that nonlocality is a real feature of our world."

You are likely to see statements like this in many different, independent articles and sources. It's a common interepretation of Bell's theorem, even by respectable physicists (those who know little of the MWI). Should you therefore believe it?

No, it's false. You could know that if you read my post
http://onqm.blogspot.com/2009/07/simple-proof-of-bells-theorem.html
in which I mention

"Note: The theorem is often said to prove that QM is nonlocal, because a reasonable local model would not allow the direction chosen for a distant measurement to influence the result of the other measurement. That is not the whole story and you should be aware of the other possibilities. In particular, Many-Worlds interpretations do not suffer this limitation because all outcomes occur and correlations might be established only after local interactions; see http://arxiv.org/abs/0902.3827"

Now here I did give an external link, because I read it and it seemed fairly reliable. You can read the linked paper and decide for yourself.

But most links I could give to discussions of QM are liable to contain misleading statements.

3) So, I could prepare the reader in advance, by telling you what to look out for at a particular link, right? Not usually practical. If I give a link, it's so that I don't have to explain the whole thing myself, but I'd practically end up having to do it anyway. In some cases though this could work, if the problem area is relatively small or obvious.

4) If you read a long link, that could take a lot of time and interrupt the flow of what I am trying to say.

5) You should know upfront, this blog is my turf. I don't claim to take a neutral stance on the issues; I just present the correct stance as I see it. This is not a public school or a newspaper.

Comments?

Top 12 things to know about physics

2009-07-31T14:20:00.011-04:00

So, lay people are naturally curious about the world, and Quantum Mechanics is one of the most interesting aspects of physics to dig into! There are counterintuitive relevations to be had, philosophical points to ponder, and in the end - though you will not have all the answers - you will be sure that the true nature of reality is very different from its surface appearance!

But if you really want to understand QM, not fully, but well enough to understand the current state of understanding - not just to get some entertainment value out of it - you should know a lot about physics. I can't teach all that stuff here.

What I will do is give a few short sentences on each topic. I'll probably edit in some references.

Top 12 things to know about physics:

1. Classical mechanics

In classical (Newtonian) mechanics, there are particles (which can be points or be extended). Each has a position and velocity. There are equations that describe how these things change as a function of time, based on force laws such as Newton's famous law of gravitation.

Key concepts: Vector, position, velocity, acceleration, mass, force, momentum, angular momentum, potential energy, kinetic energy, conservation laws, force laws

2. The harmonic oscillator

A spring-mass system is a paradigmic example of a harmonic oscillator. The restoring force increases linearly with the distance from the equilibrium point, which leads to a sinusoidal motion if it's left to oscillate freely. This oscillation is more than a little similar to that in the next subject, waves. If there is damping friction, the oscillation decays exponentially, which can be represented mathematically as a complex (real + imaginary part) frequency.

Key concepts: Linear, nonlinear, frequency, amplitude, sinusiodal, complex exponential

3. Classical waves

Sound, water waves, electromagnetic waves - these are all disturbances of some kind of field, that propagate at a specific speed. A field is a function of position, such as air pressure, that can vary with time. Disturbances propagate locally - the neighboring regions are affected first.

Key concepts: Field, amplitude, phase, frequency, wavelength, local, Doppler effect, superposition, interference

4. Special relativity

You know all that stuff you just learned about classical mechanics? Ahem ... it's all wrong! Well, not all of it, but things are a bit different.

The speed of light is found to be the same relative to any observer, when measured by observers moving at different speeds. How can that be? Well, there's time dilation ... and length contraction ... that work together to produce that result. The speed of light is a cosmic speed limit. One important consequence is that if events are simultaneous as measured by observers moving at one velocity, they will not be if measured by observers at some other velocity.

Key concepts: E = m c^2, time dilation, length contraction, light cones, relative simultaneity, spacelike seperation, Lorentz invariance

5. Electromagnetism

Electromagnetic forces hold atoms together, and electromagnetic waves have many important applications. There are positive and negative charges. Like charges repel; opposite charges attract. The force decreases as the inverse square of the distance. Magnetism is a consequence of relativity.

Key concepts: Charge, electron, proton, electric field, magnetic field, EM wave

6. General relativity

OK! So, you finally have some grasp of special relativity. You've understood the 'twin paradox', the 'barn paradox', and how it is that observers moving relative to each other each think that the other guy's clock is slower.

But wait ... there's more! Space and time are flexible. Gravity is a bending of spacetime caused by mass. In extreme cases, you might not be able to think of a flat spacetime that is bent out of shape - you might have to build it by gluing pieces together. If mass is concentrated enough, it could collapse to a black hole, from which even light can't escape - you can think of future time as pointing radially towards the center of it.

Closed timelike curves are solutions to the equations of GR in which the future loops around to become the past within those places - time travel? It has problems. There is no known way to make them, anyway.

Key concepts: Gravitational redshift, black hole, event horizon, gravitational wave, closed timelike curve, general coordinate invariance, wormhole

7. Cosmology

The universe as a whole is expanding. Looking back towards the past, it was smaller and denser, apparently having originated in a state of very high density and temperature and much smaller size - the Big Bang. According to general relativity, it would have been a state of infinite density. Looking to the future, the universe will get larger and colder; long after the last star has evaporated, it appears that it will just keep expanding. If there's a cosmological constant, it will keep growing exponentially.

Key concepts: Hubble constant, Big Bang, singularity, redshift, cosmological constant

8. Entropy and Statistical Mechanics

Disorder (entropy) tends to increase on a microscopic level. As a result, to get anything accomplished, we need a continuous flow of more-ordered stuff going in, and we dump the less-order stuff going out. For example, sunlight in (high energy per photon (particle of light), an orderly concentration) and thermal radiation out (distributing the same energy over many more, less energetic photons, which is more random since there are more ways to do it).

Statistical mechanics allows us to consider the mechanics of large numbers of particles using probability distributions rather than trying to follow each particle individually. It explains the above tendency, obtaining irreversible average behavior from time-symmetric basic equations of motion.

The thermodynamic 'arrow of time' implies that the conditions near the beginning of the universe were much lower in entropy than a typical possible microstate, which is why the entropy has a lot of room to grow.

Key concepts: Entropy, arrow of time, configuration space, probability, large number of particles

9. Quantum mechanics

That's right; before delving into interpretation of QM, it helps to know a bit of QM first.

Experiments have shown that everything has wave-like characteristics. Particles such as electrons (and in principle anything made of them, such as people) can exhibit wave-like interference effects. Waves, such as electromagnetic waves, can appear to act as though made up of individual particles (photons in the EM case).

QM explains why matter doesn't collapse: Electrons are attracted to the protons in an atom's nucleus, but if they are confined to a small place such as the nucleus, they would have a short wavelength. They would need high energy for that, and if they had it, they would zoom out. In practice there is a balance between the electron wave resisting being squeezed to a small size, versus it being attracted to the nucleus. Also, electron waves can't occupy the same state as each other, so atoms with more electrons end up with shells of electrons further out, even though they have more protons pulling the electrons tight.

As I explained in my previous post, in QM, it is not correct to say that each particle is a wave in space. Instead, there is a joint wavefunction, which lives on configuration space. I will have more to say about it in other posts.

When a measurement is made, the outcome appears to be random, with the probabilities given by the Born Rule - proportional to the square of the wavefunction. Many observables have a discrete (quantized) set of possible outcomes.

Key concepts: Matter waves, spin, quantized outcomes, wavefunction, Pauli exclusion principle, eigenstates, eigenvalues, Hamiltonian, observable, Shrodinger Picture, Heisenberg Picture, Born Rule, measurement problem, Shrodinger's Cat, entanglement, decoherence, Bell's theorem

10. Quantum field theory (QFT)

QFT is the relativistic version of quantum mechanics. Instead of particles, there are fields. There's a small set of fields for every kind of particle - electrons, photons, etc. The wavefunctional lives on the space of configurations of these fields. QFT allows the creation or destruction of 'particles' because really there are no particles - quantized excitations of the fields play the role of particles.

There are problems with QFT. It is necessary to 'renormalize' the fields because interactions can produce infinite divergences, which must be subtracted out. This can sometimes be done by assuming a minimum length scale, then taking the limit as the scale goes to zero. Even so, some problems remain but can usually be ignored by using approximations.

Key concepts: wavefunctional, field configuration, spin, boson, fermion, antiparticle, locality, renormalization

11. Quantum gravity (QG)

Nobody has yet suceeded in bringing it all together, and until they do, we won't really know what's going on. But even so, there are many important tidbits to know regarding gravity and QM.

There is a minimum length scale in quantum gravity: Make a small enough black hole, and the wavelength becomes the same size as the event horizon. Add more mass, the event horizon gets bigger; less mass, and the wavelength does. Squeeze a wave that small, and you've added the mass (E = m c^2). This scale is called the Planck length. We don't know what's going on below that scale.

Many people think that the Planck scale explains the use of renormalization in QFT. Infinite renormalization doesn't make much sense, but with a finite minimum scale, there would be just a finite scale factor. Of course, no one knows how the details would work.

It is generally thought that a fixed volume of space has a finite number of degrees of freedom in QG due to the minimum length. This affects some interpretations of QM.

Another important thing to know about is Hawking radiation. As Hawking discovered, an event horizon produces radiation. This makes black holes evaporate - on the quantum level, this is important because it means that some form of the usual information-preserving quantum mechanics might apply after all; details of what went in are encoded in the radiation.

Also, the radiation has a thermal spectrum, allowing a temperature and an entropy to be assigned to black holes. The stability of these objects is re-explained in terms of them having a huge entropy. This implies that there is some new variable that can be distributed in many ways, such as different excitations of string modes at the center (see below) or something varying near the event horizon.

QG might also forbid other weird features of GR, such as closed timelike curves.

On the cosmological level, if there is a cosmological constant, there will be Hawking radiation. In the past there appears to have been a similar large effect (inflation) but it was not a constant. In the deep future, the Hawking radiation might produce Boltzmann brains - randomly assembled brains. It's not something that would happen often but infinity is a long time and an exponentially growing universe is a big place. If over infinite time these greatly outnumber normal observers, as they would, the theory is inconsistent with our observations.

Also, if QM is applied to the equations of general relativity for a closed universe, the apparent result is that the wavefunction can not change as a function of time (total energy = zero, which leads to a static state in QM). This Wheeler-DeWitt equation is a rather controversial result.

It is not clear what the ontology of QG would be. Perhaps the wavefunctional lives on the space of possible geometries and field configurations on those.

QG is not renormalizable. String theory uses extended fundamental objects to avoid the infinities, but current work assumes a fixed background geometry, which is considered an approximation that is hard to get away from.

String theory allows many 'vaccum' solutions; our universe might decay from the current one to an uninhabitable one, perhaps helping explain the lack of Boltzmann brains over the history of the universe. Other ways out of it are for baby universes to continually form (perhaps in black holes), or for there to be no cosmological constant.

Loop quantum gravity is another approach, that avoids a fixed background, but is less developed. Other approaches include the holographic universe, in which the boundary determines everything.

Key concepts: string theory, loop quantum gravity, Plank scale, Hawking radiation, Wheeler-DeWitt equation, problem of time, baby universes, Boltzmann brains, cosmological constant, decay of the vacuum

12. The anthropic principle

Our universe seems fine-tuned for life. If parameters such as the strength of the electrical force compared to the nuclear forces were just a little different, complex chemistry would not be possible. If there were no neutrinos, supernovas could not eject the complex elements out into space. If the number of space dimensions were different, planetary systems could not form. If photons did not exist, energy could not flow from stars to plants.

There is only one reasonable explanation for this fine-tuning: there must be many universes, most of which are unsuitable for life, but some of which are. In that way, a simple overall theory can explain a complex particular universe. The anthropic principle states that the only part of reality that we find ourselves in must be such that we can exist.

The multiple universes might be the many solutions to string theory, or could maybe even be all possible mathematical stuctures - the Everything Hypothesis. (A post on it will come.)

The objection might be raised: Could not an intelligent creator explain the fine-tuning? But while a creation scenario can not be disproven, it can NOT be a fundamental explanation for the origin of life, because it merely pushes back the question, to where the intelligence of the creator would have come from. For an intelligent creator to exist, unless by a wholly unbelievable coincidence, in a simple overall theory, we must conclude that he must be a product of Darwinian evolution in his own fine tuned universe. The resulting model is much more complicated than the simple multiverse scenaio, violating Occam's Razor (the principle, used in science, that simpler explanations are more likely). This is explained in detail in Richard Dawkins' excellent book "The God Delusion".

Key concepts: Fine tuning, multiple universes, anthropic principle, God Delusion

Why MWI?

2009-07-15T16:50:00.008-04:00

Before getting into the details of the problems facing the Many-Worlds Interpretation (MWI), it's a good idea to explain why I believe the MWI in the first place ;)

MWI for the layperson:

In classical mechanics, each particle has a position and velocity. If there are N particles, the state of a physical system at a given time is given by a list of the positions (a configuration, or point in 3N-dimensional configuration space) and of the velocities for each particle:

X1 = (x1,y1,z1)

classical state: (X1(t), V1(t)), (X2(t),V2(t)), (X3(t),V3(t)), ...

In QM, there is instead the wavefunction, psi, which is a complex-number-valued function on what would classically be configuration space plus the space of spin configurations, and is a function of time:

quantum state: psi(X1,S1, X2,S2, X3,S3, ... , t)

(Spin, Si, takes on a small set of discrete values.)

This is a classic way of generalizing something: instead of a point in a space, there is a function on that space. It must be emphasized that the wavefunction is not a function on regular 3-dimensional space, but on the 3N-dimensional space of configurations. This high-dimensional arena is responsible for many of the counterintuitive properties of quantum mechanics.

If the wavefunction is somewhat sharply peaked near a configuration, though with a wavelength small compared to the width of the peak, it will behave a lot like a classical system; the peak will follow a nearly classical trajectory as a function of time. It is natural to conclude that any interesting things done by such a classical system, such as performing computations, will be done by the wavefunction. It is just like a classical world, only a little 'fuzzy' due to the finite width of the peak. Indeed, roughly this picture is probably how most people think of QM, including chemists - a classical world except that electrons and similar particles are spread out instead of being concentrated at a point.

But that is obviously not a complete picture, because the wavefunction is not concentrated around a single peak. Roughly speaking, there are many peaks, representing quite different classical configurations (e.g. the alive or dead configurations of Shrodinger's cat), and many places even away from the peaks where the wavefunction is nonzero. Yet the world we see resembles a classical, single-configuration world. How can we explain that?

There are three main approaches. The first is some modification of QM in which only one peak remains, while the others vanish - this is called 'collapse of the wavefunction'. There are three main problems with this: 1) it introduces a lot of complexity to the model which might be avoided by another approach; 2) it violates things physicists like such as conservation laws; and most importantly 3) it doesn't work because generally speaking, in proposed models that give mathematical details of 'collapse', small residues remain in the other parts of the wavefunction. Small or not, these residues still go through trajectories that should give rise to computations and thus observers - unless we have reason to believe that probability is higher in high amplitude regions; but if we do, we might as well just go with the simpler MWI since deriving that is its main problem.

The second main approach is hidden variables. As we have seen, local hidden variable models are ruled out by Bell's theorem, but nonlocal models exist that don't have that problem - most famously, the Pilot Wave Interpretation (PWI). In the PWI, a classical-like configuration point 'surfs' along the wavefunction. It has been shown that under quite general conditions, the probability distribution for the point evolves in time to match the Born Rule of QM.

Two main problems have been raised for the PWI and similar models. 1) It is nonlocal, and has a preferred reference frame contrary to the spirit of relativity. This is really a matter of taste, and I don't consider it a fatal problem at all, though I do think the nonlocality is an undesirable feature if other models can avoid it. 2) More importantly, it doesn't get rid of the other peaks in the wavefunction at all; it just adds a new trajectory of the hidden configuration point. The wavefunction is still there and should still perform all of the interesting computations as it would in the MWI. Thus, the PWI has been called 'Everett in denial'. Valentini [http://arxiv.org/abs/0811.0810] has denied that charge but his straw-like arguments are easily demolished as Brown has done [http://arxiv.org/abs/0901.1278].

I must note an important exception to the many-worlds property of the PWI: In some versions of what is proposed for quantum gravity, the wavefunction of the universe does not evolve as a function of time; this is known as the Wheeler-DeWitt equation. That would seem at first glance to rule out observers in those models (that remains to be seen even for just a wavefunction). However, the PWI hidden variables would evolve in time even though the wavefunction doesn't, making a single-world model out of it. While interesting, I find it implausible that something as complex as the wavefunction of the universe would have to be in such a model could be an initial condition.

Finally, there is the MWI itself, as first proposed by Everett and in various forms by others. In its basic form, this has the simplest mechanics as it just lets the wavefunction evolve over time, adding no hidden variables or collapse-inducing modifications to the dynamics. There are many peaks in the wavefunction which follow various trajectories and implement various computations. [Much more to be said on that rough sketch.] Each individual observer only notices a single classical-like world because that is the one associated with the motion of the peak giving rise to the computations of his own brain; the others don't have any effect on him.

This appealingly simple picture, however, raises a problem of its own: In order for our observations to be at all typical, the Born Rule (which relates probabilities to the square of the wavefunction) must hold, at least to some approximation. This means that small amplitude peaks are less probable than large amplitude peaks. Since the trajectory of a peak (which makes it perform computations and so on) does not depend on amplitude, why would that be the case in the MWI?

The possibility of derivation of the Born Rule in the MWI is the central problem in interpretation of QM. If the Born Rule does follow from the MWI, then the case for the MWI is made beyond a reasonable doubt. I will discuss in other posts various attempts to derive it.

If it does not follow, then the problem remains - what interpretations could work? Continuous collapse models and the PWI would still not work because they would still be the MWI in disguise due to having the wavefunction (with its wrong probabilities) as part of their ontology.

One possibility that looks like it should work in any case is making an honest Many-Worlds version of the PWI: having infinitely many sets of the hidden variables. The simplest version, that of having every point in configuration space sporting a wave-surfing hidden variable, is called Continuum Bohmian Mechanics (CBM). These hidden variable worlds could then outnumber the ones in the wavefunction, producing the Born Rule for typical observers. Of course, this model is more complex than the standard MWI. Also, it still would leave the question of what observers are and how to count them.

Quantum gravity remains an unsolved problem, and the solution may play a role in interpretation of QM, perhaps providing a new set of variables to work with.

Another problem is that in the long term, long after normal observers have died out, spontaneously assembled bits of random matter (which a cosmological constant would produce) would eventually include short-lived observers who would outnumber the normal ones over the history of the universe by perhaps an infinite factor. These Boltzmann Brains, and the necessity of getting rid of them in terms of their effect on typical observations, provide important constraints on what the real answers could possibly be. This deserves a post of its own, at least.

Another (and not unrelated) topic that will get its own post is the Everything Hypothesis, which postulates that every possible thing must exist as an explanation for why things are how they are.

Simple proof of Bell's Theorem

2009-07-15T14:15:00.012-04:00

I think it's worthwhile to make the proof as simple as possible, in part because some people continue to have trouble with it and continue to produce false "counterexamples" of purported local hidden variable models that violate Bell's inequalities. Please let me know via comments if you have ideas on how to make the proof even shorter and simpler.

The focus here is on QM itself. I will not discuss practicalities of experiments to test Bell's inequalities that lead to "loopholes" due to low detector efficiencies and so on in this post. Such loopholes are implausible and more recent experiments have closed some or made them even less plausible.

Thought experiment:

Any entangled pair of systems, each with at least two distinguishable states besides for position, could be used. I'll use an entangled pair of spin-1/2 particles for ease of notation.

A pair of spin-1/2 particles are generated which are in the entangled spin singlet state

|psi> = (|+,-> - |-,+>) / 2^(1/2)

One of these particles is sent to Alice; the other is sent to Bob. The two observers may be very far apart.

Alice <----------------- source ---------------> Bob

When an observer measures the component of a spin-1/2 particle's spin along any direction in space (for example, using a Stern-Gerlach device), the result of the measurement is always + or - 1/2 hbar. (hbar is Plank's constant / 2 pi)

In the state psi> above, there is a 50% chance that the spin component will be positive (+) for any direction of measurement for either particle.

In the state psi>, QM predicts that if Alice's particle is measured with the result + in direction A (call this A+, etc.), then measurement on Bob's particle in the same direction A will give the - result, A-. This kind of correllation is called an EPR correllation.

Call this pair of results (A+; A-) where A is the direction and the order indicates which Observer gets each of the results.

Let P(A+; A-) be the probability that (A+; A-) is found, and so on. Clearly, P(A+; A-) = P(A-; A+) = 1/2. There is no need to consider models that don't predict this for the proof, since they already would disagree with the predictions of QM.

Each of the Observers can choose which direction to measure the spin along. In particular, each will choose one of three directions: A, B, or C.

Define Distant Measurement Independence (DMI) as the assumption that the (singular) result of each Observer's measurement can not depend on which direction the other Observer chose to measure along.

Statement of the Theorem: DMI is not consistent with the predictions of QM.

[Note: The theorem is often said to prove that QM is nonlocal, because a reasonable local model would not allow the direction chosen for a distant measurement to influence the result of the other measurement. That is not the whole story and you should be aware of the other possibilities. In particular, Many-Worlds interpretations do not suffer this limitation because all outcomes occur and correlations might be established only after local interactions; see http://arxiv.org/abs/0902.3827]

Proof:

Assume DMI. It is possible that a model assigns certain additional properties to a particular particle that don't appear in the QM description; these are called hidden variables. These could tell the particle whether to give a "+" or a "-" result as a function of what direction its measurement is made in.

The other particle of the pair would then have to have a similar set of properties but with the opposite instructions. Such hidden variables would be required in order to produce the EPR correlations without violating DMI, because otherwise the other particle would have no way to be certain to give the opposite result when both Observers choose the same direction.

Even though only one measurement on the particle is actually made, it can be thought of as labeled by the hidden variables according to what the outcome of measurement along each of the three directions A, B, and C would have been. Consider Alice's particle to be so labeled.

Let (A+ & B-) mean that the result would be + if measured along direction A and would be - if measured along direction B, and so on. Let P(A+ & B-) be the probability that the hidden variables were such that those would be the results.

The following inequality must hold since more general cases are at least as probable as less general ones:

P(A+ & B-) = P(A+ & B- & C-) + P(A+ & B- & C+) ≤ P(A+ & C-) + P(C+ & B-)

It is not possible to measure Alice's particle along more than one direction, but Bob can help us do the next best thing; because of the EPR correlations, measuring his particle should reveal the opposite of what result Alice's particle would have given. Thus the above inequality is equivalent to

P(A+ ; B+) ≤ P(A+ ; C+) + P(C+ ; B+)

This kind of inequality is called a Bell inequality (of which there are actually several).

Quantum mechanically, P(A+ ; B+) = 1/2 sin^2 (theta(A,B)/2) where theta(A,B) is the angle between A and B; and so on.

For example, say A and B are at a right angle, with the C direction in between them.

theta(A,B) = 90 degrees, and theta(A,C) = theta(C,B) = 45 degrees.

Then P(A+ ; B+) = .25, and P(A+ ; C+) = P(C+ ; B+) = .073

Since .25 > .146, the inequality is violated. This establishes that DMI is not consistent with QM.

You can see http://arxiv.org/abs/0902.3827 for another overview of Bell's theorem.