Interlude: Anticipating the 2007 Many Worlds conference

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

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 twofold: First, to provide an easy to understand overview of many issues surrounding interpretation of quantum mechanics. That should be useful to students who intend to pursue a serious interest in philosophy of physics. Secondly, to convey my own ideas about philosophy of physics; some of that requires a lot of background in very specific issues to properly understand.

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.


Primers on Basic QM:

https://arxiv.org/abs/1803.07098

http://theoreticalminimum.com/courses/quantum-mechanics/2012/winter

Studying Quantum Mechanics: Measurement and Conservation Laws

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 received 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 negligible, 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).

See also
"WAY beyond conservation laws"

Key definitions for QM: Part 3

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

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

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

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.

Or, he can insert a 'beamsplitter' (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

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?