Why MWI cannot explain the Born rule

[RUTA]

I don't see how is that different from say, your position in the spatial dimension. You only have access to the single position, yet that doesn't stop you from using symmetries, homogeneity or isotropy.

It's very different because you have access to information from the space that surrounds you. You have no such access to "other" universes, by definition (if you have access, they're not "other," they're part of this one).

 

[Fredrik]

The existence of parallel universes DOES follow from those two axioms. But you cannot understand it without understanding the theory of decoherence (which, by the way, is not only a theory, but also an experimental fact.)

The thing is, decoherence uses more than those two axioms. It uses the Born rule implicitly, by taking the Hilbert space to be a tensor product, and by computing the "reduced" density matrix as a partial trace of the state operator of the universe.

Without the possibility to do decoherence calculations, the only way to define the worlds is to say that given a basis (any basis) for the Hilbert space of the universe, each basis vector represents a world. To go beyond that, we need the Born rule, and a way to express the Hilbert space as a tensor product. Those things make decoherence a meaningful concept.

I have previously said that decoherence defines the worlds. I no longer think that that's the most appropriate way to define the worlds. What decoherence does is to single out a basis that defines interesting worlds. If my understanding of decoherence ideas is accurate (it might not be), any other basis defines worlds where the subsystems can't contain stable records of the states of other subsystems (such as a memory in the brain of a physicist). If well-defined memory states is an essential part of what consciousness is, the worlds identified by decoherence are the only ones that can contain conscious observers.

 

[yossell]

Thanks Hurkyl

I'm neither a mathematician nor a physicist - so take what I say with a siberian salt mine. And I think I take the main point in your post.

Everything I've understood about MWI works in the same way -- "parallel worlds" and such are simply ideas built out of the quantum mechanical state space, which are used to describe the behavior of states.

Maybe this is just my own conceptual stumbling block, but I have problems seeing the parallel (no pun intended) clearly.

The ontology and ideology of Hilbert's axiomatisation of geometry is clear - the quantifiers of the theory range over points and regions and there are primitive predicates 'between' and 'congruent'. I take these predicates to be (reasonably) physically clear and meaningful, that gives the theory its *physical* content and makes it more than pure maths or logic.

From a purely logical point of view, of course, we don't really care what these predicates mean - the predicates may be replaced with formal letters xByz and xyCongzw for all we care. But if we want the theory to have more than formal properties then (I think) it is because 'point' and 'line' and 'region' correspond to genuine physical geometric objects, and 'Between', and cong to genuine physically geometric relations that we can see the theory as having some genuine physical content.

Given this, when we embed arithmetic into our Hilbertian theory is finding geometric structures which are isomorphic (in the clear model-theoretic sense) to mathematical ones. A statement in the language of Peano arithmetic becomes equivalent to a statement about geometric lines and regions. A mathematical statement, a statement in the language of Peano arithmetic, gets reintepreted as a statement about geometric entities.

Now, when it comes to mathematical and logical issues, I think this kind of thing is probably fine, because it's not clear that there's anything to our conception of mathematical objects other than something formal or structural.

But it's not clear to me how things go when we're dealing with terms that are supposed to have physical significance. The worry is this: the interpretation of the terms can play a role in solving the relevant problem and so matters of interpretation need to be tracked in a way that they don't in the more formal cases.

For instance: a no-collapse theorist may want to explain how it is that we experience a cat which is determinately dead or alive, even though, by his lights, there is no collapse. He solves his problem by postulating many worlds - there are literally two cats and two observers, each one having experience a cat in a determinate state. (Not defending this move, just noting that it offers a solution to a problem). The many worlds, the many observers, the many cats - they may not be fundamental, but they have to be there for this version of the solution to go through. But if all we're doing is dropping the collapse postulate, then I'm not sure where many anythings come in. And then, this particular solution of this problem would not be available to him.

 

[Fredrik]

Hurkyl: The axioms we're discussing aren't mathematical axioms. They are statements that describe how things in a mathematical model correspond to things and the real world. So I don't really see the point of your analogy with Euclidean geometry.

 

[Fredrik]

You get the Born rule from this by considering the "frequency operator". Only the states that have the correct statistics have a non-zero norm and they are eigenvectors of the operator. So, the "certainty rule" then implies that you will observe the statistics as given by the Born rule (at least when you consider an infinite numbers of copies of the system).

I decided to take another look at the article you referenced. First they're saying that the Born rule tells us that the probability of a sequence of measurement results i_1,\dots,i_N[/tex] is<br /> <br /> |\langle i_1|s\rangle|^2\cdots|\langle i_N|s\rangle|^2=|\langle i_1|\otimes\cdots\otimes\langle i_N|\ |s\rangle\otimes\cdots\otimes|s\rangle|^2<br /> <br /> Then they&#039;re saying that &quot;Everett noted&quot; that &quot;it follows&quot; that the probability of a particular sequence i_1,\dots,i_N[/tex] is low if |\langle i_1|\otimes\cdots\otimes\langle i_N|\ |s\rangle\otimes\cdots\otimes|s\rangle| is small, so they&amp;#039;re giving Everett credit for knowing how to take the square root of the square of a positive real number.&lt;br /&gt; &lt;br /&gt; Then they claim that in the &amp;quot;formal limit&amp;quot; N→∞, |\langle i_1|\otimes\cdots\otimes\langle i_N|\ |s\rangle\otimes\cdots\otimes|s\rangle|\rightarrow 0 if the sequence is not statistically typical. What does that even mean? If |s&amp;gt; isn&amp;#039;t orthogonal to any of the eigenstates, we&amp;#039;re just talking about a product of N numbers in the open interval (0,1) in the N→∞ limit (or whatever they have in mind when they say &amp;quot;&lt;i&gt;formal&lt;/i&gt; limit&amp;quot;). Is it even possible for the result not to go to 0? Maybe if the Nth factor goes to 1 as N→∞ limit, but in that case the probabilities of the possible results in a single experiment don&amp;#039;t add up to 1, and the |s&amp;gt; we started with has infinite norm!&lt;br /&gt; &lt;br /&gt; So after just &lt;i&gt;using&lt;/i&gt; the Born rule, and then saying that &amp;quot;this implies&amp;quot;...something that appears to be complete nonsense, they claim that this means that &amp;quot;the Born rule is a consequence of excluding zero norm states from the Hilbert space&amp;quot;! (Note that the definition of a Hilbert space already excludes them).&lt;br /&gt; &lt;br /&gt; Are you saying this is &lt;i&gt;not&lt;/i&gt; nonsense? As always, if I have misunderstood something, I&amp;#039;d like to know.&lt;br /&gt; &lt;br /&gt; I notice that you didn&amp;#039;t adress any of my arguments (&lt;a href=&quot;https://www.physicsforums.com/showthread.php?p=2495887&quot; class=&quot;link link--internal&quot;&gt;1&lt;/a&gt;, &lt;a href=&quot;https://www.physicsforums.com/showthread.php?p=2501679&quot; class=&quot;link link--internal&quot;&gt;2&lt;/a&gt;) against the whole frequency operator approach. As far as I can tell, it&amp;#039;s completely circular even for &lt;i&gt;finite&lt;/i&gt; tensor products, and the N→∞ limit isn&amp;#039;t going to make that problem go away.

 

[Demystifier]

The thing is, decoherence uses more than those two axioms. It uses the Born rule implicitly, by taking the Hilbert space to be a tensor product, and by computing the "reduced" density matrix as a partial trace of the state operator of the universe.

Without the possibility to do decoherence calculations, the only way to define the worlds is to say that given a basis (any basis) for the Hilbert space of the universe, each basis vector represents a world. To go beyond that, we need the Born rule, and a way to express the Hilbert space as a tensor product. Those things make decoherence a meaningful concept.

I have previously said that decoherence defines the worlds. I no longer think that that's the most appropriate way to define the worlds. What decoherence does is to single out a basis that defines interesting worlds. If my understanding of decoherence ideas is accurate (it might not be), any other basis defines worlds where the subsystems can't contain stable records of the states of other subsystems (such as a memory in the brain of a physicist). If well-defined memory states is an essential part of what consciousness is, the worlds identified by decoherence are the only ones that can contain conscious observers.

Fredrik, I don't agree with you that definition of subsystems in terms of tensor products is equivalent to the Born rule. After all, the former says nothing about probability per se.
And of course, one can calculate reduced density matrices without the Born rule.

 

[Fredrik]

Fredrik, I don't agree with you that definition of subsystems in terms of tensor products is equivalent to the Born rule. After all, the former says nothing about probability per se.

Consider two systems that aren't interacting with each other. If system 1 is in state |\psi\rangle when we measure A, the probability of result a is

P(a)=|\langle a|\psi\rangle|^2

If system 2 is in state |\phi\rangle when we measure B, the probability of result b is

P(b)=|\langle b|\phi\rangle|^2

According to the standard rules for probabilities, the probability of getting both of these results is

P(a,b)=P(a)P(b)=|\langle a|\psi\rangle|^2|\langle b|\phi\rangle|^2=|\langle a|\otimes\langle b|\ |\psi\rangle\otimes|\phi\rangle|^2

This means that if we use the tensor product space to represent the states of the combined system, the Born rule will hold for that space too. Can you really look at this and think that we didn't choose to use tensor product to make sure that the probabilities assigned by the Born rule satisfy P(a,b)=P(a)P(b) when the systems aren't interacting?

And of course, one can calculate reduced density matrices without the Born rule.

You might be able to calculate them, but can you really justify the use of reduced density matrices to represent states of subsystems without using the Born rule? I'm pretty sure the answer is no.

I'll try to return to both of these things with more complete answers later, but feel to investigate it yourself. My feelings won't be hurt if you post a proof of some enlightening stuff before I do.

 

[Fredrik]

An arbitrary state can be written as

\rho=\sum_i w_i|s_i\rangle\langle s_i|

If we write |s_i\rangle=|f_i\rangle\otimes|g_i\rangle, the state can be expressed as

\rho=\sum_i w_i |f_i\rangle\langle f_i|\otimes|g_i\rangle\langle g_i|

The easiest way to define the reduced density matrix, which I'll call \rho&#039;, is to use a basis |\psi_\mu\rangle for the first Hilbert space, and a basis |\phi_\alpha\rangle for the second Hilbert space, which together define a basis |\phi_\mu\phi_\alpha\rangle=|\psi_\mu\rangle\otimes|\phi_\alpha\rangle for the tensor product space. We define the operator \psi&#039; by saying that its matrix elements in the |\psi_\mu\rangle basis are

\rho&#039;_{\mu\nu}=\sum_\alpha\rho_{\mu\alpha,\nu\alpha}=\sum_\alpha\langle\psi_\mu\phi_\alpha|\rho|\psi_\nu\phi_\alpha\rangle=\sum_i w_i\langle\psi_\mu|f_i\rangle\langle f_i|\psi_\nu\rangle\sum_\alpha|\langle\phi_\alpha|g_i\rangle|^2

The sum over \alpha is =1, so

\rho&#039;_{\mu\nu}=\langle\psi_\mu|\bigg(\sum_i w_i|f_i\rangle\langle f_i|\bigg)|\psi_\nu\rangle

so we have

\rho&#039;=\sum_i w_i|f_i\rangle\langle f_i|

This is just an ordinary (mixed) state operator for the physical system associated with the first Hilbert space, so if we use a definition of QM that takes state operators to be the "states" of the theory, we don't have to justify the interpretation of the reduced density matrix as a representation of a state of a subsystem.

I'm not sure if we should be talking about the original Born rule P(a)=|\langle a|\psi\rangle|^2 or its generalization to mixed states \langle A\rangle_\rho=\mbox{Tr}(\rho A).

If we're using a definition of QM that takes unit rays to be the "states" of the theory, then the use of state operators in general needs to be justified. This is done by first noting that the average result in a series of measurements of A on identically prepared systems is

\langle A\rangle=\sum_a P(a)a=\sum_a a|\langle a|\psi\rangle|^2=\sum_a\langle a|\psi\rangle\langle\psi|A|a\rangle=\mbox{Tr}(\rho A) ...and also =\langle\psi|\Big(\sum_a|a\rangle\langle a|\Big)A|\psi\rangle=\langle\psi|A|\psi\rangle

and then noting that the average result on an ensemble with a fraction w_i of the members prepared in state |\psi_i\rangle is

\sum_i w_i\langle\psi_i|A|\psi_i\rangle=\sum_n\sum_i w_i\langle\psi_i|A|n\rangle\langle n|\psi_i\rangle=\sum_n\langle n|\Big(\sum_i w_i|\psi_i\langle\psi_i|\Big)A|n\rangle=\mbox{Tr}(\rho A)

We're using the original Born rule in the first step, so if our axioms talk about state vectors rather than state operators, we need the original Born rule to justify that expectation values can be written as \langle\psi|A|\psi\rangle, which then gives us the generalized Born rule.

When we're dealing with state operators and the generalized Born rule, the rule P(a,b)=P(a)P(b) for non-interacting systems is replaced by

\langle AB\rangle=\langle A\rangle\langle B\rangle

where "AB" is still undefined if we haven't decided to use the tensor product yet. "AB" is supposed to be the mathematical representation of the operationally defined "measure B first, then A". If we use the tensor product, the above holds true with AB defined as (A\otimes I)(I\otimes B)=A\otimes B. Can you look at this and not think that the reason we're using tensor products is that it ensures that this result holds for non-interacting systems?

I guess that answers the question of which version of the Born rule we should be talking about. The answer is that it doesn't matter. If we're talking about the original Born rule, the argument in my previous post shows that it's intimately connected to tensor products, and if we're talking about the generalized Born rule, the argument in the previous paragraph shows the same thing. (I'm leaving the proof of \mbox{Tr}(A\otimes B)=\mbox{Tr}(A)\mbox{Tr}(B) as an excercise).

 

[Demystifier]

Consider two systems that aren't interacting with each other. If system 1 is in state |\psi\rangle when we measure A, the probability of result a is

P(a)=|\langle a|\psi\rangle|^2
...

Here you assume the Born rule. Therefore, your further steps (which I don't quote) cannot be qualified as a derivation of the Born rule.

You might be able to calculate them, but can you really justify the use of reduced density matrices to represent states of subsystems without using the Born rule? I'm pretty sure the answer is no.

That's an interesting question. I will think about it.

 

[Fredrik]

Here you assume the Born rule. Therefore, your further steps (which I don't quote) cannot be qualified as a derivation of the Born rule.

That's right (and also obvious). I just proved that if we use the tensor product and the Born rule, we get P(a,b)=P(a)P(b) for non-interacting systems. (It would be a disaster to get P(a,b)≠P(a)P(b). If QM works for nuclei and electrons separately, it wouldn't work for atoms. But of course it wouldn't work for nuclei either...) The point is that quantum mechanics for individual systems, which by definition includes the Born rule, more or less forces us to us to use the tensor product to represent the states of the composite system.

I wonder if it's possible to prove that the Born rule is the only probability measure that gives us P(a,b)=P(a)P(b). That would be a derivation of the Born rule from tensor products, but it was never my goal to find a derivation. I'm just saying that the two are clearly not completely independent.

As I pointed out in #108, when we use axioms that start with state operators instead of state vectors, we should require <AB>=<A><B> instead of P(a,b)=P(a)P(b), but the conclusion is the same.

Edit: It was probably a mistake to think that we should require <AB>=<A><B> when the axioms talk about state operators and P(a,b)=P(a)P(b) when the axioms talk about state vectors. I think both identities must hold in both cases.

That's an interesting question. I will think about it.

I think #108 answers it. If we start with axioms that take state operators to be the mathematical representation of states, no justification is required. It's just an axiom. If we start with axioms that start with state vectors, the justification is given by the stuff I said about expectation values.

 

[Demystifier]

Fredrik, I mostly agree with your last post. In fact, sooner or later you and me allways arrive at an agreement, which is why I like to discuss with you.

But let me clarify one thing regarding the question whether many worlds (without the Born rule) can be derived from the Schrodinger equation. You are right that description of a subsystem by a reduced density matrix can hardly be justified without assuming the Born rule. Nevertheless, my point is that essential physical aspects of decoherence
(or at least of something closely related to decoherence) can be understood without introducing the reduced density matrix. Indeed, this stuff has been known much before the word "decoherence˝ was introduced in quantum mechanics. For example, see the Quantum Mechanics textbook written by Bohm in 1951, the chapter on the theory of quantum measurements. (This book is written in a Copenhagen style, a year before Bohm introduced his hidden-variable interpretation of QM.)

Here is the basic idea. Consider the wave function in the configuration space describing a collection of many interacting particles. Before the interaction, the total wave function is typically a product of wave functions of the non-interacting subsystems. However, after the interaction, the total wave function becomes a superposition of such products. The crucial property of this superposition is that each term of the superposition is a many-particle wave function which DOES NOT OVERLAP with any other term in the superposition. (More precisely, the overlap is negligible, due to a large number of degrees of freedom - particles.) While this lack of overlap is technically not the same as decoherence, it is closely related to it. Indeed, if you calculate the reduced density matrix from this wave function, you will obtain a decohered density matrix. But you don't need to calculate the reduced density matrix at all. This wave function itself is sufficient to understand how "many worlds˝ emerge. Since this wave function consists of many (almost) non-overlaping chanels, each chanel may be thought of as another "world˝. This is how Scrodinger equation of many degrees of freedom predicts the existence of "many worlds˝, without any additional assumptions.

 

[Hurkyl]

You are right that description of a subsystem by a reduced density matrix can hardly be justified without assuming the Born rule.

I'll take a stab at sketching it anyways. My apologies if I've missed something important.


If we start with the premise:

• We have some "full" Hilbert space
• We have some collection of observables
• Each observable A is represented as an operator A_f on the full Hilbert space
• The expectation of A on a density matrix p is Tr(p A_f)

And we can find:

• A "reduced" Hilbert space
• Each of our observables A can be represented as an operator A_r on the reduced Hilbert space
• A map that turns a density matrix p into a reduced density matrix p_r

Then all we really need to justify the reduced density matrices is the identity

Tr(p A_f) = Tr(p_r A_r)​

right?

 

[Fredrik]

he crucial property of this superposition is that each term of the superposition is a many-particle wave function which DOES NOT OVERLAP with any other term in the superposition.
...
This wave function itself is sufficient to understand how "many worlds˝ emerge.
...
This is how Scrodinger equation of many degrees of freedom predicts the existence of "many worlds˝, without any additional assumptions.

I have sort of been flip-flopping back and forth between thinking that this is right and thinking that this is wrong. I'm leaning towards wrong. Consider what I said here:

Suppose e.g. that you bet $1000 that the spin will be "up", and then you perform the measurement. The state of the system+environment will change like this:

(|↓>+|↑>)|> → |↑>|> + |↓>|>

Yes, there will be other terms, which has your memory in a superposition of "smile" and "yuck", but what decoherence does is to make the coefficients in front of them go to zero very rapidly. Now each of the remaining terms is interpreted as a "world" in which a particular result happened, and "you" (a different you in each world) remember that it happened.

Edit: This was actually a mistake. What I should have done is to define |S>=|↓>+|↑> and then said that the density matrix changes as described by

|S>|><|<S| → |↑>|><|<↑| + |↓>|><|<↓|

This is a mixed state, not a superposition.

I'm not 100% sure what you mean by "overlap", but I think you're probably talking about final state vectors like the one in the post I just quoted. We would have an "overlap" if there had also been a |↑>|> term on the right. (Technically, there always is, but we're talking about situations in which the coefficient in front of it is really small).

The process described by the first → in the quote is the development of correlations between subsystems. This is what happens when a silver atom goes through a Stern-Gerlach magnet, before we determine its position by detecting it in one location or the other. The process described by the second → in the quote is a measurement. This is what decoherence does. It turns pure states into mixed states.

I think we need the second process to define the "interesting" worlds, and I don't think we need either of them to define "worlds".

Without the possibility to do decoherence calculations, the only way to define the worlds is to say that given a basis (any basis) for the Hilbert space of the universe, each basis vector represents a world.

Edit: The first process defines some set of worlds. It's too small to be all the worlds, and too big to be all interesting worlds (with "interesting" defined as worlds in which the environment can contain stable records of the system's state), but it's certainly a set of worlds.

 

[Fredrik]

[*] The expectation of A on a density matrix p is Tr(p A_f)[/list]

Should we consider this a definition of a mathematical term, or a statement about what to expect when we perform measurements? If it's the latter, then this is the Born rule (the generalized version that works for mixed states too), which is precisely what the text you quoted (Demystifier's post) said not to use.

No time to think about the rest now. I need to get some sleep.

 

[dmtr]

It's very different because you have access to information from the space that surrounds you. You have no such access to "other" universes, by definition (if you have access, they're not "other," they're part of this one).

This argument is not a very sound one. MWI does not imply "other universes that we do not have access to", on the opposite it says that there is one universe that is defined by the universal wavefunction.

So I don't see why I can't say, that analogous to my position in space, my 'position in the probability' is not in any way preferred or unfairly sampled, and that this 'bifurcated history tree' you've mentioned is not symmetric.

 

[RUTA]

This argument is not a very sound one. MWI does not imply "other universes that we do not have access to", on the opposite it says that there is one universe that is defined by the universal wavefunction.

So I don't see why I can't say, that analogous to my position in space, my 'position in the probability' is not in any way preferred or unfairly sampled, and that this 'bifurcated history tree' you've mentioned is not symmetric.

See Adrian Kent's Perimeter presentation, "Theory Confirmation in One World and its Failure in Many," http://pirsa.org/index.php?p=speaker&name=Adrian_Kent.

 

[dmtr]

See Adrian Kent's Perimeter presentation, "Theory Confirmation in One World and its Failure in Many," http://pirsa.org/index.php?p=speaker&name=Adrian_Kent.

Well, my advice to these inhabitants would be "use the symmetry and the number of simulation copies to derive the probabilities for future events". Following this advice will help these inhabitants to predict the future better.

 

[Hurkyl]

Should we consider this a definition of a mathematical term

Certainly, but that does not exclude the possibility that it has something to say about what to expect when we perform measurements.

If it's the latter, then this is the Born rule (the generalized version that works for mixed states too), which is precisely what the text you quoted (Demystifier's post) said not to use.

I assumed that we wouldn't be worrying about reduced density matrices if we hadn't already made up our minds about density matrices.

Or, to put it differently, my argument justifies the use of reduced density matrices for studying quantitative and qualitative properties of density matrices, and is independent of whatever issues we have of the relation between density matrices and reality.

 

[jensa]

I have been meaning to ask a question related to some of the issues of this thread namely; Does decoherence require a an environment (decomposition into subsystems)?

I tend to view decoherence as an approximate feature defined as practical inability to observe a relative phase between two states. This feature does not seem to require an environment. Let's consider the prototypical case of the Schrödinger cat. The cat is a macroscopic object the state of which is determined by it's microscopic constituents. Let us assume that we can define subspaces of microscopic states where the cat is alive or dead with certainty (i.e. eigenstates of an "aliveordead operator") and the Hilbert space of the cat is a direct sum of these two subspaces. When we talk about the cat being in the state |alive> we are actually talking about a class of microscopic states belonging to the alive subspace, and similarly with the state |dead>. Now if we consider a superposition of states a|alive>+b|dead>, the relative phase \varphi=\text{arg}(b/a) is determined by the precise microscopic states. Due to the internal dynamics within each subspace the phase becomes a chaotic variable, which for all practical purposes would be impossible to observe. Note that the unobservability of the relative phase means that we could not (in practice) distinguish a pure state from a mixed state.

Regarding the MWI: Even if we accept that a macroscopic system, at least in a coarse grained view, evolves into a mixed state we still would need to somehow relate the diagonal parts of the density matrix to the probabilities of the observer experiencing a particular result. And I simply don't see how this can be done without invoking something equivalent to the born rule.

 

[Demystifier]

The process described by the second → in the quote is a measurement. This is what decoherence does. It turns pure states into mixed states.

I don't think that it is correct.

First, no physical process turns pure states into mixed states, provided that the whole system is taken into account. Such a process would contradict unitarity. Of course, if you consider a SUBsystem, then then such a process is possible.

Second, a measurement can be described even by describing the whole system. It may be impossible in practice due to a large number of the degrees of freedom, but it is possible in principle. Therefore, a measurement can be described in terms of pure states as well, at least in principle.

 

[Hurkyl]

Second, a measurement can be described even by describing the whole system. It may be impossible in practice due to a large number of the degrees of freedom, but it is possible in principle.

I am under the impression that is actually a significant point of contention between interpretations -- e.g. that Copenhagen says it's impossible in principle.

(unless, of course, you switch to a new and better physical theory)

 

[Fra]

I wonder if it's possible to prove that the Born rule is the only probability measure that gives us P(a,b)=P(a)P(b).

I wonder if you like this, more than I do?

Ariel Caticha's
Consistency, Amplitudes and Probabilities in Quantum Theory "Quantum theory is formulated as the only consistent way to manipulate probability amplitudes. The crucial ingredient is a consistency constraint: if there are two different ways to compute an amplitude the two answers must agree. This constraint is expressed in the form of functional equations the solution of which leads to the usual sum and product rules for amplitudes. A consequence is that the Schrödinger equation must be linear: non-linear variants of quantum mechanics are inconsistent. The physical interpretation of the theory is given in terms of a single natural rule. This rule, which does not itself involve probabilities, is used to obtain a proof of Born’s statistical postulate. Thus, consistency leads to indeterminism."
-- http://arxiv.org/PS_cache/quant-ph/pdf/9804/9804012v2.pdf

In effect I think he is trying to generalize Cox, Jaynes "derivation" of the rules of koglomorov's probability, not from the traditional axioms, but from some assumptions of consistency of reasoning around information.

One key assumption is that he assumes as part of the microstate of information beeing represented/quantified by a REAL numbers (or degree of beliefs). Then by a series of arguments he shows that the only consistent logical system och ang or operators etc that fulfills this (and some hidden mor or less natural assumptions) is indistinguishable from koglomorov probability axioms.

The generalisation is to isntead, assume (again key assumption) that instead the state of information is represented/quantified by a COMPLEX number. Then his idea is that similarly quantum logic emerges as the only consistent system. He argument for the Born rule is I think effectively the same as the ones trying to "count" distinguishable microstates and arrive at some kind of "frequency".

I think both of these ideas as intersting but my objection to both of those are the key assumptions of choose real or complex numbers as representation.

Ariel points out this himself:
"A second, simpler question is why do we seek a representation in terms of complex numbers? Again, no answer here; this is an unexplained feature of quantum theory. It seems that a single complex number is sufficient to convey the physically relevant information about a setup."

So I think there mere starting point of a normed linear vector space over complex numbers is quite non-trivial as well.

For me, the physical insight of the meaning of real numbers in a framework where only a finite amount of information is encodable is still mysterious. Before I understand this, I can not claim to be content with any higher level derivation which contains this as unmotivated baggage.

I'm curious if Fredrik, finds Ariels paper more satsifactory than I do?

/Fredrik

 

[Fra]

That's right (and also obvious). I just proved that if we use the tensor product and the Born rule, we get P(a,b)=P(a)P(b) for non-interacting systems. (It would be a disaster to get P(a,b)≠P(a)P(b). If QM works for nuclei and electrons separately, it wouldn't work for atoms. But of course it wouldn't work for nuclei either...) The point is that quantum mechanics for individual systems, which by definition includes the Born rule, more or less forces us to us to use the tensor product to represent the states of the composite system.

Do you picture that the notion of non-interacting systes still makes sense when you incorporate say gravity? how do you prevent the gravitational interaction?

/Fredrik

 

[Fredrik]

I wonder if you like this, more than I do?
...
I'm curious if Fredrik, finds Ariels paper more satsifactory than I do?

My impression (after spending about half an hour on it) is that he seems to have found a way to understand probability amplitudes on a slightly deeper level. It looks good, but not very significant.

Do you picture that the notion of non-interacting systes still makes sense when you incorporate say gravity? how do you prevent the gravitational interaction?

I'm not going to try to unify QM with GR today. Maybe tomorrow.

 

[RUTA]

Well, my advice to these inhabitants would be "use the symmetry and the number of simulation copies to derive the probabilities for future events". Following this advice will help these inhabitants to predict the future better.

You're still missing the point. How do you know what probability to use in a simulation if empirically all you ever have access to is one branch of the outcome? He's giving you a God's eye view so you can see that it's impossible for the individual inhabitants to do science. We're an individual inhabitant, not God.

 

[Fredrik]

I don't think that it is correct.

First, no physical process turns pure states into mixed states, provided that the whole system is taken into account. Such a process would contradict unitarity.

I meant approximately. I was just too lazy to explain that (and I thought people would understand what I meant after I had just explained it for the first type of process).

 

[Fra]

I'm not going to try to unify QM with GR today. Maybe tomorrow.

I sure won't do that neither today nor tomorrow either :)

But I guess my attitude is that I see indications that the deeper understanding of the foundations of QM, and a possible deeper motivation for QM and it's structure, might be additionally complicated by us trying to keep gravity out of it. Although on the surface foundational QG for sure looks more hairy than foundational QM, trying to find a connection as deep as possible might also enlighten us with regards to normal QM, and the structure of the standard model of particle physics in the quest for a GUT. Maybe gravity is simply the "missing link". Maybe it's a mistake to think that trying to think of gravity already from start will make the quest harder, rather than easier?

/Fredrik

 

[RUTA]

I'm not going to try to unify QM with GR today. Maybe tomorrow.

I'm hoping tomorrow ... or the day after. I have the equation, I just need a solution to get the party started

 

[Fredrik]

Maybe it's a mistake to think that trying to think of gravity already from start will make the quest harder, rather than easier?

That's certainly possible. The difference between non-relativistic QM and special relativistic QM is a different group of symmetries for the theory (or equivalently, a different algebra of observables). Is there a difference between special relativistic QM and general relativistic QM? Probably. I think that's what LQG is trying to answer. I also think that almost everything that's been written about attempts to interpret QM or its underlying mathematical structure as a description of what actually happens, have completely ignored those differences.

 

[Fredrik]

I have been meaning to ask a question related to some of the issues of this thread namely; Does decoherence require a an environment (decomposition into subsystems)?

It think the answer is yes. I don't know decoherence well, but the impression I got is that the crucial step is the calculation of a reduced density matrix, as discussed above. This isn't possible without a decomposition into subsystems. If you want a better answer, try searching arxiv.org for articles by Zurek, or buy Schlosshauer's book. (I haven't read it yet, but it's in my shopping cart).

 

[jensa]

It think the answer is yes. I don't know decoherence well, but the impression I got is that the crucial step is the calculation of a reduced density matrix, as discussed above. This isn't possible without a decomposition into subsystems. If you want a better answer, try searching arxiv.org for articles by Zurek, or buy Schlosshauer's book. (I haven't read it yet, but it's in my shopping cart).

Thank you for the response Fredrik,

I don't think that decomposition into subsystems is sufficient (maybe not even necessary) to produce decoherence. Consider two sets of interacting two-level systems. The Hilbert space of this composite system we describe as a tensor product of the individual Hilbert spaces. If we are only interested in the properties (observables) of one of the subsystems we may trace out the other one to produce a reduced density matrix. The reduced density matrix will generally be a mixed state but the coherence factors (off diagonal elements of the density matrix) need not vanish irreversibly. Most likely you would observe an oscillatory behaviour at a certain frequency. To observe irreversible behaviour you need to also assume that the system you are tracing out contains a large (infinite) number of degrees of freedom so that different frequencies add up to produce a decay on average.

It seems to me that the macroscopic nature of the environment (many degrees of freedom) is more important than the decomposition into subsystems in order to observe irreversible loss of coherence.

I hope I don't project too much of my own ignorance onto the people on this board but loss of coherence seems to be similar to the classical increase of entropy in that everybody believes it to occur but very few can actually show it (and explain what causes it). In the case of increase of classical entropy there seems to be a number of ways how to justify it. Personally I prefer the coarse grained explanation that the increase of our ignorance (entropy) comes from our mapping from the microscopic configurations towards macroscopic observables. I.e. it can be shown that when we map the many-particle phase space onto a space of macroscopic observables, and replace the distribution function \rho(X,t) where X is the coordinate in phase space, by the distribution function \tilde{\rho}(A,t) where A is a coordinate in the space of macroscopic observables, the entropy defined in terms of the macroscopic distribution function \tilde{\rho} always increases.

I believe the origin of decoherence has a similar source, namely that it is the mapping from microscopically distinguishable states onto macroscopically distinguishable observables that produces a practical impossibility to observe interference effects of macroscopic objects. As in my example with Schrödingers cat; our ignorance about the microscopic states along with the extremely short time scales of microscopic processes causes the interference effects to become "averaged out".

Of course, when we are talking about the decoherence of truly microscopic systems one needs to consider the entanglement with a macroscopic environment. Now the observables we are interested in are of course the observables associated with the original microscopic system (so they are not macroscopic observables). But still there exists a many-to-one mapping corresponding to our ignorance about the many degrees of freedom of the environment.

I wish I could be more thorough with my explanations but I have a lot of work to do. Btw, how do you guys manage to spend so much time on the boards and learn a lot of new stuff?

PS. I have Schlosshauers book and will try to read it when I get some time.

 

[Fra]

I'm hoping tomorrow ... or the day after. I have the equation, I just need a solution to get the party started

What was your general idea? (There was probably a past thread about this, but I can't remember)

/Fredrik

 

[Demystifier]

I am under the impression that is actually a significant point of contention between interpretations -- e.g. that Copenhagen says it's impossible in principle.

I guess you have in mind the idea that not everything can be described by QM, but that you also need a classical world. I would say that this is only one of several different variants of the Copenhagen interpretation. And I think that this particular variant is quite obsolete.

 

[Demystifier]

Some of you mentioned the Schlosshauer book. Let me just say that I recommend it to everyone. Today no one can say that he understands QM well without being familiar with basics of decoherence. Actually, this book contains more than just basics, but it is worth reading.

 

[Hurkyl]

I would say that this is only one of several different variants of the Copenhagen interpretation. And I think that this particular variant is quite obsolete.

My impression is that CI always has the property that collapse is real (as opposed to a mathematical technique or a change-of-frame-type thing). I know of three variants:

1. Quantum and classical mechanics tell us what's really going on¹, but QM becomes inaccurate above a certain scale and CM becomes inaccurate below a certain scale. We can effectively use the two in concert by invoking a Heisenberg cut.
2. QM doesn't tell us anything about what's really going on -- it is a theory of our ignorance² of deeper reality
3. QM will work all the way up, once we figure out what nonlinear terms should be inserted into Schrödinger's equation to make collapse occur

(note this last one still asserts that unitary evolution of states is wrong on macroscopic scales)

Is there another variant you had in mind?


1: Meaning, roughly, that the elements of the theory correspond to elements of reality
2: I don't mean this pejoratively -- I mean it as in "ignorance probabilities"[/size]

 

[Fra]

Second, a measurement can be described even by describing the whole system. It may be impossible in practice due to a large number of the degrees of freedom, but it is possible in principle.

I am under the impression that is actually a significant point of contention between interpretations -- e.g. that Copenhagen says it's impossible in principle.

(unless, of course, you switch to a new and better physical theory)

There was a recent discussion about the "definition of CI". But to leave aside that classification of views here, the reason why I think it's impossible even in principle is this:

As I see it, a given observer doesn't in general "choose" to observe this or that. The "complete" picture (as I see it) is that a given observer always "observes" (ineracts with) it's own environment.

But no given observer, can relate to, and decode all possible the degrees of freedom in the environment. So each observer sees a truncated world.

Now, I take a similar to rovelli's RQM view here, that the only way to level two "truncated worlds" against each other, is by means of the observers interacting. The view of each observer, will then be revealed by the way that act upon each other.

Like a game of poker. Each player has "a vision" of the future of the game - if my opponent folds, I have good grounds to think it's because he thinks he has slim chance to win - his vision is revealed to me. The only way to find out the opponents "visions" are to play the game.

The problem in QM, is that even though it's true that a second observer (possibly a massive one) COULD in some approximation observe the measurement process in the sense of "environment as the observer", this massive amount of information could never be conveyed back to the original observer - even if a perfect communication channel was established - simply because they are not comparable in complexity.

The analogy to say SR or GR would be perfect if we could establish the transformations that restore observer invariance here, and view these transformations in the realist sense.

But one problem is that there is no way that there can exist one-2-one transformations between structures that can't not encode the same amount of information. The transformations themselves must create and destroy information unless we have an *equilibrium situation* where the environment contains copies of the same repeating and thus "redundant" information, then we find the special case that a truncated system might contain the same information as a larger system.

But if we consider the general non-equilibrium case, I think the information preserving idea simply won't do.

I think that to understand QM, would be to find the general case, and then see why and how QM structure as we know it, does emerge as a uniqe special case. Like GR vs SR.

/Fredrik

 

[Fredrik]

There have been several discussions about the CI recently. There was a thread started by Demystifier, and this thread started by me. There are links to interesting papers in #7 and #20.

My impression is that the original CI is essentially the same as what we've been calling "the ensemble interpretation" in this forum. There's a formal difference in their definitions, but I don't see how that formal difference is an actual difference. See #33 in the thread I linked to above. Hurkyl, your #2 appears to be the same thing, but if QM doesn't tell us what actually happens, then there's nothing that suggests that "collapse" is a physical process, as you suggested at the start.

 

[RUTA]

What was your general idea?

A discrete path integral over graphs. The difference matrix K and source vector J are constructed from boundary operators in the spacetime chain complex of the graph so that Kx = J where x is the vector of nodes, links or plaquettes in the resulting scalar, vector or tensor field theory, respectively (this follows from the boundary of a boundary principle, BBP). This restricts K and J in the discrete action of the transition amplitude, Z. Note that Z is not a function of the field Q, i.e., Q is the integration variable in computing Z, yet quantum and classical field theories are all about Q. How is that?

When you ask for the probability that the k^th node, link or plaquette has the value Q_o you obtain Z(Q_k=Q_o)/Z (Z is a partition function since we're using a Euclidean path integral), which is the discrete counterpart to QFT. If you ask for the most probable value of Q_o, you find you must solve KQ_o = J, i.e., the discrete counterpart to CFT. Notice that by requiring the graphical basis satisfies the BBP, your discrete CFT automatically satisfies the BBP, which is the basis for the local conservation of momentum and energy (divergence-free stress-energy tensor). These are standard calculations, we're just proposing a different take on them that leads to a discrete CFT at odds with GR.

Anyway, we're trying to solve the tensor version of KQ_o = J for some simple situation and compare the result to GR to "get the party started." We expect differences from GR since our version of discrete tensor CFT is linear and constitutively non-local while GR is non-linear and local. Essentially, our discrete tensor CFT is Regge calculus where the nodes of the simplices are clusters of graphical elements (think of clusters of cubes joined by line segments) so there are no vacuum solutions (it's constitutively non-local aka inseparable -- no empty spacetime, just spacetimematter) and the lengths of the line segments joining the clusters are simply equal to the average values of the Q's on the shared cube faces (plaquettes) between clusters (average value = most probable value since we've a Gaussian distribution function). This resolves the problem resulting from the fact that violations of Bell's inequality imply causal and/or constitutive non-locality while GR is local on both counts, in favor of QM (GR must be revised).

We have a paper under review at Foundations of Physics. If that gets accepted and we find our classical solution, then maybe I'll start a thread We can't discuss this anymore here, it's out of context.

 

[Fredrik]

I have now read enough of this paper to see that the fact that we should be using the tensor product to represent a composite system can be derived without direct reference to the Born rule. It's based on the quantum logic approach to QM, which associates a mathematical structure with the set of statements of the form "if you measure observable A, you will get a result in the set B with probability 1". So it has some connection to probabilities, but it's not as strong as I expected. Maybe the stuff I mentioned above about how we need to use the tensor product to ensure that the Born rule satisfies P(a,b)=P(a)P(b) for non-interacting systems is in there somewhere, and I just can't see it.

 

[atyy]

What about http://arxiv.org/abs/0903.5082 which tries to derive the Born rule from

(i) States are represented by vectors in Hilbert space
(ii) Evolutions are unitary
(iii) Immediate repetition of a measurement yields the same outcome

"To derive it we cannot use reduced density matrices, Eqs. (1,2). Tracing out is averaging [25, 29, 30] - it relies on p_k = |psi_k|², Born's rule we want to derive."

 

[Fra]

A discrete path integral over graphs.
...
When you ask for the probability that the kth node, link or plaquette has the value Qo you obtain Z(Qk=Qo)/Z (Z is a partition function since we're using a Euclidean path integral), which is the discrete counterpart to QFT. I
...
We have a paper under review at Foundations of Physics. If that gets accepted and we find our classical solution, then maybe I'll start a thread We can't discuss this anymore here, it's out of context.

Thanks for the hints, just a couple of quick questions :)

I got the impression from other threads you are seeking a reconstruction of the continuum in terms of a discrete model that is more "fundamental"? I symphatise because I also find the continuum starting point inherently unphysical uncountable redundance that isn't helping at all.

Does this mean you also reconstruct a "discrete" probability theory, where the measure rather spans a discrete rational subset of [0,1], constrained by complexity?

Somehow this latter thing, is I think relevant to the discussion of defining probability since it allows to avoid the issue of "infinite measurements" and frequency limits. Instead finite information might imply that the probability measure itself is discretized and not covering a continuum. This would suggest that one could actually "count" the truncated contiuum and also define measures on stuff like the space of possibilities (which is used for feynmann summation).

/Fredrik

 

[RUTA]

Thanks for the hints, just a couple of quick questions :)

I got the impression from other threads you are seeking a reconstruction of the continuum in terms of a discrete model that is more "fundamental"? I symphatise because I also find the continuum starting point inherently unphysical uncountable redundance that isn't helping at all.

Does this mean you also reconstruct a "discrete" probability theory, where the measure rather spans a discrete rational subset of [0,1], constrained by complexity?

Somehow this latter thing, is I think relevant to the discussion of defining probability since it allows to avoid the issue of "infinite measurements" and frequency limits. Instead finite information might imply that the probability measure itself is discretized and not covering a continuum. This would suggest that one could actually "count" the truncated contiuum and also define measures on stuff like the space of possibilities (which is used for feynmann summation).

Yes, the discrete structure is fundamental to the continuum structure, not a mere approximation thereto.

Our partition function (transition amplitude, Z) is defined over a countable number of graphical elements, but each element can have an uncountable number of possible field values.

 

[Fredrik]

I don't think that decomposition into subsystems is sufficient (maybe not even necessary) to produce decoherence. Consider two sets of interacting two-level systems. The Hilbert space of this composite system we describe as a tensor product of the individual Hilbert spaces. If we are only interested in the properties (observables) of one of the subsystems we may trace out the other one to produce a reduced density matrix. The reduced density matrix will generally be a mixed state but the coherence factors (off diagonal elements of the density matrix) need not vanish irreversibly. Most likely you would observe an oscillatory behaviour at a certain frequency. To observe irreversible behaviour you need to also assume that the system you are tracing out contains a large (infinite) number of degrees of freedom so that different frequencies add up to produce a decay on average.

Sounds like you know a few things about decoherence that I don't. But you specifically mention reduced density matrices, and those can't even be defined without a tensor product decomposition. I have started reading a pdf version of Schlosshauer (I'm still buying the real one) and in the intro, he describes decoherence as the system getting more and more entangled with the environment. Everything I have seen indicates that you need to consider at least two component subsystems: "the system" and "the environment".

 

[Fredrik]

What about http://arxiv.org/abs/0903.5082 which tries to derive the Born rule from...

Thanks. I intend to check it out, but I'll probably wait until I've read some more in Schlosshauer's book. I expect that I will have objections about the use of the tensor product and the use of density matrices. The former has some connection with probabilities that I don't fully understand yet (see #139), and the latter seems impossible to justify without the Born rule (see #108).

 

[jensa]

But you specifically mention reduced density matrices, and those can't even be defined without a tensor product decomposition.

Yes of course you are right, but my point with the passage you quoted was that even if we can decompose a system into subsystems (with tensor product) and look at the reduced density matrix of the particular subsystem we are interested in, you still do not necessarily get irreversible loss of coherence! Sure, the reduced density matrix will generally be of the mixed kind but will it irreversibly go towards a total loss of coherence?

This depends on what the other subsystem is. Mostly when people talk about "environment" I believe it is implied that it consists of an infinite amount of degrees of freedom (i.e. it is macroscopic). In other words, the macroscopic nature is at least as important to the concept of decoherence as the decomposition into subsystems. My general point is that you can actually remove the decomposition as a necessity.

Let me try one last time to convince you that you can get a loss of coherence even without the decomposition feature by returning to my original example of Schrödingers cat.

Let us write the general microscopic state of the macroscopic system (cat) as:

<br /> |\psi\rangle = \sum_i c_i(t)|i\rangle<br />

where i here denotes a set of labels completely characterizing the microscopic state of all particles the cat consists of (clearly a huge amount). Let us assume that we have chosen a basis in such a way that we can clearly distinguish for which microscopic state |i\rangle the cat is either dead or alive. We can formally define a set of projection operators:

<br /> \hat{P}_\text{alive}=\sum_{i\in alive}|i\rangle\langle i|, \quad \hat{P}_\text{dead}=\sum_{i\in dead}|i\rangle\langle i|<br />

We can then associate the state \hat{P}_\text{alive}|\psi\rangle=\sum_{i\in \text{alive}}c_i(t)|i\rangle with a macroscopic state of the cat being alive and vice versa with the macroscopic "dead state". If we, for a moment, trust the conventional probability rule we have:

<br /> \text{Prob. alive}=\langle \psi|\hat{P}_\text{alive}|\psi\rangle=\sum_{i\in \text{alive}}|c_i(t)|^2, \quad \text{Prob. dead}=\langle \psi|\hat{P}_\text{dead}|\psi\rangle=\sum_{i\in \text{dead}}|c_i(t)|^2<br />

So far so good, but is it possible to define a "relative phase" between the two macroscopic states "dead" and "alive"? In principle it should be clear already here that such a feat is difficult and designing an interference experiment without knowing the microscopic configuration of the cat is pretty much impossible. However, we could analyze the operator

<br /> \hat{P}_\text{a-d}=\sum_{i\in \text{alive}}\sum_{j\in \text{dead}}|i\rangle\langle j|<br />

which connects the dead and alive subspaces and thus the object

<br /> \langle \psi|\hat{P}_\text{a-d}|\psi\rangle=\sum_{i\in \text{alive}}\sum_{j\in \text{dead}}c_i^*(t)c_j(t)<br />

is directly related to the "coherence". Now there are two issues here: 1) The c_i's are determined by the exact microscopic state (determined by initial conditions and exact many-particle hamiltonian) of which we are clearly ignorant. 2) the time scale of variation of the c_i(t) is very short compared to macroscopic time scales. We thus expect this object to fluctuate wildly (both statistically and temporally) i.e. it is essentially a chaotic variable. Performing an average (coarse graining) over this object will average it out. The probabilities on the other hand must of course add up to unity and, while it may fluctuate depending on initial conditions and as a consequence of time dependence of c_i(t), because it is always a positive quantity between 0 and 1 it will average to some constant. What this is supposed to illustrate is that the off-diagonal elements will vanish upon statistical and temporal averaging, and it is effectively impossible to create an interference experiment that may be used to observe the relative phase between the macroscopic states "dead" and "alive".

I have started reading a pdf version of Schlosshauer (I'm still buying the real one) and in the intro, he describes decoherence as the system getting more and more entangled with the environment. Everything I have seen indicates that you need to consider at least two component subsystems: "the system" and "the environment".

Yes he seems to credit the non-locality of quantum mechanics, which of course is related to system+environment. In fact most textbooks seem to use this notion to describe decoherence. However, I feel like this is only one specific type of decoherence (environmentally induced decoherence) and that decoherence in general can be described without the use of an external environment.EDIT: Sorry everyone for the long post..hope I didn't derail the discussion too much.

/Jens

 

[Fredrik]

The argument by Hartle allows you to replace the Born rule by the weaker rule that says that measuring an observable of a system if the system is in an eigenstate of that observable, will yield the corresponding eigenvalue with certainty.

I've been doing some more thinking about this. I still think that Hartle's argument is useless, and proves nothing, but I've realized that Gleason's theorem says something very similar to the above. It says that if \mu is a probability measure on the set of subspaces on a separable Hilbert space (real or complex, and at least 3-dimensional), there exists a density operator \rho such that

\mu(M)=\mbox{Tr}(\rho P_M)

where P_M is the projection operator associated with the closed subspace M.

Consider the simplest possible case, i.e. when \rho is a pure state |\psi\rangle\langle\psi|, and M is a 1-dimensional eigenspace corresponding to the eigenvalue b of an observable B. The theorem says that the only possible probability measure assigns probabilty

\mbox{Tr}(|\psi\rangle\langle\psi|b\rangle\langle b|)=\sum_{b&#039;}\langle b&#039;|\psi\rangle\langle\psi|b\rangle\langle b|b&#039;\rangle=|\langle b|\psi\rangle|^2

to that eigenspace. So it certainly looks like Gleason has derived the Born rule. There are however several subtle points worth noting here.

1. Why are we looking for probability measures on the set of closed subspaces of a separable Hilbert space? A partial answer is that the closed subspaces can be thought of as representing "properties" of physical systems in QM*. This is an axiom in the quantum logic approach to QM**, but in the traditional Hilbert space approach to QM, we would have to use the Born rule to prove that this is true.

2. Where did we "put probabilities into get probabilities out"? This is explained by item 1 and the footnotes. The probability measure (which is uniquely determined by the state) assigns non-trivial probabilities to mathematical objects (closed subspaces) that are already associated with assignments of probability 1 to possible events in the real word. (This is where it's very similar to what Count Iblis claimed above).

3. This is clearly not a derivation of the sort originally envisioned by Everett. It isn't a derivation from the assumption that the state of the universe can be described by a state vector satisfying a Schrödinger equation. Gleason didn't assume that. Instead he started with the assumption that the set of "properties" is represented by the simplest possible mathematical structure that's consistent with a set of axioms that we expect all theories to satisfy. Of course, before QM was discovered, we would have guessed that all theories must satisfy a much stronger set of axioms, so this set of axioms was chosen specifically to ensure that QM (with its Born rule) qualifies as a theory.

*) What I call "properties" goes by many names in the literature, including "propositions", "elements of reality" and "experimentally verifiable statements". The last one is probably the most appropriate, since these phrases all refer to the possible results of experiments that are assigned probability 1 by QM.

**) Technically they use another set of axioms in order to associate a mathematical structure with the set of properties, but then they define a "standard" structure as one that's isomorphic to the lattice of closed subspaces of a complex separable Hilbert space

 

[jensa]

Just to conclude my somewhat tangential series of posts: What I have been describing falls under a category which Joos calls "Fake Decoherence"*, while he prefers to restrict the word decoherence to the system+environment stuff. Personally I find the term "fake decoherence" quite misleading; the effect of practical inability to observe coherence is quite real. I would prefer to call the effect itself decoherence and then use "environmentally induced decoherence" to refer to the system+environment stuff. Of course it is quite possible that for the purposes of MWI only environmentally induced decoherence is important, although I cannot see immediately why this would be so.

*) See the book "Decoherence and the appearance of a classical world in quantum theory" by Joos, Zeh, Kiefer, Giulini, Kupsch and Stamatescu in the section entitled "True, False and Fake decoherence". I personally prefer this book over Schlosshauer's.

 

[Demystifier]

*) See the book "Decoherence and the appearance of a classical world in quantum theory" by Joos, Zeh, Kiefer, Giulini, Kupsch and Stamatescu in the section entitled "True, False and Fake decoherence". I personally prefer this book over Schlosshauer's.

This is an excellent book too. However, I cannot find the section you mention above. Can you help me (section number, page number, contributor name, or something like that)?

 

[jensa]

This is an excellent book too. However, I cannot find the section you mention above. Can you help me (section number, page number, contributor name, or something like that)?

I have the second edition where it is in Chapter 3 "Decoherence Through interaction with the Environment" by Joos, section 3.4.3. Perhaps it is absent in the first edition?

Edit: You can find the second edition by searching for it in Google Books.

 

[Demystifier]

I have the second edition where it is in Chapter 3 "Decoherence Through interaction with the Environment" by Joos, section 3.4.3. Perhaps it is absent in the first edition?

Edit: You can find the second edition by searching for it in Google Books.

Yes, I have the first edition. It does not even contain Sec. 3.4.3.

The preview by Google Books gives first 101 pages, which does not cover Sec. 3.4.3 either.