Why MWI cannot explain the Born rule

[Count Iblis]

Fredrik:

There are already no such sectors in the individual system Hilbert space, so they're clearly talking about the tensor product of infinitely many copies of that space. The removal of the zero norm "vectors" that get included by accident when we take the tensor product of infinitely many copies of a Hilbert space is necessary to ensure that the result is a Hilbert space. It doesn't have anything to do with the Born rule.

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).

 

[RUTA]

I don't see any problem, as long as the symmetric apparatus gives you symmetric probabilities (probability amplitudes to be precise).

You don't know if it's symmetric or not. You can't know. All you know is the history of your particular branch. You don't know if the statistics you observe are the overall "correct" statistics, but you can guess they're probably not.

 

[dmtr]

You don't know if it's symmetric or not. You can't know. All you know is the history of your particular branch. You don't know if the statistics you observe are the overall "correct" statistics, but you can guess they're probably not.

But I do observe the "correct" statistics. And it corresponds nicely to the symmetric apparatus. In the example above there is roughly the same number of '0's and '1's in the original sequences. I can also see that the SUM: 1 1 1 1 0 0 0 1 2 1 contains more '1's than '0's (or '2's). This corresponds nicely to the: '50/50' world is more probable than 100/0 (or 0/100).

And it doesn't look like this "correct" statistics excludes MWI in any way. Why I-who-saw-"1" should consider myself any better from I-who-saw-"0"? From the symmetry considerations I shouldn't.

 

[RUTA]

But I do observe the "correct" statistics. And it corresponds nicely to the symmetric apparatus. In the example above there is roughly the same number of '0's and '1's in the original sequences. I can also see that the SUM: 1 1 1 1 0 0 0 1 2 1 contains more '1's than '0's (or '2's). This corresponds nicely to the: '50/50' world is more probable than 100/0 (or 0/100).

And it doesn't look like this "correct" statistics excludes MWI in any way. Why I-who-saw-"1" should consider myself any better from I-who-saw-"0"? From the symmetry considerations I shouldn't.

You're missing the point. Your results are only one path in the bifurcated tree. You have no idea where your particular history resides in the tree because you only have access to your single history.

 

[dmtr]

You're missing the point. Your results are only one path in the bifurcated tree. You have no idea where your particular history resides in the tree because you only have access to your single history.

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.

 

[Demystifier]

For example it has to postulate the existence of parallel universes and that those universes are unobservable. The concept and existence of parallel universes certainly does not follow from those two axioms, it has to be postulated in their interpretation.

This is simply wrong. 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.)

 

[yossell]

The existence of parallel universes DOES follow from those two axioms.

Just a reminder of the axioms:
1. Psi is a solution of a linear deterministic equation.
2. Psi represents an objectively real entity.

Existence of many worlds simply follow from these - can't even imagine how the derivation would go unless you're packing a lot of background ASSUMPTIONS into what counts as an 'objectively real entity', or unless you mean something much less by 'many worlds' than the literal existence of many worlds. Already given example of mathematical platonist who believes psi is a solution of linear deterministic equation, and who believes psi reps an objectively real entity - but no many worlds theorist. Still waiting for the proper argument to be elaborated on.

 

[Hurkyl]

Hrm. A question to make sure everyone is on the same page.



Let's say I'm using coordinates and arithmetic as a means of studying Euclidean geometry.

Would you say that this does, or does not, require more assumptions than just the postulates of Euclidean geometry?

Would you say that the arithmetic properties of coordinates do, or do not, follow from the axioms of Euclidean geometry

 

[yossell]

Let's say I'm using coordinates and arithmetic as a means of studying Euclidean geometry.

Would you say that this does, or does not, require more assumptions than just the postulates of Euclidean geometry?

Would you say that the arithmetic properties of coordinates do, or do not, follow from the axioms of Euclidean geometry

Interesting question - I'm not sure where you're going with this but, spidey sense tingling and feeling there's a trap... I'm going to hedge an answer:

Some presentations use coordinate systems in their actual presentation of Euclidean geometry - they're built in at the start. But Euclid himself (I understand) used geometric rather than arithmetic predicates - in his system, he talked directly of lines, points and planes rather than triples of reals. Hilbert, Foundations of Geometry, gives a more modern version, involving predicates of betweenness and congruence, and modified to allow for things like the completeness of the real line to be formulated - something I'm not sure the original Euclidean system was capable of formulating.

If, by 'axioms of Euclidean geometry' you had in mind a formulation which involved only such geometric predicates, then my first reaction is to say that we did need more than just Euclidean geometry - we would also need axioms setting up the properties of real numbers, and triples of real numbers, and something linking the axioms about reals with the axioms about lines and points.

The reason, however, that I hedge, is because I believe that in Hilbert's formulation of Euclidean geometry, one can "model" the numbers and do number theory: it's possible to "define" + and x (in the sense that they have the right logical properties). If you're some kind of structuralist about numbers, this might all one needs to think one can have coordinates in Eucidean geometry. But off the top of my head, I'm not sure how much coordinate geometry can be recovered in this way.

 

[Hurkyl]

But off the top of my head, I'm not sure how much coordinate geometry can be recovered in this way.

The theory of real numbers turns out to be equivalent to the theory of Euclidean geometry, with one direction working pretty much just as you described.



(disclaimer: I am a mathematician not a physicist, and I don't have a particularly deep knowledge of what I'm going to say, so take it with a grain of salt)

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.

(and they could also be used to study arbitrary vector spaces, or state spaces of algebras, and so forth -- but I have absolutely no idea if it would ever be a useful thing to do)[/color]


The only new supposition of MWI is that the process of describing reality by quantum states evolving unitarily might continue to be applicable when we include measuring devices and observers into the quantum system -- something interpretations like Copenhagen reject a priori. Since this is a novel and powerful feature, a lot of effort is put into studying it. But if it really cannot be done, MWI would still be applicable to the domains where quantum states evolving unitarily works.



The reason I ask my question is I agree that MWI is more complex -- without simplifying assumptions like wavefunction collapse, it is technically more complicated, thus prompting new theoretical concepts to study it, and a lot of people equate these new theoretical concepts with extra assumptions and multiplying entities. (But I think that point of mine is not directed at you, but at others in the thread. I think the part of this most applicable to you is the gray parenthetical in the middle)

 

[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.