# Is Everett's relative state interpretation logically sound?

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Is Everett's relative state interpretation (Rev. Mod. Phys. 29 (1957), 454-462), usually called Many Worlds (MWI) logically sound?

Note that I am not asking about one of the many variations of the MWI formulated not in precise mathematical terms, of which it is therefore difficult to ascertain their logical status. However Everett himself tried to carefully pin down the formal meaning of his interpretation as part of his Ph.D. thesis in physics, so that the question can be meaningfully asked about his version of the MWI.

H. Stein, The Everett interpretation of quantum mechanics: Many worlds or none? Nous 18 (1984), 635-652.

A. Kent, Against many-worlds interpretations, Int. J. Mod. Phys. A5 (1990), 1745.

E.J. Squires, On an alleged 'proof' of the quantum probability law, Phys. Lett. A145 (1990), 67-68.

edited Nov 3, 2015

many "sciences" turn around the questions of the fundamental interpretations. For each, scientists, thinking having found the good formalism to ask and answer, write thesis and books, when it is not worship books, quickly popular. But, as far I know, all end in apparent circularity, which denotes the lacks of evaluation criteria or merely of any sciences basis. Physics cannot answer to anything  ( today ). History of sciences is full of similar states of knowledge before discoveries and revolutions.

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To me, it seems that the answer is no.

Everett's argumentation seems to be flawed by a well-disguised circularity:

In his original paper Rev. Mod. Phys. 29 (1957), 454-462, Everett claims that his theory is one without the need for state reduction (what he calls 'process 1'). But when he discusses observation, he brings in at the very beginning an apparently innocent additional concept, that of a 'good observation'. The formal definition of the latter is the sentence around (10)-(11) of his paper.

According to Everett, a 'good observation' is an interaction that transforms each state $\phi_S \otimes\psi_O$ such that $\phi_S$ is a fixed eigenstate of the measured variable with eigenvalue alpha into a state $\phi_S \otimes\psi_O(\alpha)$ where $\psi_O(\alpha)$ belongs to a set $X(\alpha)$ of states which belong to the awareness of $\alpha$. In particular, this entails that for different $\alpha$, the sets $X(\alpha)$ must be disjoint. Since Everett only allows the unitary dynamics (his 'process 2', see first line of his Section 3), any interaction 'in a specified period of time', must result in a unitary mapping $U$ on the state space of system plus observer.

Therefore, $U$ corresponds a 'good observation' (of the system in state $\Phi_S$) iff there is a self-mapping $\psi_O \to\psi_O(\alpha)$ of the observer state space $X$ such that
$$U(\phi_S \otimes\psi_O) = \phi_S \otimes\psi_O(\alpha) \forall \psi_O \in X.$$
From his definition, we also see that the mapping $\psi_O \to\psi_O(\alpha)$ maps $X(\alpha)$ into itself, and the whole observer state space $X$ into $X(\alpha)$. Since the interaction is unitary, it is invertible; but the restriction of $U$ to $\phi_S \otimes X$can be invertible only if $X(\alpha)=X$. But this means that there is only a single eigenvalue $\alpha$.

Therefore, under the assumptions made by Everett, there are no 'good observations', and since his analysis of the observational process depends on the latter, it is void of any meaning. Indeed, looking closer at the concept of a 'good observation', one can see that it is the projection postulate in disguise. (The above argument loses its power once one allows $U$ to be nonunitary.)

Thus Everett's analysis simply derives the projection postulate by having assumed it, without any discussion, in disguise.

answered Nov 1, 2015 by (15,767 points)

I'm not going to respond to your specific complaint with Everett's argument because you are entirely missing the point of his argument, which does not at all hinge on the particular move you take issue with. The point is very simple: is psi epistemic or ontic? If you answer "epistemic" then you don't believe in the MWI. If you answer "ontic" then the "many worlds" are unavoidable because superposition of an ontic thing is definitionally equivalent to "many worlds." In other words if you think an atom's wave function is a real "thing" and that this "thing" can be split apart into a lump that goes through a left slit and a lump that goes through a right slit, then in a clear and objective sense the atom traverses multiple paths simultaneously. We refer to such behavior as definitionally equivalent to the atom existing in multiple "worlds".

@ArnoldNeumaier OK I'll reply to your particular complaint here, since it is easy enough. Your confusion seems to be related to your previous confusion (see your question here) about how apparent collapse and subjective randomness is possible in the MWI. What you are confused about is the process which generates the self-mapping described in your post, which you complain is tautological. That process, which I explained in the linked thread, is just anthropic selection among the reference class of observers in the wave function. If we decompose the wave function of observers into a superposition indexed by $\alpha$, then there will be an observer for which everything Everett says is true. This seems to be, generally, the core conceptual confusion that most complaints about the MWI are founded on. The point is that even though the observer indexed by $\alpha$ is still in superposition with other observers, it cannot be aware of said superposition by linearity.

@ArnoldNeumaier, you don't seem to understand my comment. In your argument you say:

But this means that there is only a single eigenvalue α. Therefore...

Yes, there is only a single eigenvalue, because the observer cannot be aware of any other versions of himself that are entangled with other eigenvalues, by linearity. The process by which this one eigenvalue is selected from the spectrum of X is what I have described as the core conceptual move of the MWI: anthropic selection. That process itself isn't unitary (and it is indeed a kind of projection postulate), but the physical interaction is unitary, U is unitary, and objectively the wave function continues to evolve unitarily without collapse. This still seems to be a point of confusion for you -- the subjective (ie relative-state) vs objective (ie global wave function) dichotomy.

@ArnoldNeumaier, The entropic decoherence process is objective, doesn't require any kind of collapse whatsoever, has nothing to do with anthropics, and is not in conflict with unitarity. This is not at all controversial. The entropic growth of entanglement as a system interacts with its surroundings is a purely unitary process.

@CharlesJQuarra, that's because in MWI the various branches of the wave function can theoretically recohere. Von neumann entropy is useful for relative states, but for pure states it correctly encapsulates the fact that the dynamics are fully reversible in principle. There are plenty of other entropic definitions out there to suit whatever intuition you have about the increasing complexity of the universal wave function...

if the quantum superposition is ontic, then it is easy to create a literally ontic many-world scenario; basically make a war or not depending of a quantum measurement, divide the nation or not depending on another, and eliminate government regulations or not depending on another. Revisit 10 years later and you have vastly different versions of the world, entangled to 3 qubits.

one of the things that put me off about MWI, is that Von Neumann entropy does not seem to regard a pure state as 'many possibilities' since it says that a pure state has always zero entropy, so a 'single' possible microstate. I don't know how to reconcile the fact that a quantum superposition of two 'worlds', being pure, must have no entropy?

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I think Everett's MWI is not only logically sound, but the only logically sound interpretation of quantum physics that I have seen.

Everett says the universe evolves according to the deterministic laws of quantum physics, without collapse.

Collapse is an artifact of our limited perception, which has evolved to specialize in quick and dirty processing of things directly and urgently relevant to our survival and reproduction. The brain is a portable, low-power processing device that must focus on a small subset of inputs and discard the rest.

Every observer splits reality into manageable chunks of information and splits itself into multiple instances that focuses on one chunk (branch) each. Information is only exchanged between observer instances that share a branch.

This formulation of the MWI is often indicated as Many-Minds Interpretation (MMI) but I believe it's what Everett had in mind.

Many physicists and philosophers have tried to determine where the collapse happens and concluded that it happens in the conscious mind. But thinking that consciousness creates reality seems too anthropocentric to me. Also, how to sync different observers? In the MWI/MMI there is no collapse. The splitting that other interpretations call collapse does happen in the mind, but that doesn't imply that consciousness creates reality.

answered Nov 3, 2015 by (190 points)

Note: I am not saying that the MWI / MMI is the correct interpretation of quantum physics. I am saying that, on the basis of the current mathematical formulation of quantum physics, its proposed interpretations, and my own awareness, knowledge, and understanding thereof, at this moment the MWI / MMI make more sense to me than other interpretations. That may change, as always in science. In particular, we can't rule out changes in quantum physics itself that might radically change its interpretation.

To be taken seriously, you should give some references to justify what you think is true. Just asserting something is not enough.

Everett's MWI is not only logically sound, but the only logically sound interpretation of quantum physics

Since I pointed out a logical flaw in his reasoning, you should either tell me why my observation is wrong, or give up your claim of logical soundness.

Everett says the universe evolves according to the deterministic laws of quantum physics, without collapse. Collapse is an artifact of our limited perception

Where does Everett say that collapse is an artifact of our limited perception? Or is this your own modification of his views?

Many-Minds Interpretation (MMI) but I believe it's what Everett had in mind.

It seems to me that you are splitting Everett's mind! Or where from Everett's writing do you take your cues to support your belief?

Every observer splits reality into manageable chunks of information and splits itself into multiple instances

How often does this happen? Once a minute? Slower? faster? What decides upon the rate of splitting? How can one find out? If one cannot find out, it is no science.

@ArnoldNeumaier re "Since I pointed out a logical flaw in his reasoning, you should either tell me why my observation is wrong, or give up your claim of logical soundness."

As part of the definition of "good observation" Everett says "the observer state shall change so as to describe an observer that is 'aware' of which eigenfunction it is; that is, some properties is recorded in the memory of the observer..." - which is an operational definition of good observations, which do exist (we do remember things).

Re "Where does Everett say that collapse is an artifact of our limited perception? Or is this your own modification of his views?"

This is my own interpretation of his views. Interpretation, not modification: it seems to me that (for example) the quote above is compatible with this (re)interpretation.

Re "How often does this happen? Once a minute? Slower? faster? What decides upon the rate of splitting? How can one find out? If one cannot find out, it is no science."

It is science, but perhaps it has more to do with neurology than fundamental physics. ADDED: I guess the rate is related to the typical rate of storage of short-term memory.

The focus is made on many worlds at the same time at the same location. We know that the latter is a difficult concept, meaningless in many contexts. What about having the same at a time and location, probably different but unspecified while staying in the same universe ??? - Yes, interesting but it is just common QM without pretensions to any "world" interpretation ! Unless one having nothing interesting to do , claims that the 2 approaches are very similar and  searches what became the mwi causality ( states inheritance from parents to childs universes ) in the common theory... I don't know what may happen if the wine is good ...

@igael - that's "another many-worlds," the inflationary many-worlds. Some physicists including Susskind and Tegmark think these two many-worlds are strongly related.

https://arxiv.org/abs/1105.3796

https://en.wikipedia.org/wiki/Our_Mathematical_Universe

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