Lecture 20 Everett’s Relative-State Formulation of Quantum Mechanics PDF

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Lecture about Everett’s Relative-State Formulation of Quantum Mechanics...

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Everett’s Relative-State Formulation of Quantum Mechanics Hugh Everett III’s relative-state formulation of quantum mechanics is a proposal for solving the quantum measurement problem by dropping the collapse dynamics from the standard von Neumann-Dirac formulation of quantum mechanics. Everett intended to recapture the predictions of the standard collapse theory by explaining why observers nevertheless get determinate measurement records that satisfy the standard quantum statistics. There has been considerable disagreement over the precise content of his theory and how it was suppose to work. Here we will consider how Everett himself presented the theory, then briefly compare his presentation to the many-worlds interpretation and other no-collapse options. 1. Introduction Everett developed his relative-state formulation of quantum mechanics while a graduate student in physics at Princeton University. His doctoral thesis (1957a) was accepted in March 1957 and a paper (1957b) covering essentially the same material was published in July of the same year. DeWitt and Graham (1973) later published Everett’s longer, more detailed description of the theory (1956) in a collection of papers on the topic. The

published version was revised from a longer draft thesis that Everett had given John Wheeler, his Ph.D. adviser, in January 1956 under the title “Wave Mechanics Without Probability”. While Everett always favored the description of the theory as presented in the longer thesis, Wheeler, in part because of Bohr’s disapproval of Everett’s critical approach, insisted on the revisions that led to the much shorter thesis that Everett ultimately defended. Everett took a job outside academics as a defense analyst in the spring of 1956. While subsequent notes and letters indicate that he continued to be interested in the conceptual problems of quantum mechanics and, in particular, in the reception and interpretation of his formulation of the theory, he did not take an active role in the debates surrounding either. Consequently, the long version of his thesis (1956) is the most complete description of his theory. Everett died in 1982. See (Byrne 2010) for further biographical details and (Barrett and Byrne 2012) for an annotated collection of Everett’s papers, notes, and letters regarding quantum mechanics. See also (Osnaghi, Freitas, Freire 2009) for an excellent introduction to the history of Everett’s formulation of quantum mechanics. Everett’s no-collapse formulation of quantum mechanics was a direct reaction to the measurement problem that arises in the standard von Neumann-Dirac collapse formulation of the theory. Everett understood this problem in the context of a version of the Wigner’s Friend story. Everett’s solution to the problem was to drop the collapse postulate from the standard formulation of quantum mechanics then deduce the empirical predictions of the standard collapse theory as the subjective experiences of observers who were themselves modeled as physical systems in the theory. The result was his relative-state interpretation of pure wave mechanics. There have been many mutually incompatible presentations of Everett’s theory. Indeed, it is fair to say that most no-collapse interpretations of quantum mechanics have at one time or another either been directly attributed to Everett or suggested as charitable reconstructions. The most popular of these, the many worlds interpretation, is often simply attributed to Everett directly and without comment even when Everett himself never characterized his theory in terms of many worlds. In order to understand Everett’s proposal for solving the quantum measurement problem, one must first clearly understand what he took the quantum measurement problem to be. We will start with this, then consider Everett’s presentation of his relative-state formulation of pure wave mechanics quantum mechanics and the sense in which he took it to solve the quantum measurement problem. We will then contrast Everett’s views the many-worlds interpretation and a number of other alternatives. 2. The Measurement Problem Everett presented his relative-state formulation of pure wave mechanics as a way of avoiding conceptual problems encountered by the standard von

Neumann-Dirac collapse formulation of quantum mechanics. The main problem, according to Everett, was that the standard collapse formulation of quantum mechanics, like the Copenhagen interpretation, required observers always to be treated as external to the system described by the theory. One consequence of this was that neither the standard collapse theory nor the Copenhagen interpretation can be used to describe the physical universe as a whole. He took the von Neumann-Dirac collapse theory to be inconsistent and the Copenhagen interpretation to be essentially incomplete. We will follow the main argument of Everett’s thesis and focus here on the measurement problem as encountered by the standard collapse theory. In order to understand what Everett was worried about, one must first understand how the standard collapse formulation of quantum mechanics works. The theory involves the following principles (von Neumann, 1955): 1. Representation of States: The state of a physical system SS is represented by an element of unit length in a Hilbert space (a vector space with an inner product). 2. Representation of Observables: Every physical observable OO is represented by a Hermitian operator OO on the Hilbert space representing states, and every Hermitian operator on the Hilbert space corresponds to some observable. 3. Eigenvalue-Eigenstate Link: A system SS has a determinate value for observable OO if and only if the state of SS is an eigenstate of OO. If it is, then one would with certainty get the corresponding eigenvalue as the result of measuring OO of SS. 4. Dynamics: (a) If no measurement is made, then a system SS evolves continuously according to the linear, deterministic dynamics, which depends only on the energy properties of the system. (b) If a measurement is made, then the system SS instantaneously and randomly jumps to a state where it either determinately has or determinately does not have the property being measured. The probability of each possible post-measurement state is determined by the system’s initial state. More specifically, the probability of ending up in a particular final state is equal to the norm squared of the projection of the initial state on the final state. Everett referred to the standard von Neumann-Dirac theory the “external observation formulation of quantum mechanics” and discussed it beginning (1956, 73) and (1957, 175) in the long and short versions of his thesis respectively. While he took the standard collapse theory to encounter a serious conceptual problem, he also used it as the starting point for his presentation of pure wave mechanics, which he described as the standard collapse theory but without the collapse dynamics (rule 4b). We will briefly describe the problem with the standard theory, then turn to Everett’s

discussion of the Wigner’s Friend story and his proposal for replacing the standard theory with pure wave mechanics. According to the eigenvalue-eigenstate link (rule 3) a system would typically neither determinately have nor determinately not have a particular given property. In order to determinately have a particular property the vector representing the state of a system must be in the ray (or subspace) in state space representing the property, and in order to determinately not have the property the state of a system must be in the subspace orthogonal to it, and most state vectors will be neither parallel nor orthogonal to a given ray. The deterministic dynamics (rule 4a) typically does nothing to guarantee that a system will either determinately have or determinately not have a particular property when one observes the system to see whether the system has that property. This is why the collapse dynamics (rule 4b) is needed in the standard formulation of quantum mechanics. It is the collapse dynamics that guarantees that a system will either determinately have or determinately not have a particular property (by the lights of rule 3) whenever one observes the system to see whether or not it has the property. But the linear dynamics (rule 4a) is also needed to account for quantum mechanical interference effects. So the standard theory has two dynamical laws: the deterministic, continuous, linear rule 4a describes how a system evolves when it is not being measured, and the random, discontinuous, nonlinear rule 4b describes how a system evolves when it is measured. But the standard formulation of quantum mechanics does not say what it takes for an interaction to count as a measurement. Without specifying this, the theory is at best incomplete since it does not indicate when each dynamical law obtains. Moreover, if one supposes that observers and their measuring devices are constructed from simpler systems that each obey the deterministic dynamics, as Everett did, then the composite systems, the observers and their measuring devices, must evolve in a continuous deterministic way, and nothing like the random, discontinuous evolution described by rule 4b can ever occur. That is, if observers and their measuring devices are understood as being constructed of simpler systems each behaving as quantum mechanics requires, each obeying rule 4a, then the standard formulation of quantum mechanics is logically inconsistent since it says that the two systems together must obey rule 4b. This is the quantum measurement problem in the context of the standard collapse formulation of quantum mechanics. See the section on the measurement problem in the entry on philosophical issues in quantum theory. The problem with the theory, Everett argued, was that it was logically inconsistent and hence untenable. In particular, one could not provide a consistent account of nested measurement in the theory. Everett illustrated the problem of the consistency of the standard collapse theory in the context of an “amusing, but extremely hypothetical drama” (1956, 74–8), a story that was a few years later famously retold by Eugene Wigner.

Everett’s version of the Wigner’s Friend story involved an observer AA who knows the state function of some system SS, and knows that it is not an eigenstate of the measurement he is about to perform on it, and an observer BB who is in possession of the state function of the composite system A+SA+S. Observer AA believes that the outcome of his measurement on SS will be randomly determined by the collapse rule 4b, hence AA attributes to A+SA+S a state describing AA as having a determinate measurement result and SS as having collapsed to the corresponding state. Observer BB, however, attributes the state function of the room after AA’s measurement according to the deterministic rule 4a, hence BB attributes to A+SA+S an entangled state where, according to rule 3, neither AA nor SS even has a determinate quantum-mechanical state of its own. Everett argued that since AA and BB make incompatible state attributions to A+SA+S, the standard collapse theory yields a straightforward contradiction. It would be extraordinarily difficult in practice for BB to make a Wigner’s Friend interference measurement that would determine the state of a composite system like A+SA+S, hence the “extremely hypothetical” nature of the drama. Everett was careful, however, to explain why this was entirely irrelevant to the conceptual problem at hand. Indeed, he explicitly rejected that one might simply “deny the possibility that BB could ever be in possession of the state function of A+SA+S.” Rather, he argued, that “no matter what the state of A+SA+S is, there is in principle a complete set of commuting operators for which it is an eigenstate, so that, at least, the determination of these quantities will not affect the state nor in any way disrupt the operation of AA,” nor, he added, are there “fundamental restrictions in the usual theory about the knowability of any state functions.” And he concluded that “it is not particularly relevant whether or not BB actually knows the precise state function of A+SA+S. If he merely believes that the system is described by a state function, which he does not presume to know, then the difficulty still exists. He must then believe that this state function changed deterministically, and hence that there was nothing probabilistic in AA’s determination” (1956, 76). And, Everett argued, BB is right in so believing. That Everett took the Wigner’s Friend story, which involves an experiment that, on the basis of decoherence considerations, would be virtually impossible to perform, to pose the central conceptual problem for quantum mechanics is essential to understanding how he thought of the measurement problem and what it would take to solve it. In particular, Everett held that one only has a satisfactory solution to the quantum measurement problem if one can provide a consistent account of nested measurement. And concretely, this meant that one must be able to tell the Wigner’s Friend story consistently.

Being able to consistently tell the Wigner’s Friend story then was, for Everett, a necessary condition for any satisfactory resolution of the quantum measurement problem. 3. Everett’s Proposal In order to solve the measurement problem Everett proposed dropping the collapse dynamics (rule 4b) from the standard collapse theory and taking the resulting physical theory to provide a complete and accurate description of all physical systems in the context of all possible physical interactions. Everett called the theory pure wave mechanics. He believed that he could deduce the standard statistical predictions of quantum mechanics (the predictions that depend on rule 4b in the standard collapse formulation of quantum mechanics) in terms of the subjective experiences of observers who are themselves treated as ordinary physical systems within pure wave mechanics. Everett described the proposed deduction in the long thesis as follows: We shall be able to introduce into [pure wave mechanics] systems which represent observers. Such systems can be conceived as automatically functioning machines (servomechanisms) possessing recording devices (memory) and which are capable of responding to their environment. The behavior of these observers shall always be treated within the framework of wave mechanics. Furthermore, we shall deduce the probabilistic assertions of Process 1 [rule 4b] as subjective appearances to such observers, thus placing the theory in correspondence with experience. We are then led to the novel situation in which the formal theory is objectively continuous and causal, while subjectively discontinuous and probabilistic. While this point of view thus shall ultimately justify our use of the statistical assertions of the orthodox view, it enables us to do so in a logically consistent manner, allowing for the existence of other observers (1956, 77–8). Everett’s goal, then, was to show that the memory records of an observer as described by quantum mechanics without the collapse dynamics would agree with those predicted by the standard formulation with the collapse dynamics. More specifically, he wanted to show that observers, modeled as servomechanisms within pure wave mechanics, would have fully determinate relative measurement records and the probabilistic assertions of the standard theory will correspond to statistical properties of typical sequences of such relative records. In his version of the Wigner’s Friend story, Everett insisted on three things simultaneously: (1) there are no collapses of the quantum-mechanical state, hence BB is correct in attributing to A+SA+S a state where AA is in an entangled superposition of having recorded mutually incompatible results, (2) there is a sense in which AA nevertheless got a fully determinate

measurement result, and (3) such determinate results satisfy the standard quantum statistics. The main problem in understanding what Everett had in mind is in figuring out precisely how the correspondence between the predictions of the standard collapse theory and the pure wave mechanics was supposed to work. Part of the problem is that the former theory is stochastic with fundamentally chance events and the latter deterministic with no mention of probabilities whatsoever, but there is also a problem even accounting for determinate measurement records in pure wave mechanics. In order to see why, we will consider how Everett’s no-collapse proposal plays out in a simple interaction like AA’s measurement in the Wigner’s Friend story. Consider measuring the xx-spin of a spin-½ system. Such a system will be found to be either “xx-spin up” or “xx-spin down”. Suppose that JJ is a good observer. For Everett, being a good xx-spin observer meant that JJ has the following two dispositions (the arrows below represent the time-evolution of the composite system as described by the deterministic dynamics of rule 4a): (1)(2)|“ready”⟩J|x-spin up⟩S→|“spin up”⟩J|x-spin up⟩S|“ready”⟩J|x-spin dow n⟩S→|“spin down”⟩J|x-spin down⟩S(1)|“ready”⟩J|x-spin up⟩S→|“spin up”⟩J|xspin up⟩S(2)|“ready”⟩J|x-spin down⟩S→|“spin down”⟩J|x-spin down⟩S If JJ measures a system that is determinately xx-spin up, then JJ will determinately record “xx-spin up”; and if JJ measures a system that is determinately xx-spin down, then JJ will determinately record “xx-spin down” (and we assume, for simplicity, that the spin of the object system SS is undisturbed by the interaction). Now consider what happens when JJ observes the xx-spin of a system that begins in a superposition of xx-spin eigenstates: a|x-spin up⟩S+b|x-spin down⟩Sa|x-spin up⟩S+b|x-spin down⟩S The initial state of the composite system then is: |“ready”⟩J(a|x-spin up⟩S+b|x-spin down⟩S)|“ready”⟩J(a|x-spin up⟩S+b|x-spin d own⟩S) Here JJ is determinately ready to make an xx-spin measurement, but the object systemSS, according to rule 3, has no determinate xx-spin. Given JJ’s two dispositions and the fact that the deterministic dynamics is linear, the state of the composite system after JJ’s xx-spin measurement will be: a|“spin up”⟩J|x-spin up⟩S+b|“spin down”⟩J|x-spin down⟩Sa|“spin up”⟩J|x-spin u p⟩S+b|“spin down”⟩J|x-spin down⟩S On the standard collapse formulation of quantum mechanics, somehow during the measurement interaction the state would collapse to either the first term of this expression (with probability equal to aa squared) or to the second term of this expression (with probability equal to bb squared). In the former case, JJ ends up with the determinate measurement record “spin up”,

and in the later case JJ ends up with the determinate measurement record “spin down”. But on Everett’s proposal no collapse occurs. Rather, the postmeasurement state is simply this entangled superposition of JJ recording the result “spin up” and SS being xx-spin up and JJ recording “spin down” and SS being xx-spin down. Call this state EE. On the standard eigenvalue-eigenstate link (rule 3) EE is not a state where JJ determinately records “spin up”, neither is it a state where JJ determinately records “spin down”. So Everett’s interpretational problem is to explain the sense in which JJ’s entangled superposition of mutually incompatible records represents a determinate measurement outcome that agrees with the empirical prediction made by the standard collapse formulation of quantum mechanics when the standard theory predicts that JJ either ends up with the fully determinate measurement record “spin up” or the fully determinate record “spin down”, with probabilities equal to aa-squared and bb-squared respectively. More specifically, here the standard collapse theory predicts that on measurement the quantummechanical state of the composite system will collapse to precisely one of the following two states: |“spin up”⟩J|x-spin up⟩S or |“spin down”⟩J|x-spin down⟩S|“spin up”⟩J|x-spin u p⟩S or |“spin down”⟩J|x-spin down⟩S and that there is thus a single, simple matter of fact about which measurement result JJ recorded. Everett, then, faced two closely related problems. The determinate-record problem requires him to explain how a measurement interaction like that just described might yield a determinate record in the context of pure wa...