Quantum Mechanics in the Light of Quantum Cosmology

TLDR: In this paper, a quantum cosmology of the quasiclassical domain of familiar experience is described and the process of measurement is described. But it is not shown that all the domains are roughly equivalent or whether there are various essentially inequivalent ones.

Abstract: We sketch a quantum mechanical framework for the universe as a whole. Within that framework we propose a program for describing the ultimate origin in quantum cosmology of the quasiclassical domain of familiar experience and for characterizing the process of measurement. Predictions in quantum mechanics are made from probabilities for sets of alternative histories. Probabilities can be assigned only to sets of histories that approximately decohere. Decoherence is defined and the mechanism of decoherence is reviewed. Decoherence requires a sufficiently coarse-grained description of alternative histories of the universe. A quasiclassical domain consists of a branching set of alternative decohering histories, described by a coarse graining that is maximally refined consistent with decoherence, with individual branches that exhibit a high level of classical correlation in time. A quasiclassical domain is emergent in the universe as a consequence of the initial condition and the action function of the elementary particles. It is an important question whether all the quasiclassical domains are roughly equivalent or whether there are various essentially inequivalent ones. A measurement is a correlation with variables in a quasiclassical domain. An observer (or information gathering and utilizing system) is a complex adaptive system that has evolved to exploit the relative predictability of a quasiclassical domain. We suggest that resolution of many of the problems of interpretation presented by quantum mechanics is to be accomplished, not by further scrutiny of the subject as it applies to reproducible laboratory situations, but rather by an examination of alternative histories of the universe, stemming from its initial condition, and a study of the problem of quasiclassical domains.

Figures

Figure 4. If the completely fine grained histories arise from a single complete set of observables, say the set Q of field variables Qi at each point in space and every time, then the possible coarsegrained histories will be a subset of those illustrated in Figure 2. Maximal sets can still be defined but will, in general, differ from those of Figure 2.

Figure 3. The sum-over-histories construction of the decoherence functional.

Figure 1: The two-slit experiment. An electron gun at left emits an electron traveling towards a screen with two slits, its progress in space recapitulating its evolution in time. When precise detections are made of an ensemble of such electrons at the screen it is not possible, because of interference, to assign a probability to the alternatives of whether an individual electron went through the upper slit or the lower slit. However, if the electron interacts with apparatus which measures which slit it passed through, then these alternatives decohere and probabilities can be assigned.

Figure 2. The schematic structure of the space of sets of possible histories for the universe. Each dot in this diagram represents an exhaustive set of alternative histories for the universe. (This is not a picture of the branches defined by a given set!) Such sets, denoted by {[Pα]} in the text, correspond in the Heisenberg picture to time sequences (P 1α1(t1), P 2 α2(t2), · · · P n αn(tn)) of sets of projection operators, such that at each time tk the alternatives αk are an orthogonal and exhaustive set of possibilities for the universe. At the bottom of the diagram are the completely fine-grained sets of histories each arising from taking projections onto eigenstates of a complete set of observables for the universe at every time. For example, the set Q is the set in which all field variables at all points of space are specified at every time. This set is the starting point for Feynman’s sum-over-histories formulation of quantum mechanics. P might be the completely fine-grained set in which all field momenta are specified at each time. D might be a degenerate set of the kind discussed in Section VII in which the same complete set of operators occurs at every time. But there are many other completely fine-grained sets of histories corresponding to all possible combinations of complete sets of observables that can be taken at every time.

Frequently asked questions

Q1. What are the future works mentioned in the paper "Quantum mechanics in the light of quantum cosmology" ?

The important point is that the decoherent histories of a quasiclassical domain contain all possible choices that might be made by all possible observers that might exist, now, in the past, or in the future for that domain. This behavior, although unfortunately called “ non-local ” by some authors, involves no non-locality in the ordinary sense of quantum field theory and no possibility of signaling outside the light cone. But here, in each, there is also a correlation between the information obtained about one spin and the information that can be obtained about the other. The authors conclude that resolution of the problems of interpretation presented by quantum mechanics is not to be accomplished by further intense scrutiny of the subject as it applies to reproducible laboratory situations, but rather through an examination of the origin of the universe and its subsequent history. 

Q2. What contributions have the authors mentioned in the paper "Quantum mechanics in the light of quantum cosmology" ?

Within that framework the authors propose a program for describing the ultimate origin in quantum cosmology of the “ quasiclassical domain ” of familiar experience and for characterizing the process of measurement. A quasiclassical domain is emergent in the universe as a consequence of the initial condition and the action function of the elementary particles. The authors suggest that resolution of many of the problems of interpretation presented by quantum mechanics is to be accomplished, not by further scrutiny of the subject as it applies to reproducible laboratory situations, but rather by an examination of alternative histories of the universe, stemming from its initial condition, and a study of the problem of quasiclassical domains. This paper was the first in a series by the two authors developing a quantum mechanical framework for the universe as a whole called Decoherent Histories Quantum Mechanics, DH. The paper has not been updated or improved and the references are unchanged. The paper appeared in the Proceedings of the Santa Fe Institute Workshop on Complexity, Entropy, and the Physics of Information, May 1989 and in the Proceedings of the 3rd International Symposium on The Foundations of Quantum Mechanics in the Light of New Technology, Tokyo, Japan, August 1989. These are the action function of the elementary particles, the initial quantum state of the universe, and, since quantum mechanics is an inherently probabilistic theory, the information available about their specific history. A unified theory of the dynamics of the basic fields has long been a goal of elementary particle physics and may now be within reach. The fact that the discovery of a bird in the forest or a fossil in a cliff or a coin in a ruin implies the likelihood of discovering another similar bird or fossil or coin can not be derivable from the laws of elementary particle physics alone ; it must involve correlations that stem from the initial condition. However, during the last few years there has been increasing speculation that, even in a unified fundamental theory, free of dimensionless parameters, some of the observable characteristics of the elementary particle system may be quantum-probabilistic, with a probability distribution that can depend on the initial condition. It is not their purpose in this article to review all these developments in quantum cosmology. Rather, the authors will discuss the implications of quantum cosmology for one of the subjects of this conference — the interpretation of quantum mechanics. There has recently been much promising progress in the search for a theory of the quantum initial condition of the universe. 

Q3. What are the other requirements from probability theory?

The other requirements from probability theory are that the probability of the whole sample space be unity, an easy consequence of (11) when complete coarse graining is performed, and that the probability for an empty set be zero, which means simply that the probability of any sequence containing a projection P = 0 must vanish, as it does. 

Q4. how is the decoherence functional for coarse-grained histories obtained?

The decoherence functional for coarse-grained histories is obtained from (6) according to the principle of superposition by summing over all that is not specified by the coarse graining. 

Q5. What is the principle of summing over all possibilities for certain variables at one time?

Summing over all possibilities for certain variables at one time amounts to factoring the P ’s and eliminating one of the factors by summing over it. 

Q6. What is the class of maximal sets possible for the universe?

The class of maximal sets possible for the universe depends, of course, on the completelyfine-grained histories that are presented by the actual quantum theory of the universe. 

Q7. How can the process of prediction be organized?

By utilizing (21) the process of prediction may be organized so that for each time there is a ρeff from which probabilities for the future may be calculated. 

Q8. What is the problem of finding ordered strings of exhaustive sets of projections?

The problem of finding ordered strings of exhaustive sets of projections [Pα] so that the histories P n αn · · ·P 1 α1|Ψ > decohere according to (25) is purely algebraic and involves just subspaces of Hilbert space. 

Q9. What is the simplest model of a single oscillator?

The simplest model consists of a single oscillator interacting bilinearly with a large number of others, and a coarse graining which involves only the coördinates of the special oscillator. 

Q10. What is the problem of decoherence of strings of histories?

(f) Sets of Histories with the Same ProbabilitiesIf the projections P are not restricted to a particular class (such as projections onto ranges of Qi variables), so that coarse-grained histories consist of arbitrary exhaustive families of projections operators, then the problem of exhibiting the decohering sets of strings of projections arising from a given ρ is a purely algebraic one. 

Q11. What is the usual entropy formula for a coarse-grained set of alternative?

In order to construct that quantity the usual entropy formula is applied to sets of alternative decohering histories of the universe, rather than, as more usually, alternatives at a single time. 

Q12. What is the important theoretical construct for giving the rule that determines whether probabilities may be assigned?

The important theoretical construct for giving the rule that determines whether probabilities may be assigned to a given set of alternative histories, and what these probabilities are, is the decoherence functional D [(history)′, (history)].