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version 19 Quantum Dynamics without the Wave Function

Rafael D. Sorkin

Published Jan 5, 2015• 0 citations
OpenAlex

Abstract

When suitably generalized and interpreted, the path-integral offers an al-ternative to the more familiar quantal formalism based on state-vectors, selfadjoint operators, and external observers. Mathematically one general-izes the path-integral-as-propagator to a quantal measure µ on the space Ω of all “conceivable worlds”, and this generalized measure expresses the dy-namics or law of motion of the theory, much as Wiener measure expresses the dynamics of Brownian motion. Within such “histories-based ” schemes new, and more “realistic ” possibilities open up for resolving the philosophi-cal problems of the state-vector formalism. In particular, one can dispense with the need for external agents by locating the predictive content of µ in its sets of measure zero: such sets are to be “precluded”. But unrestricted application of this rule engenders contradictions. One possible response would remove the contradictions by circumscribing the application of the preclusion concept. Another response, more in the tradition of “quantum logic”, would accommodate the contradictions by dualizing Ω to a space of “co-events ” and effectively identifying reality with an element of this dual space. Reading the literature on “quantum foundations”, you could easily get the impression that the problems begin and end with non-relativistic quantum mechanics. Perhaps most authors have limited themselves to this special case because they thought that no essentially

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When suitably generalized and interpreted, the path-integral offers an al-ternative to the more familiar quantal formalism based on state-vectors, selfadjoint operators, and external observers. Mathematically one general-izes the path-integral-as-propagator to a quantal measure µ on the space Ω of all “conceivable worlds”, and this generalized measure expresses the dy-namics or law of motion of the theory, much as Wiener measure expresses the dynamics of Brownian motion. Within such “histories-based ” schemes new, and more “realistic ” possibilities open up for resolving the philosophi-cal problems of the state-vector formalism. In particular, one can dispense with the need for external agents by locating the predictive content of µ in its sets of measure zero: such sets are to be “precluded”. But unrestricted application of this rule engenders contradictions. One possible response would remove the contradictions by circumscribing the application of the preclusion concept. Another response, more in the tradition of “quantum logic”, would accommodate the contradictions by dualizing Ω to a space of “co-events ” and effectively identifying reality with an element of this dual space. Reading the literature on “quantum foundations”, you could easily get the impression that the problems begin and end with non-relativistic quantum mechanics. Perhaps most authors have limited themselves to this special case because they thought that no essentially
When suitably generalized and interpreted, the path-integral offers an alternative to the more familiar quantal formalism based on state-vectors, selfadjoint operators, and external observers. Mathematically one generalizes the path-integral-as-propagator to a {\it quantal measure} μμμ on the space ΩΩΩ of all ``conceivable worlds'', and this generalized measure expresses the dynamics or law of motion of the theory, much as Wiener measure expresses the dynamics of Brownian motion. Within such ``histories-based'' schemes new, and more ``realistic'' possibilities open up for resolving the philosophical problems of the state-vector formalism. In particular, one can dispense with the need for external agents by locating the predictive content of μμμ in its sets of measure zero: such sets are to be ``precluded''. But unrestricted application of this rule engenders contradictions. One possible response would remove the contradictions by circumscribing the application of the preclusion concept. Another response, more in the tradition of ``quantum logic'', would accommodate the contradictions by dualizing ΩΩΩ to a space of ``co-events'' and effectively identifying reality with an element of this dual space.
I will propose that the reality to which the quantum formalism implicitly refers is a kind of generalized history, the word history having here the same meaning as in the phrase sum-over-histories. This proposal confers a certain independence on the concept of event, and it modifies the rules of inference concerning events in order to resolve a contradiction between the idea of reality as a single history and the principle that events of zero measure cannot happen (the Kochen-Specker paradox being a classic expression of this contradiction). The so-called measurement problem is then solved if macroscopic events satisfy classical rules of inference, and this can in principle be decided by a calculation. The resulting conception of reality involves neither multiple worlds nor external observers. It is therefore suitable for quantum gravity in general and causal sets in particular. Quantum gravity and quantal reality Why, in our attempts to unify our theories of gravity and the quantum, has progress been so slow? One reason, no doubt, is that it's simply a very hard problem. Another is that we lack clear guidance from experiments or astronomical observations. But I believe that a third thing holding us back is that we haven't learned how to think clearly about the quantum world in itself, without reference to “observers” and other external agents. Because of this we don't really know how to think about the Planckian regime where quantum gravity is expected to be most relevant. We don’t know how to think about the vacuum on small scales, or about the inside of a black hole, or about the early universe. Nor do we have a way to pose questions about relativistic causality in such situations.
In this paper we address the extension problem for quantal measures of path-integral type, concentrating on two cases: sequential growth of causal sets and a particle moving on the finite lattice ℤn. In both cases, the dynamics can be coded into a vector-valued measure μ on Ω, the space of all histories. Initially, μ is just defined on special subsets of Ω called cylinder events, and we would like to extend it to a larger family of subsets (events) in analogy to the way this is done in the classical theory of stochastic processes. Since quantally μ is generally not of bounded variation, a new method is required. We propose a method that defines the measure of an event by means of a sequence of simpler events that in a suitable sense converges to the event whose measure we are seeking to define. To this end, we introduce canonical sequences approximating certain events, and we propose a measure-based criterion for the convergence of such sequences. Applying the method, we encounter a simple event whose measure is zero classically but non-zero quantally.
A model that generalizes the study of quantum Brownian motion (BM) is constructed. We consider disordered environment that may be either static (quenched), noisy or dynamical. The Zwanzig-Caldeira-Leggett BM model formally constitutes a special case where the disorder autocorrelation length is taken to be infinite. Alternatively, a localization problem is obtained if the noise autocorrelation time is taken to be infinite. Also the general case of weak nonlinear coupling to a thermal, possibly chaotic bath is handled by the same formalism. A general, Feynman-Vernon type path-integral expression for the propagator is introduced. A Wigner transformed version of this expression is utilized in order to facilitate comparison with the classical limit. It is demonstrated that nonstochastic genuine quantal manifestations are associated with the model. It is clarified that such effects are absent in the standard BM model, either the disorder or the chaotic nature of the bath are essential. Quantal correction to the classical diffusive behavior is found even in the limit of high temperatures. The suppression of interference due to dephasing is discussed, leading to the observation that due to the disorder the decay of coherence is exponential in time, and no longer depends on geometrical considerations. Fascinating non-Markovian effects due to time-correlated (colored) noise are explored. For this, a strategy is developed in order to handle the integration over paths. This strategy is extended in order to demonstrate how localization comes out from the path-integral expression.