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arXiv:2311.13972 · Reviews in Mathematical Physics

From each of Feynman’s and von Neumann’s postulates to the restricted Feynman path integrals: A mathematical theory of temporally continuous quantum measurements

Wataru Ichinose

Published Mar 28, 2024• 0 citations
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Abstract

Feynman proposed a postulate or a method of quantization in his celebrated paper in 1948. Applying Feynman’s postulate to temporally continuous quantum measurements of the positions of particles, Mensky proposed the restricted Feynman path integrals (RFPIs) for continuous quantum measurements after phenomenological considerations. Our aim in this paper is to give a rigorous proof that Mensky’s RFPIs emerge out of Feynman’s postulate under a simple approximation. In addition, it is proved that the RFPIs emerge out of von Neumann’s postulate on instantaneous measurements as well as Feynman’s postulate. The quantum systems that we study include spin systems. These results are applied to formulations of the multi-split experiments, the quantum Zeno and the Aharonov–Bohm effects.

DOI
10.1142/s0129055x2450020x

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Feynman proposed a postulate or a method of quantization in his celebrated paper in 1948. Applying Feynman’s postulate to temporally continuous quantum measurements of the positions of particles, Mensky proposed the restricted Feynman path integrals (RFPIs) for continuous quantum measurements after phenomenological considerations. Our aim in this paper is to give a rigorous proof that Mensky’s RFPIs emerge out of Feynman’s postulate under a simple approximation. In addition, it is proved that the RFPIs emerge out of von Neumann’s postulate on instantaneous measurements as well as Feynman’s postulate. The quantum systems that we study include spin systems. These results are applied to formulations of the multi-split experiments, the quantum Zeno and the Aharonov–Bohm effects.
It has been stated in the physics literature that the probability amplitude for a quantum system after a continuous position measurement can be expressed by the restricted or the weighted Feynman path integral containing information about the result and the resolution of its measurement. In the present paper it is proved that the restricted Feynman path integrals can be defined in a rigorous sense. More specifically, we consider a quantum system consisting of some particles and simultaneously perform a continuous measurement of the positions of all particles in the time interval [0,T][0,T][0,T]. Then, it is shown rigorously that the restricted Feynman path integral after the measurement can be defined in the L2L^2L2 space and satisfies the corresponding non-self-adjoint Schroeodinger equation. In addition, our results are generalized to a quantum spin system where a continuous position measurement for all spin components of all particles is simultaneously performed, though the results obtained are inadequate for general use.
In the late forties Richard Feynman (1948) proposed a path-integral form of quantum mechanics that proved to be equivalent to the conventional Schrödinger-equation form but possessed a much more heuristic force (see Feynman and Hibbs (1965) for a detailed exposition of Feynman’s theory). Our consideration of continuous measurements in the following chapters will be based on Feynman path integrals. A theory of quantum continuous measurements will be formulated directly in terms of path integrals instead of a limiting process of the type used in Chapter 2 .
Advances in technology are taking the accuracy of macroscopic as well as microscopic measurements close to the quantum limit, for example, in the attempts to detect gravitational waves. Interest in continuous quantum measurements has therefore grown considerably in recent years. Continuous Quantum Measurements and Path Integrals examines these measurements using Feynman path integrals. The path integral theory is developed to provide formulae for concrete physical effects. The main conclusion drawn from the theory is that an uncertainty principle exists for processes, in addition to the familiar one for states. This implies that a continuous measurement has an optimal accuracy-a balance between inefficient error and large quantum fluctuations (quantum noise). A well-known expert in the field, the author concentrates on the physical and conceptual side of the subject rather than the mathematical.
Feynman's formulation of quantum theory is remarkable in its combination of formal simplicity and computational power. However, as a formulation of the abstract structure of quantum theory, it is incomplete as it does not account for most of the fundamental mathematical structure of the standard von Neumann--Dirac formalism such as the unitary evolution of quantum states. In this paper, we show how to reconstruct the entirety of the finite-dimensional quantum formalism starting from Feynman's rules with the aid of a single physical postulate, the no-disturbance postulate. This postulate states that a particular class of measurements have no effect on the outcome probabilities of subsequent measurements performed. We also show how it is possible to derive both the amplitude rule for composite systems of distinguishable subsystems and Dirac's amplitude-action rule, each from a single elementary and natural assumption, by making use of the fact that these assumptions must be consistent with Feynman's rules.
Imposing restrictions on the Feynman paths of the monitored system has in the past been proposed as a universal model-free approach to continuous quantum measurements. Here we revisit this proposition and demonstrate that a Gaussian restriction, resulting in a sequence of many highly inaccurate (weak) von Neumann measurements, is not sufficiently strong to ensure proximity between a readout and the Feynman paths along which the monitored system evolves. Rather, in the continuous limit, the variations of a typical readout become much larger than the separation between the eigenvalues of the measured quantity. Thus, a typical readout is not represented by a nearly constant curve, correlating with one of the eigenvalues of the measured quantity $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}$, even when decoherence or Zeno effect is achieved for the observed two-level system, and does not point directly to the system's final state. We show that the decoherence in a ``free'' system can be seen as induced by a Gaussian random walk with a drift, eventually directing the system towards one of the eigenstates of $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}$. A similar mechanism appears to be responsible for the Zeno effect in a driven system, when its Rabi oscillations are quenched by monitoring. Alongside the Gaussian case, which can only be studied numerically, we also consider a fully tractable model with a ``hard wall'' restriction and show the results to be similar.
We demonstrate how a general von Neumann--like measurement can be analyzed in terms of histories (paths) constructed for the measured variable AAA. The Schr\"odinger state of a system in a Hilbert space of arbitrary dimensionality is decomposed into a set of substates, each of which corresponds to a particular detailed history of the system. The coherence between the substates may then be destroyed by meter(s) to a degree determined by the nature and the accuracy of the measurement(s) which may be of von Neumann, finite-time, or continuous type. The cases of a particle described by Feynman paths in the coordinate space and a qubit in a two-dimensional Hilbert space are studied in some detail.