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Physics Letters A

Stochastic physical origin of the quantum operator algebra and phase space interpretation of the Hilbert space formalism: The relativistic spin zero case

Christopher Dewdney, Peter Holland, Anastasios Kyprianidis, Z. Marić, Jean Pierre Vigier

Published Jan 1, 1986• 22 citations
Publisher
DOI
10.1016/0375-9601(86)90044-7

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The deduction by Guerra and Marra of the usual quantum operator algebra from a canonical variable Hamiltonian treatment of Nelson's hydrodynamical stochastic description of real nonrelativistic Schr\"odinger waves is extended to the causal stochastic interpretation given by Guerra and Ruggiero and by Vigier of relativistic Klein-Gordon waves. A specific representation shows that the Poisson brackets for canonical hydrodynamical observables become ``averages'' of quantum observables in the given state. Stochastic quantization thus justifies the standard procedure of replacing the classical particle (or field) observables with operators according to the scheme ${p}_{\ensuremath{\mu}}$\ensuremath{\rightarrow}-i\ensuremath{\Elzxh}${\ensuremath{\partial}}_{\ensuremath{\mu}}$ and ${L}_{\ensuremath{\mu}\ensuremath{\nu}}$\ensuremath{\rightarrow}-i\ensuremath{\Elzxh}(${x}_{\ensuremath{\mu}}$${\ensuremath{\partial}}_{\ensuremath{\nu}}$-${x}_{\ensuremath{\nu}}$${\ensuremath{\partial}}_{\ensuremath{\mu}}$).
We show how the causal interpretation provides a physical basis for the Wigner phase-space formulation of quantum mechanics by analyzing the case of a relativistic spin-zero particle in an external electromagnetic field. The Wigner function can be interpreted as both a generalized transformation and a field which defines a quantum potential in phase space. We also propose a Lorentz-vector distribution function which always associates positive probability densities with the real motions of particles and antiparticles in phase space.
In this essay, we explore the limits imposed by the impossibility of superluminal signalling on the class of physically realisable quantum operations, focusing on the difference in approaches one can take towards this problem in Hilbert space quantum theory and algebraic quantum field theory (AQFT). We review a recent proposal by Fewster and Verch for a generally covariant measurement scheme in AQFT, which we call the FV scheme; along the way, we argue that the framework as set up in its original proposal can be greatly simplified without losing generality or physical motivation. We then discuss the problem, introduced by Sorkin, that naive generalisations of the notion of operation to the relativistic context can lead to signalling between spacelike observers by introducing a third observer. We consider approaches to tackle this problem in Hilbert space quantum theory and argue that these are sometimes ambiguous, since the physical (im)possibility of operations depends in part on the specific spacetime embodiment of the quantum system under consideration, which lies outside the Hilbert space formalism. In the AQFT context, we review and generalise a recent result showing that operations induced by the FV measurement scheme do not enable superluminal signalling. We connect this result to Hilbert space by introducing a concrete model of AQFT, which we call the `hybrid model' as it provides a way to explicitly embed Hilbert space quantum theory into spacetime using the formalism of AQFT. Finally, we show that in this model, under suitable necessary and sufficient conditions, the converse result also holds: every operation that does not enable superluminal signalling can be implemented in the FV framework.