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arXiv:2204.00841 · Acta Physica Polonica B

Positivity of the Invariant Kernel Underlying Quantum Theory of the Coulomb Field

Jaroslaw Wojciech Wawrzycki

Published Jan 1, 2022• 5 citations
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Abstract

We present a proof of positivity of an invariant kernel, which is of basic importance for the Staruszkiewicz theory of the quantum Coulomb field. Presented proof of positivity is independent of the Staruszkiewicz theory and is based on the classical Schoenberg's theorem for conditionally negative definite functions, as well as on the generalized Bochner's theorem.

DOI
10.5506/aphyspolb.53.2-a5

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We present a proof of positivity of an invariant kernel, which is of basic importance for the Staruszkiewicz theory of the quantum Coulomb field. Presented proof of positivity is independent of the Staruszkiewicz theory and is based on the classical Schoenberg's theorem for conditionally negative definite functions, as well as on the generalized Bochner's theorem.
An intriguing and deep analogy between classical and quantum states is revealed using the notion of positive definite function. By Bochner's theorem, (classical) positive definite functions on phase space can be obtained by taking the Fourier transform of probability measures; similarly, quantum positive definite functions are the image via the Fourier-Plancherel operator of Wigner quasi-probability distributions. Considering the basic properties of positive definite functions — classical and quantum — one is led to define a class of semigroups of operators, the so-called classical-quantum semigroups. It is then natural to wonder whether they have any physical meaning. It turns out that the classical-quantum semigroups can also be obtained by dequantizing a certain class of quantum dynamical semigroups, namely, the classical noise semigroups. This correspondence fits in a more general group-theoretical framework in which a larger class of quantum dynamical semigroups, the twirling semigroups, can be suitably dequantized. Connections with quantum information science will be briefly discussed.
A quantum field theory is described which is a supersymmetric classical model. Supersymmetry generators of the system are used to split its Liouville operator into two contributions, with positive and negative spectrum, respectively. The unstable negative part is eliminated by a positivity constraint on physical states, which is invariant under the classical Hamiltonian flow. In this way, the classical Liouville equation becomes a functional Schrödinger equation of a genuine quantum field theory. Thus, 't Hooft's proposal to reconstruct quantum theory as emergent from an underlying deterministic system, is realized here for a field theory. Quantization is intimately related to the constraint, which selects the part of Hilbert space where the Hamilton operator is positive. This is seen as dynamical symmetry breaking in a suitably extended model, depending on a mass scale which discriminates classical dynamics beneath from emergent quantum mechanical behaviour.
In a series of papers in the 1980's Alexander Holevo proved a classification theorem for continuous quantum measurement processes, or, as they would today be called, stationary quantum trajectories in continuous time. His main tools were functional analytic in character: starting from a Bochner-type inequality he employed dilation techniques for positive definite kernels. Here we give an alternative, more probabilistic proof: we use weak convergence of measures and employ Levy's Continuity Theorem. We clarify the boundedness conditions in Holevo's theorem, and supply a simple example from quantum optics.
It is proven that a necessary and sufficient condition for an nnn-dimensional Hermitian matrix $\ensuremath{\rho}$ to be positive definite is that it be expressible in the form $\ensuremath{\rho}=OEO\ifmmode\dagger\else\textdagger\fi{}$, where OOO is a complex orthogonal matrix and EEE is a diagonal matrix with positive elements. This accomplishes a parametrization since OOO has ${n}^{2}\ensuremath{-}n$ real parameters and EEE has nnn of them. The proof is constructive, giving OOO and EEE. It is further shown that the limit forms of this expression yield all the non-negative definite matrices. The parametrization for the polarization matrix of a spin-one particle is given explicitly.
It is shown that phase-space distribution functions that characterize a single-particle quantum system obey a certain "generalized non-negativity condition," which reflects the fact that the density operator is a positive operator. A corresponding criterion is obtained for the associated characteristic functions and is found to resemble, to some extent, Bochner's theorem of classical probability theory. Necessary and sufficient conditions on a phase-space representation of quantum mechanics are also derived, which ensure that all the possible distribution functions in that representation are non-negative; but it is also shown that such distribution functions are not joint probabilities for position and momentum. In fact, our results readily provide a new proof of theorems of Wigner, and of Cohen and Margenau, which imply that quantum mechanics cannot be formulated as a stochastic theory in phase space.