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arXiv:1709.06711 · Physica Scripta

Classical states, quantum field measurement

Peter Morgan

Published Mar 4, 2019• 3 citations
arXivHTMLPDF

Abstract

Abstract Classical Koopman–von Neumann Hilbert spaces of states are constructed here by the action of classical random fields on a vacuum state in ways that support an action of the quantized electromagnetic field and of the U (1)—invariant observables of the quantized Dirac spinor field, allowing a manifestly Lorentz invariant classical understanding of the state spaces of the two field theories, generalizing the Quantum–Mechanics–Free Systems of Tsang&Caves and Quantum Non-Demolition measurements. The algebra of functions on a classical phase space is commutative but the algebra of classical observables associated with coordinate transformations is noncommutative, so that, for example, we can as much ask whether a classical state is an eigenstate of a rotation as we can in quantum mechanics and so that entangled states can be distinguished from mixed states, making classical random fields as weird as quantum fields.

DOI
10.1088/1402-4896/ab0c53

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Abstract Classical Koopman–von Neumann Hilbert spaces of states are constructed here by the action of classical random fields on a vacuum state in ways that support an action of the quantized electromagnetic field and of the U (1)—invariant observables of the quantized Dirac spinor field, allowing a manifestly Lorentz invariant classical understanding of the state spaces of the two field theories, generalizing the Quantum–Mechanics–Free Systems of Tsang&Caves and Quantum Non-Demolition measurements. The algebra of functions on a classical phase space is commutative but the algebra of classical observables associated with coordinate transformations is noncommutative, so that, for example, we can as much ask whether a classical state is an eigenstate of a rotation as we can in quantum mechanics and so that entangled states can be distinguished from mixed states, making classical random fields as weird as quantum fields.
Abstract Descriptions of classical mechanics in Hilbert space go back to the work of Koopman and von Neumann in the 1930s. Decades later, van Hove derived a unitary representation of the group of contact transformations which recently has been used to develop a novel formulation of classical mechanics in Hilbert space. This formulation differs from the Koopman-von Neumann theory in many ways. Classical observables are represented by van Hove operators, which satisfy a commutation algebra isomorphic to the Poisson algebra of functions in phase space. Moreover, these operators are both observables and generators of transformations, which makes it unnecessary to introduce unobservable auxiliary operators as in the Koopman-von Neumann theory. In addition, for consistency with classical mechanics, a constraint must be imposed that fixes the phase of the wavefunction. The approach can be extended to hybrid mixed classical-quantum systems in Hilbert space. The formalism is applied to the measurement of a quantum two-level system (qubit) by a classical apparatus.
Classical mechanics is presented here in a unary operator form, constructed using the binary multiplication and Poisson bracket operations that are given in a phase space formalism, then a Gibbs equilibrium state over this unary operator algebra is introduced, which allows the construction of a Hilbert space as a representation space of a Heisenberg algebra, giving an operator algebraic variant of the Koopman--von Neumann approach. A classically natural approach to the Poisson bracket in a Koopman--von Neumann Hilbert space formalism gives a classical logic that is the same as quantum logic: in both cases the logic of projections on Hilbert space. In this form, the measurement theory for unary classical mechanics can be the same as and inform that for quantum mechanics, expanding classical mechanics to include noncommutative operators so that it is close to quantum mechanics, instead of attempting to squeeze quantum mechanics into a classical mechanics mold. The measurement problem as it appears in unary classical mechanics suggests a classical approach that can also be successfully applied to the measurement problem of quantum mechanics.
Motivated by a recent investigation into the notion of a quantum space underlying ordinary quantum mechanics, we reformulate here the WWGM formalism with canonical coherent states and wavefunctions as expansion coefficients in terms of this basis as the starting point. It turns out that this provides us with a transparent and coherent story of simple quantum dynamics where both states (pure and mixed, making use of the Tomita representation), as wavefunctions in Hilbert spaces, and observables arise from \textit{a single space/algebra}. Altogether, putting the emphasis on building our theory out of the underlying relativity symmetry -- the centrally extended Galilean symmetry in the case at hand -- allows one to naturally derive both a kinematical and a dynamical description of a (free) quantum particle, which moreover recovers the corresponding classical picture (understood in terms of the Koopman-von Neumann formalism) in the appropriate (relativity symmetry contraction) limit. Our formulation here is the most natural framework directly connecting all of the relevant mathematical notions and we hope it may help a general physicist better visualize and appreciate the noncommutative-geometric perspective behind quantum physics.
Quantisation is considered as an action of classical point operators on field states at the Planck scale of the phase space. Emphasis is placed on the sequence of physical variables. A "state language/Dirac notation" for classical mechanics is introduced; in many cases, it replaces the principle of stationary action for conservative systems. Observed classical and quantum dynamics are considered as different vectorisations of the free semigroup contribution of time. There is a dream of describing classical dynamics as complex transformations of state vectors over such a free semigroup, rather than solving differential equations. The transition from the alphabet to states, from states to state functions, and from state functions to measurable quantities is managed by projectors. An attempt is made to provide a derivation of the Legendre transformation. The trivial multiplication of the potential energy by the wave function in the coordinate representation and its non-trivial operator action in the momentum representation are obtained without a Fourier transform nor the Planck scale. The quantisation procedure is divided into two stages: first, the action of point-like classical operators on field functions; and second, accounting for the physically finite scale of the Planck length against the backdrop of the possible infinitesimality in the classical phase space. The concept of an angle between states is introduced, defining the measure of "quantumness". The free semigroup time contribution is studied in the context of the Born rule.