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Generalized quantum dynamics as pre-quantum mechanics

Stephen L. Adler, Millard, A C

Published Aug 16, 1995• 37 citations
OpenAlex

Abstract

We address the issue of when generalized quantum dyanmics, based on a total trace Hamiltonian {\\bf H}, reduces to Heisenberg picture complex quantum mechanics. We begin by showing that when {\\bf H}={\\rm{\\bf Tr}} H_{OP}, with H_{OP} a Weyl ordered operator Hamiltonian, then the generalized quantum dynamics operator equations of motion agree with those obtained from H_{OP} in the Heisenberg picture by using canonical commutation relations. Hence generalized quantum dynamics conicides with complex quantum mechanics in this case, provided the initial values of the operators obey the canonical algebra. The remainder of the paper is devoted to a study of when this initial value condition is satisfied. We first show that for any total trace Hamiltonian which involves no noncommutative constants, there is a conserved operator \\tilde C which is closely related to the canonical algebra. We demonstrate that for a system in equilbrium, the equilbrium ensemble of initial values is governed by the distribution exp(-|\\tilde C|/\\kappa). An argument analogous to that for equipartition of energy, together with the assumption that ensemble averages of products can be replaced by products of ensemble averages, then implies that the ensemble averaged operator initial values obey the standard canonical commutation relations of complex quantum mechanics, with the imaginary unit I identified as -\\tilde C / |\\tilde C|, and with the constant \\kappa playing the role of the reduced Planck constant \\hbar. Thus there is a well--defined sense in which complex quantum mechanics and quantum field theory emerge as a statistical approximation to generalized quantum dynamics.

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We address the issue of when generalized quantum dyanmics, based on a total trace Hamiltonian {\bf H}, reduces to Heisenberg picture complex quantum mechanics. We begin by showing that when {\bf H}={\rm{\bf Tr}} H_{OP}, with H_{OP} a Weyl ordered operator Hamiltonian, then the generalized quantum dynamics operator equations of motion agree with those obtained from H_{OP} in the Heisenberg picture by using canonical commutation relations. Hence generalized quantum dynamics conicides with complex quantum mechanics in this case, provided the initial values of the operators obey the canonical algebra. The remainder of the paper is devoted to a study of when this initial value condition is satisfied. We first show that for any total trace Hamiltonian which involves no noncommutative constants, there is a conserved operator \tilde C which is closely related to the canonical algebra. We demonstrate that for a system in equilbrium, the equilbrium ensemble of initial values is governed by the distribution exp(-|\tilde C|/\kappa). An argument analogous to that for equipartition of energy, together with the assumption that ensemble averages of products can be replaced by products of ensemble averages, then implies that the ensemble averaged operator initial values obey the standard canonical commutation relations of complex quantum mechanics, with the imaginary unit I identified as -\tilde C / |\tilde C|, and with the constant \kappa playing the role of the reduced Planck constant \hbar. Thus there is a well--defined sense in which complex quantum mechanics and quantum field theory emerge as a statistical approximation to generalized quantum dynamics.
It has recently been shown, by application of statistical mechanical methods to determine the canonical ensemble governing the equilibrium distribution of operator initial values, that complex quantum field theory can emerge as a statistical approximation to an underlying generalized quantum dynamics. This result was obtained by an argument based on a Ward identity analogous to the equipartition theorem of classical statistical mechanics. We construct here a microcanonical ensemble which forms the basis of this canonical ensemble. This construction enables us to define the microcanonical entropy and free energy of the field configuration of the equilibrium distribution and to study the stability of the canonical ensemble. We also study the algebraic structure of the conserved generators from which the microcanonical and canonical ensembles are constructed, and the flows they induce on the phase space.
We consider the classical dynamics of bosonic and fermionic matrix variables in complex Hilbert space, defined by a trace action, assuming cyclic invariance under the trace and the presence of a global unitary invariance. With plausible and explicitly stated assumptions, including the existence of a large hierarchy of scale between the underlying dynamics and observed physics, we show that (1) the equilibrium statistical mechanics of this matrix dynamics, in the canonical ensemble, gives rise to an emergent quantum mechanics for many degrees of freedom, including the standard canonical commutation/anticommutation relations and the usual unitary Heisenberg and Schrödinger picture time evolutions of operators and states, and (2) the fluctuation or Brownian motion corrections to this thermodyamics lead to an energy-driven stochastic modification of the Schrödinger equation, which is known to imply state vector reduction with Born rule probabilities. Thus, quantum mechanics and its probabilistic interpretation both arise as emergent phenomena in the underlying trace dynamics.
In the first part of this series we have shown how the Schrodinger equation for a single particle and the corresponding non relativistic quantum observables can be obtained from a purely classical phonon model through the Newtonian equations of motion. In this work we focus instead on how the classical Hamiltonian formalism applied to the same phonon system allows to recover the feature of relativistic quantum mechanics for a single spinless particle. Using the classical nature of the phonon model, we naturally define continuous Poisson brackets between classical observables, which allows to recover the dynamics of such observables, i.e. the Ehrenfest relations associated to real-valued Klein-Gordon fields. The Poisson brackets also permits to obtain the generic form of constants of motions, thus generalizing the concept of inner products and momentum on Klein-Gordon fields. We then connect the formalism of real-valued classical functionals with that of hermitian operators and complex-valued wave functions. This is done through the introduction of a non-local complex-valued change of variables which allows to rewrite the real-valued Klein-Gordon equation in a form akin to the Schrodinger equation, and the classical observables as quantum expectation values. Then, we show how this change of variables allows to rewrite the classical Poisson brackets as commutators of hermitian operators. This points out the strict equivalence between the Heisenberg formalism and the formalism of classical Poisson bracket. Eventually, we illustrate how the Poisson brackets allows to recover the transformations of Poincare group in 1+1 dimension together with its algebra. The latter makes the link between the Lorentz invariant inner product of Mostafazadeh and the Casimir invariant associated to the mass of particle.
In this chapter we set up a classical Lagrangian and Hamiltonian dynamics for matrix models. The fundamental idea is to set up an analog of classical dynamics in which the phase space variables are non-commutative, and the basic tool that allows one to accomplish this is cyclic invariance under a trace. Since no assumptions about commutativity of the phase space variables (such as canonical commutators/anticommutators) are made at this stage, the dynamics that we set up is not the same as standard quantum mechanics. Quantum mechanical behavior will be seen to emerge only when, in Chapters 4 and 5, we study the statistical mechanics of the classical matrix dynamics formulated here. In Section 1.1, we introduce our basic notation for bosonic and fermionic matrices, and give the cyclic identities that will be used repeatedly throughout the book. In Section 1.2, we define the derivative of a trace quantity with respect to an operator, and give the basic properties of this definition. In Section 1.3, we use the operator derivative to formulate a Lagrangian and Hamiltonian dynamics for matrix models. In Section 1.4, we introduce a generalized Poisson bracket appropriate to trace dynamics, constructed from the operator derivative defined in Section 1.2, and give its properties and some applications. Finally, in Section 1.5 we discuss the relation between the trace dynamics time evolution equations, and the usual unitary Heisenberg picture equations of motion obtained when one assumes standard canonical commutators/anticommutators. Bosonic and fermionic matrices and the cyclic trace identities We shall assume finite-dimensional matrices, although ultimately an extension to the infinite-dimensional case may be needed. The matrix elements of these matrices will be constructed from ordinary complex numbers, and from complex anticommuting Grassmann numbers.
The Wigner problem, i.e. the investigation of general quantum mechanical commutation relations consistent with the Heisenberg evolution equations of a given shape, is studied. We follow a recently proposed generalization of this idea within which the classical analogy is postulated only for the shape of the time evolution equations and not for a Hamiltonian itself. This links our investigation to the problem of alternative Hamiltonians of classical mechanics and to canonically inequivalent phase-space descriptions of physical systems governed by the same Newton equations of motion. Warned that the time evolution and the other symmetry generators may be given ambiguously even in the formalism of classical mechanics, we do not a priori assume the shape of their quantum analogues. Instead we only require that the set of basic algebraic relations, which quantum mechanical observables are to obey, has a Lie algebra structure. Such a requirement appears to be sufficient to find solutions for simple oscillator-like dynamics. New algebras of quantum mechanical observables are not constructed as a linear envelope of the Heisenberg algebra, and their representations reflect physical results unexpected in the framework of the canonical approach. We illustrate our approach in detail for the example of the one-dimensional harmonic oscillator using the representation of the generalized coherent states.
Abstract We develop a statistical field theory for classical Nambu dynamics by partially employing the quantum field-theory method. One of the unsolved problems in Nambu dynamics has been to extend it to interacting systems without violating a generalized canonical structure characterized by the presence of multiple Hamiltonians, which together govern the dynamics of time evolution on an equal footing through a Nambu bracket as an analog to the Poisson bracket of ordinary Hamiltonian dynamics with a single Hamiltonian. In the present paper, we propose to include interactions from the standpoint of classical statistical dynamics by formulating it as a field theory on Nambu’s generalized phase space in an operator formalism. We first construct a general framework for such a field theory and its probabilistic interpretation. Then, on the basis of this framework, we give an exactly solvable model of self-interactions in a many-body Nambu system treated as a closed dynamical system satisfying the H-theorem. It is shown that a generalized microcanonical ensemble and a generalized canonical ensemble characterized by many temperatures are reached dynamically as equilibrium states starting with certain classes of initial nonequilibrium states via a continuous Markov process, which lies beyond the usual framework based on the Liouville theorem. Compared with the usual classical statistical mechanics on the basis of standard Hamiltonian dynamics, some important new features associated with Nambu dynamics will emerge, with respect to the symmetries underlying the dynamics of the nonequilibrium as well as the equilibrium states and also to some conceptual properties, such as a formulation of a generalized Kubo–Martin–Schwinger-like condition characterizing the generalized canonical equilibrium states and the “relative” nature of the temperatures.