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arXiv:hep-th/0508095 · Talk given at

A quantum field theory as emergent description of constrained supersymmetric classical dynamics

Hans-Thomas Elze

Published Aug 13, 2005• 6 citations
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Abstract

Deterministic dynamical models are discussed which can be described in quantum mechanical terms. In particular, a local quantum field theory is presented which is a supersymmetric classical model. -- The Hilbert space approach of Koopman and von Neumann is used to study the evolution of an ensemble of such classical systems. With the help of the supersymmetry algebra, the corresponding Liouville operator can be decomposed into two contributions, with positive and negative spectrum, respectively. The unstable negative part is eliminated by a constraint on physical states, which is invariant under the Hamiltonian flow. In this way, choosing suitable phase space coordinates, the classical Liouville equation becomes a functional Schroedinger equation of a genuine quantum field theory. Quantization here is intimately related to the constraint, which selects the part of Hilbert space where the Hamilton operator is positive. This is interpreted as dynamical symmetry breaking in an extended model, introducing a mass scale which discriminates classical dynamics beneath from emergent quantum mechanical behaviour.

DOI
10.48550/arxiv.hep-th/0508095

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Deterministic dynamical models are discussed which can be described in quantum mechanical terms. In particular, a local quantum field theory is presented which is a supersymmetric classical model. -- The Hilbert space approach of Koopman and von Neumann is used to study the evolution of an ensemble of such classical systems. With the help of the supersymmetry algebra, the corresponding Liouville operator can be decomposed into two contributions, with positive and negative spectrum, respectively. The unstable negative part is eliminated by a constraint on physical states, which is invariant under the Hamiltonian flow. In this way, choosing suitable phase space coordinates, the classical Liouville equation becomes a functional Schroedinger equation of a genuine quantum field theory. Quantization here is intimately related to the constraint, which selects the part of Hilbert space where the Hamilton operator is positive. This is interpreted as dynamical symmetry breaking in an extended model, introducing a mass scale which discriminates classical dynamics beneath from emergent quantum mechanical behaviour.
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.
We explore whether quantum field theory can be understood as the statistical mechanics of a time-reversal-invariant stochastic generalization of Hamiltonian dynamics. The motivation for this project, started with this paper, is to assign sharp values to all observables and thereby avoid the quantum measurement problem. In classical mechanics, motion is deterministic and corresponds to an evolution of the phase space probability density according to Liouville's equation that is governed by first derivatives of the Hamiltonian in phase space. We derive a generalization of the Liouville equation with natural constraints -- namely, reduction to classical Hamiltonian dynamics as the stochasticity parameter ℏ↦0\hbar\mapsto0ℏ↦0, Fokker-Planck form for the probability density evolution, local Hamiltonian dependence, time-reversal invariance, energy conservation, and minimality -- which turns out to be a Fokker-Planck equation with a generalized diffusion matrix that is symmetric, traceless, and constructed from the Hessian of the Hamiltonian. We then show that the Schrödinger equation in the coherent-state phase-space formulation of certain bosonic QFTs has precisely this form, with the Husimi function playing the role of the phase space probability density. The question to what extent this equation can be interpreted in terms of objective stochastic field theories is discussed in a companion paper.
Deterministic dynamical models are discussed which can be described in quantum mechanical terms. - In particular, a local quantum field theory is presented which is a supersymmetric classical model [1]. The Hilbert space approach of Koopman and von Neumann is used to study the classical evolution of an ensemble of such systems. Its Liouville operator is decomposed into two contributions, with positive and negative spectrum, respectively. The unstable negative part is eliminated by a constraint on physical states, which is invariant under the Hamiltonian flow. Thus, choosing suitable variables, the classical Liouville equation becomes a functional Schrödinger equation of a genuine quantum field theory. - We briefly mention an U(1) gauge theory with ''varying alpha'' or dilaton coupling where a corresponding quantized theory emerges in the phase space approach [2]. It is energy-parity symmetric and, therefore, a prototype of a model in which the cosmological constant is protected by a symmetry.
In this paper a new formulation of quantum dynamics of totally constrained systems is developed, in which physical quantities representing time are included as observables. In this formulation the hamiltonian constraints are imposed on a relative probability amplitude functional Ψ which determines the relative probability for each state to be observed, instead of on the state vectors as in the conventional Dirac quantization. This leads to a foliation of the state space by linear manifolds on each of which Ψ is constant, and dynamics is described as linear mappings among acausal subspaces which are transversal to these linear manifolds. This is a quantum analogue of the classical statistical dynamics of totally constrained systems developed in the previous paper. It is shown that if the von Neumann algebra C generated by constants of motion is of type I, Ψ can be consistently normalizable on the acausal subspaces on which a factor subalgebra of C is represented irreducibly, and the mappings among these acausal subspaces are conformal. How the formulation works is illustrated by simple totally constrained systems with a single constraint such as the parametrized quantum mechanics, a relativistic free particle in Minkowski and curved spacetimes, and a simple minisuperspace model. It is pointed out that the inner product of the relative probability amplitudes induced from the original Hilbert space picks up a special decomposition of the wave functions to the positive and the negative frequency modes.
Koopman and von Neumann (KvN) extended the Liouville equation by introducing a phase space function S(K)(q,p,t)S^{(K)}(q,p,t)S(K)(q,p,t) whose physical meaning is unknown. We show that a different S(q, p, t), with well-defined physical meaning, may be introduced without destroying the attractive “quantum-like” mathematical features of the KvN theory. This new S(q, p, t) is the classical action expressed in phase space coordinates. It defines a mapping between observables and operators which preserves the Lie bracket structure. The new evolution equation reduces to Schrödinger’s equation if functions on phase space are reduced to functions on configuration space. This new kind of “quantization” does not only establish a correspondence between observables and operators, but provides in addition a derivation of quantum operators and evolution equations from corresponding classical entities. It is performed by replacing ∂∂p\frac{\partial }{\partial p}∂p∂​ by 0 and p by ℏı∂∂q\frac{\hbar }{\imath } \frac{\partial }{\partial q}ℏ​∂q∂​ , thus providing an explanation for the common quantization rules.
We discuss an explicit realization of the dissipative dynamics anticipated in the proof of 't Hooft's existence theorem, which states that ' For any quantum system there exists at least one deterministic model that reproduces all its dynamics after prequantization' . – There is an energy-parity symmetry hidden in the Liouville equation, which mimics the Kaplan-Sundrum protective symmetry for the cosmological constant. This symmetry may be broken by the coarse-graining inherent in physics at scales much larger than the Planck length. We correspondingly modify classical ensemble theory by incorporating dissipative fluctuations (information loss) – which are caused by discrete spacetime continually 'measuring' matter. In this way, aspects of quantum mechanics, such as the von Neumann equation, including a Lindblad term, arise dynamically and expectations of observables agree with the Born rule. However, the resulting quantum coherence is accompanied by an intrinsic decoherence and continuous localization mechanism. Our proposal leads towards a theory that is linear and local at the quantum mechanical level, but the relation to the underlying classical degrees of freedom is nonlocal.