[Paper Review] A quantum field theory as emergent description of constrained supersymmetric classical dynamics

Hans-Thomas Elze|ArXiv.org|Aug 13, 2005

Quantum Mechanics and Applications9 references4 citations

TL;DR

This paper proposes a novel mechanism for emergent quantum field theory from constrained supersymmetric classical dynamics. By applying a Koopman-von Neumann Hilbert space formalism and imposing a supersymmetry-invariant constraint that eliminates the negative spectrum of the Liouville operator, the classical dynamics is mapped onto a genuine quantum field theory with a positive Hamiltonian and bounded energy spectrum, suggesting dynamical symmetry breaking as the origin of quantum behavior.

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.

Motivation & Objective

• To explore whether deterministic classical dynamics can give rise to quantum field theory through a constrained Hilbert space approach.
• To resolve the instability of classical Liouville dynamics—characterized by an unbounded negative spectrum—by imposing a physical state constraint.
• To demonstrate that supersymmetry and a positivity constraint can generate a stable ground state, mimicking quantum field theory.
• To interpret the emergence of quantum behavior as a consequence of dynamical symmetry breaking due to a mass scale parameter.

Proposed method

• Adopting the Koopman-von Neumann formalism to describe classical statistical mechanics in a Hilbert space with a Hermitian Liouville operator.
• Decomposing the Liouville operator into positive and negative spectral parts using the supersymmetry algebra.
• Imposing a physical state constraint that annihilates the negative spectral component, preserving invariance under Hamiltonian flow.
• Choosing phase space coordinates such that the constrained classical Liouville equation becomes a functional Schrödinger equation of a genuine quantum field theory.
• Introducing a mass scale parameter M to break supersymmetry dynamically and stabilize the spectrum.
• Constructing an extended Hamiltonian with a local order parameter to interpolate between classical and quantum regimes.

Experimental results

Research questions

• RQ1 Can a deterministic classical system with an unbounded Liouville spectrum give rise to a stable quantum field theory?
• RQ2 How does supersymmetry facilitate the decomposition and constraint of the Liouville operator to yield a positive Hamiltonian?
• RQ3 What role does dynamical symmetry breaking play in the emergence of quantum behavior from classical dynamics?
• RQ4 How does the introduction of a mass scale parameter M regulate the transition from classical to quantum behavior?
• RQ5 Can the functional Schrödinger equation of a quantum field theory be derived from a constrained classical Liouville equation?

Key findings

• The classical Liouville equation, when constrained via a supersymmetry-invariant projection, becomes equivalent to the functional Schrödinger equation of a genuine quantum field theory.
• The negative spectral component of the Liouville operator is eliminated by a physical state constraint, resulting in a positive-definite Hamiltonian and a stable ground state.
• The constraint preserves invariance under the Hamiltonian flow, ensuring consistency with classical dynamics.
• The emergence of quantum behavior is linked to dynamical symmetry breaking, where a mass scale parameter M distinguishes classical from quantum regimes.
• In the high-energy limit, the system behaves classically with unbounded spectrum; in the low-energy limit, the spectrum becomes bounded from below, exhibiting quantum behavior.
• The transition between classical and quantum regimes is regulated by a nonlinear order parameter, which modifies the dynamics through higher-order functional derivatives.