[Paper Review] Classical relativistic systems of charged particles in the front form of dynamics and the Liouville equation

V. I. Tretyak, A. V. Nazarenko|ArXiv.org|May 5, 2000

Quantum Mechanics and Applications19 references3 citations

TL;DR

This paper develops a gauge-invariant Hamiltonian formulation for classical relativistic systems of charged particles coupled to electromagnetic fields in the front form of dynamics, using Dirac's constrained Hamiltonian mechanics to eliminate gauge degrees of freedom. It derives a Liouville equation for the phase space distribution function, enabling a relativistic partition function with explicit integration over field variables, thus providing a consistent framework for relativistic statistical mechanics in a manifestly covariant gauge-fixed setting.

ABSTRACT

Classical relativistic system of point particles coupled with an electromagnetic field is considered in the three-dimensional representation. The gauge freedom connected with the chronometrical invariance of the four-dimensional description is reduced by use of the geometrical concept of the forms of relativistic dynamics. The remainder gauge degrees of freedom of the electromagnetic potential are analysed within the framework of Dirac's constrained Hamiltonian mechanics in the front form of dynamics. The results are implemented to the problems of relativistic statistical mechanics. Based on the corresponding Liouville equation the classical partition function of the system is written down in a gauge-invariant manner and an integration over field variables is performed.

Motivation & Objective

• To formulate a consistent Hamiltonian description of classical relativistic charged particles coupled to electromagnetic fields in the front form of dynamics.
• To eliminate gauge degrees of freedom arising from chronometric invariance and electromagnetic U(1) symmetry using Dirac's constrained Hamiltonian formalism.
• To construct a gauge-invariant Liouville equation for the phase space distribution function in the front form.
• To derive a relativistic partition function by integrating out field variables in a manifestly gauge-invariant way.
• To enable a consistent statistical mechanical description of relativistic systems with both particle and field degrees of freedom.

Proposed method

• The system is described using a 4D action functional with point particles and a massless vector field, preserving manifest Lorentz invariance.
• The front form of dynamics is implemented via a foliation of Minkowski space by isotropic hypersurfaces defined by σ(x) = t, fixing the time evolution.
• Constraints from the singular Lagrangian are analyzed using Dirac's method, identifying first-class constraints associated with gauge symmetries.
• Gauge freedom is removed via canonical transformations that eliminate unphysical degrees of freedom while preserving the Dirac bracket structure.
• The Liouville equation is derived for the phase space distribution function, ensuring conservation of phase space volume under time evolution.
• The partition function is constructed by integrating over field variables in the path integral, yielding a gauge-invariant statistical ensemble.

Experimental results

Research questions

• RQ1 How can gauge degrees of freedom from chronometric invariance and electromagnetic U(1) symmetry be consistently eliminated in the front form of relativistic dynamics?
• RQ2 What is the structure of the Liouville equation for a classical relativistic system of charged particles and electromagnetic fields in the front form?
• RQ3 Can a gauge-invariant relativistic partition function be derived by integrating over field variables in the front form?
• RQ4 How does the front form of dynamics simplify the integration over field degrees of freedom in statistical mechanics?
• RQ5 What is the role of the Dirac bracket in preserving phase space volume during time evolution in the constrained system?

Key findings

• The Liouville equation in the front form preserves phase space volume under time evolution, as proven via the invariance of the Jacobian of the canonical transformation generated by the Hamiltonian.
• The Hamiltonian in the front form is expressed as H = φₜ g⁻¹ [√(g(m² + p²) + (pᵢφⁱ)²) + pᵢφⁱ], which reduces to H = -½ φₜ (m² + p²)/pᵢφⁱ in the isotropic limit.
• The gauge degrees of freedom associated with the electromagnetic potential are fully eliminated through canonical transformations, leaving only physical degrees of freedom.
• The relativistic partition function is constructed in a gauge-invariant manner by integrating over field variables, with the field phase space volume preserved via the Dirac bracket.
• The equilibrium solution of the Liouville equation corresponds to the classical Gibbs ensemble, including both particle and field contributions.
• The formalism allows explicit integration over field variables in the partition function, a non-trivial result due to the structure of the front form and the constraint analysis.