Poisson brackets of the retarded and advanced potentials in the Hamiltonian theory of point electrons

Dirac's 1948 note 'III. On the theory of point electrons' sets up a Hamiltonian formulation of classical electrodynamics for a single point electron coupled to the electromagnetic field, with the electron proper time as the evolution parameter, as a reply to Lewis's critique of Dirac's earlier variational and Hamiltonian derivations of the Lorentz-Dirac radiation-reaction equation. Within this formulation Dirac postulates Poisson brackets for the electron variables and for auxiliary field variables $A_\mu(\mathbf{x};s)$, reproduces the electron's equations of motion including radiation damping, and recovers, on solutions, the standard identification of the potential as the incoming field plus half the difference of the retarded and advanced potentials. The note leaves open, however, the Poisson-bracket relations of the physical retarded and advanced potentials themselves, which are nonlocal functionals of the entire world-line and cannot be written down in closed form without first solving the coupled dynamics. The open question is to construct a phase-space formulation in which the retarded and advanced potentials appear as explicit phase-space functions with closed-form Poisson brackets with the electron and field variables.

Background

Classical electrodynamics couples point charges to the electromagnetic field, and the central dynamical puzzle of that coupling is radiation reaction (also called radiation damping or the self-force): an accelerated charge radiates energy and momentum, and the field it creates acts back on the charge itself. Dirac (1938) derived the relativistic equation of motion of a point electron, now called the Lorentz-Dirac equation, by decomposing the field on the electron's trajectory (its world-line) $z_\mu(s)$, where $s$ is proper time, the time measured by a clock co-moving with the electron and used to parametrize the world-line. Writing the electromagnetic four-potential (the relativistic potential combining the scalar and vector potential) as an incoming free field plus the electron's self-field, Dirac showed that the finite, physically meaningful combination on the world-line is

$$A_\mu = A_{\mu\,\mathrm{in}} + \tfrac{1}{2}\bigl(A_{\mu\,\mathrm{ret}} - A_{\mu\,\mathrm{adv}}\bigr),$$

where $A_{\mu\,\mathrm{ret}}(x)$ and $A_{\mu\,\mathrm{adv}}(x)$ are the retarded and advanced potentials, obtained by solving the wave equation with the electron's current as source using respectively the retarded Green function (source points on the past light cone of the field point) and the advanced Green function (source points on the future light cone), and $A_{\mu\,\mathrm{in}}$ is an arbitrary incoming solution of the source-free field equations. Substituting this combination into the Lorentz force law yields the radiation-damping term, which is proportional to the third proper-time derivative of position.

The Lorentz-Dirac equation is not a conservative system, and its potential is not an ordinary external field: the self-field part $A_{\mu\,\mathrm{ret}}-A_{\mu\,\mathrm{adv}}$ evaluated on the world-line depends on the electron's own past and future motion. This obstructs the standard variational and Hamiltonian machinery. In Hamiltonian mechanics the state of a system at one instant is a point in a phase space, and the dynamics is generated by a Hamiltonian function $F$ through the Poisson bracket, an antisymmetric bilinear operation $[u,v]$ on functions on phase space satisfying the Leibniz rule and the Jacobi identity, in terms of which the evolution of any quantity $\zeta$ is $d\zeta/ds=[\zeta,F]$. For a particle in a prescribed external potential $A_\mu(x)$ the Hamiltonian $F=-(1/2m)\{(p-eA(z))^2-m^2\}$ with canonical brackets reproduces the Lorentz force. Lewis (1945) showed that Dirac's earlier attempts (1938-1943, as characterized in that critique) to extend this to the radiating electron are not sound: the self-field term makes the effective potential a functional of the whole trajectory depending explicitly on proper time, so the identity $dA^\mu/ds=(\partial A^\mu/\partial x^\nu)v_\nu$ used in the derivations fails, and the proposed extra action term vanishes identically for a single electron, because on the world-line $A_{\mathrm{ret}}-A_{\mathrm{adv}}$ is proportional to the acceleration $\dot v_\mu$ while $\dot v\cdot v=0$. Lewis also gave a correct non-conservative variational principle for the equations of motion and questioned the uniqueness of fields defined through Dirac's limiting procedure with an auxiliary time-like four-vector, the regulator $\boldsymbol{\lambda}$.

Dirac's 1948 note 'III. On the theory of point electrons' is his reply. He first identifies the correct variational condition, $\delta(A_{\mathrm{in}}^\mu+A_{\mathrm{out}}^\mu)=0$ at fixed space-time points, and then builds a Hamiltonian formulation with proper time as the evolution parameter. The crucial move is to introduce, alongside the electron coordinates $z_\mu(s)$ and momenta $p_\mu(s)$, a family of Hamiltonian field variables $A_\mu(\mathbf{x};s)$, one variable per space-time point $\mathbf{x}$, evolving with $s$, which before the equations of motion are integrated are independent of the instantaneous particle variables. These are distinct objects from the conventional space-time potentials $A_{\mu\,\mathrm{ret}}(\mathbf{x})$ and $A_{\mu\,\mathrm{adv}}(\mathbf{x})$, which are fixed nonlocal functionals of the entire world-line and do not depend on the value of $s$ at which the phase space is taken. Dirac postulates the Poisson brackets

$$[z_\mu,z_\nu]=0,\quad [p_\mu,p_\nu]=0,\quad [p_\mu,z_\nu]=g_{\mu\nu},\quad [z_\mu,A_\nu(\mathbf{x})]=[p_\mu,A_\nu(\mathbf{x})]=0,$$ $$[A_\mu(\mathbf{x}),A_\nu(\mathbf{x}')]=\tfrac{1}{2}g_{\mu\nu}\bigl\{\Delta(\mathbf{x}-\mathbf{x}'+\boldsymbol{\lambda})+\Delta(\mathbf{x}-\mathbf{x}'-\boldsymbol{\lambda})\bigr\},$$

where $g_{\mu\nu}$ is the Minkowski metric, $\Delta$ is the invariant distribution (the difference of the retarded and advanced Green functions of the wave equation, that is, the massless Pauli-Jordan function), and $\boldsymbol{\lambda}$ is the time-like regulator inherited from his earlier limiting construction. He states the algebraic criterion that a bracket satisfying antisymmetry, additivity, the Leibniz rule, and the Jacobi identity is in principle realizable by canonical coordinates, and shows that with Lewis's Hamiltonian $F$ the evolution $d\zeta/ds=[\zeta,F]$ reproduces the particle equations including radiation damping together with a field equation for $A_\mu(\mathbf{x};s)$; integrating that field equation exhibits the integration constant as the incoming potential $A_{\mu\,\mathrm{in}}$, and evaluating at the electron position with $\boldsymbol{\lambda}\to 0$ recovers $A_\mu=A_{\mu\,\mathrm{in}}+\tfrac12(A_{\mu\,\mathrm{ret}}-A_{\mu\,\mathrm{adv}})$ on the world-line, hence the Lorentz-Dirac force.

What the 1948 analysis leaves open is precisely the brackets of the physical retarded and advanced potentials. Since $A_{\mu\,\mathrm{ret}}$ and $A_{\mu\,\mathrm{adv}}$ depend on the electron variables at all proper times, they are not functions of the phase-space variables at any fixed $s$, and their Poisson brackets with $z_\mu$, $p_\mu$, and the Hamiltonian field variables cannot be written down in closed form: one would first have to integrate the electron equations of motion, that is, solve the coupled dynamics. Completing the phase-space characterization so that the conventional retarded and advanced potentials become explicit phase-space functions is therefore the task left open by the note.

Several later developments frame this gap. Action-at-a-distance electrodynamics, the Fokker direct interparticle action in which charges interact along light cones without an independent field and whose absorber-theory elaboration by Wheeler and Feynman attributes radiation damping to the response of distant absorbing matter, keeps the half-retarded-plus-half-advanced combination at the price of delay-type equations. The no-interaction theorem of Currie, Jordan, and Sudarshan (1963) shows that a manifestly Lorentz-covariant Hamiltonian mechanics of interacting particles with canonical position variables is trivial, a structural obstruction that any covariant canonical construction must confront. Peierls (1952) defined a covariant bracket for arbitrary functionals of the fields, using the retarded and advanced Green functions of the equations linearized about a given solution; this is the basis of the modern covariant phase space formalism, in which the phase space is the space of solutions of the equations of motion. In practical radiation-reaction work the Lorentz-Dirac equation is commonly replaced by its order-reduced form, the Landau-Lifshitz equation, obtained by eliminating the higher time derivatives in favour of the external force, which is regular and admits standard Hamiltonian treatments order by order.

Problem Statement

For the classical electrodynamics of a point electron interacting with the electromagnetic field in Minkowski space, in the setting of Dirac's 1948 note (an arbitrary incoming field $A_{\mu\,\mathrm{in}}$, the Lorentz-Dirac equations of motion, proper time $s$ as the evolution parameter), construct a Hamiltonian (phase-space) formulation in which the physically defined retarded and advanced four-potentials $A_{\mu\,\mathrm{ret}}(\mathbf{x})$ and $A_{\mu\,\mathrm{adv}}(\mathbf{x})$, nonlocal functionals of the entire electron world-line, appear as explicit phase-space functions, and determine their Poisson-bracket relations with the electron variables $z_\mu$, $p_\mu$ and with the field variables in closed form, without presupposing integration of the coupled electron-field equations of motion.

Any phase space on which the potentials $A_{\mu\,\mathrm{ret}}$, $A_{\mu\,\mathrm{adv}}$ and the electron and field variables are well-defined functions is admissible; the formulation need not be restricted to the Hamiltonian coordinates of the note at fixed $s$. Whatever phase space is adopted, the bracket structure must be the source's physical one: restricted to the physical (on-shell) configurations it must reproduce Dirac's postulated brackets for the electron and field variables and generate the Lorentz-Dirac evolution, so that an off-shell or degenerate bracket that fails to reproduce the source's dynamics does not resolve the problem. The Poisson brackets sought are those of the potentials with the electron variables $z_\mu$, $p_\mu$, with the field variables, and with one another. Here 'explicit phase-space function' and 'closed form' mean a finite expression in the phase-space variables that can be evaluated at a phase-space point without first integrating the coupled electron-field equations of motion for the world-line.

The single-electron setting of the source note is the required core; extending the construction to several charges or to the action-at-a-distance (Wheeler-Feynman) formulation is a natural generalization but not a substitute for it.

References

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