Covariant Hamiltonian for the electromagnetic two-body problem

Abstract

We discuss a method to transform the covariant Fokker action into an implicit two-degreeof-freedom Hamiltonian for the electromagnetic two-body problem with arbitrary masses. This dynamical system appeared 100 years ago and it was popularized in the 1940's by the still incomplete Wheeler and Feynman program to quantize it as a means to overcome the divergencies of perturbative QED. Our finite-dimensional implicit Hamiltonian is closed and involves no series expansions. The Hamiltonian formalism is then used to motivate an EBK quantization based on the classical trajectories with a non-perturbative formula that predicts energies free of infinities.

Equations of motion for variational electrodynamics

“…(iii) Motivated by the regular orbits found in [39,40] and the relative simplicity of the equations for collinear motion, our work of Ref. [41] studied a C ∞ orbit in the case of arbitrary masses using the Hamilton-Jacobi theory. Such extension to the case of arbitrary masses has a physical application in the quantization of the electromagnetic twobody problem.…”

“…precisely the necessary Radon-Nikodym derivative for the change of variables to reduce the integrals ( 40) and ( 41) respectively to integrals ( 7) and (8). Extension to T BV ⊗ X BV is possible because of two facts: (i) the linear dependence on the first derivatives of the electromagnetic functional (40) and (41), and (ii) the only other dependence on derivatives is in the square root definition (44) of M i , which is a homogeneous function of t i (s). The above formulas reduce to the respective X BV formulas used in the main text for the special case when t i (s) ≡ s for i = 1, 2.…”

Symplectic structure for general relativity and Einstein–Brillouin–Keller quantization

“…For example, the relationship between the EBK method and the Heisenberg matrix mechanics has been discussed in reference [10] and some elucidative examples of the method have been provided in reference [11]. The EBK quantization method has also been used to analyze quantum chaos [12,13]. In this paper, we propose a new application of this method to a space-time manifold under general relativity.…”

Symplectic structure for general relativity and Einstein-Brillouin-Keller quantization

“…[11]. The EBK quantization method has also been used to analyze quantum chaos [12,13]. In this paper, we propose a new application of this method to a space-time manifold under general relativity.…”