Quantum field theory
The relativistic particle in physics is a model for the dynamics of a single particle that is propagating in a spacetime subject to forces such as gravity and (if it is charged) the electromagnetic field.
The generalization to supergeometry is the superparticle.
The relativistic particle is described by the sigma-model whose
target space is a spacetime ,
where the pseudo-Riemannian metric – or rather the Levi-Civita connection that it induces on the tangent bundle – encodes the background field of gravity acting on the particle;
worldvolume is the real line or the circle ;
background gauge field is connection on a circle group-principal bundle over , encoding a field of electromagnetism acting by Lorentz force on the particle;
configuration space is the quotient
of the space (naturally a diffeological space) of smooth functions (“trajectories”);
the exponentiated action functional is for given parameters (the particle’s mass) and (the particle’s charge)
where the first terms is the integral of the volume form of the pullback of the background metric, and where the second term is the holonomy of the circle bundle with connection around . In the case that the underlying circle-principal bundle is trivial, so that the connection is given by a 1-form , the action functional is
where the first summand is the kinetic action and the second the gauge interaction term.
The above action functional is called the Nambu-Goto action in dimension 1. Alternatively (and mandatorily for vanishing mass parameter), the kinetic action is replaced by the corresponding Polyakov action.
Covariant phase space
We determine the covariant phase space of the theory: the space of solutions to the equations of motion and the presymplectic structure.
We assume for simplicity that the class of the background circle bundle is trivial, so that the connection is equivalently given by a 1-form . Write for its curvature 2-form: the field strength of the electromagnetic field.
The variation of the gauge interaction term is
Let be a local coordinate patch with coordinates and assume that takes values in (or at least that its variation is supported there, which we can assume without restriction of generality). Then the variation is given by is
The variation of the kinetic terms is slightly subtle due to the square root in
To deal with this, we first look at variations of trajectories in a small region where is non-zero. For such we can always find a diffeomorphism such that this term is constantly in this region (recall that configurations are diffeomorphism classes of smooth curves, so we may apply such a diffeomorphism at will to compute the variation).
With the above choice diffeomorphism gauge, the equations of motion are
where is the covariant derivative with respect to the Levi-Civita connection of the metric .
Computing as before in local coordinates and parameterization such that , the variation of the kinetic terms is
(Here are the Christoffel symbols.)
This gives the equations of motion as claimed.
- Giulio Ruffini, Four approaches to quantization of the relativistic particle (arXiv:gr-qc/9806058)
- Jean M. L. Fisch and Marc Henneaux, Antibracket—antifield formalism for constrained hamiltonian systems (doi)