nLab relativistic particle

Idea

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.

Definition

The relativistic particle is described by the sigma-model whose

• target space is a spacetime $(X,g)$,

  where the pseudo-Riemannian metric $g$ – or rather the Levi-Civita connection $\nabla$ that it induces on the tangent bundle $T X$ – encodes the background field of gravity acting on the particle;

• worldvolume is the real line $\Sigma = \mathbb{R}$ or the circle $\Sigma = S^1$, the worldline;

• background gauge field is connection $\nabla$ on a circle group-principal bundle over $X$, encoding a field of electromagnetism acting by Lorentz force on the particle;

• configuration space is the quotient

  $$Conf := C^\infty(\Sigma, X)// Diff(\Sigma)$$

  of the space (naturally a diffeological space) of smooth functions $\Sigma \to X$ (“trajectories”);

• the exponentiated action functional is for given parameters $m \in \mathbb{R}$ (the particle’s mass) and $q \in \mathbb{R}$ (the particle’s charge)

  $$\exp(i S(-)) : [\gamma] \mapsto \exp( i m \int dvol(\gamma^* g)) \;\; hol(\nabla,\gamma) \,,$$

  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 $\gamma$. In the case that the underlying circle-principal bundle is trivial, so that the connection is given by a 1-form $A \in \Omega^1(X)$, the action functional is

  $$\begin{aligned} S : [\gamma] \mapsto & S_{kin}([\gamma]) + S_{gauge}([\gamma]) \\ & = \int m dvol(\gamma^* g) + \int q \gamma^* A \end{aligned} \,,$$

  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.

Properties

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 $A \in \Omega^1(X)$. Write $F = d A$ for its curvature 2-form: the field strength of the electromagnetic field.

Proposition

The variation of the gauge interaction term is

$$\delta \int_\Sigma \gamma^* A = - \int_\Sigma F(\dot \gamma, \delta \gamma) \,.$$

Proof

Let $\mathbb{R}^d \stackrel{\simeq}{\to} U \hookrightarrow X$ be a local coordinate patch with coordinates $\{x^\mu\}$ and assume that $\gamma$ takes values in $U$ (or at least that its variation is supported there, which we can assume without restriction of generality). Then the variation is given by is

$$\begin{aligned} \delta \int_\Sigma \gamma^* A & = \delta \int_\Sigma A_\mu(\gamma) \dot \gamma^\mu d \tau \\ & = \int_\Sigma \left( (\partial_{\nu} A_\mu)(\gamma) \dot \gamma^\mu - \frac{d}{d\tau} (A_\nu(\gamma)) \right) \delta \gamma^\nu d \tau \\ & = \int_\Sigma \left( (\partial_{\nu} A_\mu)(\gamma) \dot \gamma^\mu - (\partial_\mu A_\nu)(\gamma)) \dot \gamma^\mu \right) \delta \gamma^\nu d\tau \end{aligned} \,.$$

The variation of the kinetic terms is slightly subtle due to the square root in

$$dvol(\gamma^* g) = \sqrt{g(\dot \gamma, \dot \gamma)} d\tau \,.$$

To deal with this, we first look at variations of trajectories in a small region where $g(\dot \gamma, \dot \gamma)$ is non-zero. For such we can always find a diffeomorphism $\Sigma \stackrel{\simeq }{\to} \Sigma$ such that this term is constantly $= 1$ 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).

Proposition

With the above choice of diffeomorphism gauge, the equations of motion are

$$g(\nabla_{\dot \gamma} \dot \gamma / {\vert \dot \gamma}\vert ,-) = \iota_{\dot \gamma} F \,,$$

where $\nabla$ is the covariant derivative with respect to the Levi-Civita connection of the metric $g$.

Proof

Computing as before in local coordinates and parameterization such that $g(\dot \gamma, \dot \gamma) = 1$, the variation of the kinetic terms is

$$\begin{aligned} \delta \int_\Sigma dvol(\gamma^* g) & = \delta \int_\Sigma \sqrt{g(\dot \gamma, \dot \gamma)} d\tau \\ & = \int_\Sigma \left( \frac{1}{2}(\partial_\mu g_{\nu \lambda}) \dot \gamma^\nu \dot \gamma^\lambda - \frac{d}{d\tau}(g_{\mu \nu} \dot \gamma^\nu) \right) \delta \gamma^\mu d \tau \\ & = \int_\Sigma \left( \frac{1}{2}(\partial_\mu g_{\nu \lambda}) \dot \gamma^\nu \dot \gamma^\lambda - 2 (g_{\mu\nu}\ddot \gamma^\nu + (\partial_\lambda g_{\mu\nu}) \dot \gamma^\nu \dot \gamma^\lambda) \right) \delta \gamma^\mu d \tau \\ & = - \int_\Sigma g_{\mu \mu}( \ddot \gamma^\nu + \Gamma^\nu{}_{\alpha \beta} \dot \gamma^\alpha \dot \gamma^\beta ) \delta \gamma^\mu \\ & = - \int_\Sigma g_{\mu \mu}( \nabla_{\dot \gamma} \dot \gamma^\nu ) \delta \gamma^\mu \\ & = -\int_\Sigma g(\nabla_{\dot \gamma} \dot \gamma, \delta \gamma) \end{aligned} \,.$$

(Here $\Gamma^\cdot{}_{\cdot \cdot}$ are the Christoffel symbols.)

This gives the equations of motion as claimed.

Remark

The norm of the tangent vector along a physical trajectory is preserved:

$$\begin{aligned} \frac{d}{d \tau} \sqrt{g(\dot \gamma, \dot \gamma)} & \propto 2 \frac{d}{d \tau} g(\nabla_{\dot \gamma} \dot \gamma, \dot \gamma) \\ & \propto F(\dot \gamma, \dot \gamma) \\ & 0 \end{aligned} \,.$$

Therefore if the assumption $g(\dot \gamma, \dot \gamma) \neq 0$ is satisfied at one instant, is is so everywhere along the curve.

Remark

For vanishing background gauge field strength, $F = 0$, the equations of motion

$$\nabla_{\dot \gamma} \dot \gamma = 0$$

express the parallel transport of the tangent vector along a physical trajectory. This identifies these trajectories with the geodesics of $X$.

Related concepts

• sigma model, brane

  • particle,

    relativistic particle, non-relativistic particle,

    spinning particle, superparticle

  • string, spinning string, superstring

  • membrane

• Klein-Gordon equation, Dirac equation

• relativistic field theory

References

• James D. Bjorken, Sidney D. Drell: Relativistic Quantum Mechanics, McGrawHill (1964) [ark:/13960/t5fc2v05h, pdf, pdf]

• Giulio Ruffini, Four approaches to quantization of the relativistic particle [arXiv:gr-qc/9806058]

Discussion in terms of BV-formalism:

• Jean M. L. Fisch and Marc Henneaux, Antibracket—antifield formalism for constrained hamiltonian systems (doi)

• Alberto Cattaneo, Michele Schiavina, On time, Lett. Math. Phys. 107 (2017) 375-408 [doi:10.1007/s11005-016-0907-x, arXiv:1607.02412]

See also:

• Benjamin Koch, Enrique Muñoz, Path integral of the relativistic point particle in Minkowski space (arXiv:2012.15242)