relativistic particle


Quantum field theory


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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

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


The variation of the gauge interaction term is

δ Σγ *A= ΣF(γ˙,δγ). \delta \int_\Sigma \gamma^* A = - \int_\Sigma F(\dot \gamma, \delta \gamma) \,.

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

δ Σγ *A =δ ΣA μ(γ)γ˙ μdτ = Σ(( νA μ)(γ)γ˙ μddτ(A ν(γ)))δγ νdτ = Σ(( νA μ)(γ)γ˙ μ( μA ν)(γ))γ˙ μ)δγ νdτ. \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(γ *g)=g(γ˙,γ˙)dτ. 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(γ˙,γ˙)g(\dot \gamma, \dot \gamma) is non-zero. For such we can always find a diffeomorphism ΣΣ\Sigma \stackrel{\simeq }{\to} \Sigma such that this term is constantly =1= 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).


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

g( γ˙γ˙/|γ˙|,)=ι γ˙F, 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 gg.


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

δ Σdvol(γ *g) =δ Σg(γ˙,γ˙)dτ = Σ(12( μg νλ)γ˙ νγ˙ λddτ(g μνγ˙ ν))δγ μdτ = Σ(12( μg νλ)γ˙ νγ˙ λ2(g μνγ¨ ν+( λg μν)γ˙ νγ˙ λ))δγ μdτ = Σg μμ(γ¨ ν+Γ ν αβγ˙ αγ˙ β)δγ μ = Σg μμ( γ˙γ˙ ν)δγ μ = Σg( γ˙γ˙,δγ). \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 Γ \Gamma^\cdot{}_{\cdot \cdot} are the Christoffel symbols.)

This gives the equations of motion as claimed.


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

ddτg(γ˙,γ˙) 2ddτg( γ˙γ˙,γ˙) F(γ˙,γ˙) 0. \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(γ˙,γ˙)0g(\dot \gamma, \dot \gamma) \neq 0 is satisfied at one instant, is is so everywhere along the curve.


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

γ˙γ˙=0 \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 XX.


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

Discussion in terms of BV-formalism includes

Revised on January 10, 2017 16:11:42 by Urs Schreiber (