Classical Relativistic Particle

Let us consider a massive relativistic particle moving in flat D-dimensionalMinkowski space-time. The relativistic action principle, in non-relativistic notation is,

$S=-m\int dt\sqrt{1-|\dot{\vec{x}}{|}^2}$.

The canonical momenta are $\vec{p}=\frac{m\dot{\vec{x}}}{\sqrt{1-|\dot{\vec{x}}{|}^2}}$ and the energy is $E=\sqrt{|\vec{p}{|}^2+m^2}$.

The change in action is

$\delta S=-\int dt\delta \vec{x}.\dot{\vec{p}}$ and thus the equations of motion are $\dot{\vec{p}}=0$.

This formulation is asymmetric between space and time. So, to find a manifestly relativistic action principle, we introduce orthonormal coordinates $x^{\mu }\ni \mu =0,...,D-1$ onMinkowski spacetime ${\mathbb{R}}^{1,D-1}$.

The world-line of a particle is then specified by coordinate functions $x^{\mu }(\sigma )$ depending on some real parameter $\sigma$.

This can be viewed as a map

$x^{\mu }:\gamma \to {\mathbb{R}}^{1,D-1}$

$:\sigma \mapsto x^{\mu }(\sigma )$

If the particle is massive, the tangent to the world-line must be timelike:

$-{\eta }_{\mu \nu }\frac{d{x}^{\mu }}{d\sigma }\frac{d{x}^{\nu }}{d\sigma }>0\forall \sigma$ where ${\eta }_{\mu \nu }:=diag(-1,1,...,1)$.

Then, the relativistic action principle is,

$S=-m\int d\sigma \sqrt{-{\eta }_{\mu \nu }\frac{d{x}^{\mu }}{d\sigma }\frac{d{x}^{\nu }}{d\sigma }}$ which is proportional to the proper time $\Delta \tau$. Therefore, we recover the important result that the world-lines of massive particles extremise proper time. The solutions are thus straight lines in ${\mathbb{R}}^{1,D-1}$.

This action principle has two important properties:

1. Poincaré Symmetry. It is manifestly invariant under Poincaré transformations,

$x^{\prime \mu }={\Lambda }_{\nu }^{\mu }{x}^{\nu }+a^{\mu }$ where ${\Lambda }_{\rho }^{\mu }{\Lambda }_{\sigma }^{\nu }{\eta }_{\mu \nu }={\eta }_{\rho \sigma }$.

It is a global symmetry that transforms classical solutions.

2. Reparametrisation Invariance. It is independent of the choice of parameter. If we introduce a new parameter $\sigma \mapsto {\sigma }^{\prime }(\sigma )$, we have

$\frac{d{x}^{\mu }}{d\sigma }=\frac{d{x}^{\mu }}{d{\sigma }^{\prime }}\frac{d{\sigma }^{\prime }}{d\sigma }$ under which the action is invariant. Thus, we can choose the parameter $\sigma :=x^0=t$, and thereby reproduce the previous action (in non-relativistic notation).

It is a gauge redundancy that changes the parametrisation of classical solutions.