Chapter 12 Relativistic particles

When one considers relativistic particles rather than non-relativistic ones, the action

\begin{equation} \label {eq:freepart} S_P=\int dt \sum _{a=1}^N \frac 12 m_{(a)}\dot x^i_a\dot x_{ai}, \end{equation}

for \(N\) free non-relativistic particles must be replaced by

\begin{equation} \label {eq:419} S_P=-\sum _{a=1}^N m_{(a)}c\int d\tau \sqrt {-\frac {d x^\mu _a}{d \tau }\frac {dx_{a\mu }}{d\tau }}. \end{equation}

This action reduces to the non-relativistic one in the limit of small velocities as compared to the speed of light. Indeed, if we choose \(\tau =x^0=ct\),

\begin{equation} \begin{split} \label {eq:420} &-m_{(a)}c\int d\tau \sqrt {-\frac {d x^\mu _a}{d \tau }\frac {dx_{a\mu }}{d\tau }}\\&= -m_{(a)}c^{2}\int dt \sqrt {1-\frac {1}{c^2}\frac {d\vec x_a}{dt}\cdot \frac {d \vec x^a}{dt}} \approx -m_{(a)}c^2\int dt\big (1-\frac 12\frac {1}{c^2} \frac {d\vec x_a}{dt}\cdot \frac {d\vec x^a}{dt} \big ). \end {split} \end{equation}

The first term is a constant which does not contribute to the Euler-Lagrange equations and thus can be dropped, while the second terms reduces to the action for a collection of non-relativistic particles, (12.1).

Let us now set \(c=1\) and study in more details a single relativistic particle, whose worldline is described by \(x^\mu (\tau )\) in Minkowski spacetime. The associated action is

\begin{equation} \label {eq:408} S_P[x^\mu ]=\int d\tau \, L_P,\quad L_P=-m \sqrt {-\dot x^\nu \dot x_\nu }, \end{equation}

with \(\dot x^\nu =\frac {dx^\nu }{d\tau }\). For variations that vanish at the boundary of the \(\tau \) interval, the variation of the Lagrangian is

\begin{equation} \label {eq:425} \delta S_P=-m\int d\tau \, \frac {-\delta \dot x^\mu \dot x_\mu }{\sqrt {-\dot x^\nu \dot x_\nu }}= -m\int d\tau \, \delta x^\mu \frac {d}{d\tau }\Big (\frac {\dot x_\mu }{\sqrt {-\dot x^\nu \dot x_\nu }}\Big ), \end{equation}

and the Euler-Lagrange equations are given by

\begin{equation} \label {eq:426} \frac {\delta L_P}{\delta x^\mu }=-m\frac {d}{d\tau }\Big (\frac {\dot x_\mu }{\sqrt {-\dot x^\nu \dot x_\nu }}\Big )=0. \end{equation}

It is not possible to pass to the Hamiltonian formulation through a Legendre transformation: the Hessian matrix

\begin{equation} \label {eq:427} \frac {\partial ^2 L_P}{\partial \dot x^\mu \partial \dot x^\lambda } \end{equation}

is not invertibe. Indeed,

\begin{equation} \label {eq:428} \frac {\partial ^2 L_P}{\partial \dot x^\mu \partial \dot x^\lambda }= -m\frac {-\dot x^\nu \dot x_\nu \eta _{\mu \lambda }+\dot x_\mu \dot x_\lambda }{(\sqrt {-\dot x^\nu \dot x_\nu })^3},\quad \frac {\partial ^2 L_P}{\partial \dot x^\mu \partial \dot x^\lambda }\dot x^\lambda =0, \end{equation}

so that \(\dot x^\lambda \) is an eigenvector of eigenvalue \(0\).

The correct first order action principle whose equation of motion are equivalent to the Lagrangian ones in (12.7) is given by

\begin{equation} \label {eq:429} S[x^\mu ,p_\mu ,\lambda ]=\int d\tau \,\big [\dot x^\mu p_\mu -\lambda (p^\mu p_\mu +m^2)\big ]. \end{equation}

The associated Euler-Lagrange equations with respect to \(x^\mu ,p_\mu ,\lambda \) are

\begin{equation} \label {eq:430} \dot p_\mu =0,\quad \dot x^\mu -2\lambda p^\mu =0,\quad p^\mu p_\mu +m^2=0. \end{equation}

It is a “constrained Hamiltonian system” in the sense that \(\lambda (\tau )\) is a Lagrange multiplier for the constraint \(p^\mu p_\mu +m^2\) that does not involve time derivatives of the phase space variables \(x^\mu ,p_\mu \).

The general solution to the equations of motion is

\begin{equation} \label {eq:431} p_\mu (\tau )=p_\mu (0)\equiv p_\mu ,\quad x^\mu (\tau )=x^\mu (0)+2 p^\mu (0)\int _0^\tau d\tau ' \lambda (\tau '),\quad p^0=\pm \sqrt {p_ip^i+m^2}. \end{equation}

where \(p_\mu \) are \(\tau \)-independent constants. Alternatively, if \(\lambda > 0\), the equations for \(p_\mu \) may be solved as

\begin{equation} \label {eq:432} p_\mu =-\frac {1}{2\lambda }\dot x_\mu . \end{equation}

When substituting in the equation for \(\lambda \), one finds

\begin{equation} \label {eq:433} \frac {\dot x^\mu \dot x_\mu }{4\lambda ^2}+m^2=0\iff \lambda = \frac {\sqrt {-\dot x^\nu \dot x_\nu }}{2m}. \end{equation}

When substituting both of these equations into the equation for \(x^\mu \), i.e., the first equation in (12.10), one recovers the Lagrangian equations (12.6). It is in this sense that the Lagrangian equations (12.6) and the first order equations (12.10) are equivalent. Finally, note also that the first order action (12.9) reduces to the Lagrangian action (12.4) when substituting both (12.12) and (12.13).

The \(x^\mu \) can be considered as coordinates on Minkowski spacetime with metric \(\eta _{\mu \nu }={\rm diag}(-1,1,1,1)\) that is used to lower indices. Its inverse is denoted by \(\eta ^{\mu \nu }={\rm diag}(-1,1,1,1)\), \(\eta _{\mu \nu }\eta ^{\nu \rho }=\delta _\mu ^\rho \) and is used to raise indices.

Poincaré transformations are defined by

\begin{equation} \label {eq:436} x'^\mu ={\Lambda ^\mu }_\nu x^\nu +a^\mu , \end{equation}

where the matrix \({\Lambda ^\mu }_\nu \) satisfies

\begin{equation} \label {eq:437} {\Lambda ^\mu }_\nu \eta _{\mu \sigma }{\Lambda ^\sigma }_\rho =\eta _{\nu \rho } \iff \Lambda ^T\eta \Lambda =\eta \iff {\Lambda _\sigma }^\nu {\Lambda ^\sigma }_\rho =\delta ^\rho _\nu \iff \big ({\Lambda ^\sigma }_\rho \big )^{-1}={\Lambda _\sigma }^\rho , \end{equation}

The transformations with \({\Lambda ^\mu }_\nu =0\), \(x'^\mu =x^\mu +a^\mu \) are called translations, those with \(a^\mu =0\), \(x'^\mu ={\Lambda ^\mu }_\nu x^\nu \) Lorentz transformations.

For Poincaré transformations (that do not depend on \(\tau \)), the Lagrangian \(L_P\) and thus the action \(S_P\) is invariant,

\begin{equation} \label {eq:438} S_P[x'^\mu ]=S[x^\mu ]. \end{equation}

As in the discussion of Galilean invariance, infinitesimal Lorentz transformations are of the form

\begin{equation} \label {eq:439} {\Lambda ^\mu }_\nu =\delta ^\mu _\nu +{\omega ^\mu }_\nu +O(\omega ^2),\quad \omega _{\mu \nu }=-\omega _{\nu \mu }, \end{equation}

and contain \(6\) parameters, while infinitesimal translations are characterized by \(a^\mu =0+\alpha ^\mu \) and contain \(4\) parameters. The associated infinitesimal Poincaré transformations on Minkowski spacetime are given by

\begin{equation} \label {eq:440} \delta _{\omega ,\alpha } x^\mu ={\omega ^\mu }_\nu x^\nu +\alpha ^\mu . \end{equation}

According to Noether’s theorem, there are \(10\) conserved quantitites given by

\begin{equation} \label {eq:441} K_{\omega ,\alpha }=\frac {\partial L_P}{\partial \dot x^\mu }\delta _{{\omega ,\alpha }} x^\mu =m\frac {\dot x_\mu \delta _{\omega ,\alpha }x^\mu }{\sqrt {-\dot x^\nu \dot x_\nu }}. \end{equation}

The individual conserved quantities are denoted by \(P^\mu ,L^{\mu \nu }\),

\begin{equation} \label {eq:442} K_{\omega ,\alpha }=\frac 12 \omega _{\mu \nu }L^{\mu \nu }+\alpha _\mu P^\mu , \end{equation}

and are explicitly given by

\begin{equation} \label {eq:443} P^\mu =m\frac {\dot x^\mu }{\sqrt {-\dot x^\sigma \dot x_\sigma }},\quad L^{\mu \nu }=m\frac {\dot x^\mu x^\nu -\dot x^\nu x^\mu }{\sqrt {-\dot x^\sigma \dot x_\sigma }}. \end{equation}

For several particles, one finds instead,

\begin{equation} \label {eq:444} P^\mu =\sum _{a=1}^Nm_{(a)}\frac {\dot x^\mu _a}{\sqrt {-\dot x^\sigma _a\dot x_{\sigma ,a}}},\quad L^{\mu \nu }=\sum _{a=1}^Nm_{(a)} \frac {\dot x^\mu _a x^\nu _a-\dot x^\nu _a x^\mu _a}{\sqrt {-\dot x^\sigma _a\dot x_{\sigma ,a}}}. \end{equation}

relativistic invariance, to be continued