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\begin{document}

%-------------------------------------------------------------------


\section*{relativistic particle}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{quantum_field_theory}{}\paragraph*{{Quantum field theory}}\label{quantum_field_theory}

[[!include functorial quantum field theory - contents]]

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{covariant_phase_space}{Covariant phase space}\dotfill \pageref*{covariant_phase_space} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \emph{relativistic particle} in [[physics]] is a model for the dynamics of a single [[brane|particle]] that is propagating in a [[spacetime]] subject to [[force]]s such as [[gravity]] and (if it is charged) the [[electromagnetic field]].

The generalization to [[supergeometry]] is the [[superparticle]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

The \textbf{relativistic particle} is described by the [[sigma-model]] whose

\begin{itemize}%
\item [[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;


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


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


\item [[configuration space]] is the [[quotient]]

\begin{displaymath}
Conf := C^\infty(\Sigma, X)// Diff(\Sigma)
\end{displaymath}
of the space (naturally a [[diffeological space]]) of [[smooth function]]s $\Sigma \to X$ (``trajectories'');


\item 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]])

\begin{displaymath}
\exp(i S(-))
   : 
 [\gamma] 
     \mapsto
   \exp(
   i m \int dvol(\gamma^* g))
   \;\;
   hol(\nabla,\gamma)
   \,,
\end{displaymath}
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 n-bundle with connection|circle bundle with connection]] around $\gamma$. In the case that the underlying [[circle]]-[[principal bundle]] is trivial, so that the [[connection on a bundle|connection]] is given by a [[differential forms|1-form]] $A \in \Omega^1(X)$, the [[action functional]] is

\begin{displaymath}
\begin{aligned}
   S : 
   [\gamma]
    \mapsto
    & 
    S_{kin}([\gamma]) + S_{gauge}([\gamma])
    \\
    & =
    \int m dvol(\gamma^* g)
    +
    \int q \gamma^* A
  \end{aligned}
    \,,
\end{displaymath}
where the first summand is the \emph{kinetic action} and the second the \emph{gauge interaction} term.



\end{itemize}
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]].

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{covariant_phase_space}{}\subsubsection*{{Covariant phase space}}\label{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]].

\begin{prop}
\label{}\hypertarget{}{}
The [[variational calculus|variation]] of the gauge interaction term is

\begin{displaymath}
\delta \int_\Sigma \gamma^* A
  =
  - \int_\Sigma F(\dot \gamma, \delta \gamma)
  \,.
\end{displaymath}
\end{prop}
\begin{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{displaymath}
\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}
  \,.
\end{displaymath}
\end{proof}
The variation of the kinetic terms is slightly subtle due to the square root in

\begin{displaymath}
dvol(\gamma^* g) = \sqrt{g(\dot \gamma, \dot \gamma)} d\tau
  \,.
\end{displaymath}
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 [[curve]]s, so we may apply such a diffeomorphism at will to compute the variation).

\begin{prop}
\label{}\hypertarget{}{}
With the above choice of diffeomorphism gauge, the equations of motion are

\begin{displaymath}
g(\nabla_{\dot \gamma} \dot \gamma / {\vert \dot \gamma}\vert ,-)  = \iota_{\dot \gamma} F
  \,,
\end{displaymath}
where $\nabla$ is the [[covariant derivative]] with respect to the [[Levi-Civita connection]] of the metric $g$.

\end{prop}
\begin{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{displaymath}
\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}
  \,.
\end{displaymath}
(Here $\Gamma^\cdot{}_{\cdot \cdot}$ are the [[Christoffel symbols]].)

This gives the equations of motion as claimed.

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
The [[norm]] of the tangent vector along a physical trajectory is preserved:

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

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
For vanishing [[background gauge field]] strength, $F = 0$, the equations of motion

\begin{displaymath}
\nabla_{\dot \gamma} \dot \gamma = 0
\end{displaymath}
express the [[parallel transport]] of the tangent vector along a physical trajectory. This identifies these trajectories with the [[geodesics]] of $X$.

\end{remark}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[sigma model]], [[brane]]

\begin{itemize}%
\item [[particle]],

\textbf{relativistic particle}, [[non-relativistic particle]],

[[spinning particle]], [[superparticle]]


\item [[string]], [[spinning string]], [[superstring]]


\item [[membrane]]



\end{itemize}

\item [[relativistic field theory]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item Giulio Ruffini, \emph{Four approaches to quantization of the relativistic particle} (\href{http://arxiv.org/abs/gr-qc/9806058}{arXiv:gr-qc/9806058})

\end{itemize}
Discussion in terms of [[BV-formalism]] includes

\begin{itemize}%
\item Jean M. L. Fisch and [[Marc Henneaux]], \emph{Antibracket---antifield formalism for constrained hamiltonian systems} (\href{http://dx.doi.org/10.1016/0370-2693%2889%2990292-X}{doi})


\item [[Alberto Cattaneo]], Michele Schiavina, \emph{On time} (\href{http://arxiv.org/abs/1607.02412}{arXiv:1607.02412})



\end{itemize}
[[!redirects relativistic particles]]

[[!redirects quantum relativistic particle]] [[!redirects quantum relativistic particles]]



\end{document}