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\begin{document}
\title{Variational principle for the classical relativistic particle}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\author{M. Pavon\\
Dipartimento di Matematica Pura e Applicata,\\
Universit\`a di Padova,\\
via Belzoni 7,\\
and ISIB-CNR,\\
35131 Padova,\\
Italy.\\
{\tt michele.pavon@unipd.it}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\maketitle



\newpage
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\section{The classical Hamilton principle} We recall below the
hydrodynamic version of the classical Hamilton principle in the
strong form established in \cite[Section II]{P1}.

Consider a dynamical system with configuration space
$\R^n$. Let
$L(x,v):=\frac{1}{2}mv\cdot v-V(x)$ be the Lagrangian, where
$V(\cdot):\R^n\rightarrow
\R $ is of class $C^1$. Let
${\cal X}_{x_1}$ denote the class of all $C^1$ paths
$x:[t_0,t_1]\rightarrow \R^n$ such that
$x(t_1)={x_1}$. Let $\cal V$ denote the family of continuous
functions
$v:[t_0,t_1]\rightarrow
\R^n$. Consider the following problem:

Minimize over
$(x,v)\in ({\cal X}_{x_1}\times {\cal V})$
\begin{equation}\label{H1}
\int_{t_0}^{t_1}L(x(t),v(t))\,dt\:+\:S_0(x(t_0)),
\end{equation} subject to the constraint
\begin{equation}\label{H2}
\dot{x}(t)=v(t),\quad \forall t\in [t_0,t_1].
\end{equation} Notice that we have allowed a much larger class of
paths than usual, namely also trajectories solving a functional
differential equation of the form
$$\dot{x}(t)=v(\{x(\tau);t_0\le
\tau\le t_1\})(t),$$ where $v:C([t_0,t_1];\R^n)\rightarrow
C([t_0,t_1];\R^n)$. Using the method of {\it Lagrange
functionals} (see the following section), the following result
was obtained in
\cite{P1}.
\begin{theorem}Let $S(x,t)$ be a $C^1$ solution on $\R^n\times
[t_0,t_1]$ of the initial value problem
\begin{eqnarray}\label{H3} &&\frac{\partial{S}}{\partial{t}} +
\frac{1}{2m}\nabla {S} \cdot \nabla{S} + V(x) = 0,
\\ &&S(x,t_0)=S_0(x).\label{H4}
\end{eqnarray} Then any $x\in {\cal X}_{x_1}$ satisfying
\begin{equation}\label{H*}
\dot{x}(t)=\frac{1}{m}\nabla S(x(t),t),
\end{equation} solves together with $\frac{1}{m}\nabla S(x(t),t)$
the above minimum problem.
\end{theorem}

Notice that when a $C^1$ solution
$S(x,t)$ of (\ref{H3})-(\ref{H4}) exists, then there are also
solutions $x$ of the differential equation (\ref{H*}) satisfying
$x(t_1)={x_1}$, and therefore minimizing pairs. The difficulty
lies, of course, with the initial value problem
(\ref{H3})-(\ref{H4}) that in general only has a local in $t$
solution (namely, on some interval
$[t_0,\bar{t})$, $\bar{t}<t_1$).


\section{Hamilton's principle for the classical relativistic
particle} Consider a single particle acted on
by conservative forces independent of velocity.  Let
$L(x,v):=-mc^2\sqrt{1-\frac{v\cdot v}{c^2}}-V(x)$ be Einstein's Lagrangian,
where  $V(\cdot):\R^3\rightarrow \R $ is of class $C^1$. Let ${\cal X}_{x_1}$
denote the class of all $C^1$ paths $x:[t_0,t_1]\rightarrow \R^3$ such that
$x(t_1)={x_1}$. Let $\cal V$ denote the family of continuous
functions $v:\rightarrow \R^3$ such that $v(t)\cdot v(t)\le c^2, \forall t$.
  Consider the
problem: Minimize over
$(x,v)\in ({\cal X}_{x_1}\times {\cal V})$
\begin{equation}\label{R1}
J(x,v):=\int_{t_0}^{t_1}L(x(t),v(t))\,dt\:+\:S_0(x(t_0)),
\end{equation} subject to the constraint
\begin{equation}\label{R2}
\dot{x}(t)=v(t),\quad \forall t\in [t_0,t_1].
\end{equation} We proceed, as in \cite{P1}, to solve the problem
with the aid of Lagrange functionals.  A functional
$\Lambda$ defined on $({\cal X}_{x_1}\times {\cal V})$ which is
constant and finite over  the subset $C$ of pairs
$(x,v)$ satisfying the dynamical constraint (\ref{R2})  is called
a {\it Lagrange functional}. Obviously, if
$(x,v)\in C$ minimizes $J+\Lambda$ over  $({\cal X}_{x_1}\times {\cal V})$,
then it solves the original problem. We call this elementary fact {\it
Lagrange Lemma}.  We choose a sufficiently rich class of Lagrange
functionals. Let $F :
\R^3\times [t_0,t_1]\rightarrow
\R$ be any function of class $C^1$. Corresponding to such function
$F$, we define the Lagrange functional $\Lambda^F$ on
$X_{x_1}\times {\cal V}$   by
\begin{equation}         \Lambda^F(x,v) = F(x(t_1),t_1)-F(x(t_0),t_0)-
\int_{t_0}^{t_1}\left[\frac{\partial{F}}{\partial{t}}(x(t),t) +
v\cdot \nabla F(x(t),t)\right]dt.
\end{equation} The chain rule implies that $\Lambda^F(x,v) = 0$
for all pairs $(x,v)$ in
$C$. Hence, for each sufficiently smooth $F$, $\Lambda ^F$  is a
Lagrange functional for our problem. The solution procedure is
now outlined as follows.
\\{\bf Step 1.} Consider the {\it unconstrained}  minimization
\begin{equation}                 \min_{(x, v)\,\in \,(X_y\times {\cal V})}  (J
+ \Lambda ^F)(x,v).
\end{equation} We perform two-stage optimization. Namely, for
each fixed $x\in X_{x_1}$ , we try to compute an optimal control 
$v^*_x$ through {\it pointwise  minimization}  of the integrand
of $J +
\Lambda^F$.  More explicitly, consider for each $x\in X_{x_1}$  and each
$t\in [t_0,t_1]$ the finite-dimensional problem
\begin{equation}\label{R3}      \min_{|v|\le
c}\left\{-mc^2\sqrt{1-\frac{v\cdot v}{c^2}}- V(x(t)) -
\frac{\partial{F}}{\partial{t}}(x(t),t) - v\cdot \nabla
F(x(t),t)\right\}.
\end{equation} Differentiation with respect to $v$ gives the
condition
\begin{equation}\label{R4}
\frac{mv}{\sqrt{1-\frac{v\cdot v}{c^2}}}=\nabla F(x(t),t).
\end{equation} Next observe that the function
$-mc^2\sqrt{1-\frac{v^2}{c^2}} - v\cdot
\nabla F(x(t),t)$ is (strictly) convex on $|v|\le c$. Thus any
$v$ satisfying (\ref{R4})  solves (\ref{R3}). Now notice that
(\ref{R4}) gives
$$\frac{m^2v\cdot v}{1-\frac{v\cdot v}{c^2}}=\nabla
F(x(t),t)\cdot \nabla F(x(t),t). $$ This in turns yields
$$(m^2+\frac{\nabla F(x(t),t)\cdot \nabla F(x(t),t)}{c^2})v\cdot
v=\nabla F(x(t),t)\cdot
\nabla F(x(t),t),
$$ and finally
\begin{equation}\label{R5} v\cdot v=\frac{c^2 \,\nabla
F(x(t),t)\cdot \nabla F(x(t),t)}{c^2m^2+\nabla F(x(t),t)\cdot
\nabla F(x(t),t)}.
\end{equation} Conditions (\ref{R4}) and (\ref{R5}) imply
\begin{equation}\label{R6} v^*_x(t)=\frac{c\nabla 
F(x(t),t)}{\sqrt{c^2m^2+\nabla
F(x(t),t)\cdot\nabla F(x(t),t)}}, \end{equation} which clearly 
satisfies (\ref{R4}) as well
as the inequality $v\cdot v\le c^2$. We notice that
$v^*_x$ belongs to the class of admissible velocities
${\cal V}$.
\\{\bf Step 2.}  Consider the functional
$(J + \Lambda^F)(x,v^*_x)$ on the space $X_{x_1}$. We have
\begin{eqnarray}\nonumber (J +
\Lambda^F)(x,v^*_x)=S_0(x(t_0))+F(x(t_1),t_1)-F(x(t_0),t_0)+\\\int_{t_0}^{t_
1}\left[-\frac{\partial{F}}{\partial{t}}(x(t),t)
-c\sqrt{m^2c^2+\nabla F(x(t),t)\cdot\nabla
F(x(t),t)}-V(x(t))\right]\,dt,\label{R7}
\end{eqnarray} where we have used the the calculation
\begin{eqnarray}\nonumber -mc^2\sqrt{1-\frac{\nabla
F(x(t),t)\cdot \nabla F(x(t),t)}{c^2m^2 +\nabla F(x(t),t)\cdot
\nabla F(x(t),t)}}-\frac{c\nabla F(x(t),t)\cdot
\nabla F(x(t),t)} {\sqrt{c^2m^2+\nabla F(x(t),t)\cdot
\nabla F(x(t),t)}}\\\nonumber=\frac{-c^3m^2-c\nabla F(x(t),t)\cdot
\nabla F(x(t),t)} {\sqrt{c^2m^2+\nabla F(x(t),t)\cdot
\nabla F(x(t),t)}}\\\nonumber=-c\sqrt{c^2m^2+\nabla F(x(t),t)\cdot
\nabla F(x(t),t)}.
\end{eqnarray}  If we can find
$F^*$ such that $\Gamma^{F^*}(\cdot)$ is actually  {\it
constant}  on $X_{x_1}$, then the minimization with respect to
$x$ becomes trivial.  Suppose there exists a solution
$S(\cdot,\cdot)$ on $[t_0,t_1]$ of the initial value problem
\begin{eqnarray}\label{R8}
\frac{\partial{S}}{\partial{t}} + c\sqrt{m^2c^2+\nabla
S\cdot\nabla S} + V(x) =
0,\\S(x,t_0)=S_0(x)\label{R8'}\end{eqnarray}		 Then indeed
$\Gamma^S(\cdot)$ is constant on $X_{x_1}$. In fact, the
integrand is identically zero and the contribution of the
boundary terms is just $S(x(t_1),t_1)$. Hence, $\Gamma^S(x) =
S(x_1,t)$ for any  $x\in X_{x_1}$.  \\{\bf Step 3.} . Since
$\Gamma^S$ is constant over its domain, {\it any}    trajectory
$x$ in $X_{x_1}$ minimizes $\Gamma^S$ and solves therefore,
together with
$v^*_x$ the unconstrained problem. To solve the original
constrained problem, it suffices to choose an $x\in X_{x_1}$ that
respects the dynamics, namely {\it any}  solution of the boundary
value problem
\begin{eqnarray}\label{R9}	 &&\dot x(t) =\frac{c\nabla
S(x(t),t)}{\sqrt{c^2m^2+
\nabla S(x(t),t)\cdot\nabla
S(x(t),t)}},\\&&x(t_1)=x_1\end{eqnarray} We have therefore
established the following result.
\begin{theorem}Let $S(x,t)$ be a $C^1$ solution on $\R^n\times
[t_0,t_1]$ of the initial value problem (\ref{R8})-(\ref{R8'}).
Then any $x\in {\cal X}_{x_1}$ satisfying (\ref{R9}) solves, together
with its velocity, the above minimum problem.
\end{theorem}  Corresponding to a solution
$S$ of (\ref{R8})-(\ref{R8'}), let us  introduce the {\it
momentum field}
$p(x,t)$ by $p(x,t):=\nabla S(x,t)$. Then (\ref{R8})  implies the
following generalized Newton's law in local form
\begin{equation}\label{R10}
\left(\frac{\partial p}{\partial{t}}+\frac{c\nabla
S(x,t)}{\sqrt{c^2m^2+
\nabla S(x,t)\cdot\nabla S(x,t)}}\cdot \nabla
p\right)(x,t)=-\nabla V(x).
\end{equation} If $p(t):=p(x(t),t)$, where $x\in {\cal X}_{x_1}$
satisfies (\ref{R9}), then (\ref{R10}) gives
$$\frac{dp}{dt}=-\nabla V(x(t)).
$$

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\section{Covariant Lagrangian formulation for the free classical
particle} Consider the situation of the previous section with
$V\equiv 0$. Let $\tau$ be a real variable, and let
${\cal X}_{x_1}$ denote now the class of all $C^1$ paths
$x_{\nu}:[\tau_0,\tau_1]\rightarrow
\M^4=\R^3\times ic\R$ such that
$x(\tau_1)={x_1}$. Let $\cal V$ denote the family of continuous
functions on
$[\tau_0,\tau_1]$ taking values in the Minkowski space
$\M^4$ such that $v(\tau)\cdot v(\tau)\le 0, \forall
\tau\in [\tau_0,\tau_1]$.  Let
$L_{\nu}(x_{\nu},v_{\nu}):=-mc\sqrt{-v_{\nu}\cdot v_{\nu}}$ be
the Lagrangian (see e.g. \cite[p.328]{Go} for a justification of
this choice). Let
$\tilde{S}_0:\M^4\rightarrow \R$ be such that
$\tilde{S}_0(x_{\nu})$ is a world scalar. Consider the problem: Extremize over
$(x_{\nu},v_{\nu})\in ({\cal X}_{x_1}\times {\cal V})$
\begin{equation}\label{C1}
J(x_{\nu},v_{\nu}):=\int_{\tau_0}^{\tau_1}L_{\nu}
(x(\tau),v(\tau))\,d\tau\:+\:\tilde{S}_0(x_{\nu}(\tau_0)),
\end{equation} subject to the constraint
\begin{equation}\label{C2}
\frac{dx_{\nu}}{d\tau}(\tau)=v_{\nu}(\tau),\quad \forall
\tau\in [\tau_0,\tau_1].
\end{equation} Let $F : \M^4\times [\tau_0,\tau_1]\rightarrow
\R$ be a function of class $C^1$ such that $\nabla_{\nu}F(x_{\nu},\tau)$
belongs to $\M^4$,for all $(x_{\nu},\tau)$ in $(\M^4\times 
[\tau_0,\tau_1])$. Corresponding
to such function
$F$, we define the Lagrange functional $\Lambda^F$ on
$X_{x_1}\times {\cal V}$   by
\begin{eqnarray}\nonumber\Lambda^F(x_{\nu},v_{\nu}) =&&
F(x(\tau_1),\tau_1)-F(x(\tau_0),\tau_0)-\\
&&\int_{\tau_0}^{\tau_1}\left[\frac{\partial{F}}{\partial{\tau}}(x_{\n 
u}(\tau),\tau)
+ v_{\nu}(\tau)\cdot
\nabla_{\nu} F(x_{\nu}(\tau),\tau)\right]d\tau.\nonumber
\end{eqnarray}  The chain rule implies that $\Lambda^F(x,v) = 0$
for all pairs $(x_{\nu},v_{\nu})$ in
$({\cal X}_{x_1}\times {\cal V})$ satisfying (\ref{C2}). Hence,
$\Lambda ^F$  is a Lagrange functional for our problem. As usual,
consider first the unconstrained  extremization
\begin{equation}\nonumber		 {\rm
extremize}_{(x_{\nu},v_{\nu})\,\in \,(X_{x_1}\times {\cal V})}
(J +
\Lambda ^F)(x_{\nu},v_{\nu}).
\end{equation}  For each fixed $x\in X_{x_1}$, we try to compute
$v^*_{\nu}(x)$ through {\it pointwise  extremization}  of the
integrand of $J +
\Lambda^F$.  We consider namely for each $x\in X_{x_1}$  and each
$\tau\in [\tau_0,\tau_1]$ the problem
\begin{equation}\label{C3}	{\rm extremize}_{\{v_{\nu}\cdot
v_{\nu}\le 0\}}\left\{-mc\sqrt{-v_{\nu}\cdot v_{\nu}} -
\frac{\partial{F}}{\partial{\tau}}(x_{\nu}(\tau),\tau) -
v_{\nu}\cdot \nabla_{\nu} F(x_{\nu}(\tau),\tau)\right\}.
\end{equation} Setting the gradient equal to zero, we get
\begin{equation}\label{C4}
\frac{mcv^*_{\nu}(x)(\tau)}{\sqrt{-v^*_{\nu}(x)(\tau)\cdot
v^*_{\nu}(x)(\tau)}}=\nabla_{\nu}F(x_{\nu}(\tau),\tau).
\end{equation} We observe that, if $v^*_{\nu}(x)$ satisfying
(\ref{C4}) exists, then necessarily
\begin{equation}\label{C5}
\nabla_{\nu}F(x_{\nu}(\tau),\tau)\cdot\nabla_{\nu}F(x_{\nu}(\tau),\tau 
)=-m^2c^2.
\end{equation} We are then led to restrict our class of Lagrange
functionals to those induced by
$C^1$ functions $F$ satisfying identically on $\M^4\times
[\tau_0,\tau_1]$
\begin{equation}\label{C6}
\nabla_{\nu}F(x_{\nu},\tau)\cdot\nabla_{\nu}F(x_{\nu},\tau)=-m^2c^2.
\end{equation} If $F$ satisfies (\ref{C6}), then clearly
\begin{equation}\label{C7}
v^*_{\nu}(x)(\tau)=\frac{1}{m}\nabla_{\nu}F(x_{\nu}(\tau),\tau)
\end{equation} is a critical point that satisfies
\begin{equation}\label{C8} v^*_{\nu}(x)(\tau)\cdot
v^*_{\nu}(x)(\tau)=-c^2.
\end{equation} Next observe that, for such an $F$ and such a
$v^*_{\nu}(x)$,
\begin{eqnarray}\nonumber
\Gamma(x_{\nu}):=(J+\Lambda^F)(x_{\nu},v^*_{\nu}(x))=&&
\tilde{S}_0(x_{\nu}(\tau_0))+F(x(\tau_1),\tau_1)-F(x(\tau_0),\tau_0)-\\
&&\int_{\tau_0}^{\tau_1}\frac{\partial{F}}{\partial{\tau}}(x_{\nu}(\ta 
u),\tau)d\tau.\label{C9}
\end{eqnarray} In order to make $\Gamma$ constant over
$X_{x_1}$, it suffices to choose $\tilde{S}=F^*$ satisfying
\begin{equation}\label{C10}
\frac{\partial{\tilde{S}}}{\partial{\tau}}=0,\quad
\tilde{S}(x_{\nu},\tau_0)=\tilde{S}_0(x_{\nu}),
\end{equation} i.e. $\tilde{S}(x_{\nu},\tau)\equiv
\tilde{S}_0(x_{\nu})$. However, we also wanted $\tilde{S}$ to
fulfill condition (\ref{C6}). We conclude that if
$\tilde{S}_0$ satisfies
\begin{equation}\label{C11}
\nabla_{\nu}\tilde{S}_0(x_{\nu})\cdot\nabla_{\nu}\tilde{S}_0(x_{\nu})=-m^2c^2,
\end{equation} then $\tilde{S}(x_{\nu},\tau)\equiv
\tilde{S}_0(x_{\nu})$ is admissible and, in particular, makes
$\Gamma(x_{\nu})\equiv \tilde{S}_0(x_1)$ over its domain.
  Suppose henceforth that (\ref{C11}) is satisfied. To solve the
original constrained problem, choose any
$x\in X_{x_1}$ that solves the terminal value problem
\begin{eqnarray}	 &&\dot x_{\nu}(\tau)
=\frac{1}{m}\nabla_{\nu}\tilde{S}_0(x_{\nu}(\tau)),\\&&x(\tau_1)=x_1.\ 
end{eqnarray}
We call (\ref{C6}) and (\ref{C10}) the {\it Hamilton-Jacobi
equations}. Let us introduce the {\it momentum field}
$p_{\nu}(x_{\nu})$ by
$p_{\nu}(x_{\nu})=\nabla_{\nu}\tilde{S}_0(x_{\nu})$, and define
$p_{\nu}(\tau)=p_{\nu}(x_{\nu}(\tau))$. Observing that
(\ref{C11}) implies
\begin{equation}\label{C11'}
\nabla_{\nu}\cdot\nabla_{\nu}\tilde{S}_0(x_{\nu})=0,
\end{equation} (this follows from the fact that taking divergence of
both sides in (\ref{C11}) we get
$\Delta_{\nu}\tilde{S}_0 \nabla_{\nu}\tilde{S}_0 = 0$), we finally 
get the conservation law
\begin{equation}\label{C11''}
\frac{dp_{\nu}}{d\tau}=0.
\end{equation}
Our treatment of the variational principle may be compared to the usual one
\cite[Chapter 3]{R}.
Now let
$\varphi:=\exp[\frac{i}{\hbar}\tilde{S}_0]$. In view of
(\ref{C11}) and (\ref{C11'}), we get
$$
\nabla_{\nu}\cdot\nabla_{\nu}\varphi=[-\nabla_{\nu}\tilde{S}_0\cdot\na 
bla_{\nu}\tilde{S}_0+
i\hbar\nabla_{\nu}\cdot\nabla_{\nu}\tilde{S}_0]\frac{\varphi}{\hbar^2} 
=\frac{m^2c^2}{\hbar^2}\varphi.
$$ Hence, $\varphi$ is a {\it unimodular} solution of the {\it
Klein-Gordon} equation
\begin{equation}\label{KG}
\nabla_{\nu}\cdot\nabla_{\nu}\varphi=\frac{m^2c^2}{\hbar^2}\varphi.
\end{equation} Conversely, let $\varphi$ be a unimodular solution
of (\ref{KG}), and consider the real-valued function
$\tilde{S}_0:=\frac{\hbar}{i}\log\varphi$. Notice that
$\nabla_{\nu}\tilde{S}_0=\frac{\hbar}{i}\frac{\nabla_{\nu}\varphi}{\varphi}$.
We have
\begin{eqnarray}0=\nabla_{\nu}\cdot\nabla_{\nu}(\varphi\varphi^*)=
(\nabla_{\nu}\cdot\nabla_{\nu}\varphi)\varphi^*+
\varphi(\nabla_{\nu}\cdot\nabla_{\nu}\varphi^*)
+2\nabla_{\nu}\varphi\cdot\nabla_{\nu}\varphi^*\\=2[\frac{m^2c^2}{\hbar^2}
+\nabla_{\nu}\log\varphi\cdot\nabla_{\nu}\log\varphi^*]=2[\frac{m^2c^2 
}{\hbar^2}
+\frac{1}{\hbar^2}\nabla_{\nu}\tilde{S}_0\cdot\nabla_{\nu}\tilde{S}_0].
\end{eqnarray} Hence, $\tilde{S}_0$ satisfies (\ref{C11}). We
conclude that $\tilde{S}_0$ satisfies (\ref{C11}) {\it if and
only if} $\varphi=\exp[\frac{i}{\hbar}\tilde{S}_0]$ is a
unimodular solution of the Klein-Gordon equation (\ref{KG}).

Now we compare this result with the corresponding one in Section
3. We recall that, if $S$ is satisfies
\begin{equation}\label{C12}
\frac{\partial{S}}{\partial{t}} + c\sqrt{m^2c^2+\nabla
S\cdot\nabla S} = 0,\quad S(x,t_0)=S_0(x),\end{equation} then a
solution of the problem of the previous section (with
$V=0$) is given by any path $x$ satisfying
\begin{equation}\label{C13}\frac{dx}{dt} =\frac{c\nabla
S(x(t),t)}{\sqrt{c^2m^2+
\nabla S(x(t),t)\cdot\nabla S(x(t),t)}},\quad
x(t_1)=x_1,\end{equation} together with its velocity.

We remark, in passing, that a classical solution of (\ref{C12})
exists on some time interval for a large class of initial data
since the Hamiltonian $H(\nabla S)= c\sqrt{m^2c^2+\nabla
S\cdot\nabla S}$ does not depend on
$x$ and is convex in
$\nabla S$, see e.g.
\cite[Chapter 1]{PLL}. Let us introduce {\it proper time} by
\begin{equation}\label{C14} d\tau=\sqrt{1-\frac{v_x(t)\cdot
v_x(t)}{c^2}}\;dt.
\end{equation} Plugging in the expression for $v_x$, we get
\begin{equation}\label{C15} d\tau=\frac{cm
\;dt}{\sqrt{c^2m^2+\nabla S(x(t),t)\cdot\nabla S(x(t),t)}}.
\end{equation}  Notice that $\frac{d\tau}{dt}>0$, and,
consequently, there does exist a continuously differentiable inverse function
$t=t(\tau)$. Because of (\ref{C12}), we then get
\begin{equation}\label{C16} d\tau=-\frac{c^2m
dt}{\frac{\partial{S}}{\partial{t}}(x(t),t)}.
\end{equation} Let us introduce the purely imaginary variable
$x_4:=ict$, and let $x_4(\tau):=ict(\tau)$.  Also let
$$x_{\nu}=\left(\begin{array}{c}x\\x_4\end{array}\right),\quad
{\em and}\quad
x_{\nu}(\tau)=\left(\begin{array}{c}x(t(\tau))\\x_4(\tau)\end{array}\right).$$
Finally, let
$\tilde{S}(x_{\nu}):=S(x,t)$. Then (\ref{C13}), (\ref{C15}) and
(\ref{C16}), together with the definition of
$x_4$, yield
\begin{equation}\label{C16'}
\frac{dx_{\nu}}{d\tau}(\tau)=\frac{1}{m}\nabla_{\nu}\tilde{S}(x_{\nu}(\tau)).
\end{equation} Moreover, squaring (\ref{C12}), we get
\begin{equation}\label{C16''}
\nabla_{\nu}\tilde{S}(x_{\nu})\cdot\nabla_{\nu}\tilde{S}(x_{\nu})=-m^2c^2.
\end{equation} Taking
$\tilde{S}_0(x_{\nu})=\tilde{S}(x_{\nu})$, we see that we have
obtained the covariant formulation of this section. In
particular,
$\varphi:=\exp[\frac{i}{\hbar}\tilde{S}_0]$ is a unimodular
solution of the Klein-Gordon equation (\ref{KG}) or,
equivalently, that $\varphi(x,t):=\varphi(x_{\nu})$ is a
unimodular solution of
$$\Delta\varphi-\frac{1}{c^2}\frac{\partial^2}{\partial
t^2}\varphi=\frac{m^2c^2}{\hbar^2}$$ with the extra property
\begin{equation}\label{C16'''}\frac{\partial S_0}{\partial
t}:=\frac{\frac{\hbar}{i}\partial \log \varphi}{\partial t}=<0
\end{equation} on all of its domain because of (\ref{C12}).
Conversely, consider an $\M^4$-valued path satisfying
$$\frac{dx_{\nu}}{d\tau}(\tau)=\frac{1}{m}\nabla_{\nu}\frac{\hbar}{i}\log
\varphi(x_{\nu}(\tau)),$$ where $\varphi$ is a unimodular
solution of the Klein-Gordon equation fulfilling condition
(\ref{C16'''}). Then $S=\frac{\hbar}{i}\log\varphi$ satisfies
(\ref{C12}). Moreover, letting
$t:=\frac{1}{ic}x_4$, we get equation (\ref{C13}). In conclusion,
we remark that only (unimodular) solutions of the Klein-Gordon
equation satisfying (\ref{C16'''}) ({\it positive-energy
solutions}) are relevant here. Condition
(\ref{C16'''}) reflects the fact that positive increments of the proper time
$\tau$ correspond to positive increments of the time parameter
$t$.



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\end{thebibliography}


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Michele Pavon
Dipartimento di Matematica Pura ed Applicata
Universit=E0 di Padova and ISIB-CNR
via Belzoni 7
I-35131  PADOVA, ITALY


Phone:  +39-049-827 5841
=46ax:    +39-049-827 5892
E-Mail : michele.pavon@unipd.it
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