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\renewcommand{\@oddfoot}{--- ~March, 2002~ ---\hfil
\raisebox{0.3ex}{\tiny --- ~D.~A.~Arbatsky~ `` QED. II.
Arbitrariness in the choice of Lagrangian.''~
---}\hfil
--- ~\thepage~
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\title{On quantization of electromagnetic field.\\
       II. Arbitrariness in choice of Lagrangian.}
\author{D.~A.~Arbatsky\footnote{ http://daarb.narod.ru/ , http://wave.front.ru/ }}
\date{March, 2002}
\maketitle

\begin{abstract}
Here we show that addition to Lagrangian a divergence of a
function does not change the symplectic structure on invariant
phase space.
\end{abstract}

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%\myparagraph{Preliminary remarks.}
 It is well-known that the same dynamics can be obtained by
different choices of Lagrangian. The simplest replacement of this
type is a linear substitution:
\[
 L\,\longrightarrow\,L'=\alpha L   \ .
\]
So far as in these papers we study the questions of quantization
and normalize action by Planck's constant, $\hbar=1$, this
arbitrariness is absent in our research.

But there is another essential arbitrariness which is the addition
to the Lagrangian a full divergence\footnote{In principle, there
are other, more complicated substitutions which usually do not
play a practical role because of their complexity.}:
\begin{equation}
 L\,\longrightarrow\,L'=L+\partial_\mu F_\mu   \ ,
\label{LPlusDiv}
\end{equation}
where $F_\mu$ is some vector function of the coordinate $x$ and
field values $\varphi_i(x)$.

In general case the Lagrangian may contain derivatives of the
field with respect to coordinates of the order higher than first.
Here we will take into account such a possibility and we will
accept that the function $F_\mu$ may also depend on the
derivatives of the field with respect to coordinates.

So far as the substitution (\ref{LPlusDiv}) does not change
dynamics, it leaves the invariant phase space $Z$ unchanged. Now
we will show that the symplectic structure on $Z$ also remains
unchanged. In fact, this means that the whole invariant
Hamiltonian formalism does not feel such an addition. And so far
as quantization is based on invariant Hamiltonian formalism, we
get that the construction of the quantized field does not change
under the substitution~(\ref{LPlusDiv}).

It seems to be difficult to prove the invariability of the
symplectic structure using the formula for the symplectic current
as a starting point. In this connection we will remind here how
symplectic structure is defined directly in the variational terms
\cite{Witten1986,Crnkovic1987,Zucker1987,CrWitt1987,%
Crnkovic1988,BHS1991}.

Consider the action in an arbitrary region $\Omega$:
\begin{equation}
 S = \int\limits_\Omega d^4\!x\,
     L(x,\varphi_i,\partial_\mu\varphi_i,\dots,
       \partial_{\mu_1}\dots\partial_{\mu_k}\varphi_i) \ .
\label{SInOmega}
\end{equation}
Its variation, if the region is unchanged, is:
\begin{equation}
 \delta S = \int\limits_\Omega d^4\!x\,
 \left(\frac{\delta L}{\delta\varphi_i}\,\delta\varphi_i +
       \partial_\mu j_\mu\right)   \ ,
\label{VarSInOmega}
\end{equation}
Here
\[
 j_\mu =
 \frac{\delta L}{\delta(\partial_\mu\varphi_i)}\,\delta\varphi_i +
 \dots +
 \frac{\delta L}
      {\delta(\partial_\mu\partial_{\mu_2}\dots
              \partial_{\mu_k}\varphi_i)}\,
 \partial_{\mu_2}\dots\partial_{\mu_k}\delta\varphi_i   \ .
\]
Variational derivative here is understood as
\[
 \frac{\delta}{\delta a} =
 \frac{\partial}{\partial a} -
 \partial_\mu\frac{\partial}{\partial(\partial_\mu a)} +
 \partial_\mu\partial_\nu
    \frac{\partial}{\partial(\partial_\mu\partial_\nu a)} -
 \dots    \ .
\]
The action (\ref{SInOmega}) can be considered as a functional on
the set of functions $\varphi_i$. Using the identity $\delta^2=0$
we get:
\[
\int\limits_\Omega d^4\!x\,
 \left(\delta\left(\frac{\delta L}{\delta\varphi_i}\,
                   \delta\varphi_i\right) +
       \partial_\mu\delta j_\mu\right) = 0  \ .
\]
If we consider here variations only for functions $\varphi_i$ for
which action is stationary, then the first term in the last
equation becomes zero. From here we get that the symplectic
structure defined on $Z$ as
\[
 \omega = \int\limits_\Sigma d\sigma_\mu\,\delta j_\mu(x) \
\]
does not depend on the choice of the space-like surface $\Sigma$
(it is implied that $\Sigma$ behaves well enough at the infinity).

Let us consider now how the above equations are changed under the
substitution (\ref{LPlusDiv}).

Variation of the new action $S'$ can be written like in the
formula (\ref{VarSInOmega}):
\[
 \delta S' = \int\limits_\Omega d^4\!x\,
 \left(\frac{\delta L'}{\delta\varphi_i}\,\delta\varphi_i +
       \partial_\mu j'_\mu\right) =
 \int\limits_\Omega d^4\!x\,
 \left(\frac{\delta L}{\delta\varphi_i}\,\delta\varphi_i +
       \frac{\delta(\partial_\mu F_\mu)}{\delta\varphi_i}\,
              \delta\varphi_i +
       \partial_\mu j'_\mu\right)      \ .
\]
But $\delta(\partial_\mu F_\mu)/\delta\varphi_i = 0$. Therefore,
\begin{equation}
 \delta S' = \int\limits_\Omega d^4\!x\,
 \left(\frac{\delta L}{\delta\varphi_i}\,\delta\varphi_i +
       \partial_\mu j'_\mu\right)      \ .
\label{VarSDash1}
\end{equation}
On the other hand, this variation can be written in the terms of
the old action:
\begin{equation}
  \delta S' = \delta S +
     \int\limits_\Omega d^4\!x\,\partial_\mu\delta F_\mu    \ .
\label{VarSDash2}
\end{equation}
Combining (\ref{VarSDash1}) and (\ref{VarSDash2}) with
(\ref{VarSInOmega}) we get:
\[
 \int\limits_\Omega d^4\!x\,
  (\partial_\mu j'_\mu -
    \partial_\mu j_\mu - \partial_\mu\delta F_\mu) = 0 \ .
\]
So far as in this equation all three functions $j'_\mu$, $j_\mu$,
and $\delta F_\mu$ are local (i.~e. they become zero in regions
where $\delta\varphi_i=0$) we have:
\[
 \int\limits_\Sigma d\sigma_\mu\,
 (j'_\mu - j_\mu - \delta F_\mu) = 0 \ .
\]
The value $F_\mu$ can be considered as a functional on the set of
functions $\varphi_i$. Using the identity $\delta^2=0$ we get:
\[
 \int\limits_\Sigma d\sigma_\mu\,\delta j'_\mu(x) =
 \int\limits_\Sigma d\sigma_\mu\,\delta j_\mu(x)   \ .
\]
I.~e. the substitution (\ref{LPlusDiv}) leaves the symplectic
structure on $Z$ unchanged:
\[
 \omega' = \omega  \ .
\]

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\begin{thebibliography}{99}

\bibitem{Witten1986}
E.~Witten ``Interacting field theory of open superstrings'', Nucl.
Phys. {\bf B276}, 291-324 (1986).

\bibitem{Crnkovic1987}
\v{C}.~Crnkovi\'c ``Symplectic geometry and (super-) Poincar\'e
algebra in geometrical theories'', Nucl. Phys. {\bf B288}, 419-430
(1987).

\bibitem{Zucker1987}
G.~J.~Zuckerman ``Action principles and global geometry'', in {\it
Mathematical aspects of string theory}, ed. S.~T.~Yau, Singapore:
World Scientific, 259-284 (1987).

\bibitem{CrWitt1987}
\v{C}.~Crnkovi\'c, E.~Witten ``Covariant description of canonical
formalism in ge\-om\-e\-tri\-cal theories'', in {\it Three hundred
years of gravitation}, eds. S.~W.~Hawking, W.~Israel, Cambridge:
Cambridge univ. press, 676-684 (1987).

\bibitem{Crnkovic1988}
\v{C}.~Crnkovi\'c ``Symplectic geometry of the covariant
phase-space'', Class. Quant. Grav. {\bf 5}(12), 1557-1575 (1988).

\bibitem{BHS1991}
G.~Barnich, M.~Henneaux, C.~Schomblond ``Covariant description of
the ca\-non\-ic\-al formalism'', Phys. Rev. D {\bf 44}(4),
R939-R941 (1991).

\end{thebibliography}

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\end{document}
