\documentclass[12pt]{article} 

%%%%%%%%%Modifications since the submission to hep-th%%%%%%%%%%%%%%
%01/14: isotropic --> chracteristic 
 %new refs on preq schr eq. 
  %02/06 - def polysimplectic, p = -H on p.2!,  
   %03/23 - abstract: valued, covariant  
    % def 1, def 2 changed 
     %p2: such as, Thm2: potential in the sense of 2.3, Helein. 
      %note added in proofs on prequatnum schroedinger equation.      
       % >40 minor corrections in proofs. 

%\usepackage{romp33,cite,amssymb,latexsym} 
\usepackage{cite,amssymb,latexsym}

\newcounter{thMM}
\setcounter{thMM}{0}
\newcounter{leMM}
\setcounter{leMM}{0}
\newcounter{deFF}
\setcounter{deFF}{0}
\newcounter{exMP}
\setcounter{exMP}{0}
\newenvironment{theorem}[1]{\refstepcounter{thMM}\trivlist
   \item[\hskip19pt{\sc #1~\arabic{thMM}.}]\it\hskip3pt}{\endtrivlist}
\newenvironment{lemma}[1]{\refstepcounter{leMM}\trivlist
   \item[\hskip19pt{\sc #1~\arabic{leMM}.}]\it\hskip3pt}{\endtrivlist}
\newenvironment{definition}[1]{\refstepcounter{deFF}\trivlist
   \item[\hskip19pt{\sc #1~\arabic{deFF}.}]\rm\hskip3pt}{\endtrivlist}
\newenvironment{example}[1]{\refstepcounter{exMP}\trivlist
   \item[\hskip19pt{\sc #1~\arabic{exMP}.}]\rm\hskip3pt}{\endtrivlist}


\hfuzz=10pt
\textheight 235mm%216mm        
\textwidth 157mm %%125mm = Int JTP% 
%\renewcommand{\baselinestretch}{1.05} 
%%\renewcommand{\baselinestretch}{1.15} 
\oddsidemargin 3.6mm    % centered on DIN A4 paper
\evensidemargin 3.6mm  %% added for twoside 
\topmargin -11mm        % dto.


 %%% changes w.r.t. the submitted 2nd version.


\title{\normalsize\bf GEOMETRIC (PRE)QUANTIZATION IN THE POLYSYMPLECTIC 
APPROACH TO FIELD THEORY\thanks{
To appear in ``Differential Geometry and its Applications'' 
(Proc. Conf., Opava (Czech Republic), August 27-31, 2001)  
Silesian University, Opava 2002.}  
 %%% Submitted to the Proc. 
 %Based on the talk at the 8-th Int. Conf. 
%on Differential Geometry and its Applications (Opava, Czech Republic, 
%August 27-31, 2001). } 
%Geometric prequantization 
%in the polysymplectic approach to field theory 
} 

\author{
\normalsize 
 Igor V. Kanatchikov
 %%%\thanks{On leave from 
 %%%Tallinn Technical University, Tallinn, Estonia. 
 %%%E-mail: ivar@parsek.yf.ttu.ee. } 
\\
\normalsize 
II. Institut f\"ur Theoretische Physik 
der Universit\"at Hamburg \\
\normalsize      
Luruper Chaussee 149, 22761 Hamburg, Germany 
\\ 
\normalsize 
ivar@mail.desy.de\\[2ex] 
}

\date{\vspace*{-35pt}}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  Here the style of numbering and referencing is given.
%  Do not change!!!!!
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\catcode `\@=11
\@addtoreset{equation}{section}
 
\def\theequation{\arabic{section}.\arabic{equation}}
          % if you want equations to be numbered by section 
\def\section{\@startsection {section}{1}{\z@}{-3.5ex plus -1ex minus
     -.2ex}{2.3ex plus .2ex}{\normalsize\bf}}
\def\subsection{\@startsection{subsection}{2}{\z@}{-3.25ex plus -1ex minus
 -.2ex}{1.5ex plus .2ex}{\normalsize\bf}}
          % correct font size for section/subsection titles

\def\thebibliography#1{\section*{References\markboth
  {REFERENCES}{REFERENCES}}\list
  {[\arabic{enumi}]}{\settowidth\labelwidth{[#1]}\leftmargin\labelwidth
  \advance\leftmargin\labelsep
  \usecounter{enumi}}
  \def\newblock{\hskip .11em plus .33em minus -.07em}
  \sloppy
  \sfcode`\.=1000\relax}
 \let\endthebibliography=\endlist
                           % numbering of references as ``[3] Author''
 
\catcode `\@=12
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




\begin{document}

\maketitle

\begin{abstract}
The prequantization map for a Poisson-Gerstenhaber algebra 
of differential form 
  valued %% 
dynamical variables in the polysymplectic 
formulation of the De Donder--Weyl 
 covariant %%
Hamiltonian field theory is presented and the 
corresponding prequantum Schr\"odin\-ger equation is derived.  
This is the first step toward understanding the procedures of 
precanonical field quantization 
 %%put forward earlier 
from the point of view of geometric quantization. 
\medskip 

\noindent
{\bf\small Keywords:} 
{\small De Donder--Weyl formalism, polysymplectic structure, 
Poisson-Gerst\-en\-haber algebra, prequantization, 
geometric quantization, precanonical qu\-anti\-zation,   
 %superconnection, polysymplectic form, 
field quantization, prequantum Schr\"odinger equation. } 

\noindent 
{\bf\small MS classification. } {\small 53D50, 81S10, 81T70, 70S05.} 
%53D50 Geometric quantization
%70S05 Lagrangian formalism and Hamiltonian formalism
%%70Sxx  Classical field theories 
%81S10 Geometry and quantization, symplectic methods
%81T70 Quantization in field theory; cohomological methods 

\noindent 
{\bf\small PACS. } {\small 03.70, 11.10, 02.40.Yy, 03.65.Ca.}
%03.70.+k Theory of quantized fields
%11.10.-z Field theory
%11.10.Ef Lagrangian and Hamiltonian approach 
%02.10.Xm Multilinear algebra 
%02.40.Yy Geometric mechanics
%45.20.Jj Lagrangian and Hamiltonian mechanics  
%03.65.Ca   Formalism (... of Quantum mechanics) 
\end{abstract}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%               %%%%%%%%%%%% File ncom.tex %%%%%%%%%%%%%%%              %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\newcommand{\beq}{\begin{equation}}
\newcommand{\eeq}{\end{equation}}
\newcommand{\beqa}{\begin{eqnarray}}
\newcommand{\eeqa}{\end{eqnarray}}
\newcommand{\nn}{\nonumber}

\newcommand{\half}{\frac{1}{2}}

\newcommand{\xt}{\tilde{X}}

\newcommand{\uind}[2]{^{#1_1 \, ... \, #1_{#2}} }
\newcommand{\lind}[2]{_{#1_1 \, ... \, #1_{#2}} }
\newcommand{\com}[2]{[#1,#2]_{-}} 
\newcommand{\acom}[2]{[#1,#2]_{+}} 
\newcommand{\compm}[2]{[#1,#2]_{\pm}}

\newcommand{\lie}[1]{\pounds_{#1}}
\newcommand{\co}{\circ}
\newcommand{\sgn}[1]{(-1)^{#1}}
\newcommand{\lbr}[2]{ [ \hspace*{-1.5pt} [ #1 , #2 ] \hspace*{-1.5pt} ] }
\newcommand{\lbrpm}[2]{ [ \hspace*{-1.5pt} [ #1 , #2 ] \hspace*{-1.5pt}
 ]_{\pm} }
\newcommand{\lbrp}[2]{ [ \hspace*{-1.5pt} [ #1 , #2 ] \hspace*{-1.5pt} ]_+ }
\newcommand{\lbrm}[2]{ [ \hspace*{-1.5pt} [ #1 , #2 ] \hspace*{-1.5pt} ]_- }

 %%%%%%%%%adapted to 11pt%%%%%%%%% 
\newcommand{\pbr}[2]{ \{ \hspace*{-2.2pt} [ #1 , #2\hspace*{1.5 pt} ] 
\hspace*{-2.3pt} \} }
%\newcommand{\nbr}[2]{ [ \hspace*{-2.0pt} [ #1 , #2\hspace*{1.4 pt} ] 
%\hspace*{-2.0pt} ] }
\newcommand{\nbr}[2]{ [ \hspace*{-1.5pt} [ #1 , #2 \hspace*{0.0pt} ] 
\hspace*{-1.5pt} ] }

\newcommand{\we}{\wedge}
\newcommand{\dv}{d^V}
\newcommand{\nbrpq}[2]{\nbr{\xxi{#1}{1}}{\xxi{#2}{2}}}
\newcommand{\lieni}[2]{$\pounds$${}_{\stackrel{#1}{X}_{#2}}$  }
%\newcommand{\dgh}{D_{G\wedge H}}

\newcommand{\rbox}[2]{\raisebox{#1}{#2}}
\newcommand{\xx}[1]{\raisebox{1pt}{$\stackrel{#1}{X}$}}
\newcommand{\xxi}[2]{\raisebox{1pt}{$\stackrel{#1}{X}$$_{#2}$}}
\newcommand{\ff}[1]{\raisebox{1pt}{$\stackrel{#1}{F}$}}
\newcommand{\dd}[1]{\raisebox{1pt}{$\stackrel{#1}{D}$}}
 %\newcommand{\nbr}[2]{{\bf[}#1 , #2{\bf ]}}
\newcommand{\der}{\partial}
\newcommand{\oo}{$\Omega$}
\newcommand{\Om}{\Omega}
\newcommand{\om}{\omega}
\newcommand{\eps}{\epsilon}
\newcommand{\si}{\sigma}
\newcommand{\Lm}{\bigwedge^*}

\newcommand{\inn}{\hspace*{2pt}\raisebox{-1pt}{\rule{6pt}{.5pt}\hspace*
{0pt}\rule{.5pt}{8pt}\hspace*{4pt}}}
\newcommand{\sro}{Schr\"{o}dinger\ }
\newcommand{\bm}{\boldmath}
\newcommand{\vol}{\omega}%%{\widetilde{vol}}                              
               \newcommand{\dvol}[1]{\der_{#1}\inn \vol}

\newcommand{\bd}{\mbox{\bf d}}
\newcommand{\bder}{\mbox{\bm $\der$}}
\newcommand{\bI}{\mbox{\bm $I$}}


%\renewcommand{\mu}{i}
%\renewcommand{\nu}{j}
\newcommand{\be}{\beta} 
\newcommand{\ga}{\gamma} 
\newcommand{\de}{\delta} 
%\newcommand{\si}{\sigma} 
\newcommand{\Ga}{\Gamma} 
\newcommand{\gmu}{\gamma^\mu}
\newcommand{\gnu}{\gamma^\nu}
\newcommand{\ka}{\kappa}
\newcommand{\hka}{\hbar \kappa}
\newcommand{\al}{\alpha}
\newcommand{\lapl}{\bigtriangleup}
\newcommand{\psib}{\overline{\psi}}
\newcommand{\Psib}{\overline{\Psi}}
\newcommand{\derts}{\stackrel{\leftrightarrow}{\der}}
\newcommand{\what}[1]{\widehat{#1}}

\newcommand{\bx}{{\bf x}}
\newcommand{\bk}{{\bf k}}
\newcommand{\bq}{{\bf q}}

\newcommand{\omk}{\omega_{\bf k}} 
\newcommand{\lpl}{\ell}
\newcommand{\zb}{\overline{z}} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\normalsize 
 
\section{Introduction}

The idea of quantization of fields based on a manifestly 
covariant version of the Hamiltonian formalism in field theory 
known 
in the calculus of variation of multiple integrals 
 %\cite{volterra,dedonder,weyl,caratheodory} 
%\cite{dedonder,weyl,rund,dedecker} 
\cite{dedonder,dedecker} 
has been proposed 
for several times throughout the last century dating back to 
M. Born and H. Weyl \cite{bw}. 
 %good,dedecker,marx,garcia,hermann,kijowski,
 %gunther,gotay,sardanashvily,navarro
The mathematical study of 
 %the %??  
geometrical structures underlying the related aspects of the 
calculus of variations and classical field theory 
has  been undertaken recently by several groups of authors 
%\cite{sternberg,garcia,binz,gimm,carinena,deleon,roman-roy,sardan }, 
\cite{gimm,norris,deleon,roemer,paufler1,roman,sardan,helein} 
including Demeter Krupka's group \cite{krupka} 
 %%%here 
in the Czech Republic. 
One of the central issues for the purposes 
of quantization of fields is a proper definition of Poisson brackets 
within the covariant Hamiltonian formalism in field theory. 
This has been accomplished  in our earlier papers
\cite{romp98,bial96,goslar96} 
which are based on the notion of the {\em polysymplectic} form as 
a field theoretic analogue of the symplectic form in mechanics 
and present a construction of Poisson brackets 
 %defined on 
of   differential forms 
 %and 
 leading to a Poisson-Gerstenhaber algebra 
structure generalizing   a Poisson algebra in mechanics. 
The corresponding {\em precanonical\/} quantization of 
field theories was developed heuristically in 
\cite{qs96,bial97,lodz98}, its relation to the standard 
quantum field theory was considered in \cite{pla2001}, 
and a possible  application to quantum gravity 
was discussed in \cite{ijtp2001}. 
In this paper we present 
 %the %%% 
elements of geometric prequantization 
in field theory  
based on the abovementioned Poisson-Gerstenhaber brackets 
and derive the  corresponding prequantum analogue 
of the Schr\"odinger equation. 
The main purpose of our consideration is to pave a way to a better 
understanding of the procedures of precanonical field quantization 
 from the point of view of the principles of geometric quantization 
 %\cite{sniat,woodh,weinst,rom,tuyn}.  
\cite{sniat}. 

%as an intermediate  step towards the genuine covariant Schr\"odinger 
%equation of the precanonical field quantization approach. 



\section{Polysymplectic structure and the Poisson-Gerstenhaber brackets} 

Let us briefly describe the polysymplectic structure  
\cite{romp98,goslar96}
which underlies the 
De Donder--Weyl (DW) Hamiltonian form of the field equations 
 %\cite{dedonder,weyl,rund} 
\cite{dedonder} 
\beq
\der_\mu y^a (x) = {\der H}/{\der p^\mu_a},  
\quad 
\der_\mu p^\mu_a (x) = - {\der H}/{\der y^a} ,  
\eeq 
where $p^\mu_a := \frac{\der L}{\der y^a_\mu}$, 
called {\em polymomenta\/}, 
and $H:= y^a_\mu p^\mu_a - L = H(y^a, p^\mu_a, x^\mu)$, 
called the {\em DW Hamiltonian function\/},  
are determined by the first order Lagrangian density 
$L = L(y^a, y^a_\mu, x^\nu)$.  These equations are 
known to be equivalent to the Euler-Lagrange field equations if 
$L$ is regular in the sense that 
$$\det \left ( \left |\left|  \frac{\der^2 L}{\der y^a_\mu \der y^b_\nu}
\right |\right | \right ) \neq 0.$$ 

Let us view classical fields $y^a= y^a(x)$ as sections in the 
{\em covariant configuration bundle} $Y\rightarrow X$ 
over an oriented $n$-dimensional space-time manifold $X$ with the volume form 
$\omega$. The local coordinates in 
 %$Y\rightarrow X$
 $Y$ are $(y^a,x^\mu)$. 
Let $\bigwedge{}^p_q(Y)$ denotes the space of 
$p$-forms on $Y$ which are annihilated by $(q+1)$ arbitrary vertical 
vectors of $Y$. 




The space $\bigwedge^n_1(Y)\rightarrow Y$,    
which generalizes the cotangent bundle,    
is a model of 
 %what is often called the 
a %??the   
{\em multisymplectic phase space\/} \cite{gimm} 
 %and 
which possesses the canonical structure 
\beq 
\Theta_{MS} = p_a^\mu dy^a \we \omega_\mu + p\, \omega , 
\eeq
where 
 %$\omega$ is the volume form on $X$ and 
$\omega_\mu := \der_\mu\inn\omega$ 
are the basis of $\bigwedge^{n-1} T^*X$.  
The section $p= - H(y^a,p^\mu_a,x^\nu)$ gives the 
multidimensional 
{\em Hamiltonian Poincar\'e-Cartan form\/}  $\Theta_{PC}$.  
 
For the purpose of introducing the Poisson brackets 
which reflect the dynamical structure of DW Hamiltonian 
equations (1) we need a structure which is independent 
of $p$ or a choice of $H$: 

\begin{definition}{Definition} \label{def1} 
The {\em extended polymomentum phase space\/} is the quotient bundle 
$Z$: $\bigwedge^{n}_1(Y) / \bigwedge^{n}_0(Y) \!\rightarrow \! Y.  
%\!\rightarrow \!X
$ 
\end{definition} 
The local coordinates on $Z$ are $(y^a,p_a^\nu,x^\nu)$. %=: (z^v, x^\mu)$. 
A canonical structure on $Z$ 
 can be understood as 
an equivalence class 
of forms 
$\Theta := [p_a^\mu dy^a\we \omega_\mu \quad {\rm mod} \bigwedge{}^{n}_0(Y)]$. 

\begin{definition}{Definition} \label{def2}  
The {\em polysymplectic structure\/} on $Z$ is an equivalence class 
 %$\Omega = [d\Theta \quad mod \bigwedge{}_1^{n+1}(Y)]$.  
 %\end{definition} 
 %
 %A suitable local coordinate expression of the polysymplectic form 
of 
  %the %?? - printed version 
forms $\Omega$ given by 
%\beq
%\Omega := [d\Theta \quad {\rm mod}\left . \bigwedge\right .^{n+1}_1(Y)] 
%= [- dy^a\we dp^\mu_a \we \omega_\mu \quad {\rm mod} 
%\left . \bigwedge\right .^{n+1}_1(Y)] . 
%\eeq 
\beq
\Omega := [d\Theta \quad {\rm mod} \;\mbox{$\bigwedge^{n+1}_1(Y)$}] 
= [- dy^a\we dp^\mu_a \we \omega_\mu \quad {\rm mod}  
\;\mbox{$\bigwedge^{n+1}_1(Y)$}] . 
\eeq 
\end{definition} 





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\texxx}{ %submitted text and hep-th%%%%%%%%%%% 

\begin{definition}{Definition} \label{def1} 
The %composed 
quotient 
bundle 
$Z$: $\bigwedge^{n}_1(Y) / \bigwedge^{n}_0(Y) \!\rightarrow \! Y 
%\!\rightarrow \!X
$ 
 %with the local coordinates $(y^a,p_a^\nu,x^\nu)$ 
is called the {\em extended polymomentum phase space\/}. 
\end{definition} 
 %
The local coordinates on $Z$ are $(y^a,p_a^\nu,x^\nu) =: (z^v, x^\mu)$. 
 
\begin{definition}{Definition} \label{def2}  
The {\em polysymplectic structure\/} on $Z$ is an equivalence class 
of closed non-degenerate forms $\Omega\in \bigwedge^{n+1}_2(Z)$ 
modulo $\bigwedge^{n+1}_1(Z)$.  
\end{definition} 
%Here the non-degeneracy is meant in the standard sense that 
%no non-vanishing vector field annihilates $\Omega$. 
A suitable local coordinate expression of the polysymplectic form 
$\Omega$ is given by 
\beq
\Omega = - dy^a\we dp^\mu_a \we \omega_\mu 
\eeq 
which is understood as a representative in the 
 %equivalence class 
quotient 
$\bigwedge^{n+1}_2(Z)/\bigwedge^{n+1}_1(Z)$. 
  } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 


The equivalence classes are introduced as an alternative to 
the explicit introduction of a non-canonical connection 
on the multisymplectic phase space in order to define 
the polysymplectic structure as a ``vertical part'' of the 
multisymplectic form $d\Theta_{MS}$\cite{paufler1}. 
The fundamental constructions, 
such as 
the Poisson bracket below,  are 
required to be independent of the choice of representatives in the 
equivalence classes, as they are expected to be independent of the 
choice of a connection. 


\begin{definition}{Definition} \label{def3}
A multivector field of degree $p$, $\xx{p}\in \bigwedge^p TZ$,  
is called {\em vertical\/} 
if $\xx{p}\inn F  = 0$ for any form $ F \in \bigwedge^{*}_0(Z)$. 
\end{definition} 

The polysymplectic form establishes a map of 
horizontal forms   of degree $p$, 
$\ff{p}$$\in$$\bigwedge^p_0(Z)$, $p=0,1,..., (n-1)$,
to vertical multivector fields 
of degree $(n-p)$, $\xx{n-p}{}_F$,  called {\em Hamiltonian\/}: 
\beq
\xx{n-p}{}_F\inn \Omega = d \ff{p}.  
\eeq 
More precisely, 
horizontal forms forms are mapped to the {\em equivalence 
classes\/} of Hamiltonian multivector fields 
modulo the {\em characteristic\/}   
multivector fields $\xx{p}_0$: $\xx{p}_0\inn\Omega = 0$, $p=2,...,n$. 
The forms for which the map (4) exists are also called {\em Hamiltonian\/}. 
It is easy to see that the space of Hamiltonian forms is not stable 
with respect to the exterior product of forms. However, 

\begin{lemma}{Lemma} \label{lem1} 
The space of Hamiltonian forms is closed with respect to the 
graded commutative, associative {\em co-exterior\/} product 
 \beq
\ff{p}\bullet \ff{q} := *^{-1}(*\ff{p}\we *\ff{q}) .  
\eeq 
\end{lemma} 

{\sc Proof: } A straightforward proof  is to solve (4) to see that 
Hamiltonian $p$-forms are restricted to specific  $(n-p)$-polylinear 
forms in $p^\mu_a$, and then to check that the $\bullet-$product 
preserves the space of these forms (see \cite{bial96,paufler1}). 
%Another way 
%is to observe that the map (4) can be preserved if both sides 
%are contracted with multivectors $\alpha \in \bigwedge^* TX$. 
% .......... 
{\hfill $\Box$}  

\medskip 

Note that  
 the definition of the  $\bullet$-product requires only 
 %%%the  $\bullet$-product requires only 
the volume form $\omega$ on the space-time, not the metric structure. 
Given $\omega$ a $p$-form $F \!\in\! \bigwedge^p T^*X$ can be mapped to 
an $(n-p)-$multivector $X_F\!\in\! \bigwedge^{n-p} TX$: 
$X_F\inn \omega=F$. Then the 
exterior product of multivectors $(\wedge)$ induces   the $\bullet$-product 
of forms in $\bigwedge^* T^*X$. 
   %which can be lifted to forms in $\bigwedge^*_0(Z)$. 
The construction 
   %,  which can be lifted to forms in $\bigwedge^*_0(Z)$, 
is given by 
the %??
commutative diagram 
$$
\begin{array}{ccc}
& \quad \bullet \quad & \vspace*{-6pt }\\
\mbox{$\bigwedge^p$} T^*X \otimes  \mbox{$\bigwedge^q$}T^*X& 
$\rightarrowfill$ & \mbox{$\bigwedge^{p+q-n}$} T^*X \vspace*{+5pt }
\\
\!\!\!\!\! \omega\ \Big\downarrow  & &\Big\downarrow \ \omega 
\\
& \quad \wedge \quad \vspace*{-5pt }&
\\
\mbox{$\bigwedge^{n-p}$}TX \otimes\mbox{$\bigwedge^{n-q}$}TX  
& $\rightarrowfill$ & \mbox{$\bigwedge^{n-p+n-q}$} TX
\end{array}
$$                                                            
and can be lifted to forms in $\bigwedge^*_0(Z)$. 



The Poisson bracket of Hamiltonian forms
 $\pbr{\;}{\,}$  is induced by the 
Schouten-Nijenhuis bracket $\nbr{\;}{\,}$ 
of the corresponding 
Hamiltonian multivector fields: 
\beq
%\nbr{\xx{n-p}}{\xx{n-q}} \inn \Omega 
%= : 
- d \pbr{\ff{p}}{\ff{q}} := 
\nbr{\xx{n-p}}{\xx{n-q}} \inn \Omega . 
\eeq
As a consequence, 
\beq
\pbr{\ff{p}{}_1}{\ff{q}{}_2} = (-1)^{(n-p)} \xx{n-p}{}_1 \inn d \ff{q}{}_2 
 %= (-1)^{(n-p)} \pounds_{\small \xx{n-p}{}_1} (\ff{q}{}_2) 
= (-1)^{(n-p)} \xx{n-p}{}_1 \inn \xx{n-q}{}_2 \inn \Omega , 
\eeq 
whence the independence of the definition of the choice of 
representatives  in the equivalence classes of $X_F$ and $\Omega$ 
is obvious. The algebraic properties of the bracket 
are given by the following 

\begin{theorem}{Theorem} \label{the1} 
The space of Hamiltonian forms with the 
operations $\pbr{\;}{\,}$ and $\bullet$  
is  a { (Poisson-)Gerstenhaber algebra\/}, i.e. 
\beqa
\pbr{\ff{p} }{\ff{q} } &=& -(-1)^{g_1 g_2}
\pbr{\ff{q}}{\ff{p}}, \nn \\ 
\mbox{$(-1)^{g_1 g_3} \pbr{\ff{p}}{\pbr{\ff{q}}{\ff{r}}}$} 
&\!+\!& 
\mbox{$(-1)^{g_1 g_2} \pbr{\ff{q}}{\pbr{\ff{r}}{\ff{p}}}$} 
  \\
 %&&\; + \quad \!
&& \hspace*{15pt}+ \quad \! 
\mbox{$(-1)^{g_2 g_3} \pbr{\ff{r}}{\pbr{\ff{p}}{\ff{q}}} 
 %\quad \!=  \quad \!0,$}  \\ 
 \; = \;0, $}  
 \nn \\
\pbr{\ff{p}}{\ff{q}\bullet \ff{r}} 
&=& 
\pbr{\ff{p}}{\ff{q}}\bullet \ff{r}
+ (-1)^{g_1(g_2+1)} \ff{q}\bullet\pbr{\ff{p}}{\ff{r}},   
\nn 
\eeqa 
where $g_1 = n-p-1$,  $g_2 = n-q-1$,  $g_3 = n-r-1$.  
\end{theorem} 

{\sc Proof: } The graded Lie algebra properties are a 
straightforward consequence of (2.6) and the graded Lie nature of the 
Schouten-Nijenhuis bracket. The graded Leibniz property can be 
seen as a consequence of the Fr\"olicher-Nijenhuis theorem \cite{fn}
%on classification of graded derivations 
 %on the exterior algebra 
adapted to the algebra of forms equipped with the co-exterior 
product. 
{\hfill $\Box$}  



\section{Prequantization map} 

Having in our disposal a generalization of the symplectic structure and 
a Poisson algebra to 
 %field theory within 
the DW Hamiltonian formalism  
of field theory        
it is natural to ask if geometric quantization 
can be generalized to this framework. The first step in this 
direction would be a generalization of the {\em prequantization map\/}  
%\cite{sniat,woodh,weinst,rom,tuyn} 
 \cite{sniat}
$F \rightarrow O_F$ 
which maps dynamical variables $F$ on the classical phase 
space to the first order (prequantum) operators $O_F$ 
on (prequantum) Hilbert space and fulfills three properties: 
  \begin{quote}
(Q1) the map $F\rightarrow O_F$ is linear; \\
(Q2) if $F$ is constant, then $O_F$ is the corresponding multiplication 
operator; \\
(Q3) the Poisson bracket of dynamical variables is 
related to the commutator of the corresponding operators 
as follows: 
\end{quote} 
\beq 
[O_{F_1}, O_{F_2}]=-i\hbar O_{\{F_1, F_2\}}.   
\eeq  
In the case of a Poisson-Gerstenhaber algebra we expect that the commutator 
(1) is replaced by the {\em graded\/} commutator 
$[A,B] := A\co B - (-1)^{\deg A \deg B} B\co A.$ 

\begin{theorem}{Theorem} \label{TH3}
The prequantum operator of a differential form 
dynamical variable $F$ 
is given by the formula 
\beq
O_{F}= i\hbar \pounds_{X_F} 
+ X_F{} \inn \Theta \bullet + F\bullet ,  
\eeq 
where $\pounds_{X_F} := [X_F, d]$  and 
$\Theta$ is a 
(local) polysymplectic potential 
 in the sense of (2.3). 
 %: $\Omega=: [d\Theta]$.  
 %and   $\pounds_{X_F} := [X_F, d]$.   
 \end{theorem}

{\sc Proof: } See a straightforward calculation in 
\cite{torun2001}. {\hfill $\Box$}  \\
 
The most intriguing aspect of the representation (2) is that the 
prequantum operator $O_F$ is non-homogeneous: 
for an $f$-form $F$ the degree of the first term in (2) 
is  $(n-f-1)$ and the degree of the other two terms is $(n-f)$. 
This 
 fact 
suggests that  prequantum wave functions are 
 complex 
non-homogeneous horizontal differential forms, 
 %which are 
i.e. 
 %the %%?? 
sections of the complexified bundle 
\mbox{$\bigwedge^*_0(Z)^{\mathbb C} \rightarrow Z$}. 
The corresponding (graded) prequantum Hilbert space 
 will be considered in \cite{torun2001} (see also \cite{mg9a}).  

Note that formulas (1) and (2) imply  that one can introduce a formal 
non-homogeneous ``supercovariant derivative'' with respect to a 
multivector field 
$X$: $\nabla_X:= \pounds_X -\frac{i}{\hbar} X\inn\Theta\bullet$ 
with the curvature of the corresponding ``superconnection'' $\nabla$ 
(cf. \cite{quillen})  
\beq
\Omega(X_1, X_2) := -i\hbar \left ( 
[\nabla_{X_1}, \nabla_{X_2}] - \nabla_{\nbr{X_1}{X_2}} \right ) 
\eeq 
coinciding with the polysymplectic form. 

One of the important questions is what is the dynamical equation 
for the wave functions. Let us consider how geometric 
prequantization can help us to find an answer. 


\section{Prequantum Schr\"odinger equation} 
 
The origin of the Schr\"odinger equation in quantum mechanics 
from the point of view of geometric (pre)quantization can be understood 
as follows. The classical equations of motions are incorporated in the 
vector field $X_*$ which annihilates the exterior differential of the 
(Hamiltonian) Poincare-Cartan form 
\beq
\Theta = p dq - H(p,q) dt , 
\eeq 
i.e. 
\beq
X_*\inn d\Theta =0. 
\eeq
The classical trajectories in the phase space are known to be 
the integral curves of $X_*$. 

Now, if we think of geometric prequantization based on the presymplectic 
structure given by $d\Theta$ we notice 
 %%the non-trivial aspect of it 
that the zero ``observable'' has a non-trivial 
(presymplectic) prequantum operator: 
\beq 
 0 \rightarrow O_0 = i\hbar \pounds_{X_*} + X_*\inn \Theta , 
\eeq 
where 
$ %\beq
X_*=X^t\der_t + X^q\der_q + X^p\der_p  
$ %\eeq
   %with 
and 
\beq
X^q = \der_p H, \quad X^p = -\der_q H , 
\eeq 
as it follows from (2) under the assumption $X^t=1$ (which is just a 
choice of time parametrization). The obvious consistency requirement 
then is that $O_0$ vanishes on prequantum wave functions 
$\Psi=\Psi (p,q,t)$, i.e. 
\beq   
%O_0 |\psi \left . \right > = 0 
O_0 (\Psi) = 0 . 
\eeq 
Using the explicit form of the operator $O_0$ derived from (3), (4): 
\beq
O_0=  i\hbar \der_t + i\hbar (\der_p H \der_q - \der_q H \der_p) 
+ p\der_p H - H(p,q) 
\eeq 
one can write (5) in the form of the 
{\em prequantum Schr\"odinger equation\/}
\beq
i\hbar \der_t \Psi = O_H \Psi , 
\eeq 
where $O_H$ is the (symplectic) prequantum operator of 
the Hamilton canonical function:  
\beq
O_H = - i\hbar (\der_p H \der_q - \der_q H \der_p) 
 - p\der_p H + H(p,q)  . 
\eeq 
 %Using (3)-(5) the explicit form of this condition takes the form: 
 %\beq
 %%\left ( i\hbar \der_t + i\hbar (\der_p H \der_q - \der_q H \der_p)) 
 %%+ p\der_p H - H(p,q) \right ) \Psi (p,q,t) = 0 . 
 % i\hbar \der_t \Psi = \left \{ 
 %- i\hbar (\der_p H \der_q - \der_q H \der_p)) 
 %- p\der_p H + H(p,q)  \right \} \Psi , 
 %\eeq
 %where in the right hand side of (7) 
 % one recognizes the (symplectic)  
 %prequantum operator of the Hamilton canonical function: $O_H$. 

The above consideration demonstrates 
the origin of the Schr\"odinger equation in 
 %the geometric formalism of Hamiltonian mechanics, specifically in 
the classical relation (2) . The subsequent 
steps of quantization just reduce the Hilbert space of the wave 
functions (by choosing a {\em polarization}) 
and construct a proper operator of $H$ on this Hilbert space, 
%with 
the form of the Schr\"odinger equation (7) remaining intact.   
This observation motivates our consideration of the field 
theoretic prequantum Schr\"odinger equation in the following 
section: having obtained it on the  level of prequantization 
one may have a better idea as to what is the covariant 
Schr\"odinger equation for quantum fields within the approach 
based on DW Hamiltonian formulation (2.1). 

 %As far as I know, 
%Let us remark that 
There has been a little discussion of the 
prequantum Schr\"o\-dinger equation in the literature  
(cf. \cite{preschr})  
 %The main reason is that
for the reason that it works on a wrong 
Hilbert space of functions over the phase space, thus 
contradicting the uncertainty principle. 
 %%% 
It can serve, therefore, only as an intermediate step toward 
the true quantum mechanical Schr\"odinger equation. 

Let us note that 
eqs. (7), (8) recently 
 %were derived 
appeared within the hypothetical 
framework  of ``subquantum mechanics'' proposed by J. Sou\v cek 
\cite{soucek} 
 %deterministic QM  
whose starting point was quite different from geometric quantization.  
 %employed completely independent arguments based on the 
 %path integral to derive them.   
 %%%It might be of interest, therefore,  to explore a possible 
A possible 
connection between the  ``subquantum mechanics''  and geometric 
prequantization could be an interesting subject to study, 
particularly in connection with the question recently 
revisited by G. Tuynman \cite{tuynman2} as to ``were 
there is the border between classical and quantum mechanics in geometric 
quantization?''  




\section{Prequantum Schr\"odinger equation in field theory} 

In this section we present a field theoretic generalization of the 
 %arguments of the previous section.  
above derivation of the prequantum Schr\"odinger equation. 

It is known \cite{roemer,roman,romp98}  
that the classical field equations 
 in the form (2.1) 
are encoded in the multivector field 
of degree $n$, $\xx{n}{}_* \in \bigwedge^n TZ$, 
which annihilates the exterior differential 
of the multidimensional Hamiltonian Poincare-Cartan $n$-form 
\beq
\Theta_{PC} = p^\mu_a dy^a \we \omega_\mu - H(y^a, p_a^\mu) \omega , 
\eeq
i.e. 
\beq
\xx{n}{}_* \inn d\Theta_{PC}= 0.  
\eeq
 %%d\Theta - multisymplectic form 

%\beq
%\xx{n}{}_* = \xx{n}{}^h_* + \xx{n}{}^V_* 
 %+ \xx{n}{}^{VV}_* + ... , 
%\eeq 
%with the notation introduced: 
%\beqa 
%\xx{n}{}^h_* &:=& X\uind{\mu}{n} \der\lind{\mu}{n} , \nn \\
% \xx{n}{}^V_* &:=&X^v{}\uind{\mu}{n-1} \der_v{}\lind{\mu}{n} , \nn \\
% \xx{n}{}^{VV}_*  &:=& 
% X^{v_1 v_2}{}\uind{\mu}{n-1} \der_{v_1 v_2}{}\lind{\mu}{n}, 
%\quad \mbox{\rm  etc. } \nn 
%\eeqa 

Let us extend the geometric prequantization map (3.2) to the 
case of the ``pre-polysymplectic'' form $d\Theta_{PC}$,  
 usually called {\em multisymplectic\/}. 
Again, the feature of this extension is that there is a 
non-trivial prequantum operator corresponding to the 
zero function on the polymomentum phase space:  
 %which is identically zero: 
\beq 
O_0 = i\hbar [\xx{n}{}_*, d] + \xx{n}{}_* \inn \Theta_{PC}\bullet. 
\eeq 
Therefore, the consistency requires the prequantum wave 
function $\Psi = \Psi(y^a, p^\mu_a,x^\mu)$ to obey the condition 
\beq 
O_0 (\Psi) =0 
\eeq 
which is expected to yield the field theoretic 
prequantum Schr\"odinger equation. 

   
It is easy to see that the operator (3) is non-homogeneous: the first 
term has the degree $-(n-1)$ while the last one has the degree $-n$. 
Therefore, the prequantum wave function in (4) is a 
 horizontal 
non-homogeneous form 
 %$\Psi \in \bigwedge^{n-1}_0(Z) \oplus \bigwedge^{n}_0(Z)$: 
$$
\Psi = \psi \omega + \psi^\nu\omega_\nu,  
$$ 
a section of the bundle  
$ \left ( \bigwedge^{n-1}_0(Z) \oplus \bigwedge^{n}_0(Z) 
\right ){}^{\mathbb C}   \rightarrow Z$ which generalizes the complex line bundle 
over the symplectic phase space used in the usual geometric quantization. 


In terms of the Hamiltonian vector field associated with $H$: 
$$  
\xx{n}{}_H \inn d\Theta = dH, 
$$ 
where $\Theta$ is a potential of the polysymplectic form $\Omega$,
the vertical part of $\xx{n}{}_*$ takes the form: 
$$  
\xx{n}{}_*^V= (-1)^n (\xx{n}{}_*\inn \omega) \xx{n}_H. 
$$
Then (5.4) yields the prequantum Schr\"odinger equation in 
the form: 
\beq
i\hbar (\der_\mu \psi^\mu - 
(-1)^n \der_\mu \psi \, dx^\mu)  
=  - (-1)^{n} \left ( i\hbar \xx{n}{}_H \inn d \Psi 
+ \xx{n}{}_H \inn \Theta  \bullet \Psi \right )    
+ H\bullet \Psi.  
\eeq
   
\newcommand{\oldtextl}{
Assuming $\xx{n}_*\inn \omega = const$ and introducing the 
vertical Hamiltonian multivector field associated with $H$: 
$$  
\xx{n}{}_H \inn d\Theta = dH, 
$$ 
where $\Theta$ is a potential of the polysymplectic form $\Omega$, 
the prequantum Schr\"odinger equation (4) takes the form: 
\beq
i\hbar (\der_\mu \psi^\mu - 
(-1)^n \der_\mu \psi \, dx^\mu)  
=  - (-1)^{n} i\hbar \xx{n}{}_H \inn d \Psi 
+ %(-1)^{n-1} 
(-1)^{n} \xx{n}{}_H \inn \Theta  \bullet \Psi   
+ H\bullet \Psi.  
\eeq
 }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   


For {\em odd\/} $n$ the right hand side of (5) is identified with 
 %$$ O_H = i\hbar\pounds{\small {}_{\small \xx{n}{}_H}} + 
 %\xx{n}{}_H \inn \Theta \bullet + H\bullet, $$ 
the  (polysymplectic) prequantum operator of $H$ (see (3.2)),   
and (5) takes a particularly appealing form (cf. \cite{bial94}) 
\beq 
i\sigma \hbar \, d\bullet \Psi = O_H (\Psi) , 
\eeq 
where $\sigma = \pm 1$ for Euclidian/Lore\-ntzian spacetimes 
(in our conventions $*\omega=\sigma$), 
and $d\bullet$ is the {\em co-exterior\/} differential \cite{bial97} 
which is non-vanishing only on the subspace of $(n-1)$- and $n$-forms: 
 %e.g. 
$$d\bullet (\psi^\nu \omega_\nu) = 
\der_\mu \psi^\nu dx^\mu\bullet \omega_\nu = \sigma \der_\nu \psi^\nu, 
\quad 
d\bullet (\psi \omega) = \sigma \der_\mu \psi dx^\mu.$$ 

For {\em even\/} $n$ the right hand side of (5) 
 %cannot be written as 
is not $O_H$ because of the wrong sign 
 %$(-1)^n$ 
in front of the first two terms. The left hand side 
is also different from the one in (6). 
 %which is likely to be an artefact of the present 
 %formulation of geometric prequantization.  
%A distinction between even and odd dimensions 
%is likely to be an artefact of prequantization formula 
%which should disappear  at least 
%on the level of quantization.\footnote{See Note added in proofs.} 
 A distinction between even and odd space-time dimensions is a problematic 
 feature of the present derivation 
 %which, however, can be avoided 
 %by a proper modification of the prequantization 
 %map.\footnote{See note added in proofs.} 
based on a specific prequantization 
formula (3.2).\footnote{See Note added in proofs.} 

The meaning of our discussion in this section is that it provides 
a hint to the actual form of the covariant Schr\"odinger 
equation 
in field theory 
which one can expect within 
 the approach to 
field quantization 
based on 
 %De Donder-Weyl 
the covariant DW Hamiltonian formalism.  


%In components ... 




\newcommand{\oldtextf}{
Assuming for simplicity that 
\beq
X\uind{\mu}{n} \der\lind{\mu}{n} \inn \omega = 1 
\eeq

we obtain: 
\beqa 
[\xx{n}{}^h_*, d] \Psi &=& \der_\mu \psi^\mu - 
(-1)^n \der_\mu \psi \, dx^\mu, 
\nn \\ 
{}[ \xx{n}{}^V_*, d ] \Psi &=& (-1)^{n-1} \xx{n}{}^v{}\uind{\mu}{n-1} 
\der\lind{\mu}{n-1} \inn (\der_v \psi \,\omega + \der_v \psi^\mu \,\omega_\mu) , 
\nn \\ 
{}[\xx{n}{}^{VV}_*, d] \Psi &=& 0,  
\quad \mbox{\rm  etc. } 
\nn 
%{}\xx{n}{}^{}_* \inn \Theta_{PC} = {}\xx{n}{}^{V}_* \inn \Theta_{PC} -
\eeqa
Besides, 
$$
\xx{n}{}^{}_* \inn \Theta_{PC} = 
{}\xx{n}{}^{V}_* \inn \Theta_{PC} - H .  
$$ 
} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Discussion} 

We presented a formula of prequantum operators  
corresponding to Hamiltonian forms. It realizes a 
representation of the Poisson-Gerstenhaber algebra 
of Hamiltonian forms by operators acting 
on prequantum 
%Hilbert space of 
wave functions given by 
nonhomogeneous forms  $\Psi$, the sections of 
$\bigwedge^*_0(Z)^{\mathbb C}   \rightarrow Z$.  
We also argued that these wave functions  
fulfill the prequantum Schr\"odinger equation (5.5). 

%hilbert space 
%scalar product 
%superconnection 

The next step in geometric quantization would be to reduce the 
prequantum Hilbert space by introducing a {\em polarization\/} in the 
polymomentum phase space.   
 % definition? not 1/2  
A generalization of the {\em vertical\/} polarization 
 %\der^a_mu \Psi = 0 ... 
reduces the space of wave functions 
to the functions depending on 
field variables and space-time variables: $\Psi (y^a, x^\mu)$. 
A construction of 
 %the quantum operator $\what{H}$ 
quantum operators on the new Hilbert space 
of quantum wave functions requires further generalization 
of the techniques of geometric quantization,  
  %\cite{geomquant}, 
such as the notion of the metaplectic correction and the 
Blattner-Kostant-Sternberg pairing,  
 %both being well 
%which is beyond the scope of this paper. 
which is not developed yet. 

However, the quantum operator $\what{H}$ 
is already known from the heuristic 
procedure of ``precanonical quantization'' 
\cite{qs96,bial97,lodz98,pla2001,ijtp2001}  
based on  quantization of a small 
Heisenberg-like subalgebra of brackets of 
differential forms generalizing the canonical variables.  
Within  precanonical quantization it was found suitable 
to work in terms of the space-time Clifford algebra valued 
operators and wave functions, rather than in terms of 
non-homogeneous forms and the graded endomorphism valued 
operators acting on them. In general, 
a relation between the two formulations is given by the 
``Chevalley quantization'' map from the 
{\em co-\/}exterior algebra to the Clifford algebra: 
$\omega_\mu\bullet \rightarrow 
- \frac{1}{\varkappa} \gamma_\mu,$ where the constant $\kappa$ 
is introduced to match the physical dimensions 
($1/\varkappa \sim$ length{}$^{n-1}$). 
The  corresponding Clifford product of forms is  
given by (cf. \cite{joos}) 
$$\omega_\mu\vee\omega_\nu = 
\omega_\mu\bullet\omega_\nu + \varkappa^{-2} \eta_{\mu\nu}.
$$
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%\newcommand{\newtexta}{ 
Note that the appearence of the metric $\eta_{\mu\nu}$ 
at this stage is related to the fact that a definition 
of the scalar product of wave functions 
represented by non-homogeneous forms, i.e. their probabilistic 
interpretation, requires a space-time metric. 

Under the above ``Cliffordization'' and the vertical polarization 
the wave function becomes Clifford valued: $\Psi=\Psi(y^a,x^\mu)$ 
and the left hand side of  (5.5), (5.6) can be expressed in terms of the 
Dirac operator acting on $\Psi=\psi + \psi_\nu\gamma^\nu$;  
in particular, $d\bullet \sim \gamma\gamma^\nu\der_\nu$, 
where 
 %$\gamma \sim \prod_{\nu=1}^{\nu=n} \psi_\nu$ 
 $\gamma \sim \gamma_1\gamma_2...\gamma_n$ 
corresponds to the Hodge 
duality operator $*$.  
 %so that the left hand side of  (5.5) can be expressed in terms of the 
 %Dirac operator acting on $\Psi=\psi + \psi_\nu\gamma^\nu$, and 
Similarly, 
the operator of $H\bullet$  is represented as $\sim \!\gamma \what{H}$. 
The coefficients not specified here are fixed by the requirement that 
the resulting Dirac-like equation is causal and consistent, 
thus leading to the covariant Schr\"odinger equation for 
quantum fields in the form 
\beq
i\hbar\varkappa \gamma^\mu\der_\mu \Psi = \what{H}\Psi . 
\eeq 
A similar reasoning leads to the represenation of polymomenta: 
$\hat{p}{}_a^\mu = -i\hbar\varkappa \gamma^\mu {\der}/{\der y^a}$.  
These results have been anticipated  within precanonical 
field quantization earlier \cite{qs96,bial97,lodz98} 
 %OLD TEXT, replaced May 2002 %%%%  
 %(see also \cite{navarro} where similar relations were postulated).  
 %A derivation of the explicit form of $\what{H}$ requires  
 %a generalization of more advanced aspects of geometrical 
 %quantization which is not yet developed. 
 %However,  $\what{H}$ can be found using 
 %heuristic %%% 
 %precanonical quantization. 
(see also \cite{navarro} where similar relations were postulated). 
This approach also allows us to derive the explicit form of 
$\what{H}$. For example,  
in the 
%particular 
case of interacting scalar fields $y^a$ one can show that \cite{bial97} 
$$ 
\what{H} = - \mbox{\large $\frac{1}{2}$} \hbar^2\varkappa^2 
\lapl + V(y), 
%\frac{\der^2}{\der y^2} + V(y), 
$$ 
where $\lapl$ is the Laplace operator in the space of field 
variables. 


%} 
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\oldtextk}{
Under this ``Cliffordization'' 
the Schr\"odinger equation on $\Psi=\Psi(y^a,x^\mu)$ 
 %assumes 
can be argued to take the  form  (cf. (5.6)): 
\beq
i\hbar\varkappa \gamma^\mu\der_\mu \Psi = \what{H}\Psi 
\eeq 
and the polymomenta are represented as follows: 
$\hat{p}{}_a^\mu = -i\hbar\varkappa \gamma^\mu {\der}/{\der y^a}.$  
In the particular case of interacting 
scalar fields $y^a$, one can show that \cite{bial97} 
%and the DW Hamiltonian operator, in the case of interacting 
%scalar fields $y^a$, reads: 
$$ 
\what{H} = -\mbox{$ \frac{1}{2}$} \hbar^2\varkappa^2 
\lapl + V(y), 
%\frac{\der^2}{\der y^2} + V(y), 
$$ 
where $\lapl$ is the Laplace operator in the space of field 
variables. 
 } 
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

What we have arrived at is a multidimensional hypercomplex 
generalization of the Schr\"odinger equation from quantum mechanics 
to field theory, where the space-time Clifford algebra, which 
arose from quantization of differential forms, generalizes the 
algebra of the complex numbers in quantum mechanics, and the notion of the 
unitary time evolution is replaced by the 
space-time propagation  governed by the Dirac operator. 
In \cite{pla2001} we discussed how this description of quantum 
fields can be related to the standard description in the 
functional Schr\"odinger representation. In doing so the 
Schr\"odinger wave functional arises as a specific composition 
of amplitudes given by Clifford-valued wave functions of the 
precanonical approach, and the 
parameter $\varkappa$ appears to be related to the 
ultra-violet cutoff. 

Obviously, in this presentation we have left untouched a lot 
of important issues both on the level of prequantization 
and on the level of quantization.  A development of 
 %the elements 
the present version of geometric quantization in field theory 
 %presented in this paper 
would  further clarify the mathematical foundations of 
precanonical quantization of fields and also advance its 
understanding and applications.   The whole field appears 
to us as appealing, mathematically rich and unexplored 
as the field of 
 the %%?? 
geometric quantization 
 %of mechanical systems 
approach to quantum mechanics 
was 25-30 years ago. 

\bigskip 


{ %\footnotesize

{\bf Note added in proofs (May, 2002)}   

\medskip 

A distinction between odd and even $n$ in Sect. 5 can be avoided 
by noticing that the prequantization map (3.2) can be modified 
as follows: 
\beq
O'_{F}= (-1)^{(n-f-1)}\left ( i\hbar \pounds_{X_F} 
+  X_F{} \inn \Theta \bullet \right ) 
+ F\bullet ,
\eeq
where $(n-f)$ is the co-exterior degree of $F$. 
Then the right hand side of 
(5.5) is identified with $O'_{H}(\Psi)$ for any $n$. The left 
hand side of (5.5) also can be written in a universal form for 
any $n$ using the 
reversion anti-automorphism $\beta$
in a co-exterior Grassmann algebra:   
%co-exterior Grassmann algebra anti-automorphism 
%(the reversion) $\beta$: 
$\beta(F) := (-1)^{\frac{1}{2}(n-f)(n-f-1)}F$. 
Then the prequantum Schr\"odinger equation (5.5) 
can be written as follows: 
\beq
i\sigma\hbar (-1)^{\frac{1}{2}n(n-1)}\beta(d\bullet\Psi) = O'_{H} (\Psi). 
\eeq
Note that a choice between two representations (3.2) and (6.2) 
can be made once the scalar product is specified. 

} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bigskip 

\medskip 

{\bf Acknowledgments. } 

\medskip 

The work has been supported by  
the 
%Graduate School at the University of Hamburg 
%``Future Developments in Particle Physics'' 
{\em Graduiertenkolleg\/} 
``Future Developments in Particle Physics'' 
%``Zuk\"unftige Entwicklungen in der Teilchenphysik" 
at the University of Hamburg 
to which I express my sincere gratitude. 




\begin{thebibliography}{99} 

%\footnotesize 
\small 

\bibitem{dedonder} Th. De Donder, 
{\em Th\'eorie invariantive du calcul des
variations,\/}  
 Nuov. \'{e}d.  
Gauthier-Villars 1935, Paris;    
        
%\bibitem{weyl} 
H. Weyl,   
{Geodesic fields in the calculus of variations, } 
 {\em Ann. Math. (2)}  {\bf 36} (1935) 607-29;  
        %\bibitem{rund} 

H. Rund, {\em The Hamilton-Jacobi theory 
in the calculus of variations,\/} D. van Nostrand, Toronto 1966; 

H. Kastrup, Canonical theories of Lagrangian dynamical systems in physics, 
{\em Phys. Rep.\/} {\bf 101} (1983) 1-167. 

\bibitem{dedecker} P. Dedecker,  
 {On the generalizations of symplectic geometry to multiple integrals 
 in the calculus of variations, }  
in:  {\em Lect. Notes. Math.\/} vol. {\bf 570} p. 395-456,  
Springer, Berlin 1977.  %395-456

\bibitem{bw} M. Born, 
On the quantum theory of the electromagnetic field, 
{\em Proc. Roy. Soc. (London)} {\bf A143}  (1934) 410-37; 
 
H. Weyl, 
 { Observations on Hilbert's independence theorem and 
 Born's quantization of field equations, }
{\em Phys. Rev.} {\bf 46}  (1934) 505-8.  

\bibitem{gimm} M.J. Gotay,   J. Isenberg and J. Marsden, 
{\sl Momentum  maps and classical relativistic fields\/},  
%(Berkeley preprint 1998, various versions exist since 1985),    
Part I:  Covariant field theory,   
 %{\em Preprint}  
{\tt physics/9801019}  (and references therein). 

\bibitem{norris}R.O. Fulp, J.K. Lawson, and L.K. Norris, 
Generalized symplectic geometry as a covering theory
for the Hamiltonian theories of classical particles and fields,  
{\em J. Geom. Phys.\/} {\bf 20} (1996) 195-206;  
 
L.K. Norris, 
$n$-symplectic algebra of observables in covariant Lagrangian field theory, 
{\em J. Math. Phys. } {\bf 42} (2001) 4827--4845 (and references therein).   

\bibitem{deleon} M. de Le\'on, E. Merino, M. Saldago, 
$k$-cosymplectic manifolds and Lagrangian field theories, 
{\em J. Math. Phys.\/} {\bf 42} (2001) 2092-104 (and references therein). 

\bibitem{roemer} 
M. Forger and H. R\"omer, 
A Poisson bracket on multisymplectic phase space, 
{\em Rep. Math. Phys.\/} {\bf 48} (2001) 211-8, 
{\tt math-ph/0009037}; 

C. Paufler, H. R\"omer, 
De Donder-Weyl equations and multisymplectic geometry,  
%to appear in {\em Rep. Math. Phys.\/ } 
{\tt math-ph/0107019}.  

\bibitem{paufler1} C. Paufler, 
A vertical exterior derivative in multisymplectic 
geometry and a graded Poisson bracket for nontrivial geometries, 
{\em Rep. Math. Phys.} {\bf 47} (2001) 101-19, 
{\tt math-ph/0002032}. 

%Cornelius Paufler, Hartmann Romer
%Geometry of Hamiltonean n-vectors in Multisymplectic Field Theory 
%{\tt math-ph/0102008}. 
%Cornelius Paufler 
%On The Geometry of Field Theoretic Gerstenhaber Structures
%{\em Rep. Math. Phys.\/} {\bf 48} (2001) 203-10, 
%math-ph/0102012 
 %

\bibitem{roman} 
A. Echeverr{\'\i}a-Enr{\'\i}quez and   
M.C.  Mu\~noz-Lecanda, 
Variational calculus in several variables: a Hamiltonian approach, 
{\em Ann. Inst. Henri Poincar\'e } {\bf 56}  (1992) 27-47;  

A. Echeverr{\'\i}a-Enr{\'\i}quez,  
M.C.  Mu\~noz-Lecanda and   N. Roman-Roy,
{On the multimomentum bundles and the Legendre maps in field theories, }
{\em Rep. Math. Phys.\/} {\bf 45}  (2000) 85-105, 
{\tt math-ph/9904007};  

A. Echeverr{\'\i}a-Enr{\'\i}quez,  
M.C.  Mu\~noz-Lecanda and   N. Roman-Roy, 
Multivector field formulation of Hamiltonian field theries: 
equations and symmetries, 
{\em J. Phys.\/} {\bf A32} (1999) 8461-84, 
 {\tt math-ph/9907007}; 

A. Echeverr{\'\i}a-Enr{\'\i}quez,  
M.C.  Mu\~noz-Lecanda and   N. Roman-Roy, 
Geometry of multisymplectic Hamiltonian first-order field theories,  
{\em J. Math. Phys.\/} {\bf 41} (2000) 7402--44, 
{\tt math-ph/0004005}.  
%(and references therein).   
 %%

\bibitem{sardan} G. Giachetta, L. Mangiarotti  
and  G. Sardanashvily,  
 {\em New Lagrangian and Hamiltonian Methods in Field Theory\/},  
 World Scientific,  Singapore 1997. 
 % 

\bibitem{helein} F. H\'elein and J. Kouneiher, 
Finite dimensional Hamiltonian formalism for gauge and field theories,  
{\em J.~Math.~Phys. \/} {\bf 43} (2002) 2306-2347, 
{\tt math-ph/0010036}. 

\bibitem{krupka} 
O. Krupkov\'a, Hamiltonian field theory, 
{\em J.~Geom.~Phys.\/} (2002) in print;  

O. Krupkov\'a and D. Smetankov\'a, 
{Legendre transformation for regularizable Lagrangians in field theory},  
{\em Lett. Math. Phys.\/} {\bf 58} (2001) 189--204,  
{\tt math-ph/0111004} (and references therein). 
 %Letters in Math. Phys., in print 

\bibitem{romp98} I.V. Kanatchikov, 
Canonical structure of classical 
field theory in the polymomentum phase space,    
{\em Rep. ~Math.~Phys.\/} {\bf 41}   (1998) 49-90,   
{\tt hep-th/9709229}.  

\bibitem{bial96} I.V.~Kanatchikov, 
 On field theoretic generalizations of a Poisson algebra,   
{\em Rep. Math. Phys.\/} {\bf 40} (1997)  225-34,     
{\tt hep-th/9710069}. 

\bibitem{goslar96} I.V.~Kanatchikov, 
Novel algebraic structures from the 
polysymplectic form in field theory,   
 in: {\em GROUP21, Physical Applications and Mathematical 
 Aspects of Geometry, Groups and Algebras } 
 vol. 2, eds.  H.-D. Doebner et. al., 
 Singapore 1997, World Scientific,  894-9     
 {\tt hep-th/9612255}.  

\bibitem{qs96} I.V.~Kanatchikov, 
{ Toward the Born-Weyl quantization of fields, }
{\em Int. J. Theor. Phys.\/} {\bf 37}  (1998) 333-42,   
{\tt quant-ph/9712058}. 

\bibitem{bial97}  I.V. Kanatchikov,  
{De Donder-Weyl theory and a hypercomplex 
extension of quantum mechanics to field theory, } 
 {\em Rep. Math. Phys.\/} {\bf 43} (1999) 157-70,    
{\tt hep-th/9810165}.   

\bibitem{lodz98} I.V. Kanatchikov,  
On quantization of field theories in polymomentum variables,    
in: 
{\sl Particles, Fields and Gravitation, }   
 (Proc. Int. Conf. \L\'od\'z, Poland, Apr. 1998)   
 ed.  J. Rembielinski    
{\em AIP Conf. Proc.\/} vol. {\bf 453}, p. 356-67,     
 %(1998) 356-67, 
Amer. Inst. Phys., Woodbury (NY) 1998, 
  {\tt hep-th/9811016}. 

\bibitem{pla2001} I.V. Kanatchikov, 
 Precanonical quantization and the Schr\"odinger wave functional, 
{\em Phys. Lett.\/} {\bf A283} (2001) 25-36,  
{\tt hep-th/0012084}. 

\bibitem{ijtp2001} I.V. Kanatchikov, 
Precanonical quantum gravity: quantization without 
the space-time decomposition, 
{\em Int. J. Theor. Phys.\/} {\bf 40} (2001) 1121-49,  
{\tt gr-qc/0012074}.  
 %

\bibitem{sniat}  J. \'Sniatycki, {\em Geometric quantization and quantum 
mechanics,\/}  Springer-Verlag, New York 1980;  
 %\bibitem{woodh} 

N.M.J. Woodhouse, {\em Geometric Quantization,\/} 2nd ed., 
Clarendon Press, Oxford 1991;  
 %\bibitem{weinst} 
 
S. Bates and A. Weinstein, 
{\em Lectures on Geometry of Quantization,\/} 
 %Amer. Math. Soc., 
Berkeley Math. Lect. Notes vol. 8,  
%Berkeley Center for Pure and Appl. Math. 1997.  
Berkeley 1997; 

  %\bibitem{tuyn} 
G.M. Tuynman, Geometric Quantization, 
in {\em Proc. Seminar 1983-1985 Mathematical Structures in field 
theories, vol. 1,\/} 
p. 1-157. CWI Syllabus vol. 8, Center for Math. and Comp. Sci., 
Amsterdam 1985;  

 %\bibitem{rom} 
 A. Echeverr\'\i a-Enr\'\i quez, M.C. Mu\~nos-Lecanda, 
N. Rom\'an-Roy and C. Victoria-Monge, 
 Mathematical foundations of geometric quantization, 
{\em Extracta Math.\/} {\bf 13} (1998) 135-238,  
{\tt math-ph/9904008};  

%
\bibitem{fn}  A. Fr\"olicher and A. Nijenhuis, 
Theory of vector-valued differential forms. 
I. Derivations of the graded ring of differential forms, 
{\em Indag. Math.\/} {\bf 18} (1956) 338--59;  

 %F. Mimura, T. Sakurai, T. N\^ono, 
 %Extended derivations associated with vector-valued differential forms. 
 %{\em Tensor (N.S.)\/} {\bf 51} (1992) 193--204

I. Kol\'a\v r, P.W. Michor, J. Slov\'ak, 
{\em Natural operations in differential geometry\/}, 
Springer-Verlag, Berlin 1993.

\bibitem{torun2001} I.V. Kanatchikov, 
 Geometric quantization of Hamiltonian forms in field theory, 
{\em in preparation\/}. 

\bibitem{mg9a} I.V. Kanatchikov, 
 Covariant geometric prequantization of fields, 
in : {\em Proc. IX Marcel Grossmann Meeting\/}, p. 1065, 
World Scientific, Singapore 2001, 
{\tt gr-qc/0012038}. 

\bibitem{quillen} D. Quillen, 
Superconnections and Chern character, 
{\em Topology\/} {\bf 24} (1985) 89-95; 

V. Mathai and D. Quillen, 
Superconnecitons, Thom clases 
and equivariant differential forms, 
{\em Topology\/} {\bf 25} (1986) 85-110. 

\bibitem{preschr} 
B. Kostant,  
Line bundles and prequantized Schr\"odinger equation, 
in: {\em Proc. CNRS Coll. on Group Theor. Methods in Physics\/}, 
p. IV1-22, Marseille 1972 (cited according to \'Sniatycki in [21]);  

D.J. Simms, 
On the Schr\"odinger equation given by geometric quantization, 
in: {\em Lect. Notes Math.\/} {\bf v. 676} p. 351-356, Springer, N.Y. 1978;   

Qian-Shu Li and Xu-Guang Hu, 
On the quantum mechanical representation in phase space, 
{\em Physica Scripta\/} {\bf 51} (1995) 417-422;  

%Hu, Xu-Guang(PRC-JIL-TK); Li, Qian-Shu(PRC-JIL-TK); Tang, Au-Chin(PRC-JIL-TK) 
%A new formulation of the potential scattering in quantum phase space. 
%Phys. Scripta 54 (1996) 129--136.  

L. Charles, 
 Feynman path integral and Toeplitz quantization, 
{\em Helv. Phys. Acta\/} {\bf 72} (1999) 341-355.  



\bibitem{soucek} J. Sou\v cek, 
Subquantum models: basic principles, effects and tests, 
{\tt quant-ph/0107040}.  

\bibitem{tuynman2} G. Tuynman,  
Super symplectic geometry, 
talk at the Bia\l owie\.za Workshop 2001, {\em in preparation};   

G. Tuynman,  The Lagrangean in symplectic mechanics, 
in: {\em Proc. Int. Conf.  in Honor of Jean Leray\/}, 
to be published by Kluwer Acad. Publ. 




\bibitem{bial94} I.V. Kanatchikov, 
From the Poincar\'e-Cartan form 
to a Gerstenhaber algebra of Poisson   
brackets in field theory, 
{in}: {\em Coherent States, and 
Complex Structures\/},  
J.-P. Antoine e.a. eds., Plenum Press, N.Y. 1995, 
p. 173-83,  
{\tt hep-th/9511039}.   
%pp. 173-183; 



\bibitem{joos} E. K\"ahler, 
 Der innere Differentialkalk\"ul,  
{\em  Rend. Mat. e Appl. (5)\/} {\bf 21} (1962) 425-523; 

E. K\"ahler, 
Innerer und \"asserer Differentialkalk\"ul,  
{\em  Abh. Deutsch. Akad. Wiss. Berlin Kl. Math.
Phys. Tech.\/} No. 4 (1960) 32 pp.;  

W. Graf, 
Differential forms as spinors, 
{\em Ann. Inst. H. Poincar\'e\/} {\bf A29} (1978) 85-109; 

P. Becher and H. Joos, 
The Dirac-K\"ahler equation on the lattice, 
{\em Z. Phys.\/} {\bf C15} (1982) 343-65;   

I.M. Benn and R.W. Tucker, 
 Fermions without spinors, 
{\em Comm. Math. Phys.\/} {\bf 89} (1983) 341--62. 

\bibitem{navarro} M. Navarro, 
 Toward a finite-dimensional formulation of quantum field theory, 
{\em Found. Phys. Lett.\/} {\bf 11} (1998) 585-593, 
{\tt quant-ph/9805010}. 
 
 %Quantum and Classical Fields in the Finite-Dimensional Formalism
 %hep-th/0110078  


\end{thebibliography}

\end{document}

\\ 
Title: Geometric (pre)quantization in the polysymplectic approach
to field theory 
Authors: I.V. Kanatchikov (II ITP, Univ. of Hamburg)
Comments: 12 pages, LaTeX. V2: minor corrections, important Note 
added in proofs, refs detailed.    
\\ 
The prequantization map for a Poisson-Gerstenhaber algebra 
of dynamical variables represented by differential forms within 
the polysymplectic formulation of the De Donder--Weyl covariant 
Hamiltonian field theory is presented and the corresponding 
prequantum Schroedinger equation for a non-homogeneous form 
valued wave function is derived.  This is the first step toward
understanding the procedures of covariant precanonical field
quantization from the point of view of geometric quantization.  
\\



bi-connection theory of gravity 

89d:83004 83A05 (53B50 83C99) 
Chernikov, N. A.(2-JINR) 
Difficult questions of relativity theory. 
Soviet J. Particles and Nuclei 18 (1987), no. 5, 427--440 (1988); translated from Fiz. lementar. Chastits i Atom. Yadra 18 (1987), no. 5,
1000--1034, 1166 (Russian. English summary) 

References: 0
                          Reference Citations: 0
                                                                  Review Citations: 0


The "difficult questions" referred to in this title are old ones, primarily concerned with interpretations and semantics. The discussion makes
extensive use of historical references, especially Russian ones not generally noted in Western literature on the subject. The definition and
meaning of many concepts are considered, including: invariance and covariance, coordinate systems and reference frames, the principles of
special and general relativity, and coordinate conditions. More emphasis is placed on the significance of coordinate conditions, especially the
harmonic one advocated by Fock, than is usual in current works in which the significance of their choice is relegated to that of computational
convenience. This paper touches on a wide range of topics, and includes a brief description of the local differential formalism used in general
relativity.



88h:83001 83-00 
Chernikov, N. A. 
On the necessity of a dictionary for gravitationists. (Russian) 
Problemy Teor. Gravitatsii i lement. Chastits No. 17, (1986), 24--33. 

References: 0
                          Reference Citations: 0
                                                                  Review Citations: 0


The author calls for a dictionary of gravitation (general relativity): "to enable specialists to communicate, beginners to dispel misunderstanding
and philosophers to draw the right conclusions". There exist dictionaries of pure mathematics but one of a great theory of physics would face
a problem of its own: the peculiar point of view of its compilers. Nevertheless, the article draws attention to a frequently neglected question of
the relationship between science and philosophy. A theory of the former almost invariably reflects some initial philosophical standard of its
architects, yet philosophy of science only turns its attention to an already constructed edifice. Misunderstanding is a common outcome.

                                          Reviewed by A. H. Klotz 

 
88a:58004 58A05 (53B05 82-02) 
Chernikov, N. A. 
{\cyr Konspekt teorii afinno\u\i svyaznosti dlya gravitatsionistov}. (Russian) [A review of the theory of affine connection for
gravitationalists] 
With an English summary. {\cyr Soobshcheniya Ob\cdprime edinennogo Instituta Yadernykh Issledovani\u\i. Dubna} [Communications of the
Joint Institute for Nuclear Research. Dubna], R2-86-207. 
Joint Inst. Nuclear Res., Dubna, 1986. 16 pp. 0.21 r. 

References: 0
                          Reference Citations: 0
                                                                  Review Citations: 0


The author defines and discusses several notions from the theory of affine connections, taken from A. P. Norden's book [Spaces with affine
connection (Russian), second edition, "Nauka", Moscow, 1976; MR 57 #7421].
  


87m:83050 83C99 (31D05 53B50) 
Chernikov, N. A.(2-JINR) 
Difficulties in the relativistic theory of gravitation. (Russian) 
Proceedings of the VII international conference on the problems of quantum field theory (Russian) (Alushta, 1984), 382--410, 
Ob\cdprime ed. Inst. Yadernykh Issled. Dubna, {rm D}2-84-366, 
Ob\cdprime ed. Inst. Yadernykh Issled., Dubna, 1984. 

References: 0
                          Reference Citations: 0
                                                                  Review Citations: 0


General remarks about relativity, non-Euclidean geometry and the problem of defining gravitational energy are followed by a brief exposition of
Riemannian geometry. 

\{For the entire collection see MR 87k:81004.\}

                                       Reviewed by Andrzej Trautman 


80h:53018 53B20 (83C99) 
\v Cernikov, N. A. 
An affinely connected space with a nonsymmetric tensor field. (Russian) 
Izv. Vyssh. Uchebn. Zaved. Mat. 1978, no. 5(192), 114--123. 

References: 0
                          Reference Citations: 0
                                                                  Review Citations: 0


Let $g\sb {\alpha\beta}(x)$ be a nonsymmetric tensor field on a smooth manifold. Affine connections $\Gamma$ defined as the solutions of
$(*)$ $\partial\sb \gamma g\sb {\alpha\beta}-g\sb {\alpha\beta}\Gamma\sb \alpha{}\sp \sigma{}\sb \gamma-g\sb {\alpha\sigma}\Gamma\sb
\gamma{}\sp \sigma{}\sb \beta=0$ and appearing in Einstein's unified field theory are investigated. Explicit expressions are given for the
contracted mean connection and for the curvature vector. Some of the conditions for the unambiguous solvability of $(*)$ are found. Also a
relation to the Born-Infeld equation is detected.

                                          Reviewed by L. Tamssy 


