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\begin{document}
\title{On Quantization of Field Theories in 
Polymomentum Variables\thanks{Submitted July 1998. To 
appear in: {\em Particles, Fields and Gravitation,} Proc. Int. Conf.,  
%in memorial of R. Raczka, 
Lodz 1998, eds. K. Smolinski and J. Rembielinski (AIP Proc., 1998).}
}  
\author{Igor V. Kanatchikov$^*$
%\thanks{On leave from Tallin Technical 
%University, Tallin, Estonia}
}  
%\dagger
\address{$^*$ Laboratory of Analytical Mechanics and Field Theory \\
Institute of Fundamental Technological Research \\
Polish Academy of Sciences \\ 
\'Swi\c etokrzyska 21, Warsaw PL-00-049, Poland \\ 
{ \footnotesize \tt e-mail: ikanat@ippt.gov.pl} }
 
%\lefthead{LEFT head}
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\vspace*{-74mm}
\hbox to 5.4truein{%\tiny %
\vspace*{-1mm}\footnotesize 
%AIP Proceedings vol.     
\hfil 
\hbox to 0 truecm{ 
\hss \normalsize hep-th/9811016} 
}
\vspace*{70mm}

 
\begin{abstract}
Polymomentum canonical theories, 
%in particular, the De Donder-Weyl theory, 
which are manifestly covariant 
multi-parameter generalizations 
of the Hamiltonian formalism  to field theory,  
are 
considered 
as a possible basis of quantization.  
We arrive at a  multi-parameter hypercomplex 
%covariant, multi-parameter, ...  
generalization of 
quantum mechanics to field theory 
in which 
the algebra of 
complex numbers and a time parameter 
are replaced 
 %respectively 
by 
the space-time Clifford algebra 
and space-time variables  
treated in a manifestly covariant fashion. 
%appearing  on equal footing.   
The corresponding  covariant   
%hypercomplex 
%multi-parameter 
generalization of 
the Schr\"odinger equation 
% for the wave function over a finite dimensional 
%covariant=multi-parameter analogue of the configuration space 
 %is shown to fulfill 
 %fulfills 
is shown to be consistent 
with several aspects of the correspondence 
 % principle, such as a relation to 
principle such as  a relation to 
the 
%polymomentum 
De Donder-Weyl 
Hamilton-Jacobi theory  in the classical limit 
and the Ehrenfest theorem. 
  %Within  the ultra-local approximation a relation of 
A relation of 
the corresponding wave function 
(over 
%%? the 
a finite dimensional configuration space 
of field and space-time variables)      
with    the Schr\"odinger wave functional in quantum field theory 
  %is examined. 
%using  
is examined 
  %within  
     in  the ultra-local approximation. 
\end{abstract} 

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\newcommand{\sometext}{
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The Hamiltonian formalism is a basis of 
most of the quantization procedures, the stochastic quantization 
and the path integral quantization being the 
remarkable exceptions. %BRST in Lagrangian setting.
The conventional Hamiltonian formalism in field theory requires a 
singling out of a time parameter. This implies the global 
hyperbolicity of the underlying space-time manifold. 
When applying this framework 
to the problem of quantization of General Relativity 
a  concern  about its  applicability arises.   
In fact, already qualitative considerations demonstrate that on 
quantum level the space-time undergoes intricate  fluctuations of 
the metric and topology leading to the so-called space-time foam, 
for which  
%the underlying  requirement 
%of the Hamiltonian formulation for the space-time to be  globally 
%hyperbolic cannot be fulfilled. 
no global hyperbolicity can be expected. 
}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%{\sl PACS classification:} 03.50, 02.40}
%\thanks{
%{\sl  AMS classification:} 70 G 50, 58 F 05, 53 C 80}
%\thanks{
%{\sl Keywords}:  
%De Donder--Weyl theory, Hamiltonian formalism, quantization}
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%\section*{}

\section*{Introduction }   


The canonical quantization is based on the Hamiltonian 
formalism. 
The conventional Hamiltonian formalism in field theory is 
 an infinite dimensional version of the one in mechanics. 
As a result, 
the  %%?  
quantum field theory based on it is 
essentially 
the %%? 
quantum mechanics of systems with an infinite number 
of degrees of freedom. 
Most of the difficulties and ambiguities 
of quantum field theory are due to this infinite dimensionality.  
%%%the mode expansion - free theory and its pertubative vicinity
%%%curved s-t - no particle concept 
However, should  quantum fields always be understood in this 
way? Does this picture exhaust all aspects of quantum fields? 
Is there a  
%%ISKONNO. ISKONNYJ %% 
``genuine quantum field theory'' %%?better? 
%``genuine quantum theory of fields''  
%%``Iskonnaya Kvantovaja Teoriya Polya'' 
%%``Iskonno polevaya kv teoriya polya'' 
%% I mean: ``genuinely field-theoretic quantum theory of fields'' 
more general  
that just 
%%?a  
quantum mechanics applied to fields?   
It is clear that in 
%%?the 
pertubative regime, i.e. in 
the %%? 
vicinity 
of a free field theory 
which can be 
represented as a continuum of 
harmonic oscillators,   the above picture can work well, and it 
really does as 
the 
%%?an  
experimental triumph of pertubative 
quantum field theory  demonstrates. 
However, applicability of this picture 
in non-pertubative domain 
and in curved space-time, where no natural particle concept exists 
in general, 
\nopagebreak[2]can   
be more limited. 
%\pagebreak   
%in ... this picture can be of more limited usefulness.  

A  %%possible   
conceivable %=imaginable 
approach to the above posed  questions can be 
based on the (not  widely 
acknowledged yet)  
fact that 
the conventional 
 %generalization of 
 %extension of 
version of 
the Hamiltonian formalism in field theory is not 
the only one possible. 
% the only possible way of generalizing the 
%  Hamiltonian formalism to field theory. 
%%==> one=Hamiltonian formalism in field theory??   
 In fact, there exist  different alternative extensions 
of the Hamiltonian  formulation 
 %from mechanics 
to field theory   
%in the sense 
%that all of them reduce  
which all reduce 
to the Hamilton  formalism in mechanics 
if the number of space-time  dimensions equals to one. 
These extensions originate from the calculus of variations 
of multiple integrals \cite{dw,rund,giaquinta,kastrup}.   
Unlike the conventional Hamiltonian formalism, 
%? comma   
all these formulations are constructed in a manifestly covariant way 
not requiring any singling out of a time dimension. They can 
be applied even if the signature of the space-time is not 
Minkowskian. 
This is achieved by assigning the canonical momentum like variables, 
which we called {\em polymomenta} \cite{ikanat0},   
to the whole set of space-time 
derivatives of a field:   
$\der_\mu y^a \rightarrow p^\mu_a$\footnote{Throughout the paper 
$y^a$ denote 
field variables, $x^\mu$ are space-time variables 
($\mu=1,...,n$),  
$\der_\mu y^a$ are 
space-time derivatives (or first jets) of field variables,  
$p_a^\mu$ denote polymomenta.}.   
An analogue of the phase space is then a  
finite dimensional phase space of variables 
$(y^a, p_a^\mu, x^\nu)$  
 %to be referred to as 
which we call  the {\em polymomentum phase space}.  
Corresponding 
generalizations of the canonical formalism will be referred to 
as  {\em polymomentum canonical theories}. 
In the geometric (Cartan's) approach to the calculus of variations  
these theories 
(a version of which is also known 
as the multisymplectic formalism \cite{gimm}) 
appear as 
a %?the 
result of a certain choice of the so-called  Lepagean 
equivalents of 
%%?the 
a 
field-theoretic (multidimensional) analogue 
of the Poincar\'e-Cartan form \cite{kastrup,lepage,gimm,gotay,dedecker}.    
%%? \cite{dedecker}   
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\ignore}{  
Some mathematical issues related to this kind of 
theories, also known as the multisymplectic formalism \cite{gimm},   
have been studied recently 
%\cite{cantrijn,ibort,spain,carinena,deleon} 
\cite{math} 
from various points of view and 
using a language different from that we adopted in the present paper. 
}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%% For rare applications in physics see 
%%\cite{kastrup,gimm,nambu,vonrieth,grigore,ikanat,...}  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
Unfortunately, applications of these theories in 
physics have been so far rather rare (see for references 
\cite{kastrup,ikanat0,gimm}). 

The simplest example of a  polymomentum canonical theory 
is 
%given by 
the so-called  De Donder-Weyl (DW) 
theory\cite{dw,rund,kastrup,gimm}.  
%in the calculus of variations. 
Given a Lagrangian density 
$L=L(y^a, \der_\mu y^a, x^\nu)$, 
the polymomenta are introduced by 
the formula $p_a^\mu:=\der L/ \der (\der_\mu y^a)$. An analogue of the 
Hamilton canonical function defined as  
$H:= \der_\mu y^a p_a^\mu - L$ is referred to as 
the {\em DW Hamiltonian function} in what follows. 
Note, that $H$  is a function 
%on the finite dimensional {\em polymomentum phase space} 
of variables $(y^a, p_a^\mu,x^\mu)=:z^M$.  
In  these variables the Euler-Lagrange field 
equations can be rewritten in 
%a  manifestly covariant first order form 
the form of {\em DW Hamiltonian field equations}  
\beq
\der_\mu y^a = \der H / \der p^\mu_a, 
\quad \der_\mu p^\mu_a = - \der H/ \der y^a . 
\eeq
Clearly, this formulation reproduces the standard Hamiltonian formulation 
in mechanics at $n=1$. At $n>1$ it provides us with a kind of 
multi-parameter, or ``multi-time'',  
manifestly covariant 
generalization of the Hamiltonian formalism. 
In doing so fields are treated not as 
infinite dimensional mechanical systems 
evolving with time, but rather as systems varying in space-time, 
with the DW Hamiltonian function controlling such a variation 
(similarly to the usual Hamiltonian controlling the time evolution). 

The objective of the present contribution is to discuss an approach 
to quantization of fields based on polymomentum canonical theories. 
Although we confine ourselves exclusively to the approach based on the 
DW theory, we believe that basic ideas presented in what follows 
can be  extended 
%also 
to more general polymomentum theories. 


\section*{Graded Poisson bracket and quantization}

The canonical quantization in mechanics is essentially 
based on the algebraic structure given by the Poisson bracket. 
One of the reasons why  polymomentum canonical theories 
have not been used as a basis of quantization was   
%most likely 
the lack of an appropriate generalization of the Poisson bracket.  
In \cite{ikanat0} we proposed such a generalization 
%for 
within 
the  DW theory. The bracket is defined  on horizontal differential 
forms 
%($0\leq p \leq n$) 
%\[
$F=\frac{1}{p!}F\lind{\mu}{p}(z^M)    %(y^a,p_a^\mu,x^\nu)
dx^{\mu_1}\we ... \we dx^{\mu_p} $
%\] 
of various degrees $p$ ($0\leq p \leq n$),  which   
play the role of dynamical variables 
(instead of  functions in mechanics or functionals in the 
conventional Hamiltonian formalism in  field theory). 
It leads to graded analogues of the Poisson algebra 
structure\cite{ikanat0,bial96}. 
More specifically, the bracket on differential forms in 
DW theory leads to generalizations of 
the  so-called Gerstenhaber algebra \cite{gerst} 
(a graded analogue of the Poisson 
algebra with the grade of an element of the algebra 
with respect to the bracket 
differing 
%%?different 
by one from its grade 
with respect to the multiplication).    
For the purposes of the present paper it suffices to know 
a small subalgebra of the canonical brackets and 
a %%?  
representation of the field equations in terms 
of the bracket operation\footnote{For the reason of a limited space 
we avoid discussing properties of graded Poisson bracket in 
DW theory in details. In what follows we simply chose facts 
which we need and refer the interested reader for more details to 
\cite{ikanat0,bial96,bial97}. }. 

Using the notation $\om_\mu:= (-1)^{(\mu-1)} 
dx^1\we ... \we \what{dx^\mu}\we ... \we dx^n$ 
the canonical brackets in the (Lie) subalgebra of 
forms of degree $0$ and $(n-1)$ read \cite{ikanat0}  
$$\pbr{p_a^\mu\omega_\mu}{y^b}
= 
\delta^b_a , \quad 
\pbr{p_a^\mu\omega_\mu}{y^b\omega_\nu}
=
\delta^b_a\omega_\nu, \quad 
\pbr{p_a^\mu}{y^b\omega_\nu}
= 
\delta^b_a\delta^\mu_\nu , 
\refstepcounter{equation} 
\eqno {  (\theequation a,b,c) } 
$$ 
with  other  brackets vanishing.  
%Notice that arguments of the brackets are in general 
%forms of different degrees. 
All brackets in (2) reduce to 
the canonical bracket in mechanics when $n=1$; 
in this sense they are canonical and can be viewed  as 
a starting point of  quantization. 


Let us adopt the Dirac correspondence rule that 
Poisson brackets go over to commutators divided by 
$i\hbar$ 
%% , ?? 
and apply it to the canonical brackets (2). 
Note that this is just an assumption: while this 
principle proved to work well for 
the %%? 
usual Poisson 
bracket its precise form and  applicability to 
graded Poisson bracket in DW theory has to be confirmed. 
By quantizing (2a) we immediately conclude that  
$$
\what{p_a^\mu \om_\mu}=i\hbar \der_a,  
$$ 
where $\der_a$ is a partial derivative with respect to 
the 
field variables. 
The commutator corresponding to (2c) leads to  
%%?the 
a realization of $\what{\om}_\mu$  and $\what{p}{}^\mu_a$  
in terms of Clifford imaginary units, 
or Dirac matrices, 
under the assumption that the 
law of composition of operators is the symmetrized Clifford (=matrix) 
product\cite{qs96,bial97} 
\beq
\hat{p}{}^\nu_a = - i\kappa \ga^\nu\der_a ,
\quad 
\widehat{ \omega}_\nu = - \kappa^{-1} \ga_\nu . 
\eeq 
The quantity 
$\kappa$  of the dimension [{\em length}]$^{n-1}$  
appears here on dimensional grounds. Due to the infinitesimal 
nature of the volume element $\om_\mu$ we expect 
the absolute value of $\kappa$   to be ``very large''. 
Hence its relation to the ultra-violet cutoff scale \cite{qs96} can be 
anticipated 
%(see also sect. IV).    
(see also the last section before Conclusion). 


Note that  
%%%%text from potsdam98.tex. MODIFY!! 
the realization of operators in terms of 
        %Dirac matrices 
Clifford imaginary units  
implies a 
certain 
generalization of the formalism of quantum mechanics. 
%which we have arrived at here. 
Namely, whereas  the conventional quantum mechanics 
is built up on complex numbers    
%Namely, similar to the complex numbers,   
%which play the key role in the formalism of quantum mechanics and  
which are  
essentially the Clifford numbers corresponding to the 
{\em one}-dimensional space-time (= the time dimension in mechanics),  
 the present  approach to  quantization of fields 
viewed as 
%%``multi-time'' 
multi-parameter  Hamiltonian systems 
(of the De Donder-Weyl type) 
makes use of  the hypercomplex (Clifford) algebra of 
the underlying space-time manifold\cite{bt,hestenes}. 




In order to guess the form of 
%the Schr\"dinger equation 
quantum equations of motions within the present approach it is 
important to know how the field equations are 
represented  in terms of the bracket operation and what is the meaning 
of the bracket with the DW Hamiltonian function. 
In fact, the bracket with $H$ exists only for forms of degree 
higher than $(n-1)$ \cite{ikanat0}. Using $(n-1)$-form 
canonical variables appearing in (2) 
DW Hamiltonian equations (1) can be written in Poisson bracket formulation 
as follows \cite{ikanat0} (cf. \cite{bial97})  
\beq
\bd (y^a\om_\mu)=*\pbr{H}{y^a\om_\mu} = * \der H/\der p^\mu_a, 
\quad 
\bd (p_a^\mu \om_\mu) = *\pbr{H}{p_a^\mu\om_\mu} = - * \der H / \der y^a ,  
\eeq 
where $*$ is the Hodge duality operator acting on horizontal forms,  
and $\bd$ is the total exterior differential 
$
\bd F := \frac{1}{p!}\der_M F\lind{\mu}{p}\der_\mu z^M 
dx^\mu\we dx^{\mu_1} \we ... \we dx^{\mu_p}, 
$ 
with $z^M$ denoting the set of variables $(y^a,p_a^\mu,x^\mu)$. 
%To formulate the equations of motion of 
For  more general dynamical variables represented by $p$-forms $F$ 
we need a notion of the bracket with an $n$-form $H\om$, where 
$\om:=dx^1\we ... \we dx^n$, 
which allows us to write the equations of motion in 
%%?a 
the symbolic form \cite{ikanat0}   
$$
\bd F = \pbr{H\om}{F} + d^h F,
$$ 
where $d^h$ is the exterior differential with respect to the 
space-time (=horizontal) variables. 
Hence, we 
%reach the conclusion 
conclude that the DW Hamiltonian %function 
``generates'' infinitesimal  space-time variations 
of dynamical variables 
corresponding to  the total exterior differentiation, 
much like the Hamilton 
function in mechanics generates the infinitesimal evolution 
along the time dimension. 



Now, an analogue of the 
Schr\"odinger equation 
can be expected to have 
%%? the 
a form 
$ \hat{\i } \hat{d } \Psi \sim \what{H} \Psi $, 
where $\hat{\i }$ and $\hat{d }$ denote appropriate analogues 
of the imaginary unit and the exterior differentiation respectively. 
Keeping in mind the above remark on a hypercomplex generalization 
of quantum mechanics appearing here, an analogy between the 
exterior differential and the Dirac operator 
(in fact, the latter is $d - *^{-1}d*$ \cite{bt}), 
and natural 
requirements 
imposed by the correspondence principle,  
the following generalization of the  Schr\"odinger equation 
can be 
% written down 
formulated 
\cite{qs96,bial97,firststeps}   
\beq
\label{seqcl}
i \hbar \kappa \gamma^\mu \der_\mu \Psi = \what{H} \Psi, 
\eeq
where $\widehat{H}$ is the operator corresponding to the 
DW Hamiltonian function, the constant $\kappa$ 
of dimension [{\em length}]$^{-(n-1)}$ appears again on dimensional 
grounds, and $\Psi=\Psi(y^a,x^\mu)$ is a wave function over the 
 configuration space of  field and space-time variables. 
In the following section we demonstrate that this 
equation fulfills several aspects of the correspondence principle. 
Note  also that it  reproduces the quantum mechanical Schr\"odinger 
equation at $n=1$. 

%the wave function ... 

Let us construct the DW Hamiltonian operator 
%using an example of the 
for the system of interacting scalar fields $y^a$ in flat space-time 
given by the Lagrangian density 
\beq
L= \half \der_\mu y^a \der^\mu y_a - V(y). 
\eeq
Then the polymomenta and the DW Hamiltonian function are given by 
\beq 
p^a_\mu= \der_\mu y^a, \quad 
H= \half p^a_\mu p_a^\mu + V(y).  
\eeq
DW Hamiltonian field equations take the form 
\beq
\der_\mu y^a =  p_\mu^a,  
\quad \der_\mu p^\mu_a = - \der V/ \der y^a, 
\eeq 
which is essentially  a first order form of 
a system of coupled Klein-Gordon equations. 

By quantizing the bracket 
\beq
\pbr{ p^\mu_a p^a_\mu}{y^b \omega_\nu}= 2 p^b_\nu   
\eeq
we obtain \cite{bial97}  
$$\widehat{p^\mu_a p^a_\mu}= -\hbar^2 \kappa^2 \lapl ,$$ 
where  $\lapl:= \der_a \der^a$ 
is the Laplacian operator in the  space of field variables. 
Thus the DW Hamiltonian operator of  the system of 
interacting scalar fields takes the form  
\beq
\widehat{H} = -\half \hbar^2 \kappa^2 \lapl + V(y) .
\eeq 
%It is interesting to note 

Note that for 
a free scalar field 
$V(y)= (1/2 h^2) m^2 y^2,$ 
so that the DW Hamiltonian operator becomes similar 
to the Hamiltonian operator of the 
harmonic oscillator in the space of field variables.  
Its eigenvalues divided by $\kappa$ 
read 
%\chi_{{}_N}=:\kappa m_N=\kappa m (N+\half)% 
$m_N=m (N+\half)$. 
Separating variables $\Psi(y,x^\mu)= \Phi(x) f(y)$ 
from (5) we obtain 
$$\what{H} f_N = \kappa m_N f_N, \quad 
%and 
i\hbar\ga^\mu\der_\mu\Phi= m_N \Phi .$$
%Klein=Gordon $
    %conclude that the space-time behavior is given by 
 %the spinor 
Then for a free scalar field any solution of (5) 
is a linear combination of 
\beq
\Psi_{N,\bk, r}(y,\bx,t)  = u_{N,r}({\bk}) 
f_N(y) e^{\epsilon_r (i\omega_{N,\bk}t - i\bk\cdot\bx)}  ,  
\eeq
where 
%$\omega_{N,\bk}=({\bk^2 + m_N^2/\hbar^2})^{1/2}$. 
$\omega_{N,\bk}:=\sqrt{\bk^2 + m_N^2/\hbar^2}$,  
 $u_{N,r}({\bk})$ is a properly normalized constant spinor, 
$\epsilon_r=+1(-1)$ for positive (negative) energy solutions, and  
$f_N$ are eigenfunctions of the harmonic oscillator in $y$-space. 
As a consequence, any Green  function 
of (5) is given by \cite{firststeps}  
\beq 
K^{} (y',\bx{}',t'; y,\bx{},t) 
=  \sum_{N=0}^{\infty}  \bar{f}_N(y') f_N(y) D_N^{} (\bx{}'-\bx, t'-t),
\eeq
where  $D_N^{}$ denotes a Green function of 
the spinor field of  mass $m_N$. In doing so the type 
of the Green   function $D$ should coincide  with the 
type of the Green function $K$. 
Note that at large space-time 
separations $|x'-x|\gg \hbar/m$ 
%only the contribution of the term with $N=0$ survives
the contribution of the term with $N=0$ dominates, so that 
the asymptotic space-time behavior of corresponding Green functions 
is that of a spinor particle with mass $\half m$.    
%(as a matter of fact, the multiple $\half$ here depends on the 
%ordering of operators in $H$).   
We hope to present a more detailed analysis elsewhere. 




%\section*{Generalized Schr\"odinger Equation}

\section*{The Correspondence Principle}
 
In this section we discuss three properties 
of Eq. (5) which make it a proper candidate 
to the  Schr\"odinger equation 
within 
the %%?
polymomentum quantization. 
All three are in fact different aspects 
of the correspondence principle. 
 
Let us recall  first that the DW canonical theory leads 
to its own field theoretic generalization of the Hamilton-Jacobi 
theory \cite{rund,kastrup}. The corresponding Hamilton-Jacobi equation 
is a partial differential equation 
%for 
on $n$ functions 
$S^\mu=S^\mu(y^a,x^\nu)$ 
   %on the configuration space 
$$%\beq
\der_\mu S^\mu+ H(x^\mu, y^a, p_a^\mu = \der S^\mu / \der y^a )=0.
$$%\eeq
In a simple example of scalar fields (6)  the 
DW  Hamilton-Jacobi  equation 
reads 
%takes the form 
\beq
\der_\mu S^\mu = -\half \der_a S^\mu \der_a S_\mu 
- \half \frac{m^2}{\hbar^2} y^2. 
\eeq
Now, if we substitute 
(a hypercomplex analogue of) 
the quasiclassical ansatz 
\beq 
\Psi = R \, \exp (iS^\mu \ga_\mu / \hka ) \eta, 
\eeq  
where $\eta$ is a constant reference spinor,  
to  
 %Eq. (5) with (10) as the DW Hamiltonian operator,  
(5) and (10) 
 we obtain 
a set of equations 
which can be transformed to 
the form  \cite{bial97}
%(for more details see \cite{bial97})  
\beq
\der_\mu S^\mu = -\half \der_a S^\mu \der_a S_\mu 
- \half \frac{m^2}{\hbar^2} y^2 
+\half \hbar^2 \kappa^2 \frac{\lapl R }{R}, 
\eeq
$$
\der_a S^\mu \der^a S_\mu=\der_a|S| \der^a |S| ,  
\quad    
\der_\mu S^\mu = \frac{S^\mu}{|S|} \der_\mu |S| . 
 \refstepcounter{equation} 
\eqno {  (\theequation a,b) }
$$
In the  first of these we recognize the  
DW Hamilton-Jacobi equation 
(13) with an additional 
term $\half \hbar^2 \kappa^2 \lapl R/{R}$   
which is similar to the 
so-called {\em quantum potential} known in quantum mechanics \cite{bohm} 
and vanishes in the classical limit $\hbar\rightarrow 0$.  
Last two equations are supplementary conditions which  appear 
most likely  due to the fact that the quasiclassical 
ansatz (14) does not represent a most general spinor, thus 
imposing  certain restrictions on  dynamics of the wave function. 
Note that in the 
case of quantum mechanics, $n=1$, conditions (16a,b) reduce 
to trivial identities. 
%interpret of conditions (16a,b)! 

Thus, it is argued 
that 
in the classical limit equation (5) leads to the DW Hamilton-Jacobi 
equation (with two supplementary conditions 
which are specific to field theory 
and probably are due to restrictions 
imposed by the chosen in (14) analogue of the quasiclassical ansatz).   

 
%Second 
Another aspect of the correspondence principle 
we are  to consider 
is the Ehrenfest theorem. Let us assume that  
%%? the  
expectation values of operators are given by 
\beq
\left < {} \what{O}{}\right > 
:= \int dy  \, \overline{\Psi}\what{O}{\Psi} , 
\eeq 
where $\overline{\Psi}$ is the Dirac conjugate of $\Psi$. 
These expectation values depend on 
%%?the 
space-time points 
as the averaging is performed only over the field space.   
Using 
%%?the 
generalized Schr\"odinger equation (5) 
with the DW Hamiltonian (10)  
we can show that \cite{bial97}  
%%(see \cite{bial97} for more details) 
\beq 
\der_\mu \left < {}\hat{p}{}_a^\mu {}\right > 
=- \left < {}\der_a \what{H} {}\right > ,  \\ 
\quad \der_\mu\left  <\widehat{y_a \omega^\mu} {}\right > 
= \left  < \what{p^\mu_a \omega_\mu}{}\right > . 
\eeq 
By comparing (18) with  DW Hamiltonian field 
equations (8) we conclude that the latter are fulfilled 
"in average" as a consequence of the representation of operators (3), 
%%the? 
generalized Schr\"odinger equation (5),  
and the definition of 
%%?the 
expectation values (17). 
However, it should be noted that 
this property is fulfilled only for specially 
chosen operators (try e.g. to evaluate $\der_\mu \left < {} y^a {}\right >$ 
to see that this will not yield  the desired result 
$\left < {}\hat{p}{^a_\mu}{}\right >$ for scalar fields).  
Moreover, the scalar product 
$
%\left < {} \what{O}{}\right > 
\int dy  \, \overline{\Psi}{\Psi} 
$ 
implied by definition (17) in general is 
not  positive definite 
 %constant on the space-time . 
and depends on points of the space-time. 
Therefore, it can not be used for a probabilistic 
interpretation. 
These drawbacks 
urge us 
to look for a more 
appropriate  version of the Ehrenfest theorem. 
 
An alternative is suggested by the fact that 
generalized Schr\"odinger  
equation (5) possesses 
a %%the?  
positive definite and time independent 
scalar product %\cite{bial97} 
\beq
 \int d\bx \int dy \Psib \beta \Psi  , 
\eeq 
where we introduced the notation 
%%$\ga^\mu = (\ga^i,\ga^t)$ $(i,j = 1,...,n-1)$. 
$\ga^\mu =: (\ga^i,\beta)$ $(i,j = 1,...,n-1)$, thus 
explicitly singling out the time variable $t:=x^n$ and the 
time component of $\ga$-matrices: $\beta:=\ga^t$ ($\beta^2=1$).  
The existence of the satisfactory scalar product 
of this kind
%, which 
%fulfills the requirements of the probabilistic interpretaion, 
%and  
%the nonexistence of the  satisfactory scalar product 
%which  involves only integration over the field space, 
implies  that the probabilistic interpretation of the wave function 
which fulfills 
generalized Schr\"odinger equation (5) is 
possible only if a time dimension is singled out. 
The wave function $\Psi(y^a,\bx,t)$ is  interpreted then 
as a probability amplitude of 
obtaining the field value $y$
%in the infinitesimal vicinity of the point  $y$ 
 %when measuring  
in the  
%infinitesimal vicinity of the 
space point $\bx$ 
in the moment of time $t$. 
%? at time $t$. 
%Hence 
As a result, the theory becomes very much similar to 
usual quantum mechanics of 
a fictious (spinor) particle in the space of variables $(y^a, \bx)$. 

Now, 
new (global) expectation values of operators 
can be defined by  
\beq  
\left  < \what{O} {}\right > 
:= \int dy \int d\bx \Psib \beta  \what{O} \Psi . 
\eeq 
These expectation values depend only on time. 
Using  definition (20) and 
generalized Schr\"odinger equation (5) written in the 
form 
\beq
i\hbar \der_t \Psi = - i \hbar \al^i \der_i \Psi  +  
\frac{1}{\kappa}\beta \what{H} \Psi , 
\eeq 
where 
%$\beta:=\ga^t$, 
$ \al^i := \beta \ga^i$,   
we obtain 
\beq
\der_t \left < {}y^a {}\right > =  \left < {} \what{p}^a_t {}\right >, 
\quad  
\der_t  \left < {} \what{p}_a^t {}\right > 
= - \left < {} \what{\der_i p^i_a} {}\right > 
- \left < {} \der_a \what{H} {}\right > . 
\eeq  
Note that in (22) we identified 
%%we need to identify 
$\what{\der_i p^i_a}$ with 
$-2i\hbar\kappa \ga^i \der_a \der_i $. 
 %Below it will become clear that 
This identification is 
consistent with yet another 
aspect  of the  correspondence principle 
a discussion of which follows. 

 
This  aspect is a relation 
between the classical equations of motion and the 
Heisenberg equations of motion of operators.  
From (21) it follows that the time evolution is 
given by the operator 
\beq
\what{\mbox{$\cal E$}}:= 
- i \hbar \al^i \der_i   + \frac{1}{\kappa}\beta \what{H} . 
\eeq 
Then, proceeding according to 
the standard quantum mechanics 
we obtain 
\beq
\der_t y^a = \frac{i}{\hbar} [\Eop,y^a] = \hat{p}^a_t , 
\quad  
\der_t \what{p}^t_a = \frac{i}{\hbar} [\Eop,\what{p}^t_a] 
= - \what{\der_i p^i_a} - \der_a\what{H} ,     
\eeq  
where we  assumed  as before 
\beq
\what{\der_i p^i_a} =  
-2i\hbar\kappa \ga^i \der_a \der_i . 
\eeq
 Hence, 
%we have shown that 
as a consequence of generalized 
Schr\"odinger equation (5) and the representation of operators (3),   
the Heisenberg equations of motion 
%of operators 
have the same form as 
%%?the 
classical  DW Hamiltonian equations (1) 
written in the form with a singled out time dimension. 

%the property familiar in quantum mechanics.  



%\section*{$\Psi(\lowercase{y,x^i, t})$ \lowercase{vs.} 
%$\Psi(  [\lowercase{y(x^i)}],\lowercase{t})$} 

\section*{Relation to the Schr\"odinger wave functional} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%A challenge to the scheme presented in previous sections is 
%the relation to the conventional quantum field theory. 
%Understanding  this relationship is important 
%for clarifying  a physical content of the scheme 
%and for its  possible   
%practical applications.  
%%%%%PUT TO CONCLUSION%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 


In this section a possible relationship between the    
Schr\"odinger wave functional in 
%the standard functional formulation of 
quantum field theory\cite{hatfield} 
and our 
%hypercomplex 
wave function is 
%discussed 
examined\footnote{The presentation  
here essentially follows an 
unpublished preprint by the author\cite{firststeps}.}. 
We confine ourselves  to the simplest example of 
a free real scalar field. For the seek of simplicity we 
henceforth put $n=3+1$ and $\hbar=1$. 

The idea is as follows. On the one hand, 
the  Schr\"odinger wave functional 
$\Psi([y(\bx)],t)$ is known to be a probability amplitude of 
the field configuration $y=y(\bx)$ to be observed in the moment 
of time $t$. 
On the other hand, 
our wave function $\Psi(y,\bx,t)$  
%has a meaning of 
can be interpreted as a probability 
amplitude of finding the value $y$ of the field 
in the point $\bx$ in the moment of time $t$. 
Hence, the wave functional could in principle be related to 
a certain composition of single amplitudes given by our 
wave function. 

Let us consider the Schr\"odinger functional 
corresponding to the vacuum state of 
a free scalar field\cite{hatfield} 
\beq   
\Psi_0([y(\bx )], t)= \eta
 \exp \left( 
i E_0 t 
-
\frac{1}{2 {}}  \int \! 
\frac{d \bk}{(2 \pi)^{3}}
\ \omk \, \tilde{y}{}(\bk) \tilde{y}{}(-\bk) 
\right) , 
\eeq
where the Fourier expansion  
$y(\bx) = \int \! \frac{d \bk}{(2 \pi)^{3}}\,y(\bk)
e^{i\bk \bx } $
is used, 
$\eta$ is a normalization factor, 
 $ \omega_\bk := \sqrt{m^2 + \bk^2}$,  
 and 
$E_0$ is the 
%(divergent) 
vacuum state energy 
\beq
E_0= \lim_{V\rightarrow\infty \atop Q\rightarrow\infty} 
%%\int d\bx g(\bx,\bx) = 
\half  \int_V d\bx  \int_Q \frac{d\bk }{(2\pi)^3}\omega_\bk 
%%=\lim_{V\rightarrow\infty} 
%%\half V \int \frac{d\bk }{(2\pi)^3} \ \omega_\bk . 
\eeq 
which is divergent if either the ultraviolet cutoff $Q$ of the 
volume of integration in  $\bk$-space 
or the infrared cutoff $V$ of the volume of integration over 
$\bx$-space  go to infinity. The symbol $\lim$ has a formal meaning 
throughout. 


By replacing the Fourier integral 
by the Fourier series according to 
the rule $\int \frac{d \bk}{(2 \pi)^{3}} \rightarrow 
\lim_{V\rightarrow\infty}\frac{1}{ V } 
\sum_{[\bk]}$, $[\bk] \in {\ZZ}^3$,  
the Schr\"odinger vacuum state functional can be written in the 
form of an infinite product 
of the harmonic oscillator ground state wave functions  
over all cells in $\bk$-space 
\beq
\Psi_0([y(\bx )], t) = \eta 
\lim_{V\rightarrow\infty} \prod_{[\bk] } 
\exp \, \half \left(  i  \omk t 
-  \frac{1}{ V {} } \ \omk {y}{}^2(|\bk |)  \right) . 
\eeq


Now, let us  consider the ground state ($N=0$) wave functions 
(cf. Eq. (11)) 
of generalized Schr\"odinger equation (5) 
for a free scalar field 
\beq
\Psi_{N=0,\bk} (y_\bx,\bx,t) = u_{N=0}(\bk) 
e^{  i  \omega_{0,\bk} t -i \bk \cdot \bx }
e^{-\frac{m}{2 \kappa} y_{}^2} , %y_{\bx }^2} .  
\eeq
where    
$\omega_{0,\bk}=\sqrt{(\frac{m}{2})^2 + \bk^2}$.   
 %They are probability amplitudes of obtaining 
 %the value $y_{\bx}$ when measuring the field in the 
 %point $\bx$ at time $t$.  
To simplify a subsequent analysis, 
which is in any case of preliminary character,  
we  ignore in what follows the spinor nature 
of the wave function encoded in  $u_{N=0}(\bk)$. 
 Taking into consideration the probabilistic interpretation 
of solutions (29) 
and assuming that 
%in the approximation that 
there are no correlations 
between the field 
  %%amplitudes 
values in space-like separated points,  
%(which physically corresponds to the ``ultra-local''  limit 
%in which the Feynman propagator between space-like separated 
%points  vanishes) 
the amplitude of funding in the vacuum state 
the whole configuration 
$y=y(\bx)$  
can be represented as an infinite product 
of single amplitudes given by 
the ground state solutions (29) with $y_{}=y(\bx)$ 
over all points $\bx$ of the space.  
In order to ensure the spatial  isotropy and homogeneity 
which are expected for the 
vacuum state    
we also have to take  a product over all possible 
values of wave numbers 
because    each  
%, because ... 
separate mode with a wave number $\bk$  violates 
these properties. 
This also 
%conforms 
agrees  with an idea of the vacuum state in 
which all possible $\bk$-states are filled. 
Hence, the following symbolic formula for 
the 
%ultra-local limit of 
approximate 
composed  vacuum  amplitude 
can be written (up to a normalization)
\beq 
\prod_{\bk } \prod_{\bx  } 
%%\left( 
e^{  i  \omega_{0,\bk} t -i \bk \cdot \bx } 
%
e^{-\frac{m}{2 \kappa {}} y(\bx)^2} .  %%y_{\bx }^2} 
%%\right)  
\eeq 
This 
%%symbolic 
expression can be 
assigned a meaning if a certain discretization in both 
$\bx$- and $\bk$-spaces is assumed. 
This discretization can be related to finite values of 
%%?the 
cutoff parameters $V$ and $Q$ which imply  
 %the availability of 
minimal 
volume elements in $\bk$-space and  in $\bx$-space to be, 
respectively, $(2\pi)^3/V=:\xi^3$ and   $(2\pi)^3/Q=:\lambda^3$.  
Then coordinates in $\bx$- and $\bk$-space are 
integers $[\bx]\in \ZZ^3$ and  $[\bk]\in \ZZ^3$ 
such that  $\bx=[\bx]\lambda$, $\bk=[\bk]\xi$.   
The continuum limit formally corresponds to 
$V\rightarrow\infty$ and $Q\rightarrow\infty$,  
however, an analysis of its existence 
 %%of such a limit 
in mathematical sense is beyond the scope of the present 
consideration.   
Using this discretization,  
the obvious identity 
$\prod_{\bk} e^{i\bk\cdot\bx}=1$, 
and the Fourier series expansion 
$y ({\bx}) =\frac{1}{V}\sum_{[\bk]}y_{\bk}e^{i\bk\cdot\bx},$ 
we obtain 
\beqa
&& \prod_{\bk }  
e^{  i  \omega_{0,\bk} t
} %-i \bk \cdot \bx } 
\prod_{\bx  }
e^{-\frac{m}{2 \kappa {}} y({\bx })^2} 
\nn \\
&=& 
\lim_{V\rightarrow\infty \atop Q\rightarrow\infty} 
\prod_{[\bk] } e^{  i  \omega_{0,\bk} t } 
%%%-i \bk \cdot \bx } 
 \prod_{[\bx]} 
\exp\left( -\frac {m}{2 \kappa {}} \frac{1}{V^2}\sum_{[\bq']}\sum_{[\bq'']} 
y_{\bq'}y_{\bq''}e^{i(\bq'+\bq'')\cdot \bx}   \right) 
\nn \\
&=&
\lim_{V\rightarrow\infty \atop Q\rightarrow\infty} 
\prod_{[\bk] } e^{  i  \omega_{0,\bk} t}   
\exp\left( -\frac {m}{2 \kappa {}} 
\sum_{[\bx]}  
\frac{1}{V^2}\sum_{[\bq']}\sum_{[\bq'']} 
y_{\bq'}y_{\bq''}e^{i(\bq'+\bq'')\cdot \bx}   \right) 
\nn \\
&=&
\lim_{V\rightarrow\infty \atop Q\rightarrow\infty} 
\prod_{[\bk] } e^{  i  \omega_{0,\bk} t  }
 \exp\left( -\frac {m}{2 \kappa {}} 
%\frac{VQ}{(2\pi)^3} 
\frac{QV}{(2\pi)^3}\frac{1}{V^2} 
\sum_{[\bq]} y_{\bq}y_{-\bq}\right) 
\nn \\
&=&
\lim_{V\rightarrow\infty \atop Q\rightarrow\infty} 
\prod_{[\bk] } \exp 
\left( {  i  \omega_{0,\bk} t} 
-\frac {m}{2 \kappa {}V} \frac{Q}{(2\pi)^3}y_{\bk}y_{-\bk} \right) 
,  
%\nn \\&=&
%\lim_{Q\rightarrow\infty}
%\exp \left( i  \omega_{0,\bk} t
%-\frac {m Q}{2 (2\pi)^3 \kappa {} } 
%y_{\bk}y_{-\bk} \right)  
\eeqa
where in passing to the fourth  line we have taken into  
account that the number of cells both in 
$\bx$- and $\bk$-space 
%%is 
is equal to $QV/(2\pi)^3$.   



Let us compare the composed amplitude 
(31) with the 
standard vacuum functional in the form 
(28).  
%Two principal  discrepancies can be noticed.  
%The first one is that 
Two additional parameters 
$\kappa$ and  $Q$  
 %are present 
appear in (31):  
$Q$ 
is 
an  (infinitely large)  ultra-violet cutoff of the volume in 
$\bk$-space, while  $\kappa$ 
%has appeared in (3.5....)   
is essentially the inverse 
of an  infinitesimal (or very small) volume element 
in $\bx$-space (cf. Eq. (3)), 
i.e a kind of fundamental length to the power 3. 
 {}From the physical point of view 
it is  quite natural to relate 
the inverse of the  fundamental 
length to the ultraviolet  cutoff. We thus 
 %%can 
identify  $\kappa=Q/(2\pi)^{3}$ 
%%(at least in the continuum limit) 
%and obtain the following expression  for the composed amplitude 
obtaining  the composed amplitude 
\beq
\lim_{V\rightarrow\infty}  %% \atop Q\rightarrow\infty} 
\prod_{[\bk] } \exp 
\left( {  i  \omega_{0,\bk} t} 
-\frac {m}{2 V} y_{\bk}y_{-\bk} \right) 
\eeq 
which is similar to (28) 
except that in (32) the proper mass $m$ appears instead 
of the frequency $\omega_{\bk}=\sqrt{m^2 + \bk^2}$ 
 %in (28), 
and $\omega_{0,\bk}$ replaces $\half \omega_{\bk}$.%in (28). 


It is easy to see that the discrepancy between 
(28) and (32) 
%vanishes 
disappears in the 
%very long wavelength, or ultra-local, 
ultra-local limit $|\bk| \ll m$.   
In this limit the two-point Wightman function 
$\left < {} y(\bx_1) y(\bx_2){}\right >$ 
between space-like separated points 
$\bx_1$ and  $\bx_2$ vanishes,   
so that there are no correlations between 
the field values in these points. 
This is, however, exactly the assumption 
which we made when 
writing the approximate composed amplitude in the form (30). 
Hence, in the ultra-local limit 
the composed amplitude constructed from the 
the ground state wave functions obeying generalized 
Schr\"odinger  equation (5) 
is consistent with  
the Schr\"odinger 
wave functional of the vacuum state (28).  
%%(of a free scalar field).   
%Note that 
Unfortunately, an attempt to extend this correspondence 
beyond the ultra-local limit 
leads to  a difficulty of 
writing an  expression for the composed amplitude 
similar to (30) which would account for all relevant 
correlations between the field values in space-like  
separated points. 
 
Note, that another important byproduct of our analysis 
in this section  
is a conclusion  that the constant $\kappa$ which appeared in 
(3) and (5) 
on purely dimensional grounds has to be identified 
with an  ultraviolet cutoff scale quantity. 


\section*{Conclusion}

Field theories can be viewed as multi-parameter 
Hamiltonian-like systems in which space-time variables 
appear on equal footing as analogues of 
the %%?a  
time parameter in mechanics. 
%The De Donder-Weyl theory is an example of such a formulation. 
A quantization 
%based on 
of such a version of the Hamiltonian formalism leads 
to an 
%%multi-parameter 
extension of the formalism of quantum mechanics in which the 
Clifford algebra of underlying space-time manifold 
plays a key role similar to that of complex numbers 
in quantum mechanics.  The latter thus appears as a special 
case of a theory with a single (time) parameter. 
In this formulation a description of 
quantized fields is achieved in terms of  
a (spinor) wave function on a finite dimensional analogue 
of the configuration space 
(the space of field and space-time variables). 
The wave function  satisfies a multi-parameter 
covariant generalization of the Schr\"odinger 
equation, Eq. (5), which is a partial derivative equation 
%very much 
similar to the Dirac equation with the mass term replaced 
by an operator corresponding to a multi-parameter (polymomentum) 
analogue of Hamilton's canonical function. 
%; this operator is 
%a differential operator in the space of field variables. 
%Our choice of the Schr\"odinger equation is consistent 
%with certain requirements of the correspondence principle. 
Note that despite the dynamics is formulated in a manifestly 
covariant manner 
%an evidence based on 
the consideration of 
scalar products 
%has been found 
suggests that a proper probabilistic 
interpretation of the wave function 
still may require a time parameter to be singled out. 


The  description outlined above appears to be very different from  
that  known in contemporary  quantum field theory.  
%Hence, a
A relation to the latter is a challenge  to the theory presented here.  
In this paper we pointed out  a relation 
to the Schr\"odinger wave functional 
which can thus far be followed only in  ultra-local approximation.  
However, the latter is too rough for the real physics.  
 %and 
Hence, further efforts are required to clarify possible 
connections with the standard quantum field theory.  

Note in conclusion, that 
 %if the problems with the physical interpretation 
 %of the present formalism and its relationship to the standard 
 %theory will be resolved it 
the present approach may have interesting applications to the 
problem of quantization of gravity and 
field theories on non-Lorentzian  space-times 
if  the problems with the physical interpretation   
%of the present formalism and its 
and the relationship to the standard 
quantum field theory are  resolved. 
Further discussion  can be found  in \cite{potsdam98} 
were a sketch of an approach to quantization of general relativity 
based on      the present framework is presented.   


%\begin{thebibliography}{99}
%\newcommand{\bib}[1]{\bibitem{#1}}


\begin{references}
\newcommand{\bib}[1]{\bibitem{#1}} 

%\footnotesize

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%\bib{Weyl35} 
H. Weyl, 
%{Geodesic fields in the calculus of variations, } 
 {\em Ann. Math. (2)}  {\bf 36}, 607 (1935). %%607-629 
 


\bib{rund} H. Rund,  
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\bib{giaquinta} M. Giaquinta, and S. Hilderbrandt,  
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\bib{gimm}M.J. Gotay, J. Isenberg, and J. Marsden,   
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\bib{gotay} M.J. Gotay,
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\bib{dedecker} P. Dedecker, 
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%(see also  his papers cited there) 


 
\bib{bial96} I.V. Kanatchikov, 
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\bib{gerst} M. Gerstenhaber, 
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\bib{qs96} I.V. Kanatchikov, 
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{\em Int. J. Theor. Phys.} {\bf37}, 333 (1998), {\tt quant-ph/9712058}.   %333-342 
%see also I.V. Kanatchikov, 
%Hypercomplex wave functions and the Born-Weyl quantization in 
%field theory, prep.rint March 1996, 16pp, 
%(presented as a part of a poster at the GROUP21 Colloquium, 
%Goslar, Germany 1996), unpublished 

\bib{bial97} I.V. Kanatchikov, 
{\em De Donder-Weyl theory and a hypercomplex 
extension of quantum mechanics to field theory},  
%preprint, May (1998), 
to appear in {\em Rep. Math. Phys.} (1998), 
{\tt hep-th/9810165}.   




\bib{bt} I.\,M. Benn and R.\,W. Tucker, 
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\bib{hestenes} D. Hestenes,  
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\bib{bohm} D. Bohm, B.J. Hiley,  and P.N. Kaloyerou, 
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%Holland, book\\
%Bohm and Hiley, book  \\


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\end{document} 

to appear in {\em Proc. Int. Seminar on Mathematical Cosmology,}   
Potsdam March 30 - April 4, 1998,   
eds.  M. Rainer, and H.-J. Schmidt,  
 Singapore: World Sci., Singapore, 1998. 



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\section*{Quantization of Gravity: A Proposal} 

The scheme of the polymomentum quantization described in previous sections 
can be applied 
to the problem of quantization of General Relativity. 
The first step would be the DW-like formulation. 
For the metric General Relativity the suitable formulation 
was presented by Ho\v rava \cite{horava}. 
Let the field variables be the components of the metric density 
$ h^{\alpha\beta}:= \sqrt{|g|}g^{\al\beta}$. 
Let us introduce the quantities 
\beq 
Q^\al_{\beta\gamma} := 
\half[\delta^\alpha_\beta\Gamma^\delta_{\gamma\delta} 
+ \delta^\alpha_\gamma\Gamma^\delta_{\beta\delta} ]
  - \Gamma^\al_{\beta\gamma} ,  
\eeq 
which will play the role of polymomenta and 
write the DW Hamiltonian in terms of polymomenta 
\beq
H(h^{\alpha\beta}, Q^\al_{\beta\gamma}) := 
h^{\alpha\ga} [ Q^\delta_{\al\beta}Q^\beta_{\ga\delta}+ 
\frac{1}{1-n}Q^\beta_{\al\beta}Q^\delta_{\ga\delta} ] , 
\eeq  

the Einstein field equations can be 
written 
%cast 
in 
the 
DW Hamiltonian form  (cf. Eq.~(1)) 
\beq    
\der_\al h^{\beta\ga}
= 
\der H / \der Q^\al_{\beta\ga} , 
\quad 
\der_\al Q^\al_{\beta\ga}
= 
- \der H / \der h^{\beta\ga}  . 
\eeq  
The first of Eqs. (19) is equivalent  to the well-known expression 
of the Christoffel symbols in terms of a metric. 
The second one yields the Einstein equations for the 
Christoffel symbols. 


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