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\begin {document}

\title{Dynamical Symmetries of Planar Field Configurations}
\tocauthor{Kh. S. Nirov}
\author{Khazret S. Nirov{\footnote{E--mail: 
nirov@ms2.inr.ac.ru}}\\
{\small \it Institute for Nuclear Research of the Russian 
Academy of Sciences,}\\ 
{\small\it 60th October Anniversary Prospect, 7a, 117312 Moscow, Russia}}

\date{}
\maketitle

\begin{abstract}
Hidden symmetries induced by spin dynamics of parity and
time-reversal invariant three-dimensional field configurations 
are discussed. The problem of lifting the dynamical symmetries
off the one-particle consideration onto the quantum field theory
level is solved.
\end{abstract}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}

Interest in planar, or three-dimensional, field theories is basically
motivated by perspectives they can give us for better understanding
critical phenomena generic to four-dimensional physics \cite{Ref1}. 
Indeed, the materials revealing critical phenomena, such as the 
high-temperature superconductivity and the quantum Hall effect, 
at certain conditions have quasi-planar structures \cite{Ref2}, 
so that, they can effectively be described by three-dimensional 
models. Note that such models are restricted to be parity and 
time-reversal invariant, because no violation of the discrete 
$P$- and $T$-symmetries has been so far observed in the related 
experiments \cite{exp}. 

The actual relevance of three-dimensional field systems to critical 
phenomena observed in the real world might be explained by some 
{\it hidden}, or {\it dynamical}, {\it symmetries}. The notion of 
dynamical symmetries was introduced by A. O. Barut in the middle of 
60s \cite{BaRa}. One of the classical examples of systems possessing 
dynamical symmetries is the well-known non-relativistic Kepler problem, 
with the $SO(4)$ group for the bound states, and $SO(3,1)$ for the 
scattering states of a particle in the Kepler potential. 

In many field systems, hidden symmetries are induced by specific 
{\it spin dynamics}, and so, the problem of revealing such symmetries 
and investigating their structures can be reduced to an appropriate 
description of the corresponding spin degrees of freedom. The problem 
of describing spin dynamics of field systems is carried out within the 
framework of the {\it pseudoclassical mechanics} \cite{pm}. This approach 
generalizes the concept of the classical mechanics to purely algebraic 
constructions of rings with commuting and anticommuting elements and 
graded Lie algebras, and thus considers elements of Grassmann 
algebras as classical dynamical variables (functions on the phase space). 
Explicitly, it allows one to describe spin degrees of freedom of 
quantum-mechanical and field systems by means of various superparticle 
(pseudoclassical) Lagrangians, so just at the classical mechanical level. 

In refs. \cite{NiPl1,NiPl2,NiPl3}, we proposed and studied a new 
pseudoclassical model by means of which the spin dynamics of the 
simplest parity and time-reversal conserving system of Chern--Simons 
fields was described. Actually, topologically massive $U(1)$ gauge 
fields are claimed to be the basic tool for constructing effective 
models for critical phenomena mentioned above \cite{who}. This 
justifies our choice of the object under investigation. An analysis 
of the local, discrete (space-time) and global symmetries of our 
particle-mechanical model allowed us to uncover dynamical $U(1,1)$ 
symmetry and $S(2,1)$ supersymmetry hidden in the topologically 
massive gauge fields' theory at the {\it one-particle level}. 
It was shown \cite{NiPl3} also that the dynamical $S(2,1)$
supersymmetry leads to a non-standard super-extension of the
three-dimensional Poincar\'e group labeled by the zero eigenvalue
of the corresponding superspin operator (the term `non-standard' 
here is to regard the feature of the supersymmetry generators whose
anticommutators result in an operator being different from the
one-particle Hamiltonian of the system). 

In this talk, we will discuss a natural way to lift the dynamical 
$U(1,1)$ symmetry, revealed at the one-particle level, onto the 
{\it quantum field theory} level: Such a lifting can be achieved 
while investigating dynamical symmetries of the reduced phase space 
Hamiltonian of the constrained field system in question.   


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The field theory in question}

Topologically massive gauge fields \cite{CS} can be presented in
terms of a self-dual free massive field theory \cite{self}. Along
this line of approach, the field system in question -- a simplest
combination of spin-up and spin-down Chern--Simons $U(1)$ gauge
fields -- is described by a doublet of vector fields 
${\cal F}^\mu_+$, ${\cal F}^\mu_-$ satisfying the linear first order
differential equations
\begin{equation}
{\cal L}^\pm_{\mu\nu} {\cal F}^\nu_\pm = 0, \qquad
{\cal L}^\pm_{\mu\nu} \equiv \varepsilon_{\mu\nu\lambda} 
\partial^\lambda \pm m \eta_{\mu\nu}, 
\label{eq1}
\end{equation}
where the metric and the totally antisymmetric tensors of the
three-dimen\-si\-o\-nal Minkowski space-time are fixed as
$\eta_{\mu\nu} = diag(-1,+1,+1)$ and $\varepsilon^{012} = 1$. 
Due to the equations (\ref{eq1}), the fields ${\cal F}^\mu_\pm$
obey also the Klein--Gordon equation 
$(-\partial^2 + m^2) {\cal F}^\mu_\pm = 0$
and the transversality condition
$\partial_\mu {\cal F}^\mu_\pm = 0$.
Hence, the vector fields ${\cal F}^\mu_\pm$ are carrying massive
irreducible representations of the spins $s = \mp 1$ of the
three-dimen\-si\-o\-nal Poincar\'e group, respectively.

Introducing the notation
$\Phi = {{\cal F_+} \choose {\cal F_-}}$ 
for the doublet of vector fields and using standard conveniences
with the Pauli matrices, we can rewrite the basic equations (\ref{eq1})
in the form
\begin{equation}
D \Phi = 0, \qquad
D \equiv PJ \otimes 1 + m\cdot 1\otimes \sigma_3, 
\label{eq2}
\end{equation}
where $P_\mu = -i\partial_\mu$, and 
$(J_\mu)^\alpha_\beta = -i{\varepsilon^\alpha}_{\mu\beta}$
are generators of the three-dimen\-si\-o\-nal Lorentz group in the
vector representation, 
$[J_\mu,J_\nu] = -i\varepsilon_{\mu\nu\lambda} J^\lambda$,
$J_\mu J^\lambda = -2$. 
Then, by definition \cite{BaRa}, the operators commuting with
the differential operator $D$ form the set of dynamical symmetry
group generators of the field theory with the action functional
\begin{equation}
{\cal A} = \int d^3x \Phi^T(x) D \Phi(x)
\label{A}
\end{equation} 
at the {\it one-particle level}. The explicit forms of such operators
and their (super)algebras were obtained in ref. \cite{NiPl3}. And so,
they represented hidden $U(1,1)$ symmetry and $S(2,1)$
supersymmetry groups of the parity and time-reversal 
conserving system of Chern--Simons fields (\ref{A}). 
It was also shown in ref. \cite{NiPl3} that the one-particle states of
the field system (\ref{A}) realize an irreducible representation of a 
non-standard super-extension of the $(2+1)$-dimensional Poincar\'e group,
$ISO(2,1|2,1)$, labeled by the zero eigenvalue of the corresponding
superspin operator. 

To see that these dynamical (super)symmetries are caused by certain 
spin dynamics, it is worthwhile noting that the field system 
(\ref{eq1})--(\ref{A}) is the quantum-mechanical counterpart 
of the pseudoclassical model with the Lagrangian 
\cite{NiPl1,NiPl2,NiPl3}
\begin{equation}
L_T = \frac{1}{2e} \left( \dot{x}_\mu 
- \frac{i}{2} v \varepsilon_{\mu\nu\lambda} \xi^\nu_a \xi^\lambda_a 
\right)^2 - \frac{1}{2} e m^2 - iqmv \xi^\mu_1 \xi_{2\mu} 
+ \frac{i}{2} \xi^\mu_a \dot{\xi}_{a\mu}.
\label{LT}
\end{equation}
The configuration space of this particle-mechanical system is represented
by the set of even, $x_\mu$, $e$, $v$, and odd, $\xi^\mu_a$, variables,
$\mu=0,1,2$, $a=1,2$. In this, $x_\mu$ are the space-time coordinates of
the particle, $e$ and $v$ make sense as Lagrange multipliers, with $e$
being the world-line metric, and the odd variables $\xi^\mu_a$ are
incorporated specifically for spin degrees of freedom of the 
three-dimen\-si\-o\-nal field theory anticipated as quantum-mechanical
counterpart of the pseudoclassical model (\ref{LT}). Additionally,
$q$ is a real $c$-number parameter kept in order to demonstrate 
symmetry properties of the model and its such an interesting feature 
as revealed in ref. \cite{NiPl1} and called there {\it the classical 
quantization of the model parameter}. 
To find the field system corresponding to the pseudoclassical model 
(\ref{LT}), its general quantum state $\Psi(x)$ can be realized over 
the vacuum as an expansion into the complete set of eigenvectors of 
the fermion number operator \cite{NiPl1,NiPl2,NiPl3}. The coefficients 
of this expansion will belong to a rigged Hilbert space. 
The latter is because of the mass-shell, $\phi \approx 0$, and quadratic 
in spin variables nilpotent, $\chi \approx 0$, constraints following from 
the Lagrangian $L_T$. The quantum counterparts of these first class 
constraints are to single out the physical subspace of the theory. 

Further, the initial point in constructing the topologically
massive fields' system (\ref{A}) out of the particle-mechanical 
model (\ref{LT}) is just an extract from the postulates of quantum
theory: If the linear (Hilbert) space framework provides the kinematical
principle of quantum theory, the explicit calculation of states and their
scalar products, that is the concrete realization of the linear space, form
the dynamical part of quantum theory \cite{BaRa}. Having this viewpoint, 
the transition from the quantum-mechanical counterpart of the pseudoclassical 
model to the corresponding field theory can be performed through such a 
realization of the Hilbert state space and scalar product. Then, the 
quantum-mechanical quantities averaged over the state space (physical
subspace) with respect to an appropriate scalar product will appear to 
be the desirable field theory's objects (in particular, hermiticity
of physical operators must be considered as the fundamental criterion
to propose the scalar product). 

Specifying, the doublet of vector fields satisfying the equations 
(\ref{eq1}) turns out to be the consistent solution to the quantum 
constraints $\widehat\phi \Psi = \widehat\chi \Psi = 0$. This is a 
subspace of the total state space related to the special values 
$|q| = 2$ of the model parameter. In fact, the constraint $\chi$
plays the role of Hamiltonian for the spin variables in the model
(\ref{LT}), and the operator $D$ is its quantum analog on the
physical subspace \cite{NiPl3}. Exactly this observation allows
one to link our pseudoclassical model to the field system in question. 
Dynamical (super)symmetries of the field theory (\ref{A}) are
generated by average values of the quantum counterparts of
the integrals of motion of the system (\ref{LT}) with $|q| = 2$.
The corresponding quantum mechanical nilpotent operators
\cite{NiPl3} realize mutual transformation of the physical 
states of spins $+1$ and $-1$.

Strikingly enough, the procedure just described is a resemblance to 
the idea of constructing a string field theory via BRST formalism
put forward by W. Siegel \cite{Siegel} and subsequently developed by 
E. Witten \cite{Witten}. There, an object of the form 
$A = \int d\mu \langle\Psi\vert\Omega\vert\Psi\rangle$,
with a BRST operator $\Omega$ singling out physical states 
and $d\mu$ being an integration measure, was proposed as a 
string field theory action. Also, a scalar product 
$\langle\vert\vert\rangle$ was fixed to ensure hermiticity 
for the BRST operator. The underlying idea was taken from the 
observation that the functional $A$ is extremal on the physical
subspace: the variational principle applied to the ``action'' 
$A$ reproduces ``quantum equations of motion'' encoded in the 
BRST condition $\Omega\vert\Psi\rangle = 0$, and besides, it keeps 
symmetries of the initial first-quantized theory. We dealt with an 
analogous construction, while having the constraint operator 
$\widehat\chi$ instead of the BRST-charge $\Omega$ and the finite-mode 
decomposition of the general state vector $\Psi(x)$ where 
the expansion coefficients were, in particular, topologically massive 
$U(1)$ gauge fields. As well as in the string field theory 
\cite{Siegel,Witten}, the action ${\cal A}$ detains symmetries 
of its first-quantized counterpart.

Now, to generalize dynamical symmetries to the {\it quantum field theory
level}, we need to quantize the system with the action functional (\ref{A}), 
already treating the doublet ${\cal F}^\mu_\pm$ not as a wave function piece, 
but as dynamical field variables. We will concern this problem in the 
next section within the framework of the Hamiltonian formalism. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Hamiltonian description of the field system}

To construct the Hamiltonian description of the field theory 
(\ref{A}), let us introduce, as usual, the phase space with 
the generalized momenta ${\cal P}^\mu_\epsilon(\vec{x},t)$ 
canonically conjugate to the generalized coordinates 
${\cal F}^\mu_\epsilon(\vec{x},t)$, $\epsilon = +,-$, 
\begin{equation}
\{ {\cal F}^\mu_\epsilon(\vec{x},t) , 
{\cal P}^\nu_{\epsilon^\prime}(\vec{y},t) \} = 
\delta_{\epsilon\epsilon^\prime} \eta^{\mu\nu}
\delta(\vec{x} - \vec{y}).
\label{pois}
\end{equation}
There are primary constraints in the system,
\begin{equation}
{\cal P}^0_\epsilon \approx 0, \qquad
{\cal P}^i_\epsilon 
+ \varepsilon^{0ij} {\cal F}^j_\epsilon \approx 0,
\label{prim}
\end{equation}
and hence, the total Hamiltonian 
$H_T(t) = \int d^2\vec{x}\; h_T(\vec{x},t)$
contains Lagrange multipliers
$\lambda^0_\epsilon$ and $\lambda^i_\epsilon$; we have
\[
h_T = h + \sum_{\epsilon = +,-} 
\lambda^0_\epsilon {\cal P}^0_\epsilon + \lambda^i_\epsilon 
\left( {\cal P}^i_\epsilon 
+ \varepsilon^{0ij} {\cal F}^j_\epsilon \right),
\]
where the density of the canonical Hamiltonian $h(\vec{x},t)$ 
is given by the expression
\[
h = \sum_{\epsilon = +,-} \varepsilon^{0ij} \left(
{\cal F}^j_\epsilon \partial_i {\cal F}^0_\epsilon
- {\cal F}^0_\epsilon \partial_i {\cal F}^j_\epsilon \right)
+ \epsilon m \left( {\cal F}^0_\epsilon {\cal F}^0_\epsilon 
- {\cal F}^i_\epsilon {\cal F}^i_\epsilon \right).
\]
Under subsequent application of the Dirac-Bergmann formalism 
\cite{DB}, secondary constraints arise in the system,
\begin{equation}
\varepsilon^{0ij} \partial_i {\cal F}^j_\epsilon 
- \epsilon m {\cal F}^0_\epsilon \approx 0, 
\label{sec}
\end{equation}
and the Lagrange multipliers get fixed,
$\lambda^0_\epsilon = - \partial_i {\cal F}^i_\epsilon$,
$\lambda^i_\epsilon = - \partial^i {\cal F}^0_\epsilon 
+ \epsilon m \varepsilon^{0ij} {\cal F}^j_\epsilon$.
The complete set of the constraints (\ref{prim}), (\ref{sec}) is
of the second class. We see that the initial phase space can be
reduced to the corresponding constraint surface. Restricting the
Poisson brackets (\ref{pois}) onto this reduced phase space, one
obtains the related Dirac brackets. The physical degrees of freedom,
i.e. the points of the reduced phase space, may now be described
by the set of the field variables ${\cal F}^i_\epsilon(\vec{x},t)$,
$i=1,2$, $\epsilon=+,-$, having the Dirac brackets
\begin{equation}
\{ {\cal F}^i_\epsilon(\vec{x},t) , 
{\cal F}^j_{\epsilon^\prime}(\vec{y},t) \}^* = 
\frac{1}{2} \delta_{\epsilon\epsilon^\prime} \varepsilon^{0ij} 
\delta(\vec{x} - \vec{y})
\label{Db}
\end{equation}
and satisfying the equations of motion
\[
\dot{\cal F}^i_\epsilon = -\epsilon \varepsilon^{0ij}\,
\Delta_{jk}\, {\cal F}^k_\epsilon, 
\qquad 
\Delta_{ij} = m \delta_{ij} - \frac{1}{m} 
\varepsilon_{0ik} \partial^k \varepsilon_{0jl} \partial^l.
\]
It is worthwhile noting that one has also the equations
$\dot{\cal F}^0_\epsilon = - \partial_i {\cal F}^i_\epsilon$
confirming the above mentioned transversality condition. 
In terms of these variables, converted into the representation
\begin{equation}
{\cal F}^i_\epsilon(\vec{x},t) = \frac{1}{2\pi} \int d^2\vec{p}\;
{\rm e}^{i\vec{p}\vec{x}}\, \tilde{\cal F}^i_\epsilon(\vec{p},t),
\qquad
\overline{\tilde{\cal F}}{}^i_\epsilon(\vec{p},t) 
= \tilde{\cal F}^i_\epsilon(-\vec{p},t), 
\label{arep}
\end{equation}
the Hamiltonian of the system on the reduced phase space takes 
the form
\begin{equation}
H^* = - \int d^2\vec{p} \sum_{\epsilon=+,-} \epsilon \;
\tilde{\cal F}^i_\epsilon\, \tilde\Delta_{ij}\, 
\tilde{\cal F}^j_\epsilon,
\qquad
\tilde\Delta_{ij} \equiv m \delta_{ij} 
+ \frac{1}{m} \varepsilon_{0ik} p^k 
\varepsilon_{0jl} p^l.
\label{hamr}
\end{equation}
Seeing that the operator $\tilde\Delta_{ij}$ allows for the
representation
\[
\tilde\Delta_{ij} = \omega_{ik} {\omega^k}_j, \qquad
\omega_{ij} \equiv \sqrt{m} \delta_{ij} + \frac{1}{\sqrt{m}}
\frac{\varepsilon_{0ik}p^k\varepsilon_{0jl}p^l}{\sqrt{m^2+\vec{p}^2}+m},
\]
so that the new fields 
$u^\epsilon_i = \omega_{ij} \tilde{\cal F}^j_\epsilon$ 
are well-defined, the Hamiltonian $H^*$ can be given another 
convenient form
\[
H^* = \int d^2\vec{p} \left( u^-_i u^-_i - u^+_i u^+_i \right).
\]
Finally, passing on to the complex variables
$z_\epsilon = u^\epsilon_1 + i u^\epsilon_2$ 
with the canonical brackets
\[
\{ z_\epsilon(p) , \bar{z}_{\epsilon^\prime}(q) \}^*
= - i |p^0| \delta_{\epsilon\epsilon^\prime}
\delta(\vec{p} + \vec{q}), 
\qquad
|p^0| \equiv \sqrt{m^2 + \vec{p}^2},
\]
we get for the reduced phase space Hamiltonian the expression
\begin{equation}
H^* = \int d^2\vec{p} \left( |z_-|^2 - |z_+|^2 \right).
\label{H*}
\end{equation}
The latter is exactly the well-known \cite{BaRa} defining quadratic 
form for the symmetry group $U(1,1)$ with the non-compact 
group manifold ${\cal M}(U(1,1)) \cong S^1 \times R^2$. 
Note that if one rescales the complex variables $z_\pm$ as
${|p^0|^{1/2}} z_\epsilon \rightarrow z_\epsilon$, they become 
the classical counterparts of the oscillator annihilation-creation 
operators obeying the standard commutation relations
\begin{equation}
[\widehat{z}_\epsilon(p),\widehat{\bar z}{}_{\epsilon^\prime}(q)]
= \delta_{\epsilon\epsilon^\prime} \delta(\vec{p} + \vec{q}).
\label{1}
\end{equation}
Let us introduce a positive-definite scalar product $( \, , )$
on the corresponding state space and denote the respective Hermitian 
conjugation by the dagger $\dagger$. Then the operators 
$\widehat{z}_\epsilon$ and $\widehat{\bar z}{}_\epsilon$ are mutually 
conjugate with respect to the scalar product $( \,, )$. We have
\begin{equation}
{(\widehat{z}_\epsilon)}^\dagger = \widehat{\bar z}{}_\epsilon.
\label{2}
\end{equation}
The explicit forms of the dynamical symmetry group generators can 
easily be found (see e.g. ref. \cite{BaRa}). They are simply the 
conserved charges of the field theory with the Hamiltonian being 
the quantum counterpart of $H^*$. Their densities are given by the
expressions
$$
u = \frac{1}{2}\left( \widehat{\bar z}{}_-\widehat{z}_- 
- \widehat{\bar z}{}_+\widehat{z}_+ \right), \quad
q_0 = \frac{1}{2}\left( \widehat{\bar z}{}_-\widehat{z}_- 
+ \widehat{\bar z}{}_+\widehat{z}_+ + 1 \right), \quad
$$
\begin{equation}
q_+ = \widehat{z}_+\widehat{z}_-, \quad
q_- = \widehat{\bar z}{}_-\widehat{\bar z}{}_+ .
\end{equation}
In this, $u$ is the density of the $U(1)$ generator, 
while the rest fulfil the $su(1,1)$ algebra 
\[
[ q_+ , q_- ] = 2 q_0, \qquad
[ q_\pm , q_0 ] = \pm q_\pm.
\]
The operators $u$ and $q_0$ are Hermitian, and $q_+$ and $q_-$
are mutually conjugate with respect to the scalar product $(\,,)$,
${(u)}^\dagger = u$, ${(q_0)}^\dagger = q_0$, 
${(q_\pm)}^\dagger = q_\mp$.
We have thus reproduced the dynamical 
$U(1,1) = U(1)\times SU(1,1)$ symmetry of the 
$P$,$T$-invariant system of topologically massive vector fields 
at {\it the quantum field theory level}. Certainly, one could 
gain the same ends while using not the auxiliary representation
(\ref{arep}), but the ordinary decomposition of the fields
${\cal F}^i_\epsilon$ over annihilation and creation operators.

Instead of considering the action ${\cal A}$, one might deal with
another action functional
\begin{equation}
{\cal A}^\prime = - \int d^3x \Phi^T(x) D^\prime \Phi(x),
\qquad
D^\prime = (1\otimes\sigma_3) D
\label{A'}
\end{equation}
leading to {\it the same equations} (\ref{eq1}), (\ref{eq2}) for the 
Chern-Simons vector fields. Constructing the reduced phase space 
Hamiltonian now reads:
\begin{equation}
H^{\prime *} = \int d^2\vec{p} \sum_{\epsilon=+,-}
\tilde{\cal F}^i_\epsilon\, \tilde\Delta_{ij}\, 
\tilde{\cal F}^j_\epsilon = 
\int d^2\vec{p} \left( |z_-|^2 + |z_+|^2 \right).
\label{H'}
\end{equation}
In contrast with the preceding analysis, one should thus find the
dynamical symmetry group of the field theory with ${\cal A}^\prime$
to be $U(2)$ having the compact group manifold 
${\cal M}^\prime(U(2)) \cong S^3$, 
as it seems obvious from the defining quadratic form (\ref{H'}). 
However, the dynamical symmetry group is again $U(1,1)$.
This result on continuous global symmetries hidden in (\ref{A}) 
and (\ref{A'}) looks somewhat enigmatic, if one takes into account 
only the equations of motion and ignores the corresponding discrete 
symmetries. Indeed, the action ${\cal A}$ is odd under the parity
and time-reversal transformations, 
$P,T: {\cal A} \rightarrow -{\cal A}$, 
although it describes physical states with the spins $-1$ and $+1$,
being thus respective to a $P,T$-invariant system. The price for such
a disorder is that the reduced phase space Hamiltonian $H^*$ is not 
positive-definite. On the other hand, ${\cal A}^\prime$ is parity 
and time-reversal invariant, and its reduced phase space Hamiltonian
$H^{\prime *}$ is positive-definite, but now the oscillator-like
operators have the following commutation relations:
\begin{equation}
[\widehat{z}_\epsilon(p),\widehat{\bar z}{}_{\epsilon^\prime}(q)]
= - \epsilon \delta_{\epsilon\epsilon^\prime} 
\delta(\vec{p} + \vec{q}).
\label{1'}
\end{equation}  
As a consequence, we get 
\begin{equation}
{(\widehat{z}_\epsilon)}^\dagger = 
- \epsilon \widehat{\bar z}{}_\epsilon
\label{2'}
\end{equation} 
for the Hermitian conjugation with respect to our positive-definite
scalar product $(\,,)$. The last two relations are essentially 
different from the corresponding eqs. (\ref{1}) and (\ref{2}). 
Taking into account these properties, we obtain the densities of the
dynamical symmetry group $U(1,1)$ generators of the system (\ref{A'}),
(\ref{H'}) to be of the form
$$
u^\prime = \frac{1}{2}\left( \widehat{\bar z}{}_-\widehat{z}_- 
+ \widehat{\bar z}{}_+\widehat{z}_+ \right), \quad
q^\prime_0 = \frac{1}{2}\left( \widehat{\bar z}{}_-\widehat{z}_- 
- \widehat{\bar z}{}_+\widehat{z}_+ + 1 \right), \quad
$$
\begin{equation}
q^\prime_+ = i \widehat{z}_+\widehat{z}_-, \quad
q^\prime_- = i \widehat{\bar z}{}_-\widehat{\bar z}{}_+ .
\end{equation}
They satisfy the same relations as do the above described 
(non-primed) $U(1,1)$ generators. 

We have observed that the hidden symmetries of the planar free
field system (\ref{A}), revealed when explicitly modelling the 
underlying spin dynamics at the one-particle level, appeared to 
be the dynamical symmetries of the corresponding field theory's 
reduced phase space Hamiltonian. And again, as well as for the 
dynamical picture of the pseudoclasical gauge system 
\cite{NiPl1,NiPl2,NiPl3}, 
the discrete symmetries turned out to be of crucial importance 
for the continuous global symmetries of this field theory. 
However, we must mention that it is rather difficult, if only 
feasible, to explain the hidden {\it supersymmetries} of the 
system under consideration working on exclusively the field 
theory level (say, performing fermionization). It seems 
very likely that in order to see the dynamical symmetries to be 
accompanied by the supersymmetries leading to non-standard 
super-extensions of the Poincar\'e group \cite{NiPl3}, it 
is necessary to investigate the corresponding pseudoclassical 
models. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion}

We have discussed dynamical symmetries uncovered in the simplest
parity and time-reversal invariant system of Chern--Simons fields.
Revealed first at the one-particle level by means of the corresponding 
pseudoclassical particle model, the dynamical symmetries have been 
lifted onto the quantum field theory level.  

Note that similar analysis can be provided for the $P,T$-invariant
three-dimensional fermion system \cite{GPS}, too. The motivation for 
research into these field systems is usually based on the claim about 
their relevance to critical phenomena in planar physics. Therefore, 
the actual problem to be further concerned in view of the present 
analysis would be to find any possible development of the discussed 
dynamical (super)symmetries in relation to physical processes. However,
these systems must be regarded as toy models only, while being 
useful for the matter of applying methods and proving their power.
Indeed, the systems turned out to have a fatal disadvantage: 
either the corresponding field theories' reduced phase space Hamiltonians 
are not positively definite, so that, in the quantum theory their spectra 
are not bounded from below, or a half of the reduced phase space variables, 
meant as the physical variables, are quantized with an indefinite metric. 
This circumstance is a result of the requirement for the systems under 
consideration to have the discrete space-time symmetries. It can mean, 
for example, that the parity and time-reversal invariance should be 
broken in more `realistic' models, although experiments have not 
revealed such a violation so far. 




\small

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\end{thebibliography}


\end{document}



