%Paper: hep-th/9409117
%From: Vladimir V. Tovstiak <tovstiak@pem.kharkov.ua>
%Date: Tue, 20 Sep 94 12:46 GMT+0300

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%COVARIANT QUANTIZATION%%%%%%%%%%%%%%%%%%%%%%
%%%%OF $d=4$ BRINK-SCHWARZ SUPERPARTICLE%%%%%%%%%%%%%%%
%%%%%%%%%%%WITH LORENTZ HARMONICS%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%S.O. Fedoruk, V.G. Zima%%%%%%%%%%%%%%%%%%%%%%












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\begin{center}
{\bf{\Large COVARIANT QUANTIZATION OF $d=4$ BRINK-SCHWARZ SUPERPARTICLE
WITH LORENTZ HARMONICS}}
\vspace{1cm}

\centerline{\bf Fedoruk S. O., Zima V. G.}

\begin{center}{\it\small  Kharkov State
University,\\ Svobody Sq., 4, Kharkov, 310077, Ukraine.}
\end{center}\bigskip\nopagebreak


\end{center}

\begin{abstract}
         The first and second covariant quantization of a free massless $d=4$
superparticle with pure gauge auxiliary spinor variables have been carried out.
Arised chiral harmonic superfield corresponds to superparticle with finite
superspin and it is interpreted as Radon transformation of free massless finite
superspin superfield. A prescription for the second quantization procedure of
harmonic fields has been formulated. Microcausality is achievable only under
description of the particles by cohomology class of harmonic superfields with
integer negative homogeneity index and ordinary connection between spin and
statistics.
\end{abstract}

\vspace{1cm}
        Fermion constraints of Brink-Schwarz superparticle [1] are a mixture
of first and second class constraints. Their covariant separation
which is necessary for carrying out Dirac quantization procedure [2] is
impossible without use of auxiliary variables. Introduction of the last ones
must be accompanied by introduction of gauge symmetries which exclude these
variables under
a transition to the physical gauge. Among similar variables the Lorentz
harmonics [3-8] are especially attractive ones in consequence of
transparent group structure of corresponding symmetries and clearness of
physical sence (covariantization of light-cone gauge). Theories with
Lorentz harmonics in contrast to other have variables where
at classical level under first quantization it is placed an information
concerning
Lorentz group representation by which the correspond superfield is
transformed. Lorentz harmonics may be added to any particle model. So their
application to Brink- Schwarz superparticle may be considered as an
example of their using.

      Operator quantization, which we choose, is convenient for studying of
spectrum (one from the purposes of our work). Say, BFV-BRST quantization [9],
in general it does not involve difficulties, should supplement the
investigation by superfluous technical steps and tangle the
situation by excessive gauge variables.

     To avoid complications associated with infinite connectivity of spinor
group and corresponding problem of statistics in space-time dimension $d=3$
or with nonconcidence of spinor group and any classical group in $d=10$
let us restrict ourself to well investigated case of $d=4$, where the
equivalence of known approaches is traced easily [4-6,10].

        Noncompactness of the gauge group admiting existence of reducible
but operator-irreducible representation requires to supplement Dirac
quantization
prescription by a step which associates wave function with a cohomology class
appearing as a result of direct application of the prescription. To this end it
is built in general bijective translationally invariant correspondence between
harmonic superfields and those nongauge ones which are transformed under
irreducible representation of Lorentz group. The correspondence is realized
by an integral transformation [11] which is Radon's transformation.
In case of finite-dimensional representation the
classes of harmonic fields take a part in the correspondence.

        Twistorial representation for energy-momentum vector of massless
superparticle
appearing naturally in the approach limits its spin by finite values. So the
representations
with infinite spin are excluded. In addition the microcausality condition
eliminates
infinite-component fields which are admissible in principle.

        Substitution of 4-momentum written in adapted to it light-cone basis
in second quantized nongauge massless finite spin superparticle field
represents it as a result of above-mentioned integral transformation
of corresponding harmonic superfield. This result obtained for scalar
field in [8] makes possible to formulate for harmonic superfields a
prescription of second  quantization.

        Equations of motion for massless fields are dual to harmonicity
condition.

        Harmonic space of superparticle is compact one.

        Lorentz harmonics are commuting Weyl spinors $v^\pm$ which can be
considered as columns of $2\times2$ matrix $v$. By kinematic constraint
 $v^-v^+=1$ this matrix is turned into an element of
 $SL(2,C)$ group and pair of harmonic spinors
becomes a normalized basis of spinor space. It is trivially written a system
of first class constraints which are bilinear on harmonics $v^\pm$ and their
canonically conjugated momenta $p^\mp_v$, form algebra of the group and
preserve
the normalization condition. The unique diagonal
operator of mentioned type not entered into the algebra is the generator
 $\chi\equiv v^-p^-_v+v^+p^-_v$ of dilation in spinor space. Its equality to
zero gives gauge condition for
kinematic constraint with regard to which this constraint is considered as
fulfiled in
strong sense. Two diagonal operators from the algebra, namely real
 $d^{+-}=2Re(v^+p^-_v-v^-p^+_v)$ and imaginary $d^0=2iIm(v^+p^-_v-v^-p^+_v)$
ones, generate transformations from subgroups of group
 $SL(2,C)$ which correspond to subgroups
 $SO\uparrow(1,1)$ and
 $SO(2)$ of Lorentz group. Thus the full
preimages of these subgroups of Lorentz group under covering homomorphism act
on spinors
Eigenvalues of diagonal operators are called conformal and spin weights.
Harmonic spinors $v^\pm$ are their common eigenvectors, both weights of which
are equal to $\pm 1$. Therefore harmonic indices $\pm$ are collective ones. For
harmonic spinor they indicate values of both weights. Under conjugation spin
weight  of quantities, in particular of Lorentz harmonics, changes sign
whereas conformal weight remains invariable. By definition
 $\overline {(v^{\pm})}=\bar {v}^{\pm}$,
 $\overline {(p^{\pm}_v)}=\bar {p}^{\pm}_v$.

       Harmonic spinors define vierbein of light vectors
 $u_{\mu}^{\pm\pm}=v^{\pm}\sigma_{\mu}\bar {v}^{\pm}$,
 $u_{\mu}^{\pm\mp}=v^{\pm}\sigma_{\mu}\bar {v}^{\mp}$ which are normalized by
conditions $u^{--}u^{++}=-2$, $u^{+-}u^{-+}=+2$. This vierbein connects
orthonormalized frame with light one under preserving volume and not outputing
time axis from future light cone.

       Let us written the covariant form in superspace with coordinates
 $z^M=(x^{\mu},\theta^{\alpha},\bar {\theta}^{\dot {\alpha}})$ as
 $\omega_{\mu}=dx_{\mu}-\imath\,d\theta\sigma_{\mu}\bar
{\theta}+i\theta\sigma_{\mu}\,d\bar {\theta}$.
On superparticle world line $\omega=\dot{\omega}\,d\tau$, where $\tau$ is
development parameter. In Lagrangian
 $L=p\dot\omega-\frac e2 p^2$ the einbein $e$ is Lagrange multiplier, $p$ is
4-momentum. Constraints are $p^2\approx 0$ (masslessness),
 $d_{\theta}\equiv p_{\theta}+i\hat {p}\overline {\theta}\approx 0$
and c. c., where $\hat {p}\equiv p^{\mu}\sigma_{\mu}$. Their nonzero Poisson
brackets are $\{d_{\theta},\bar {d_{\theta}}\}=2i\hat p$.

        Lorentz harmonics are introduced by adding to superparticle Hamiltonian
the linear combination of constraints generating transformations from harmonic
 $SL(2,C)$. Coefficients of this linear combination
are Lagrange multipliers. In resulting phase space let us pass from
orthonormalized coordinates to light cone ones
 $x^{\pm\pm}=x^{\mu}u_{\mu}^{\pm\pm}$, $p^{\pm\mp}=p^{\mu}u_{\mu}^{\pm\mp}$,
 $\theta^\pm=\theta v^\pm$, $\bar p_{\theta}^\pm=\mp\bar p_\theta\bar
 v^\pm$ and similarly for other. At this canonical transformation the
harmonic spinors are invariable but harmonic constraints obtain terms
providing their action on light coordinates.

        Now we can, without loss of covariance, adapt light basis to
superparticle
momentum by putting the gauge conditions $p^{\pm\mp}\approx 0$. For nonzero
4-momentum, when at least one from the longitidual light coordinates is
nonzero,
say $p^{--}\neq 0$,these conditions are admissible for the boost harmonic
constraints $d_c^{--}$ and $\bar{d_c}^{--}$. Here index ``$c$'' denotes the
writing of constraints in light coordinates and, for example $d^{--}\equiv
v^-p_v^-$. In
this case the fermion constraints $d_\theta^+$, $\bar d_\theta^+$ are
first class but $d_\theta^-$, $\bar d_\theta^-$ are second class.
Remaining harmonic gauge subgroup is the minimal parabolic (Borel)
subgroup of the group $SL(2,C)$. This circumstance is interpreted as
compactness of the used harmonics, following from symmetry of the task and
not being a consequence of arbitrary choice.

        It is important that along with transformation from Borel subgroup the
considered gauge is preserved also under the discrete transformations, forming
the group $Z_2$ of rotations about the angle $\pi$ around an axis which is
perpendicular to the momentum. Choosing this axis as the second one we obtain
transformation being equivalent to rising - lowering of the harmonic spinor
indices. For harmonic spinors this means replacement of index ``$+$'' onto
index
``$-$'', and conversely, with simultaneous change of sign at one of this
replacement. This makes possible to assume that massless condition
 $p^2\approx 0$ at $p\neq 0$ is equivalent to condition $p^{++}\approx 0$ but
already without indicated discrete symmetry.

        General solution of kinematic constraint has the following form
 $v^-=v^-(v^+)+\xi^{--}v^+$, where
 $v^-(v^+)\equiv -\sigma_0\bar{v}^+ / (v^+\sigma_0\bar{v}^+)$. Independent
complex
variable $\xi^{--}$ transforms ingomogeneously under Lorentz transformations.
General solution of gauge condition $\chi=0$ for kinematic constraint is
 $p_v^+ =(v^+ p_v^-)v^+p_{\xi}^{++}v^-$, where $p_{\xi}^{++}$ is
canonically conjugated momentum to $\xi^{--}$. The last is obviously if
there is considered canonical transformation which connects harmonic variables
 $v^{\pm}$, $p_v^{\pm}$, satisfying kinematic constraint and gauge condition
for it, with independent variables $\xi^{--}$ and $p_{\xi}^{++}$, $v^+$ and
their new canonically conjugated momenta again denoted by $p_v^-$. In new
variables the harmonic constraints $d^{++}\equiv v^+ p^+$ and c. c. have the
form $d^{++}=p_{\xi}^{++}$, $\bar{d}^{++}=\bar{p}_{\xi}^{++}$. They make
possible to exclude variables $\xi^{--}$ and $p_{\xi}^{++}$ by imposing of
gauge condition $\xi^{--}=const$. Relevant Dirac brackets for remaining
quantities coincide with Poisson brackets for them which where up to exclusion.
Thus the spinor $v^-$ is excluded and in the beginning of the first
quantization procedure one has only one harmonic spinor $v^+$ in term of which
the remaining conformal and spin constraints have the forms
 $d^{+-}=2Re(v^+ p_v^-)$, $d^0=2iIm(v^+ p_v^-)$.


        Not all from the coordinates $x^{++}$, $\theta^{\pm}$,
 $\bar{\theta}^{\pm}$, $v^+$, $\bar{v}^+$, remaining after
exception of coordinates $x^{\pm\mp}$ by putting the gauge conditions, are
permutative owing to second class Fermi-constraints contribution in Dirac
brackets. Commutativity is restored by means of the transition to chiral
light coordinates $x_L^{++}\equiv x^{++}-2\imath\theta^+\bar\theta^+$,
 $x_R^{--}\equiv x^{--}+2i\theta^-\bar\theta^-$ (canonical transformation).
 It should be observed that such transformation is possible also
before intake of harmonics. Momenta $p_L^{--}$, $p_R^{++}$, canonically
conjugated to chiral coordinates, commute themselves and with all from
considered, nonconjugated for them, coordinates. Therefore under quantization
in the coordinate representation where coordinate operators are diagonal and
momenta are realized in terms of the differentiation operators we set
 $p_L^{--}=2i{\partial\over{\partial x_L^{++}}}$,
 $p_R^{++}=2i{\partial\over{\partial x_R^{--}}}$. Solution of masslessness
condition, imposed on wave function as first class constraint, does not
depend on the right chiral coordinate $x_R^{--}$.

       Excepting $\bar\theta^+$ and $\bar p_\theta^-$ by means of
the use of second class Fermi-constraints, we conclude that wave function,
defined by Dirac prescription, depends on the (anti)commuting coordinates
 $x_L^{++}$, $\theta^{\pm}$, $\bar\theta^-$, $v^+$, $\bar v^+$.
It is subjected to quantum analogs of the first class constraints
 $d_\theta^+$, $\bar d_\theta^+$, $d_c^{+-}$, $d_c^0$. Under linear
quantization the quantum
analogs of the first class constraints are built easily in term of the
differentiation operators.

       Ordering constants, named conformal $c$ and spin $s$ weights and been,
in general, complex-valued, arise in operators, associated to diagonal
constraints $d_c^{+-}$ and $d_c^0$, which contain the products of the
noncommuting canonically conjugated variables.

       Owing to Fermi-constraints the wave function $\Psi$ does not depend on
 $\theta^{-}$ and $\bar\theta^{-}$. In consequence of diagonal harmonic
constraints the wave function is multiplied by $e^{2ca}$ and $e^{2is\varphi}$
under the transformation of light coordinates from
 $SO\uparrow(1,1)$ ( $v^{\pm}\to e^{\pm a}v^{\pm}$, $a\in R$ ) and from
 $SO(2)$ ($v^{\pm}\to e^{\pm i\phi}v^{\pm}$,
 $0\leq\phi\leq4\pi$) respectively. By these it is justified a weighting of
suitable indices on it, $\Psi\equiv\Psi^{c,s}$.

       One-valuedness condition for the wave function under the
 $SL(2,C)$ transformations from indicated rotation
subgroup means the quantization of spin weight so that $2s\in Z$.

       Let us pass in the central basis, i. e. express light coordinates in
terms
of usual ones and harmonics. Then, the wave function as a function on $v^+$
is a homogeneous function of the index $\chi=(n_1,n_2)$, where
 $n_{1,2}=c\pm s+1$. Such infinite-differentiated functions are well studied
[11].
They correspond bijectively infinite-differentiable functions on the celestial
sphere, playing a role of superparticle harmonic space $C P^1$, which
is parametrized by auxiliary coordinates.

       In locally convex vector topological space $D_{\chi}$ of the
homogeneous infinite-differentiable functions of homogeneity index $\chi$
in complex affine plane $\stackrel {o} {{C}^2}\equiv C^2\setminus\{0\}$,
running by harmonic spinor $v^+$, it is realized the infinite-dimensional,
in general irreducible, representation of group
 $SL(2,C)$. Representations with opposite values
of homogeneity index are equivalent. Only in integer points, when both integer
or both half-integer $c$ and $s$ satisfy the condition $|c+1|>|s|$, in the
space
 $D_{\chi}$ there is invariant subspace with irreducible representation on it.
It is subspace $E_{\chi}$ of dimension $n_1 n_2$ in positive integer points,
when $c\geq 0$, and it is subspace $F_{\chi}$ of codimension $n_1 n_2$ in
negative integer points, when $c<0$. For integer positive index $\chi$
one have isomorphysms
 $D_{\chi}/E_{\chi}\simeq F_{-\chi}\simeq D_{(-n_1,n_2)}\simeq D_{(n_1,-n_2)}$,
 $E_{\chi}\simeq D_{-\chi}/F_{-\chi}$ of spaces and respective equivalences of
the
representations.

        Thus as a result of the first operator quantization of massless
superparticle
we obtain always infinite-component superfield (an homogeneous
infinite-differentiable function of index $\chi$ on harmonic spinor
 $v^+\in \stackrel{o}{C^2}$) which is subjected to harmonicity conditions,
guaranteing its dependence on supercoordinates via $x_L^{++}$ and $\theta^+$,
$$
  v^+\hat\partial_L\,\Psi^{c,s}=\hat\partial_L\overline v^+\,\Psi^{c,s}=0,
  \ v^+D_L\,\Psi^{c,s}=\bar D_L\,\Psi^{c,s}=0,
$$
where $D_L$, $\bar D_L$ are covariant spinor derivatives in chiral basis.
These equations are a part of those which are written in [6] for the wave
function in central basis. In our approach the exclusion of $v^-$ take a place
up to quantization and therefore the rest of equations from [6] is absent.
Independence of wave function on $v^-$ is discussed also in [12].

        It should be noted that values of even variables $x^{++}$ and $v^+$,
on which the harmonic superfield is dependent, by means of determining relation
 $x^{++}=x^\mu u_\mu^{++}$  define an isotropic hyperplane in space-time,
hence the harmonic superfield is actually the function on this isotropic
hyperplane. Therefore for passage to usual superfield, depending on
superspace point, it is necessary to perform (inverse) Radon
transformation [11]. As is known, the last realizes passage from
functions, given on set of some geometric objects (here they are
space-time points) to functions, given on set of another geometric objects
(here they are isotropic hyperplanes) with simultaneous transformation
of function supply. Of course the harmonic superfield being homogeneous
does not have in general the right homogeneity degree of Radon transformation,
therefore it is necessary to have a multiplicator  i. e. some
homogeneous function correct homogeneity degree of harmonic
superfield up to relevant.  Fixing of certain type multiplicator
defines object, compared to the harmonic superfield in result of the second
quantization. We shall take such as usual superfield with irreducible
representation of Lorentz group, realizing on its index. Then Radon
transformation can be found by direct calculations as a constituent of
Fourier transformation, which distinguishes many-dimensional variant of
last from one-dimensional one. Lower, after analysis of spectrum limiting
consideration by massless superfields of finite superspin, this
calculation is carryied out for second quantized fields.

         One can easily calculate that a supersymmetric generalization of
Pauli-Lubanski pseudovector is proportional to the 4-momentum on harmonic
superfield. Therefore a superhelicity is finite. It is opposite to the spin
weight $\Lambda=-s$; field components in expansion in $\theta^+$ describe
massless particles with helicities $\Lambda$ and $\Lambda+1/2$. Spin
finiteness in theories with twistor representation of 4-momentum (here
 $p_\mu=-\frac 12 p^{--}u_\mu^{++}$) is well known [13,14]. In contrast to the
twistor approach where energy positiveness is postulated, in harmonic one the
energy sign of usual (not super-) particle coincides with the sign of nonzero
light component $p^{--}$ and in case superparticle energy positiveness is
provided by supersymmetry.

         In contrast to the spin weight $s$ the conformal weight $c$ of
harmonic
superfield does not connect with characteristic of field particles. It
defines only concrete way of their field description, i. e. the representation
of group $SL(2,C)$ realized on superfield index. Infinite-dimensionness of
this representation in case field, obtained by quantization of harmonic
superparticle, assumes investigation of locality and feasibility, connected
with it, of the theorem about connection between spin and statistics, which
does not valid
to the infinite spin massless particles [15], describing by infinite-component
fields. This task forces to turn to the adequate description of second
quantized fields.

          Lorentz-covariant (super)field is transformed according to of
equality
\begin{equation}
  U(A)\Phi_i(x,\theta)U(A)^{-1}=(T(A^{-1})\Phi)_i(\Lambda(A)x,\theta A^{-1}),
\end{equation}
where $U(A)$ is unitary operator representing element $A$ of quantum mechanical
Lorentz group $SL(2,C)$ in Hilbert space of states, $T(A)$ is the some
representation of group $SL(2,C)$, acting only on index $i$,
 $x^\prime=\Lambda(A)x$, $\hat x^\prime=A\hat x A^+$. The same (super)particle
described by means of irreducible representation of Poincare (super)group can
be described by different types of fields, being characterized by different
representation $T(A)$. To have possibility for description of field, on index
of which is realized arbitrary (not necessarily finite-dimensional) irreducible
representation of Lorentz group, it is introduced an ``index spinor''
 $\zeta\neq 0$, running complex affine plane $\stackrel {o} {{C}^2}$. Then
the finite-component fields with irreducible representation $(j,k)$ on index
are bijectively compared the homogeneous polynomials $\Phi(\zeta;x,\theta)$
of the homogeneity degree $(2j+1,2k+1)$ in $\zeta$, obtaining by means of
convolution of spinor indices of superfield $\Phi_i(x,\theta)$, replacing
in total the collective index $i$, with index spinor $\zeta$ and with its
conjugate one $\bar\zeta$. On language of index spinor the superfield
transformation law takes the form
$$
  U(A)\Phi(\zeta;x,\theta)U(A)^{-1}=
  \Phi(\zeta A^{-1};\Lambda(A)x,\theta A^{-1}).
$$
All operator-irreducible representations of group $SL(2,C)$ are realized on
suitable spaces of homogeneous functions with respect to pair of complex
variables $\zeta=(\zeta^1,\zeta^2)\in\stackrel {o} {{C}^2}$.Therefore
superfield $\Phi_i(x,\theta)$ with arbitrary irreducible representation of
Lorentz group on index $i$ can be considered as superfield
$\Phi(\zeta;x,\theta)$
being homogeneous function of corresponding homogeneity degree on index spinor
 $\zeta$. Unless the equivalent representations being distincted we can
restrict
ourself to consideration only noninteger points, when representations with
indices $\chi$ and $-\chi$ are equivalent, and positive integer points, where
the functions from invariant subspace of homogeneous polunomials are taken.

        It is sufficient to consider the usual free massless fields satisfying
Klein-Gordon equation. Second quantization of such fields is carried out in
frames of usual formalizm. Annihilation field of particle with helicity
 $\lambda$ is
$$
  \varphi_\chi^{(+)} (\zeta,x)=(2\pi)^{-\frac 32}
  \int\,d^4 p\delta_+(p^2)e^{ipx}u_\chi(\zeta,p;\lambda)a(\vec p,\lambda) ,
$$
where $\delta_+(p^2)\equiv\theta(p^0)\delta(p^2)$, $u_\chi$ is Wigner wave
function (WWF) in momentum representation, $a(\vec p,\lambda)$ is annihilation
operator, $\chi$ is homogeneity bidegree in index spinor. In creation field
 $\varphi_\chi^{(-)}(\zeta,x)$, in just the same way transforming under
Poincare
transformations, the product
 $e^{-ipx}v_\chi(\zeta,p;\lambda^\prime)b^+(\vec p,\lambda^\prime)$ with
operator $b^+(\vec p,\lambda^\prime)$ of the creation of helicity
 $\lambda^\prime$ antiparticle and WWF $v_\chi(\zeta,p;\lambda^\prime)$ is
integrated in the same measure. Creation-annihilation operators of particles
and antiparticles with the opposite helicities are transformated with respect
to one-dimensional contragredient representations of Lorentz group
$$
  U(A)a(\vec p,\lambda)U(A)^{-1}=
  e^{-i\lambda\Theta(R)}a(\vec p^\prime,\lambda),
$$
where $R$ is transformation from little group $E(2)$ of the standard momentum
 $\stackrel {o}{p}=(k,0,0,k)$,$k>0$, corresponding transformation $A$ at
fixed Wigner operator $B_p$ ; $R=B_{p^\prime}^{-1}AB_p$. Rotation angle in
Euclidean plane is defined by condition $exp(i\Theta(R)/2)\hat\sigma=
\hat\sigma R\hat\sigma$, where $\hat\sigma=(1+\sigma_3)/2$. Wigner operator
is taken in form of product of the rotation $R(\vec p)$, transforming basic
vector of third axis into unit vector $\vec p/|\vec p|$ of momentum, and the
pure Lorentz transformation along third axis
 $H(|\vec p|)=exp(\frac 12\ln\frac{|\vec p|}k\sigma_3)$,
 $B_p=R(\vec p)H(|\vec p|)$.

            Wave function of arbitrary momentum is obtained from wave function
of standard momentum by the transformation only of index spinor
\begin{equation}
  u_\chi(\zeta,p;\lambda)=u_\chi(\zeta B_p,\stackrel {o} {p};\lambda),
\end{equation}
so that the transformation low of standard momentum wave function is
\begin{equation}
  (T(R)u_\chi)(\zeta,\stackrel {o} {p};\lambda)  =
  e^{i\lambda\Theta(R)}u_\chi(\zeta,\stackrel {o} {p};\lambda).
\end{equation}

            The little group is generated by components of Pauli-Lubanski
vector $w^\lambda=\frac 12\epsilon^{\lambda\mu\nu\rho}M_{\mu\nu}P_\rho=
k(I_3,L_1,L_2,I_3)$ in standard momentum system, which form algebra of group
 $E(2)$ of Euclidean plane motion. A role of the rotation generator is played
by $I_3$. $L_{1,2}=I_{1,2}\mp K_{2,1}$ are operators of ``translations'',
where rotation operators $I_j=\frac 12\epsilon_{jkl}M_{kl}$ and boosts
 $K_j=M_{0j}$ can be combinated in generators
 $A_j=I_j+iK_j=\frac 12\zeta\sigma_j\frac\partial{\partial\zeta}$ and
 $B_j=I_j-iK_j=-\frac 12\bar\zeta\sigma_j^T\frac\partial{\partial\bar\zeta}$
of two $SU(2)$ algebras, commuting among themselves. Roles of corresponding
angular momenta values are played by the constituents of homogeneity bidegree
of field $\nu_1/2$ and $\nu_2/2$, $\nu_{1,2}=n_{1,2}-1$.

            According to (3) we have $\Theta(R)\simeq\vartheta$ up to higher
order infinitesimals in neighborhood of identity, here $\vartheta$ is rotation
parameter from $E(2)$. Therefore
\begin{equation}
  L_1u_\chi(\zeta,\stackrel {o} {p};\lambda)=
  L_2u_\chi(\zeta,\stackrel {o} {p};\lambda)=
  (I_3-\lambda)u_\chi(\zeta,\stackrel {o} {p};\lambda)=0.
\end{equation}
First two equalities mean, that WWF does not depend on $\zeta^2$ and
 $\bar\zeta^2$ in standard momentum system if $\zeta^1\neq 0$. The last
reservation is unessential since $\zeta^1$ is not an invariant of little
group.

           For such wave function it is obvious that eigen-values of operators
 $A_3$ and $B_3$, i. e. $\nu_1/2$ and $\nu_2/2$, define homogeneity index.
Thus spin weight $s$ of nongauge field, described massless particles of finite
spin, necessary coincides with their helicity $\lambda$, $s=\lambda$. For
finite-component fields with $c\geq|s|$ this statement has been obtained by
Weinberg [16]. WWF of arbitrary momentum
\begin{equation}
  u_\chi(\zeta,p;\lambda)=
  {[\zeta B_p]}^{c+s} {\overline{[\zeta B_p]}}
  ^{c-s} / \Gamma(c+|s|+1) ,
\end{equation}
where $[\zeta B_p]\equiv(\zeta B_p)^1$, is analitic function on $c$ at
fixed $s$.

           Dependence WWF on the index spinor $\zeta$ is defined by four
conditions. Two of them must provide its dependence only on the first
component in the standard momentum system and two determine the homogeneity
degree.
Under consideration of the finite-component fields with Lorentz indices
the spin and conformal weights are fixed by indication of index set. One of
two remaining conditions can be unnecessary if the field has indices only of
one type, for example, only undotted ones. In explicitly Lorentz-covariant
form for WWF the mentioned conditions (equations of motion) have form
\begin{equation}
  \frac\partial{\partial\zeta}\hat p \, u_\chi(\zeta,p;\lambda)=
  \hat p\frac\partial{\partial\bar\zeta} \, u_\chi(\zeta,p;\lambda)=0.
\end{equation}

           WWF of the antiparticle is
\begin{equation}
  v_\chi(\zeta,p;\lambda^\prime)=
  \overline{u_{\bar\chi}(\zeta,p;-\lambda)} ,
\end{equation}
where $\bar\chi\equiv(\bar c,-s)$, therefore its helicity
$\lambda^\prime=-s$ is opposite to helicity of particle.

           Quantum field $\varphi_\chi(\zeta,x)$ is combination of the
positive- and negative-frequency parts with arbitrary in general complex
weights $\xi_\chi$, $\tilde\xi_\chi$ :
\begin{equation}
  \varphi_\chi(\zeta,x)=\xi_\chi\varphi_\chi^{(+)}(\zeta,x)+
  \tilde\xi_\chi\varphi_\chi^{(-)}(\zeta,x).
\end{equation}
Phases of weight multipliers $\xi_\chi$ and $\tilde\xi_\chi$ can be excluded by
the redefinition of creation-annihilation operators of particles and
antiparticles.
Along with massless Klein-Gordon equation the field satisfies the
equations of motion
\begin{equation}
  \frac\partial{\partial\zeta} \hat\partial \, \varphi_\chi(\zeta,x)=
  \hat\partial\frac\partial{\partial\bar\zeta} \, \varphi_\chi(\zeta,x)=0
, \end{equation} following from (6).

           Spinor basis in the standard momentum system is connected with
basis in adapted to particle 4-momentum light-cone system by means of
following transformations. By the boost in direction of standard momentum,
transforming standard light coordinate into given one. By rotation, matching
first component phase of the first basic spinor with the first component
phase of harmonic spinor $v^+$. By nilpotent transformation from
stability group of standard momentum, identifying the second basis spinor
with harmonic ones $v^-(v^+)$. Composition of these transformation gives
 Wigner operator in light basis. To obtain its presentation in the usual
basis we can use harmonic matrix as a bridge, connecting usual spinor
basis with light one. It gives expression
\begin{equation}
  B_p=\left(v_\alpha^+{\left(\frac{p^{--}}{2k}\right)}^{\frac 12}
  {\left(\frac{v_1^+}{\bar v_1^+}\right)}^{-\frac 12} \, ,  \,
  v_\alpha^-(v^+){\left(\frac{p^{--}}{2k}\right)}^{-\frac 12}
  {\left(\frac{v_1^+}{\bar v_1^+}\right)}^{\frac 12}\right),
\end{equation}
where is omited, in general arbitrary, phase of first basic spinor component in
the standard momentum system.

            Now WWF is
\begin{equation}
  u_\chi(\zeta,p;\lambda)=K_\chi(\zeta v^+)
  \left(\frac{p^{--}}{2k}\right)^c\left(\frac{v_1^+}{\bar v_1^+}\right)^{-s}.
\end{equation}
Here
\begin{equation}
  K_\chi(z)=|z|^{2c}\exp(2is\arg z)/ \Gamma(c+|s|+1)
\end{equation}
is homogeneous generalized function of index $\chi$ on complex variable $z$,
which is analitic with respect to $c$ at fixed (half)integer $s$ [11].
In the light basis, adapted to the particle 4-momentum, the
Lorentz-invariant measure on light cone is
 $d^3p/|\vec p|=-i v^+dv^+\bar v^+d\bar v^+p^{--}dp^{--}$, where the end of
harmonic spinor runs a closed piecewise smooth surface in complex affine plane,
enclosing origin and intersecting with each, passing through origin, straight
line in one point. Annihilation field of particle with helicity $\lambda=s$ is
presented as surface integral
\begin{equation}
  \varphi_\chi^{(+)}(\zeta,x)=\frac 1{2i}
  \int v^+dv^+\bar v^+d\bar v^+ K_\chi(\zeta v^+)
  \psi_{-\chi}^{(+)}(x^{++},v^+)
\end{equation}
over the mentioned surface of operator function
\begin{equation}
  \psi_{-\chi}^{(+)}(x^{++},v^+)=\frac 1{(2k)^c(2\pi)^{\frac 32}}
  \int_0^\infty\,dp^{--}e^{-\frac i2 p^{--}x^{++}}(p^{--})^{c+1}
  \left(\frac{v_1^+}{\bar v_1^+}\right)^{-s}a(\vec p,s)
\end{equation}
which is homogeneous of index $-\chi$ :
\begin{equation}
  \psi_{-\chi}^{(+)}(|a|^2 x^{++},av^+)=|a|^{-2(c+2)}
  \exp(-2is\arg a)\psi_{-\chi}^{(+)}(x^{++},v^+).
\end{equation}

        It is convinient to rewrite the surface integral in terms of integral
over all affine plane $\stackrel{o}{C^2}$ with using of measure
 $[\omega\wedge\bar\omega](v^+)\equiv\omega\wedge\bar\omega H(v^+)$, where
 $H(v^+)$ is finitary infinite-differentiable function on
 $\stackrel{o}{C^2}$, satisfying
 $\int_{\stackrel{o}{C^2}}\,H(av^+)\frac{|dad\bar a|}{|a|^2}=1 \,
\forall\, v^+\in\stackrel{o}{C^2}$ ; $|dad\bar a|=da\wedge d\bar
a/2i$.  A residue of the homogeneous function with bidegree $(-2,-2)$
does not depend on the choice of $H$ [17]. We have \begin{equation}
  \varphi_\chi^{(+)}(\zeta,x)=\int[\omega\wedge\bar\omega](v^+)
  K_\chi(\zeta v^+)\psi_{-\chi}^{(+)}(x^{++},v^+)  .
\end{equation}
This form representation takes place for the creation field
 $\varphi_\chi^{(+)}(\zeta,x)$ as well, after substitution $a(\vec p,\lambda)$
on $b^+(\vec p,-\lambda)$ and positive-frequency exponential on
negative-frequency
one, and for the full massless field.

          For noninteger $\chi$ [11] the integral operator with kernel
 $K_\chi(\zeta_1\zeta_2)$, which is permutable with representations of
 $SL(2,C)$, is defined uniquely and it is reversible such that the composition
of
transformations with kernels $K_{-\chi}$ and $K_\chi$ is multiple of the
identical transformation with coefficient
 $\pi^2(-1)^{2s}/\Gamma(c+|s|+1)\Gamma(-c+|s|)$. In integer points the kernel
 $K_\chi$ need extension of the definition, but transformation defined by it
annuls the invariant subspace. Therefore in the integer points the massless
field is associated with equivalence class of harmonic fields with respect to
addition
of the function from invariant subspace.

          We shall understand microcausality (locality) as equality to zero of
field permutation function in the space-like region. Admiting only two types
of the statistics (Fermi or Bose) we shall consider permutation function of
the form
\begin{equation}
  i\Delta_\chi(x-y;\zeta,\omega)=
  [\varphi_\chi(\zeta,x)\,,\,\varphi_\chi(\omega,y)^+]_\sigma ,
\end{equation}
where $\sigma=\pm$ indicates (anti)commutator.

          Permutation function $\Delta=\Delta^{(+)}+\Delta^{(-)}$ and its
positive- $\Delta^{(-)}$ and negative- $\Delta^{(+)}$ frequency parts are
separately homogeneous of bidegree $\chi$ in $\zeta$ and bidegree
$\bar\chi$ in $\omega$, Lorentz-invariant and satisfy the same equations,
 that rearranged fields. In consequence of a principle of spectral
property ( $p^2=0,p^0>0$ ) the parts of definite frequency
$\Delta_\chi^{(\pm)}(x;\zeta,\omega)$ as functions on $x$ are holomorphic
in tube of past(future) $T^{(\mp)}=\{ x\in C^4| Im \,
x\in\stackrel{o}{V}_{(\mp)} \}$ where $\stackrel{o}{V}_{(\mp)}$ is
interior of forward(backward) light cone.  Therefore it is sufficient to
define expression for $\Delta_\chi^{(+)}(x;\zeta,\omega)$ at $x\in T^{(-)}$
 with positive time-like real part and small imaginary part only in time
 coordinate but for remaining points of tube to take of analitic
continuation, which not presents any complication since the result is
expressed in terms of the well known functions. It is convenient to lead
calculations in special frame of reference, where $\zeta=(\alpha,0)$,
$\omega=(0,\alpha)$, which is possible when $\zeta\omega\neq 0$.

         Permutation function in real space-time points is obtained as boundary
value of function, obtained in the past tube
\begin{equation}
  \Delta_\chi(x;\zeta,\omega)=\lim(\Delta_\chi^{(+)}(x^\prime;\zeta,\omega)+
  \sigma\Delta_\chi^{(+)}(-\overline{x^\prime};\omega,\zeta)),
\end{equation}
where limit $x^\prime\rightarrow x$ is taken by the points of past tube and
here
is used a connection between different frequency parts. Second term in perlimit
expression is found by analytic continuation of the first one from $x$ to
 $-\bar x$ across space-lice real part region with keeping small imaginary part
in time coordinate $x^0$. Such path permits to pass around necessary cuts in
planes of composite variables, on which many-valued multipliers
in obtained expression for $\Delta_\chi^{(+)}(x;\zeta,\omega)$ are depended
$$
  i\Delta_\chi^{(+)}(x;\zeta,\omega)=
  |\xi_\chi|^2 \frac 1{(2\pi)^{2(1+Re c)}}
  e^{-i\pi(Re c+1)}|\zeta\omega|^{2 Re c}e^{2is\arg(\zeta\hat x\bar\omega)}
\times
$$
$$
  \left(\frac{\zeta\hat x\bar\zeta}{\omega\hat x\bar\omega}\right)^{i Im c}
  (\nu^2)^{(Re c+1)}(1+\nu^2 x^2)^{|s|} \times  \qquad     \eqno{(19)}
$$
$$
 {}_2 F_1(c+|s|+1,\bar c+|s|+1;2|s|+1;1+\nu^2 x^2)/\Gamma(2|s|+1) ,
 \qquad
$$

where $\nu^2\equiv|\zeta\omega|^2/(\zeta\hat x\bar\zeta)
(\omega\hat x\bar\omega)$. Depending on $x$ multipliers are many-valued. In
special frame of reference $(\zeta\hat x\bar\zeta)/(\omega\hat x\bar\omega)=
\nu^2(x^0+x^3)^2$, $\nu^{-2}=x_\bot^2-x^2$, $1+\nu^2 x^2=\nu^2 x_\bot^2$,
where $x_\bot^2=x_1^2+x_2^2$. Cuts in planes of these variables, if they are
nesessary, are drawn along the positive real semiaxis (for hypergeometric
function from $+1$). Path of analitic continuation remains on the one hand of
cut in the plane of the first considered composite variable and bypasses cut in
planes of the other ones.

          To microcausality takes place, hypergeometric function must be
turned in itself at the going around of cut that is possible only in degenerate
case, which correspondes to positive integer values of homogeneity index
$\chi$.
When this condition is fulfilled, the permutation function in space-like region
turns out proportional to $|\xi_\chi|^2+\sigma e^{2\pi is}|\tilde\xi_\chi|^2$.
Now the microcausality requires fulfilment of equality $|\xi_\chi|=
|\tilde\xi_\chi|$ that is ``crossing'' [16] and the usual connection between
spin
and statistics, $(-1)^{2s}=-\sigma$. ``Crossing'' allow to regard
 $\xi_\chi=\tilde\xi_\chi=1$. Thus massless nongauge quantum fields of finite
spin, transformed with by irreducible representation of Lorentz group, for
which locality takes place are finite-component fields with integer positive
index of homogeneity and usual connection between spin and statistics. Absence
of locality for fields with continuous spin is well known [15].

          As a first step to construction of quantum theory of interaction it
should be considered description of superparticle in Yang-Mills and/or
gravitational background. Introduction of harmonics makes possible ``feels''
not only vielbein coefficients but also Lorentz connection, more exactly,
in the spirit of Kaluza-Klein theories [18] in harmonic superspace among
vielbein components, at the same time with ``initial'' and introduced for
harmonic sector, there is a place for ``initial'' Lorentz connection. Imposing
suitably generalized nonconventional constraints [19] on vielbein coefficients
we can, by imposition of conventional constraints, express remaining
coefficients
of vielbein and connection in terms of the selected basic ones and get
rid of Weil transformation compensator as independent superfield. Solution
of the found constraints will have given the formulation similar to [20],
where real physical superspace is hypersurface in complex superspace and
precisely harmonic coordinates, obtained under quantization of harmonic
superparticle in flat space, play the role of Gaussian coordinates on surface.
At present these questions are under consideration.

           Authors have acknowledge to I. A. Bandos for interest to work and
numerous useful discussion.



\vspace{2cm}
{\Large{\bf Reference}}

\vspace{1cm}

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\end{document}
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