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

\begin{abstract}
We introduce a notion  of induced symplectic representation of the
Poincare group. Classical relativistic fields are considered as
such representations. We describe the method of investigation of
these fields in the sense of their reducibility. We introduce the
notion  of the field oscillator as an inducing Hamiltonian system.
\end{abstract}

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\myparagraph{Symplectic representations.}
 By Poincare group we will call a connected continuous
ten-parameter group of transformations of space-time including
both homogeneous transformations (the Lorentz group) and
space-time shifts. We will denote the Poincare group by the
symbol~$\mathcal P$.

So far as in these papers we study fields that are relativistic
invariant, every solution of equations of the motion is
transformed to a solution by a transformation from the Poincare
group. So, this defines the action of the Poincare group on the
invariant phase space~$Z$.

In a more concrete way this action can be described in the
following way. Let we have a transformation $g\in\mathcal P$. It
acts on points of space-time in the following way:
\[
 x \ \to\ x'=gx \ .
\]
In the space, where the values of the field function~$\varphi_i$
lie, the transformation~$g$ acts as:
\[
 \varphi_i \ \to \ \varphi'_i= \Lambda_{ij}\,\varphi_j \ .
\]
The action of $g$ in the space~$Z$ can be described by the
formulas:
\begin{equation}
 \underline c \,\longrightarrow\,\underline{\smash{gc}} \ ,
 \qquad
 \varphi_i(x)^{\,\underline{\smash{gc}}} =
          \Lambda_{ij}\,\varphi_j(g^{-1}x)^{\,\underline c}    \ .
\label{Poin}
\end{equation}

As we have said in the paper~[I], we will consider here only
linear fields. The action of the Poincare group~$\mathcal P$
obviously preserve the linear structure in the space~$Z$. So, in
the space~$Z$ we have a linear representation of the Poincare
group.

Furthermore, the Poincare group preserves in the space~$Z$ the
symplectic structure. So, we come to linear {\it symplectic}
representations of the Poincare group $\mathcal P$. The connection
of symplectic representations with unitary will become clear after
we define the operation of quantization in the paper~[VI]. It will
be shown there that in the field theory symplectic representations
apparently play a role that is not less important than that of
unitary. Anyway, they are connected with the construction of
quantized field more closely.

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\myparagraph{``Reduction'' of field representation.}
\label{reduction}
 Consider now the question of reducibility of field
representations of the Poincare group~$\mathcal P$. We will always
imply here reducibility in the complex sense. In order to not
complexify the space~$Z$ let us consider the conjugate
space~$Z^*_{\mathbb C}$. In~$Z^*_{\mathbb C}$, obviously, acts the
conjugate representation of the group~$\mathcal P$. In order to
define this action we just need to read the formula~(\ref{Poin}) a
little differently:
\begin{equation}
 \varphi_i(x)^{\,\underline \cdot} \,\longrightarrow\,
     \big(g\,\varphi_i(x)\big)^{\,\underline \cdot} \ ,
 \qquad
 \big(g\,\varphi_i(x)\big)^{\,\underline c} =
          \Lambda_{ij}\,\varphi_j(g^{-1}x)^{\,\underline c}    \ ,
\label{Poinadj}
\end{equation}
i.~e. we connect the symbol~$g$ not with an element of~$Z$, but
with an element of~$Z^*_{\mathbb C}$. Below by a field
representation we will always imply the action of the Poincare
group by formula~(\ref{Poinadj}) in the complexified conjugate
space~$Z^*_{\mathbb C}$.

For infinite-dimensional representations the notion of reduction
requires specification. Now we will explain the structure of field
representations under consideration and simultaneously specify in
what sense we will talk about their reducibility.

In the case of unitary representations the main tool of
investigation is Mackey theorem (see, for example,
\cite{Mackey1968}). This theorem brings investigation of a
representation of the Poincare group~$\mathcal P$ to investigation
of the corresponding little Lorentz group~$\mathcal L_k$. The
construction of Mackey induction can be generalized to any linear
representations. But a general linear representation (including
symplectic) of the group~$\mathcal P$ can be not induced from the
little group. Nevertheless, representations induced in Mackey
sense constitute a very wide class, and in fact, we will restrict
ourself with them.

Let us make a Fourier transformation of the field and write the
result in the form:
\begin{equation}
 \widetilde\varphi_i(k) = 2\pi\,\delta(k^2-m^2)\cdot a_i(k)  \ .
\label{fourdec}
\end{equation}
It is natural to call the scalar~$m$ a mass of the field, like in
quantum theory.

On the grounds of the formula~(\ref{fourdec}) it is natural to say
that the field representation decomposes into the direct sum of
positive- and negative-frequency subrepresentations:
\[
 Z^*_{\mathbb C} = Z^{*\,(+)}_{\mathbb C} \oplus
                   Z^{*\,(-)}_{\mathbb C} \ .
\]
These two representations we will also call field representations.
The projectors to the positive- and negative-frequency subspaces
in the Fourier representation are just operators of multiplication
by the functions $\theta(+k)$ and $\theta(-k)$, correspondingly.

So far as we consider here real fields, $a_i(-k)=a^*_i(k)$. So, it
is natural to suppose that both subspaces $Z^{*\,(+)}_{\mathbb C}$
and $Z^{*\,(-)}_{\mathbb C}$ are reducible in the same degree. In
this connection we can restrict ourself with consideration of the
positive-frequency subspace.

Consider now a fixed vector~$k^{(0)}$ on the mass surface:
$(k^{(0)})^2=m^2$. And consider the subgroup of the Lorentz group
that leaves this vector unchanged, i.~e. the so called little
group of this vector. Let us denote this
 group~$\mathcal L_{k^{(0)}}$. Obviously, the set of values
$a_i(k^{(0)})$ corresponding to different values of the index~$i$
is transformed linearly under action of transformations
from~$\mathcal L_{k^{(0)}}$. This complex representation of the
little group we will suppose to be finite-dimensional. Like in the
unitary case, we will say that the field representation under
consideration is induced by this representation of the little
group.

Let us, in analogy with the unitary case, suppose that the further
reducibility of the field representation (i.~e. reduction of its
positive-frequency part) is fully defined by reducibility of the
inducing representation.

Let us note here also that in the case of massive fields, i.~e.
when $m>0$, the little group~$\mathcal L_{k^{(0)}}$ is the group
of three-dimensional rotations $SO(3)$. This group is compact, and
therefore, its representation can be made unitary by introducing a
proper scalar product. Therefore, the inducing representation
turns out to be fully reducible, and the equivalence classes of
irreducible components, like in the unitary case, are defined by a
one integer or half-integer number. It is natural to call this
number, like in the quantum case, a {\it spin} of the irreducible
component under consideration.

In the case of zero-mass field the little group, as we know, is
the group of motions of a Euclidian plane~$E(2)$. This group is
not compact, and situation becomes much more complicated than in
the unitary case. We will see it soon with example of
electromagnetic field.

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

\myparagraph{Field oscillator as inducing Hamiltonian system.}
 It is easy to notice that there is a great similarity in
description of the harmonic oscillator and the scalar field. Now
we will research this phenomenon in detail.

Consider any real field that is written in the Fourier
representation as~(\ref{fourdec}). Let us suppose that the Poisson
brackets of the field values in the Fourier representation have
the form:
\begin{equation}
 \{\,\widetilde\varphi_i(k)\,,\,\widetilde\varphi_j(k')\,\} =
 B_{ij}(k)\cdot
 \Big[\,i\,\varepsilon(k)\cdot 2\pi\,\delta(k^2-m^2)\,\Big]\cdot
 (2\pi)^4\delta(k+k') \ .
\label{generalPoiss}
\end{equation}
Here $B_{ij}(k)$ is some tensor function. In the cases of the
scalar field and the non-physical electromagnetic field, in
accordance with the formulas of the paper [I], this function is
just a constant. But in general case it is not so. So, let us
consider its properties with more details.

First, so far as $B_{ij}(k)$ is multiplied by
$2\pi\,\delta(k^2-m^2)$, we can admit that it is defined only on
the mass surface $k^2=m^2$. Second, using antisymmetry of the
Poisson bracket, we get:
\begin{equation}
 B_{ij}(k)=B_{ji}(-k) \ .
\label{asymmpoiss}
\end{equation}

Furthermore, requirements of relativistic invariance make very
hard restrictions for $B_{ij}(k)$. Let us fix some point~$k^{(0)}$
on the mass surface. If we know the value of the function
$B_{ij}(k)$ in this point, then, using relativistic invariance, we
can define it in the other points. The value $B_{ij}(k^{(0)})$ is
not arbitrary also: it must be invariant under action of the
little group $\mathcal L_{k^{(0)}}$.

The fact that the Poisson bracket is a complexified real bracket
makes another restriction for $B_{ij}(k)$:
$B^*_{ij}(k)=B_{ij}(-k)$. Taking into account~(\ref{asymmpoiss})
we can write it also as $B^*_{ij}(k)=B_{ji}(k)$.

Consider now the values $a_i(+k^{(0)})$ and $a_i(-k^{(0)})$. As it
was shown in the section~\ref{reduction}, they form a
representation of the little group $\mathcal L_{k^{(0)}}$. It is
obvious also that these values form a representation of the group
of shifts\footnote{Speaking more generally, these values form a
representation of the seven-parameter subgroup of the Poincare
group that includes the little group $\mathcal L_{k^{(0)}}$ and
shifts.}. A shift by 4-vector $l$ can be described by the
formulas:
\[
 a_i(k^{(0)})  \,\to\, a_i(k^{(0)}) \cdot e^{+ik^{(0)}l} \ ,
 \qquad
 a_i(-k^{(0)}) \,\to\, a_i(-k^{(0)})\cdot e^{-ik^{(0)}l} \ .
\]
Let us introduce a notation $\tau=k^{(0)}l$. Then we can think
that the values $a_i(+k^{(0)})$ and $a_i(-k^{(0)})$ form a
representation of the group $\mathbb R\times\mathcal L_{k^{(0)}}$,
where $\mathbb R$ is the additive group of real numbers,
parameterized by the number~$\tau$.

Let us consider now the values $a_i(+k^{(0)})$ and $a_i(-k^{(0)})$
as dynamical variables of some new system. We will call this
system a {\it field oscillator}. In order to economize notations
we will write variables of the field oscillator with the same
symbols, but we will write as an argument not the vector~$k^{(0)}$
but a real number~$\omega$; and $\omega=+1$ corresponds to
$k=+k^{(0)}$, and $\omega=-1$ corresponds to $k=-k^{(0)}$. The
phase space of the field oscillator can be constructed by
realification of the space of inducing representation. The Poisson
bracket can be defined by the formulas:
\[
 \{\,a_i(+1)\,,\,a_j(+1)\,\} = \{\,a_i(-1)\,,\,a_j(-1)\,\} = 0 \ ,
\]
\[
 \{\,a_i(+1)\,,\,a_j(-1)\,\} =\,iB_{ij}(+1)   \ .
\]
Assuming that the Poisson bracket is not degenerate we can, as
usually, calculate the symplectic structure.

The action of the group $\mathbb R\times\mathcal L_{k^{(0)}}$ on
the phase space of the field oscillator leaves the Poisson
brackets invariant. Therefore it is symplectic.

Now we can interpret the action of a transformation from $\mathbb
R$, characterized by the parameter~$\tau$, as a time shift by the
time period~$\tau$ (the ``time'' of the field oscillator is a
non-dimensional parameter).

We can even define ``coordinates'' of the field oscillator at
``time'' $t$:
\[
 \textstyle
 \varphi_i(t)=
    \frac1{\sqrt2}\,
    \big(\,  a_i(+1)\cdot e^{-it} +
             a_i(-1)\cdot e^{+it}    \,\big) \ .
\]
We see that the oscillator of a real field has real coordinates.

If we consider the little group $\mathcal L_{k^{(0)}}$ as an
abstract group, then it does not depend on the choice of the
vector $k^{(0)}$. If we describe the oscillator by internal
notions of the Hamiltonian formalism (phase space, symplectic
structure, symplectic action of the group $\mathbb R\times\mathcal
L_{k^{(0)}}$), then its construction does not depend on the choice
of the vector $k^{(0)}$ also. So, for any $\mathcal P$-invariant
field we can uniquely juxtapose
 $\mathbb R\times\mathcal L_{k^{(0)}}$- invariant field
oscillator.

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

\myparagraph{Scalar field representation.} \label{sfrepr}
 Let us apply now the described scheme to the scalar field. The
inducing representation in this case is the one-dimensional
identity representation. The field oscillator is just the usual
real oscillator with one degree of freedom.

So, with respect to the action of the Poincare group, the
space~$Z^*_{\mathbb C}$ decomposes into the direct sum of the
positive- and the negative-frequency reducing subspaces, and each
of these subspaces is irreducible. These two subspaces we will
denote $Z^{*\,(+)}_{\mathbb C}$ and $Z^{*\,(-)}_{\mathbb C}$. The
projections of elements of the space~$Z^*_{\mathbb C}$ to these
two spaces we will provide with labels~${}^{(+)}$ or ${}^{(-)}$,
correspondingly:
\[
 \widetilde\varphi^{\,(\pm)}(k) =
       \theta(\pm k)\,\widetilde\varphi(k) \ ,
 \qquad
 \varphi^{\,(\pm)}(x)=
       \int \frac{d^4\!k}{(2\pi)^4}\,e^{-ikx}\,
       \widetilde\varphi^{\,(\pm)}(k)   \ .
\]

Using formulas from the paper [I], we can calculate the Poisson
brackets of these projections:
\[
 \{\,\widetilde\varphi^{\,(\pm)}(k) \,,\,
     \widetilde\varphi^{\,(\pm)}(k')\,\} = 0 \ ,
\]
\[
 \{\,\widetilde\varphi^{\,(\pm)}(k) \,,\,
     \widetilde\varphi^{\,(\mp)}(k')\,\} =
 -\, \Big[\, i\,\theta(\pm k)\,\varepsilon(k)
             \cdot 2\pi\,\delta(k^2-m^2) \,\Big]
     \cdot (2\pi)^4\delta(k+k')       \ .
\]

In the coordinate representation we have, correspondingly:
\[
 \{\,\varphi^{\,(\pm)}(x)\,,\,\varphi^{\,(\pm)}(x')\,\} = 0 \ ,
\]
\[
 \{\,\varphi^{\,(\pm)}(x)\,,\,\varphi^{\,(\mp)}(x')\,\} =
   - D^{\,(\pm)}_m(x-x') \ .
\]
Here symbols $D^{\,(+)}_m(y)$ and $D^{\,(-)}_m(y)$ denote the
positive and the negative-frequency parts of the function
$D_m(y)$:
\begin{equation}
 D^{\,(\pm)}_m(y)=\int \frac{d^4\!k}{(2\pi)^4} \, e^{-iky} \cdot
 \Big[\, i\,\theta(\pm k)\,\varepsilon(k)
         \cdot 2\pi\,\delta(k^2-m^2) \,\Big]     \ .
\label{Dpmfunc}
\end{equation}

The given formulas turn out to be very useful for calculation of
propagators of the quantized field. But they naturally follow from
the classical theory.

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

\myparagraph{Electromagnetic field representation.}
\label{emrepres}
 According to the section~\ref{reduction}, the representation of
the non-physical electromagnetic field decomposes also into the
direct sum of positive- and negative-frequency representations.
The formulas of the section~\ref{sfrepr} can be rewritten in the
following way:
\[
 \widetilde A_\mu^{\,(\pm)}(k) = \theta(\pm k)\,\widetilde A_\mu(k) \ ,
 \qquad
 A_\mu^{\,(\pm)}(x)= \int \frac{d^4\!k}{(2\pi)^4}\,e^{-ikx}\,
              \widetilde A_\mu^{\,(\pm)}(k)   \ ,
\]
\[
 \{\,\widetilde A_\mu^{\,(\pm)}(k) \,,\,
     \widetilde A_\nu^{\,(\pm)}(k')\,\} = 0 \ ,
\]
\[
 \{\,\widetilde A_\mu^{\,(\pm)}(k) \,,\,
     \widetilde A_\nu^{\,(\mp)}(k')\,\} =
 g_{\mu\nu}\cdot
 \Big[\, i\,\theta(\pm k)\,\varepsilon(k)
         \cdot 2\pi\,\delta(k^2) \,\Big]
 \cdot (2\pi)^4\delta(k+k')       \ ,
\]
\[
 \{\,A_\mu^{\,(\pm)}(x)\,,\,A_\nu^{\,(\pm)}(x')\,\} = 0 \ ,
\]
\[
 \{\,A_\mu^{\,(\pm)}(x)\,,\,A_\nu^{\,(\mp)}(x')\,\} =
   g_{\mu\nu}\, D^{\,(\pm)}_0(x-x') \ .
\]

Here $D^{\,(\pm)}_0(y)$ are functions~(\ref{Dpmfunc}) with $m=0$.
In this case they can be written also as:
\[
 D^{\,(\pm)}_0(y) =
    \frac1{i\,(2\pi)^2} \cdot
    \frac1{\pm\, y^2-i\,0\cdot\varepsilon(y)} =
    \frac1{4\pi}\,\varepsilon(y)\,\delta(y^2) \pm
       \frac1{i\,(2\pi)^2}\, \mathcal P \,\frac1{y^2}       \ .
\]

The inducing representation in this case is the complex vector
representation of the little group~$\mathcal L_{k^{(0)}}$ with
$(k^{(0)})^2=0$. A more detailed description of this
representation is given in the paper~[V]. It has two composition
series:
\[
 \{0\}= M^0_{\mathbb C} \subset M^\parallel_{\mathbb C} \subset
 M^{(+1)}_{\mathbb C} \subset M^\perp_{\mathbb C} \subset
 M^4_{\mathbb C} = M_{\mathbb C} \ ,
\]
\[
 \{0\}= M^0_{\mathbb C} \subset
 M^\parallel_{\mathbb C} \subset
 M^{(-1)}_{\mathbb C} \subset
 M^\perp_{\mathbb C} \subset
 M^4_{\mathbb C} = M_{\mathbb C} \ .
\]
Under induction these two composition series turn into composition
series of the field representation.
\[
 \{0\}= Z^{*\,(+)\,0}_{\mathbb C}         \subset
        Z^{*\,(+)\,\parallel}_{\mathbb C} \subset
        Z^{*\,(+)\,(+1)}_{\mathbb C}      \subset
        Z^{*\,(+)\,\perp}_{\mathbb C}     \subset
        Z^{*\,(+)\,4}_{\mathbb C} =
               Z^{*\,(+)}_{\mathbb C} \ ,
\]
\[
 \{0\}= Z^{*\,(+)\,0}_{\mathbb C}         \subset
        Z^{*\,(+)\,\parallel}_{\mathbb C} \subset
        Z^{*\,(+)\,(-1)}_{\mathbb C}      \subset
        Z^{*\,(+)\,\perp}_{\mathbb C}     \subset
        Z^{*\,(+)\,4}_{\mathbb C} =
               Z^{*\,(+)}_{\mathbb C} \ .
\]
Here the notations for the subspaces are in natural concordance
with the notations of the paper [I].

The field oscillator in this case turns out to be a system with
eight-dimensional phase space.

In connection with the division of the field into the positive-
and the negative-frequency parts, let us notice also that so far
as electromagnetic field is real, the Lorentz condition for
scattered states can be written in three equivalent forms:
\[
 -\,i\,k_\mu\, a_\mu(k)^{\,\underline{\mathrm{rad}}} = 0 \ ,
\]
\[
 -\,i\,k_\mu\, a_\mu^{(+)}(k)^{\,\underline{\mathrm{rad}}} = 0 \ ,
\]
\[
 -\,i\,k_\mu\, a_\mu^{(-)}(k)^{\,\underline{\mathrm{rad}}} = 0 \ .
\]

So far as positive- and negative-frequency parts of the field play
different roles in quantization, in the paper~[VI] we will see
that in quantum theory there is an analog of only the second form
of this condition.

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\begin{thebibliography}{99}

\bibitem{Mackey1968}
G.~W.~Mackey {\it\rql Predstavleniya grupp v gilbertovom
prostranstve\rqr}, appendix in the book~\cite{Segal1968}.
 [G.~W.~Mackey ``Group representations in Hilbert space'',
appendix in the book~\cite{Segal1968}.]

\bibitem{Segal1968}
I.~E.~Segal {\it\rql Matematicheskie problemy relyativistskoy
fiziki\rqr}, M.: Mir (1968).
 [I.~E.~Segal {\it ``Ma\-the\-ma\-ti\-cal problems of relativistic
physics''}, Providence, Rhode island: AMS (1963).]

\end{thebibliography}

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\end{document}
