# The explicit form of the unitary representation of the Poincaré group for
vector-valued wave functions (massive and massless), with applications to
photon’s localization and position operators

Arkadiusz Jadczyk<EMAIL_ADDRESS>Ronin Institute, Montclair, NJ 07043

###### Abstract

We geometrically derive the explicit form of the unitary representation of the
Poincaré group for vector-valued wave functions and use it to apply speed-of-
light boosts to simple polarization basis to end up with Hawton-Baylis photon
position operator with commuting components. We give explicit formulas for
other photon boost eigenmodes. We investigate the underlying affine
connections on the light cone in momentum space and find that while Pryce
connection is metric semi-symmetric, the flat Hawton-Baylis connection is not
semi-symmetric. Finally we discuss localizability of photon states on closed
loops and show that photon states on the circle, both unnormalized improper
states and finite norm wave packet smeared over washer-like regions are
strictly localized not only with respect to Hawton-Baylis operators with
commuting components but also with respect to the noncommutative Jauch-Piron-
Amrein POV measure.

###### keywords:

photon wave function; unitary representation; Poincaré group; boost
eigenmodes; position operator; POV measure; Riemannian metric; affine
connection; semi-symmetric metric connection; loop localized photon states

##  1 Introduction

In classical relativistic field theory the electric and magnetic field
strengths are combined into a closed two-form $F$ over the Minkowski
spacetime. Since $F$ is closed, $dF=0,$ it admits a four-vector potential $A,$
(a 1-form) so that $F=dA.$ While in classical theory $A$ is convenient, but
not strictly necessary, when we move to quantum theory $A$ is almost
unavoidable. There, when describing the interaction of the electromagnetic
field with charged matter (particles or fields), $A$ has a simple geometrical
interpretation as a connection one-form in a complex line bundle, and $F$
becomes a curvature of this connection (physicists sometimes like to use the
term “a non-integrable phase factor" for the fact that we are dealing with a
connection form of a no-zero curvature). This simple and beautiful geometrical
picture gets, however, somewhat lost when we move to the quantum theory of
electromagnetic field itself (both first and second quantized).111A promising
new way of keeping the geometry alive also in quantum field theory (within the
algebraic framework) has been suggested by D. Buchholz et al. - c.f. [1] and
references therein. Photons - the quanta of the electromagnetic field, are
treated there as relativistic elementary particles, and as such are described
by irreducible unitary representations of the Poincaré group. The whole
machinery of Lie algebras, Casimir operators, Lie groups, induced
representations and “little groups" is being brought forward, and geometry
gets almost completely forgotten. Physicists construct one-particle Hilbert
spaces and multi-particle Fock spaces (while the first quantization is a
‘miracle’, the second quantization is a functor), move into the algebra of
operators, and this is a whole new world, with little place for differential
geometry. V.S. Varadarajan in his classic monograph “Geometry of Quantum
Theory”, chapter “Representations in vector bundles and wave equations” comes
very close to fulfilling this task, unfortunately, when it comes to photons,
the chapter ends with the sentence “We do not get into these ideas here." -
[2, p. 371].

The group-theoretical analysis of elementary relativistic quantum systems lead
to the concept of imprimitivity systems, developed by G.W. Mackey (cf. e.g.
[2, Ch. VI] and references therein), and to the associated concept of the
localization of elementary quantum particles. A.S. Wightman [3] applied these
concepts to the study of localizability of quantum mechanical systems and came
to conclusion confirming the previous analysis of T.D. Newton and E.P Wigner
[4], namely that photons (as well as other particles of rest mass zero and
helicity $\geq 1$) are covariantly non-localizable in a strict sense of an
imprimitivity system based on the $3$-d Euclidean group acting on
$\mathbb{R}^{3}.$.

J.M. Jauch and C. Piron [5], developed a concept of “weak localizability”
replacing projection-valued measure by POV (positive operator-valued)
measures, and A.O. Amrein [6] proved that there exist photon states strictly
POV-localized in arbitrarily small regions of space, while, more recently, I.
and Z. Bialynicki-Birula [7] argued that photons cannot be sharply localized
because of a kind of complementarity between magnetic and electric energy
localization.

Closely related to the problem of a photon’s localization is the problem of
existence of the photon position operator ${\bf Q}.$ The problem is not
exactly the same since a given vector-valued operator may have infinitely many
representations in terms of POV measures (except when $Q_{i}$’s commute, and
then there is a distinguished projection-valued spectral measure). It is known
[8] that the standard requirements of the covariance with respect to the
Euclidean group and inversions lead to a unique ${\bf Q},$ \- known as the
Pryce photon position operator222The proof of uniqueness provided in this
reference has a hole, as it requires an additional restriction on the form of
the operator. But the hole in the proof can be completed, and no extra
assumptions are in fact necessary., the trouble is that the components $Q_{i}$
do not commute, which makes the simple probabilistic interpretation for the
photon’s localization problem impossible.333While $Q_{i}$ admits a natural
decomposition with respect to a POV measure, K. Kraus [9] has shown that there
exists more than one such measure, so that the question appears which one of
them is more natural than others, and why? B.S. Skagerstam [10] interpreted
noncommutativity of components of the Pryce operators $Q_{i}^{\mbox{PR}}$ in
terms of the curvature of a connection in a photon’s momentum
space444Skagerstam is using Jackiw’s concept of “three-cocycle” [11],
developed before under a different name (generalized imprimitivity system) by
the present author [12]., and applied it to derivation of the Berry phase for
a photon [13]. The same idea has been discussed before by I. and Z.
Bialynicki-Birula [14], except that in their paper the same connection has
been derived independently of the photon’s position operator question.

Abandoning the requirement of a covariance with respect to the $3D$ rotation
group opens the way towards the construction of a huge family of “position
operators" for photons, including those with commuting components and
therefore admitting a unique spectral decomposition. M. Hawton [15] has
started a whole series of works in this direction. Commuting of the components
of the position operator implies a flat (curvature zero) connection on the
positive light cone in momentum space. Connections of this type (no curvature,
but torsion) has been long ago investigated by A. Staruszkiewicz [16, 17], who
required that the parallel transport preserves the natural (degenerate)
metric, the volume form, and that the connection is semi-symmetric (a
constraint on the torsion). Hawton and Baylis [18] investigated a particular
photon position operator ${\bf Q}^{\mbox{HB}}$ with commuting components, the
Hawton-Baylis operator , with axial symmetry. 555If Berry phase is related to
a non-zero curvature of the connection responsible for the parallel transport,
as it is usually assumed, then any such position operator must lead to a
vanishing Berry phase. Recently these ideas have been further developed by
Dobrski et al. [19, 20].

The present work started with the realization that there is an apparent
discrepancy between the geometrical picture of the photon wave function as a
section of the tangent or cotangent bundle666Whenever a (non-degenerate)
metric is available, there is no need to distinguishes between tangent and
cotangent bundles. The distinction between the two becomes relevant only in
premetric formulations of electrodynamics - c.f. [21] and references therein.,
and the way the Lorentz boosts act on vector-valued functions within the
unitary representation of the Poincaré group considered in all papers dealing
with the photon wave function and with photon position operators.777For the
photon wave function see e.g. [22], while for photon position operators
discussed in a similar context as that taken in the present paper see [20] and
references therein. In the present paper we start with a geometrical
description of massive vector fields transforming naturally under the Poincaré
group. We work in the momentum space888A good introduction to photon’s wave
mechanics, in both momentum and position space, as well as to the photon’s
localization problem, can be found in the review article [23], and in the
monograph [24], by Ole Keller. and discuss the natural unitary representation
acting on the Hilbert space of sections of the tangent bundle of the positive
mass hyperboloid, square integrable with respect to the natural Lorentz
invariant measure. The natural Riemannian metric appearing there becomes
degenerate in the limit $m=0.$ For $m>0,$ taking the positive square root of
this metric we split the tangent space at each point into two mutually
orthogonal parts: the longitudinal-transversal split with respect to the mass-
independent standard Euclidean metric. This enables us to twist the unitary
representation along the longitudinal part. We obtain an explicit form of so
obtained irreducible unitary representation and then take the limit $m=0$,
which now is finite (though not irreducible). Only then the Lie algebra of the
Poincaré group acquires the standard form.

We then use the explicit form of the boost unitary operators to obtain rather
unexpected result: the polarization basis used for constructing the
teleparallel connection by Hawton and Baylis [18] can be obtained by taking
the speed-of-light limit in the $z$-direction of simple Hertz-type potentials
${\bf e}_{1}\sim{\mathbf{p}}\times{\mathbf{w}}$ and ${\bf
e}_{2}\sim{\mathbf{p}}\times({\mathbf{p}}\times{\mathbf{w}}),{\mathbf{w}}=(0,0,1).$
We then discuss the Hawton-Baylis connection and the associated photon
position operator with commuting components, and compare it to the Pryce
connection with non-zero curvature and torsion. In particular we find that the
Pryce connection is metric semi-symmetric, while Hawton-Baylis connection does
not have the semi-symmetry property.

In the last part of this paper we analyze photon states ${\mathbf{f}}_{\ell}$
localized on loops $\ell$ in photon’s position space. They are given as simple
superpositions of plane waves localized at the points of the loop. For the
particular case when the loop is a circle on a $z=0$ plane we show that
${\mathbf{f}}_{l}$ are localized on the circle not only with respect to the
Jauch-Piron-Amrein POV measure $F(\Delta)$ (what was known before), but also
(which came as a surprise), with respect to ${\bf Q}^{\mbox{HB}}.$

We construct simple wave packets made of these circle states that provide
normalized photon states localized in arbitrarily small washer-like regions of
space, again both with respect to ${\bf Q}^{\mbox{HB}}$ and $F(\Delta).$

Notation
We work in the momentum representation, spacetime signature $(-+++).$.
Coordinates $(p^{0},p^{1},p^{2},p^{3})=(p^{0},{\mathbf{p}}),$
$p^{2}={\mathbf{p}}^{2}-(p^{0})^{2}.$ We will write $\mathbb{R}^{3}$ to denote
the $3$-dimensional real vector space of the momentum vectors ${\mathbf{p}}.$
Greek indices $\mu,\nu,\rho,\sigma$ run from $0$ to $3$. Latin indices
$i,j,k,l$ run from $1$ to $3$. We use the Greek letter $\alpha,$ also running
from $1$ to $3$ for numbering the basis vectors in momentum space. Summation
over repeated indices is implied. Only positive energies, $p^{0}>0,$ are being
used. We will use the notation $\boldsymbol{\pi}$ to denote the dimensionless
unit vector in the direction of the momentum ${\mathbf{p}}$:
$\boldsymbol{\pi}={\mathbf{p}}/|{\mathbf{p}}|.$

##  2 Massive vector field

For $m>0,$ we denote by $V^{+}_{m}$ the hyperboloid $p^{2}+m^{2}=0,\,p^{0}>0,$
i.e.

$p^{0}=+\sqrt{{\mathbf{p}}^{2}+m^{2}}.$ (1)

Later on we will be interested in the limit $m\rightarrow 0.$ The mass
hyperboloid $V^{+}_{m}$ is globally parametrized by
${\mathbf{p}}\in\mathbb{R}^{3}.$ In the limit $m\rightarrow 0,$ $V^{+}_{m}$
becomes the positive cone $V^{+}_{0}$, and we will remove the origin
${\mathbf{p}}=0.$

###  2.1 The tangent bundle

Let $TV^{+}_{m}$ be the tangent bundle of $V^{+}_{m}.$ Let $p^{\mu}(t)$ be a
(differentiable) path in $V_{m}^{+},$ with
$(p^{\mu}(0))=(p^{0}=\sqrt{{\mathbf{p}}^{2}+m^{2}},{\mathbf{p}}).$ For each
$t$ we have

$p^{0}(t)^{2}={\mathbf{p}}(t)^{2}+m^{2}.$ (2)

Differentiating at $t=0$ and setting $v^{\mu}=(dp^{\mu}(t)/dt)|_{t=0},$ we get

$v^{0}p^{0}={\mathbf{v}}\cdot{\mathbf{p}},$ (3)

therefore, in $TV^{+}_{m},$ at ${\mathbf{p}},$ we can use only the coordinates
$v^{i}$, the coordinate $v^{0}$ being given by

$v^{0}({\mathbf{p}})=\frac{{\mathbf{v}}\cdot{\mathbf{p}}}{p^{0}}.$ (4)

###  2.2 Action of the Lorentz group

Let $\eta=(\eta_{\mu\nu})$ be the matrix $\eta=\mbox{diag}(-1,+1,+1,+1).$ The
inverse matrix $\eta^{-1}=(\eta^{\mu\nu})$ has the same matrix elements.
Indices are being raised and lowered with the matrices $\eta^{-1}$ and $\eta$
respectively. In particular $p_{i}=p^{i},$ and $p_{0}=-p^{0}.$999We will never
use $p_{0}$.

Let ${\mathcal{L}^{\uparrow}}$ be the orthochronous Lorentz group, that is the
group of those $4\times 4$ real matrices $L=({L^{\mu}}_{\nu})$ satisfying
$\eta^{-1}L^{T}\eta=L^{-1},\,{L^{0}}_{0}>0.$

The group ${\mathcal{L}^{\uparrow}}$ acts on $V^{+}_{m}$ via $p\mapsto Lp,$
$(Lp)^{\mu}={L^{\mu}}_{\nu}p^{\nu}.$ On the mass hyperboloid $V^{+}_{m},$
$p^{0}$ is determined by ${\mathbf{p}}.$ Therefore, on $V^{+}_{m},$ we can
write $(Lp)^{i}={L^{i}}_{j}p^{j}+{L^{i}}_{0}\,p^{0},$ where
$p^{0}=\sqrt{{\mathbf{p}}^{2}+m^{2}}.$ Let ${\mathbf{p}}\mapsto L{\mathbf{p}}$
denote this action:

$(L{\mathbf{p}})^{i}={L^{i}}_{j}\,p^{j}+(L^{i})_{0}\,p^{0}.$ (5)

It induces the action on the tangent bundle $TV_{m}^{+}$ as follows.

If $v^{\mu}$ is a vector tangent to $V^{+}_{m}$ at ${\mathbf{p}},$ then
$(Lv)^{\mu}={L^{\mu}}_{\nu}v^{\nu}$ is tangent to $V^{+}_{m}$ at $Lp.$ Using
now Eq. (4) we obtain

$(Lv)^{i}=({L^{i}}_{j}+\frac{{L^{i}}_{0}p_{j}}{p^{0}})v^{j}.$ (6)

We set

${\tilde{D_{(m)}}^{i}}_{j}(L,{\mathbf{p}})={L^{i}}_{j}+\frac{{L^{i}}_{0}p_{j}}{p^{0}}.$
(7)

Wave functions of the massive spin $1$ particle are sections of the tangent
bundle $TV^{+}_{m}.$ We denote by $\operatorname{Sec}(TV^{+}_{m})$ the space
of these sections. If ${\mathbf{p}}\mapsto{\mathbf{f}}({\mathbf{p}})$ is in
$\operatorname{Sec}(TV^{+}_{m})$, then $L{\mathbf{f}}$ is defined through the
formula

$(L{\mathbf{f}})(L{\mathbf{p}})=L({\mathbf{f}}({\mathbf{p}})),$ (8)

or, in coordinates:

$(L{\mathbf{f}})^{i}(L{\mathbf{p}})={\tilde{D_{(m)}}^{i}}_{j}(L,{\mathbf{p}})f^{j}({\mathbf{p}}).$
(9)

Replacing $L{\mathbf{p}}$ with ${\mathbf{p}}$ and ${\mathbf{p}}$ with
$L^{-1}{\mathbf{p}}$, we get

$(L{\mathbf{f}})^{i}({\mathbf{p}})={\tilde{D_{(m)}}^{i}}_{j}(L,L^{-1}{\mathbf{p}})f^{j}(L^{-1}{\mathbf{p}}).$
(10)

The formula (10) defines a natural linear action of ${\mathcal{L}^{\uparrow}}$
on $\operatorname{Sec}(TV^{+}_{m}).$

It can be verified by a direct calculation that the matrices
$\tilde{D_{(m)}}(L,{\mathbf{p}})$ satisfy the following “cocycle relations”:

$\tilde{D_{(m)}}(L_{1}L_{2},{\mathbf{p}})=\tilde{D_{(m)}}(L_{1},L_{2}{\mathbf{p}})\tilde{D_{(m)}}(L_{2},{\mathbf{p}}),$
(11)

which is equivalent to

$(L_{1}L_{2}){\mathbf{f}}=L_{1}(L_{2}{\mathbf{f}}),$ (12)

which means, that we have a linear representation of the group
${\mathcal{L}^{\uparrow}}$ on $\operatorname{Sec}(TV^{+}_{m}).$ So far no
restriction on sections ${\mathbf{p}}\mapsto f({\mathbf{p}})$ are needed. The
minimal assumption is that they are (Borel) measurable. Notice also that
$f({\mathbf{p}})$ can be assumed to be real. As long as we are interested only
in pure Lorentz transformations, and not in spacetime translations, there is
no need to complexify $TV^{+}_{m}$.

###  2.3 Riemannian metric on $V^{+}_{m}.$

For $m>0$ the fibers of $TV^{+}_{m}$ carry a natural Lorentz invariant
Riemannian structure: the scalar product $v\cdot
v^{\prime}={\mathbf{v}}\cdot{\mathbf{v}}^{\prime}-v^{0}{v^{\prime}}^{0}$ is
evidently Lorentz-invariant. Substituting the expression (4) we obtain the
coordinate expression for the Riemannian metric $g_{(m)ij}({\mathbf{p}})$

$g_{(m)ij}({\mathbf{p}})=\delta_{ij}-\frac{p_{i}p_{j}}{|{\mathbf{p}}|^{2}+m^{2}}.$
(13)

Its inverse is given by

$g_{(m)}^{ij}=\delta^{ij}+\frac{p^{i}p^{j}}{m^{2}}.$ (14)

Since the flat metric $\eta$ is invariant under the linear action of the
Lorentz group, the induced metric $g_{m}$ is invariant under the induced
action. We have

$\tilde{D_{(m)}}(L,{\mathbf{p}})^{T}g_{(m)}(L{\mathbf{p}})\tilde{D_{(m)}}(L,{\mathbf{p}})=g_{(m)}({\mathbf{p}}).$
(15)

In what follows we will need the positive square root of $g_{(m)}.$ Since, as
long as $m>0,$ $g_{(m)}$ is positive definite, there exists a unique positive-
definite square root  $h_{(m)}=({h^{(m)i}}_{j})$ of $g_{(m)}.$ It can be
verified that $h_{(m)}$ is given by the following explicit expression:

${{h_{(m)}}^{i}}_{j}({\mathbf{p}})=\delta^{i}_{j}+\lambda({\mathbf{p}})\pi^{i}\pi_{j},$
(16) $\lambda({\mathbf{p}})=\frac{m}{p^{0}}-1.$ (17)

We will also need its inverse $(h_{(m)})^{-1},$ which is given by

${{{h_{(m)}}^{-1}}^{i}}_{j}({\mathbf{p}})=\delta^{i}_{j}+\mu({\mathbf{p}})\pi^{i}\pi_{j},$
(18) $\mu({\mathbf{p}})=\frac{p^{0}}{m}-1.$ (19)

######  Remark 1.

We have two scalar products in $\mathbb{R}^{3}_{\mathbf{p}}$, the standard,
Euclidean one, $\delta_{ij}$, and the one determined by the metric
$g_{(m)ij}.$ In the following whenever we write the dot product or raise or
lower the space index $(i,j,\ldots),$ we will always use the standard
Euclidean metric.

###  2.4 The Hilbert space and unitary representation

We have

$\sqrt{\det g_{(m)}}=\frac{m}{p^{0}},$ (20)

therefore, since $g_{(m)}$ is Lorentz invariant, $d^{3}p/p^{0}$ is a Lorentz
invariant measure on $V^{+}_{m}.$ We define the Hilbert space
${\tilde{\mathcal{H}}}_{m}$ as the Hilbert space of sections
${\mathbf{f}}({\mathbf{p}})$ of $TV^{+}_{m}$ square integrable with respect to
the scalar product

$({\mathbf{f}},{\mathbf{f}}^{\prime})_{m}=\int
g_{(m)ij}({\mathbf{p}})\bar{f}^{i}({\mathbf{p}})f^{\prime
j}({\mathbf{p}})\frac{d^{3}p}{p^{0}}.$ (21)

By construction the formula

$(\tilde{U}(L){\mathbf{f}})({\mathbf{p}})\doteq(L{\mathbf{f}})({\mathbf{p}})=\tilde{D_{(m)}}(L,L^{-1}{\mathbf{p}}){\mathbf{f}}(L^{-1}{\mathbf{p}})$
(22)

defines a unitary representation of ${\mathcal{L}^{\uparrow}}$ on
${\tilde{\mathcal{H}}}_{m}.$

The unitary representations of ${\mathcal{L}^{\uparrow}}$ for different values
of $m$ have the same form but act in Hilbert spaces with different scalar
products in the fibers. In order to be able to take the limit $m=0$ it is
convenient to use just one standard fiber scalar product, independent of the
value of $m$, but make the form of the representation $m$-dependent. To this
end let ${\mathcal{H}}_{m}$ be the Hilbert space of sections of $TV^{+}_{m}$
square integrable with respect to the standard Hermitian scalar product

$<{\mathbf{f}},{\mathbf{f}}^{\prime}>_{m}=\int\bar{{\mathbf{f}}}({\mathbf{p}})\cdot{\mathbf{f}}^{\prime}({\mathbf{p}})\frac{d^{3}p}{p^{0}},$
(23)

where $\bar{{\mathbf{f}}}$ denotes the complex conjugate (not needed if
${\mathbf{f}}$ is real). Then the map

$h_{(m)}:{\mathbf{f}}({\mathbf{p}})\mapsto{h_{(m)}}({\mathbf{p}}){\mathbf{f}}({\mathbf{p}}),$
(24)

where $h_{(m)}$ is given by Eq. (16), is an isometry from
${\tilde{\mathcal{H}}}_{m}$ to ${\mathcal{H}}_{m}.$ Correspondingly we have a
unitary representation $L\mapsto
U_{m}(L)=h_{(m)}\circ\tilde{U}(L)\circ{h_{(m)}}^{-1}$ of
${\mathcal{L}^{\uparrow}}$ on ${\mathcal{H}}_{m}.$ It follows then from the
definition that $U(L)$ can be written as

$(U_{m}(L)f)({\mathbf{p}})=D_{m}(L,L^{-1}{\mathbf{p}})f(L^{-1}{\mathbf{p}}),$
(25)

where

$D_{m}(L,{\mathbf{p}})=h_{(m)}(L{\mathbf{p}})\tilde{D_{(m)}}(L,{\mathbf{p}})h_{(m)}^{-1}({\mathbf{p}}).$
(26)

By using Eq. (15) and the definition of $h_{(m)}$ we find that the matrices
$D_{m}(L,{\mathbf{p}})$ are orthogonal, which makes the property of unitarity
of $U_{m}(L)$ evident. 101010Using the terminology of [2, p. 175] one says
that the cocycles $\tilde{D_{(m)}}$ and $D_{m}$ are strictly cohomologous.

####  2.4.1 The longitudinal-transversal split

Assuming ${\mathbf{p}}\neq 0,$ the eigenvalue equation
$g_{(m)}{\mathbf{f}}=\lambda{\mathbf{f}}$ for the real symmetric matrix
$g_{(m)}$ reads

$f^{i}({\mathbf{p}})-\frac{{\mathbf{p}}\cdot{\mathbf{f}}({\mathbf{p}})}{|{\mathbf{p}}|^{2}+m^{2}}p^{i}=\lambda
f^{i}({\mathbf{p}}),$

i.e.

$(1-\lambda)f^{i}({\mathbf{p}})=\frac{{\mathbf{p}}\cdot{\mathbf{f}}({\mathbf{p}})}{|{\mathbf{p}}|^{2}+m^{2}}p^{i}.$
(27)

Thus either $\lambda=1,$ and then
${\mathbf{p}}\cdot{\mathbf{f}}({\mathbf{p}})=0,$ or $\lambda\neq 1,$ and then
${\mathbf{f}}({\mathbf{p}})$ is proportional to ${\mathbf{p}}.$ For each
${\mathbf{p}}\neq 0$ let $\mathcal{P}_{0}({\mathbf{p}})$ be the orthogonal
projection (in $\mathbb{R}^{3}$ endowed with the Euclidean metric
$\delta_{ij}$) on the one-dimensional subspace consisting of vectors
proportional to ${\mathbf{p}}$ :

$\mathcal{P}_{0}({\mathbf{p}})\,{\mathbf{f}}({\mathbf{p}})=(\boldsymbol{\pi}\cdot{\mathbf{f}}({\mathbf{p}}))\boldsymbol{\pi}.$
(28)

Then $\mathcal{P}_{0}({\mathbf{p}})$ projects onto the eigenspace of
$g_{(m)}({\mathbf{p}})$ belonging to the eigenvalue
$m^{2}/(|{\mathbf{p}}|^{2}+m^{2}).$ Let
$\mathcal{P}_{1}({\mathbf{p}})=I-\mathcal{P}_{0}({\mathbf{p}})$ be the
orthogonal projection on the complementary subspace of vectors corresponding
to the eigenvalue $1$:

$\mathcal{P}_{1}({\mathbf{p}}){\mathbf{f}}({\mathbf{p}})={\mathbf{f}}({\mathbf{p}})-(\boldsymbol{\pi}\cdot{\mathbf{f}}({\mathbf{p}}))\boldsymbol{\pi},$
(29)

so that

$g_{(m)}({\mathbf{p}})=\frac{m^{2}}{|{\mathbf{p}}|^{2}+m^{2}}\mathcal{P}_{0}({\mathbf{p}})+\mathcal{P}_{1}({\mathbf{p}}).$
(30)

Then we immediately get

$h_{(m)}({\mathbf{p}})=\frac{m}{\sqrt{|{\mathbf{p}}|^{2}+m^{2}}}\mathcal{P}_{0}({\mathbf{p}})+\mathcal{P}_{1}({\mathbf{p}}),$
(31)

and

${h_{(m)}}^{-1}({\mathbf{p}})=\frac{\sqrt{|{\mathbf{p}}|^{2}+m^{2}}}{m}\mathcal{P}_{0}({\mathbf{p}})+\mathcal{P}_{1}({\mathbf{p}}).$
(32)

It should be noticed, however, that, as long as $m>0,$ the split above is not
invariant under the action of boosts of the Lorentz group on $V_{m}^{+}.$

###  2.5 The limit $m=0$

For $m=0$ the matrices $\tilde{D_{(m)}}({\mathbf{p}})$ are still given by Eq.
(7), but now $p^{0}=|{\mathbf{p}}|.$ It is less obvious that the matrices
$D_{m}$ given by Eq. (26) remain finite in the zero mass limit. In Proposition
2 below we show that, for all $L$ in ${\mathcal{L}^{\uparrow}}$ and
${\mathbf{p}}$ in $V_{m}^{+},$ the limit

$D_{0}(L,{\mathbf{p}})=\lim\limits_{m=0}D_{m}(L,{\mathbf{p}})$ (33)

exists, and we provide its explicit form.

######  Proposition 2.

$D_{0}(L,{\mathbf{p}})$ is finite and it is given by the following explicit
formula:

${{D_{0}}^{i}}_{j}(L,{\mathbf{p}})={L^{i}}_{j}+{L^{i}}_{0}\pi_{j}-{\pi^{\prime}}^{i}{L^{0}}_{j}+{\pi^{\prime}}^{i}\pi_{j}(1-{L^{0}}_{0}),$
(34)

where $\boldsymbol{\pi}^{\prime}=\frac{L{\mathbf{p}}}{|L{\mathbf{p}}|}.$ For
each $L\in{\mathcal{L}^{\uparrow}}$ and ${\mathbf{p}}\neq 0$ the matrix
$D_{0}(L,{\mathbf{p}})$ is orthogonal

$D_{0}(L,{\mathbf{p}})^{T}=D_{0}(L,{\mathbf{p}})^{-1}.$ (35)

The representation

$(U_{0}(L){\mathbf{f}})({\mathbf{p}})\doteq
D_{0}(L,L^{-1}{\mathbf{p}}){\mathbf{f}}(L^{-1}{\mathbf{p}})$ (36)

of ${\mathcal{L}^{\uparrow}}$ on the Hilbert space
${\mathcal{H}}_{0}=L^{2}(\mathbb{R}^{3}_{\mathbf{p}},d^{3}p/|{\mathbf{p}}|)\otimes\mathbb{C}^{3}$
is unitary.

###### Proof.

Substituting Eqs. (31) and (32) into Eq. (26) we obtain for
$D_{m}(L,{\mathbf{p}})$ the sum of four terms $t_{1},t_{2},t_{3},t_{4},$ where

$\displaystyle t_{1}(m)$ $\displaystyle=$
$\displaystyle\frac{\sqrt{|{\mathbf{p}}|^{2}+m^{2}}}{\sqrt{|L{\mathbf{p}}|^{2}+m^{2}}}\mathcal{P}_{0}(L{\mathbf{p}})\tilde{D_{(m)}}(L,{\mathbf{p}})\mathcal{P}_{0}({\mathbf{p}}),$
(37) $\displaystyle t_{2}(m)$ $\displaystyle=$
$\displaystyle\mathcal{P}_{1}(L{\mathbf{p}})\tilde{D_{(m)}}(L,{\mathbf{p}})\mathcal{P}_{1}({\mathbf{p}}),$
(38) $\displaystyle t_{3}(m)$ $\displaystyle=$
$\displaystyle\frac{m}{\sqrt{|L{\mathbf{p}}|^{2}+m^{2}}}\mathcal{P}_{0}(L{\mathbf{p}})\tilde{D_{(m)}}(L,{\mathbf{p}})\mathcal{P}_{1}({\mathbf{p}}),$
(39) $\displaystyle t_{4}(m)$ $\displaystyle=$
$\displaystyle\frac{\sqrt{|{\mathbf{p}}|^{2}+m^{2}}}{m}\mathcal{P}_{1}(L{\mathbf{p}})\tilde{D_{(m)}}(L,{\mathbf{p}})\mathcal{P}_{0}({\mathbf{p}}).$
(40)

Noticing that the orthogonal projections $\mathcal{P}_{0}$ and
$\mathcal{P}_{1}$ do not depend of $m$ we see that $t_{1}(0)$ and $t_{2}(0)$
are finite, and are given by

$t_{1}(0)=\frac{|{\mathbf{p}}|}{|L{\mathbf{p}}|}\mathcal{P}_{0}(L{\mathbf{p}})\tilde{D_{(0)}}(L,{\mathbf{p}})\mathcal{P}_{0}({\mathbf{p}}),$
(41)
$t_{2}(0)=\mathcal{P}_{1}(L{\mathbf{p}})\tilde{D_{(0)}}(L,{\mathbf{p}})\mathcal{P}_{1}({\mathbf{p}}),$
(42)

while the third term vanishes

$t_{3}(0)=0.$ (43)

We will now show that, surprisingly, the fourth term $t_{4}(m)$ also vanishes
in the limit $m=0.$ To this end we consider first the product

$D\mathcal{P}_{0}(m)\doteq\tilde{D_{(m)}}(L,{\mathbf{p}})\mathcal{P}_{0}({\mathbf{p}})$
(44)

of its last two factors. Using the definitions we have

${{D\mathcal{P}_{0}(m)}^{i}}_{k}=\left({L^{i}}_{j}+\frac{{L^{i}}_{0}p_{j}}{p^{0}}\right)\frac{p^{j}p_{k}}{|{\mathbf{p}}|^{2}}=\left(\frac{{L^{i}}_{j}p^{j}}{|{\mathbf{p}}|^{2}}+\frac{{L^{i}}_{0}}{p^{0}}\right)p_{k}.$
(45)

Setting $p^{\prime}=Lp,$ we have ${L^{i}}_{j}p^{j}=p^{\prime
i}-{L^{i}}_{0}p^{0},$ therefore

${{D\mathcal{P}_{0}(m)}^{i}}_{k}=\frac{p^{\prime
i}p_{k}}{|{\mathbf{p}}|^{2}}+{L^{i}}_{0}\left(\frac{1}{p^{0}}-\frac{p^{0}}{|{\mathbf{p}}|^{2}}\right)p_{k}=\frac{p^{\prime
i}p_{k}}{|{\mathbf{p}}|^{2}}-m^{2}\frac{{L^{i}}_{0}p_{k}}{p^{0}|{\mathbf{p}}|^{2}}.$
(46)

Now, the projection $\mathcal{P}_{1}({\mathbf{p}}^{\prime})$ vanishes on the
first term $\frac{p^{\prime i}p_{k}}{|{\mathbf{p}}|^{2}},$ as the range of
this matrix is in the longitudinal subspace at ${\mathbf{p}}^{\prime}$, and
$\frac{\sqrt{|{\mathbf{p}}|^{2}+m^{2}}}{m}\mathcal{P}_{1}(L{\mathbf{p}})$
acting on the second term vanishes linearly in $m.$ Therefore also
$t_{4}(0)=0,$ and so

$D_{0}(L,{\mathbf{p}})=\frac{|{\mathbf{p}}|}{|{\mathbf{p}}^{\prime}|}\mathcal{P}_{0}({\mathbf{p}}^{\prime})\tilde{D_{(0)}}(L,{\mathbf{p}})\mathcal{P}_{0}({\mathbf{p}})+\mathcal{P}_{1}({\mathbf{p}}^{\prime})\tilde{D_{(0)}}(L,{\mathbf{p}})\mathcal{P}_{1}({\mathbf{p}}).$
(47)

Since the subbundle of longitudinal vectors is invariant under the action of
$\tilde{D_{(0)}},$ we have
$\mathcal{P}_{1}({\mathbf{p}}^{\prime})\tilde{D_{(0)}}(L,{\mathbf{p}})\mathcal{P}_{0}({\mathbf{p}})=0$
and
$\mathcal{P}_{0}({\mathbf{p}}^{\prime})\tilde{D_{(0)}}(L,{\mathbf{p}})\mathcal{P}_{0}({\mathbf{p}})=\tilde{D_{(0)}}(L,{\mathbf{p}})\mathcal{P}_{0}({\mathbf{p}}).$
Therefore Eq. (47) simplifies to

$D_{0}(L,{\mathbf{p}})=\tilde{D_{(0)}}(L,{\mathbf{p}})+\frac{|{\mathbf{p}}|}{|{\mathbf{p}}^{\prime}|}\tilde{D_{(0)}}(L,{\mathbf{p}})\mathcal{P}_{0}({\mathbf{p}})-\mathcal{P}_{0}({\mathbf{p}}^{\prime})\tilde{D_{(0)}}(L,{\mathbf{p}}).$
(48)

The matrix $\tilde{D_{(0)}}(L,{\mathbf{p}})\mathcal{P}_{0}({\mathbf{p}})$ we
have already calculated - we just set $m=0$ in Eq. (46). Using now the fact
that $L^{T}\eta L=\eta,$ a straightforward algebra gives us

${(\mathcal{P}_{0}({\mathbf{p}}^{\prime})\tilde{D_{(0)}}(L,{\mathbf{p}}))^{i}}_{j}={\pi^{\prime}}^{i}({L^{0}}_{j}+\pi_{j}{L^{0}}_{0}),$
(49)

and leads to the formula (34) of Proposition 2.111111Notice that
$\tilde{D_{(0)}}(L,{\mathbf{p}})$ gives the first two terms of the right hand
side of (34). The matrices $D_{0}(L,{\mathbf{p}})$ are orthogonal as limits of
a continuous family of orthogonal matrices, and, since the measure
$d^{3}p/p^{0}$ is Lorentz invariant, the representation $U_{0}$ is unitary. ∎

We notice that for pure space rotations:
${L_{i}}^{0}={L_{0}}^{j}=0,{L_{0}}^{0}=1,$ we have
${{D_{0}}_{i}}^{j}={L_{i}}^{j}.$

###  2.6 The Poincaré group Lie algebra

So far we have discussed only the homogeneous transformations form the
Poincaré group - the Lorentz transformations from ${\mathcal{L}^{\uparrow}}.$
Now we add translations. They are implemented by complex phase rotations:

$(U_{0}(a){\mathbf{f}})({\mathbf{p}})=e^{ia\cdot
p}{\mathbf{f}}({\mathbf{p}})=e^{i(a^{1}p^{1}+a^{2}p^{2}+a^{3}p^{3}-a^{0}p^{0})}{\mathbf{f}}({\mathbf{p}}).$
(50)

These transformations are evidently unitary on ${\mathcal{H}}_{0}.$ Having the
unitary representation of the whole group we will calculate now the self-
adjoint infinitesimal generators. For translations we define
$P_{\mu}=-i(dU(a)/da^{\mu})|_{a=0},$ obtaining

$P^{i}{\mathbf{f}}({\mathbf{p}})=p^{i}{\mathbf{f}}({\mathbf{p}}),\quad
P^{0}{\mathbf{f}}({\mathbf{p}})=|{\mathbf{p}}|{\mathbf{f}}({\mathbf{p}}).$
(51)

The Lie algebra of the orthogonal group $SO(4)$ consists of antisymmetric
matrices $\tilde{m}_{\mu\nu}$ given by

$(\tilde{m}_{\mu\nu})^{\sigma\rho}=\delta_{\mu}^{\sigma}\delta_{\nu}^{\rho}-\delta_{\nu}^{\sigma}\delta_{\mu}^{\rho}.$
(52)

The matrices $m^{\mu\nu}=\tilde{m}^{\mu\nu}\eta$ form then a basis in the Lie
algebra of the Lorentz group

${(m_{\mu\nu})^{\sigma}}_{\rho}=\delta_{\mu}^{\sigma}\eta_{\nu\rho}-\delta_{\nu}^{\sigma}\eta_{\mu\rho}.$
(53)

Defining ${\bf m}=(m^{i})$ and ${\bf n}=(n^{i})$ through
$m^{i}=\frac{1}{2}\epsilon^{ijk}m_{jk},\,n^{i}=m_{0i},$ we obtain the
commutation relations

$[m^{i},m^{j}]=-\epsilon_{ijk}m^{k},\,[m^{i},n^{j}]=-\epsilon_{ijk}n^{k},\,[n^{i},n^{j}]=\epsilon_{ijk}m^{k}.$
(54)

Then the self-adjoint operators $M^{i},N^{i}$ are defined as

$M^{i}=-idU(\exp(t\,m^{i}))/dt|_{t=0},\,N^{i}=-idU(\exp(t\,n^{i}))/dt|_{t=0}.$
(55)

Using our explicit formulas, we obtain:

$\displaystyle{\bf M}$ $\displaystyle=$ $\displaystyle{\bf L}+{\bf s},$ (56)
$\displaystyle{\bf N}$ $\displaystyle=$ $\displaystyle{\bf K}+{\bf k},$ (57)

where

${\bf L}=-i({\mathbf{p}}\times\partial/\partial{\mathbf{p}},\quad{\bf
K}=i|{\mathbf{p}}|\partial/\partial{\mathbf{p}},\quad{\bf
k}=\boldsymbol{\pi}\times{\bf s},$ (58) ${\bf s}=-i{\bf m}.$ (59)

The commutation relations obtained from the definitions are the standard ones:

$[N^{i},P^{j}]=i\delta^{ij}P^{0},\,[N^{i},P^{0}]=iP^{i},\,[M^{i},P^{j}]=i\epsilon_{ijk}P^{k},\,[M^{i},P^{0}]=0,$
(60)
$[N^{i},N^{j}]=-i\epsilon_{ijk}M^{k},\,[M^{i},N^{j}]=i\epsilon_{ijk}N^{k},\,[M^{i},M^{j}]=i\epsilon_{ijk}M^{k}.$
(61)

Replacing $\bf{M}$ by $\bf{L}$ and $\bf{N}$ by $\bf{K}$ we obtain the same
commutation relations.

######  Remark 3.

The representation $U_{0}$ defined by (36) is unitary on the Hilbert space
${\mathcal{H}}_{0}$ of vector-valued functions square integrable with respect
to the scalar product

$<{\mathbf{f}},{\mathbf{f}}^{\prime}>_{0}=\int{\mathbf{f}}^{\dagger}({\mathbf{p}}){\mathbf{f}}^{\prime}({\mathbf{p}})\frac{d^{3}p}{|{\mathbf{p}}|}.$
(62)

In Sec. 4 we will use a different scalar product using a non-Lorentz-invariant
measure $d^{3}p.$ To make the representation $U_{0}$ unitary with respect to
this scalar product, we need to adjust $D_{0}$ by introducing an extra scaling
factor and defining

$D(L,{\mathbf{p}})=\sqrt{\frac{|{\mathbf{p}}|}{|L{\mathbf{p}}|}}D_{0}(L,{\mathbf{p}}).$
(63)

The expression for infinitesimal generators for this unitary representation
remains the same, except that in the definition of ${\bf K}$ we must replace
$i|{\mathbf{p}}|\partial/\partial{\mathbf{p}}$ by
$i|{\mathbf{p}}|\partial/\partial{\mathbf{p}}+\frac{1}{2}\boldsymbol{\pi}.$

Notice however, that with the scalar product (62), and with $D_{0}$ replaced
by $D$, the transversal components of ${\mathbf{f}}({\mathbf{p}})$ cannot be
directly interpreted as tangent vectors, since they do not have the correct
transformation properties under boosts.

The unitary space inversion operator $\Pi$ and antiunitary time inversion
operator $\Theta$ for this representation are given by

$(\Pi{\bf f})({\mathbf{p}})={\bf f}(-{\mathbf{p}}),\quad(\Theta{\bf
f})({\mathbf{p}})={\bf f}^{*}(-{\mathbf{p}}).$ (64)

For a general representation of the Poincaré group one defines the four-
dimensional Pauli-Lubanski pseudovector $W^{\mu}$ as

$W_{\mu}=\frac{1}{2}\epsilon_{\nu\rho\sigma\mu}\,P^{\nu}M^{\rho\sigma}.$ (65)

We have

$W^{0}={\bf P}\cdot{\bf M},\,{\bf W}=P^{0}\,{\bf M}-{\bf P}\times{\bf N}.$
(66)

It follows from the very definition that $\eta_{\mu\nu}P^{\mu}W^{\nu}=0.$ For
mass zero representations $P^{\mu}$ is lightlike, and therefore $W^{\mu}$ is
proportional to $P^{\mu}$:

$W^{\mu}=\Lambda P^{\mu}.$ (67)

The proportionality operator $\Lambda$ commutes with all the generators and is
called the helicity operator (see e.g. [25, p.64]). In our case one finds that
for generators $P^{\mu},\bf{L},\bf{K}$ we have $W^{\mu}=0$, therefore
$\Lambda=0,$ while for the generators $P^{\mu},\bf{M},\bf{N}$ we have

$\Lambda=\boldsymbol{\pi}\cdot{\bf M}=\boldsymbol{\pi}\cdot\bf{s},$ (68)

or explicitly, using Eq. (59),

$(\Lambda{\mathbf{f}})({\mathbf{p}})=i\boldsymbol{\pi}\times{\mathbf{f}}({\mathbf{p}}).$
(69)

The spectrum of $\Lambda$ is discrete and consists of three points
$\lambda=\pm 1$ and $0.$ Therefore $\Lambda^{2}$ is a projection onto the
subspace ${\mathcal{H}}_{ph}$ of ${\mathcal{H}}_{0}.$ ${\mathcal{H}}_{ph}$ is
a direct sum of eigenspaces ${\mathcal{H}}_{\pm}$ of $\Lambda$ corresponding
to eigenvalues $\lambda=\pm 1.$ The photon states are represented by vectors
in ${\mathcal{H}}_{ph}.$ The orthogonal complement ${\mathcal{H}}_{l}$ of
${\mathcal{H}}_{ph}$ in ${\mathcal{H}}_{0}$ describes a spinless particle. It
follows from these definitions that

${\mathcal{H}}_{ph}=\\{{\mathbf{f}}\in{\mathcal{H}}_{0}:\,{\mathbf{p}}\cdot{\mathbf{f}}({\mathbf{p}})=0\\}$
(70)

and that

${\mathcal{H}}_{l}=\\{{\mathbf{f}}\in{\mathcal{H}}_{0}:\,{\mathbf{f}}({\mathbf{p}})=c({\mathbf{p}}){\mathbf{p}}\\}$
(71)

for some scalar function $c({\mathbf{p}}).$ Since $\Lambda$ commutes with all
the generators of the Poincaré group, they leave the subspaces
${\mathcal{H}}_{ph}$ and ${\mathcal{H}}_{l}$ invariant.

##  3 Application of the explicit form: photon polarization vectors boosted to
the speed of light

###  3.1 Action of pure boosts

We will be interested in one-parameter subgroups of Lorentz transformations
$L({\mathbf{w}},s),\\\
{\mathbf{w}}\in\mathbb{R}^{3},{\mathbf{w}}^{2}=1,\,s\in\mathbb{R},$ defined by

$L({\mathbf{w}},s)=\exp(-s\,{\mathbf{w}}\cdot{\bf
n})=\exp\left(s\left(\begin{smallmatrix}-w_{1}&0&0&0\\\ -w_{2}&0&0&0\\\
-w_{3}&0&0&0\\\ 0&-w_{1}&-w_{2}&-w_{3}\end{smallmatrix}\right)\right).$ (72)

For the matrix ${L({\mathbf{w}},s)^{\mu}}_{\nu}$ we then obtain121212The
parameter $s$ is known as rapidity , $s=\mbox{atanh}(\beta),$ $\beta=v/c.$

$\displaystyle{L^{0}}_{i}$ $\displaystyle=$
$\displaystyle{L^{i}}_{0}=-w_{i}\sinh(s),$ $\displaystyle{L^{i}}_{j}$
$\displaystyle=$ $\displaystyle\delta^{i}_{j}+w^{i}w_{j}(\cosh(s)-1),$ (73)
$\displaystyle{L^{0}}_{0}$ $\displaystyle=$ $\displaystyle\cosh(s).$

For a fixed ${\mathbf{w}}$ we have the group property:

$L({\mathbf{w}},s_{1})L({\mathbf{w}},s_{2})=L({\mathbf{w}},s_{1}+s_{2}).$ (74)

In particular we have $L({\mathbf{w}},s)^{-1}=L({\mathbf{w}},-s).$
For some special sections ${\mathbf{f}}({\mathbf{p}})$ of $TV_{0}^{+}$ we will
be interested in calculating the limits

$\lim\limits_{s\rightarrow\pm\infty}U_{0}(L({\mathbf{w}},s)){\mathbf{f}}.$
(75)

Having in mind Eq. (36) let

$D_{0}^{\prime}({\mathbf{w}},s,{\mathbf{p}})\doteq
D_{0}(L({\mathbf{w}},s),L({\mathbf{w}},s)^{-1}{\mathbf{p}}).$ (76)

The following Proposition shows that the limits

$D_{0}^{\prime}({\mathbf{w}},{\mathbf{p}})_{\pm}=\lim\limits_{s\rightarrow\pm\infty}D_{0}^{\prime}({\mathbf{w}},s,{\mathbf{p}})$
(77)

exist, and provides their explicit form.

######  Proposition 4.

For all ${\mathbf{p}}$ not parallel to ${\mathbf{w}},$ we have

$D_{0}^{\prime}({\mathbf{w}},{\mathbf{p}})_{\pm}=\mathbbm{1}\pm\boldsymbol{\pi}{\mathbf{w}}^{T}-\frac{{\mathbf{w}}{\mathbf{w}}^{T}-(\boldsymbol{\pi}\cdot{\mathbf{w}})\boldsymbol{\pi}{\mathbf{w}}^{T}+\boldsymbol{\pi}\boldsymbol{\pi}^{T}\pm{\mathbf{w}}\boldsymbol{\pi}^{T}}{1\pm\boldsymbol{\pi}\cdot{\mathbf{w}}}.$
(78)

where for any two vectors ${\bf a},{\bf b}$ we denote by ${\bf a}{\bf b}^{T}$
the matrix with components $a_{i}b^{j}.$

###### Proof.

The proof is by a straightforward (though tedious) algebra. From the
definition of $D^{\prime}_{0}$ and Eq. (34) we have the following explicit
expression for $D_{0}(L,L^{-1}{\mathbf{p}})$:

${{D_{0}(L,L^{-1}{\mathbf{p}})}^{i}}_{j}(L,{\mathbf{p}})={L^{i}}_{j}+{L^{i}}_{0}\pi^{\prime}_{j}-{\pi}^{i}{L^{0}}_{j}+{\pi}^{i}\pi^{\prime}_{j}(1-{L^{0}}_{0}),$
(79)

where now
$\boldsymbol{\pi}^{\prime}=\frac{L^{-1}{\mathbf{p}}}{|L^{-1}{\mathbf{p}}|}.$
Substituting there $L({\mathbf{w}},s)$ for $L,$ we obtain fractions containing
$\cosh(s),\sinh(s),$ their squares and products. The terms with
$\sinh(s)\cosh(s)$ cancel out, while the terms with
$\cosh^{2}(s),\sinh^{2}(s)$ collect so that we can apply the identity
$\cosh^{2}(s)-\sinh^{2}(s)=1$. Dividing the numerators and denominators (which
are now linear in $\cosh(s)$ and $\sinh(s)$) by $\cosh(s)$ we use the fact
that $\lim\limits_{s\rightarrow\pm\infty}\tanh(s)=\pm 1$ to obtain the result.
∎

###  3.2 The polarization basis

Using the Euclidean metric we will silently rise and lower the space indices
with the Kronecker deltas $\delta^{ij}$ and $\delta_{ij}.$

Let ${\mathbf{a}}$ be a unit (${\mathbf{a}}\cdot{\mathbf{a}}=1$) vector in
$\mathbb{R}^{3}$, and let ${\mathbf{p}}$ be a non-zero vector in
$\mathbb{R}^{3}$, ${\mathbf{p}}$ not parallel to ${\mathbf{a}}.$ Define the
following two vectors
$e^{\mathbf{a}}_{1}({\mathbf{p}}),e^{\mathbf{a}}_{2}({\mathbf{p}})$ in
$\mathbb{R}^{3}$ as131313Cf. Ref. [26, Eq. (3.3)]

$\displaystyle e^{\mathbf{a}}_{1}({\mathbf{p}})$ $\displaystyle=$
$\displaystyle\frac{{\mathbf{p}}\times{\mathbf{a}}}{|{\mathbf{p}}\times{\mathbf{a}}|},$
(80) $\displaystyle e^{\mathbf{a}}_{2}({\mathbf{p}})$ $\displaystyle=$
$\displaystyle\frac{{\mathbf{p}}\times({\mathbf{p}}\times{\mathbf{a}})}{|{\mathbf{p}}\times({\mathbf{p}}\times{\mathbf{a}})|}.$
(81)

Adding the third vector field $e_{\parallel}$ defined as

$e_{\parallel}({\mathbf{p}})=\boldsymbol{\pi},$ (82)

we obtain an orthonormal basis, which can be considered as an orthonormal
moving frame in the (real) tangent bundle of the positive light cone in
momentum space. The sections $e^{\mathbf{a}}_{1}$ and $e^{\mathbf{a}}_{2}$ of
the tangent bundle $TV^{+}_{0}$ become singular on the straight line
determined by ${\mathbf{a}}.$

######  Remark 5.

One can check that $\Lambda e_{\parallel}({\mathbf{p}})=0,$ and that the
states $e^{\mathbf{a}}_{1}({\mathbf{p}})\pm ie^{\mathbf{a}}_{2}({\mathbf{p}})$
are eigenstates of the helicity operator to the eigenvalue $\pm 1:$

$\Lambda(e^{\mathbf{a}}_{1}\pm ie^{\mathbf{a}}_{2})=\pm(e^{\mathbf{a}}_{1}\pm
ie^{\mathbf{a}}_{2}).$ (83)

######  Proposition 6.

The limits
${e^{\mathbf{a}}_{\alpha}}^{\pm}=\lim\limits_{s\rightarrow\pm\infty}U_{0}(L({\mathbf{w}},t))\,e^{\mathbf{a}}_{\alpha},\,\alpha=1,2,$
exist, and are given by

${e^{\mathbf{a}}_{1}}^{\pm}=\frac{\pm
1}{\sqrt{1-({\mathbf{w}}\cdot{\mathbf{a}})^{2}}}\left({\mathbf{w}}\times{\mathbf{a}}-\frac{\boldsymbol{\pi}\cdot({\mathbf{w}}\times{\mathbf{a}})}{1\pm\boldsymbol{\pi}\cdot{\mathbf{w}}}(\boldsymbol{\pi}\pm{\mathbf{w}})\right),$
(84)
${e^{\mathbf{a}}_{2}}^{\pm}=\frac{1}{\sqrt{1-({\mathbf{w}}\cdot{\mathbf{a}})^{2}}}\left({\mathbf{w}}\times({\mathbf{w}}\times{\mathbf{a}})-\frac{\boldsymbol{\pi}\cdot({\mathbf{w}}\times({\mathbf{w}}\times{\mathbf{a}}))}{1\pm\boldsymbol{\pi}\cdot{\mathbf{w}}}(\boldsymbol{\pi}\pm{\mathbf{w}})\right).$
(85)

Moreover, the vector fields $e_{\mathbf{a}}^{\alpha}$ are invariant under the
boosts in ${\mathbf{w}}$-direction. We have

$({\mathbf{w}}\cdot{\bf N})\,e^{\mathbf{a}}_{\alpha}=0,$ (86)

where ${\bf N}$ is the boost generator (57).

###### Proof.

We know from Proposition 4 that the limits
$D_{0}^{\prime}({\mathbf{w}},{\mathbf{p}})_{\pm}$ exist. On the other hand,
using the definitions of $e^{\mathbf{a}}_{\alpha}$ and
$L({\mathbf{w}},s)^{-1}=L({\mathbf{w}},-s),$ as well as the fact that
$\lim\limits_{s\rightarrow\pm\infty}\tanh(s)=\pm 1,$ we obtain

$\lim\limits_{s\rightarrow\pm\infty}e^{\mathbf{a}}_{1}(L({\mathbf{w}},s)^{-1}{\mathbf{p}})=\frac{\pm{\mathbf{w}}\times{\mathbf{a}}}{\sqrt{1-({\mathbf{w}}\cdot{\mathbf{a}})^{2}}},$
(87)
$\lim\limits_{w\rightarrow\pm\infty}e^{\mathbf{a}}_{2}(L({\mathbf{w}},w)^{-1}{\mathbf{p}})=\frac{{\mathbf{w}}\times({\mathbf{w}}\times{\mathbf{a}})}{\sqrt{1-({\mathbf{w}}\cdot{\mathbf{a}})^{2}}}.$
(88)

The result follows then by an application of the matrix of
$D_{0}^{\prime}({\mathbf{w}},{\mathbf{p}})_{\pm}$ given by formula (78) to the
vectors (87) and (88). The last statement of the proposition is the immediate
consequence of the definitions. ∎

####  3.2.1 Lorentz-boost eigenmodes

Let us take for ${\mathbf{w}}$ the vector

${\mathbf{w}}=(0,0,1),$

(boost in the direction of the third axis), and for ${\mathbf{a}}$ the vector

${\bf a}=(0,1,0).$

We consider the case of $\lim\limits_{w\rightarrow-\infty}.$ Using the
formulas (87) and (88) we obtain the following two mutually orthogonal unit
vector fields:

$e_{1}({\mathbf{p}})=\frac{1}{1-\pi^{3}}\begin{pmatrix}1-\pi^{3}-(\pi^{1})^{2}\\\
-\pi^{1}\pi^{2}\\\ (1-\pi^{3})\pi^{1}\end{pmatrix},\quad
e_{2}({\mathbf{p}})=\frac{1}{1-\pi^{3}}\begin{pmatrix}\pi^{1}\pi^{2}\\\
-(1-\pi^{3})+(\pi^{2})^{2}\\\ -(1-\pi^{3})\pi_{2}\end{pmatrix}.$ (89)

|  |
---|---|---
Table 1: The vector fields
$e_{1}({\mathbf{p}}),e_{2}({\mathbf{p}}),e_{3}({\mathbf{p}})$ plotted at the
unit sphere in the momentum space.

These two vector fields are transversal, they define a photon polarization
basis.141414Notice that while each of the original vector fields
$e^{\mathbf{a}}_{1},e^{\mathbf{a}}_{2}$ has two singular points on the unit
sphere, in their speed-of-light limits these two singularities merge into just
one - the minimum required by the ‘hairy ball theorem’ of algebraic topology.
Together with the third vector field

$e_{3}({\mathbf{p}})=\boldsymbol{\pi},$ (90)

which is longitudinal and spans the helicity zero subspace, they form an
orthonormal basis in $T_{\mathbf{p}}V_{0}^{+}$ \- c.f. Table 1.

It is easily verified that

$M^{3}e_{1}=-ie_{2},\,M^{3}e_{2}=ie_{1},\,M^{3}e_{3}=0,$ (91)

and

$N^{3}e_{1}=N^{3}e_{2}=N^{3}e_{3}=0.$ (92)

######  Remark 7.

The vector fields $e_{1},e_{2}$ are real. Introducing their complex
combinations

$e_{\pm}=\frac{1}{\sqrt{2}}(e_{2}\pm ie_{1}),$ (93)

we obtain two unit complex vector fields of definite helicity and the third
component of angular momentum:

$M^{3}e_{\pm}=\mp e_{\pm},\,\Lambda e_{\pm}=\pm e_{\pm}.$ (94)

Evidently we also have

$N^{3}e_{\pm}=0.$ (95)

The basis $e_{\alpha},$ is naturally embedded into a one-parameter family of
Lorentz-boost eigenmodes151515For the scalar wave equation some of their
properties have been discussed in Ref. [27]. $e^{(\lambda)}_{\alpha}$ defined
by the formula below:

$e^{(\lambda)}_{\alpha}({\mathbf{p}})=\exp\left(i\lambda\log\sqrt{\frac{|{\mathbf{p}}|-p_{3}}{|{\mathbf{p}}|+p_{3}}}\right)\,e_{\alpha}({\mathbf{p}}),\,(\alpha=1,2,3),\,\lambda\in\mathbb{R}.$
(96)

A straightforward verifications shows that $e^{(\lambda)}_{\alpha}$ still
satisfies Eq. (91), but Eq. (92) is replaced by

$N^{3}e^{(\lambda)}_{\alpha}=\lambda\,e^{(\lambda)}_{\alpha}.$ (97)

######  Remark 8.

In fact one can show that every solution of the eigenvalue problem

$N^{3}f=\lambda f$ (98)

is of the form

$f({\mathbf{p}})=\sum_{\alpha=1}^{3}c_{\alpha}(p_{1},p_{2})e^{(\lambda)}_{\alpha}({\mathbf{p}}).$
(99)

###  3.3 The teleparallel connection

In the following we will use the notation, assumptions and results of Sec.
3.2.1.

####  3.3.1 Stereographic coordinates on the lightcone

We will use the moving frame $e_{\alpha}({\mathbf{p}}),\,(\alpha=1,2,3)$ on
$V_{0}^{+}$ to define teleparallel affine connection on $V_{0}^{+}.$ The frame
has a singularity at the points ${\mathbf{p}}=(0,0,t),\,t>0,$ and the natural
rotational covariance with respect to rotations around the third axis. It
takes especially simple form in a coordinate system $X=(x,y,\omega)$ using the
stereographic projection from the unit sphere in the momentum space.
161616Stereographic projection coordinates are also being used, in a similar
context, in Refs. [16],[17], but the coordinates $x,y$ in these papers are
twice ours $x,y$, as the projection plane in these papers is positioned at the
bottom of the unit sphere, and not through the center.

We set

$\displaystyle p^{1}$ $\displaystyle=$
$\displaystyle\frac{2x\omega}{x^{2}+y^{2}+1},$ (100) $\displaystyle p^{2}$
$\displaystyle=$ $\displaystyle\frac{2y\omega}{x^{2}+y^{2}+1},$ (101)
$\displaystyle p^{3}$ $\displaystyle=$
$\displaystyle\frac{x^{2}+y^{2}-1}{x^{2}+y^{2}+1}\,\omega,$ (102)

and the inverse transform

$\displaystyle x$ $\displaystyle=$ $\displaystyle\frac{\pi^{1}}{1-\pi^{3}},$
(103) $\displaystyle y$ $\displaystyle=$
$\displaystyle\frac{\pi^{2}}{1-\pi^{3}},$ (104) $\displaystyle\omega$
$\displaystyle=$ $\displaystyle|{\mathbf{p}}|.$ (105)

Given a coordinate system $X^{i}$ and a vector $\xi$ with coordinates
$\xi^{i},$ we have the standard transformation law to another coordinate
system $X^{i^{\prime}}$:

$\xi^{i^{\prime}}=\frac{\partial X^{i^{\prime}}}{\partial X^{i}}\xi^{i}.$
(106)

Applying this law to the vectors $e_{1},e_{2},e_{3}$ we obtain their
components in stereographic coordinates $\omega,x,y$;171717Surprisingly they
happen to essentially coincide with the basis
$\vec{E}_{1},\vec{E}_{2},\vec{E}_{3}$ considered in Ref. [19, Eq. (2.42)]:
$e_{1}=-\vec{E}_{1},\,e_{2}=-\vec{E}_{2},\,e_{3}=\vec{E}_{3}.$ In fact we have
$e_{1}=\widehat{\partial_{x}},\,e_{2}=-\widehat{\partial_{y}},\,e_{3}=\widehat{\partial_{\omega}}=\partial_{\omega},$
where the hat over a vector denotes the unit vector in its direction.

$e_{1}=\begin{pmatrix}\frac{1+x^{2}+y^{2}}{2\omega}\\\ 0\\\
0\end{pmatrix},\,e_{2}=\begin{pmatrix}0\\\ -\frac{1+x^{2}+y^{2}}{2\omega}\\\
0\end{pmatrix},\,e_{3}=\begin{pmatrix}0\\\ 0\\\ 1\end{pmatrix}.$ (107)

The Euclidean metric in stereographic coordinates has the form

$g=(g_{ij})=\begin{pmatrix}\frac{4\omega^{2}}{(1+x^{2}+y^{2})^{2}}&0&0\\\
0&\frac{4\omega^{2}}{(1+x^{2}+y^{2})^{2}}&0\\\ 0&0&1\end{pmatrix},$ (108)
$g^{-1}=(g^{ij})=\begin{pmatrix}\frac{(1+x^{2}+y^{2})^{2}}{4\omega^{2}}&0&0\\\
0&\frac{(1+x^{2}+y^{2})^{2}}{4\omega^{2}}&0\\\ 0&0&1\\\ \end{pmatrix},$ (109)

while the Lorentz invariant degenerate metric
${g_{0}}_{ij}=\delta_{ij}-\pi_{i}\pi_{j}$ is obtained from $g_{ij}$ by
replacing $1$ with $0$ in the right bottom corner. We can use the metric
$g_{ij}$ to obtain the corresponding dual basis $e^{1},e^{2},e^{3}$ in the
cotangent bundle:

$e^{1}=\begin{pmatrix}\frac{2\omega}{1+x^{2}+y^{2}}\\\ 0\\\
0\end{pmatrix},\,e^{2}=\begin{pmatrix}0\\\ -\frac{2\omega}{1+x^{2}+y^{2}}\\\
0\end{pmatrix},\,e^{3}=\begin{pmatrix}0\\\ 0\\\ 1\end{pmatrix}.$ (110)

The Lorentz invariant volume form $vol=d^{3}p/|{\mathbf{p}}|$ becomes

$vol=\frac{4\omega}{(1+x^{2}+y^{2})^{2}}\,d\omega\,dx\,dy.$ (111)

####  3.3.2 The connection coefficients

The moving frame $e_{\alpha}$ defines a unique flat affine connection on
$V_{0}$ in which the vector fields $e_{\alpha}$ are parallel:

$\nabla_{i}e_{\alpha}^{k}\doteq\partial_{i}e_{\alpha}^{k}+\Gamma_{ij}^{k}\,e_{\alpha}^{j}=0.$
(112)

The coefficients $\Gamma_{ij}^{k}$ of this teleparallel connection, in the
coordinate system $(X^{i})=(x,y,\omega)$ are then given by

${\Gamma_{i}}_{j}^{k}=e_{\alpha}^{k}\frac{\partial e_{j}^{\alpha}}{\partial
X^{i}},$ (113)

where $e^{\alpha}$ is the dual basis, thus
${e_{i}}^{\alpha}{e^{j}}_{\alpha}=\delta^{j}_{i}.$

A straightforward calculation gives then the following expressions:

$\Gamma_{1}=\frac{-2x}{a}\begin{pmatrix}1&0&0\\\ 0&1&0\\\
0&0&0\end{pmatrix},\,\Gamma_{2}=\frac{-2y}{a}\begin{pmatrix}1&0&0\\\ 0&1&0\\\
0&0&0\end{pmatrix},\Gamma_{3}=\frac{1}{\omega}\begin{pmatrix}1&0&0\\\ 0&1&0\\\
0&0&0\end{pmatrix},\,$ (114)

where $a=1+x^{2}+y^{2}.$ Our connection $\Gamma$ has the properties
$\nabla_{i}g=\nabla_{i}g_{0}=0,$ and $\nabla_{i}vol=0,$ therefore it should
have come under the scope of affine connections discussed by Staruszkiewicz in
Refs. [16, 17]. However, as we show in the paragraph below, it is not ‘semi-
symmetric’ – an extra condition imposed on the class of connections analyzed
by Staruszkiewicz.

##### The connection $\Gamma_{ij}^{k}$ given by Eq. (114) is not semi-
symmetric.

Let $M$ be an $n$-dimensional manifold with a coordinate system $x^{i}.$ A
linear connection $\nabla$ with connection coefficients $\Gamma_{ij}^{k}$ is
said to be semi-symmetric if its torsion tensor
$T_{ij}^{k}=\Gamma_{ij}^{k}-\Gamma_{ji}^{k}$ is of the form

$T_{ij}^{k}=\delta_{i}^{k}\tau_{j}-\delta_{j}^{k}\tau_{i},$ (115)

$\tau$ being a $1$-form. If that is the case, contracting the indices $i,k$ we
get $\tau_{j}=\frac{1}{n-1}T_{ij}^{i}.$ In our case $n=3$, therefore

$\tau_{j}=\frac{1}{2}T_{ij}^{i}.$ (116)

For the connection given by Eq. (114) we get, for instance,
$\tau_{2}=-y/(1+x^{2}+y^{2}).$ But for a semi-symmetric connection we should
have, for instance,
$T_{12}^{1}=\delta_{1}^{1}\tau_{2}-\delta_{2}^{1}\tau_{1}=\tau_{2},$ while
from Eq. (114) we have $T_{12}^{1}=\Gamma_{12}^{1}-\Gamma_{21}^{1}=0-0=0.$
Thus our connection is not semi-symmetric.

##  4 Photon position operator with commuting components and axial symmetry

To discuss the photon localization it is more convenient to work in
representation in which the scalar product in the Hilbert space of sections of
the tangent bundle $TV_{0}^{+}$ is given by the formula (c.f. Remark 3.)

$(f,f^{\prime})=\int_{\mathbb{R}^{3}}f({\mathbf{p}})^{\dagger}f^{\prime}({\mathbf{p}})d^{3}p.$
(117)

Since the measure $d^{3}p$ is not Lorentz-invariant the formula for the boost
operator gets now an extra term (comparing to Eq. (57), and takes the form

${\bf N}={\bf K}+\frac{i}{2}\boldsymbol{\pi}+{\bf n}.$ (118)

As a consequence the sections ${\mathbf{p}}\mapsto e_{\alpha}({\mathbf{p}})$
do not any longer satisfy Eq. (92) - but
${\mathbf{p}}\mapsto|{\mathbf{p}}|^{-1/2}e_{\alpha}({\mathbf{p}})$ do.

In what follows we will be using the scalar product (117). We will denote
${\mathcal{H}}$ the corresponding Hilbert space:
${\mathcal{H}}=L^{2}(\mathbb{R}^{3},d^{3}p)\otimes\mathbb{C}^{3}.$

The map $f\mapsto|{\mathbf{p}}|^{1/2}f$ is an isometry between the two Hilbert
spaces, the space ${\mathcal{H}}$ with the scalar product defined with the
measure $d^{3}p,$ and ${\mathcal{H}}_{0},$ with the scalar product
$d^{3}p/p_{0}.$

###  4.1 Position operators as covariant derivatives

There is a straightforward relation between the position operator and a
covariant derivative concept in vector bundles. Indeed, the main property
required from any position operator $Q_{i}$ is the property of satisfying the
canonical commutation relations with the momentum operators $P_{i}$:

$[Q_{j},P^{k}]=i\delta_{j}^{k}\,I.$ (119)

In our case, it implies that for any section ${\mathbf{f}}({\mathbf{p}})$ of
$T(V_{0}^{+})$ and any scalar function $\phi({\mathbf{p}})$ we have

$-i\,Q_{j}(\phi{\mathbf{f}})({\mathbf{p}})=\frac{\partial\phi({\mathbf{p}})}{\partial
p_{j}}\,{\mathbf{f}}({\mathbf{p}})+\phi({\mathbf{p}})(Q_{j}{\mathbf{f}})({\mathbf{p}}).$
(120)

Thus the operators $-iQ_{i}$ have the Leibniz rule property, the main property
defining a covariant derivative $\nabla_{i}$ in a vector bundle (cf. e.g. [28,
p. 89, Eq. (1.1)]). For $Q_{i}$ to be Hermitian, $\nabla_{i}$ must be anti-
Hermitian, and for this to be the case, the (linear) connection determined by
$\nabla_{i}$ should preserve the fiber scalar product
$({\mathbf{f}}({\mathbf{p}}),{\mathbf{f}}^{\prime}({\mathbf{p}}))_{\mathbf{p}}={\mathbf{f}}({\mathbf{p}})^{\dagger}{\mathbf{f}}^{\prime}({\mathbf{p}}).$
Then

${\bf Q}=i\boldsymbol{\nabla},$ (121)

or, explicitly

$(Q_{j}{\mathbf{f}})^{k}({\mathbf{p}})=i\partial_{j}f^{k}({\mathbf{p}})+i\Gamma_{il}^{k}({\mathbf{p}})f^{l}({\mathbf{p}}).$
(122)

###  4.2 The Pryce connection and operator - geometric construction

There is a standard construction in the differential geometry of vector
bundles that results in a canonical connection adapted to a split of a trivial
vector bundle into a direct sum of its two vector sub-bundles (cf. e.g. [29,
p. 319, 4.], [30, Exercise 10]). We adapt this standard construction to our
purpose as follows. First, using the global coordinates $p^{i}$, we realize
the tangent bundle $TV_{0}^{+}$ as a trivial product bundle
$V_{0}^{+}\times\mathbb{R}^{3}.$ In the trivial bundle we have a canonical
covariant derivative $\partial_{i}=\frac{\partial}{\partial p^{i}}.$ Our
trivial bundle is naturally split into a direct sum of the helicity zero sub-
bundle and the helicity $\pm 1$ sub-bundle of photon states. Let
$P({\mathbf{p}})$ denote the orthogonal projection on the helicity zero states

$(P({\mathbf{p}})f)^{i}({\mathbf{p}})=\pi^{i}\pi_{j}f^{j}({\mathbf{p}}).$
(123)

Then $I-P$ is the orthogonal projection on the complementary sub-bundle of
photon states. The natural covariant derivative adapted to this splitting is
then defined by the formula181818Cf. [31, Eq. 2]. We use the label
${}^{\mbox{PR}}$ to mean either “Projection” or “Pryce”.

$\nabla_{i}^{\mbox{PR}}=Pd_{i}P+(I-P)\partial_{i}(I-P).$ (124)

Using the idempotent property $P^{2}=P,$ we find a simpler form:

$\nabla_{i}^{\mbox{PR}}=\partial_{i}+[P,\partial_{i}(P)],$ (125)

while substituting the explicit form (123) of $P$ leads to

$(\nabla_{i}^{\mbox{PR}}{\mathbf{f}})^{j}=\partial_{i}f^{j}+\frac{1}{|{\mathbf{p}}|}\left(\pi^{j}\delta_{ik}-\pi_{k}\delta_{i}^{j}\right)f^{k},$
(126)

thus the corresponding connection coefficients are given by

${\Gamma^{\mbox{PR}}}_{ik}^{j}={{[P,\partial_{i}(P)]}^{j}}_{k}=\frac{1}{|{\mathbf{p}}|}\left(\pi^{j}\delta_{ik}-\pi_{k}\delta_{i}^{j}\right).$
(127)

####  4.2.1 The Pryce connection is metric semi-symmetric

A metric connection  is semi-symmetric - cf. e.g. [32, 33] and references
therein - if its connection coefficients $\Gamma_{ij}^{k}$ are of the form

$\Gamma_{ik}^{j}={\overline{\Gamma}}_{ik}^{j}+\delta_{i}^{j}\tau_{k}-g_{ik}\tau^{j},$
(128)

where ${\overline{\Gamma}}_{ik}^{j}$ are the coefficients os the Levi-Civita
connection of the metric, and where $\tau_{i}$ and $\tau^{i}=g^{ij}\tau_{j}$
are covariant and contravariant components of a vector field, respectively. In
that case the curvatures of the two connections are related by191919For the
curvature tensor components of a connection $\nabla$ we use the convention
${{R_{ij}}^{k}}_{l}\xi^{l}=\left((\nabla_{i}\nabla_{j}-\nabla_{j}\nabla_{i})\xi\right)^{k}.$

${{R_{ij}}^{k}}_{l}={{\overline{R}_{ij}}^{k}}_{l}+\delta_{j}^{k}\tau_{il}-\delta_{i}^{k}\tau_{jl}+\delta_{il}{\tau_{j}}^{k}-\delta_{jl}{\tau_{i}}^{k},$
(129)

where

$\tau_{ij}=\nabla_{i}\tau_{j}-\tau_{i}\tau_{j}+\frac{1}{2}g_{ij}\tau^{2},$
(130)

and $\tau^{2}=g^{ij}\tau_{i}\tau_{j}.$

In our case $g_{ij}=\delta_{ij}$ and the connection coefficients
${\overline{\Gamma}}_{ik}^{j}$ are all zero. Comparing Eqs. (128) and (127) we
can see that the connection $\nabla^{\mbox{PR}}$ is metric semi-symmetric,
with

$\tau_{i}=-\pi_{i}/|{\mathbf{p}}|,.$ (131)

We then get

$\pi_{ij}=\pi_{ji}=\frac{1}{|{\mathbf{p}}|^{2}}\left((\Sigma^{2})_{ij}+\frac{1}{2}\delta_{ij}\right),$
(132)

where

$\Sigma_{ij}=-\epsilon_{ijk}\pi^{k},$ (133)

and the curvature tensor simplifies to

$R^{\mbox{PR}}_{ijkl}=\frac{1}{|{\mathbf{p}}|^{2}}\Sigma_{ij}\Sigma_{kl}.$
(134)

From Eq. (128) we get for the torsion

${T^{\mbox{PR}}}_{ij}^{k}=\delta_{i}^{k}\tau_{j}-\delta_{j}^{k}\tau_{i},$
(135)

where $\tau_{i}$ is given by (131).

####  4.2.2 The difference between the teleparallel and the Pryce connection

The difference of two connections is a tensor. In our case it is a matter of a
straightforward calculations to find this tensor for the teleparallel
connection $\Gamma$ defined by Eq. (114) and the Pryce connection given by Eq.
(126). We obtain

$\Gamma_{ij}^{k}-{\Gamma^{\mbox{PR}}}_{ij}^{k}=b_{i}{\Sigma^{k}}_{j},$ (136)

where

${\bf
b}=\frac{\boldsymbol{\pi}\times{\bf{\mathbf{w}}}}{|{\mathbf{p}}|-{\mathbf{p}}\cdot{\bf{\mathbf{w}}}},\quad{\mathbf{w}}=(0,0,1).$
(137)

####  4.2.3 The Pryce operator

We now use Eqs. (121) and (124). Denoting

${\bf X}=i\frac{\partial}{\partial{\mathbf{p}}},$ (138)

the position operator corresponding to $\nabla^{\mbox{PR}}$ is

${\bf Q}^{\mbox{PR}}\doteq i\boldsymbol{\nabla}^{\mbox{PR}}=P{\bf
X}P+(I-P){\bf X}(I-P),$ (139)

or, explicitly,

$({\bf
Q}^{\mbox{PR}}_{j}{\mathbf{f}})^{k}=i\partial_{j}f^{k}+\frac{i}{|{\mathbf{p}}|}\left(\pi^{k}\delta_{jl}-\pi_{l}\delta_{j}^{k}\right)f^{l}.$
(140)

Using Eqs. (59),(53) it can be also written as

${\mathbf{Q}}^{\mbox{PR}}={\mathbf{X}}+\frac{({\mathbf{p}}\times{\bf
s})}{|{\mathbf{p}}|^{2}},$ (141)

and, in this form, it is known as the Pryce position operator. The following
alternative formula for this operator, expressing it in terms of the Poincaré
group generators, is easily verified and is well known

${\bf Q}^{\mbox{PR}}=\frac{1}{2}\left(\frac{1}{P^{0}}{\bf N}+{\bf
N}\frac{1}{P^{0}}\right),$ (142)

where $N^{i}$ are the boost generators (118). It should be mentioned that even
though the bundle $TV_{0}^{+}$ is trivial, the non-triviality of its splitting
defined by the helicity-related projection $P$ is reflected in the fact that
the adapted connection has a non-zero curvature, and, as a consequence, the
components of the Pryce operator do not commute.

######  Remark 9.

The flat covariant derivative $d$ does not mix the two photon $\pm 1$
helicities. As a consequence the Pryce operator commutes with the helicity
operator $\Lambda,$ and not only with its square $\Lambda^{2}=I-P.$

Using now Eq. (134) we get

$([Q_{i},Q_{j}]{\mathbf{f}})^{k}=-\frac{1}{|{\mathbf{p}}|^{2}}\Sigma_{ij}{\Sigma^{k}}_{l}f^{l},$
(143)

which is usually written as (see e.g. [34, Eq. (5,9), p. 40])202020In Ref.
[34] helicity is defined with the opposite sign.

$[Q_{i},Q_{j}]=i\epsilon_{ijk}\frac{p_{k}}{|{\mathbf{p}}|^{3}}\Lambda,$ (144)

where $\Lambda=i\Sigma$ is the helicity operator.

###  4.3 The Hawton-Baylis operator

The teleparallel connection described in Sec. 3.3 leads to the Hawton-Baylis
photon position operator $Q^{\mbox{HB}}_{i}$ with commuting components

$Q^{\mbox{HB}}_{i}=i\nabla_{i},$ (145)

with the defining relations

$\nabla_{i}e_{\alpha}=0,\,(i=1,2,3),\,(\alpha=1,2,3),$ (146)

where $e_{\alpha}$ are given by Eqs. (89)-(90)

From Eqs. (136),(137) we then get212121C.f. [18, Eq. 53] and [19, Eq. 2.49]).

$Q^{\mbox{HB}}_{i}{\mathbf{f}}=Q^{\mbox{PR}}_{i}{\mathbf{f}}+b_{i}\Lambda{\mathbf{f}}.$
(147)

In particular

$Q^{\mbox{HB}}_{3}=Q^{\mbox{PR}}_{3}.$ (148)

Using this last equality, together with Eqs. (142) and (118) we recover the
fact, mentioned after Eq. (118), that the states
$\tilde{e}_{\alpha}=|{\mathbf{p}}|^{-1/2}e_{\alpha}$ are invariant under
boosts in the direction of the third axis. Indeed, from the Leibniz property
(120) and from (146) we get

$Q^{\mbox{HB}}_{i}\tilde{e}_{\alpha}=-\frac{i\pi_{i}}{2|{\mathbf{p}}|}\tilde{e}_{\alpha}.$
(149)

Thus also

$Q^{\mbox{PR}}_{3}\tilde{e}_{\alpha}=-\frac{i\pi_{3}}{2|{\mathbf{p}}|}\tilde{e}_{\alpha}.$
(150)

On the other hand, using (142) and the Lie algebra commutation relation
$[N^{3},P^{0}]=iP^{3}$, $Q^{\mbox{PR}}_{3}$ can be written as

$Q^{\mbox{PR}}_{3}=\frac{1}{|{\mathbf{p}}|}N^{3}-\frac{i\pi_{3}}{2|{\mathbf{p}}|},$
(151)

which, together with (150), leads to
$(1/|{\mathbf{p}}|)N^{3}\tilde{e}_{\alpha}=0,$ and thus
$N^{3}\tilde{e}^{\alpha}=0.$ But this reasoning does not give us the clue as
how the basis $e_{\alpha}$ can be obtained by taking the speed of light limit
of the simple polarization basis (81).

It is straightforward to verify that the operators $Q_{i}=Q^{\mbox{HB}}_{i}$
have the axial symmetry:

$[M^{3},Q_{1}]=iQ_{2},\,[M^{3},Q_{2}]=-iQ_{1},\,[M^{3},Q_{3}]=0.$ (152)

####  4.3.1 Photon states localized on circles

Since the three components of $Q^{\mbox{HB}}$ commute, they can be
simultaneously diagonalized, and the simultaneous eigenvalue equation
$Q^{\mbox{HB}}_{i}{\mathbf{f}}=q_{i}{\mathbf{f}}$ has three independent
solutions

${\mathbf{f}}_{\alpha,{\bf q}}({\mathbf{p}})=\exp(-i{\bf
q}\cdot{\mathbf{p}})e_{\alpha}({\mathbf{p}}),\,({\mathbf{q}}\in\mathbb{R}^{3},\alpha=1,2,3).$
(153)

The states ${\mathbf{f}}_{1,{\bf q}}\pm i{\mathbf{f}}_{2,{\bf q}}$ describe
photons localized at ${\bf q}\in\mathbb{R}^{3},$ of helicity $\pm 1,$ and
${\mathbf{f}}_{3,{\bf q}}$ is of helicity $0.$ However, since the states
$e_{\alpha}({\mathbf{p}})$ have a rather complicated
${\mathbf{p}}$-dependence, and owing to the axial symmetry of $Q^{\mbox{HB}},$
it is natural to look for simple axially symmetric simultaneous eigenvalue
equations for $Q=Q^{\mbox{HB}}$:

$\displaystyle M^{3}\,{\mathbf{f}}$ $\displaystyle=$ $\displaystyle 0,$ (154)
$\displaystyle Q_{3}\,{\mathbf{f}}$ $\displaystyle=$ $\displaystyle 0,$ (155)
$\displaystyle(Q_{1}^{2}+Q_{2}^{2})\,{\mathbf{f}}$ $\displaystyle=$
$\displaystyle R^{2}\,{\mathbf{f}},\,R>0.$ (156)

These would give us states localized on the circle $x^{2}+y^{2}=R^{2},\,z=0,$
in the Cartesian coordinates $(x,y,z)$ in the photon’s position space. To this
end it is convenient to introduce cylindrical coordinates $(\rho,\phi,p_{3}),$
$\rho=\sqrt{(}p_{1}^{2}+p_{2}^{2}).$ One can then verify that states of the
form

${\mathbf{f}}(\rho,\phi,p_{z})=F(\rho)\begin{pmatrix}-\sin(\phi)\\\
\cos(\phi)\\\ 0\end{pmatrix}$ (157)

are simple solutions of Eqs. (154) and (155), while Eq. (156) imposes a second
order ordinary differential equation on $F(\rho)$:

$\rho^{2}F^{\prime\prime}(\rho)+\rho
F^{\prime}(\rho)+(R^{2}\rho^{2}-1)F(\rho)=0.$ (158)

with the following solution finite at the origin [35]:

$F(\rho)=cJ_{1}(R\rho),$ (159)

where $J_{1}$ is the Bessel function of the first kind, and $c$ is a constant
- see Fig. 1.

Figure 1: Bessel function $J_{1}(\rho).$

For reasons that will be clear from the next paragraph we choose for the
constant $c$ the value:

$c=-iR.$ (160)

##### Loop states

Let $\ell$ be a closed loop in $\mathbb{R}_{\mathbf{x}}^{3}$ defined by a
function ${\mathbf{r}}(t),\quad 0\leq t\leq 2\pi.$ Following Ref. [31] let us
define the state ${\mathbf{f}}_{\ell}$ by

${\mathbf{f}}_{\ell}({\mathbf{p}})=\frac{1}{2\pi}\int_{0}^{2\pi}e^{-i{\mathbf{p}}\cdot{\mathbf{r}}(t)}\,d{\mathbf{r}}(t).$
(161)

Then ${\mathbf{f}}_{l}({\mathbf{p}})$ is a superposition of simultaneous
eigenstates of $X_{i}$ \- see Eq. (138) - ${\bf X}$-localized in
$\mathbb{R}_{\mathbf{x}}^{3}$ at the points of the loop $\ell$. Therefore it
is ${\bf X}$-localized on $\ell.$ The fact that the loop is closed, i.e.
${\mathbf{r}}(2\pi)={\mathbf{r}}(0),$ implies that the state
${\mathbf{f}}_{\ell}({\mathbf{p}})$ is an (improper) element of
${\mathcal{H}}_{ph},$ i.e that

${\mathbf{p}}\cdot{\mathbf{f}}_{\ell}({\mathbf{p}})=0.$ (162)

Taking the Fourier transform ${\mathbf{f}}_{\ell}({\mathbf{x}})$ of
${\mathbf{f}}_{\ell}({\mathbf{p}})$ we get

${\mathbf{f}}_{\ell}({\mathbf{x}})=\frac{1}{(2\pi)^{3}}\int_{\mathbb{R}}^{3}e^{i{\mathbf{p}}{\mathbf{x}}}{\mathbf{f}}_{\ell}({\mathbf{p}})d^{3}p=\frac{1}{2\pi}\int_{0}^{2\pi}\delta\left({\mathbf{x}}-{\mathbf{r}}(t)\right)d{\mathbf{r}}(t),$
(163)

and it is clear ${\mathbf{f}}_{\ell}({\mathbf{x}})$ has its support on the
loop in the position coordinates space - it vanishes at all points
${\mathbf{x}}$ outside the loop. It is also clear that for two non-
intersecting loops $\ell,\ell^{\prime}$ the states ${\mathbf{f}}_{\ell}$ and
${\mathbf{f}}_{\ell^{\prime}}$ are orthogonal to each other.222222Loops may
form topologically inequivalent knots. In this respect the loop states
discussed above are similar to knotted solutions of Maxwell equations
discussed in Ref. [26, Sec. 7].

As an example let us take for $\ell$ the circle in $(x,y)$ plane of radius $R$
given by the parametric equations

$x=R\cos t,\,y=R\sin t,\,z=0.$ (164)

A straightforward calculation in cylindrical coordinates $(\rho,\phi,p_{4})$
(in the momentum space) leads then to

${\mathbf{f}}_{x^{2}+y^{2}=R^{2}}({\mathbf{p}})=-i\frac{R}{\rho}\,J_{1}(R\rho)\begin{pmatrix}-p_{2}\\\
p_{1}\\\ 0\end{pmatrix}.$ (165)

where $J_{1}$ is the Bessel function and
$\rho=\sqrt{(p^{1})^{2}+(p^{2})^{2}},$ which coincides with the state
${\mathbf{f}}$ given by the formula (157).

##### Amrein’s washer photon states

We use Eq. (163) for a circle of radius $R$ at $z=z_{0}$, thus

$\begin{split}x(t)&=R\cos t,\\\ y(t)&=R\sin t,\\\ z&=z_{0}.\end{split}$ (166)

Thus

${\mathbf{f}}_{R,z_{0}}=\frac{1}{2\pi}\int_{0}^{2\pi}\delta(x-R\cos t,y-R\sin
t,z-z_{0})\begin{pmatrix}-\sin t\\\ \cos t\\\ 0\end{pmatrix}dt.$ (167)

We will use cylindrical coordinates $(r,\phi,z)$ in the position space. In
these coordinates

$\delta(x-x^{\prime},y-y^{\prime},z-z^{\prime})=\frac{1}{r}\delta(r-r^{\prime})\delta(\phi-\phi^{\prime})\delta(z-z^{\prime}),$
(168)

while

$d^{3}x=rdr\,d\phi\,dz.$ (169)

Thus

${\mathbf{f}}_{R,z_{0}}(r,\phi,z)=\frac{1}{2\pi}\int_{0}^{2\pi}\frac{\delta(r-R)}{r}\delta(z-z_{0})\delta(\phi-t)\begin{pmatrix}-\sin
t\\\ \cos t\\\ 0\end{pmatrix}dt.$ (170)

After integrating over $dt$ we get

${\mathbf{f}}_{R,z_{0}}(r,\phi,z)=\frac{1}{2\pi}\frac{\delta(r-R)}{r}\delta(z-z_{0})\begin{pmatrix}-\sin\phi\\\
\cos\phi\\\ 0\end{pmatrix}.$ (171)

We now take a continuous superposition of these states for $R$ varying between
$R_{1}$ and $R_{2}>R_{1}$ end $z_{0}$ varying between $z_{1}$ and
$z_{2}>z_{1}$. As a superposition of photon states, it will be still a photon
state. Let $\chi_{R_{1},R_{2}}(r)$ be the function equal $1$ for $R_{1}\leq
r\leq R_{2}$ and zero otherwise, and, similarly, let $\chi_{z_{1},z_{2}}$ be
the function equal $1$ for $z_{1}\leq z\leq z_{2}$ and zero otherwise. Taking
the integral

$\int_{-\infty}^{\infty}dz\int_{0}^{\infty}\chi_{R_{1},R_{2}}(R)\chi_{z_{1},z_{2}}(z){\mathbf{f}}_{R,z_{0}}(r,\phi,z)\,dR\,dz$
(172)

we obtain

${\mathbf{f}}_{R_{1},R_{2},z_{1},z_{2}}(r,\phi,z)=\frac{1}{2\pi
r}\chi_{R_{1},R_{2}}(r)\chi_{z_{1},z_{2}}(z))\begin{pmatrix}-\sin\phi\\\
\cos\phi\\\ 0\end{pmatrix}.$ (173)

The function is evidently square integrable with respect to
$d^{3}x=rdr\,d\phi\,dz$ and its probability density is zero everywhere except
for the bolt washer-like region $R_{1}\leq r\leq R_{2}$, $z_{1}\leq z\leq
z_{2}$ \- see Fig. 2.

Figure 2: Amrein’s washer state (171) is strictly (weakly) localized in the
region $1\,\mbox{nm}\leq R\leq\sqrt{2}\,\mbox{nm},\,1\,\mbox{nm}\leq z\leq
1.4\,\mbox{nm}.$

We call it Amrein’s state, as the existence of such states was first proved in
1968 by A.O. Amrein [6]. These states are also strictly localized with respect
to the commuting position operators $Q_{i}^{\mbox{HB}}.$

###  4.4 POV measure photon’s localization

Every self-adjoint operator admits a spectral decomposition. Usually we write
it as:

$A=\int\lambda dE(\lambda).$ (174)

More generally, given a family of commuting observables, we have a spectral
measure on the common spectrum of these observables. Here we have operators
$X_{i}=i\partial/\partial p^{i}$ defined on ${\mathcal{H}},$ with commuting
components $[X_{i},X_{j}]=0,$ and we have a unique spectral measure on
$\mathbb{R}^{3}$ such that

$X_{i}=\int_{\mathbb{R}^{3}}x_{i}dE({\mathbf{x}}).$ (175)

Then for every Borel set $\Delta\subset\mathbb{R}^{3}$ the operator

$E(\Delta)=\int_{\Delta}dE({\mathbf{x}})$ (176)

is a projection operator on the subspace of states localized in $\Delta,$ for
any two sets $\Delta$ i $\Delta^{\prime}$ the operators $E(\Delta)$ i
$E(\Delta^{\prime}$ commute.

However the operators $X_{i}$ do not leave the subspace ${\mathcal{H}}_{ph}$
invariant, therefore we have introduced $Q_{i}^{\mbox{PR}}$

$Q_{i}^{\mbox{PR}}=PX_{i}P+(I-P)X_{i}(I-P),$ (177)

where $(I-P)$ is the orthogonal projection operator on ${\mathcal{H}}_{ph}.$
Thus we have

$Q_{i}^{\mbox{PR}}=\int_{\mathbb{R}^{3}}x_{i}dF({\mathbf{x}}),$ (178)

where

$F(\Delta)=PE(\Delta)P+(I-P)E(\Delta)(I-P).$ (179)

Now $F(\Delta)$ are not any longer projection operators, and for different
$\Delta$ they do not commute. Nevertheless they are non-negative operators
$0\leq F(\Delta)\leq 1$ and

$\int_{\mathbb{R}}^{3}dF({\mathbf{x}})=I,$ (180)

Therefore we have a POV - positive operator valued measure. Jauch and Piron
[5] called a photon state ${\mathbf{f}}$ weakly localized in $\Delta$ if
$F(\Delta){\mathbf{f}}={\mathbf{f}},$ and conjectured existence of such
states. Amrein [6] provided a rigorous proof of their existence and has shown
how to construct them. Our formula (171) provides a rich explicit family of
such states. Notice however that the circle states
${\mathbf{f}}_{x^{2}+y^{2}=R^{2}}({\mathbf{p}})$ while satisfying

$(Q^{\mbox{HB}}_{1})^{2}+(Q^{\mbox{HB}}_{2})^{2})\,{\mathbf{f}}_{x^{2}+y^{2}=R^{2}}=R^{2}{\mathbf{f}}_{x^{2}+y^{2}=R^{2}},$
(181)

they do not (c.f. [36]) satisfy the equation

$(Q^{\mbox{PR}}_{1})^{2}+(Q^{\mbox{PR}}_{2})^{2})\,{\mathbf{f}}_{x^{2}+y^{2}=R^{2}}=R^{2}{\mathbf{f}}_{x^{2}+y^{2}=R^{2}}.$
(182)

Instead, these states are eigenstates to the eigenvalue $R^{2}$ of the
operator $Q_{x^{2}+y^{2}}\neq(Q^{\mbox{HB}}_{1})^{2}+(Q^{\mbox{HB}}_{2})^{2})$
defined as

$\begin{split}Q_{x^{2}+y^{2}}&=P((X_{1})^{2}+(X_{2})^{2})P+(I-P)((X_{1})^{2}+(X_{2})^{2})(I-P)\\\
&=\int(x^{2}+y^{2})\,dF({\mathbf{x}}),\end{split}$ (183)

where $dF({\mathbf{x}})$ is the POV measure defined by Eq. (179).

The circle localized states ${\mathbf{f}}_{x^{2}+y^{2}=R^{2}}$ are
superpositions of helicity $+1$ and helicity $-1$ states, in agreement with
Theorem 2 of Ref. [6]. Their projections on definite helicity subspaces, $+1$
or $-1$, are still circle-localized with respect to the Hawton-Baylis
operator, but they are not weakly localized in the sense of the POV measure
$F(\Delta)$ \- in agreement with Theorem 1 of Ref. [6]

##  5 Conclusions

In conclusion, using differential geometric structures on the mass hyperboloid
and on the light cone in momentum space we derived the explicit form of the
unitary representation of the Poincaré group for helicity zero and helicity
$\pm 1$ massless particles. Using this explicit form for the boost in the
direction of the third axis we have found that simple photon polarization
states based on Hertz-type potentials survive the light speed limit and
generate a polarization basis used in the construction of photon position
operators with commuting components. We have compared these operators, as well
as the underlying affine connections, to the classical Pryce operator and
connection, and have found that the Pryce connection, Eq. (125), non-flat, but
with rotational symmetry, is metric semi-symmetric, while the Hawton-Baylis
flat, but only axially symmetric, connection, Eq. (114), does not have this
property. We have constructed finite-norm photon states localized in bolt
washer-like regions and proved that they are strictly localized in these
regions with respect to Hawton-Baylis position operators and also with respect
to the Jauch–Piron–Amrein POV measures. They are also $z$-localized (but not
radius-localized) with respect to the Pryce photon position operator.

## Acknowledgements

Thanks are due to M. Schlichtinger and J. Szulga for the discussion of the
preliminary version of the part of Sec. 3.1, as well as to G. Koczan, P.
Lescaudron and R. Coquereaux for their interest and for a stimulating
discussion on related subjects. I wish to thank my wife, Laura, for reading
the manuscript and for her constant support.
This work was supported by Quantum Future Group Inc.

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