  *

# Isomorphism between the Białynicki-Birula and the Landau-Peierls Fock space
quantization of the electromagnetic field in position representation

M. Federico, H. R. Jauslin  Laboratoire Interdisciplinaire Carnot de
Bourgogne, UMR 6303 CNRS
\- Université de Bourgogne Franche-Comté, BP47870, 21078 Dijon, France
<EMAIL_ADDRESS>
###### Abstract

We first present a summary of the quantization of the electromagnetic field in
position space representation, using two main approaches: the Landau-Peierls
approach in the Coulomb gauge and the Białynicki-Birula approach, based on the
Riemann-Silberstein vector. We describe both in a framework that starts with a
classical Hamiltonian structure and builds the quantum model in a bosonic Fock
space by a precisely defined principle of correspondence. We show that the two
approches are completly equivalent. This is formulated by showing that there
is a unitary map between the Fock spaces that makes them isomorphic. Since all
the physically measurable quantities can be expressed in terms of scalar
products, this implies that the two quantizations lead to exactly the same
physical properties. We show furthemore that the isomorphism is preserved in
the time evolutions. To show the equivalence, we use the concepts of helicity
and frequency operators. The combination of these two operators provides a
formulation that allows one to make the link between these two methods of
quantization in a precise way. We also show that the construction in the
Białynicki-Birula quantization that avoids the presence of negative
eigenvalues in the Hamiltonian, in analogy with the one for the Dirac equation
for electrons and positrons, can be performed through an alternative choice of
the canonical variables for Maxwell’s equations.

Keywords: Fock space quantization, electromagnetic field, photon states,
photon wave function, negative energy states, helicity operator, frequency
operator.

##  1 Introduction

In the past few years, many headways have been made in single-photon
production, such as the designing of the pulse shapes [1, 2, 3], which
provides a powerful tool for the development of new quantum technologies. This
feature has brought back the interest of the description of photons in the
position space representation [4, 5, 6, 7, 8] and of photon localization [9,
10, 11, 12].

The concept of photon emerges from the quantization of the classical
electromagnetic field theory which can be carried through a systematic scheme
consisting in the following steps: (a) Formulate Maxwell’s equations in a
Hamiltonian form, identifying a pair of canonical variables, defining a phase
space such that the corresponding Hamilton equations are equivalent to
Maxwell’s equations. (b) Construct a complex Hilbert space $\mathcal{H}$ of
classical configurations and a complex representation of the Hamiltonian
structure. (c) Build a bosonic Fock space [13, 14] from the classical Hilbert
space as

$\displaystyle\mathbb{F}^{\mathfrak{B}}(\mathcal{H}):=\bigoplus_{n=0}^{\infty}\mathcal{H}^{\otimes_{S}n},$
(1)

where $\mathcal{H}^{\otimes_{S}n}$ is $n$-times the symmetrized tensor product
of $\mathcal{H}$ with itself. (d) Construct quantum observables from their
classical equivalent using a correspondence principle. (e) Define the
generator of the quantum dynamics in Fock space from the single-photon
dynamics, which is determined by the classical Hamiltonian dynamics. The
choice of the classical Hamiltonian structure to build the quantized theory is
thus central and allows to describe photons in different ways. The momentum
description is often used since it diagonalizes the Hamiltonian of the system
[15, 16, 17, 18, 19]; however, it does not provide a direct description of the
photon states in position space, in particular for photon pulses.

In this article, we will discuss two position space versions of the
construction explained above and the relation between them. We will start with
the standard quantization in the Coulomb gauge, using the vector potential and
the electric field as canonical variables and the so called Landau-Peierls
(LP) field as complex representation [20, 21]. Then we will introduce the
quantization formulated by Białynicki-Birula (BB) [22, 23, 24] using the
electric and magnetic fields as canonical variables and based on the Riemann-
Silberstein complex vector [25]. The main goal is to show that the two
approaches are equivalent in the sense of Dirac’s theory of transformations
[26], by showing that there is an isomorphism between their respective Hilbert
spaces of states and between the observables. This implies that all the
physical predictions are the same. We will construct the quantum theory in the
Schrödinger picture, but once the model is set up, one can construct the
corresponding Heisenberg and interaction pictures. In the present formulation,
we do not invoke the concept of photon wave function but only the concepts of
states and observables, which are the two essential concepts needed in the
quantum field theory of the electromagnetic field. Although the different
possible representations are equivalent as we will show, they each have some
advantages depending on the aspect of the theory one wants to look at. The BB
quantization for instance does not require to choose a particular gauge which
makes it a theoretically powerfull construction and it is also Lorentz
invarient as it was shown in [22, 7, 27, 8]. In particular, the BB
representation is well adapted to study the electromagnetic energy in position
space at the price of using some less usual tools in its contstruction such as
a weighted scalar product.

All along the article, we will use the concept of helicity which is central in
the construction of the BB classical Hamiltonian formalism and to make the
link with the LP field. One can introduce this concept through the helicity
operator $\Lambda$ defined by

$\displaystyle\Lambda$ $\displaystyle:=(-\Delta)^{-1/2}\nabla\times$ (2a)
$\displaystyle=(\nabla\times)(-\Delta)^{-1/2}$ (2b)
$\displaystyle=c\Omega^{-1}\nabla\times$ (2c) $\displaystyle={\rm
sign}(\nabla\times),$ (2d)

where $\Delta=\nabla^{2}$ is the Laplace operator. We have used the notation

$\displaystyle\Omega:=c(-\Delta)^{1/2},$ (2c)

which denotes the frequency operator defined as the unique positive operator
satisfying $\Omega^{2}=-c^{2}\Delta$ [28, 29, 30], we remark that
$\left[\Lambda,\Omega\right]=0$. Since the curl operator can be written as

$\displaystyle\nabla\times=\left[\begin{array}[]{ccc}0&-\partial_{3}&\partial_{2}\\\
\partial_{3}&0&-\partial_{1}\\\
-\partial_{2}&\partial_{1}&0\end{array}\right]=-i(\vec{S}\cdot\nabla),$ (2g)

with $\vec{S}=(S^{1},S^{2},S^{3})$, $S^{j}_{kl}=-i\epsilon_{jkl}$, and
$\epsilon_{jkl}$ being the totally antisymmetric Levi-Cività tensor, i.e.

$\displaystyle S^{1}=\left[\begin{array}[]{ccc}0&0&0\\\ 0&0&-i\\\
0&i&0\end{array}\right],\qquad S^{2}=\left[\begin{array}[]{ccc}0&0&i\\\
0&0&0\\\ -i&0&0\end{array}\right],\qquad
S^{3}=\left[\begin{array}[]{ccc}0&-i&0\\\ i&0&0\\\ 0&0&0\end{array}\right].$
(2q)

We remark that the matrices $S^{j}$ are hermitian and they satisfy the spin-1
commutation relations $[S^{1},S^{2}]=iS^{3}$ (completed by cyclic
permutations). Defining the operator $\vec{\hat{p}}:=-i\hbar\nabla$, in
analogy with the momentum operator of quantum mechanics, one can write also
$\nabla\times=(\vec{S}\cdot\vec{\hat{p}})/\hbar$,
$|\vec{\hat{p}}|=(\vec{\hat{p}}\cdot\vec{\hat{p}})^{1/2}=\hbar(-\Delta)^{1/2}$,
and the helicity operator

$\displaystyle\Lambda=\vec{S}\cdot\frac{\vec{\hat{p}}}{|\vec{\hat{p}}|}.$ (2r)

This formula leads to the formal interpretation of the helicity operator as
the projection of the spin $\vec{S}$ on the “direction of the motion” defined
by $\vec{\hat{p}}/|\vec{\hat{p}}|$. This interpretation is based on the
analogy of these expressions with the ones of a quantum particle of spin 1. We
emphasize however that at this stage we are dealing only with some operators,
$\nabla\times$ and $\Omega$, that appear in complex representations of the
classical Maxwell equations.

We will extensively use these operators to construct the LP field as well as
to write the isomorphism making the link with the BB representation, which is
the main result of this work. We remark that circular polarization plane waves

$\vec{\phi}_{\vec{k},\pm}=\frac{1}{(2\pi)^{3/2}}\vec{\epsilon}_{\pm}(\vec{k})e^{i\vec{k}\cdot\vec{x}},$
(2s)

with

$\vec{\epsilon}_{+}(\vec{k})=\frac{1}{\sqrt{2}|\vec{k}|\sqrt{k_{x}^{2}+k_{y}^{2}}}\left(\begin{array}[]{c}-k_{x}k_{z}+i|\vec{k}|k_{y}\\\
-k_{y}k_{z}-i|\vec{k}|k_{x}\\\ k_{x}^{2}+k_{y}^{2}\end{array}\right),\hskip
14.22636pt\vec{\epsilon}_{-}(\vec{k})=\vec{\epsilon}_{+}(\vec{k})^{\star},$
(2t)

are eigenfunctions of the three operators with eigenvalues
$\nabla\times\vec{\phi}_{\vec{k},\pm}=\pm|\vec{k}|\,\vec{\phi}_{\vec{k},\pm}$,
$\Lambda\vec{\phi}_{\vec{k},\pm}=\pm\vec{\phi}_{\vec{k},\pm}$ and
$\Omega\vec{\phi}_{\vec{k},\pm}=\omega_{\vec{k}}\vec{\phi}_{\vec{k},\pm}$,
where $\omega_{\vec{k}}=c|\vec{k}|>0$. Using the spectral theorem, one can
properly define any power $\Omega^{\alpha}$ for a real number
$\alpha\in[-2,2]$ by giving its action on the plane waves basis

$\Omega^{\alpha}\vec{\phi}_{\vec{k},\pm}=\omega^{\alpha}_{\vec{k}}\vec{\phi}_{\vec{k},\pm}.$
(2u)

The relations (2a)-(2d) can be shown by applying them to the basis functions
(2s). One can also show that any transverse field $\vec{v}$ can be decomposed
into a sum of a positive and a negative helicity part $\vec{v}^{(h\pm)}$

$\displaystyle\vec{v}=\vec{v}^{(h+)}+\vec{v}^{(h-)},$ (2v)

where $\Lambda\vec{v}^{(h\pm)}=\pm\vec{v}^{(h\pm)}$. The positive and negative
helicity parts can be constructed by applying the following projectors
$\mathbb{P}^{(h\pm)}:=(1\pm\Lambda)/2$, i.e.
$\vec{v}^{(h\pm)}=\mathbb{P}^{(h\pm)}\vec{v}$.

In the following, we will start by a summary of the canonical quantization of
the free electromagnetic field in the Coulomb gauge using the LP field in
position space and its equivalent in momentum space to make a link with the
standard notations of the quantum optics literature [15, 16, 17, 18, 19]. Then
we will introduce the BB quantization using a slightly different notation than
the one used originaly by Białynicki-Birula, which is more convenient to write
the isomorphism linking the LP and BB quantizations. After giving the explicit
formulation of this isomorphism, we will show through a few examples that the
two quantized theories are indeed equivalent and that the passage from one to
the other is explicitely given by the isomorphism. Finally, we will discuss
the time evolution within both theories, by explicitly linking their quantum
Hamiltonian operators which represent the total energy of the system and the
generator of the dynamics. This step ensures that the equivalence is preserved
at any time.

##  2 Canonical quantization in the Coulomb gauge

Maxwell’s equations in free space can be written in terms of the vector
potential $\vec{A}$ in the Coulomb gauge as

$\displaystyle\frac{\partial^{2}\vec{A}}{\partial t^{2}}$
$\displaystyle=c^{2}\Delta\vec{A},$ (2wa) $\displaystyle\nabla\cdot\vec{A}$
$\displaystyle=0,$ (2wb) $\displaystyle\vec{B}$
$\displaystyle=\nabla\times\vec{A},$ (2wc) $\displaystyle\vec{E}$
$\displaystyle=-\frac{\partial\vec{A}}{\partial t},$ (2wd)

with $c$ the speed of light in vaccum. In what follows, we will rewrite this
set of equations to give it a Hamiltonian structure.

###  2.1 Real canonical variables in position space

By choosing as canonical variables $\vec{A}$ and its canonically conjugate
variable $\vec{\Pi}=\varepsilon_{0}\frac{\partial\vec{A}}{\partial
t}=-\varepsilon_{0}\vec{E}$, the wave equation (2wa) can be written in
Hamiltonian form

$\displaystyle\frac{\partial\vec{A}}{\partial t}$ $\displaystyle=\frac{\delta
H}{\delta\vec{\Pi}}=\frac{\vec{\Pi}}{\varepsilon_{0}},$ (2wxa)
$\displaystyle\frac{\partial\vec{\Pi}}{\partial t}$
$\displaystyle=-\frac{\delta
H}{\delta\vec{A}}=\varepsilon_{0}c^{2}\Delta\vec{A}=-\varepsilon_{0}\Omega^{2}\vec{A},$
(2wxb)

where the Hamilton function $H$ is given by

$\displaystyle H$
$\displaystyle=\int_{\mathbb{R}^{3}}d^{3}x\left(\frac{1}{2\varepsilon_{0}}\vec{\Pi}\cdot\vec{\Pi}+\frac{\varepsilon_{0}}{2}\vec{A}\cdot\Omega^{2}\vec{A}\right).$
(2wxy)

The Hamilton equations (2wxa), (2wxb) and the constraints

$\displaystyle\nabla\cdot\vec{A}$ $\displaystyle=0,$ (2wxza)
$\displaystyle\nabla\cdot\vec{\Pi}$ $\displaystyle=0,$ (2wxzb)

are equivalent to Maxwell’s equations (2wa)-(2wd).

Remark: The Hamilton function $H$ has a structure similar to a one-dimensional
harmonic oscillator where $\varepsilon_{0}$ takes the place of the mass, and
the operator $\Omega$ the place of the frequency. Because of this analogy, it
has been stated that Maxwell’s equations look like an infinite dimensional
harmonic oscillator [15, 16, 17, 18, 19]. This image can be used as a
guideline for the construction of the quantized theory in terms of a Fock
space for the states.

###  2.2 Complex representation of Maxwell’s equations in position space — The
Landau-Peierls field

We define the following complex field, called the Landau-Peierls (LP) field
[20, 21, 31, 30] as

$\displaystyle\vec{\psi}:=\frac{1}{\sqrt{2\hbar}}\left[(\varepsilon_{0}\Omega)^{1/2}\vec{A}+i(\varepsilon_{0}\Omega)^{-1/2}\vec{\Pi}\right],$
(2wxzaa)

in the Hilbert space

$\displaystyle\mathcal{H}_{LP}:=\left\\{\vec{\psi}(\vec{x})\Big{|}\nabla\cdot\vec{\psi}=0,\
\langle{\vec{\psi}}|\vec{\psi}\rangle_{LP}<\infty\right\\},$ (2wxzab)

with the scalar product

$\displaystyle\langle\vec{\psi}|\vec{\psi}^{\prime}\rangle_{LP}:=\int_{\mathbb{R}^{3}}d^{3}x\
\vec{\psi}^{\star}(\vec{x})\cdot\vec{\psi}^{\prime}(\vec{x}).$ (2wxzac)

The Hamilton function in terms of these complex variables takes the form

$\displaystyle H$ $\displaystyle=\hbar\int_{\mathbb{R}^{3}}d^{3}x\
\vec{\psi}^{\star}\cdot\Omega\vec{\psi},$ (2wxzad)

and the corresponding Hamilton equations are

$i\frac{\partial\vec{\psi}}{\partial t}=\frac{1}{\hbar}\frac{\delta
H}{\delta\vec{\psi}^{\star}}=\Omega\vec{\psi}.$ (2wxzae)

Together with the transversality constraint $\nabla\cdot\vec{\psi}=0$, they
are equivalent to Maxwell’s equations. We remark that equation (2wxzae) has
the form of a Schrödinger equation where $\Omega$ is the generator of the
dynamics. Electromagnetic fields can be recovered from $\vec{\psi}$ by
inverting (2wxzaa),

$\displaystyle\vec{A}(\vec{x})$
$\displaystyle=\sqrt{\frac{\hbar}{2\varepsilon_{0}}}\Omega^{-1/2}\left(\vec{\psi}(\vec{x})+\vec{\psi}^{\star}(\vec{x})\right),$
(2wxzafa) $\displaystyle\vec{E}(\vec{x})$
$\displaystyle=i\sqrt{\frac{\hbar}{2\varepsilon_{0}}}\Omega^{1/2}\left(\vec{\psi}(\vec{x})-\vec{\psi}^{\star}(\vec{x})\right),$
(2wxzafb) $\displaystyle\vec{B}(\vec{x})$
$\displaystyle=\sqrt{\frac{\hbar}{2\varepsilon_{0}}}\Omega^{-1/2}\nabla\times\left(\vec{\psi}(\vec{x})+\vec{\psi}^{\star}(\vec{x})\right).$
(2wxzafc)

Using the definition of the helicity operator (2c), one can write the inverse
link between the Coulomb vector potential $\vec{A}$ and the magnetic field
$\vec{B}=\nabla\times\vec{A}$ as

$\vec{A}=c\Omega^{-1}\Lambda\vec{B}.$ (2wxzafag)

###  2.3 Momentum space formulation

A similar procedure can be done in the reciprocal momentum (or Fourier) space.
In this section we will only give the main results of that description and the
link with the position description. For a more detailed overview of this
formulation, we refer to the standard quantization of the free electromagnetic
field of the quantum optics literature [15, 16, 17, 18, 19].

We use the orthonormal basis of circular polarization plane waves (2s) to
decompose $\vec{\psi}$ as

$\displaystyle\vec{\psi}(\vec{x})=\int_{\mathbb{R}^{3}}d^{3}k\sum_{\sigma=\pm}\vec{\phi}_{\vec{k},\sigma}(\vec{x})z(\vec{k},\sigma),$
(2wxzafah)

where
$z(\vec{k},\sigma):=\langle\vec{\phi}_{\vec{k},\sigma}|\vec{\psi}\rangle_{LP}\in\mathbb{C}$.
We define the momentum Hilbert space as

$\displaystyle\mathcal{H}_{m}:=\\{z(\vec{k},\sigma)\ |\ \langle
z|z\rangle_{m}<\infty\\},$ (2wxzafai)

endowed with the scalar product

$\displaystyle\langle
z|z^{\prime}\rangle_{m}:=\int_{\mathbb{R}^{3}}d^{3}k\sum_{\sigma=\pm}z^{\star}(\vec{k},\sigma)z^{\prime}(\vec{k},\sigma).$
(2wxzafaj)

The denomination of this space as momentum space is anticipated from the
quantized theory since in that context, the variable $\vec{k}$ is
proportionnal to the momentum of monochromatic photons [15, 16, 17, 18, 19].
It is also refered to as the Fourier space representation since it is related
to the position space formulation by a decomposition into the circular plane
waves basis which is very close to a Fourier transform. Indeed, one can pass
from the position space description to the momentum description using the map

$\displaystyle\mathcal{M}:$ $\displaystyle\mathcal{H}_{LP}$
$\displaystyle\rightarrow\mathcal{H}_{m}$ (2wxzafak)
$\displaystyle\vec{\psi}(\vec{x})$ $\displaystyle\mapsto
z(\vec{k},\sigma)=\int_{\mathbb{R}^{3}}d^{3}x\
\vec{\phi}^{\star}_{\vec{k},\sigma}(\vec{x})\cdot\vec{\psi}(\vec{x}),$

which is a bounded unitary symplectic transformation providing an isomorphism
between the two Hilbert spaces. The inverse map $\mathcal{M}^{-1}$ is given by
(2wxzafah), and the equivalence of the scalar products reads

$\displaystyle\langle
z|z^{\prime}\rangle_{m}=\langle\mathcal{M}\psi|\mathcal{M}\vec{\psi}^{\prime}\rangle_{m}=\langle\psi|\vec{\psi}^{\prime}\rangle_{LP},$
(2wxzafal)

which is proven using the completeness of the basis
$\\{\vec{\phi}_{\vec{k},\sigma}\\}$. The Hamilton function expressed in the
new variables takes the form

$\displaystyle
H=\int_{\mathbb{R}^{3}}d^{3}k\sum_{\sigma=\pm}\hbar\omega_{\vec{k}}\
z^{\star}(\vec{k},\sigma)z(\vec{k},\sigma),$ (2wxzafam)

and can be put in a harmonic oscillator form

$\displaystyle
H=\int_{\mathbb{R}^{3}}d^{3}k\sum_{\sigma=\pm}\left(\frac{1}{2\varepsilon_{0}}p^{2}_{\vec{k},\sigma}+\frac{1}{2}\varepsilon_{0}\omega_{\vec{k}}^{2}q^{2}_{\vec{k},\sigma}\right),$
(2wxzafan)

by defining the real variables

$\displaystyle p_{\vec{k},\sigma}$
$\displaystyle:=-i\sqrt{\frac{\hbar\varepsilon_{0}\omega_{\vec{k}}}{2}}(z-z^{\star}),$
(2wxzafaoa) $\displaystyle q_{\vec{k},\sigma}$
$\displaystyle:=\sqrt{\frac{\hbar}{2\varepsilon_{0}\omega_{\vec{k}}}}(z+z^{\star}),$
(2wxzafaob) $\displaystyle z(\vec{k},\sigma)$
$\displaystyle=:\frac{1}{\sqrt{2\hbar}}\left((\varepsilon_{0}\omega_{\vec{k}})^{1/2}q_{\vec{k},\sigma}+i(\varepsilon_{0}\omega_{\vec{k}})^{-1/2}p_{\vec{k},\sigma}\right).$
(2wxzafaoc)

###  2.4 Quantization using a correspondence principle

From the classical Hilbert spaces defined above, one can construct two bosonic
Fock spaces defined by

$\displaystyle\mathbb{F}^{\mathfrak{B}}(\mathcal{H})=\bigoplus_{n=0}^{\infty}\mathcal{H}^{\otimes_{S}n}.$
(2wxzafaoap)

A strength of this construction is that $\mathcal{H}$ stands for either
$\mathcal{H}_{LP}$ or $\mathcal{H}_{m}$. In the same spirit, the definition of
creation-anihilation operators within the Fock space has the same form for
both choices of the classical Hilbert space used:

$\displaystyle\hat{B}_{\eta}:\mathcal{H}^{\otimes_{S}l}\to\mathcal{H}^{\otimes_{S}(l-1)},\qquad\qquad\qquad\hat{B}^{\dagger}_{\eta}:\mathcal{H}^{\otimes_{S}l}\to\mathcal{H}^{\otimes_{S}(l+1)},$
(2wxzafaoaq)

with their action [13, 14] on the tensor product monomials defined by

$\displaystyle\hat{B}^{\dagger}_{\eta}|\eta_{1}\otimes\ldots\otimes\eta_{l}\rangle$
$\displaystyle:=$
$\displaystyle\sqrt{l+1}~{}\hat{S}_{l+1}|\eta\otimes\eta_{1}\otimes\ldots\otimes\eta_{l}\rangle,$
(2wxzafaoara)
$\displaystyle\hat{B}_{\eta}|\eta_{1}\otimes\ldots\otimes\eta_{l}\rangle$
$\displaystyle:=$
$\displaystyle~{}\frac{1}{\sqrt{l}}\sum_{j=1}^{l}\langle\eta|\eta_{j}\rangle_{\mathcal{H}}~{}\hat{S}_{l-1}~{}|\eta_{1}\otimes\ldots\otimes\xcancel{\eta}_{j}\ldots\otimes\eta_{l}\rangle,$
(2wxzafaoarb)

where $\eta_{j}$ are elements in $\mathcal{H}$, the notation
$\xcancel{\eta}_{j}$ indicates that this term is missing and $\hat{S}_{l}$ are
symmetrization operators. The bosonic creation-anihilation operators satisfy
the following commutation relations [19, equation (3.192)],[30, 32]

$\displaystyle\left[\hat{B}_{\eta_{A}},\hat{B}^{\dagger}_{\eta_{B}}\right]$
$\displaystyle=\langle{\eta_{A}}|{\eta_{B}}\rangle_{\mathcal{H}},$
(2wxzafaoarasa)
$\displaystyle\left[\hat{B}_{\eta_{A}},\hat{B}_{\eta_{B}}\right]$
$\displaystyle=0=\left[\hat{B}^{\dagger}_{\eta_{A}},\hat{B}^{\dagger}_{\eta_{B}}\right],$
(2wxzafaoarasb)

where $\langle\cdot|\cdot\rangle_{\mathcal{H}}$ denotes either
$\langle\cdot|\cdot\rangle_{LP}$ or $\langle\cdot|\cdot\rangle_{m}$. The
second ingredient to fully describe the quantum field theory is the
representation of each observable by a selfadjoint operator acting on
$\mathbb{F}^{\mathfrak{B}}(\mathcal{H})$. Which operators are to be associated
to the physical observables have to be postulated. For a system like the
Maxwell field that have a quadratic Hamiltonian, it can be done through a
principle of correspondence, suggested by the fact that in the momentum space,
the classical Hamilton function (2wxzafan) has the form of an infinite
collection of independent harmonic oscillators.

####  2.4.1 Quantization in the momentum space $\mathcal{H}_{m}$

By this analogy, one postulates the following rule of correspondence, first
defined in the momentum Fock space
$\mathbb{F}^{\mathfrak{B}}(\mathcal{H}_{m})$:

$\displaystyle{\rm Quantization~{}map:}\hskip 42.67912pt$ $\displaystyle
z(\vec{k},\sigma)$ $\displaystyle\mapsto\hat{B}_{\varphi_{\vec{k},\sigma}},$
(2wxzafaoarasata) $\displaystyle z^{\star}(\vec{k},\sigma)$
$\displaystyle\mapsto\hat{B}^{\dagger}_{\varphi_{\vec{k},\sigma}}.$
(2wxzafaoarasatb)

where $\varphi_{\vec{k},\sigma}$ are the basis eigenfunctions in
$\mathcal{H}_{m}$ given by

$\displaystyle\vec{\phi}_{\vec{k},\sigma}(\vec{x})\mapsto\mathcal{M}\vec{\phi}_{\vec{k},\sigma}$
$\displaystyle=\varphi_{\vec{k},\sigma}(\vec{k}^{\prime},\sigma^{\prime})$
(2wxzafaoarasataua)
$\displaystyle=\delta_{\sigma,\sigma^{\prime}}\delta(\vec{k}-\vec{k}^{\prime}).$
(2wxzafaoarasataub)

####  2.4.2 Quantization in the Landau-Peierls space $\mathcal{H}_{LP}$

By the isomorphism $\mathcal{M}$, the correspondence rule can be translated to
the position LP Fock space $\mathbb{F}^{\mathfrak{B}}(\mathcal{H}_{LP})$ using
(2wxzafah):

$\displaystyle{\rm Quantization~{}map:}\hskip 28.45274pt$
$\displaystyle\vec{\psi}(\vec{x})$
$\displaystyle\mapsto\int_{\mathbb{R}^{3}}d^{3}k\sum_{\sigma=\pm}\vec{\phi}_{\vec{k},\sigma}(\vec{x})\hat{B}_{\vec{\phi}_{\vec{k},\sigma}}=:\vec{\hat{\Psi}}(\vec{x}),$
(2wxzafaoarasatauava) $\displaystyle\vec{\psi}^{\star}(\vec{x})$
$\displaystyle\mapsto\int_{\mathbb{R}^{3}}d^{3}k\sum_{\sigma=\pm}\vec{\phi}^{\star}_{\vec{k},\sigma}(\vec{x})\hat{B}^{\dagger}_{\vec{\phi}_{\vec{k},\sigma}}=:\vec{\hat{\Psi}}^{\dagger}(\vec{x}),$
(2wxzafaoarasatauavb)

which defines the field operators $\vec{\hat{\Psi}}(\vec{x})$ and
$\vec{\hat{\Psi}}^{\dagger}(\vec{x})$. From these maps, one can write the
electromagnetic field operators following (2wxzafa)-(2wxzafc)

$\displaystyle\vec{\hat{A}}(\vec{x})$
$\displaystyle=\sqrt{\frac{\hbar}{2\varepsilon_{0}}}\Omega^{-1/2}\left(\vec{\hat{\Psi}}(\vec{x})+\vec{\hat{\Psi}}^{\dagger}(\vec{x})\right),$
(2wxzafaoarasatauavawa) $\displaystyle\vec{\hat{E}}(\vec{x})$
$\displaystyle=i\sqrt{\frac{\hbar}{2\varepsilon_{0}}}\Omega^{1/2}\left(\vec{\hat{\Psi}}(\vec{x})-\vec{\hat{\Psi}}^{\dagger}(\vec{x})\right),$
(2wxzafaoarasatauavawb) $\displaystyle\vec{\hat{B}}(\vec{x})$
$\displaystyle=\sqrt{\frac{\hbar}{2\varepsilon_{0}}}\Omega^{-1/2}\nabla\times\left(\vec{\hat{\Psi}}(\vec{x})+\vec{\hat{\Psi}}^{\dagger}(\vec{x})\right).$
(2wxzafaoarasatauavawc)

Remark: We point out that the interpretation of the field operators
$\vec{\hat{\Psi}}^{\dagger}$ and $\vec{\hat{\Psi}}$ is not to be confused with
that of the creation-anihilation operators $\hat{B}^{\dagger}_{\vec{\psi}}$
and $\hat{B}_{\vec{\psi}}$ for $\vec{\psi}\in\mathcal{H}_{LP}$. Indeed, when
applied to the vacuum state, $\hat{B}^{\dagger}_{\vec{\psi}}$ creates a photon
carried by the classical solution $\vec{\psi}$ of Maxwell’s equations [30],
while each component $\hat{\Psi}_{j}^{\dagger}$ is an operator-valued
distribution which has to be integrated over the whole space acting on a test
function to properly create a photon.

###  2.5 Equivalence between the LP and momentum quantizations

In this section, we show that the isomorphism $\mathcal{M}$ given by equations
(2wxzafah), (2wxzafak) and (2wxzafal), provides a direct relation between the
quantized theories we have constructed above. Indeed, since the central
concept of bosonic creation-anihilation operators, which allows one to write
any state and observable in the theory, are explicitely linked by the
classical isomorphism $\mathcal{M}$, all physical predictions made with the
two representations are the same.

If one considers e.g. a single-photon state in the position space LP
representation which reads as

$\displaystyle\hat{B}_{\vec{\psi}}^{\dagger}|\varnothing\rangle_{LP}=|\vec{\psi}\rangle_{LP},$
(2wxzafaoarasatauavawax)

with $\vec{\psi}\in\mathcal{H}_{LP}$. One can use the isomorphism
$\mathcal{M}$ to write it in terms of a creation operator in the momentum
space as follows

$\displaystyle|\vec{\psi}\rangle_{LP}$
$\displaystyle=|\mathcal{M}^{-1}z\rangle_{LP}$ (2wxzafaoarasatauavawaya)
$\displaystyle=\mathcal{M}^{-1}|z\rangle_{m}$ (2wxzafaoarasatauavawayb)
$\displaystyle=\mathcal{M}^{-1}\hat{B}_{z}^{\dagger}|\varnothing\rangle_{m}$
(2wxzafaoarasatauavawayc)
$\displaystyle=\mathcal{M}^{-1}\hat{B}_{z}^{\dagger}\mathcal{M}|\varnothing\rangle_{LP},$
(2wxzafaoarasatauavawayd)

from which we deduce that
$\hat{B}_{\vec{\psi}}^{\dagger}=\mathcal{M}^{-1}\hat{B}_{z}^{\dagger}\mathcal{M}=\mathcal{M}^{-1}\hat{B}_{\mathcal{M}\vec{\psi}}^{\dagger}\mathcal{M}$,
where
$\hat{B}_{z}^{\dagger}:\mathcal{H}_{m}^{\otimes_{S}l}\to\mathcal{H}_{m}^{\otimes_{S}(l+1)}$
is the creation operator acting on the momentum Fock space
$\mathbb{F}^{\mathfrak{B}}(\mathcal{H}_{m})$. We have extended here the
isomorphism $\mathcal{M}$ to the whole Fock space by defining its action on
the vacuum state:
$\mathcal{M}|\varnothing\rangle_{LP}=|\varnothing\rangle_{m}$. The same state
can then be described in one or the other representation without ambiguity.
Regarding the observables, one can for instance express the electric field in
terms of creation-anihilation operators in the LP position representation as

$\displaystyle\vec{\hat{E}}_{LP}(\vec{x})$
$\displaystyle=i\sqrt{\frac{\hbar}{2\varepsilon_{0}}}\Omega^{1/2}\left(\vec{\hat{\Psi}}(\vec{x})-\vec{\hat{\Psi}}^{\dagger}(\vec{x})\right)$
(2wxzafaoarasatauavawayaza)
$\displaystyle=i\sqrt{\frac{\hbar}{2\varepsilon_{0}}}\int_{\mathbb{R}^{3}}d^{3}k\sum_{\sigma=\pm}\omega_{\vec{k}}^{1/2}\left(\vec{\phi}_{\vec{k},\sigma}(\vec{x})\hat{B}_{\vec{\phi}_{\vec{k},\sigma}}-\vec{\phi}^{\star}_{\vec{k},\sigma}(\vec{x})\hat{B}^{\dagger}_{\vec{\phi}_{\vec{k},\sigma}}\right),$
(2wxzafaoarasatauavawayazb)

which is expressed in the momentum representation by

$\displaystyle\vec{\hat{E}}_{m}(\vec{x})$
$\displaystyle=\mathcal{M}\vec{\hat{E}}_{LP}\mathcal{M}^{-1}$
(2wxzafaoarasatauavawayazbaa)
$\displaystyle=i\sqrt{\frac{\hbar}{2\varepsilon_{0}}}\int_{\mathbb{R}^{3}}d^{3}k\sum_{\sigma=\pm}\omega_{\vec{k}}^{1/2}\left(\vec{\phi}_{\vec{k},\sigma}(\vec{x})\mathcal{M}\hat{B}_{\vec{\phi}_{\vec{k},\sigma}}\mathcal{M}^{-1}-\vec{\phi}^{\star}_{\vec{k},\sigma}(\vec{x})\mathcal{M}\hat{B}^{\dagger}_{\vec{\phi}_{\vec{k},\sigma}}\mathcal{M}^{-1}\right)$
(2wxzafaoarasatauavawayazbab)
$\displaystyle=i\sqrt{\frac{\hbar}{2\varepsilon_{0}}}\int_{\mathbb{R}^{3}}d^{3}k\sum_{\sigma=\pm}\omega_{\vec{k}}^{1/2}\left(\vec{\phi}_{\vec{k},\sigma}(\vec{x})\hat{B}_{\varphi_{\vec{k},\sigma}}-\vec{\phi}^{\star}_{\vec{k},\sigma}(\vec{x})\hat{B}^{\dagger}_{\varphi_{\vec{k},\sigma}}\right)$
(2wxzafaoarasatauavawayazbac)
$\displaystyle=i\sqrt{\frac{\hbar}{2\varepsilon_{0}}}\int_{\mathbb{R}^{3}}d^{3}k\sum_{\sigma=\pm}\omega_{\vec{k}}^{1/2}\left(\vec{\phi}_{\vec{k},\sigma}(\vec{x})\hat{a}_{\vec{k},\sigma}-\vec{\phi}^{\star}_{\vec{k},\sigma}(\vec{x})\hat{a}^{\dagger}_{\vec{k},\sigma}\right),$
(2wxzafaoarasatauavawayazbad)

where we have used in the last equality the usual notation
$\hat{a}_{\vec{k},\sigma}=\hat{B}_{\delta_{\sigma,\sigma^{\prime}}\delta(\vec{k}-\vec{k}^{\prime})}$
and
$\hat{a}^{\dagger}_{\vec{k},\sigma}=\hat{B}^{\dagger}_{\delta_{\sigma,\sigma^{\prime}}\delta(\vec{k}-\vec{k}^{\prime})}$
[15, 16, 17, 18, 19]. Furthemore, if one plugs in the expression of the
circular plane wave eigenfunctions (2s), we obtain the usual expression of the
quantum optics literature [15, 16, 17, 18, 19]

$\displaystyle\vec{\hat{E}}_{m}(\vec{x})$
$\displaystyle=i\sqrt{\frac{\hbar}{2\varepsilon_{0}(2\pi)^{3}}}\int_{\mathbb{R}^{3}}d^{3}k\sum_{\sigma=\pm}\omega_{\vec{k}}^{1/2}\left(\vec{\epsilon}_{\sigma}(\vec{k})e^{i\vec{k}\cdot\vec{x}}\hat{a}_{\vec{k},\sigma}-\vec{\epsilon}_{\sigma}(\vec{k})^{\star}e^{-i\vec{k}\cdot\vec{x}}\hat{a}^{\dagger}_{\vec{k},\sigma}\right).$
(2wxzafaoarasatauavawayazbabb)

With this example, we have shown how one can pass from one representation to
the other using directly the classical isomorphism $\mathcal{M}$ in a standard
way, i.e. directly applied to the states and through a similarity relation for
creation-anihilation operators or observables. This simple relation guarantees
that the predictions one can make with one theory is consistent with what is
obtained with the other.

##  3 Białynicki-Birula’s complex formulation of Maxwell’s equations — Choice
of the canonical variables

We consider now another approach for the quantization which has been
developped mainly by Białynicki-Birula [22, 23, 24]. One of the advantages of
this construction is that it does not require to choose any particular gauge
since it starts by defining the canonical variables to be directly
proportional to the electric and magnetic fields as [24, p.25, Section 10]

$\displaystyle\vec{\mathcal{P}}_{RS}$
$\displaystyle:=\frac{\vec{B}}{\sqrt{\mu_{0}}},$
(2wxzafaoarasatauavawayazbabca) $\displaystyle\vec{\mathcal{Q}}_{RS}$
$\displaystyle:=\sqrt{\varepsilon_{0}}\vec{E}.$
(2wxzafaoarasatauavawayazbabcb)

The Hamilton function is defined by

$\displaystyle K_{RS}$
$\displaystyle:=\frac{c}{2}\int_{\mathbb{R}^{3}}d^{3}x\left(\vec{\mathcal{P}}_{RS}\cdot\nabla\times\vec{\mathcal{P}}_{RS}+\vec{\mathcal{Q}}_{RS}\cdot\nabla\times\vec{\mathcal{Q}}_{RS}\right)$
(2wxzafaoarasatauavawayazbabcbda)
$\displaystyle=\frac{c}{2}\int_{\mathbb{R}^{3}}d^{3}x\left(\varepsilon_{0}\vec{E}\cdot\nabla\times\vec{E}+\frac{1}{\mu_{0}}\vec{B}\cdot\nabla\times\vec{B}\right),$
(2wxzafaoarasatauavawayazbabcbdb)

yielding the following Hamilton equations

$\displaystyle\frac{\partial\vec{\mathcal{P}}_{RS}}{\partial t}$
$\displaystyle=-\frac{\delta
K_{RS}}{\delta\vec{\mathcal{Q}}_{RS}}=-c\nabla\times\vec{\mathcal{Q}}_{RS},$
(2wxzafaoarasatauavawayazbabcbdbea)
$\displaystyle\frac{\partial\vec{\mathcal{Q}}_{RS}}{\partial t}$
$\displaystyle=\frac{\delta
K_{RS}}{\delta\vec{\mathcal{P}}_{RS}}=c\nabla\times\vec{\mathcal{P}}_{RS},$
(2wxzafaoarasatauavawayazbabcbdbeb)

that are equivalent to Maxwell’s equations. We denote the Hamilton function by
$K_{RS}$ and not by $H$, to emphasize that it is not the total electromagnetic
energy [16, 33], which reads

$\displaystyle\mathcal{E}_{tot}$
$\displaystyle=\frac{1}{2}\int_{\mathbb{R}^{3}}d^{3}x\left(\varepsilon_{0}\vec{E}\cdot\vec{E}+\frac{1}{\mu_{0}}\vec{B}\cdot\vec{B}\right)$
(2wxzafaoarasatauavawayazbabcbdbebfa)
$\displaystyle=\frac{1}{2}\int_{\mathbb{R}^{3}}d^{3}x\left(\vec{\mathcal{P}}_{RS}\cdot\vec{\mathcal{P}}_{RS}+\vec{\mathcal{Q}}_{RS}\cdot\vec{\mathcal{Q}}_{RS}\right).$
(2wxzafaoarasatauavawayazbabcbdbebfb)

###  3.1 Complex representation of Maxwell’s equations in position space — the
Riemann-Silberstein vector

From the canonical variables defined above one can construct a complex
representation of the field called the Riemann-Silberstein (RS) vector [24,
25]

$\displaystyle\vec{F}_{RS}:=\frac{1}{\sqrt{2}}\left(\vec{\mathcal{Q}}_{RS}+i\vec{\mathcal{P}}_{RS}\right)=\sqrt{\frac{\varepsilon_{0}}{2}}\left(\vec{E}+ic\vec{B}\right).$
(2wxzafaoarasatauavawayazbabcbdbebfbg)

Maxwell’s equations can be written in the RS complex representation as

$\displaystyle i\frac{\partial\vec{F}_{RS}}{\partial t}=-\frac{\delta
K_{RS}}{\delta\vec{F}_{RS}^{\star}}=c\nabla\times\vec{F}_{RS},$
(2wxzafaoarasatauavawayazbabcbdbebfbh)

with

$\displaystyle K_{RS}=c\int_{\mathbb{R}^{3}}d^{3}x\
\vec{F}_{RS}^{\star}\cdot\nabla\times\vec{F}_{RS}.$
(2wxzafaoarasatauavawayazbabcbdbebfbi)

In analogy with what we did for the Coulomb gauge, we define the symplectic
unitary map

$\displaystyle\vec{F}_{RS}\mapsto
z_{RS}(\vec{k},\sigma)=\int_{\mathbb{R}^{3}}d^{3}x\
\vec{\phi}_{\vec{k},\sigma}^{\star}(\vec{x})\cdot\vec{F}_{RS}(\vec{x}),$
(2wxzafaoarasatauavawayazbabcbdbebfbj)

from which we deduce the Hamilton function in terms of the RS momentum
variable $z_{RS}$

$\displaystyle K_{RS}=\int_{\mathbb{R}^{3}}d^{3}k\left[\omega_{\vec{k}}\
z_{RS}^{\star}(\vec{k},+)z_{RS}(\vec{k},+)-\omega_{\vec{k}}\
z_{RS}^{\star}(\vec{k},-)z_{RS}(\vec{k},-)\right].$
(2wxzafaoarasatauavawayazbabcbdbebfbk)

Written in terms of the real variables defined as

$\displaystyle p^{RS}_{\vec{k},\sigma}$
$\displaystyle:=-i\sqrt{\frac{\varepsilon_{0}\omega_{\vec{k}}}{2}}(z_{RS}-z_{RS}^{\star}),$
(2wxzafaoarasatauavawayazbabcbdbebfbla) $\displaystyle
q^{RS}_{\vec{k},\sigma}$
$\displaystyle:=\sqrt{\frac{1}{2\varepsilon_{0}\omega_{\vec{k}}}}(z_{RS}+z_{RS}^{\star}),$
(2wxzafaoarasatauavawayazbabcbdbebfblb) $\displaystyle z_{RS}(\vec{k},\sigma)$
$\displaystyle=:\frac{1}{\sqrt{2}}\left((\varepsilon_{0}\omega_{\vec{k}})^{1/2}q^{RS}_{\vec{k},\sigma}+i(\varepsilon_{0}\omega_{\vec{k}})^{-1/2}p^{RS}_{\vec{k},\sigma}\right),$
(2wxzafaoarasatauavawayazbabcbdbebfblc)

the Hamilton function becomes

$\displaystyle
K_{RS}=\int_{\mathbb{R}^{3}}d^{3}k\left[\left(\frac{(p_{\vec{k},+}^{RS})^{2}}{2\varepsilon_{0}}+\frac{\varepsilon_{0}}{2}\omega^{2}_{\vec{k}}(q^{RS}_{\vec{k},+})^{2}\right)-\left(\frac{(p^{RS}_{\vec{k},-})^{2}}{2\varepsilon_{0}}+\frac{\varepsilon_{0}}{2}\omega^{2}_{\vec{k}}(q^{RS}_{\vec{k},-})^{2}\right)\right].$
(2wxzafaoarasatauavawayazbabcbdbebfblbm)

We see here that it looks like two infinite collections of independent
harmonic oscillators but the second one appears with a minus sign. This
situation is similar to the one encountered with the Dirac equation, in which
the electrons could have positive and negative energies. It poses a difficulty
for the construction of the quantized model. However, this difficulty can be
avoided by making a different choice for the canonical variables, as we
describe in the next section. The construction we are going to introduce is
strongly inspired by the work of Białynicki-Birula (BB) [22] which is the
reason why we will call it the BB complex representation, although we
formulate it in a slightly different form.

###  3.2 Alternative choice of canonical variables that avoid the negative
eigenvalues in the Hamiltonian

To construct canonical variables that avoid the difficulty of the negative
eigenvalues describe above, we have to locate the variables that lead to
negative contributions. This can be done using the notion of helicity which
allows us to decompose the RS canonical variables into their positive and
negative helicity parts $\vec{F}_{RS}=\vec{F}_{RS}^{(h+)}+\vec{F}_{RS}^{(h-)}$
or equivalently in terms of the real variables
$\vec{\mathcal{P}}_{RS}=\vec{\mathcal{P}}_{RS}^{(h+)}+\vec{\mathcal{P}}_{RS}^{(h-)}$
and
$\vec{\mathcal{Q}}_{RS}=\vec{\mathcal{Q}}_{RS}^{(h+)}+\vec{\mathcal{Q}}_{RS}^{(h-)}$,
in terms of which the Hamilton function takes the form

$\displaystyle K_{RS}$
$\displaystyle=\frac{c}{2}\int_{\mathbb{R}^{3}}d^{3}x\left[\left(\vec{\mathcal{P}}_{RS}^{(h+)}\cdot\nabla\times\vec{\mathcal{P}}_{RS}^{(h+)}+\vec{\mathcal{Q}}_{RS}^{(h+)}\cdot\nabla\times\vec{\mathcal{Q}}_{RS}^{(h+)}\right)\right.$
$\displaystyle\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\left.+\left(\vec{\mathcal{P}}_{RS}^{(h-)}\cdot\nabla\times\vec{\mathcal{P}}^{(h-)}+\vec{\mathcal{Q}}_{RS}^{(h-)}\cdot\nabla\times\vec{\mathcal{Q}}_{RS}^{(h-)}\right)\right]$
$\displaystyle=\frac{1}{2}\int_{\mathbb{R}^{3}}d^{3}x\left[\left(\vec{\mathcal{P}}_{RS}^{(h+)}\cdot\Omega\vec{\mathcal{P}}_{RS}^{(h+)}+\vec{\mathcal{Q}}_{RS}^{(h+)}\cdot\Omega\vec{\mathcal{Q}}_{RS}^{(h+)}\right)\right.$
$\displaystyle\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\left.-\left(\vec{\mathcal{P}}_{RS}^{(h-)}\cdot\Omega\vec{\mathcal{P}}_{RS}^{(h-)}+\vec{\mathcal{Q}}_{RS}^{(h-)}\cdot\Omega\vec{\mathcal{Q}}_{RS}^{(h-)}\right)\right],$
(2wxzafaoarasatauavawayazbabcbdbebfblbnb)

where we have used $c\nabla\times=\Omega\Lambda$ to obtain the second
expression and the fact that $\Lambda\vec{v}^{(h\pm)}=\pm\vec{v}^{(h\pm)}$. It
shows that the negative contributions identified in the momentum
representation come from the negative helicity part of the field. To
circumvent this problem, one can define new canonical variables that we call
the BB variables

$\displaystyle\vec{\mathcal{Q}}_{BB}$
$\displaystyle:=\vec{\mathcal{Q}}_{RS},\hskip 56.9055pt\vec{\mathcal{P}}_{BB}$
$\displaystyle:=\Lambda\vec{\mathcal{P}}_{RS},$
(2wxzafaoarasatauavawayazbabcbdbebfblbnbo)

and a new classical Hamilton function

$\displaystyle K_{BB}$
$\displaystyle:=\frac{1}{2}\int_{\mathbb{R}^{3}}d^{3}x\left[\left(\vec{\mathcal{P}}_{BB}^{(h+)}\cdot\Omega\vec{\mathcal{P}}_{BB}^{(h+)}+\vec{\mathcal{Q}}_{BB}^{(h+)}\cdot\Omega\vec{\mathcal{Q}}_{BB}^{(h+)}\right)\right.$
(2wxzafaoarasatauavawayazbabcbdbebfblbnbp)
$\displaystyle\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\left.+\left(\vec{\mathcal{P}}_{BB}^{(h-)}\cdot\Omega\vec{\mathcal{P}}_{BB}^{(h-)}+\vec{\mathcal{Q}}_{BB}^{(h-)}\cdot\Omega\vec{\mathcal{Q}}_{BB}^{(h-)}\right)\right],$

which is positive since $\Omega$ is a positive operator, as opposed to
$\nabla\times$. This alternative choice of the canonical variables thus avoids
the problem of negative eigenvalues. We emphasize that $K_{BB}$ is not the
Hamilton function $K_{RS}$ expressed in the new variables since the relation
(2wxzafaoarasatauavawayazbabcbdbebfblbnbo) is not a canonical transformation.

###  3.3 The Białynicki-Birula complex representation

We choose a different Hamiltonian structure that gives the same classical
Maxwell equations. For the complex representation, we define a modification of
the RS vector adapted to the new canonical variables which we call the
Białynicki-Birula vector

$\displaystyle\vec{F}_{BB}:=\frac{1}{\sqrt{2}}\left(\vec{\mathcal{Q}}_{BB}+i\vec{\mathcal{P}}_{BB}\right)=\sqrt{\frac{\varepsilon_{0}}{2}}\left(\vec{E}+ic\Lambda\vec{B}\right),$
(2wxzafaoarasatauavawayazbabcbdbebfblbnbq)

which can be decomposed in the six-component field (bispinor) notation used
originally in [22]

$\Psi_{BB}:=\left(\begin{array}[]{c}\vec{F}_{BB}^{(h+)}\\\
\vec{F}_{BB}^{(h-)}\\\
\end{array}\right)=\left(\begin{array}[]{c}\vec{F}_{RS}^{(h+)}\\\
\vec{F}_{RS}^{(h-)\star}\\\ \end{array}\right).$
(2wxzafaoarasatauavawayazbabcbdbebfblbnbr)

Remarks:

  * •

The BB vector $\vec{F}_{BB}$ that we introduced here was not mentionned
explicitely in [22, 24] but is a completely equivalent formulation of the
original formulation using the six-component field $\Psi_{BB}$. We prefer,
however, to use $\vec{F}_{BB}$ since it is a more compact notation which
allows us to write simpler formulas especially when we will introduce the
isomorphism with the LP field.

  * •

In the original works of BB [22], the six-component field $\Psi_{BB}$ was
defined using the positive and negative frequency parts of the RS field which
we denote by $\vec{F}_{RS}^{(f\pm)}$, instead of the positive and negative
helicity parts as we have done here. This difference does not change anything
for the definition of $\Psi_{BB}$ since
$\vec{F}_{RS}^{(h\pm)}=\vec{F}_{RS}^{(f\pm)}$. However, we stress out that
this property is in general not true, e.g.
$\vec{\psi}^{(h\pm)}\neq\vec{\psi}^{(f\pm)}$ since $\vec{\psi}^{(f-)}=0$ for
any $\vec{\psi}\in\mathcal{H}_{LP}$ while $\vec{\psi}^{(h-)}\neq 0$ in
general. Other examples are the electric and the magnetic fields for which
$\vec{E}^{(h\pm)}\neq\vec{E}^{(f\pm)}$ and
$\vec{B}^{(h\pm)}\neq\vec{B}^{(f\pm)}$.

  * •

The structure of the Białynicki-Birula six-component spinor can be illustrated
by the description of the general elliptically polarized plane waves
propagating in the $z$-direction, i.e. $\vec{k}=(0\ 0\ k)^{T}$ with $k>0$. The
electric $\vec{E}^{\pm}$ and the magnetic $\vec{B}^{\pm}$ fields of a
left/right cirlcularly polarized plane wave propagating in direction $z$ can
be written as [33, p.299, eq. (7.21)]

$\displaystyle\vec{E}^{\pm}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{\varepsilon_{0}(2\pi)^{3}}}\left(\begin{array}[]{c}\cos(kz-\omega
t)\\\ \mp\sin(kz-\omega t)\\\ 0\end{array}\right),$
(2wxzafaoarasatauavawayazbabcbdbebfblbnbsd) $\displaystyle\vec{B}^{\pm}$
$\displaystyle=$
$\displaystyle\frac{1}{c\sqrt{\varepsilon_{0}(2\pi)^{3}}}\left(\begin{array}[]{c}\pm\sin(kz-\omega
t)\\\ \cos(kz-\omega t)\\\ 0\end{array}\right).$
(2wxzafaoarasatauavawayazbabcbdbebfblbnbsh)

We remark that the left/right circularly polarized plane waves have
respectively $+/-$ helicity, since they are eigenfunctions of the curl
operator $\nabla\times$ with positive and negative eigenvalues.

The plane wave with left circular polarization is represented by the
Białynicki-Birula six-component spinor

$\displaystyle\Psi_{BB}^{\rm(left~{}circular)}=\sqrt{\frac{\varepsilon_{0}}{2}}\left(\begin{array}[]{c}\vec{E}^{+}+ic\vec{B}^{+}\\\
\vec{0}\end{array}\right)=(2\pi)^{-3/2}e^{i(kz-\omega
t)}\left(\begin{array}[]{c}\vec{\epsilon}_{+}\\\ \vec{0}\end{array}\right),$
(2wxzafaoarasatauavawayazbabcbdbebfblbnbsbx)

with $\vec{\epsilon}_{+}=\frac{1}{\sqrt{2}}\left(1\ i\ 0\right)^{T}$. The
plane wave with right circular polarization is represented by

$\displaystyle\Psi_{BB}^{\rm(right~{}circular)}=\sqrt{\frac{\varepsilon_{0}}{2}}\left(\begin{array}[]{c}\vec{0}\\\
\vec{E}^{-}-ic\vec{B}^{-}\end{array}\right)=(2\pi)^{-3/2}e^{i(kz-\omega
t)}\left(\begin{array}[]{c}\vec{0}\\\ \vec{\epsilon}_{-}\end{array}\right),$
(2wxzafaoarasatauavawayazbabcbdbebfblbnbscc)

with $\vec{\epsilon}_{-}=\frac{1}{\sqrt{2}}\left(1\ -i\ 0\right)^{T}$.

The single-photon plane wave with general elliptic polarization is represented
by the linear combinations

$\displaystyle\Psi_{BB}^{\rm(elliptic)}$ $\displaystyle=$
$\displaystyle\alpha_{+}\Psi_{BB}^{\rm(left~{}circular)}+\alpha_{-}\Psi_{BB}^{\rm(right~{}circular)}$
(2wxzafaoarasatauavawayazbabcbdbebfblbnbscd) $\displaystyle=$
$\displaystyle(2\pi)^{-3/2}e^{i(kz-\omega
t)}\left(\begin{array}[]{c}\alpha_{+}\vec{\epsilon}_{+}\\\
\alpha_{-}\vec{\epsilon}_{-}\end{array}\right),$
(2wxzafaoarasatauavawayazbabcbdbebfblbnbscg)

where $\alpha_{\pm}\in\mathbb{C}$, $|\alpha_{+}|^{2}+|\alpha_{-}|^{2}=1$.

The Hamilton function $K_{BB}$ in the BB complex representation reads

$\displaystyle K_{BB}$ $\displaystyle=\int_{\mathbb{R}^{3}}d^{3}x\
\vec{F}_{BB}^{\star}\cdot\Omega\vec{F}_{BB},$
(2wxzafaoarasatauavawayazbabcbdbebfblbnch)

and the corresponding Hamilton equations can be written as

$i\frac{\partial\vec{F}_{BB}}{\partial t}=\frac{\delta
K_{BB}}{\delta\vec{F}_{BB}^{\star}}=\Omega\vec{F}_{BB}.$
(2wxzafaoarasatauavawayazbabcbdbebfblbnci)

Using the six-component notation $\Psi_{BB}$ the last two equations are
equivalent to

$\displaystyle K_{BB}=\int_{\mathbb{R}^{3}}d^{3}x\
\Psi_{BB}^{\star}\cdot\Omega\Psi_{BB},$
(2wxzafaoarasatauavawayazbabcbdbebfblbncj)

and

$i\frac{\partial\Psi_{BB}}{\partial t}=\frac{\delta
K_{BB}}{\delta\Psi_{BB}^{\star}}=\Omega\Psi_{BB}.$
(2wxzafaoarasatauavawayazbabcbdbebfblbnck)

Remark: The construction introduced here is similar to what is done for the
Dirac equation to remove the negative energy solutions and which led to the
interpretation of anti-matter. However, in our case the negative contributions
are not negative energies since $K_{RS}$ is not the energy of the field. We
remove them because it is more convenient for the quantization as we will see
later, and it allows to make the link with the quantization in the Coulomb
gauge explicit.

###  3.4 Quantization in the Białynicki-Birula representation

We associate to the BB vector, a classical Hilbert space $\mathcal{H}_{BB}$
defined as

$\displaystyle\mathcal{H}_{BB}$
$\displaystyle:=\left\\{\vec{F}_{BB}(\vec{x})\Big{|}\langle\vec{F}_{BB}|\vec{F}_{BB}\rangle_{BB}<\infty\right\\}$
(2wxzafaoarasatauavawayazbabcbdbebfblbncl)

with the following weighted scalar product [22, 23, 24]

$\displaystyle\langle\vec{F}_{BB}|\vec{F}^{\prime}_{BB}\rangle_{BB}=\frac{1}{\hbar}\int_{\mathbb{R}^{3}}d^{3}x\
\vec{F}_{BB}(\vec{x})\cdot\Omega^{-1}\vec{F}^{\prime}_{BB}(\vec{x}).$
(2wxzafaoarasatauavawayazbabcbdbebfblbncm)

The main motivation for the choice of this weighted scalar product is that it
is Lorentz invariant [22, 7, 8, 27]. We will see that it is also an essential
ingredient for the isomorphism with the standard quantization in the Coulomb
gauge.

Remark: Fields with different helicities are orthogonal to each other with
respect to both scalar products $\langle\cdot|\cdot\rangle_{LP}$ and
$\langle\cdot|\cdot\rangle_{BB}$.

Classical electromagnetic fields can be expressed in terms of the BB vector by
inverting equation (2wxzafaoarasatauavawayazbabcbdbebfblbnbq) which gives

$\displaystyle\vec{E}(\vec{x})$
$\displaystyle=\frac{1}{\sqrt{2\varepsilon_{0}}}\left(\vec{F}_{BB}(\vec{x})+\vec{F}_{BB}^{\star}(\vec{x})\right),$
(2wxzafaoarasatauavawayazbabcbdbebfblbncna) $\displaystyle\vec{B}(\vec{x})$
$\displaystyle=\frac{-i}{\sqrt{2\varepsilon_{0}c^{2}}}\Lambda\left(\vec{F}_{BB}(\vec{x})-\vec{F}_{BB}^{\star}(\vec{x})\right).$
(2wxzafaoarasatauavawayazbabcbdbebfblbncnb)

From this classical Hilbert space $\mathcal{H}_{BB}$, one can construct the
bosonic Fock space $\mathbb{F}^{\mathfrak{B}}(\mathcal{H}_{BB})$ by the
general procedure (2wxzafaoaq)-(2wxzafaoarasb). The quantum observables
associated to the classical physical quantities are then obtained by the
following correspondence principle in analogy with what was done for the
Coulomb gauge and with the harmonic oscillator structure of the Hamiton
function. Indeed, one can develop the BB vector into

$\displaystyle\vec{F}_{BB}(\vec{x})$
$\displaystyle=\int_{\mathbb{R}^{3}}d^{3}k\sum_{\sigma=\pm}\vec{g}_{\vec{k},\sigma}(\vec{x})z_{BB}(\vec{k},\sigma),\;\;\;\;z_{BB}(\vec{k},\sigma)=\langle\vec{g}_{\vec{k},\sigma}|\vec{F}_{BB}\rangle_{BB},$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncoa)
$\displaystyle\vec{F}^{\star}_{BB}(\vec{x})$
$\displaystyle=\int_{\mathbb{R}^{3}}d^{3}k\sum_{\sigma=\pm}\vec{g}^{\star}_{\vec{k},\sigma}(\vec{x})z^{\star}_{BB}(\vec{k},\sigma),$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncob)

where the functions $\\{\vec{g}_{\vec{k},\sigma}\\}$ are an orthonormal basis
of $\mathcal{H}_{BB}$, which can be taken as
$\vec{g}_{\vec{k},\sigma}=i\sqrt{\hbar\omega_{\vec{k}}}\vec{\phi}_{\vec{k},\sigma}$.
The quantization map of the BB representation is thus defined by

$\displaystyle z_{BB}$
$\displaystyle\mapsto\hat{C}_{\vec{g}_{\vec{k},\sigma}},$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpa) $\displaystyle z^{\star}_{BB}$
$\displaystyle\mapsto\hat{C}^{\dagger}_{\vec{g}_{\vec{k},\sigma}},$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpb)

where $\hat{C}_{\vec{g}_{\vec{k},\sigma}}$ and
$\hat{C}^{\dagger}_{\vec{g}_{\vec{k},\sigma}}$ are creation-anihilation
operators in the BB Fock space $\mathbb{F}^{\mathfrak{B}}(\mathcal{H}_{BB})$,
defined by the general construction (2wxzafaoara) and (2wxzafaoarb). We define
also BB field operators as

$\displaystyle\vec{\hat{F}}_{BB}(\vec{x})$
$\displaystyle:=\int_{\mathbb{R}^{3}}d^{3}k\sum_{\sigma=\pm}\vec{g}_{\vec{k},\sigma}(\vec{x})\hat{C}_{\vec{g}_{\vec{k},\sigma}},$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqa)
$\displaystyle\vec{\hat{F}}^{\dagger}_{BB}(\vec{x})$
$\displaystyle:=\int_{\mathbb{R}^{3}}d^{3}k\sum_{\sigma=\pm}\vec{g}^{\star}_{\vec{k},\sigma}(\vec{x})\hat{C}^{\dagger}_{\vec{g}_{\vec{k},\sigma}}.$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqb)

Quantized electromagnetic observables are given directly by the operators

$\displaystyle\vec{\hat{E}}(\vec{x})$
$\displaystyle=\frac{1}{\sqrt{2\varepsilon_{0}}}\left(\vec{\hat{F}}_{BB}(\vec{x})+\vec{\hat{F}}^{\dagger}_{BB}(\vec{x})\right),$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcra)
$\displaystyle\vec{\hat{B}}(\vec{x})$
$\displaystyle=\frac{-i}{\sqrt{2\varepsilon_{0}c^{2}}}\Lambda\left(\vec{\hat{F}}_{BB}(\vec{x})-\vec{\hat{F}}^{\dagger}_{BB}(\vec{x})\right).$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrb)

One can check that it corresponds to the quantization proposed by BB, as
written e.g. in [23] by computing the form of the RS field operator:

$\displaystyle\vec{\hat{F}}_{RS}$
$\displaystyle=\sqrt{\frac{\varepsilon_{0}}{2}}\left(\vec{\hat{E}}+ic\vec{\hat{B}}\right)=\vec{\hat{F}}_{BB}^{(h+)}+\vec{\hat{F}}_{BB}^{(h-){\dagger}}$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcsa) $\displaystyle=\int
d^{3}k\sum_{\sigma=\pm}\left[\left(\frac{1+\Lambda}{2}\right)\vec{g}_{\vec{k},\sigma}(\vec{x})\hat{C}_{\vec{g}_{\vec{k},\sigma}}+\left(\frac{1-\Lambda}{2}\right)\vec{g}_{\vec{k},\sigma}^{\star}(\vec{x})\hat{C}^{\dagger}_{\vec{g}_{\vec{k},\sigma}}\right]$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcsb) $\displaystyle=\int
d^{3}k\sum_{\sigma=\pm}\left[\mathbb{P}^{(h+)}\vec{g}_{\vec{k},\sigma}(\vec{x})\hat{C}_{\vec{g}_{\vec{k},\sigma}}+\mathbb{P}^{(h-)}\vec{g}_{\vec{k},\sigma}^{\star}(\vec{x})\hat{C}^{\dagger}_{\vec{g}_{\vec{k},\sigma}}\right]$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcsc) $\displaystyle=\int
d^{3}k\left[\vec{g}_{\vec{k},+}(\vec{x})\hat{C}_{\vec{g}_{\vec{k},+}}+\vec{g}_{\vec{k},-}^{\star}(\vec{x})\hat{C}^{\dagger}_{\vec{g}_{\vec{k},-}}\right]$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcsd) $\displaystyle=\int
d^{3}k\left[i\sqrt{\frac{\hbar\omega_{\vec{k}}}{(2\pi)^{3}}}\vec{\epsilon}_{+}(\vec{k})e^{i\vec{k}\cdot\vec{x}}\hat{C}_{\vec{g}_{\vec{k},+}}+i\sqrt{\frac{\hbar\omega_{\vec{k}}}{(2\pi)^{3}}}\vec{\epsilon}_{-}(\vec{k})^{\star}e^{-i\vec{k}\cdot\vec{x}}\hat{C}^{\dagger}_{\vec{g}_{\vec{k},-}}\right]$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcse) $\displaystyle=\int
d^{3}k\sqrt{\frac{\hbar\omega_{\vec{k}}}{(2\pi)^{3}}}\vec{e}(\vec{k})\left[e^{i\vec{k}\cdot\vec{x}}\hat{a}(\vec{k})+e^{-i\vec{k}\cdot\vec{x}}\hat{b}^{\dagger}(\vec{k})\right],$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcsf)

where we have used the notation
$\vec{e}(\vec{k})=\vec{\epsilon}_{+}(\vec{k})=\vec{\epsilon}_{-}(\vec{k})^{\star}$
and the creation-anihilation operators are linked through
$\hat{a}(\vec{k})=i\hat{C}_{\vec{g}_{\vec{k},+}}$ and
$\hat{b}^{\dagger}(\vec{k})=i\hat{C}^{\dagger}_{\vec{g}_{\vec{k},+}}$. The
expression (2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcsf) coincides
with [23, equation (7)] which confirms that the quantization we have defined
in
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpa)-(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpb)
coincides with the one used by BB e.g. in [22, 23, 24, 34, 35, 36, 37].

The representation of the energy observable and the generator of the time
evolution will be discussed in Section 5.

##  4 Equivalence between the LP and the BB quantizations — isomorphism

In Section 2.5, we have shown that the Coulomb gauge quantizations in position
and momentum representations are equivalent and linked through explicit
relations given by the classical isomorphism $\mathcal{M}$. In the same
spirit, we will now introduce a classical isomorphism $\mathcal{I}$ between
the LP and the BB Hilbert spaces in order to show that the two constructed
quantized theories are equivalent. We will also show how one can pass from one
theory to the other through explicit transformations involving the classical
isomorphism.

We start by defining the map

$\displaystyle\mathcal{I}:\mathcal{H}$ LP
$\displaystyle\rightarrow\mathcal{H}_{BB}$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcsct)
$\displaystyle\vec{\psi}$
$\displaystyle\mapsto\vec{F}_{BB}=\mathcal{I}\vec{\psi}:=i\sqrt{\hbar}\Omega^{1/2}\vec{\psi},$

which is extended to the Fock space by also defining
$\mathcal{I}|\varnothing\rangle_{LP}=|\varnothing\rangle_{BB}$. The inverse
transformation is given by

$\displaystyle\mathcal{I}^{-1}:\mathcal{H}$ BB
$\displaystyle\rightarrow\mathcal{H}_{LP}$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscu)
$\displaystyle\vec{F}_{BB}$
$\displaystyle\mapsto\vec{\psi}=\mathcal{I}^{-1}\vec{F}_{BB}:=-\frac{i}{\sqrt{\hbar}}\Omega^{-1/2}\vec{F}_{BB}.$

One can check that $\mathcal{I}$ indeed links the two representations by
computing directly

$\displaystyle i\sqrt{\hbar}\Omega^{1/2}\vec{\psi}$
$\displaystyle=\frac{i}{\sqrt{2}}\Omega^{1/2}\left(\sqrt{\varepsilon_{0}}\Omega^{1/2}\vec{A}-i\sqrt{\varepsilon_{0}}\Omega^{-1/2}\vec{E}\right)$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscva)
$\displaystyle=i\sqrt{\frac{\varepsilon_{0}}{2}}\left(\Omega\vec{A}-i\vec{E}\right)$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscvb)
$\displaystyle=\sqrt{\frac{\varepsilon_{0}}{2}}\left(ic\Lambda\nabla\times\vec{A}+\vec{E}\right)$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscvc)
$\displaystyle=\frac{1}{\sqrt{2}}\left(\sqrt{\varepsilon_{0}}\vec{E}+\frac{i}{\sqrt{\mu_{0}}}\Lambda\vec{B}\right)$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscvd)
$\displaystyle=\frac{1}{\sqrt{2}}\left(\vec{\mathcal{Q}}_{BB}+i\vec{\mathcal{P}}_{BB}\right)=\vec{F}_{BB},$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscve)

where we have used the relation $\Omega=c\Lambda\nabla\times$ and
$\Lambda^{-1}=\Lambda$. The equivalence of their respective scalar products
follows immedialtely from the definitions

$\displaystyle\langle\mathcal{I}\vec{\psi}|\mathcal{I}\vec{\psi}^{\prime}\rangle_{BB}$
$\displaystyle=\frac{1}{\hbar}\int_{\mathbb{R}^{3}}d^{3}x\
(\mathcal{I}\vec{\psi})^{\star}\cdot\Omega^{-1}\mathcal{I}\vec{\psi}^{\prime}$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscvcwa)
$\displaystyle=\int_{\mathbb{R}^{3}}d^{3}x\
\Omega^{1/2}\vec{\psi}^{\star}\cdot\Omega^{-1}\Omega^{1/2}\vec{\psi}^{\prime}$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscvcwb)
$\displaystyle=\int_{\mathbb{R}^{3}}d^{3}x\
\vec{\psi}^{\star}\cdot\vec{\psi}^{\prime}=\langle\vec{\psi}|\vec{\psi}^{\prime}\rangle_{LP},$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscvcwc)

where we have used the selfadjoint property of $\Omega^{1/2}$ to obtain
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscvcwc).

We can express the link between the creation-anihilation operators of both
representations by starting e.g. with a single-photon state in the LP
representation

$\displaystyle\hat{B}_{\vec{\psi}}^{\dagger}|\varnothing\rangle_{LP}=|\vec{\psi}\rangle_{LP},$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscvcwcx)

which can be rewritten using the isomorphism $\mathcal{I}$ as

$\displaystyle|\vec{\psi}\rangle_{LP}$
$\displaystyle=|\mathcal{I}^{-1}\vec{F}_{BB}\rangle_{LP}$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscvcwcya)
$\displaystyle=\mathcal{I}^{-1}|\vec{F}_{BB}\rangle_{BB}$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscvcwcyb)
$\displaystyle=\mathcal{I}^{-1}\hat{C}_{\vec{F}_{BB}}^{\dagger}|\varnothing\rangle_{BB}$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscvcwcyc)
$\displaystyle=\mathcal{I}^{-1}\hat{C}_{\vec{F}_{BB}}^{\dagger}\mathcal{I}|\varnothing\rangle_{LP}.$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscvcwcyd)

From this one can identify

$\displaystyle\hat{B}^{\dagger}_{\vec{\psi}}=\mathcal{I}^{-1}\hat{C}^{\dagger}_{\vec{F}_{BB}}\mathcal{I}=\mathcal{I}^{-1}\hat{C}^{\dagger}_{\mathcal{I}\vec{\psi}}\mathcal{I},$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscvcwcycz)

and therefore the other relations are

$\displaystyle\hat{C}^{\dagger}_{\vec{F}_{BB}}=\mathcal{I}\hat{B}^{\dagger}_{\vec{\psi}}\mathcal{I}^{-1},\>\>\>\>\>\>\>\>\>\hat{B}_{\vec{\psi}}=\mathcal{I}^{-1}\hat{C}_{\vec{F}_{BB}}\mathcal{I},\>\>\>\>\>\>\>\>\>\hat{C}_{\vec{F}_{BB}}=\mathcal{I}\hat{B}_{\vec{\psi}}\mathcal{I}^{-1}.$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscvcwcyda)

The equivalence of electromagnetic quantities can be verified for instance for
the electric field

$\displaystyle\mathcal{I}\vec{\hat{E}}_{LP}\mathcal{I}^{-1}$
$\displaystyle=i\sqrt{\frac{\hbar}{2\varepsilon_{0}}}\int_{\mathbb{R}^{3}}d^{3}k\sum_{\sigma=\pm}\omega_{\vec{k}}^{1/2}\left(\vec{\phi}_{\vec{k},\sigma}(\vec{x})\mathcal{I}\hat{B}_{\vec{\phi}_{\vec{k},\sigma}}\mathcal{I}^{-1}-\vec{\phi}^{\star}_{\vec{k},\sigma}(\vec{x})\mathcal{I}\hat{B}^{\dagger}_{\vec{\phi}_{\vec{k},\sigma}}\mathcal{I}^{-1}\right)$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscvcwcydbb)
$\displaystyle=i\sqrt{\frac{\hbar}{2\varepsilon_{0}}}\int_{\mathbb{R}^{3}}d^{3}k\sum_{\sigma=\pm}\omega_{\vec{k}}^{1/2}\left(\vec{\phi}_{\vec{k},\sigma}(\vec{x})\hat{C}_{\mathcal{I}\vec{\phi}_{\vec{k},\sigma}}-\vec{\phi}^{\star}_{\vec{k},\sigma}(\vec{x})\hat{C}^{\dagger}_{\mathcal{I}\vec{\phi}_{\vec{k},\sigma}}\right)$
$\displaystyle=\frac{1}{\sqrt{2\varepsilon_{0}}}\int_{\mathbb{R}^{3}}d^{3}k\sum_{\sigma=\pm}\left(\vec{g}_{\vec{k},\sigma}(\vec{x})\hat{C}_{\vec{g}_{\vec{k},\sigma}}+\vec{g}^{\star}_{\vec{k},\sigma}(\vec{x})\hat{C}^{\dagger}_{\vec{g}_{\vec{k},\sigma}}\right)$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscvcwcydbc)
$\displaystyle=\frac{1}{\sqrt{2\varepsilon_{0}}}\left(\vec{\hat{F}}_{BB}(\vec{x})+\vec{\hat{F}}^{\dagger}_{BB}(\vec{x})\right)\equiv\vec{\hat{E}}_{BB}(\vec{x}),$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscvcwcydbd)

where we have used the basis
$\vec{g}_{\vec{k},\sigma}=i\sqrt{\hbar\omega_{\vec{k}}}\vec{\phi}_{\vec{k},\sigma}$
of $\mathcal{H}_{BB}$ and we have identified $\vec{\hat{F}}_{BB}$ and
$\vec{\hat{F}}_{BB}^{\dagger}$ by their definitions
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqa),(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqb).
This expression indeed coincides with
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcra) for the electric field.

However, one has to be careful when transforming other objects like the field
operators $\vec{\hat{\Psi}}$ and $\vec{\hat{F}}_{BB}$. Indeed,
$\vec{\hat{F}}_{BB}\neq\mathcal{I}\vec{\hat{\Psi}}\mathcal{I}^{-1}$ since if
one decomposes the LP field operator in terms of creation-anihilation
operators, it yields

$\displaystyle\mathcal{I}\vec{\hat{\Psi}}(\vec{x})\mathcal{I}^{-1}$
$\displaystyle=\int_{\mathbb{R}^{3}}d^{3}k\sum_{\sigma=\pm}\vec{\phi}_{\vec{k},\sigma}(\vec{x})\mathcal{I}\hat{B}_{\vec{\phi}_{\vec{k},\sigma}}\mathcal{I}^{-1}$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscvcwcydbdca)
$\displaystyle=\int_{\mathbb{R}^{3}}d^{3}k\sum_{\sigma=\pm}\vec{\phi}_{\vec{k},\sigma}(\vec{x})\hat{C}_{\vec{g}_{\vec{k},\sigma}}$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscvcwcydbdcb)
$\displaystyle=-i\int_{\mathbb{R}^{3}}d^{3}k\sum_{\sigma=\pm}(\hbar\omega_{\vec{k}})^{-1/2}\vec{g}_{\vec{k},\sigma}(\vec{x})\hat{C}_{\vec{g}_{\vec{k},\sigma}}$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscvcwcydbdcc)
$\displaystyle=\frac{-i}{\sqrt{\hbar}}\Omega^{-1/2}\vec{\hat{F}}_{BB}.$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscvcwcydbdcd)

##  5 Dynamics and BB quantum Hamiltonian

The Hamiltonian in a quantum theory plays two roles: it is the operator
associated to the total energy observable of the system and it is the
generator of the dynamics of the states in the Schrödinger representation. The
construction of the dynamics within the Fock space formulation is done as
follows: one starts with the one-quantum subspace, which coincides with the
classical Hilbert space $\mathcal{H}$, and whose time evolution is given by
the classical one. From this, a one-quantum operator is identified as the
generator of the dynamics. For Maxwell’s equations, the generator of the
dynamics is $\hbar\Omega$ for all representations we discussed before ($\hbar$
has been added here in relation to (2wxzae),
(2wxzafaoarasatauavawayazbabcbdbebfblbnci) or
(2wxzafaoarasatauavawayazbabcbdbebfblbnck) in order to match the dimension
required by a Schrödinger equation in the quantized model). The next step is
to extend this operator $\hbar\Omega$, which is defined in the single-photon
subspace, to the entire Fock space $\mathbb{F}^{\mathfrak{B}}(\mathcal{H})$.
This can be done in general with the map denoted $\rmd\Gamma(\hbar\Omega)$
[14] and defined by its action on the monomials
$|\eta_{1}\otimes_{S}\eta_{2}\otimes_{S}\ldots\otimes_{S}\eta_{n}\rangle$:

$\displaystyle\rmd\Gamma(\hbar\Omega)~{}|\eta_{1}\otimes_{S}\eta_{2}\otimes_{S}\ldots\otimes_{S}\eta_{n}\rangle$
$\displaystyle:=|\hbar\Omega\eta_{1}\otimes_{S}\eta_{2}\otimes_{S}\ldots\otimes_{S}\eta_{n}\rangle$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscvcwcydbdcdd)
$\displaystyle\>\>+|\eta_{1}\otimes_{S}\hbar\Omega\eta_{2}\otimes_{S}\ldots\otimes_{S}\eta_{n}\rangle$
$\displaystyle\>\>+\dots+$ $\displaystyle\>\>+|\eta_{1}\otimes_{S}\
\eta_{2}\otimes_{S}\ldots\otimes_{S}\hbar\Omega\eta_{n}\rangle.$

If $\phi_{\kappa}$ is an orthonormal basis, with respect to
$\langle\cdot|\cdot\rangle_{\mathcal{H}}$, of eigenfunctions of
$\Omega\phi_{\kappa}=\omega_{\kappa}\phi_{\kappa}$, the map
$\rmd\Gamma(\hbar\Omega)$ can be expressed as [14, p. 440]

$\rmd\Gamma(\hbar\Omega)=\sum_{\kappa}\hbar\omega_{\kappa}\hat{B}^{\dagger}_{\phi_{\kappa}}\hat{B}_{\phi_{\kappa}}.$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscvcwcydbdcde)

We apply this procedure to the different Hilbert spaces defined in the
preceding sections starting with the LP representation

$\displaystyle\rmd\Gamma_{LP}(\hbar\Omega)$
$\displaystyle=\int_{\mathbb{R}^{3}}d^{3}k\sum_{\sigma=\pm}\hbar\omega_{\vec{k}}\hat{B}^{\dagger}_{\vec{\phi}_{\vec{k},\sigma}}\hat{B}_{\vec{\phi}_{\vec{k},\sigma}}.$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscvcwcydbdcdfa)

We see that it is equal to the quantized total energy operator which can be
written as

$\displaystyle\hat{H}_{LP}$ $\displaystyle=\hbar\int_{\mathbb{R}^{3}}d^{3}x\
\vec{\hat{\Psi}}^{{\dagger}}\cdot\Omega\vec{\hat{\Psi}}=\int_{\mathbb{R}^{3}}d^{3}k\sum_{\sigma=\pm}\hbar\omega_{\vec{k}}\hat{B}^{\dagger}_{\vec{\phi}_{\vec{k},\sigma}}\hat{B}_{\vec{\phi}_{\vec{k},\sigma}},$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscvcwcydbdcdfdg)

showing that the total energy is the generator of the dynamics in the Coulomb
gauge formulation of the quantized electromagnetic field.

One can apply the same procedure to the BB formulation using the basis
$\\{\vec{g}_{\vec{k},\sigma}\\}$ and one obtains

$\displaystyle\rmd\Gamma_{BB}(\hbar\Omega)$
$\displaystyle=\int_{\mathbb{R}^{3}}d^{3}k\sum_{\sigma=\pm}\hbar\omega_{\vec{k}}\hat{C}^{\dagger}_{\vec{g}_{\vec{k},\sigma}}\hat{C}_{\vec{g}_{\vec{k},\sigma}},$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscvcwcydbdcdfdha)

which coincides with the total energy

$\displaystyle\hat{\mathcal{E}}_{tot}$
$\displaystyle=\int_{\mathbb{R}^{3}}d^{3}x\
\vec{\hat{F}}_{BB}^{{\dagger}}\cdot\vec{\hat{F}}_{BB}$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscvcwcydbdcdfdhdia)
$\displaystyle=\sum_{\kappa,\kappa^{\prime}}\left(\int_{\mathbb{R}^{3}}d^{3}x\
\vec{g}^{\star}_{\kappa}(\vec{x})\cdot\vec{g}_{\kappa^{\prime}}(\vec{x})\right)\hat{C}^{\dagger}_{\vec{g}_{\kappa}}\hat{C}_{\vec{g}_{\kappa^{\prime}}}$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscvcwcydbdcdfdhdib)
$\displaystyle=\hbar\sum_{\kappa,\kappa^{\prime}}\langle\vec{g}_{\kappa}|\Omega\vec{g}_{\kappa^{\prime}}\rangle_{BB}\
\hat{C}^{\dagger}_{\vec{g}_{\kappa}}\hat{C}_{\vec{g}_{\kappa^{\prime}}}$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscvcwcydbdcdfdhdic)
$\displaystyle=\sum_{\kappa}\hbar\omega_{\kappa}\
\hat{C}^{\dagger}_{\vec{g}_{\kappa}}\hat{C}_{\vec{g}_{\kappa}}\equiv\hat{H}_{BB},$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscvcwcydbdcdfdhdid)

where we have introduced the more compact notations
$\kappa\equiv(\vec{k},\sigma)$ and
$\sum_{\kappa}\equiv\int_{\mathbb{R}^{3}}d^{3}k\sum_{\sigma=\pm}$. This
confirms that in the quantized theory, the generator of the dynamics is the
quantized operator of the total energy in the BB representation and not the
quantization of the classical Hamilton function $K_{BB}$
(2wxzafaoarasatauavawayazbabcbdbebfblbnbp). The time evolution of the states
is given by the unitary operator $\hat{U}_{BB}(t)=e^{-i\hat{H}_{BB}t}$, and it
leads e.g. when applied to single-photon states to the expressions of the form
[22, equation (4.20)]. One can further check that the Hamiltonians written in
their respective representations can be recovered from one to the other using
the isomorphism (see
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscvcwcydbdcd))

$\displaystyle\mathcal{I}\hat{H}_{LP}\mathcal{I}^{-1}$
$\displaystyle=\hbar\int_{\mathbb{R}^{3}}d^{3}x\
\mathcal{I}\vec{\hat{\Psi}}^{{\dagger}}\mathcal{I}^{-1}\cdot\Omega\mathcal{I}\vec{\hat{\Psi}}\mathcal{I}^{-1}$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscvcwcydbdcdfdhdidja)
$\displaystyle=\int_{\mathbb{R}^{3}}d^{3}x\
\Omega^{-1/2}\vec{\hat{F}}_{BB}^{{\dagger}}\cdot\Omega^{1/2}\vec{\hat{F}}_{BB}$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscvcwcydbdcdfdhdidjb)
$\displaystyle=\int_{\mathbb{R}^{3}}d^{3}x\
\vec{\hat{F}}_{BB}^{{\dagger}}\cdot\vec{\hat{F}}_{BB}=\hat{H}_{BB}.$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscvcwcydbdcdfdhdidjc)

The equivalence of the generators of the dynamics guarantees that the
isomorphism is preserved during the time evolution and that the two theories
are completely equivalent.

##  6 Concluding remarks

  * •

In the literature, the relation between the LP field and the RS vector has
been discussed for instance in [22, section 5.3] where it was stated that the
two objects can be linked but only from BB to LP i.e. for every
$\vec{F}_{BB}\in\mathcal{H}_{BB}$, there should exist a well defined
$\vec{\psi}\in\mathcal{H}_{LP}$ but a doubt was expressed about the reverse.
The singularity of the inverse transformation and the nonintegrability of a
kernel of the form $|\vec{x}-\vec{x}^{\prime}|^{-7/2}$ could prevent a
bijection between the two representations. Our present formulation of the
isomorphism shows that there is no such difficulty. Our argument is based on
the fact that $\Omega$ is a positive selfadjoint operator and both
$\Omega^{\pm 1/2}$ are well defined and selfadjoint in the Hilbert space.
Concerning the representation as integral operators, it was shown e.g. in [38,
39, 40, 41] that they can be written for any field $\vec{v}(\vec{x})$ as

$\left[\Omega^{-1/2}\vec{v}\right](\vec{x})=\frac{\pi}{\sqrt{c}}\int_{\mathbb{R}^{3}}d^{3}x^{\prime}\frac{\vec{v}(\vec{x}^{\prime})}{\left(2\pi|\vec{x}-\vec{x}^{\prime}|\right)^{5/2}},$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscvcwcydbdcdfdhdidjdk)

and

$\left[\Omega^{1/2}\vec{v}\right](\vec{x})=\frac{3\sqrt{2c}}{16\pi^{3/2}}\mathcal{P}\int_{\mathbb{R}^{3}}d^{3}x^{\prime}\frac{\vec{v}(\vec{x})-\vec{v}(\vec{x}^{\prime})}{|\vec{x}-\vec{x}^{\prime}|^{7/2}},$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscvcwcydbdcdfdhdidjdl)

where $\mathcal{P}$ stands for the Cauchy principal value. Consequently, the
isomorphism guarantees a one-to-one correspondence between the states from one
Hilbert space to the other.

  * •

In a recent article [42], the concept of fidelity for photons was discussed
using the BB quantization in both position and momentum space representations.
The BB momentum representation can be written from the BB position
representation as follows

$\displaystyle\tilde{f}_{m}(\vec{k},\sigma)$
$\displaystyle:=\sqrt{\frac{(2\pi)^{3}}{\hbar c}}\int_{\mathbb{R}^{3}}d^{3}x\
\vec{\phi}_{\vec{k},\sigma}^{\star}(\vec{x})\cdot\vec{F}_{BB}(\vec{x}),$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscvcwcydbdcdfdhdidjdma)
$\displaystyle\vec{F}_{BB}(\vec{x})$ $\displaystyle:=\sqrt{\frac{\hbar
c}{(2\pi)^{3}}}\int_{\mathbb{R}^{3}}d^{3}k\sum_{\sigma=\pm}\tilde{f}_{m}(\vec{k},\sigma)\vec{\phi}_{\vec{k},\sigma}(\vec{x}),$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscvcwcydbdcdfdhdidjdmb)

and it defines a BB momentum Hilbert space

$\mathcal{H}_{BBm}:=\\{\tilde{f}_{m}(\vec{k},\sigma)|\langle\tilde{f}_{m}|\tilde{f}_{m}\rangle_{BBm}<\infty\\},$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscvcwcydbdcdfdhdidjdmdn)

with the BB momentum weighted scalar product [42, equation (2)]

$\langle\tilde{f}_{m1}|\tilde{f}_{m2}\rangle_{BBm}:=\int_{\mathbb{R}^{3}}\frac{d^{3}k}{(2\pi)^{3}|\vec{k}|}\sum_{\sigma=\pm}\tilde{f}_{m1}^{\star}(\vec{k},\sigma)\tilde{f}_{m2}(\vec{k},\sigma).$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscvcwcydbdcdfdhdidjdmdo)

One can check that equations
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscvcwcydbdcdfdhdidjdma),(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscvcwcydbdcdfdhdidjdmb)
and (2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscvcwcydbdcdfdhdidjdmdn)
provide a Hilbert space isomorphism between $\mathcal{H}_{BB}$ and
$\mathcal{H}_{BBm}$. Consequently, if one defines the fidelity in momentum
space to be [42, equation (18)])

$\mathfrak{F}_{m}:=\frac{|\langle\tilde{f}_{m1}|\tilde{f}_{m2}\rangle_{BBm}|^{2}}{\langle\tilde{f}_{m1}|\tilde{f}_{m1}\rangle_{BBm}\langle\tilde{f}_{m2}|\tilde{f}_{m2}\rangle_{BBm}},$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscvcwcydbdcdfdhdidjdmdp)

the isomorphism implies immediately that

$\displaystyle\mathfrak{F}_{m}=\frac{|\langle\vec{F}_{BB1}|\vec{F}_{BB2}\rangle_{BB}|^{2}}{\langle\vec{F}_{BB1}|\vec{F}_{BB1}\rangle_{BB}\langle\vec{F}_{BB2}|\vec{F}_{BB2}\rangle_{BB}}=:\mathfrak{F}_{BB},$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscvcwcydbdcdfdhdidjdmdq)

where $\mathfrak{F}_{BB}$ is the fidelity in position space. In [42], a
comparison is made with another conceivable definition of fidelity

$\displaystyle\mathfrak{F}^{\prime}_{BB}:=\frac{|\langle\vec{F}_{BB1}|\vec{F}_{BB2}\rangle|^{2}}{\langle\vec{F}_{BB1}|\vec{F}_{BB1}\rangle\langle\vec{F}_{BB2}|\vec{F}_{BB2}\rangle},$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscvcwcydbdcdfdhdidjdmdr)

with a scalar product $\langle\cdot|\cdot\rangle$ without the weight
$\Omega^{-1}$

$\langle\vec{F}_{BB1}|\vec{F}_{BB2}\rangle:=\int_{\mathbb{R}^{3}}d^{3}x\
\vec{F}_{BB1}^{\star}(\vec{x})\cdot\vec{F}_{BB2}(\vec{x})\neq\langle\vec{F}_{BB1}|\vec{F}_{BB2}\rangle_{BB}.$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscvcwcydbdcdfdhdidjdmds)

The results discussed in the present work indicate that $\mathfrak{F}_{m}$ and
$\mathfrak{F}_{BB}$ should be taken as the measure of fidelity and not
$\mathfrak{F}^{\prime}_{BB}$. The main point is that the scalar product of the
Hilbert space $\mathcal{H}_{BB}$ should be used if one works in the BB
representation. Furthemore, through the isomorphism
$\mathcal{I}:\mathcal{H}_{LP}\rightarrow\mathcal{H}_{BB}$ the fidelities
$\mathfrak{F}_{m}$ and $\mathfrak{F}_{BB}$ coincide with the one defined in
the LP as well as in the standard momentum representation.

If one uses the Hilbert space $\mathcal{H}_{BB}$ that we defined in
(2wxzafaoarasatauavawayazbabcbdbebfblbncl), the fidelity for photons is the
same whether it is measured in position or momentum space as expected for a
physical property which should always be independent of the chosen
representation for the states.

  * •

The isomorphism between the LP and the BB representations can also be
interpreted in a reversed way. If we start with the LP representation it has
two weak features: it relays on the choice of a particular gauge and it is not
manifestly Lorentz invariant. The isomorphism $\mathcal{I}$ provides a simple
way to modify the formulation to obtain a gauge independent and manifestly
Lorentz invariant representation.

  * •

The relation between $\vec{F}_{BB}$ and $\vec{F}_{RS}$ (and thus also with
$\vec{E}$ and $\vec{B}$) is non-local, since it involves the projection into
the positive and negative helicity parts, which is a non-local operation. The
non-local character, which was considered a weak feature of the LP
representation, is thus also present in the BB representation. One might
surmise that it is likely a general feature of the quantum field theories for
photons.

  * •

The field $\Omega^{1/2}\vec{\psi}$ had been already introduced in the book by
L. Mandel and E. Wolf [15, equation (12.11-32) ], and called the “energy wave
function”, as well as in [4, equation (2.15)]. This denomination has been also
adopted for the six component vector $\Psi_{BB}$ in [7, eq (10.8)],[8, p.3].
These denominations are however somewhat misleading. As it was shown in [23,
equation (12)], the mean value of the local energy density can be expressed as

$\displaystyle\langle\hat{\mathcal{E}}_{elm}\rangle$
$\displaystyle=|\vec{F}_{RS}^{(h+)}|^{2}+|\vec{F}_{RS}^{(h-)}|^{2}=|\vec{F}_{BB}^{(h+)}|^{2}+|\vec{F}_{BB}^{(h-)}|^{2}$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscvcwcydbdcdfdhdidjdmdta)
$\displaystyle=|\Psi_{BB}|^{2}.$
(2wxzafaoarasatauavawayazbabcbdbebfblbncncocpcqcrcscvcwcydbdcdfdhdidjdmdtb)

However, $|\Psi_{BB}|^{2}$ is not a Born rule probability density for the
local energy observable. It appears naturally in the mean value of the local
energy, but it is not its probability. Since the concept of wave function
carries most often the connotation of a Born rule probability, we prefer to
use the denomination Białynicki-Birula bispinor for $\Psi_{BB}$ and
Białynicki-Birula vector  for $\vec{F}_{BB}$.

##  7 Conclusion

In this article, we have explored different approaches of the quantization of
the free electromagnetic field in position space that use the LP field and the
BB vector. The key result was to show that they are linked by an isomorphism
of their respective Hilbert spaces that we explicitely wrote using the
frequency and the helicity operators. With this relation we were able to show
that the quantum theories built from them through a Fock space quantization
are equivalent, meaning that they predict the same physical results. This
equivalence is preserved during the time evolution. In the literature, both
the LP field and BB vectors were used for different practical problems and
using the results of the present work, any calculation made from one
representation can be easily translated into the other using the isomorphism
$\mathcal{I}$.

This work was supported by the “Investissements d’Avenir” program, project
ISITE-BFC / IQUINS (ANR-15-IDEX-03), QUACO-PRC (ANR-17-CE40-0007-01) and the
EUR-EIPHI Graduate School (17-EURE-0002). We also acknowledge support from the
European Union’s Horizon 2020 research and innovation program under the Marie
Sklodowska-Curie grant agreement No. 765075 (LIMQUET).

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