however. For example, the operators $\widetilde{a}_{s\lambda}(k,t)$ and
$\widetilde{a}_{-s\lambda}(-k,t)$ both describe photons whose coherent states
have exactly the same electric field expectation values. The two operators
evolve with different dynamics, however, and whereas the
$\widetilde{a}_{s\lambda}(k,t)$ operators describes photons propagating in the
$s$ direction, the $\widetilde{a}_{-s\lambda}(-k,t)$ operators describe
photons propagating in the $-s$ direction. Although the standard
$a_{k\lambda}$ operators can be used to construct wave packets with electric
field expectation values of any shape, they cannot be used to construct
localised wave packets of an arbitrary shape that propagate at the speed of
light in a fixed direction.

###### Photon states

To construct the relevant Hilbert space of the quantised EM field, we start
again by defining a vacuum state $\ket{0}$. Applying any annihilation operator
to the vacuum state destroys the state:

$\displaystyle\widetilde{a}_{s\lambda}(k,t)\,\ket{0}$ $\displaystyle=$
$\displaystyle 0\,.$ (4.33)

By applying both the left- and right-hand sides of Eq. (4.32) to the vacuum
state, one can see that the vacuum states in the position and momentum
representations coincide.

By applying a creation operator $\widetilde{a}^{\dagger}_{s\lambda}(k,t)$ to
the vacuum state an arbitrary number of times we may generate the full
symmetric Fock space of states containing any number of photons. A state
containing $n$ identical photons is denoted

$\displaystyle\ket{n_{s\lambda}(k,t)}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{n!}}\,{\widetilde{a}^{\dagger}_{s\lambda}(k,t)}^{n}\ket{0}.$
(4.34)

When $n=1$ we shall refer to these states as single-photon states.

#####  4.3.2 The photon commutation relations

###### Commutation relations

In this section we look at the commutation relations for the photon creation
and annihilation operators $\widetilde{a}^{\dagger}_{s\lambda}(k,t)$ and
$\widetilde{a}_{s\lambda}(k,t)$ defined in the previous section. As photons
are bosons, many-photon states are unchanged when two or more photons are
exchanged. This leads us to the typical commutation relation for bosonic
particles:

$\displaystyle\Big{[}\widetilde{a}_{s\lambda}(k,t),\widetilde{a}_{s^{\prime}\lambda^{\prime}}(k^{\prime},t^{\prime})\Big{]}=\Big{[}\widetilde{a}^{\dagger}_{s\lambda}(k,t),\widetilde{a}^{\dagger}_{s^{\prime}\lambda^{\prime}}(k^{\prime},t^{\prime})\Big{]}=0\,$

for any $s$, $s^{\prime}$, $\lambda$, $\lambda^{\prime}$, $k$, $k^{\prime}$,
$t$ and $t^{\prime}$. These relations are consistent with Eq. (4.13). In order
to determine the commutation relation between photon creation and annihilation
operators we use the Fourier transformations of the photon operators given in
Eq. (4.31). Using the blip commutation relation given in Eq. (4.17) we find
that

$\displaystyle\big{[}\widetilde{a}_{s\lambda}(k,t),\widetilde{a}^{\dagger}_{s^{\prime}\lambda^{\prime}}(k^{\prime},t)\big{]}=\delta(k-k^{\prime})\,\delta_{s,s^{\prime}}\,\delta_{\lambda,\lambda^{\prime}}\,.$
(4.36)

Using this commutator one may also show that single-photon states are
orthogonal to one another as

$\displaystyle\braket{1_{s\lambda}(k,t)}{1_{s^{\prime}\lambda^{\prime}}(k^{\prime},t)}$
$\displaystyle=$
$\displaystyle\braket{0}{\big{[}\widetilde{a}_{s\lambda}(k,t),\widetilde{a}^{\dagger}_{s^{\prime}\lambda^{\prime}}(k^{\prime},t)\big{]}}{0}$
(4.37) $\displaystyle=$
$\displaystyle\delta(k-k^{\prime})\,\delta_{s,s^{\prime}}\,\delta_{\lambda,\lambda^{\prime}}\,.$

As one would expect, the photons that we consider in this section are bosonic
particles.

###### The fundamental equation of motion

By using the Fourier expressions of the blip operators we may also determine a
fundamental equation of motion for the photon operators. By substituting the
right-hand side of Eq. (4.32) into Eq. (4.12) one finds that the photon
operators satisfy the equation

$\frac{\text{d}}{\text{d}t}\widetilde{a}_{s\lambda}(k,t)=-ikc\,\widetilde{a}_{s\lambda}(k,t).$
(4.38)

The solution of this equation is

$\widetilde{a}_{s\lambda}(k,t)=e^{-ikct}\,\widetilde{a}_{s\lambda}(k,0).$
(4.39)

This equation is the usual equation of motion of the quantised EM field in
momentum space and shows that photons oscillate with an angular frequency
$\omega=kc$. Since $k$ now takes all values along the real line, $\omega$ can
now take all real values. As we have seen already and shall see again,
however, this is not a problem because the energy observable is always
positive. This is possible because the energy observable and the generator of
the system’s dynamics are no longer the same.

#####  4.3.3 Field observables in the momentum representation

###### The field observables

In this section we consider how to express the complex electric and magnetic
field observables in their momentum representations. As in Section 2.1, we
denote these fields $\widetilde{\mathbf{E}}(k,t)$ and
$\widetilde{\mathbf{B}}(k,t)$. These fields are three-vector valued and
parametrised by a time $t$ and a wave number $k$. As we have seen in Section
4.2.3, the electric and magnetic field vectors are a linear superposition of
travelling wave solutions with a fixed direction of propagation and
polarisation. As the Fourier transform is linear, the same applies in the
momentum representation. Thus

$\widetilde{\mathbf{O}}(k,t)=\sum_{s=\pm
1}\sum_{\lambda={\mathsf{H},\mathsf{V}}}\,\widetilde{\mathbf{O}}_{s\lambda}(k,t)$
(4.40)

where $\widetilde{\mathbf{O}}=\widetilde{\mathbf{E}},\widetilde{\mathbf{B}}$.
As in Eqs. (4.31) and (4.32), we include a factor of $s$ in the definition of
the wave vector. Hence

$\displaystyle\mathbf{O}_{s\lambda}(x,t)=\int_{-\infty}^{\infty}\frac{\text{d}k}{\sqrt{2\pi}}\;e^{iskx}\,\widetilde{\mathbf{O}}_{s\lambda}(k,t)$
$\displaystyle\widetilde{\mathbf{O}}_{s\lambda}(k,t)=\int_{-\infty}^{\infty}\frac{\text{d}x}{\sqrt{2\pi}}\;e^{-iskx}\,\mathbf{O}_{s\lambda}(x,t).$
(4.41)

Like the photon operators themselves, the field observables in the momentum
representation satisfy the equation

$\frac{\text{d}}{\text{d}t}\widetilde{\mathbf{O}}_{s\lambda}(k,t)=-ikc\,\widetilde{\mathbf{O}}_{s\lambda}(k,t).$
(4.42)

In analogy to standard quantisations of the EM field, in which the system can
be described as a collection of simple harmonic oscillators (see Section 2.1.6
and Ref. [123]), and in agreement with the transformations given in Eqs.
(4.31) and (4.32), we assume that the complex electric and magnetic field
observables can be expressed as a linear combination of the photon
annihilation operators:

$\displaystyle\widetilde{\mathbf{E}}(k,t)$ $\displaystyle=$
$\displaystyle\sum_{s=\pm
1}c\,\Omega(k)\,e^{i\varphi(k)}\left[\widetilde{a}_{s{\sf
H}}(k,t)\,\widehat{\boldsymbol{y}}+\widetilde{a}_{s{\sf
V}}(k,t)\,\widehat{\boldsymbol{z}}\right]\,,$
$\displaystyle\widetilde{\mathbf{B}}(k,t)$ $\displaystyle=$
$\displaystyle\sum_{s=\pm
1}s\,\Omega(k)\,e^{i\varphi(k)}\left[\widetilde{a}_{s{\sf
H}}(k,t)\,\widehat{\boldsymbol{z}}-\widetilde{a}_{s{\sf
V}}(k,t)\,\widehat{\boldsymbol{y}}\right].$ (4.43)

Here $\Omega(k)$ is a $k$-dependent numerical factor. By introducing the
$k$-dependent phase $\varphi(k)$ we may assume that $\Omega(k)$ is real. By
substituting Eq. (4.39) into Eq. (4.3.3) one can show that Eq. (4.42) is
satisfied.

###### The field amplitudes

The equation of motion (4.42), however, does not determine the particular
value of $\Omega(k)\,e^{i\varphi(k)}$. The factor $\Omega(k)$ is a function
that determines the amplitude of the field observables and, therefore, the
energy of a single photon in the $(k,s,\lambda)$ mode. Due to the isotropy of
free space, we can safely assume that $\Omega(k)$ does not have any dependence
on either $s$ or $\lambda$. Furthermore, to ensure the symmetry of the field
excitations, $\Omega(k)$ must be an even function. For the time being we shall
not specify $\Omega(k)$ any further, but shall return to this function in
Section 4.4.1.

#####  4.3.4 Energy in the momentum representation

###### The energy observable

For completeness we now also derive the energy observable of the quantised EM
field $H_{\text{energy}}(t)$ in the momentum representation. Taking as our
starting point the expression for the classical field energy in Eq. (2.21),
and substituting in for the classical fields the position-dependent field
observables in their Fourier representations, one finds that

$\displaystyle H_{\text{energy}}(t)$ $\displaystyle=$
$\displaystyle{\varepsilon_{0}Ac^{2}\over 4}\sum_{s=\pm 1}\sum_{\lambda={\sf
H},{\sf V}}\int_{-\infty}^{\infty}\text{d}k\,$ (4.44)
$\displaystyle\times\big{\|}\Omega(k)\,e^{i\varphi(k)}\,\widetilde{a}_{s\lambda}(k,t)+\Omega^{*}(-k)\,e^{-i\varphi(-k)}\,\widetilde{a}^{\dagger}_{s\lambda}(-k,t)\big{\|}^{2}\,.\leavevmode\nobreak\
\leavevmode\nobreak\ $

The expectation value of the energy observable above is again positive which
is guaranteed by the squared expression in the integrand.

In Eq. (4.44), the energy observable has an explicit dependence on the choice
of phase $\varphi(k)$. When we restrict this theory to positive wave numbers
only, this dependence disappears, but we cannot make that assumption here. The
absolute phase of a field is not observable, and therefore should not have an
explicit dependence in the energy of the field. Therefore, we must impose the
following condition:

$\varphi(k)=-\varphi(-k).$ (4.45)

One can see that the phase gained by evolving the system in time is of this
form. By substituting Eq. (4.39) into Eq. (4.44) one may verify that the
energy observable is time-independent.

###### Interference effects

Unlike the usual Hamiltonian for a system of uncoupled harmonic oscillators,
the above energy observable contains additional terms consisting of pairs of
annihilation or creation operators only. These terms have a natural
explanation as they are necessary in order to accommodate interference effects
that may arise in this new model. In particular, the real field expectation
values associated with photon states occupying the positive-frequency or
negative-frequency part of the Fourier spectrum of a blip can interfere with,
and even cancel, each other when characterised by the same values of $s$ and
$\lambda$.

To see this consider the operator

$D=D_{k}(\alpha)\,D_{-k}(-\alpha^{*}).$ (4.46)

Here $D_{k}(\alpha)$ is the displacement operator defined in Eq. (3.23) for
states with a single polarisation and direction of propagation. The subscript
$k$ is to remind us that this is a coherent superposition of states with a
single frequency $k$ rather than of normalised wave packets. By again using
Eq. (3.23) and the expression for the field observables in Eq. (4.3.3), one
may show that the operator $D$ defined in Eq. (4.46) commutes with both the
electric and magnetic field observables. It follows therefore that

$\displaystyle\braket{\alpha_{k},-\alpha^{*}_{-k}}{\textbf{O}(x,t)}{\alpha_{k},-\alpha^{*}_{-k}}$
$\displaystyle=$ $\displaystyle\braket{0}{D^{\dagger}\mathbf{O}(x,t)D}{0}$
(4.47) $\displaystyle=$
$\displaystyle\braket{0}{D^{\dagger}D\,\mathbf{O}(x,t)\ket{0}}$
$\displaystyle=$ $\displaystyle\braket{0}{\mathbf{O}(x,t)}{0}$
$\displaystyle=$ $\displaystyle 0$

where $\mathbf{O}=\mathbf{E},\mathbf{B}$. The fourth line follows from the
third by the unitarity of displacement operators. In the above we see that,
whereas a single coherent state $\ket{\alpha_{k}}$, for example, has a non-
zero field expectation value, by introducing to the system the additional
state $\ket{-\alpha^{*}_{-k}}$, the field expectation values will vanish
everywhere.

When we consider wave packets containing an integer number of photons
characterised by a single polarisation and value of $s$, the positive-
frequency and negative-frequency components of the wave packet cannot
interfere with each other either totally constructively or totally
destructively because their wave fronts propagate in opposite directions. On
the other hand, when, as in Eq. (4.47), we calculate the expectation value of
the classical field observables with respect to a coherent state, a state
containing contributions from the positive part of the Fourier spectrum of a
blip will contribute to both the positive-frequency part of the field
expectation value and its negative-frequency complex conjugate. Similarly, a
state containing photons belonging to the negative-frequency part of a blip
will contribute to both the negative-frequency part of the expectation value
and its positive-frequency complex conjugate.

As a result of this, by introducing negative-frequency excitations we are able
to construct distinct states that have opposing field expectation values and
therefore interfere destructively. Similarly the state
$\ket{\alpha_{k},\alpha^{*}_{-k}}$ has an energy four times that of either
coherent state independently, which is due to constructive interference.
Without the additional terms, this would not be correctly represented in the
energy expectation value of the state. This means, however, that in the
classical limit of this theory there is a redundancy between the positive- and
negative-frequency photons characterised by $\omega>0$ and $\omega<0$
respectively. In both cases it is possible to construct identical field
expectation values that evolve identically, but both are needed to construct
localised single-photon wave packets that propagate without dispersion. In
future, in order to avoid any problems that may arise due to this redundancy,
it may be required that the classical observables are projected onto a
subspace of the total Hilbert space containing the usual positive-frequency
excitations only.

#####  4.3.5 The dynamical Hamiltonian

Eq. (4.38) shows that the n-photon states $\ket{n_{s\lambda}(k,t)}$ are, up to
the accumulation of a time-dependent phase factor, invariant under the
dynamical evolution of the EM field. They are therefore eigenstates of the
dynamical Hamiltonian. Using the photon commutation relations in Eq. (4.36) it
can be shown that

$\displaystyle H_{\text{dyn}}(t)$ $\displaystyle=$ $\displaystyle\sum_{s=\pm
1}\sum_{\lambda={\sf H},{\sf V}}\int_{-\infty}^{\infty}\text{d}k\,\hbar
ck\,\widetilde{a}^{\dagger}_{s\lambda}(k,t)\widetilde{a}_{s\lambda}(k,t)\,.$
(4.48)

Using the transformations given in Eqs. (4.31) and (4.32), the above
Hamiltonian can be shown to be identical to that given in Eq. (4.29).

In Section 4.2.4 we pointed out that the dynamical Hamiltonian for blip states
had both positive and negative eigenvalues. This ensured that the dynamics of
localised light pulses was reversible: light moving to the left is
indistinguishable from light moving to the right when time is reversed. Here
we have found that, if our system contains photons of negative $k$, then the
dynamical Hamiltonian in this representation also possesses positive and
negative eigenvalues. In the momentum representation, the dynamical
Hamiltonian takes a much simpler form than the equivalent expression in the
position representation (cf. Eqs. (4.29) and (4.48)).

####  4.4 The importance of the position and momentum representations, and the
role of the regularisation operator

In this section let us emphasize that both the position and momentum space
quantisation approaches are important in obtaining a complete picture of the
quantised EM field. For example, studying the EM field in position space has
helped us to identify an otherwise hidden degree of freedom: the parameter $s$
which characterises the direction of propagation of wave packets of light. In
Section 4.2 we determined the Hilbert space for modelling localised wave
packets in one dimension. By solving Maxwell’s equations we were also able to
determine a set of field observables in addition to deriving a dynamical
Hamiltonian that describes the evolution of the system. However, we were not
able to determine the particular value of the regularisation operator
$\mathcal{R}$ which establishes a relationship between the local blip
operators and the electric and magnetic field observables.

Although both the position and momentum representations are essentially
equivalent, it is easier to determine these regularisation factors in momentum
space. In the following we shall determine this factor. The Fourier transforms
in Eqs. (4.31) and (4.32) allow us to alternate freely between the position
and momentum representations. Once we have identified $\Omega(k)$ we will then
be able to determine the effect of the regularisation $\mathcal{R}$ on the
blip operators. Later in this section we shall look at the relationship
between this and the standard description of the quantised EM field in
momentum space. It is shown that the standard description emerges when we
restrict the photon annihilation operators that we consider to a certain
subset of operators.

#####  4.4.1 Lorentz covariance

In Section 4.3.3, we were only able to define the electric and magnetic field
observables up to a $k$-dependent factor $\Omega(k)$ which was shown to be
directly related to the energy of a photon. One way of determining this factor
is to take into account that spontaneous emission from an individual atom or
ion results in exactly one photon. This assumption is in good agreement with
quantum optics experiments which generate single energy quanta on demand [156,
157, 158, 159]. These behave as monochromatic waves with energies and
frequencies determined by the atomic transition frequency $\omega_{0}$. A
different but equivalent method is to ensure that the electric and magnetic
field observables transform correctly under the orthochronous Lorentz
transformations. In this section we shall follow this latter approach.

For light propagating along the $x$ axis, the possible transformations are
translations in $x$ and $t$, and rotations about and boosts along the $x$
axis. A rotation changes the polarisation of the $\mathbf{E}$ and $\mathbf{B}$
fields, which are oriented in the $y$-$z$ plane, but leaves the direction of
propagation unchanged. The inner product between two states is a Lorentz
scalar, and is therefore unchanged by a Lorentz transformation. We shall
denote a single such transformation by the Greek letter $\Lambda$. Naturally,
these changes of reference frame induce a unitary operation $U(\Lambda)$.
Under a translation or a rotation, the photon creation operator gains only a
phase factor. The Lorentz boosts along the $x$ axis, however, involve a more
interesting transformation. Let us choose the particular transformation
$\Lambda$ that causes a Doppler shift of the wave number $k$ to the new wave
number $p$ without changing the polarisation or direction of the photon. By
taking into account Eq. (4.36) and using the Lorentz invariant measure
$\text{d}k/|k|$ one may show that the single-photon inner product given in Eq.
(4.37) is both Lorentz- and form-invariant only when

$\displaystyle
U(\Lambda)\,\widetilde{a}_{s\lambda}(k)\,U^{\dagger}(\Lambda)=\sqrt{\left|\frac{p}{k}\right|}\,\widetilde{a}_{s\lambda}(p).$
(4.49)

When a change of reference frame is made, either by moving the field or moving
ourselves, the new field is some transformed value of the original field
strength measured at the untransformed position. In classical
electromagnetism, the electric and magnetic field vectors transform as the
components of an antisymmetric rank-2 tensor. We should expect the expectation
values of the quantised field observables to transform in just the same way.
By unitarily transforming the position-dependent field observables in their
momentum representations, one finds that the correct transformation occurs
when

$\displaystyle\Omega(k)=\sqrt{|k|}\,\Omega_{0}.$ (4.50)

This result is calculated in three dimensions in Appendix (A), but the one-
dimensional result is analogous. Furthermore, if we want the expectation
values of the energy observable $H_{\text{energy}}(t)$ in Eq. (4.44) and of
the dynamical Hamiltonian $H_{\text{dyn}}(t)$ in Eq. (4.48) to be the same, at
least in some cases, we must also choose

$\displaystyle\left|\Omega_{0}\right|^{2}$ $\displaystyle=$
$\displaystyle{2\hbar\over\varepsilon_{0}Ac}\,.$ (4.51)

The latter equivalence only holds for states with positive values of $k$. In
general, the above choice of $\Omega_{0}$ implies that the energy expectation
value of a single photon in the $(s,k,\lambda)$ mode equals $\hbar c|k|$.

#####  4.4.2 The regularisation operator revisited

In Section 4.3.1 we demonstrated how to transform between the blip and photon
operators by means of a Fourier transform. By combining Eqs. (4.31) and (4.32)
with Eq. (4.2.3) one will arrive at the expressions

$\displaystyle\textbf{E}(x,t)$ $\displaystyle=$ $\displaystyle\sum_{s=\pm
1}\int^{\infty}_{-\infty}\frac{\text{d}k}{\sqrt{2\pi}}\;c\,{\rm e}^{{\rm
i}(skx+\varphi(k))}\left({\cal R}\left[\widetilde{a}_{s{\sf
H}}\right](k,t)\,\widehat{\boldsymbol{y}}+\mathcal{R}\left[\widetilde{a}_{s{\sf
V}}\right](k,t)\,\widehat{\boldsymbol{z}}\right),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $ $\displaystyle\textbf{B}(x,t)$
$\displaystyle=$ $\displaystyle\sum_{s=\pm
1}\int^{\infty}_{-\infty}\frac{\text{d}k}{\sqrt{2\pi}}\;s\,{\rm e}^{{\rm
i}(skx+\varphi(k))}\left({\cal R}\left[\widetilde{a}_{s{\sf
H}}\right](k,t)\,\widehat{\boldsymbol{z}}-\mathcal{R}\left[\widetilde{a}_{s{\sf
V}}\right](k,t)\,\widehat{\boldsymbol{y}}\right)\,.$ (4.52)

Moreover, by substituting Eq. (4.50) into the Fourier transforms (see. Eq.
(4.3.3)) of the electric and magnetic field observables given in Eq. (4.3.3),
one finds that

$\displaystyle\textbf{E}(x,t)$ $\displaystyle=$ $\displaystyle\sum_{s=\pm
1}\int^{\infty}_{-\infty}\frac{\text{d}k}{\sqrt{2\pi}}\;c\,\sqrt{\frac{2\hbar|k|}{\varepsilon_{0}Ac}}\,{\rm
e}^{{\rm i}(skx+\varphi(k))}\left(\widetilde{a}_{s{\sf
H}}(k,t)\,\widehat{\boldsymbol{y}}+\widetilde{a}_{s{\sf
V}}(k,t)\,\widehat{\boldsymbol{z}}\right)\,,$ $\displaystyle\textbf{B}(x,t)$
$\displaystyle=$ $\displaystyle\sum_{s=\pm
1}\int_{-\infty}^{\infty}\frac{\text{d}k}{\sqrt{2\pi}}\;s\,\sqrt{\frac{2\hbar|k|}{\varepsilon_{0}Ac}}\,{\rm
e}^{{\rm i}(skx+\varphi(k))}\left(\widetilde{a}_{s{\sf
H}}(k,t)\,\widehat{\boldsymbol{z}}-\widetilde{a}_{s{\sf
V}}(k,t)\,\widehat{\boldsymbol{y}}\right).$ (4.53)

This equivalence between the complex field observables above allows us to
determine the action of the superoperator $\mathcal{R}$ on a photon
annihilation operator. We find that

$\displaystyle{\cal R}\left[\widetilde{a}_{s\lambda}\right](k,t)$
$\displaystyle=$
$\displaystyle\sqrt{2\hbar|k|\over\varepsilon_{0}Ac}\,\widetilde{a}_{s\lambda}(k,t)\,.$
(4.54)

In the momentum representation, in order to regularise the photon operators we
multiply them by $(2\hbar|k|/\varepsilon_{0}Ac)^{1/2}$. As expected from the
symmetries of the quantised EM field, the superoperator ${\cal R}$ depends on
neither $x$, $t$, $s$ nor $\lambda$.

In Eq. (4.4.2) we assumed that the regularised blip operators, which before
regularisation are the Fourier transforms of the photon operators, are the
Fourier transforms of the regularised photon operators. The action of the
regularisation superoperator $\mathcal{R}$ on a blip annihilation operator can
therefore be determined by calculating the Fourier transform of Eq. (4.54).
One will find that the action of the regularisation operator on a blip
annihilation operator is of the type given in Eq. (4.19) where the
$R(x-x^{\prime})$ is given by

$\displaystyle\mathcal{R}(x-x^{\prime})=\int_{-\infty}^{\infty}\frac{\text{d}k}{2\pi}\;e^{ik(x-x^{\prime})}\sqrt{\frac{2\hbar|k|}{\varepsilon_{0}Ac}}.$
(4.55)

The above trick of introducing the superoperator ${\cal R}$ allows us to
describe the quantised EM field in position space in terms of local bosonic
blip operators without having to sacrifice the Lorentz covariance of the
electric and magnetic field observables $\textbf{E}(x,t)$ an
$\textbf{B}(x,t)$. Importantly, however, this distribution is not local in the
same way that the Dirac delta function is. We can see this by evaluating
$\mathcal{R}(x-x^{\prime})$ at values $x\neq x^{\prime}$:

$\displaystyle\mathcal{R}(x-x^{\prime})=-\sqrt{\frac{\hbar}{4\pi\varepsilon_{0}Ac}}\frac{1}{|x-x^{\prime}|^{3/2}}.$
(4.56)

This means that the field observables are not a simple linear superposition of
blip operators defined at the same point (see Fig. 4.2 (c)). However, since it
is easier to work with bosonic annihilation and creation operators, we can
perform all calculations in the Hilbert space created by the local bosonic
operators. As a final point we may mention that, due to the equivalence of the
field observables, the two representations of the energy observable are equal.

#####  4.4.3 Comments on field and blip localisation

As was mentioned in the Introduction, some authors will prefer to work with
non-locally acting operators that have a closer link to the field observables
[35, 53, 54, 55]. Such operators are given by

$\displaystyle A_{s\lambda}(x,t)$ $\displaystyle=$ $\displaystyle{\cal
R}[a_{s\lambda}](x,t)$ (4.57)

and describe excitations that share a vacuum with the blip operators. The
reason we differentiate between blip operators $a_{s\lambda}(x,t)$ and the
field excitations $A_{s\lambda}(x,t)$ is that the blip operators possess a set
of bosonic commutation relations (Eq. (4.17)) with respect to the conventional
inner product. In contrast to this, the commutation relation for the
$A_{s\lambda}(x,t)$ operators is

$\displaystyle\big{[}A_{s\lambda}(x,t),A^{\dagger}_{s^{\prime}\lambda^{\prime}}(x^{\prime},t)\big{]}$
$\displaystyle=$
$\displaystyle\int^{\infty}_{-\infty}\frac{\text{d}k}{2\pi}\,\frac{2\hbar|k|}{\varepsilon_{0}Ac}\,{\rm
e}^{{\rm
i}sk(x-x^{\prime})}\,\delta_{ss^{\prime}}\,\delta_{\lambda\lambda^{\prime}}\,.$

Nevertheless, one can see from Eq. (4.2.3), for example, that the energy
quanta associated with the $A_{s\lambda}(x,t)$ operators can be more easily
associated with the electric and magnetic field observables.

With respect to the usual inner product, the single-excitation states
$\ket{1^{A}_{s\lambda}(x,t)}=A^{\dagger}_{s\lambda}(x,t)\ket{0}$ are not
orthogonal to one another. Using Eq. (4.4.3), one can show that

$\braket{1^{A}_{s\lambda}(x,t)}{1^{A}_{s^{\prime}\lambda^{\prime}}(x^{\prime},t)}=\braket{0}{\big{[}A_{s\lambda}(x,t),A^{\dagger}_{s^{\prime}\lambda^{\prime}}(x^{\prime},t)\big{]}}{0}\neq
0.$ (4.59)

This is related to the well-known problem that the measurable field
observables have the units of $(\text{energy density})^{1/2}$ and not
$(\text{probability density})^{1/2}$ as would be expected for a wave function
compatible with the Born rule.

Figure 4.2: [55] This figure compares the locality of different excitations
under the standard inner product and the inner product denoted A. (a) A
monochromatic photon is equally likely to be found anywhere along the $x$ axis
using either inner product. (b) Under the standard inner product a localised
blip, illustrated by the white particle, carries with it a non-local field
whose relation to the position of the blip is given by Eq. (4.56). (c) Under
the standard inner product, a localised field excitation, which is generated
by the operators $A^{\dagger}_{s\lambda}(x,t)$ operator, is constructed from a
non-local superposition of blips. Under the biorthogonal inner product, the
blips and fields can be localised simultaneously.

Although we shall not make use of it here, for completeness it is worth
describing a method by which we may impose bosonic commutation relations on
the $A_{s\lambda}(x,t)$ operators. This is achieved by introducing a new inner
product – labelled by superscript $A$ – such that:

$\displaystyle\braket{1^{A}_{s\lambda}(x,t)}{1^{A}_{s^{\prime}\lambda^{\prime}}(x^{\prime},t)}^{A}$
$\displaystyle=$
$\displaystyle\braket{1_{s\lambda}(x,t)}{1_{s^{\prime}\lambda^{\prime}}(x^{\prime},t)}$
(4.60) $\displaystyle=$ $\displaystyle\delta(x-x^{\prime}).$

Under this new inner product the single-photon field excitations form an
orthogonal basis of states in the position representation. This is shown in
Fig. 4.2 (d). However, this new inner product drastically alters the Hilbert
space of the quantised EM field. One consequence of this change, for instance,
is that some previously Hermitian operators become non-Hermitian, whereas
other previously non-Hermitian operators may become Hermitian [60, 61, 62].
For this reason, this aspect of quantum physics is known as “biorthogonal” or
“pseudo-Hermitian” quantum mechanics. It is an interesting area of physics
that has attracted a lot of attention in the field of local quantum
electrodynamics (QED) [35, 45, 53, 54, 151, 160]. Although this is a very
elegant way of restoring orthogonality, as we have investigated in Ref. [55],
introducing a biorthogonal system introduces complexities that are by no means
necessary to our understanding of the local dynamics of photons.

#####  4.4.4 The relation to standard descriptions

In this final subsection, we compare the description of the EM field in
Section 4.3 with the standard description of the quantised EM field in
momentum space, and ask which additional assumptions have to be made for the
latter to emerge. Looking at Eqs. (4.44) and (4.48), and using the value of
$\Omega(k)$ given by Eqs. (4.50) and (4.51), we can see that the energy
observable and dynamical Hamiltonian coincide when the negative-frequency
modes are excluded and we restrict ourself to positive values of $k$ only. As
one can check, in this case the field observables in Eqs. (4.4.2) and (4.4.2)
become the same as the local field observables in the standard description of
the quantised EM field [131]. As we know, the positive-frequency photon states
provide a complete description of the quantised EM field in the sense that
they can be superposed to reproduce the right electric and magnetic field
expectation values of wave packets of any shape. However, as we have seen in
the introduction to this chapter, they are not sufficient to generate the
quantum versions of all possible solutions of Maxwell’s equations, like
highly-localised wave packets that remain localised [72, 73].

####  4.5 Discussion

In standard quantisations of the free EM field in one dimension, the basic
excitations of the field are characterised by a wave number $k$, a frequency
$\omega=|k|c$ and a polarisation $\lambda$. These degrees of freedom, however,
are not sufficient for constructing single-photon wave packets that are both
localised and propagate at the speed of light without dispersion. In this
chapter we have constructed a complete description of the quantised EM field
in one dimension by assuming that the basic quanta of the field are blips,
which have a well-defined position, polarisation and direction of propagation.
Using these conditions, we then identified the Schrödinger equation for this
system and constructed electric and magnetic field observables
$\mathbf{E}(x,t)$ and $\mathbf{B}(x,t)$ that are consistent with Maxwell’s
equations. The Lorentz covariance of these field operators is achieved with
the help of a regularisation superoperator ${\cal R}$. Although the blips
themselves are local, they generate non-local fields which can be felt in a
region of the space surrounding them (see Fig. 4.2 (b)).

By writing the blip excitations as superpositions of monochromatic photons, we
have also shown that our approach is consistent with the standard theory of
the quantised EM field with the addition of countable negative-frequency
states. Previously, these states have been widely overlooked but the concept
of adopting them to negate the consequences of Hegerfeldt’s theorem [72] is
not new. For example, Allcock pointed out that negative-frequency modes are
necessary for defining states that have a well-defined and measurable time of
arrival [161, 162, 163]. It is also possible to define a time of arrival for
blip states. Negative-frequency solutions have also been considered in Refs.
[31, 52, 101, 106, 164, 165, 166, 167]. In Refs. [53, 146, 168, 169] negative
frequencies were also considered in the context of constructing real field
excitations. In this chapter, by introducing local particles of light with a
given direction of propagation we have clarified how these solutions arise
naturally in a covariant quantised theory. In addition, we have exposed some
consequences of a theory containing these states such as the difference
between the energy observable $H_{\text{energy}}(t)$ and the generator for
time translations, that is, the dynamical Hamiltonian.

In classical electrodynamics, a local description of the EM field is often
preferable to a non-local description. Similarly, we expect that the modelling
of the quantised EM field in terms of blip states will sometimes be preferable
to the standard description in terms of monochromatic photon states. For
example, this theory should provide an extremely useful tool for the modelling
of the quantised EM field in inhomogeneous dielectric media and on curved
space-times [111]. Other potential applications include modelling linear
optics experiments with ultra-broadband photons [147, 148, 149, 150] or local
light-matter interactions. For example, in Ref. [110], we used the blip
annihilation operators $a_{s\lambda}(x,t)$ to construct locally-acting mirror
Hamiltonians. This work might also inspire other areas of quantum physics
where there is a need for a local description of quantum fields, for instance,
the modelling of electrons in quantum transport problems. Later in this thesis
we shall show how the results of this paper also provide novel insight into
the Fermi problem and the Casimir effect.

###  Chapter 5 Local photons in three dimensions

This chapter provides a complete quantisation of the free EM field in three
dimensions in terms of particles that are, like in one dimension, perfectly
localised and propagate at the speed of light. In Section 5.2 we initially
take a naïve approach to quantisation by describing light as rays, but later
find that these excitations form an over-complete basis of the Hilbert space.
A complete basis is constructed in terms of plane wave excitations. In Section
5.3 we show that a reduced and complete Hilbert space can still be constructed
in terms of perfectly localised excitations, and in Section 5.4 we use these
excitations to provide a new quantisation of the free EM field in three
dimensions that is analogous to the results of the previous chapter. Here, as
before, we construct a set of field observables and determine a dynamical
Hamiltonian for the system which differes from the usual energy observable.

####  5.1 Introduction: Lessons from one dimension

In the previous chapter we quantised the free EM field in one dimension in
terms of localised energy quanta, so-called blips, that have a well-defined
direction of propagation and move without dispersion. The purpose of this
chapter is to extend the work of the previous chapter and determine a more
complete quantisation of the free EM field that describes all possible
solutions of Maxwell’s equations in three dimensions. A three-dimensional
description of the EM field will have the benefit of providing a general
formalism for describing light in a much more expansive array of problems, for
example, predicting experimental verifications of the Casimir effect. This
theory will hopefully provide a greater set of tools that will enable new
investigations into many different topics.

In Section 2.1.4, we showed that the one-dimensional wave equation, satisfied
by both $\mathbf{E}(x,t)$ and $\mathbf{B}(x,t)$, can be factorised into two
first-order equations which solve for fields propagating either to the left or
the right at the speed of light. This enabled us to immediately construct a
first-order differential equation of motion for the basic excitations of the
EM field. In three dimensions this is not possible as solutions of the wave
equation (2.16) cannot all be said to propagate in a given direction, such as
left or right, but may propagate in all sorts of directions. This makes it
difficult to identify the basic solutions of Maxwell’s equations in the
position representation. In the past, some authors have used a massless Dirac-
like equation as the fundamental equation of motion for photons in the
position representation. In this chapter, however, we shall instead follow the
one-dimensional approach as closely as possible by identifying the solutions
of Maxwell’s equations that have a fixed direction of propagation.

In Section 2.1.4, we considered waves that propagate along the $x$ axis only.
As the three-dimensional wave equation is rotationally invariant, any rotation
of these solutions will provide a further set of solutions to Maxwell’s
equations. Taking this into account and letting $\mathbf{s}$ be a constant
unit vector, any one-dimensional solution of the wave equation propagating in
the $\mathbf{s}$ direction will satisfy the equation

$\left[\mathbf{s}\cdot\nabla+\frac{1}{c}\frac{\partial}{\partial
t}\right]\mathbf{O}_{\mathbf{s}}(\mathbf{r},t)=0$ (5.1)

where
$\mathbf{O}_{\mathbf{s}}=\mathbf{E}_{\mathbf{s}},\mathbf{B}_{\mathbf{s}}$ are
the solutions of the wave equation that propagate in the $\mathbf{s}$
direction only. Here $\mathbf{s}$ is assumed to be constant with respect to
all coordinates.

Using the equation of motion (5.1), it is also possible to show that the
three-dimensional electric and magnetic field vectors
$\mathbf{E}_{\mathbf{s}}(\mathbf{r},t)$ and
$\mathbf{B}_{\mathbf{s}}(\mathbf{r},t)$ are orthogonal to both $\mathbf{s}$
and each other. To show that this is the case, one must first note that, for a
divergence-less vector field $\textbf{O}_{\mathbf{s}}$, the following relation
holds:

$\nabla\times(\mathbf{s}\times\mathbf{O})=\mathbf{s}\cdot\nabla\mathbf{O}_{\mathbf{s}}.$
(5.2)

Using the above relation in conjunction with Faraday’s and the Maxwell-Ampère
law given by Eqs. (2.10) and (2.11), one may show that Eq. (5.1) is satisfied
when

$\textbf{E}_{\mathbf{s}}=c\left(\mathbf{s}\times\textbf{B}_{\mathbf{s}}\right)$
(5.3)

and

$\textbf{B}_{\mathbf{s}}=-\frac{1}{c}\left(\mathbf{s}\times\textbf{E}_{\mathbf{s}}\right).$
(5.4)

Thus $\textbf{E}_{\mathbf{s}}$, $\textbf{B}_{\mathbf{s}}$ and $\mathbf{s}$
form an orthogonal triad. Moreover, since $\mathbf{E}_{\mathbf{s}}$ and
$\mathbf{B}_{\mathbf{s}}$ can be rotated about $\mathbf{s}$ without changing
their relative orientations, the electric and magnetic fields propagating in
the direction of $\mathbf{s}$ may be further characterised by two linear
polarisations orthogonal to $\mathbf{s}$. Note also that $\mathbf{s}$ is
parallel to the Poynting vector defined in Eq. (2.20).

The considerations above provide a good starting point for quantising the
free, three-dimensional EM field in position space. We have seen, for
instance, that there are solutions of Maxwell’s equations that propagate in a
fixed direction $\mathbf{s}$, have two distinct polarisations, and satisfy a
first-order equation of motion. In the remainder of this chapter, we shall
build upon these observations to develop a description of the three-
dimensional quantised EM field in the position and momentum representations.

####  5.2 A naïve approach to quantisation in the position representation

In this section we describe the free EM field in terms of localised
excitations that can each travel in any single direction, and we define some
of their basic properties. We also define an appropriate set of field
observables. Towards the end of this section we shall demonstrate that these
excitations provide an over-complete basis of the Hilbert space and that plane
wave excitations present a more suitable set of principal excitations.

#####  5.2.1 The Hilbert space

###### Blip states

As in the one-dimensional theory, we assume here that the fundamental building
blocks of the electric and magnetic field observables are a set of spatially
localised particles. We shall again refer to these excitation as blips. In
three dimensions, blips are localised to a single point characterised by the
three-dimensional position vector $\mathbf{r}\in\mathbb{R}^{3}$. As discussed
in the preceding introduction, blips have a unique direction of propagation
which is no longer fixed along a specific axis but may be oriented in any
constant direction of our choosing. The direction of propagation is
characterised by the three-dimensional unit vector $\mathbf{s}\in S^{2}$ which
is oriented in the direction of propagation. Here $S^{2}$ is the unit two-
sphere which consists of all possible unit vectors. Blip states are also
characterised by a polarisation $\lambda\in\\{\mathsf{H},\mathsf{V}\\}$ that
specifies the orientation of the associated electric and magnetic field
observables.

In this description, the basic elements of the Hilbert space are localised
energy quanta that move with a fixed direction tracing out rays as illustrated
in Fig. 5.1. The reader may like to argue that light does not form perfectly
narrow rays in reality. This, however, would only be a consequence of having
an over-complete Hilbert space rather than an under-complete one. For now we
shall proceed by assuming that the basic excitations of the field propagate
along these rays and later determine any constraints on the arrangement of
these rays that would effectively reduce the size of our Hilbert space.

Figure 5.1: The figure shows a single blip propagating in the direction of
$\mathbf{s}$. The blip remains localised as it propagates along the trajectory
characterised by $\mathbf{s}$ and indicated by the dotted line. The parameters
$\mathbf{r}$ and $\mathbf{s}$ determine this trajectory uniquely. An identical
blip characterised by a vector $-\mathbf{s}$ would propagate along the same
path in the opposite direction. The coordinates $(q_{1},q_{2},q_{3})$ form a
right-handed coordinate system where the $q_{1}$ axis is aligned with
$\mathbf{s}$.

###### Annihilation and creation operators

As in previous sections we define a set of creation and annihilation operators
that add and remove blips from the system. We denote the blip annihilation
operators $a_{\mathbf{s}\lambda}(\mathbf{r},t)$ in the Heisenberg picture and
$a_{\mathbf{s}\lambda}(\mathbf{r},0)$ in the Schrödinger picture. The vacuum
state $\ket{0}$ of the system, which contains exactly zero blip excitations,
is defined such that

$a_{\mathbf{s}\lambda}(\mathbf{r},t)\ket{0}=0$ (5.5)

for all $\mathbf{r},t,\mathbf{s}$ and $\lambda$. The vacuum state is
normalised to unity: $\braket{0}{0}=1$.

As before, a single-blip state is generated by applying the creation operator
$a^{\dagger}_{\mathbf{s}\lambda}(\mathbf{r},t)$ to the vacuum state:

$\ket{1_{\mathbf{s}\lambda}(\mathbf{r},t)}=a^{\dagger}_{\mathbf{s}\lambda}(\mathbf{r},t)\ket{0}.$
(5.6)

The single-blip Hilbert space is therefore spanned by such states for all
$\mathbf{r}$, $\mathbf{s}$ and $\lambda$ at a fixed time. By applying a
creation operator to the vacuum state repeatedly we can generate states that
contain any number of blips. In analogy to Eqs. (3.12) and (4.10), a state
that contains $n$ identical blips at a time $t$ is denoted

$\ket{n_{\mathbf{s}\lambda}(\mathbf{r},t)}=\frac{1}{\sqrt{n!}}a^{\dagger}_{\mathbf{s}\lambda}(\mathbf{r},t)^{n}\,\ket{0}.$
(5.7)

The total Hilbert space is the symmetric Fock space of the single-blip Hilbert
space.

###### A fundamental equation of motion

In the introduction to this chapter we pointed out that components of the
electric and magnetic field vectors that propagate in a space-fixed direction
$\mathbf{s}$ will satisfy the first-order equation of motion (5.1). The
solutions to Eq. (5.1) describe waves that propagate in the direction of
$\mathbf{s}$ at the speed of light. As blips represent the basic excitations
of the field, we propose here that blips characterised by a vector
$\mathbf{s}$ will also propagate in this direction at the speed of light. This
condition places a constraint on the expectation values of the blip operators,
and therefore the operators themselves, which leads us to the following
equation of motion:

$\left[\mathbf{s}\cdot\nabla+\frac{1}{c}\frac{\partial}{\partial
t}\right]\,a_{\mathbf{s}\lambda}(\mathbf{r},t)=0.$ (5.8)

Although a blip state is characterised by three spatial coordinates that
specify its position in space, as blips only move in a single direction
(indicated by $\mathbf{s}$), the dynamics of a blip will only depend on a
single coordinate. The remaining two coordinates define the position of this
one-dimensional trajectory. To help us describe the solutions to Eq. (5.8),
let us introduce a new Cartesian coordinate system
$\mathbf{r}=(q_{1},q_{2},q_{3})$ where the $q_{1}$ axis is oriented in the
direction of $\mathbf{s}$ as illustrated in Fig. 5.1. Although this coordinate
system depends on the orientation of $\mathbf{s}$, there is still a well-
defined way of constructing this coordinate system for each choice of
$\mathbf{s}$. We shall assume that all three coordinate axes are orthogonal.
The solutions to Eq. (5.8) are then given by

$a_{\mathbf{s}\lambda}(q_{1},q_{2},q_{3};t)=a_{\mathbf{s}\lambda}(q_{1}-ct,q_{2},q_{3};0).$
(5.9)

This equation of motion is analogous to Eq. (4.11).

#####  5.2.2 Blip commutation relations

###### Commutation relations

As blips are bosons, a state containing many particles is unchanged by the
exchange of any two blips within that state. Consequently, all blip creation
and annihilation operators commute with one another:

$\left[a^{\dagger}_{\mathbf{s}\lambda}(\mathbf{r},t),a^{\dagger}_{\mathbf{s}^{\prime}\lambda^{\prime}}(\mathbf{r}^{\prime},t^{\prime})\right]=\Big{[}a_{\mathbf{s}\lambda}(\mathbf{r},t),a_{\mathbf{s}^{\prime}\lambda^{\prime}}(\mathbf{r}^{\prime},t^{\prime})\Big{]}=0.$
(5.10)

for all $\mathbf{r}$, $\mathbf{r}^{\prime}$, $t$ , $t^{\prime}$, $\mathbf{s}$,
$\mathbf{s}^{\prime}$, $\lambda$ and $\lambda^{\prime}$. As in Eq. (3.15), the
commutation relation between the blip creation and annihilation operators is
determined by the inner product between two single-blip states:

$\braket{1_{\mathbf{s}\lambda}(\mathbf{r},t)}{1_{\mathbf{s}^{\prime}\lambda^{\prime}}(\mathbf{r}^{\prime},t^{\prime})}=\Big{[}a_{\mathbf{s}\lambda}(\mathbf{r},t),a^{\dagger}_{\mathbf{s}^{\prime}\lambda^{\prime}}(\mathbf{r}^{\prime},t^{\prime})\Big{]}.$
(5.11)

This inner product must now be determined.

###### Orthogonal states

As blips are entirely localised at a point specified by the vector
$\mathbf{r}$, blips defined at different positions must all be orthogonal to
one another at a given fixed time. Moreover, as polarisation is a measurable
quantity and blips characterised by different values of $\mathbf{s}$ have
different dynamics, they must all be distinguishable and therefore belong to
different Hilbert spaces. Hence, in the following we find that

$\displaystyle\braket{1_{\mathbf{s}\lambda}(\mathbf{r},t)}{1_{\mathbf{s}^{\prime}\lambda^{\prime}}(\mathbf{r}^{\prime},t^{\prime})}$
$\displaystyle=$
$\displaystyle\braket{0}{\Big{[}a_{\mathbf{s}\lambda}(\mathbf{r},t),a^{\dagger}_{\mathbf{s}^{\prime}\lambda^{\prime}}(\mathbf{r}^{\prime},t^{\prime})\Big{]}}{0}$
(5.12) $\displaystyle=$
$\displaystyle\delta^{3}(\mathbf{r}-\mathbf{r}^{\prime})\,\delta^{2}(\mathbf{s}-\mathbf{s}^{\prime})\,\delta_{\lambda,\lambda^{\prime}}.$

###### Single-blip wave packets

As we have seen above, the single-blip states characterised by a single value
of the continuous variables $\mathbf{s}$ and $\mathbf{r}$ are therefore non-
normalisable. Physical states of the system, on the other hand, are given by
finitely normalised wave packets as were described in Section 3.2.2. Hence, a
single-photon wave packet is given by the superposition

$\ket{\psi(t)}=a^{\dagger}(t)\,\ket{0}=\sum_{\lambda=\mathsf{H},\mathsf{V}}\int_{S^{2}}\text{d}^{2}\mathbf{s}\int_{-\infty}^{\infty}\text{d}^{3}\mathbf{r}\;\psi_{\mathbf{s}\lambda}(\mathbf{r})\ket{1_{\mathbf{s}\lambda}(\mathbf{r},t)}$
(5.13)

where $a^{\dagger}(t)$ is the single-photon creation operator for the wave
packet with position wave function $\psi_{\mathbf{s}\lambda}(\mathbf{r})$.

To ensure that this state is normalised and that
$\left[a(t),a^{\dagger}(t)\right]=1$, the wave function
$\psi_{\mathbf{s}\lambda}(\mathbf{r})$ must satisfy the condition

$\sum_{\lambda=\mathsf{H},\mathsf{V}}\int_{S^{2}}\text{d}^{2}\mathbf{s}\int_{-\infty}^{\infty}\text{d}^{3}\mathbf{r}\;|\psi_{\mathbf{s}\lambda}(\mathbf{r})|^{2}=1.$
(5.14)

Eq. (5.13) above will help us determine the complete physical Hilbert space of
states later in this chapter.

#####  5.2.3 Field observables in the position representation

###### Field observables

In this section we shall determine a set of expressions for the complex three-
dimensional field observables $\mathbf{E}(\mathbf{r},t)$ and
$\mathbf{B}(\mathbf{r},t)$ in terms of the blip operators
$a_{\mathbf{s}\lambda}(\mathbf{r},t)$ and
$a^{\dagger}_{\mathbf{s}\lambda}(\mathbf{r},t)$. Analogous to the results
found in Sections 3.2 and 4.2.3, we shall assume that the complex field
observables are given by a linear superposition of blip annihilation
operators. Since both the components of the electric and magnetic field
observables and the blip operators obey Eq. (2.16), we assume that

$\displaystyle\mathbf{E}(\mathbf{r},t)$ $\displaystyle=$
$\displaystyle\int_{S^{2}}\text{d}^{2}\mathbf{s}\;c\,\left\\{\boldsymbol{\cal{R}}_{\mathbf{s}\mathsf{H}}[a_{\mathbf{s}\mathsf{H}}](\mathbf{r},t)+\boldsymbol{\cal{R}}_{\mathbf{s}\mathsf{V}}[a_{\mathbf{s}\mathsf{V}}](\mathbf{r},t)\right\\}$
$\displaystyle\mathbf{B}(\mathbf{r},t)$ $\displaystyle=$
$\displaystyle\int_{S^{2}}\text{d}^{2}\mathbf{s}\;\left\\{\boldsymbol{\cal{R}}_{\mathbf{s}\mathsf{H}}[a_{\mathbf{s}\mathsf{H}}](\mathbf{r},t)+\boldsymbol{\cal{R}}_{\mathbf{s}\mathsf{V}}[a_{\mathbf{s}\mathsf{V}}](\mathbf{r},t)\right\\}\times\mathbf{s}.$
(5.15)

The above operator $\boldsymbol{\cal{R}}_{\mathbf{s}\lambda}$ denotes a
translation-invariant superoperator that shall again be referred to as the
regularisation operator. These field observables have been chosen such that
the relations given in Eq. (5.3) are satisfied.

###### The regularisation operator

As in Section 4.2.3, we have introduced a position-independent regularisation
operator in order to relate the electric and magnetic field observables to the
local blip operators. This regularisation operator will again allow us to
ensure that the field observables have the correct dimensionality and
transformation properties without having to sacrifice the orthogonality of the
blip operators. Unlike the regularisation operator defined in Section 4.2.3,
the regularisation operator is now vector valued. In one dimension, light is
always polarised in the $y$-$z$ plane, and the constant polarisation vectors
can therefore be forgotten. In three dimensions, however, the polarisation
vectors, like $\mathbf{s}$, can be oriented in any direction, so must be
included in the definition of $\boldsymbol{\mathcal{R}}_{\mathbf{s}\lambda}$.

The regularisation operator can again be understood in a distributional sense
in the following way

$\displaystyle\boldsymbol{\mathcal{R}}_{\mathbf{s}\lambda}\Big{[}a_{\mathbf{s}\lambda}\Big{]}(\mathbf{r},t)$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{3}}\text{d}^{3}\mathbf{r}^{\prime}\;\boldsymbol{\cal{R}}_{\mathbf{s}\lambda}(\mathbf{r}-\mathbf{r}^{\prime})\,a_{\mathbf{s}\lambda}(\mathbf{r}^{\prime},t)$
$\displaystyle\boldsymbol{\mathcal{R}}^{*}_{\mathbf{s}\lambda}\Big{[}a^{\dagger}_{\mathbf{s}\lambda}\Big{]}(\mathbf{r},t)$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{3}}\text{d}^{3}\mathbf{r}^{\prime}\;\boldsymbol{\cal{R}}_{\mathbf{s}\lambda}^{*}(\mathbf{r}-\mathbf{r}^{\prime})\,a^{\dagger}_{\mathbf{s}\lambda}(\mathbf{r}^{\prime},t)$
(5.16)

In order to satisfy Eq. (5.3), the distributions
$\boldsymbol{\mathcal{R}}_{\mathbf{s}\lambda}(\mathbf{r}-\mathbf{r}^{\prime})$
must be oriented in a direction orthogonal to $\mathbf{s}$. Furthermore, a
change in the polarisation will rotate
$\boldsymbol{\mathcal{R}}_{\mathbf{s}\lambda}(\mathbf{r}-\mathbf{r}^{\prime})$
about the direction of propagation. The orientation of
$\boldsymbol{\mathcal{R}}_{\mathbf{s}\lambda}(\mathbf{r}-\mathbf{r}^{\prime})$
therefore depends on both $\mathbf{s}$ and $\lambda$.

#####  5.2.4 Physical particle states

###### Field expectation values

In Section 5.2.3 we constructed a set of field observables that satisfy both
the equation of motion 5.1 for light propagating in one direction and the
field relations specified in Eq. (5.3). We must now ensure that the
expectation values of these field observables always satisfy Maxwell’s
equations (2.8)-(2.11). The main outcome of this section will be to determine
which wave functions $\psi_{\mathbf{s}\lambda}(\mathbf{r})$ correspond to
physical solutions of Maxwell’s equations. This will enable us later on to
reduce a possibly over-complete Hilbert space to a complete set of states
describing all possible physical solutions of Maxwell’s equations.

Since field expectation values calculated with respect to a state containing
an integer number of blips are always zero, we shall first construct a
coherent state $\ket{\alpha(t)}$ from the normalised wave packets defined in
Eq. (5.13) in the manner described in Section 3.2.2. With the help of Eqs.
(3.20), (3.21), (5.12) and (5.13), one may then show that

$a_{\mathbf{s}\lambda}(\mathbf{r},t)\ket{\alpha(t)}=\alpha\,\psi_{\mathbf{s}\lambda}(\mathbf{r})\ket{\alpha(t)}.$
(5.17)

Hence, the state $\ket{\alpha(t)}$ is an eigenstate of the blip annihilation
operator. It follows from the equation above that the expectation values of
the (complex) electric and magnetic field observables,
$\mathbf{E}(\mathbf{r},t)$ and $\mathbf{B}(\mathbf{r},t)$, calculated with
respect to $\ket{\alpha(t)}$ are given by

$\displaystyle\braket{\mathbf{E}(\mathbf{r},t)}$ $\displaystyle=$
$\displaystyle\alpha\sum_{\lambda={\mathsf{H},\mathsf{V}}}\int_{S^{2}}\text{d}^{2}\mathbf{s}\;c\,\boldsymbol{\cal{R}}_{\mathbf{s}\lambda}\left[\psi_{\mathbf{s}\lambda}\right](\mathbf{r})$
$\displaystyle\braket{\mathbf{B}(\mathbf{r},t)}$ $\displaystyle=$
$\displaystyle\alpha\sum_{\lambda={\mathsf{H},\mathsf{V}}}\int_{S^{2}}\text{d}^{2}\mathbf{s}\;\boldsymbol{\cal{R}}_{\mathbf{s}\lambda}\left[\psi_{\mathbf{s}\lambda}\right](\mathbf{r})\times\mathbf{s}.$
(5.18)

In the above expectation values the notation
$\boldsymbol{\cal{R}}_{\mathbf{s}\lambda}\left[\psi_{\mathbf{s}\lambda}\right](\mathbf{r})$
is shorthand for the quantity

$\boldsymbol{\cal{R}}_{\mathbf{s}\lambda}\left[\psi_{\mathbf{s}\lambda}\right](\mathbf{r})=\int_{\mathbb{R}^{3}}\text{d}^{3}\mathbf{r}^{\prime}\;\boldsymbol{\cal{R}}_{\mathbf{s}\lambda}(\mathbf{r}-\mathbf{r}^{\prime})\,\psi_{\mathbf{s}\lambda}(\mathbf{r}^{\prime}).$
(5.19)

###### Characteristics of physical wave packets

The field observables (5.2.3) have been oriented such that the relations given
in Eq. (5.3) are satisfied. Therefore, as all field and blip operators satisfy
Eq. (5.1) (see Eq. (5.8)), all field expectation values are guaranteed to
satisfy Faraday’s law and the Maxwell-Ampère law provided that the field
expectation values are divergence-less. Hence, in order to ensure that the
field expectation values obey Maxwell’s equations, we must only determine for
which wave functions Gauss’s laws for electric and magnetic fields, Eqs. (2.8)
and (2.9), are satisfied.

With this in mind let us first calculate the divergence of the field
expectation values given in Eq. (5.2.4). Using the spatial independence of
$\boldsymbol{\mathcal{R}}_{\mathbf{s}\lambda}$, one may show that Gauss’s laws
are satisfied only when the gradient of the wave function
$\psi_{\mathbf{s}\lambda}(\mathbf{r})$ is oriented in a direction orthogonal
to both $\boldsymbol{\mathcal{R}}_{\mathbf{s}\lambda}$ and
$\boldsymbol{\mathcal{R}}_{\mathbf{s}\lambda}\times\mathbf{s}$. This result is
determined in App. (B). Since $\boldsymbol{\mathcal{R}}_{\mathbf{s}\lambda}$
is orthogonal to $\mathbf{s}$, the gradient of the wave function must lie
either parallel or anti-parallel to $\mathbf{s}$. Referring to the coordinate
system $(q_{1},q_{2},q_{3})$ defined in Section 5.2.1, the wave functions that
correspond to solutions of Maxwell’s equations depend only on the coordinate
$q_{1}$, but not on either $q_{2}$ or $q_{3}$. This result implies that not
all choices of wave function represent physical solutions to Maxwell’s
equations and, therefore, that our blip Hilbert space is over-complete. In the
next section we shall determine a set of annihilation and creation operators
for the physical single-photon states that we have found in this section.

#####  5.2.5 A complete Hilbert space

###### Plane wave excitations

Let us look again at the single-photon wave packet defined in Eq. (5.13). We
now only consider the physical wave functions, which do not depend on either
$q_{2}$ or $q_{3}$. The single-photon wave function then takes the form

$\ket{\psi(t)}=\sum_{\lambda=\mathsf{H},\mathsf{V}}\int_{S^{2}}\text{d}^{2}\mathbf{s}\int_{-\infty}^{\infty}\text{d}q_{1}\;\psi_{\mathbf{s}\lambda}(q_{1})\ket{1_{\mathbf{s}\lambda}(q_{1},t)}$
(5.20)

where

$\ket{1_{\mathbf{s}\lambda}(q_{1},t)}=a^{\dagger}_{\mathbf{s}\lambda}(q_{1},t)\ket{0}=\frac{1}{\sqrt{L_{2}L_{3}}}\int_{L_{2}}\text{d}q_{2}\int_{L_{3}}\text{d}q_{3}\;\,a^{\dagger}_{\mathbf{s}\lambda}(\mathbf{r},t)\ket{0}.$
(5.21)

In the above we must normalise the state
$\ket{1_{\mathbf{s}\lambda}(q_{1},t)}$ by the lengths $L_{2}$ and $L_{3}$ in
the $q_{2}$ and $q_{3}$ directions in order to ensure that Eq. (5.14) is still
satisfied. $L_{2}$ and $L_{3}$ are assumed to be very large. The single-blip
states defined in Eq. (5.21) are a superposition of blips characterised by a
single $q_{1}$ for all values of $q_{2}$ and $q_{3}$. Hence, such states
represent plane waves that propagate at the speed of light along the $q_{1}$
axis, normal to the plane itself, in the direction of $\mathbf{s}$. These
states are illustrated in Fig. 5.2. Plane wave states generated from the
vacuum by the $a^{\dagger}_{\mathbf{s}\lambda}(q_{1},t)$ operators are
characterised by a polarisation $\lambda$, a direction of propagation
$\mathbf{s}$ and a position along the $q_{1}$ axis.

Figure 5.2: The figure shows a single plane wave excitation propagating in the
direction of $\mathbf{s}$. The excitation is completely localised in the
$q_{1}$ direction and completely delocalised in the $q_{2}$ and $q_{3}$
directions. The plane propagates along the dotted trajectory at the speed of
light in the direction of the red arrow without dispersion or rotation.

We have seen above that all physical single-photon states in our description
can be constructed by superposing the plane wave excitations given in Eq.
(5.21). Although the localised blip excitations can also be used to construct
all physical single-photon wave packets, they can be used to construct non-
physical ones too. For this reason our original Hilbert space was over-
complete. A better way of quantising the free EM field is to treat the plane
wave excitations generated by the $a_{\mathbf{s}\lambda}(q_{1},t)$ operators
as the basic excitations of the free EM field. This is in close analogy to the
results in Chapter 4, where the basic excitations are plane waves propagating
along the $x$ axis only. Furthermore, in the classical theory, plane waves
form a complete basis for the solutions of the vector wave equation [170],
which is satisfied by both the electric and magnetic fields. Later, in Section
5.3, we shall see that there is an equivalent description of the reduced and
complete Hilbert space in terms of localised excitations rather than plane
waves. Before we get to this, in the remainder of this section we shall
examine some important properties of the plane wave excitations.

###### A fundamental equation of motion

As both $q_{2}$ and $q_{3}$ are orthogonal to the direction of propagation,
the blip equation of motion given in Eq. (5.8), which contains a derivative in
the $q_{1}$ direction only, acts identically upon the plane wave excitations
as it does upon the blip excitations. We find, therefore, that

$\left[c\mathbf{s}\cdot\nabla+\frac{\partial}{\partial
t}\right]\,a_{\mathbf{s}\lambda}(q_{1},t)=0.$ (5.22)

Analogous to the dynamical solutions of the one-dimensional excitations given
in Eq. (4.11), the solutions of this equation are

$a_{\mathbf{s}\lambda}(q_{1},t)=a_{\mathbf{s}\lambda}(q_{1}-ct,0)$ (5.23)

In the above we must remember that the $q_{1}$ axis is oriented in the
direction of $\mathbf{s}$ and is therefore reversed when the direction of
propagation is reversed. Both left- and right- propagating plane wave
solutions are therefore present in Eq. (5.23).

###### Commutation relations

As we have done for other excitations, the commutation relation for plane wave
excitations can be determined by calculating the inner product between two
single-particle states. Using Eqs. (5.12) and (5.21) one may show that the
inner product between two plane wave excitations is given by

$\displaystyle\braket{1_{\mathbf{s}\lambda}(q_{1},t)}{1_{\mathbf{s}^{\prime}\lambda^{\prime}}(q_{1}^{\prime},t)}$
$\displaystyle=$
$\displaystyle\left[a_{\mathbf{s}\lambda}(q_{1},t),a^{\dagger}_{\mathbf{s}^{\prime}\lambda^{\prime}}(q_{1}^{\prime},t)\right]$
(5.24) $\displaystyle=$
$\displaystyle\delta(q_{1}-q_{1}^{\prime})\,\delta^{2}(\mathbf{s}-\mathbf{s}^{\prime})\,\delta_{\lambda,\lambda^{\prime}}.$

This determines the plane wave commutation relation. One may also show that,
as usual, any creation or annihilation operator will commute with any other
creation or annihilation operator respectively:

$\Big{[}a_{\mathbf{s}\lambda}(q_{1},t),a_{\mathbf{s}^{\prime}\lambda^{\prime}}(q_{1}^{\prime},t^{\prime})\Big{]}=\Big{[}a^{\dagger}_{\mathbf{s}\lambda}(q_{1},t),a^{\dagger}_{\mathbf{s}^{\prime}\lambda^{\prime}}(q_{1}^{\prime},t^{\prime})\Big{]}=0.$
(5.25)

for all $q_{1}$, $q^{\prime}_{1}$, $\mathbf{s}$, $\mathbf{s}^{\prime}$,
$\lambda$, $\lambda^{\prime}$, $t$ and $t^{\prime}$.

####  5.3 Localised states in three dimensions

In this section we show that there is an alternative and complete set of basis
states for the free EM field that are localised, and can therefore replace
plane waves as the principal excitations of the field. We also introduce a set
of vector-valued excitations that have the same dimensions as blips, but
relate much more easily to the field observables which are distinct from the
blips.

#####  5.3.1 Blips in the position representation

###### Blip states

In the previous section we have shown that when light propagates in a fixed
and constant direction, the basic excitations of the free EM field are plane
waves. Such excitations are not localised to a point, but, as the name
suggests, localised to a plane and therefore completely delocalised in two out
of three dimensions. In the position representation of the EM field, we must
be able to describe the fields at all points in space and time. In order to
make such a construction possible, however, it is necessary that the basic
energy quanta of the field are point-like, and not stretched over a plane as
the $a_{\mathbf{s}\lambda}(q_{1},t)$ energy quanta are. The purpose of this
section is to show that there is an alternative complete basis of states in
three dimensions that are completely localised in space.

Let us postulate the existence of localised energy quanta characterised by a
position $\mathbf{r}$ and time $t$, a polarisation $\lambda$ and an additional
discrete parameter $s=\pm 1$. Here we reduce the Hilbert space that was
considered earlier on in this chapter by making the transition from
$\mathbf{s}\in S^{2}$ to $s\in\\{-1,+1\\}$. From the last section we saw that
the plane wave excitations can still propagate in two directions along the
$q_{1}$ axis. In analogy to the one-dimensional blips, the parameter $s$
introduced here is intended to play a similar role and characterise the
forwards and backwards motion of the plane. In other words, a state
characterised by a particular value $s$ behaves just like an identical state
characterised by $-s$ if time is reversed. As these particles are localised we
shall refer to them as blips, and we will see later on that they provide a
complete basis of the single-photon Hilbert space. We denote the annihilation
operators for these excitations $a_{s\lambda}(\mathbf{r},t)$ and demand that
they share a vacuum state with the blip operators defined in Section 5.2.1.
Finally, we also assume that these excitations obey the wave equation (2.16):

$\left[\mathbf{\nabla}^{2}-\frac{1}{c^{2}}\frac{\partial^{2}}{\partial
t^{2}}\right]a_{s\lambda}(\mathbf{r},t)=0.$ (5.26)

###### Blip commutation relations

By applying the blip creation operators $a^{\dagger}_{s\lambda}(\mathbf{r},t)$
repeatedly to the vacuum state we can generate states containing any number of
identical blips. We denote the $n$-blip states

$\ket{n_{s\lambda}(\mathbf{r},t)}=\frac{1}{\sqrt{n!}}\,a^{\dagger}_{s\lambda}(\mathbf{r},t)^{n}\,\ket{0}.$
(5.27)

When $n=1$ we refer to these states as single-blip states. Since blips are
localised, single-blip states defined at different positions at a fixed time
are orthogonal to one another. We also assume here that states characterised
by different values of $s$ and $\lambda$ are distinguishable. This implies
that such states are also orthogonal to one another. Hence, the single-blip
inner product is given by

$\displaystyle\braket{1_{s\lambda}(\mathbf{r},t)}{1_{s^{\prime}\lambda^{\prime}}(\mathbf{r}^{\prime},t)}=\delta^{3}(\mathbf{r}-\mathbf{r}^{\prime})\,\delta_{s,s^{\prime}}\,\delta_{\lambda,\lambda^{\prime}}.$
(5.28)

Taking into account that the bosonic creation and annihilation operators
commute with all other creation and annihilation operators respectively, the
inner product defined above in Eq. (5.28) implies the following commutation
relations for the blip operators:

$\displaystyle\Big{[}a_{s\lambda}(\mathbf{r},t),a^{\dagger}_{s^{\prime}\lambda^{\prime}}(\mathbf{r}^{\prime},t)\Big{]}$
$\displaystyle=$
$\displaystyle\delta^{3}(\mathbf{r}-\mathbf{r}^{\prime})\,\delta_{s,s^{\prime}}\,\delta_{\lambda,\lambda^{\prime}},$
$\displaystyle\Big{[}a^{\dagger}_{s\lambda}(\mathbf{r},t),a^{\dagger}_{s^{\prime}\lambda^{\prime}}(\mathbf{r}^{\prime},t^{\prime})\Big{]}$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\Big{[}a_{s\lambda}(\mathbf{r},t),a_{s^{\prime}\lambda^{\prime}}(\mathbf{r}^{\prime},t^{\prime})\Big{]}$
$\displaystyle=$ $\displaystyle 0.$ (5.29)

These expressions hold for all $\mathbf{r}$, $\mathbf{r}^{\prime}$, $s$,
$s^{\prime}$, $\lambda$, $\lambda^{\prime}$, $t$ and $t^{\prime}$.

#####  5.3.2 Blips in the momentum representation

###### Photon states

An alternative but equivalent representation of these local excitations can be
found by expressing the blip operators in their Fourier representations as was
done for the one-dimensional blips in Section 4.3.1. By applying a Fourier
transform to the blip operators $a_{s\lambda}(\mathbf{r},t)$ we obtain the
operators $\widetilde{a}_{s\lambda}(\mathbf{k},t)$. These are defined as

$\widetilde{a}_{s\lambda}(\mathbf{k},t)=\int_{\mathbb{R}^{3}}\frac{\text{d}^{3}\mathbf{r}}{(2\pi)^{3/2}}\;e^{-is\mathbf{k}\cdot\mathbf{r}}\,a_{s\lambda}(\mathbf{r},t).$
(5.30)

We assume in the following, and will see later on, that the
$\widetilde{a}_{s\lambda}(\mathbf{k},t)$ operators represent the annihilation
operators of monochromatic photons. The inverse transformation is given by

$a_{s\lambda}(\mathbf{r},t)=\int_{\mathbb{R}^{3}}\frac{\text{d}^{3}\mathbf{k}}{(2\pi)^{3/2}}\;e^{is\mathbf{k}\cdot\mathbf{r}}\,\widetilde{a}_{s\lambda}(\mathbf{k},t).$
(5.31)

As in Section 4.3.1, the parameter $s$ is included in the exponent of the
Fourier transformation as a convenient convention so that by reversing the
sign of $s$ we simultaneously invert the orientation of the wave vector. Using
Eq. (5.30) we can see that the photon vacuum state coincides with the blip
vacuum state.

###### Photon commutation relations

Since there is a strict invertible relationship between the blip and photon
operators given by Eqs. (5.30) and (5.31), we can determine the commutation
relations for the latter by direct calculation. Using the blip commutation
relations given in Eq. (5.3.1), one finds that

$\Big{[}\widetilde{a}_{s\lambda}(\mathbf{k},t),\widetilde{a}^{\dagger}_{s^{\prime}\lambda^{\prime}}(\mathbf{k}^{\prime},t)\Big{]}=\delta^{3}(\mathbf{k}-\mathbf{k}^{\prime})\,\delta_{s,s^{\prime}}\,\delta_{\lambda,\lambda^{\prime}}.$
(5.32)

One may also show that the photon creation and annihilation operators commute
with all other creation and annihilation operators respectively:

$\Big{[}\widetilde{a}_{s\lambda}(\mathbf{k},t),\widetilde{a}_{s^{\prime}\lambda^{\prime}}(\mathbf{k}^{\prime},t^{\prime})\Big{]}=\Big{[}\widetilde{a}^{\dagger}_{s\lambda}(\mathbf{k},t),\widetilde{a}^{\dagger}_{s^{\prime}\lambda^{\prime}}(\mathbf{k}^{\prime},t^{\prime})\Big{]}=0$
(5.33)

for all $\mathbf{k}$, $\mathbf{k}^{\prime}$, $s$, $s^{\prime}$, $\lambda$,
$\lambda^{\prime}$, $t$ and $t^{\prime}$.

Using Eq. (5.31), we can now solve Eq. (5.26) which determines the time-
dependence of the photon operators. By substituting Eq. (5.31) into Eq. (5.26)
and solving the resulting equation one finds that

$a_{s\lambda}(\mathbf{r},t)=\int_{\mathbb{R}^{3}}\frac{\text{d}^{3}\mathbf{k}}{(2\pi)^{3/2}}\;e^{is\mathbf{k}\cdot\mathbf{r}-i\omega
t}\,\widetilde{a}_{s\lambda}(\mathbf{k},0).$ (5.34)

In the above $\omega$ is a real number satisfying the relation
$\omega^{2}=|\mathbf{k}|^{2}c^{2}$; however, $\omega$ must take both positive
and negative values in order to ensure that the transformation $s\mapsto-s$
always reverses the dynamics of the blip. In Chapter 4, when the parameter $s$
is defined as the direction of propagation, there were only two ways in which
the frequency $\omega$ could be continuously related to the wave vector $k$.
These were $\omega=\pm kc$. In three dimensions, however, where single-photon
wave packets can propagate in all sorts of directions, there are many
different possible choices of how to relate $\omega$ to $\mathbf{k}$ and $s$,
and different choices will be more suitable than others depending on the wave
packet we are describing.

#####  5.3.3 A complete basis of blip states

###### The relation between plane waves and blips

In this section we shall look at the relationship between the blip operators
$a_{s\lambda}(\mathbf{r},t)$ introduced in Section 5.3.1 and the plane wave
operators $a_{\mathbf{s}\lambda}(q_{1},t)$ introduced in Section 5.2.5. Both
excitations are characterised by an identical number of parameters which
suggests that a relationship exists. First, let us consider the following wave
packet:

$\ket{1_{s\lambda}(q_{1},t)}=\int_{-\infty}^{\infty}\frac{\text{d}k_{1}}{\sqrt{2\pi}}\;e^{isk_{1}q_{1}}\,\widetilde{a}^{\dagger}_{s\lambda}(\mathbf{k},t)\ket{0}.$
(5.35)

In the above we shall choose the coordinate $q_{1}$ such that its axis lies
either parallel or anti-parallel to the vector $\mathbf{k}$ in such a way that
$\mathbf{k}\cdot\mathbf{r}=k_{1}q_{1}$ where $k_{1}$ takes all real values. By
applying the above definitions and substituting into Eq. (5.35) the time-
dependence of the photon operator given in Eq. (5.34), the state given in Eq.
(5.35) simplifies to

$\ket{1_{s\lambda}(q_{1},t)}=\int_{-\infty}^{\infty}\frac{\text{d}k_{1}}{\sqrt{2\pi}}\;e^{isk_{1}(q_{1}-sct)}\,\widetilde{a}^{\dagger}_{s\lambda}(\mathbf{k},0)\ket{0}.$
(5.36)

In the above expression we have made the particular choice $\omega=k_{1}c$
which satisfies $\omega^{2}=|\mathbf{k}|^{2}c^{2}$; although, as mention
earlier, other choices are possible. This wave packet contains only a single
blip in a superposition over the region spanned by the coordinates $q_{2}$ and
$q_{3}$ and has no dependence on either coordinate. This region has the
structure of a plane whose normal is parallel to the $q_{1}$ axis. We can
think of this wave packet, therefore, as a plane wave. By noting the exponent
in Eq. (5.36), one can see that the plane wave excitation above propagates
along the $q_{1}$ axis in a direction determined by the parameter $s$. This is
the exact same dynamics that were displayed by the plane wave solutions in
Section (5.2.5). Moreover, in fact, when
$\mathbf{s}=s\,\widehat{\mathbf{q}}_{1}$ where $\widehat{\mathbf{q}}_{1}$ is a
unit vector oriented in the direction of increasing $q_{1}$, the operator
$a_{s\lambda}(q_{1},t)$ associated with the state in Eq. (5.36) satisfies Eq.
(5.8). Here $\widehat{\mathbf{q}}_{1}$ lies on the unit half-sphere.

###### Plane wave commutation relations

We have determined that we can construct plane wave operators
$a_{s\lambda}(q_{1},t)$ with the correct dynamics by superposing monochromatic
photons over a single axis. With the help of the photon commutation relations
in Eq. (5.32), we can also determine a set of commutation relations for the
plane wave excitations constructed in Eq. (5.35). To determine this
commutation relation we first look at the inner product between the two plane
wave excitations defined at equal times. Using Eq. (5.36) we find that the
equal-time inner product between two plane wave packets with possibly
different normals is given by the expression

$\displaystyle\braket{1_{s\lambda}(q_{1},t)}{1_{s^{\prime}\lambda^{\prime}}(q_{1}^{\prime},t)}$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int_{-\infty}^{\infty}\text{d}k_{1}\int_{-\infty}^{\infty}\text{d}k^{\prime}_{1}\;e^{-i(sk_{1}q_{1}-s^{\prime}k^{\prime}_{1}q^{\prime}_{1})}$
(5.37)
$\displaystyle\times\braket{0}{\Big{[}\widetilde{a}_{s\lambda}(\mathbf{k},t),\widetilde{a}^{\dagger}_{s^{\prime}\lambda^{\prime}}(\mathbf{k}^{\prime},t)\Big{]}}{0}.$

To continue we note that the photon commutator (5.32), which appears in Eq.
(5.37), can be written in a slightly different form. As the vector
$\mathbf{k}$ satisfies $\mathbf{k}=k_{1}\widehat{\mathbf{q}}_{1}$, the photon
commutator can be expressed as

$\displaystyle\Big{[}\widetilde{a}_{s\lambda}(\mathbf{k},t),\widetilde{a}^{\dagger}_{s^{\prime}\lambda^{\prime}}(\mathbf{k}^{\prime},t)\Big{]}$
$\displaystyle=$
$\displaystyle\delta^{2}(\widehat{\mathbf{q}}_{1}-\widehat{\mathbf{q}}^{\prime}_{1})\,\delta(k_{1}-k^{\prime}_{1})\,\delta_{s,s^{\prime}}\,\delta_{\lambda,\lambda^{\prime}}$
(5.38) $\displaystyle=$
$\displaystyle\delta^{2}(\mathbf{s}-\mathbf{s}^{\prime})\,\delta(k_{1}-k^{\prime}_{1})\,\delta_{\lambda,\lambda^{\prime}}.$

In this final line we have used the definition
$\mathbf{s}=s\hat{\mathbf{q}}_{1}$ from the previous section. Here
$\mathbf{s}$ is the direction of propagation. As $\hat{\mathbf{q}}_{1}$ lies
on the unit half-sphere, $\mathbf{s}$ lies on the unit sphere. By substituting
Eq. (5.38) into Eq. (5.37) one finds that

$\displaystyle\Big{[}a_{s\lambda}(q_{1},t),a^{\dagger}_{s^{\prime}\lambda^{\prime}}(q_{1}^{\prime},t)\Big{]}$
$\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\frac{\text{d}k_{1}}{2\pi}\;e^{-isk_{1}(q_{1}-q_{1}^{\prime})}\delta^{2}(\mathbf{s}-\mathbf{s}^{\prime})\,\delta_{\lambda,\lambda^{\prime}}$
(5.39) $\displaystyle=$
$\displaystyle\delta_{s}(q_{1}-q_{1}^{\prime})\,\delta^{2}(\mathbf{s}-\mathbf{s}^{\prime})\,\delta_{\lambda,\lambda^{\prime}}.$

The commutator coincides exactly with that for the plane wave quanta in
Section 5.2.5. Finally, let us add for completeness that planes wave
annihilation and creation operators, commute amongst themselves:

$\Big{[}a^{\dagger}_{s\lambda}(q_{1},t),a^{\dagger}_{s^{\prime}\lambda^{\prime}}(q_{1}^{\prime},t^{\prime})\Big{]}=\Big{[}a_{s\lambda}(q_{1},t),a_{s^{\prime}\lambda^{\prime}}(q_{1}^{\prime},t^{\prime})\Big{]}=0$
(5.40)

for all $q_{1}$, $q_{1}^{\prime}$, $s$, $s^{\prime}$, $\lambda$,
$\lambda^{\prime}$, $t$ and $t^{\prime}$ in complete analogy to Eq. (4.13).

###### Blips as a complete basis

The localised blips defined in Section 5.3.1 can be used to construct single-
photon plane wave states that have a well-defined direction of propagation in
addition to a polarisation about and position along the photon’s trajectory.
These states were constructed in Eq. 5.35. What is more, the plane wave
operators constructed in Section 5.3.3 are characterised by an identical set
of parameters, an identical set of commutation relations, and governed by the
same equation of motion as the plane wave operators defined in Section 5.2.5.
Both excitations also share a vacuum state. As the commutation relations
define the Hilbert space of our system, we find that the Hilbert spaces of the
two systems are identical, and governed by the same dynamical Hamiltonian. The
main result of this section, then, is that the system of localised blips
defined in Section 5.3.1, or alternatively the system of monochromatic photons
defined in Section 5.3.2, provide a complete description of the free quantised
EM field in three dimensions.

Unlike the plane wave energy quanta, blips are localised to a point. We are
now in a position to quantise the free EM field in the position
representation. This will be the topic of Section 5.4 of this Chapter. In the
standard theory of the quantised EM field, photon states could be
characterised at most by a position and a polarisation. In this section we
have introduced an additional parameter $s=\pm 1$, thus doubling the usual
Hilbert space. In one-dimension, this additional parameter could be
interpreted as a direction of propagation. In three dimensions, however, this
is not necessarily the case. Here $s$ characterises states that are the time-
reversal of each other. As discussed in Section 5.3.2, there are many
different ways to introduce the parameter $s$; for instance, it can be
interpreted as the direction of propagation of a plane wave. What this
parameter implies, however, is that for every particle in the Hilbert space,
the time-reversal of that particle is also a member of the Hilbert space.
Moreover, these excitations are orthogonal to each other.

Finally, we would like to mention that it is perhaps not entirely surprising
that the plane wave excitations form a complete basis of states. In Chapter 4,
the one-dimensional blip states were localised to a point along the $x$ axis
only. These excitations are essentially also plane wave solutions as they are
totally delocalised in the $y$-$z$ plane. We can see therefore that the
results of this section are consistent with the results of Chapter 4, the
difference being that in one dimension we only consider propagation along the
$x$ axis; in three dimensions we consider propagation in all directions.

#####  5.3.4 Vector excitations

###### Vector operators

Blips form a complete basis of Hilbert space and are perfectly localised in
space. We need not go any further; we have what we are looking for, and can
construct all possible states from these blip excitations. For practical
reasons, however, in this section we shall introduce one more type of
excitation: one that is vector valued.

The original reason for constructing blips was to describe the quantised
solutions of Maxwell’s equations. The solutions of Maxwell’s equations are
vector fields; the states generated by blips, on the other hand, are not. In
general the orientation of the field will have an effect on the type of wave
packets that can be constructed and their resulting dynamics, particularly if
the orientation of the fields is not position-independent. For example,
according to the so-called “Hairy Ball Theorem,” perfectly isotropic
electromagnetic spherical waves do not exist (see Fig. 5.3). On the other
hand, blip states localised at a position $\mathbf{r}$ can be superposed into
a sphere without difficulty. For this reason, it becomes difficult to relate
the vector-valued field observables to the scalar-valued blip operators that
we have considered so far.

Figure 5.3: For any continuous vector field tangent to the surface of a sphere
there must be at least one point where the vector field vanishes. This is
known as the “Hairy-ball” theorem. The red arrow in the figure indicates the
point at which the vector field disappears. As the EM field is oriented
tangentially to its direction of propagation, it is impossible to construct
continuous and spherically symmetric solutions to Maxwell’s equations.

In general, we may wish to consider wave packets that have different
structures and propagate in position-dependent directions. To accommodate
these possibilities, and more closely relate the field observables and
particle excitations, in this section we shall use the photon operators
defined in Section 5.3.2 to construct vector-valued blip operators. This
particular choice of notation results in simpler expressions for the field
observables later on in this chapter. The purpose of this section is to
introduce tools that will help us when applying this formalism to real
applications in future.

Let us introduce a new vector-valued annihilation operator
$\boldsymbol{a}_{s}(\mathbf{r},t)$. In terms of the photon operators
$\widetilde{a}_{s\lambda}(\mathbf{k},t)$, the vector operator
$\boldsymbol{a}_{s}(\mathbf{r},t)$ is given by the formula

$\boldsymbol{a}_{s}(\mathbf{r},t)=\sum_{\lambda={\mathsf{H},\mathsf{V}}}\int_{\mathbb{R}^{3}}\frac{\text{d}^{3}\mathbf{k}}{(2\pi)^{3/2}}\;e^{i(s\mathbf{k}\cdot\mathbf{r}-\omega
t)}\,\widetilde{a}_{s\lambda}(\mathbf{k})\,\boldsymbol{e}_{s\lambda}(\mathbf{k}).$
(5.41)

Here the vectors $\boldsymbol{e}_{s\mathsf{H}}(\mathbf{k})$ and
$\boldsymbol{e}_{s\mathsf{V}}(\mathbf{k})$ are two real polarisation vectors
orthogonal both to each other and the vector $\mathbf{k}$ for any
$\mathbf{k}$, $s$ and $\lambda$. These operators are closely related to the
electric field observable; for instance, one may show that
$\boldsymbol{a}_{s}(\mathbf{k},t)$ is, like the electric field, divergence-
less: $\mathbf{\nabla}\cdot\boldsymbol{a}_{s}(\mathbf{k},t)=0$. To specify the
polarisation vectors more completely we demand that the unit vectors
$(\widehat{\mathbf{q}}_{1},\boldsymbol{e}_{s\mathsf{H}}(\mathbf{k}),\boldsymbol{e}_{s\mathsf{V}}(\mathbf{k}))$
form a right-handed coordinate system. The horizontally and vertically
polarised components of the vector annihilation operator are given by the
localised blip operators $a_{s\mathsf{H}}(\mathbf{r},t)$ and
$a_{s\mathsf{V}}(\mathbf{r},t)$ respectively.

###### A fundamental equation of motion

From Section 5.3.1, we know that the blip operators
$a_{s\lambda}(\mathbf{r},t)$ satisfy the second order wave equation for waves
propagating at the speed of light (see Eq. (5.26)). This equation is unchanged
by the transformation $s\mapsto-s$, and therefore does not distinguish a
difference in the dynamics of light characterised by excitations with
different values of $s$. As the $\boldsymbol{a}_{s}(\mathbf{r},t)$ operators
are closely related to the electric field observable, we can introduce another
vector-valued operator $\boldsymbol{b}_{s}(\mathbf{r},t)$ that is closely
related to the magnetic field observable. We define the
$\boldsymbol{b}_{s}(\mathbf{r},t)$ operators to be those satisfying the
following equation of motion:

$\mathbf{\nabla}\times\boldsymbol{a}_{s}(\mathbf{r},t)=-\frac{s}{c}\frac{\partial}{\partial
t}\,\boldsymbol{b}_{s}(\mathbf{r},t).$ (5.42)

Unlike the wave equation, this equation contains only first-order derivatives
with respect to time and the role of the parameter $s$ can be seen. The
relationship between $\boldsymbol{a}_{s}(\mathbf{r},t)$ and
$\boldsymbol{b}_{s}(\mathbf{r},t)$ given in Eq. (5.42) can also be represented
by means of a position-independent superoperator $\chi$ where
$\boldsymbol{b}_{s}(\mathbf{r},t)=-i\chi[\boldsymbol{a}_{s}](\mathbf{r},t)$.
The operator $\chi$ acts on the vector operators in a distributional sense
just like the regularisation operator in Eq. (4.19). The operator $\chi$,
however, mixes together the different components of the vector operators. In
Ref. [50], $\chi$ is referred to as the helicity operator. By calculating the
divergence of both sides of Eq. (5.42) one can show that, like the
$\boldsymbol{a}_{s}(\mathbf{r},t)$ operators, the
$\boldsymbol{b}_{s}(\mathbf{r},t)$ operators are divergence-less too.

Although different from the $\boldsymbol{a}_{s}(\mathbf{r},t)$ operators we
should bear in mind that the $\boldsymbol{b}_{s}(\mathbf{r},t)$ operators are
not independent of the $\boldsymbol{a}_{s}(\mathbf{r},t)$ operators. Both of
these operators can be represented in terms of the
$a_{s\lambda}(\mathbf{r},t)$ blip operators. Consequently, any state
containing $\boldsymbol{a}_{s}(\mathbf{r},t)$ energy quanta also contains
$\boldsymbol{b}_{s}(\mathbf{r},t)$ energy quanta. Any light pulse containing
an electric component also contains a magnetic component. In particular,
whereas the horizontally and vertically polarised components of
$\boldsymbol{a}_{s}(\mathbf{r},t)$ are superpositions of horizontally and
vertically polarised blips respectively, the horizontally and vertically
polarised components of the $\boldsymbol{b}_{s}(\mathbf{r},t)$ operators are
superpositions of vertically and horizontally polarised blips respectively. In
addition, by calculating the curl of Eq. (5.42) and using Eq. (5.26) one can
show that $\boldsymbol{a}_{s}(\mathbf{r},t)$ and
$\boldsymbol{b}_{s}(\mathbf{r},t)$ satisfy the equation

$\mathbf{\nabla}\times\boldsymbol{b}_{s}(\mathbf{r},t)=\frac{s}{c}\frac{\partial}{\partial
t}\,\boldsymbol{a}_{s}(\mathbf{r},t).$ (5.43)

Using the $\chi$ notation, the above equation may be written
$\boldsymbol{a}_{s}(\mathbf{r},t)=i\chi[\boldsymbol{b}_{s}](\mathbf{r},t)$.
Hence together Eqs. (5.42) and (5.43) form a closed set of first-order
equations for the $\boldsymbol{a}_{s}(\mathbf{r},t)$ and
$\boldsymbol{b}_{s}(\mathbf{r},t)$ operators and $\chi^{2}=-1$. We shall use
these equations in Section 5.4.3 to construct a dynamical Hamiltonian in the
position representation.

For a given single-blip state $\ket{\psi(t)}$ we may define the following
vector wave functions:

$\displaystyle\boldsymbol{\psi}^{E}_{s}(\mathbf{r},t)$ $\displaystyle=$
$\displaystyle\braket{0}{\boldsymbol{a}_{s}(\mathbf{r},0)}{\psi(t)}$
$\displaystyle\boldsymbol{\psi}^{B}_{s}(\mathbf{r},t)$ $\displaystyle=$
$\displaystyle\braket{0}{\boldsymbol{b}_{s}(\mathbf{r},0)}{\psi(t)}.$ (5.44)

We may refer to $\boldsymbol{\psi}^{E}_{s}(\mathbf{r},t)$ and
$\boldsymbol{\psi}^{B}_{s}(\mathbf{r},t)$ as the electric and magnetic wave
functions respectively. Both are divergence-less and represent the probability
distribution of a particular electric and magnetic field configuration.

###### Are blips local?

The blip creation operators $a^{\dagger}_{s\lambda}(\mathbf{r},t)$ defined at
different positions are orthogonal to one another and therefore define
perfectly localised particles. One the other hand, no two components
$a_{si}(\mathbf{r},t)$ or $b_{si}(\mathbf{r},t)$ of the
$\boldsymbol{a}_{s}(\mathbf{r},t)$ and $\boldsymbol{b}_{s}(\mathbf{r},t)$
operators (where $i=1,2,3$ refers to a component in a constant Cartesian
basis) possess a bosonic commutation relation. This result can be shown using
Eqs. (5.41) and 5.42. It seems, therefore, that the components of the vector
operators cannot be localised in space. Does this mean that photons cannot be
localised and that we are unable to give an exact description of photons in
the position representation? The answer is emphatically “no.”

The components of the $\boldsymbol{a}_{s}(\mathbf{r},t)$ and
$\boldsymbol{b}_{s}(\mathbf{r},t)$ operators do not have a local bosonic
commutation relation precisely because the operators possess only transverse
polarisations. It is sometimes thought that this implies that the photon
cannot be localised. In this quantisation, however, the principle excitations
of the electromagnetic field in three-dimensions are the blip operators
$a_{s\lambda}(\mathbf{r},t)$ which do have a local commutation relation as all
momenta are taken into account for each polarisation (see Eq. (5.3.1)). This
is reminiscent of the position operator introduced by Hawton [34] which takes
into consideration longitudinally as well as transversely polarised field
components.

By representing any quantum state in a basis of blip states, we can define its
position-dependent wave function. The vector excitations
$\boldsymbol{a}_{s}(\mathbf{r},t)$ and $\boldsymbol{b}_{s}(\mathbf{r},t)$ are
non-local, but the position of these non-local excitations is well-defined.
The non-local commutation relations for the $\boldsymbol{a}_{s}(\mathbf{r},t)$
and $\boldsymbol{b}_{s}(\mathbf{r},t)$ operators therefore do not represent a
quantum problem founded in an apparent inability to localise the photon; it is
a classical problem brought about by the fact that electromagnetic waves do
not form themselves into point-like objects. In this sense, the energy quanta
of the field behave very much like the classical solutions of Maxwell’s
equations. Shortly we shall see that the EM field observables are closely
related to the vector excitations.

####  5.4 The EM field observables

Hitherto we have constructed a complete Hilbert space for the carrier
particles of the quantised EM field in position space. These are the localised
blips, which are characterised by a position $\mathbf{r}$, a time $t$, a
polarisation $\lambda$ and an additional discrete parameter $s$. Furthermore,
we have shown that these blip excitations can be used to construct vector-
valued operators that, like the electric and magnetic field observables, are
divergence-less and obey the wave equation for vector fields. In this next
section we shall complete the development of the three-dimensional theory of
the free EM field by constructing the electromagnetic field observables that
act on this Hilbert space and a dynamical Hamiltonian that determines the
evolution of the blips.

#####  5.4.1 Observables in the position representation

###### Field observables in the position representation

Consistent with the field observables given in Eq. (5.2.3) and the vector
operator defined in Eq. (5.41), we assume that the complex electric and
magnetic field observables are a linear superposition of the
$\boldsymbol{a}_{s}(\mathbf{r},t)$ and $\boldsymbol{b}_{s}(\mathbf{r},t)$
operators respectively:

$\displaystyle\mathbf{E}(\mathbf{r},t)$ $\displaystyle=$
$\displaystyle\sum_{s=\pm
1}\,c\,\mathcal{R}\left[\boldsymbol{a}_{s}\right](\mathbf{r},t)$
$\displaystyle\mathbf{B}(\mathbf{r},t)$ $\displaystyle=$
$\displaystyle\sum_{s=\pm
1}\,s\,\mathcal{R}\left[\boldsymbol{b}_{s}\right](\mathbf{r},t).$ (5.45)

Here as in previous definitions of the field observables, the blip operators
are related to the field observables by a position-independent regularisation
operator $\mathcal{R}$. Using the equations of motion (5.42) and (5.43) for
the vector operators, it can be shown that the field observables defined in
Eq. (5.4.1) satisfy all four of Maxwell’s equations (2.8)-(2.11).

###### Field observables in the momentum representation

The field observables in the momentum representation are given by the
expressions

$\displaystyle\mathbf{E}(\mathbf{r},t)$ $\displaystyle=$
$\displaystyle\sum_{s=\pm
1}\sum_{\lambda={\mathsf{H},\mathsf{V}}}\int_{\mathbb{R}^{3}}\frac{\text{d}^{3}\mathbf{k}}{(2\pi)^{3/2}}\;c\,\Omega(\mathbf{k})e^{i(s\mathbf{k}\cdot\mathbf{r}+\varphi(k))}\,\widetilde{a}_{s\lambda}(\mathbf{k},t)\,\boldsymbol{e}_{s\lambda}(\mathbf{k},t)$
$\displaystyle\mathbf{B}(\mathbf{r},t)$ $\displaystyle=$
$\displaystyle\sum_{s=\pm
1}\sum_{\lambda={\mathsf{H},\mathsf{V}}}\int_{\mathbb{R}^{3}}\frac{\text{d}^{3}\mathbf{k}}{(2\pi)^{3/2}}\;s\,\frac{\Omega(\mathbf{k})}{\omega}e^{i(s\mathbf{k}\cdot\mathbf{r}+\varphi(k))}\,\widetilde{a}_{s\lambda}(\mathbf{k},t)\,\left(\mathbf{k}\times\boldsymbol{e}_{s\lambda}(\mathbf{k},t)\right).$
(5.46)

This is in agreement with Eqs. (5.41) and (5.42) and the expressions for the
field observables in Eq. (5.4.1).

###### The regularisation operator

Unlike the regularisation operator defined in Section 5.2.3, in Eq. (5.4.1)
the regularisation operator is not vector valued. Instead, it acts on the
vector blip operators to produce the also vector-valued electric and magnetic
field observables. The regularisation operator can be understood again in a
distributional sense such that

$\displaystyle\mathcal{R}\left[\boldsymbol{a}_{s}\right](\mathbf{r},t)=\int_{\mathbb{R}^{3}}\text{d}^{3}\mathbf{r}^{\prime}\;\mathcal{R}(\mathbf{r}-\mathbf{r}^{\prime})\,\boldsymbol{a}_{s}(\mathbf{r}^{\prime},t).$
(5.47)

Due to the isotropy of free space, the regularisation operator
$\mathcal{R}(\mathbf{r}-\mathbf{r}^{\prime})$ is independent of $\mathbf{s}$
and $\lambda$, and symmetric in its two arguments:
$\mathcal{R}(\mathbf{r}-\mathbf{r}^{\prime})=\mathcal{R}(\mathbf{r}^{\prime}-\mathbf{r})$.

#####  5.4.2 The energy observable

###### Energy in the position representation

Now that we have expressions for the electric and magnetic field observables,
by substituting these expressions into the classical energy (2.21) we can
determine an expression for the energy observable of the system. After
carrying out this substitution one finds that

$\displaystyle H_{\text{energy}}(t)$ $\displaystyle=$
$\displaystyle\sum_{s=\pm
1}\int_{\mathbb{R}^{3}}\text{d}^{3}\mathbf{r}\;\frac{\varepsilon_{0}c^{2}}{8}$
(5.48)
$\displaystyle\times\left\\{\|\mathcal{R}[\boldsymbol{a}_{s}](\mathbf{r},t)+\text{H.c.}\|^{2}+\|\mathcal{R}[\boldsymbol{b}_{s}](\mathbf{r},t)+\text{H.c.}\|^{2}\right\\}.$

Due to the complicated commutation relations between the vector blip
operators, this operator is not in a form that is of much practical use to us.
Fortunately we may simplify this expression further. We should recall that the
horizontally and vertically polarised components of the
$\boldsymbol{a}_{s}(\mathbf{r},t)$ operators are given by the horizontally and
vertically polarised blip operators respectively. Consequently, in Eq. (5.48),
the product
$\boldsymbol{a}_{s}(\mathbf{r},t)\cdot\boldsymbol{a}_{s^{\prime}}(\mathbf{r},t)$
is given by the expression
$\sum_{\lambda=\mathsf{H},\mathsf{V}}a_{s\lambda}(\mathbf{r},t)a_{s^{\prime}\lambda}(\mathbf{r},t)$
where $a_{s\lambda}(\mathbf{r},t)$ are the usual local blip operators.
Moreover, using the property $\chi^{2}=-1$ it may be shown that the total
energy of the magnetic field is equal to the total energy of the electric
field. Putting both of these observations together one finds that the energy
observable can be written in the simpler form

$H_{\text{energy}}(t)=\sum_{s=\pm
1}\sum_{\lambda={\mathsf{H},\mathsf{V}}}\frac{\varepsilon_{0}c^{2}}{4}\int_{\mathbb{R}^{3}}\text{d}^{3}\mathbf{r}\;\left\\{\mathcal{R}\left[a_{s\lambda}\right](\mathbf{r},t)+\text{H.c.}\right\\}^{2}.$
(5.49)

This expression is entirely analogous to the energy observable derived for the
one dimensional fields in Eq. (4.22). Like the expression for the energy
observable in Eq. (4.22), the above energy observable is always positive,
which is ensured by the square in the integrand.

###### The regularisation operator revisited

As in one dimension, the regularisation operator plays an important role in
specifying the energy of a blip excitation. In order to determine the exact
form of the energy observable, we must now determine what effect the
regularisation operator has on a blip, or in other words, what is the exact
expression for the distribution $\mathcal{R}(\mathbf{r}-\mathbf{r}^{\prime})$.
In Chapter 4, the regularisation operator was determined by ensuring that the
field observables transformed correctly under Lorentz boosts. We proceed
analogously in this section and first determine the effect of the
regularisation operator on the photon operator
$\widetilde{a}_{s\lambda}(\mathbf{k},t)$. The distribution
$\mathcal{R}(\mathbf{r}-\mathbf{r}^{\prime})$ is then determined by Fourier
transforming any multiplicative factor gained in the regularisation of the
photon operators.

Consider again a pure Lorentz boost in a given direction denoted by the Greek
letter $\Lambda$. This transformation shifts all momenta $\mathbf{k}$ to the
new momenta $\mathbf{p}$. As before, the corresponding transformation on the
states of the system is implemented by the unitary transformation $U(\Lambda)$
which ensures that all transition amplitudes are invariant under the change of
reference frame. Using the Lorentz invariant measure
$\text{d}^{3}\mathbf{k}/|\mathbf{k}|$ and the photon commutation relation
given in Eq. (5.32), one can show that the inner product between two
normalised photon wave packets is both Lorentz- and form-invariant only when

$U(\Lambda)\,\widetilde{a}_{s\lambda}(\mathbf{k},0)\,U^{\dagger}(\Lambda)=\sqrt{\frac{|\mathbf{p}|}{|\mathbf{k}|}}\,\widetilde{a}_{s\lambda}(\mathbf{p})$
(5.50)

up to a unitary rotation of the polarisation states. I shall ignore this
rotation in the above transformation as the anti-symmetric field tensor
transforms correctly to accommodate this rotation and we are only interested
in the overall normalisation of the field. See Section 5.9 of Ref. [127] for
more details. Using this result one can again show that the fields transform
correctly only when the regularised photon operator is given by

$\mathcal{R}[\widetilde{a}_{s\lambda}](\mathbf{k},t)=\Omega_{0}\,\sqrt{|\mathbf{k}|}\,\widetilde{a}_{s\lambda}(\mathbf{k},t)$
(5.51)

for some real and constant numerical factor $\Omega_{0}$. See again Appendix.
A for more details.

As we shall see in Section 5.4.3, when restricted to positive-frequency
excitations the energy observable and dynamical Hamiltonian coincide when
$\Omega_{0}$ is given by

$\Omega_{0}=\sqrt{\frac{2\hbar}{\varepsilon_{0}c}}.$ (5.52)

For this choice of $\Omega_{0}$ the energy expectation value of a single-
photon state in the $(\mathbf{k},s,\lambda)$ mode is $\hbar|\mathbf{k}|c$.
Moreover, since the regularised blip operator is just the Fourier transform of
the regularised photon operator we can determine that

$\mathcal{R}(\mathbf{r}-\mathbf{r}^{\prime})=\int_{\mathbb{R}^{3}}\text{d}^{3}\mathbf{k}\;(2\pi)^{-3}\,e^{i\mathbf{k}\cdot(\mathbf{r}-\mathbf{r}^{\prime})}\,\sqrt{\frac{2\hbar|\mathbf{k}|}{\varepsilon_{0}c}}=-\frac{3}{8\pi^{3/2}}\sqrt{\frac{\hbar}{c\varepsilon_{0}}}\frac{1}{|\mathbf{r}-\mathbf{r}^{\prime}|^{7/2}}.$
(5.53)

The final equality sign has been determined, for example, in Refs. [28, 69]
after an exponential regularisation has been used. The result above again
shows that in three dimensions the field observables are non-locally related
to the blip operators. Given Eq. (5.53), we have completely determined the
field observables and the energy observable in the position representation.

#####  5.4.3 The dynamical Hamiltonian

###### An equation of motion for blips

The two equations of motion (5.42) and (5.43) describe completely the dynamics
of the blip operators in three-dimensions. As these equations are first-order
in time they provide a suitable starting point for our determining a dynamical
Hamiltonian that satisfies Heisenberg’s equation. The commutation relations
for the vector operators $\boldsymbol{a}_{s}(\mathbf{r},t)$ and
$\boldsymbol{b}_{s}(\mathbf{r},t)$ are not simple and it is more appropriate
therefore for us to express the dynamical Hamiltonian in terms of the locally
bosonic blip operators $a_{s\lambda}(\mathbf{r},t)$. Hence, to continue, we
must find a way to express Eq. (5.42) in terms of the local blip operators.

In Eqs. (5.42) and (5.43) there are two different types of operator: there are
the $\boldsymbol{a}_{s}(\mathbf{r},t)$ operators, and there are the
$\boldsymbol{b}_{s}(\mathbf{r},t)$ operators. By comparing Eqs. (5.41) and
(5.31) one can see that in the $(q_{1},q_{2},q_{3})$ basis the
$\boldsymbol{a}_{s}(\mathbf{r},t)$ have components
$(0,a_{s\mathsf{H}}(\mathbf{r},t),a_{s\mathsf{V}}(\mathbf{r},t))$. Hence, it
is straightforward to represent the $\boldsymbol{a}_{s}(\mathbf{r},t)$
operators in terms of blip operators. The $\boldsymbol{b}_{s}(\mathbf{r},t)$
operators on the other hand are defined entirely by the equation Eq. (5.42),
or, equivalently, related to the $\boldsymbol{a}_{s}(\mathbf{r},t)$ operators
by a helicity operator $\chi$. This relation is not necessarily local as it
depends upon the relationship between the wave vector $\mathbf{k}$ and the
frequency $\omega$ for which there are many different choices. I shall proceed
as in Section 5.3.3 where the direction of propagation is indicated by the
vector $\mathbf{s}=s\widehat{\mathbf{q}}_{1}$. As the electric and magnetic
fields are therefore related by a rotation about the $q_{1}$ axis, in the
$(q_{1},q_{2},q_{3})$ basis the components of the
$\boldsymbol{a}_{s}(\mathbf{r},t)$ and $\boldsymbol{b}_{s}(\mathbf{r},t)$
operators are related by a rotation. In particular, using the right hand rule
and the expressions for the field observables given in Eq. (5.4.1), the
components of the $\boldsymbol{b}_{s}(\mathbf{r},t)$ operators are given by
$(0,-a_{s\mathsf{V}}(\mathbf{r},t),a_{s\mathsf{H}}(\mathbf{r},t))$.

By substituting the above expressions for the vector operators back into Eq.
(5.42) one finds that

$\mathbf{\nabla}\times\begin{pmatrix}0\\\ a_{s\mathsf{H}}(\mathbf{r},t)\\\
a_{s\mathsf{V}}(\mathbf{r},t)\end{pmatrix}=-\frac{s}{c}\frac{\partial}{\partial
t}\begin{pmatrix}0\\\ -a_{s\mathsf{V}}(\mathbf{r},t)\\\
a_{s\mathsf{H}}(\mathbf{r},t).\end{pmatrix}$ (5.54)

This equation can be solved component-wise by remembering that
$(\widehat{\mathbf{q}}_{1},\boldsymbol{e}_{\mathsf{H}},\boldsymbol{e}_{\mathsf{V}})$
forms a right-handed coordinate system and that the blip operators depend on
the $q_{1}$ coordinate only. The above equation (5.54) then reduces to the
simpler equation

$\frac{\partial}{\partial
t}a_{s\lambda}(\mathbf{r},t)=-sc\frac{\partial}{\partial
q_{1}}a_{s\lambda}(\mathbf{r},t).$ (5.55)

This equation is completely analogous to the equation of motion (4.12) for
light propagating in one dimension. In Eq. (4.12), the space derivative is
with respect to the $x$ coordinate axis. Here the space derivative is with
respect to the $q_{1}$ axis lying either parallel or antiparallel to the
direction of propagation. This equation is general when the direction of
propagation is given by $\mathbf{s}=s\widehat{\mathbf{q}}_{1}$.

###### The dynamical Hamiltonian

By using the equation of motion (5.55) for localised blip excitations we can
now construct a dynamical Hamiltonian for light in three dimensions. We
proceed as before by assuming that the dynamical Hamiltonian is an exchange
Hamiltonian for blips. This operator takes the form

$\displaystyle H_{\text{dyn}}(t)$ $\displaystyle=$ $\displaystyle i\sum_{s=\pm
1}\sum_{\lambda=\mathsf{H},\mathsf{V}}\int_{\mathbb{R}^{3}}\text{d}^{3}\mathbf{r}^{\prime}\int_{\mathbb{R}^{3}}\text{d}^{3}\mathbf{r}^{\prime\prime}\;\hbar
sc\,f_{s\lambda}(\mathbf{r}^{\prime},\mathbf{r}^{\prime\prime})$ (5.56)
$\displaystyle\times
a^{\dagger}_{s\lambda}(\mathbf{r}^{\prime},t)a_{s\lambda}(\mathbf{r}^{\prime\prime},t).$

In the above $f_{s\lambda}(\mathbf{r}^{\prime},\mathbf{r}^{\prime\prime})$ is
complex. To determine
$f_{s\lambda}(\mathbf{r}^{\prime},\mathbf{r}^{\prime\prime})$ we use Eq.
(5.56) to calculate the time evolution of a blip operator and compare the
result with the blip equation of motion: in this case Eq. (5.55). Using
Heisenberg’s equation and the blip commutation relations given in Eq. (5.3.1)
we find that

$\frac{\text{d}}{\text{d}t}a_{s\lambda}(\mathbf{r},t)=sc\int_{\mathbb{R}^{3}}\text{d}^{3}\mathbf{r}^{\prime\prime}\;f_{s\lambda}(\mathbf{r},\mathbf{r}^{\prime\prime})\,a_{s\lambda}(\mathbf{r}^{\prime\prime},t).$
(5.57)

Hence we may verify that the dynamical Hamiltonian is given by

$\displaystyle H_{\text{dyn}}(t)$ $\displaystyle=$ $\displaystyle-i\sum_{s=\pm
1}\sum_{\lambda={\mathsf{H},\mathsf{V}}}\int_{\mathbb{R}^{3}}\text{d}^{3}\mathbf{r}^{\prime}\int_{\mathbb{R}^{3}}\text{d}^{3}\mathbf{r}^{\prime\prime}\;\hbar
sc\,\delta^{\prime}(\mathbf{r}^{\prime}-\mathbf{r}^{\prime\prime})$ (5.58)
$\displaystyle\times
a^{\dagger}_{s\lambda}(\mathbf{r}^{\prime},t)a_{s\lambda}(\mathbf{r}^{\prime\prime},t).$

Here $\delta^{\prime}(\mathbf{r}^{\prime}-\mathbf{r}^{\prime\prime})$ denotes
the derivative of a delta function with respect to $q_{1}$. By recalling that
$\mathbf{k}\cdot\mathbf{r}=k_{1}q_{1}$, Eq. (5.58) can be written in the
alternative form

$\displaystyle H_{\text{dyn}}(t)$ $\displaystyle=$ $\displaystyle\sum_{s=\pm
1}\sum_{\lambda={\sf H},{\sf
V}}\int_{\mathbb{R}^{3}}\text{d}^{3}\mathbf{r}^{\prime}\int_{\mathbb{R}^{3}}\text{d}^{3}\mathbf{r}^{\prime\prime}\int_{\mathbb{R}^{3}}\text{d}^{3}\mathbf{k}\;\frac{\hbar
k_{1}c}{(2\pi)^{3}}$ (5.59) $\displaystyle\times{\rm
e}^{is\mathbf{k}\cdot(\mathbf{r}^{\prime}-\mathbf{r}^{\prime\prime})}\,a^{\dagger}_{s\lambda}(\mathbf{r}^{\prime},t)\,a_{s\lambda}(\mathbf{r}^{\prime\prime},t).$

As one can see above, the dynamical Hamiltonian in three dimensions also has
positive and negative eigenvalues. This differentiates it from the dynamical
Hamiltonian in standard quantisations that has only positive eigenvalues.

In Eq. 5.59, the frequency of a photon with wave vector $s\mathbf{k}$ is given
by $k_{1}$ where $k_{1}$ is the $q_{1}$ component of $\mathbf{k}$. This is a
slightly odd choice of frequency because, as mentioned in Section 5.3.3, the
unit vector $\widehat{\mathbf{q}}_{1}$ lies on the half-sphere, and there are
many such half-spheres to choose from. This choice of dynamical Hamiltonian is
therefore most appropriate when there is a clear overall direction of
propagation, as there is, for instance, for plane waves. One may instead wish
to redefine $\widehat{\mathbf{q}}_{1}$ such that it covers the entire unit
sphere, which is much more general. Consider the alternative description where
$\widehat{\mathbf{q}}_{1}$ covers the unit sphere, the wave vector is given by
$\mathbf{k}=|k|\widehat{\mathbf{q}}_{1}$, and the direction of propagation is
given by $\mathbf{s}=s\widehat{\mathbf{q}}_{1}$. In such circumstances, while
Eq. (5.58) is still satisfied, the frequency of a monochromatic photon is
given by $s|k|c$. From this alternative form of the dynamical Hamiltonian we
can see that the negative frequency contribution to the dynamical Hamiltonian
arises due to the introduction of the new parameter $s$. Furthermore, when
restricted to positive frequencies only, the energy observable (5.49) and
dynamical Hamiltonian (5.59) coincide. One can also check that the energy
observable commutes with the dynamical Hamiltonian and is therefore conserved.

####  5.5 Discussion

In Chapter 4, the basic excitations of the free EM field were a set of bosonic
particles localised to a position along the $x$ axis that propagate either to
the left or the right at the speed of light. In this chapter we have
generalised the results of Chapter 4 in order to determine a complete set of
basis states for the free EM field in three dimensions that are both localised
and propagate at the speed of light. After an initial attempt to quantise the
field in terms of localised excitations that have a well-defined direction of
propagation, which we found to be an over-complete description, we introduced
a complete set of blip annihilation operators $a_{s\lambda}(\mathbf{r},t)$.
The corresponding blip states are locally orthogonal to one another and, in
addition to polarisation, are characterised by a discrete parameter $s=\pm 1$
which doubles the usual Hilbert space.

Unlike in one dimension, where the localised blips have a clear direction of
propagation, in three dimensions a localised blip state contains contributions
from photons propagating along all axes through the point of localisation with
a forwards or backwards motion determined by the choice of $s$. This feature
is reminiscent of the Huygens-Fresnel principle in which each point on a wave
front acts as a source of spherical waves. This does mean, however, that
highly-localised wave packets will spread out, but they do so only in
accordance with Maxwell’s equations.

In Section 5.4.1 we derived expressions for the electric and magnetic field
observables in terms of the blip operators by means of a position-independent
regularisation operator $\mathcal{R}$ and a set of non-local and vector-valued
operators $\boldsymbol{a}_{s}(\mathbf{r},t)$ and
$\boldsymbol{b}_{s}(\mathbf{r},t)$. By introducing vector operators we are
more easily able to relate the EM fields to the blips or carriers of the
field. These vector operators are similar to those defined in Ref. [50], but
differ in their dimension, as we make a clear distinction between the fields
and the carriers. As in Chapter 4, the regularisation operator allows us to
introduce particle excitations without sacrificing the Lorentz covariance of
the field observables. The expression given in Eq. (5.53) shows that the
relationship between the field observables and blip operators is non-local
with infinite tails. This was also the case for the one-dimensional fields in
Chapter 4. Hence, a single blip generates a non-local field that can be felt
across all of space.

By expressing the blip operators in terms of monochromatic photon operators
$\widetilde{a}_{s\lambda}(\mathbf{k},t)$, we have also shown that our approach
is consistent with the standard theory up to the addition of negative-
frequency states. The Hilbert space in three dimensions, as in one dimension,
is only doubled, and this doubling takes into account the negative-frequency
states that are usually overlooked. As states of the quantised EM field now
evolve with negative as well as positive frequencies, the dynamical
Hamiltonian and the energy observable are no longer equal, and the two
operators only coincide when restricted to the subset of positive-frequency
states.

The energy observable (5.49) is strictly positive due to the square in the
integrand. This observable also contains pure annihilation and pure creation
components that do not occur in the standard energy observable. As in one
dimension, these extra terms determine the interference that occurs between
the more traditional positive-frequency excitations and the new negative-
frequency excitations. The addition of these terms also means that single-
photon states are not eigenstates of the energy observable. Single-photon
states are, however, eigenstates of the dynamical Hamiltonian (5.59). This is
so because the dynamical Hamiltonian is a recreation operator that annihilates
exactly one excitation and replaces it at a new location. Unlike the energy
observable, the dynamical Hamiltonian has both positive and negative
eigenvalues. One can see that by changing the value of $s$, the dynamical
Hamiltonian changes sign and reverses the apparent flow of time.

##  Part III Applications

###  Chapter 6 A Fermi problem for light

Fermi’s two-atom problem [1] provides an interesting model for studying
causality in quantum electrodynamics. The problem is complicated, however, by
the precise manner in which the atoms and fields are coupled together. In this
chapter we study a simpler but equivalent problem that does not use atoms,
involving a beam-splitter and a pair of photon detectors, in order to
determine whether the field propagates causally or not. In Section 6.2 we
describe this experiment in more detail and study the propagation of light
through the new system. In Section 6.3 we construct a pair of detection
operators representing each detector in the experiment, and show that a
causality violation takes place for light described by the standard theory. In
Section 6.4 we show that there is no causality violation for light when we
introduce negative-frequency photons. We conclude this chapter with a
discussion in Section 6.5.

####  6.1 Introduction: Fermi’s original two-atom problem

In empty space, light propagates at a constant and finite speed. The speed of
light is the same for all time-like observers, irrespective of their motion
relative to one another or to that of the light source. In order to ensure the
validity of relativity theory, it is crucial that the speed of light cannot be
exceeded by any signal capable of relaying causal information, or, in other
words, any signal capable of relaying to the receiver some knowledge of the
actions of the transmitter. Naturally, this speed limit introduces a lower
bound on the time it takes for particles to propagate from one place to
another, and it is important that this lower bound be apparent in any quantum
theory.

In 1932, Fermi published a paper on the quantisation of the EM field and its
interactions with matter [1]. Here he considered the following problem.
Consider two identical atoms separated by some large distance $R$. Suppose
also that one of these atoms is in its first excited state whilst the other
atom and the pervading EM field are in their ground states. After a negligible
time the excited atom will decay emitting a single photon that then propagates
towards the second atom. Being in its ground state, the second atom will
absorb the incoming photon and “jump” to its excited state. How long should
one have to wait after the photon is first emitted until it is absorbed again
by the second atom? In an intuitive picture of this interaction, the energy
transferred from the first atom is carried by a “flying” photon, as
illustrated in Fig. 6.1, that propagates from the first atom to the second.
Taking causality into consideration, the second atom should not become excited
until a time $R/c$ has passed.

Figure 6.1: A illustration of the expected behaviour in Fermi’s two-atom
problem. An atom initially in its excited state drops to its ground state
generating a single photon from the vacuum. The photon propagates to a
distant, unexcited atom causing it to transition into its excited state. The
second atom is expected to be found in an excited state only after a time
$R/c$ has passed.

In Ref. [1], Fermi used a perturbative approach to determine the probability
of finding the second atom in an excited state, and found that this
probability remains strictly zero for all times less than $R/c$. This is the
desired and expected result. In Fermi’s calculations, however, an
approximation was used that extended the frequency of photon states into the
negative frequency domain. Shirokov [90] and others [94, 95, 96, 97] have
since pointed out that without making this assumption the solution to Fermi’s
problem results in a small but non-zero probability for the second atom to be
found in an excited state before the expected time, thereby violating Einstein
causality.

In the years since, many attempts have been made to reconcile the predictions
of special relativity and quantum optical models for the experimental scenario
illustrated in Fig. 6.1, [99, 100, 101, 102, 103, 104, 105, 106, 107, 108]. In
some of these cases, however, a causal result is only found due to various
approximations and modifications of the standard theory [99, 107], or by
looking at the problem from a different view point, for example, Refs. [100,
105] look at the problem from the view point of algebraic QFT rather than of
photon wave packets. Other authors have found that there is no strict Einstein
causality [89, 91, 92, 93, 94, 95, 96, 97], and that there is always a non-
zero probability of immediately exciting the second atom. In these cases,
however, there is no superluminal signalling as this probability does not
depend on the presence of the first atom, but is instead attributed to vacuum
correlations between the two atoms [91, 92, 94, 96]. Hence, there is no causal
relationship between measurements at the two atoms. Such a result, however, is
closely related to the particular choice of the coupling between the EM field
and the two atoms, or upon the final states of the field and source atom [91,
94]. For instance, In Ref. [91], causality is preserved only when one refrains
from applying the rotating wave approximation, the validity of which is also
investigated in [107, 171]. Unfortunately, due to the expected non-causal
effects in the original two-atom Fermi problem being very weak, it is very
difficult to verify their presence experimentally [98].

By taking a wider view of the Fermi problem and not focussing solely on the
atom-field interaction, Hegerfeldt demonstrated that the violation of strict
causality is a result of the fact that the system Hamiltonian is bounded below
[89, 93]. This is a direct result of Hegerfeldt’s theorem [72], which states
that, under a projective measurement, a single-photon wave packet initially
localised to a finite region will spread out immediately. The work in Ref.
[93] has been criticised in relation to the validity of the measuring process
and the fact that possible loopholes must be treated with adequate care [100,
103, 105]. For instance, in Ref. [103], the probability of exciting the second
atom is not represented as a projective measurement and Hegerfeldt’s
conclusions are therefore avoided. Nevertheless, in a position wave function
approach to quantum optics, Hegerfeldt’s theorem appears to demonstrate that
photons can propagate at superluminal speeds.

In summary, although there appears to be no superluminal signalling between
the two atoms in Fig. 6.1, it is unclear whether there is any evidence of a
causality violation or not. The results on this topic vary and are intimately
related to the particular way in which the atom and field are coupled
together; however, the existence of superluminal correlations would seem to
suggest that photon wave packets may be able to propagate at speeds exceeding
the speed of light. The work of Hegerfeldt suggests that the presence of atoms
is not crucial to the calculations that predict non-causal behaviour. In this
chapter, therefore, we analyse a simpler but analogous problem to Fermi’s two-
atom problem involving a beam-splitter and two spatially localised detectors.
We find in this chapter that single-photon wave packets described in the
standard quantum theory do not propagate as would be expected for a localised
light pulse. When expressed as a superposition of blip operators, however,
localised single-photon wave packets propagate causally.

####  6.2 An alternative experiment

The alternative experimental setup that we shall consider in this chapter is
illustrated in Fig. 6.2. In this section we shall construct an analogy between
Fermi’s original two-atom system and our experiment in order to determine how
a causality violation may be verified. We shall also describe the propagation
of light through the beam-splitter. Initially we shall only consider positive-
frequency photon states, as are present in standard quantisations, but later
in this chapter we shall consider the propagation of both the positive-
frequency and negative-frequency photons described in Chapter 4.

#####  6.2.1 Our experiment

The interferometer in Fig. 6.2 consists of a source of spatially-localised
ultra-broadband light pulses [147, 148, 149, 150] aimed at a beam-splitter,
and a pair of photon detectors positioned along the output trajectories of the
beam-splitter. We denote the detector at the end of the vertical arm Detector
1 and the detector at the end of the horizontal arm Detector 2. We assume that
the separation between the beam-splitter and Detector 2 is much greater than
that between the beam-splitter and Detector 1.

Figure 6.2: An interferometer consisting of a beam-splitter and two photon
detectors. A short pulse of light approaches the beam-splitter from the left
and is then diverted towards the two detectors. The distance from the beam-
splitter to Detector 1 is $L_{1}$ and the distance from the beam-splitter to
Detector 2 is $L_{2}\gg L_{1}$. In relation to Fig. 6.1, region 1 corresponds
to the first atom whereas region 2 corresponds to the second atom.

In the original formulation of Fermi’s two-atom problem, the photon leaves the
first atom at the time the system is turned on and the atom begins to decay.
The photon then arrives at the second atom the moment we first register a non-
zero probability of finding the second atom in an excited state. The
difference between these times is the photon travel time from which a speed is
calculated. In the interferometer system illustrated in Fig. 6.2, all photons
will initially be emitted by a source to the left of the beam-splitter. Upon
reaching the beam-splitter, some photons will be directed towards Detector 1
and the rest towards Detector 2. Due to the presence of the beam-splitter, the
positions of photons propagating along the vertical and horizontal arms of the
interferometer are highly correlated, and a signal at Detector 1 will notify
us of the position of the remaining photons in the horizontal arm to within a
width of the wave packet.

After passing through the beam-splitter, the two output wave packets have the
same wave function as the initial wave packet entering the beam-splitter. If
Detector 1 has a particular probability of clicking at a time $t_{1}$, then,
assuming that all wave packets propagate at the speed of light, the
probability for a subsequent click at Detector 2 at a time $t_{2}$ must be
directly proportional to the initial probability for Detector 1 to click after
taking into account a short time delay caused by the difference in the
distances from the beam-splitter to the detectors. More specifically, when the
photon wave packets propagate at the speed of light, the two probabilities
must be directly proportional when

$t_{2}=t_{1}+(L_{2}-L_{1})/c$ (6.1)

Furthermore, there must be a zero probability for Detector 2 to click before a
sufficient time has elapsed after Detector 1 has clicked.

This type of causality, in which correlations cannot be generated at speeds
exceeding the speed of light, is known as Einstein Causality. From the point
of view of localised photon wave packets, if a wave packet propagates at the
speed of light, correlations cannot be generated at speeds exceeding the speed
of light. This experiment specifically tests Einstein causality, hence a
verification of causality is possible by ensuring that the probabilities for
Detectors 1 and 2 to click obey the relationship described above. In this
chapter we carry out these calculations for photons in both the standard and
blip descriptions of light.

The model we propose here offers a simpler theoretical model for studying
possible causality violations in a way that can be realised in a laboratory.
In this experiment there is a clear signature for the beginning and the end of
the photon’s journey in the form of the detector clicks. This is well defined
in the theory and in a possible experiment. Furthermore, the nature of the
coupling between photons and atoms does not need to be discussed or justified.

#####  6.2.2 Light in the interferometer

We next describe the propagation of a short photon wave packet through the
beam-splitter illustrated in Fig. 6.2 by introducing a unitary scattering
operator $S(t)$. We use the standard description of photon wave packets in
this section, but a generalisation to our new quantisation is easily made
later on.

###### One-dimensional systems

In the experimental setup illustrated in Fig. 6.2, we consider light
propagating along two perpendicular arms of an interferometer. In each arm of
the interferometer, light propagates either towards or away from the beam-
splitter independently of what is going on in the other; if we ignore the
effect of the beam-splitter that is. In reality, the photon wave packets
travelling along each arm would be localised in the plane orthogonal to their
direction of propagation, but as there is no interference between light in the
two arms of the interferometer except at the beam-splitter, it is sufficient
for our purposes to treat light in the interferometer as two distinct one-
dimensional systems: one system describing light propagating along the
vertical arm, and the other describing light propagating along the horizontal
arm. States in different arms occupy distinct regions of the total Hilbert
space. We shall proceed next to construct wave packets of light in one
dimension in preparation for our discussion on the beam-splitter in Section
6.2.4.

###### Single-photon wave packets

In the quantised theory of the free EM field, the stationary solutions of
Maxwell’s equations are associated with a set of bosonic particles which form
the basic energy quanta of the system. In Section 3.2, these energy quanta are
associated with the running wave solutions of Maxwell’s equations in one
dimension. This approach is particularly well suited for modelling light in
the experimental setup considered in Fig. 6.2 which is concerned with light
propagating in one direction only. In this chapter we shall consider again the
set of creation and annihilation operators associated with light propagating
in one dimension with a single frequency and a given polarisation,
$a^{\dagger}_{k\lambda}$ and $a_{k\lambda}$. The magnitude and sign of $k$
denote the frequency divided by the speed of light and the direction of
propagation respectively.

As in Section 3.2.2, a single-photon wave packet is a linear superposition of
the single-photon excitations generated by applying the
$a^{\dagger}_{k\lambda}$ creation operators to the vacuum state. Hence,

$\ket{\psi}=\sum_{\lambda={\mathsf{H},\mathsf{V}}}\int_{-\infty}^{\infty}\text{d}k\;\psi_{k\lambda}\,a^{\dagger}_{k\lambda}\ket{0}$
(6.2)

with complex coefficients $\psi_{k\lambda}$. This state is normalised when the
following condition holds:

$\sum_{\lambda={\mathsf{H},\mathsf{V}}}\int_{-\infty}^{\infty}\text{d}k\;|\psi_{k\lambda}|^{2}=1.$
(6.3)

###### Many-photon wave packets

The single-photon creation operator that generates the above wave packet from
the vacuum is given by $a^{\dagger}$ where

$a=\sum_{\lambda={\mathsf{H},\mathsf{V}}}\int_{-\infty}^{\infty}\text{d}k\;\psi^{*}_{k\lambda}\,a_{k\lambda}$
(6.4)

Using Eq. (6.4) and the commutation relation (3.14) one can check that $a$
satisfies the single-photon commutation relation
$\big{[}a,a^{\dagger}\big{]}=1$ when the state is properly normalised
according to Eq. (6.3). To construct a many-photon state we proceed as in
Section 3.2.2. In this chapter we consider wave packets for any general set of
coefficients $c_{n}$. We can then afterwards examine particular choices of
these coefficients.

#####  6.2.3 The dynamics of photon wave packets

The dynamics of a state in the Hilbert space can be determined by proceeding
as in Section 3.1 and solving the Schrödinger or Heisenberg equation. In
standard descriptions of the quantised EM field, the Hamiltonian is given by
the energy observable (3.16). This is in contrast to the quantisations
developed in Chapters 4 and 5 where the dynamical Hamiltonian determines the
time-evolution of the system. For simplicity, we work in the Heisenberg
picture that was described in Section 3.1.3.

By Solving Heisenberg’s equation with the Hamiltonian given in Eq. (3.16), the
single-photon annihilation operator for a time-dependent state is given by

$\displaystyle a(t)=U(t,0)\,a\,U^{\dagger}(t,0)$ $\displaystyle=$
$\displaystyle\sum_{\lambda={\sf H},{\sf
V}}\int_{-\infty}^{\infty}\text{d}k\,\psi^{*}_{k\lambda}\,{\rm e}^{{\rm
i}c|k|t}\,a_{k\lambda}.$ (6.5)

Here $U(t,0)=\exp(-{\rm i}Ht/\hbar)$ is the usual free-space time-evolution
operator of the quantised EM field.

Suppose that the EM field has initially been prepared in the many-particle
state $\ket{\psi}$, which we introduced in Eq. (3.20) for the single-photon
operators defined in Eq. (6.5). The state at a later time $t$ is found in an
equivalent superposition of number states constructed in an analogous way
using the time-evolved annihilation operator $a(t)$ defined in Eq. (6.5). To
show this we substitute the identity $\text{id}=U^{\dagger}(t,0)U(t,0)$
between each time-independent creation operator in the many-particle state and
use the relation $U(t,0)\ket{0}=\ket{0}$. This leads to the expression

$\displaystyle U(t,0)\,|\psi\rangle$ $\displaystyle=$