$\displaystyle\sum_{n=0}^{\infty}{c_{n}\over
n!}\left(U(t,0)\,a^{\dagger}\,U^{\dagger}(t,0)\right)^{n}|0\rangle$ (6.6)
$\displaystyle=$ $\displaystyle\sum_{n=0}^{\infty}{c_{n}\over
n!}\left(a^{\dagger}(t)\right)^{n}|0\rangle.$

#####  6.2.4 The beam-splitter

###### The Hilbert space of the beam-splitter system

The purpose of the beam splitter in Fig. 6.2 is to split any incoming wave
packets into two parts that will propagate along the vertical and horizontal
arms of the interferometer respectively. To distinguish the photons in the
vertical arm from those in the horizontal arm, we denote the annihilation
operators of the latter by $b_{k\lambda}$ and the former by $c_{k\lambda}$.
The total single-photon Hilbert space is then given by the direct sum of
$\mathcal{H}^{b}$ and $\mathcal{H}^{c}$,
$\mathcal{H}^{b}\bigoplus\mathcal{H}^{c}$, which are the single-photon Hilbert
spaces for photons in the horizontal and vertical arms respectively. Since the
single-photon states
$|1^{(b)}_{k\lambda}\rangle=b^{\dagger}_{k\lambda}|0\rangle$ and
$|1^{(c)}_{k\lambda}\rangle=c^{\dagger}_{k\lambda}|0\rangle$ evolve
independently, they must be orthogonal to one another. This implies that

$\displaystyle\langle
1^{i}_{k\lambda}|1^{i}_{k^{\prime}\lambda^{\prime}}\rangle=\big{[}b_{k\lambda},b^{\dagger}_{k^{\prime}\lambda^{\prime}}\big{]}=\big{[}c_{k\lambda},c^{\dagger}_{k^{\prime}\lambda^{\prime}}\big{]}=\delta_{\lambda,\lambda^{\prime}}\,\delta(k-k^{\prime})$
(6.7)

where $i=b,c$. All other commutators vanish. Thus the $b_{k\lambda}$ and
$c_{k\lambda}$ operators obey the usual bosonic commutation relations.

Next, in analogy to Eq. (6.4), we construct annihilation operators $b$ and $c$
for normalised wave packets travelling along each arm of the interferometer,

$\displaystyle b$ $\displaystyle=$ $\displaystyle\sum_{\lambda={\sf H},{\sf
V}}\int_{-\infty}^{\infty}\text{d}k\,\psi^{(b)*}_{k\lambda}\,b_{k\lambda}\,,$
$\displaystyle c$ $\displaystyle=$ $\displaystyle\sum_{\lambda={\sf H},{\sf
V}}\int_{-\infty}^{\infty}\text{d}k\,\psi^{(c)*}_{k\lambda}\,c_{k\lambda}\,.$
(6.8)

The time evolution of the corresponding single-photon operators can be
determined by solving Heisenberg’s equation (4.23). The Hamiltonian of the
total system is given by the sum of two harmonic oscillator Hamiltonians of
the type given in Eq. (3.16): one in terms of the $b_{k\lambda}$ operators,
and the other in terms of the $c_{k\lambda}$ operators.

###### Scattering operators

As an initial state passes through the beam-splitter, a superposition of $b$
and $c$ states will be produced. The effect of this splitting on an initial
state $|\psi_{\text{in}}(0)\rangle$ can be described with the help of a
scattering operator $S(t,0)$. After passing through the beam-splitter, the
state of the outgoing light at a time $t$ is given by

$\displaystyle\ket{\psi_{\text{out}}(t)}$ $\displaystyle=$ $\displaystyle
S(t,0)\,\ket{\psi_{\text{in}}(0)}.$ (6.9)

As the time-evolution of a state before and after passing through the beam
splitter is the same, it does not matter when the wave packet is split into
two parts. Without any restrictions, we can therefore assume that

$\displaystyle S(t,0)$ $\displaystyle=$ $\displaystyle SU(t,0)\,,$ (6.10)

where $S$ satisfying $SS^{\dagger}=S^{\dagger}S=1$ is a unitary scattering
operator. In the case of a 50:50 beam-splitter, by denoting the annihilation
operators for a photon that enters the beam-splitter along the $x$ axis
$a_{k\lambda}$, $S$ changes the single-photon excitations such that [137, 138]

$\displaystyle Sa_{k\lambda}S^{\dagger}$ $\displaystyle=$
$\displaystyle{1\over\sqrt{2}}\left(b_{k\lambda}+{\rm
i}c_{k\lambda}\right)\,.$ (6.11)

Taking this into account we find that the annihilation operator $a$ with the
complex coefficients $\psi^{*}_{k\lambda}$ introduced in Eq. (6.5) transforms
such that

$\displaystyle Sa(t)S^{\dagger}$ $\displaystyle=$
$\displaystyle{1\over\sqrt{2}}\sum_{\lambda={\sf H},{\sf
V}}\int_{-\infty}^{\infty}\textrm{d}k\,\psi^{*}_{k\lambda}\,{\rm e}^{{\rm
i}c|k|t}\left(b_{k\lambda}+{\rm i}c_{k\lambda}\right)$ (6.12) $\displaystyle=$
$\displaystyle{1\over\sqrt{2}}\left(b(t)+{\rm i}c(t)\right).$

Suppose that the light source in Fig. 6.2 prepares the light in a
superposition state such that $|\psi_{\text{in}}(0)\rangle=|\psi\rangle$. We
shall assume that the second input port remains empty. Using Eq. (6.12), the
light exiting the beam-splitter at a time $t$ will be given by

$\displaystyle\ket{\psi_{\text{out}}(t)}$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}{c_{n}\over
n!}\left({1\over\sqrt{2}}\left(b^{\dagger}(t)+{\rm
i}c^{\dagger}(t)\right)\right)^{n}\ket{0}.$ (6.13)

The right hand side of Eq. (6.13) is determined using the unitarity of $S$ and
substituting $S^{\dagger}\,S$ between each input creation operator.

####  6.3 Photon detection in the interferometer

In this section we construct two detection operators that represent the
measurement of a photon by Detectors 1 and 2 in Fig. 6.2. Interpreting the
photon wave function as the probability amplitude for detection, Detectors 1
and 2 are represented by a pair of projection operators. Later in this section
we shall calculate and compare the probabilities for Detector 1 to click at a
time $t_{1}$, and for Detector 2 to click at a later time $t_{2}$ conditional
to the earlier click at Detector 1. We then show that a causality violation
takes place in the standard theory of the quantised EM field. For simplicity,
and since it does not affect our conclusions, from now on we shall only
consider light with a single fixed polarisation $\lambda$. This could be
achieved by placing a polarising filter in front of the detectors or behind
the source.

#####  6.3.1 Detection operators

When light enters a detection device, it will be absorbed by the device and
cause a click at a given time $t$ with a particular probability
$P_{\text{click}}(t)$. The act of detecting a photon is represented by a
detection operator acting on the Hilbert space. In this chapter we shall
assume that the detection operator represents an ideal measurement on the
field and that we therefore do not need to pay any attention to the internal
mechanism of the detecting device. Furthermore, we shall assume that our
photon detector is a perfect detector, although the experiment would work
equally well with a less efficient detector. This implies that a light pulse
entering the detector will be detected with absolute certainty, and that the
detector will never produce a false reading. When a detector makes a
measurement on a field, it does so with respect to a particular measurable
characteristic of the field. In this chapter, the characteristic of interest
is the position of a photon. The detector is designed to respond to any photon
that comes within the bounds of the detecting equipment and not otherwise.

As localised particles are represented by the locally bosonic position
eigenstates of a system, the detector responds to the position eigenstates of
the field that lie within the detecting region. In spite of this, the
detection operator does not usually coincide with the position observable in
that region. Rather the detection operator projects onto the eigenstates of
the position observable with a particular probability. This probability
represents the belief that the projection onto a position eigenstate indicates
a photon at that position [172]. Since we are assuming that our detectors are
perfect detectors, this belief is an absolute certainty. We therefore
associate a unit probability with each projector. The resulting mathematical
structure of our detection operator is a projection valued measure associated
with every possible region of space. A projection valued measure is suitable
for a detection operator because, for any excited state, the probability of a
detection is always positive and there is a unit probability of finding a
photon somewhere in space.

In the following the probability for a detector to click $P_{\text{click}}(t)$
at a time $t$ is determined by calculating the expectation value of the
projection operator $P_{\text{detect}}(t)$ also at a time $t$ with respect to
the state of the light incident on the detector. As the detection operator is
positive, any chance to detect a photon always results in a non-zero
expectation value. Using the idempotent property of the projection operator,
$P^{2}=P$, the expectation value of the detection operator with respect to a
state $\ket{\psi}$ is equal to the square magnitude of the projection operator
acting on that state. Hence, the probability of the detector clicking at a
time $t$ is given by

$\displaystyle P_{\rm click}(t)$ $\displaystyle=$ $\displaystyle\left\|P_{\rm
detect}(t)\,\ket{\psi(0)}\right\|^{2}=\left\|P_{\rm
detect}\,\ket{\psi(-t)}\right\|^{2}$ (6.14)

where the projection operator $P_{\text{detect}}$ is defined at $t=0$. In the
remainder of this section we identify appropriate detection operators
$P^{(1)}_{\rm detect}$ and $P^{(2)}_{\rm detect}$ for Detectors 1 and 2 in the
setup shown in Fig. 6.2 and calculate their respective detection rates.

#####  6.3.2 Detection operators in the interferometer

###### Localised single-photon wave packets

The first step in constructing the appropriate detection operators is to
identify a set of position eigenstates for light in the two arms of the
interferometer onto which states will be projected. As discussed in the
introduction of this thesis, the task of constructing localised photon states
in quantum theory has been beset with difficulties. Indeed, Part II of this
thesis is dedicated to constructing a local theory of the free quantised EM
field. In this section, we shall construct localised states in the standard
Hilbert space (Section 3.2) in the manner of Mandel [68, 69] and Cook [66,
67]. There are several reasons for considering these states as the localised
photon states. First of all, these states are locally orthogonal to one
another, which is consistent with our definition of localised states in
Chapters 4 and 5. Furthermore, their inner product has the correct dimensions
to be interpreted as a probability density, and, in one dimension, the
position-dependent number operator represents exactly the photon number
density operator [68].

According to this definition, we define a pair of localised single-excitation
states $\ket{1_{b}(x)}$ and $\ket{1_{c}(y)}$ in the following way:

$\displaystyle|1_{b}(x)\rangle$ $\displaystyle=$
$\displaystyle{1\over\sqrt{2\pi}}\int_{-\infty}^{\infty}\textrm{d}k\,{\rm
e}^{ikx}\,b^{\dagger}_{k\lambda}\,\ket{0}\,,$ $\displaystyle|1_{c}(y)\rangle$
$\displaystyle=$
$\displaystyle{1\over\sqrt{2\pi}}\int_{-\infty}^{\infty}\textrm{d}k\,{\rm
e}^{iky}\,c^{\dagger}_{k\lambda}\,\ket{0}\,$ (6.15)

Here $\ket{0}$ is the vacuum state for the total combined system
$\ket{0}=\ket{0_{b}}\bigotimes\ket{0_{c}}$. These states relate to the non-
local single-photon states $b^{\dagger}_{k\lambda}\ket{0}$ and
$c^{\dagger}_{k\lambda}\ket{0}$ via a Fourier transform and describe localised
single-photon wave packets at positions $x$ and $y$ along the horizontal and
vertical axes respectively. There are no other field excitations anywhere else
in the system.

###### Number-resolving detectors

For simplicity, in the setup shown in Fig. 6.2 we shall assume that our
detectors are number-resolving detectors and designed such that they only
react to single-excitation states, i.e. states which can be obtained by
applying a local photon creation operator, Eq. (6.3.2), to the vacuum state.
This experiment could also be done with non-number resolving detectors, but
since contributions from different number states are purely additive and
positive, any violation of causality that occurs for the number-resolving
detectors would also be present for more general single-photon detection
operators.

We shall define a pair of detectors that each have a finite length $D$ and
respond only to photons within this range. The detection operators project
onto the single-photon subspace of photons spanned by the states (6.3.2) that
lie within the region of the detectors. For convenience we also suppose that
the beam-splitter is positioned at the origin of both the $x$ and $y$
coordinate axes. Consequently, Detector 1 extends from $L_{1}$ to $L_{1}+D$
and Detector 2 extends from $L_{2}$ to $L_{2}+D$. The detection operators may
therefore be written as

$\displaystyle P^{(1)}_{\rm detect}$ $\displaystyle=$
$\displaystyle\int_{L_{1}}^{L_{1}+D}\textrm{d}y\,|1_{c}(y)\rangle\langle
1_{c}(y)|\,,$ $\displaystyle P^{(2)}_{\rm detect}$ $\displaystyle=$
$\displaystyle\int_{L_{2}}^{L_{2}+D}\textrm{d}x\,|1_{b}(x)\rangle\langle
1_{b}(x)|\,.$ (6.16)

Using the definition of the localised excitations in Eq. (6.3.2), one may
check that $(P^{(i)}_{\rm detect})^{2}=P^{(i)}_{\rm detect}$ for both $i=1,2$.

###### Signalling between detectors

Using the commutation relations in Eq. (6.7) one may verify that any operator
that acts only on the $b$-photons in the horizontal arm will always commute
with any operator that acts only on the $c$-photons in the vertical arm. It is
important that observables on the different arms of the interferometer commute
with each other as this implies that the probability of measuring a particular
outcome of an experiment on the $b$-photons is statistically independent of
any operation carried out on the $c$-photons and vice-versa. In other words,
the frequency of a particular result occurring when the $b(c)$-photons are
measured does not depend on any experiment performed on the $c(b)$-photons.
This includes the measurements performed by Detectors $1$ and $2$.

The statistical independence of measurements on the two arms of the
interferometer does not, however, prohibit correlations existing between the
$b$\- and $c$-photons. In fact, this experiment relies on the strong
correlations between photons in each arm as these correlations provide a
relation between the expected times of arrival of a photon in each of the
detectors. When one detector clicks, if the light pulse is very short, a click
at the second detectors cannot be far behind. The fact that the position
observables in the two arms of the interferometer commute is important
because, although there are correlations between the $b$\- and $c$-photons, it
means that a detection in one detector cannot either hurry along or hold off a
detection in the other: the probability of detection in each will be
unchanged. Therefore, by making a measurement at Detector $1$ we set for
ourselves an expected time of arrival for a photon to arrive at Detector $2$
(assuming propagation at $c$), but do not change the independent trajectory of
light towards that detector. This ensures that we are calculating the speed of
a light pulse when unobserved.

#####  6.3.3 Probability of a detection

###### Detector 1

We now consider the probability for Detector 1 to detect exactly one photon at
some time $t_{1}$ and the probability for Detector 2 to detect exactly one
photon at a later time $t_{2}$ conditional on the earlier click at Detector 1
at time $t_{1}$. In the following, the probability for Detector 1 to click
will be denoted $P^{(1)}_{\text{click}}(t_{1})$. The probability for Detector
2 to click will be denoted $P^{(2)}_{\text{click}}(t_{2})$.

As the scattered light leaves the beam-splitter, the quantum state for the
component of light seen by Detector 1 at a time $t_{1}$ is given by the
projection of the detection operator $P^{(1)}_{\text{detect}}$ onto the output
state $\ket{\psi_{\text{out}}(-t_{1})}$ defined in Eq. (6.13). The probability
for Detector 1 to click is then, according to Eq. (6.14), given by the square
magnitude of the resulting state. Using the expression for the output state
$\ket{\psi_{\text{out}}(-t_{1})}$ in Eq. (6.13) and the detection operator for
Detector 1 in Eq. (6.3.2), one finds that the probability for Detector 1 to
click at a time $t_{1}$ is given by

$P^{(1)}_{\rm click}(t_{1})={1\over 2}\sum_{n=0}^{\infty}{|c_{n+1}|^{2}\over
2^{n}n!}\,\int_{L_{1}}^{L_{1}+D}\textrm{d}y\,\|\braket{1_{c}(y)}{c^{\dagger}(-t_{1})}{0_{c}}\|^{2}.$
(6.17)

When the coefficients $c_{n}$ are chosen for a coherent state (see Eq.
(3.21)), one may show that
$\sum_{n=0}|c_{n+1}|^{2}/2^{n}n!=\alpha^{2}\,\text{exp}[-|\alpha|^{2}/2]$.

###### Detector 2

The state seen by Detector 1 was given by $\ket{\psi_{\rm out}(-t_{1})}$
specified in Eq. (6.13). On the other hand, once a click has been registered
by Detector 1, the state seen by Detector 2 is given by

$\displaystyle\ket{\psi_{\rm out}(-t_{2})}$ $\displaystyle=$ $\displaystyle
U(-t_{2},-t_{1})\,{P^{(1)}_{\rm detect}\,\ket{\psi_{\rm
out}(-t_{1})}\over\|P^{(1)}_{\rm detect}\,\ket{\psi_{\rm
out}(-t_{1})}\|^{1/2}}.$ (6.18)

This state is normalised. To determine the probability for Detector 2 to
click, one proceeds analogously to the previous calculation by applying the
detection operator $P^{(2)}_{\text{detect}}$ to the output state
$\ket{\psi_{\text{out}}(-t_{2})}$. The squared magnitude of the resulting
state gives the probability of detection. Using Eqs. (6.18) and (6.13), and
both detection operators in Eq. (6.3.2) one finds that

$P^{(2)}_{\rm
click}(t_{2})=\frac{|c_{2}|^{2}}{4}\left(\sum_{n=0}^{\infty}{|c_{n+1}|^{2}\over
2^{n}n!}\right)^{-1}\,\int_{L_{2}}^{L_{2}+D}\textrm{d}x\,\|\braket{1_{b}(x)}{b^{\dagger}(-t_{2})}{0_{b}}\|^{2}.$
(6.19)

This probability depends solely on the two-photon contribution of the initial
output state in Eq. (6.13). This is because both detectors will only register
a photon if there is exactly one photon in each arm of the interferometer. It
is therefore only the two-photon part of the output state that contributes to
Eq. (6.19).

###### Transition amplitudes

By expressing the single-photon creation operators and the localised photon
states in their Fourier representations, we may show that the transition
amplitudes within the modulus signs in Eqs. (6.17) and (6.19) are equivalent
to the integral expressions

$\displaystyle\braket{1_{b}(x)}{b^{\dagger}(-t_{2})}{0_{b}}$ $\displaystyle=$
$\displaystyle{1\over\sqrt{2\pi}}\int_{-\infty}^{\infty}{\rm d}k\,{\rm
e}^{-i(kx-c|k|t_{2})}\,\psi_{k\lambda}\,,$
$\displaystyle\braket{1_{c}(y)}{c^{\dagger}(-t_{1})}{0_{c}}$ $\displaystyle=$
$\displaystyle{1\over\sqrt{2\pi}}\int_{-\infty}^{\infty}{\rm d}k\,{\rm
e}^{-i(ky-c|k|t_{1})}\,\psi_{k\lambda}$ (6.20)

for the particular $\lambda$ that we have chosen for this output state. If the
initially prepared incoming wave packet contains photons with positive $k$
only, the above transition amplitudes depend on $y-ct_{1}$ and $x-ct_{2}$
respectively. This is as one would expect for light moving towards the two
detectors.

#####  6.3.4 Causality violations in the standard theory

We now study our expressions for the detection probabilities (6.17) and (6.19)
and show that a causality violation takes place when using the standard
description of single-photon wave packets.

###### Expected results

The probability for Detector 2 to click at a time $t_{2}$ conditional on a
previous click by Detector 1 at a time $t_{1}$ was given in Eq. (6.19). In
order to determine whether causality is satisfied, let us look more closely at
the first transition amplitude in Eq. (6.3.3) that appears in Eq. (6.19). By
decomposing the transition amplitude into its Fourier components, and making
the substitutions $x=y+L_{2}-L_{1}$ and $t_{2}=t_{1}+(L_{2}-L_{1})/c$, we find
that

$\braket{1_{b}(x)}{b^{\dagger}(-t_{2})}{0_{b}}={1\over\sqrt{2\pi}}\int_{-\infty}^{\infty}{\rm
d}k\,{\rm e}^{-i(ky-c|k|t_{1})}\,e^{-i(k-|k|)(L_{2}-L_{1})}\,\psi_{k\lambda}$
(6.21)

By comparing this result with the transition amplitude in Eq. (6.20) we can
see that the two expressions are almost identical. The only difference here is
that the amplitude in Eq. (6.21) contains an additional exponential term.

Recall here that causality applies when the probabilities
$P^{(1)}_{\text{click}}(t_{1})$ and
$P^{(2)}_{\text{click}}(t_{1}+(L_{2}-L_{1})/c)$ are proportional to each
other. This occurs only when the second line of Eq. (6.20) and Eq. (6.21) are
also proportional to each other. As the Fourier transform of any absolutely
integrable function is unique [155], we can conclude that the two transition
amplitudes in Eq. (6.20) are proportional, and in fact equal, only when the
second exponential in Eq. (6.21) vanishes altogether. This occurs on only two
occasions: when $L_{1}=L_{2}$, or when $k=|k|$. Next we shall examine these
conditions in more detail in order to determine the characteristics of the
system or state under which causality is preserved.

###### An inconsistency with Einstein’s relativity

Let us first consider the condition that $L_{2}=L_{1}$. It was asserted at the
beginning of the chapter that the horizontal arm of the interferometer was
much longer than the vertical arm. This implies that $L_{2}\gg L_{1}$ and
therefore that the detectors do not click at the same time. Hence, we can
immediately disregard this condition and assume that $L_{1}\neq L_{2}$. The
second condition states that $k=|k|$ and, therefore, that $k$ takes positive
values only. In the standard theory of the quantised EM field, the sign of $k$
indicates the direction of propagation. In this experiment, light propagates
in a single direction away from the beam-splitter and towards the detectors.
The above condition, then, is only the statement that the state under
consideration ought to contain contributions only from photons propagating
towards rather than away from the detectors. On first inspection this
condition seems to be one that we can accept perfectly well. We should not,
however, be too hasty with our conclusions. Let us have a closer look at Eq.
(6.2) which is the general expression for a single-photon wave packet.

The photon wave packet propagates causally only when we neglect the negative
$k$ terms. These wave packets are defined in Eq. (6.2) by changing the limits
of the integral from $(-\infty,\infty)$ to $[0,\infty)$ or by replacing the
momentum-dependent wave function $\psi_{k\lambda}$ with
$\theta(k)\,\psi_{k\lambda}$ where $\theta(k)$ is the Heaviside step-function.
Using the definition of perfectly localised states given in Eq. (6.3.2), the
position-dependent wave function of a state is given by the Fourier transform
of the momentum-dependent wave function. The Fourier transform of the wave
function $\theta(k)\,\psi_{k\lambda}$ is non-zero at all positions for all
allowed choices of $\psi_{k\lambda}$. Consequently, a single-photon wave
packet that propagates in a single direction cannot be localised completely to
a finite region, but must be spread out across all available space.

At the beginning of this chapter we asserted that the photons entering the
interferometer were initially localised somewhere to the left of the beam-
splitter. This is a condition that could be imposed by introducing a shutter
mechanism in front of the beam-splitter. Since this assumption implies that
the photon wave packet contains contributions from all wave numbers, we must
also disregard the second causality condition. Thus, we find that, in the
standard description of the quantised EM field, localised wave packets do not
propagate at the speed of light, and correlations do form at superluminal
speeds. In the language of single-photon wave functions, this implies that
photon wave packets may propagate at speeds exceeding the speed of light, and
consequently that either causality is violated or there does not exist a wave
function that represents the position of a single photon propagating through
space. In the following section we shall examine whether this dilemma may be
resolved by describing the light propagating along each arm of the
interferometer as a system of blips.

####  6.4 Resolving causality issues with blips

By carrying out similar calculations as in the previous section, in this
section we demonstrate that localised photon wave packets propagate causally
when described as a system of blips. We also discuss causality of the field
observables.

#####  6.4.1 An argument for negative-frequency photons

In the previous section we have shown that, in the standard theory of the
quantised EM field, single-photon wave packets cannot simultaneously be
localised and propagate at the speed of light. This problem arises because
causality applies only when the frequency of all photons involved is equal to
their wave number multiplied by the speed of light. Since localised wave
packets necessarily contain photons with both positive and negative wave
numbers, and frequency is strictly positive, this can never be the case. In
Chapter 4 of this thesis we introduced blip excitations in one dimension. We
discovered, however, that blip excitations contain contributions from photons
that evolve with negative as well as positive frequencies. This suggests that,
if the quantised field is described in terms of blip excitations, causality
may be preserved for local excitations.

Hence, in this section, we shall consider again the interferometer experiment
described in the introduction of this chapter. This time, however, we shall
model the EM field propagating along the arms of the interferometer as two
distinct systems of one-dimensional blips. To continue we shall first describe
the scattering of blip excitations by the beam-splitter and construct
detection operators appropriate for detecting blips at the desired locations.
We shall then repeat the calculations carried out in Section 6.3.3 in order to
determine whether the blip wave packets propagate causally or not.

#####  6.4.2 Scattering operators for blips

In this experiment we shall, as before, assume that the photon is localised
somewhere to the left of the interferometer. The initial state of the system
is given by

$\ket{\psi_{\text{in}}(0)}=\int_{-\infty}^{\infty}\text{d}x\;\psi_{s\lambda}(x)\,a^{\dagger}_{s\lambda}(x,0)\ket{0}$
(6.22)

Since the wave packet approaches the beam-splitter from the left we shall
assume that $\psi_{s\lambda}(x)=0$ when $s=-1$, or, in other words, the
initial wave packet only contains blips that propagate to the right. We shall
also assume that the wave function $\psi_{s\lambda}(x)$ is non-zero only in a
small region to the left of the beam-splitter and for a single polarisation
$\lambda$.

As in Eq. (6.9), the output state of the beam-splitter is determined by acting
upon the input state with a unitary scattering operator $S$ that splits the
incoming wave packet into two components. The annihilation operators for blips
propagating along the horizontal arm are denoted $b_{s\lambda}(y)$, whereas
those propagating along the vertical arm are denoted $c_{s\lambda}(x)$. As
before, these excitations are distinct, and all commutators between $b$ and
$c$ creation and annihilation operators are zero. The scattering operator is
modelled completely analogously to Eq. (6.11) which was defined for
monochromatic photon states. This transformation leaves both the wave number
and polarisation unchanged, but we shall now also assume that the parameter
$s$ is unchanged by the scattering transformation. The two output wave packets
therefore propagate away from the beam-splitter. The output state is then
given by Eq. (6.13). In this case, however, the time-dependent excitations are
given by

$\displaystyle b(t)$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\text{d}x\;\psi_{s\lambda}^{*}(x)\,b_{s\lambda}(x,t)$
$\displaystyle c(t)$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\text{d}y\;\psi_{s\lambda}^{*}(y)\,c_{s\lambda}(y,t)$
(6.23)

respectively. The $b_{s\lambda}(x,t)$ and $c_{s\lambda}(x,t)$ operators
independently satisfy the blip commutation relations given in Eq. (4.17).

#####  6.4.3 Detecting blips in the interferometer

###### Detection operators

We consider now a pair of number-resolving detection operators that respond to
single-blip states located within the detecting regions $L_{1}$ to $L_{1}+D$
up the vertical arm and $L_{2}$ to $L_{2}+D$ along the horizontal arm. Here we
shall drop the $s$ and $\lambda$ subscripts because we assume that all the
light reaching the detector propagates away from the beam-splitter and is
polarised in a single direction. The assumption that light reaching the
detectors propagates in only one direction is based upon the fact that the
photon source is to the left of the beam-splitter. As the blip excitations are
locally orthogonal to one another, the appropriate detection operators are
given just as in Eq. (6.3.2). Here, however, the single-excitation states
$\ket{1_{b}(x)}$ and $\ket{1_{c}(y)}$ are generated by acting upon the vacuum
state with the blip operators $b_{s\lambda}^{\dagger}(x,0)$ and
$c_{s\lambda}^{\dagger}(y,0)$ for the values $s$ and $\lambda$ selected by the
wave function $\psi_{s\lambda}(x)$.

###### Detection probabilities

We are now in a position to calculate the probabilities for Detectors 1 and 2
to click at the times $t_{1}$ and $t_{2}$ respectively. As before, the
probability for Detector 1 to click is evaluated by first evolving the beam-
splitter output state backwards in time by a time $t_{1}$, and then taking the
expectation value of the new projection operator $P^{(1)}_{\text{detect}}$.
The resulting probability $P^{(1)}_{\text{click}}$ is given by Eq. (6.17). In
the new expression, however, the local states and operators within the
transition amplitude are the local blip states and operators, and not the
local states defined in Eq. (6.3.2). Using the time-dependent inner product
given in Eq. (4.15) and the annihilation operators given in Eq. (6.4.2), the
probability of finding a blip at Detector 1 at a time $t_{1}$ is now given by

$P^{(1)}_{\rm click}(t_{1})={1\over 2}\sum_{n=0}^{\infty}{|c_{n+1}|^{2}\over
2^{n}n!}\,\int_{L_{1}}^{L_{1}+D}\textrm{d}y\,\|\psi(y-ct_{1})\|^{2}.$ (6.24)

We can now see, by looking at the square of the wave function, that the
probability of detecting a blip is equal to the probability of finding a blip
in a region of length $D$ shifted along from Detector 1 by a length $ct_{1}$.
Thus, there is only a non-zero probability to detect a blip at Detector 1 if
the wave packet is initially a distance of $ct_{1}$ away from the detector, as
we would expect.

Next, let us calculate the probability of finding a second blip at Detector 2
at a time $t_{2}$. After a detection has been made at Detector 1, the
remaining output state is again given by Eq. (6.18). After evolving the state
backwards to a time $-t_{2}$, we then calculate the expectation value of the
second detection operator $P^{(2)}_{\text{detect}}$. The conditional
probability for Detector 2 to click at a time $t_{2}$ is given by

$P^{(2)}_{\rm
click}(t_{2})=\frac{|c_{2}|^{2}}{4}\left(\sum_{n=0}^{\infty}{|c_{n+1}|^{2}\over
2^{n}n!}\right)^{-1}\,\int_{L_{2}}^{L_{2}+D}\textrm{d}x\,\|\psi(x-ct_{2})\|^{2}.$
(6.25)

This expression can be written in an alternative form by making the
substitution $y=x-L_{2}+L_{1}$, which is always true due to our choice of
detector placement. We shall also examine the particular case where
$t_{2}=t_{1}+(L_{2}-L_{1})/c$. We choose this time because it is the time we
would expect a light signal to take to travel from a distance $L_{1}$ to a
distance $L_{2}$. The resulting expression for $P^{(2)}_{\text{click}}(t_{2})$
is given by

$P^{(2)}_{\rm
click}(t_{2})=\frac{|c_{2}|^{2}}{4}\left(\sum_{n=0}^{\infty}{|c_{n+1}|^{2}\over
2^{n}n!}\right)^{-1}\,\int_{L_{1}}^{L_{1}+D}\textrm{d}x\,\|\psi(y-ct_{1})\|^{2}.$
(6.26)

One can see by cross-examining Eqs. (6.24) and (6.26), that the probabilities
of finding a blip in Detector 1 at a time $t_{1}$ and a blip in Detector 2 at
a time $t_{2}=t_{1}+(L_{2}-L_{1})/c$ are directly proportional to each other.
In both cases the overlap between the wave function and the detector regions
are identical, as we would expect after taking causality into consideration.
Thus, when the quantised EM field in one dimension is described using blips,
the position-dependent wave functions all propagate at the speed of light.

#####  6.4.4 Causality of field observables

In Section 6.3.4 we demonstrated that, when described using the standard
theory (see Section 3.2), single-photon wave functions cannot be
simultaneously localised to a finite region of space and propagate at the
speed of light. This is a slightly puzzling and apparently contradictory
result considering that the electric and magnetic field observables are
solutions of the wave equation for waves propagating at $c$ (Eq. (2.16)). In
fact, some authors claim that, in the context of algebraic quantum field
theory, the fact that the field observables obey this wave equation is enough
to demonstrate that no causality issues can occur [100, 105]. The problem of
infinite propagation speeds, however, is not directly related to the
propagation of the electric and magnetic field observables.

In the Heisenberg picture, the electric and magnetic field observables in one
dimension are given, as in Eqs. (3.31) and (3.32), by the expressions

$\displaystyle\mathbf{E}(x)$ $\displaystyle=$ $\displaystyle
c\int_{-\infty}^{\infty}\text{d}k\,\sqrt{\frac{\hbar|k|}{4\pi
Ac\varepsilon_{0}}}\,e^{ikx-i|k|ct}\,\left[a_{k\mathsf{H}}\,\widehat{\boldsymbol{y}}+a_{k\mathsf{V}}\,\widehat{\boldsymbol{z}}\right]+\text{H.c.}\,,$
$\displaystyle\mathbf{B}(x)$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\text{d}k\,\sqrt{\frac{\hbar|k|}{4\pi
Ac\varepsilon_{0}}}\,e^{ikx-i|k|ct}\,\text{sgn}(k)\left[a_{k\mathsf{H}}\,\widehat{\boldsymbol{z}}-a_{k\mathsf{V}}\,\widehat{\boldsymbol{y}}\right]+\text{H.c.}$
(6.27)

Let us now calculate the expectation value of these field observables with
respect to a horizontally polarised coherent state containing on average
$|\alpha|^{2}$ photons defined by the wave function $\psi_{k\mathsf{H}}$.
Taking into account Eqs. (3.20) and (3.21), one can show that

$\displaystyle\Braket{\mathbf{E}(x,t)}$ $\displaystyle=$ $\displaystyle
c\alpha\int_{-\infty}^{\infty}\textrm{d}k\,\sqrt{\frac{\hbar|k|}{4\pi
Ac\varepsilon_{0}}}\,{\rm
e}^{ikx-i|k|ct}\,\psi^{*}_{k\mathsf{H}}\,\widehat{\boldsymbol{y}}+\text{c.c.}\,,$
$\displaystyle\Braket{\mathbf{B}(x,t)}$ $\displaystyle=$
$\displaystyle\alpha\int_{-\infty}^{\infty}\textrm{d}k\,\sqrt{\frac{\hbar|k|}{4\pi
Ac\varepsilon_{0}}}\,{\rm
e}^{ikx-i|k|ct}\,\text{sgn}(k)\,\psi^{*}_{k\mathsf{H}}\,\widehat{\boldsymbol{z}}+\text{c.c.}$
(6.28)

Restricting the wave function to a single polarisation does not affect the
generality of this result.

In these expressions, as before, positive and negative $k$ correspond to left-
and right-travelling waves. In these expressions one can see that at a time
$t=0$ the expectation values of the field observables can have any
distribution in space we like, even localised ones. This is the case because
there are no restrictions on $\psi_{k\mathsf{H}}$ for any $k$. What is less
obvious is that these expectation values can have any distribution in space at
any time and propagate at the speed of light even if $k$ is restricted to only
positive or negative values. Unlike the expectation values of the projection
operators, when restricted to only positive values of $k$ we find that the
additional complex conjugate term completes the missing frequency terms to
create localised wave packets with a definite direction of propagation that do
not disperse as time evolves. Hence, the expectation values of the electric
and magnetic fields are complete solutions of Maxwell’s equations that
propagate both to the left and the right at a speed $c$.

Although the electric and magnetic field expectation values propagate causally
for positive frequency states, the expectation values of the detection
operators do not because they measure only the complex part of the field. They
do, however, propagate causally for states containing both positive- and
negative-frequency photons. To see that this is so consider the position
operator acting on the single-blip Hilbert space below:

$X=\sum_{s=\pm
1}\sum_{\lambda=\mathsf{H},\mathsf{V}}\int_{-\infty}^{\infty}\text{d}x\;x\,\ket{1_{s\lambda}(x)}\bra{1_{s\lambda}(x)}\,.$
(6.29)

The blip states are the eigenvalues of this observable. After evolving this
operator in time with respect to the dynamical Hamiltonian (4.29), the time-
dependent position operator is given by

$X(t)=\sum_{s=\pm
1}\sum_{\lambda={H,V}}\int_{\infty}^{\infty}\text{d}x\;(x+sct)\,\ket{1_{s\lambda}(x)}\bra{1_{s\lambda}(x)}.$
(6.30)

The eigenvalues of the position operator are the same at all times. As time
advances they are displaced by a distance $sct$ as would be expected for a
localised photon.

If the position states in Eq. (6.29) are replaced by the localised states
given in Eq. (6.3.2) and the system evolved forward in time using the one-
dimensional generalisation of Eq. (3.16), the time-dependent position operator
is given by

$X(t)=\sum_{\lambda={H,V}}\int_{-\infty}^{\infty}\text{d}k_{1}\int_{-\infty}^{\infty}\text{d}k_{2}\;g(k_{1},k_{2};t)\ket{1_{\lambda}(k_{1})}\bra{1_{\lambda}(k_{2})}.$
(6.31)

where

$g(k_{1},k_{2};t)=\int_{-\infty}^{\infty}\text{d}x\;x\,e^{i(k_{1}-k_{2})x+i(|k_{1}|-|k_{2}|)ct}.$
(6.32)

The localised states $\ket{1_{\lambda}(x)}$ are not the eigenstates of the
above operator for any non-zero time.

####  6.5 Discussion

When two identical atoms, one in an excited state and one in its ground state,
are coupled to the electromagnetic field, the excited atom will emit a photon
that is absorbed at a later time by the second atom. It is often found that
there is a non-zero probability for the second atom to become excited
immediately after the coupling is turned on, but as this probability is
independent of the first atom, it is not usually interpreted as evidence of
superluminal propagation from the first atom to the second. This result,
however, does not depend on the use of atoms. For this reason, in this chapter
we have considered an alternative causality experiment that uses a beam-
splitter and a pair of number-resolving photon detectors positioned along the
output ports of the beam-splitter as illustrated in Fig. 6.2.

The results of this chapter show that in the standard theory of the quantised
EM field a state initially localised to the left of the beam-splitter has a
non-zero probability of causing a click at either of the detectors at a later
time. In fact, Detector 2 may register a click long before a photon is
received by Detector 1. In this experiment a click at either detector must
correspond to a measurement of the state initially localised to the left of
the beam-splitter. In the standard quantisation of the free EM field,
therefore, localised photon states propagate at speeds exceeding the speed of
light. This is in agreement with the results of Hegerfeldt [72] and Malament
[73]. As the two detection operators defined in Eq. (6.3.2) commute with one
another it is impossible for an observer at Detector 1 to send a message to an
observer at Detector 2 by taking measurements with his detector. It is then
possible to say that there is no causality violation because this experiment
does not examine the propagation of causal information. Nevertheless, the
results in this chapter clearly demonstrate that the localised states defined
in Eq. (6.3.2) disperse filling all space immediately. This result cannot be
accepted if we are to construct a local theory of photons.

We have also shown in this chapter that when the field is modelled in terms of
blips, localised photon wave packets propagate at the speed of light. In
particular, the probability of detecting a photon at Detector 2 conditioned on
an earlier detection made by Detector 1 is directly proportional to the
initial probability of Detector 1 clicking once an appropriate delay has been
taken into consideration. In Section 6.3.4 we found that causality only
applies when the photon wave number (multiplied by $c$) and frequency are
equal. The problem of infinite propagation speeds can therefore be overcome by
removing the lower bound on the Hamiltonian, as was done in Chapters 4 and 5
of this thesis, but also in Refs. [53, 55, 110, 161, 162, 163, 164, 168, 173].
In spite of this, theories that are restricted to positive field Hamiltonians
are by far the most prevalent in standard quantum physics textbooks [123, 124,
127, 130, 131]. In Ref. [91], the superluminal contribution to the excitation
probability of the second atom was attributed, not to a “flying” photon, but
to the vacuum fluctuations of the EM field. Our results are in partial
agreement with this conclusion, as we have also shown that such correlations
have nothing to do with the use of atoms but with the current description of
the EM field. Here, however, we do not see this as a reason to excuse these
correlations, but as a justification that an alternative description of the
field is needed.

This chapter has provided an important verification that the blip theory
offers a more intuitive means of describing photon wave packets than the
standard theory in realistic experimental scenarios. In particular the results
of this chapter confirm that negative-frequency states are a necessary part of
the construction of photon wave packets. The existence of negative-frequency
modes is fundamental to a variety of amplification effects including Penrose
superradiance [174], which has recently been measured in a non-linear optical
system [175], and the production of Hawking radiation by a black hole [176] or
Unruh radiation in an accelerating reference frame [177]. In these instances,
however, amplification occurs due to a conversion from modes with a negative
norm to those with a positive norm. The positive-frequency and negative-
frequency parts of the blip’s Fourier spectrum discussed in this chapter both
have a positive norm and so do not relate to these effects. Modes with a
negative norm do still exist in our new theory in addition to the newly
introduced negative-frequency photon states. In order to study effects such as
the Unruh effect we would need to study whether negative-norm states can be
converted to positive-norm states in this new theory.

The results of this chapter provide a good indication that the blip theory
will prove a useful tool for studying quantum effects in the position
representation. In this next chapter we shall investigate the behaviour of
blips in a one-dimensional cavity and a three-dimensional cavity. By studying
light inside a cavity in the position representation we hope to gain a new
perspective on the origin of the Casimir effect.

###  Chapter 7 The Casimir effect

In this chapter we provide an alternative derivation of the Casimir force
between two parallel, perfectly reflecting mirrors in both one and three
dimensions using the local theory of photons constructed in Chapters 4 and 5.
We recover the usual textbook expressions for the Casimir force as an exact
result without the need for regularisation procedures. In Section 7.2 we
investigate the propagation of blips near a perfect cavity in one dimension
and construct a corresponding set of field observables. In Section 7.3 we show
that the zero-point energy of the field inside the cavity differs from its
usual expression by a convergent term that depends on the width of the cavity.
This additional term leads to an attractive force between the mirrors. In
Section 7.4 we repeat these calculations for the case of a perfectly
reflecting cavity in three dimensions. We similarly find that there is an
attractive force between the two plates in agreement with standard results. We
conclude this chapter with a discussion in Section 7.5.

####  7.1 Introduction: The Casimir effect

The Casimir effect refers to the attractive force between two parallel and
highly reflecting mirrors in a vacuum [178]. Since its initial discovery the
Casimir effect has received a lot of attention in the literature [179, 180,
181, 182, 183, 184], and, despite not having a counterpart in classical
electrodynamics, recent experiments confirm its predictions [185, 186, 187,
188, 189, 190]. Nevertheless, there is still some controversy surrounding the
origin of this effect [191, 192, 193]. For example, the standard derivation of
the Casimir effect requires regularisation procedures to be used before
identifying a finite contribution to the zero-point energy of the EM field
that depends on the mirror separation $D$ [128].

Moreover, the standard derivation simply assumes that the plates restrict the
quantised EM field inside the cavity to standing waves with a discrete set of
resonant frequencies. In these standing wave mode models one cannot take into
account from which direction light enters an optical cavity and therefore
cannot reproduce the typical behaviour of Fabry-Perot cavities which have
maximum transmission rates for resonant light [194]. Discrete mode models also
imply that no light is permitted inside a cavity with mirror separations well
below typical optical wavelengths, which would appear to contradict recent
experiments with nano-cavities [195]. Special care must be taken in these
circumstances, however, as the modes excited inside the cavity are often
described as nano-gap plasmonic modes, which lie on the surface of the cavity
walls. The plasmon wavelengths are always smaller than the free-space optical
wavelengths allowing an enhancement of light inside very small cavities. To
fully capture all aspects of optical cavities, an alternative approach to the
Casimir effect is needed.

In Chapters 4 and 5 of this thesis we introduced a quantisation of the one-
and three-dimensional EM field in the position representation. A description
such as this lends itself naturally to the modelling of locally interacting
quantum systems, like a cavity, and other quantum optics experiments [149,
196]. In the blip descriptions, the EM field is viewed as a collection of
localised energy quanta described by the local operators $a_{s\lambda}(x,t)$
and $a_{s\lambda}(\mathbf{r},t)$. These operators avoid the previous
localisability issues as the associated energy quanta are completely localised
in space (see Eqs. (4.17) and (5.3.1)). Moreover, as was shown in Chapter 6
for the one-dimensional excitations, because these excitations evolve
according to a dynamical Hamiltonian that has both positive and negative
eigenvalues, these states propagate at the speed of light without dispersion.

The purpose of this chapter is to use the local theories developed in Chapters
4 and 5 to model the behaviour of the quantised EM field between two parallel
mirrors and re-derive the Casimir effect in both one and three dimensions.
Instead of imposing boundary conditions by demanding that the electric field
amplitudes vanish at the mirror surfaces and restricting the EM field inside
the cavity to standing waves, in this chapter we realise boundary conditions
in a dynamical fashion by looking at how blips propagate near a cavity. The
results of this chapter will provide a more intuitive description of the
Casimir effect and will support the validity of the local field quantisation
approach to light-matter systems. As the calculations involved in the
derivation of the Casimir effect in one and three dimensions are very similar,
the most part of this chapter will be dedicated to the one-dimensional effect.
The derivation of the three-dimensional effect will be shorter as many of the
relevant discussions carry over.

####  7.2 Light inside an optical cavity

In this section we describe the propagation of blips both inside and outside a
one-dimensional cavity using the mirror image method of classical optics. By
applying the regularisation operator of Chapter 4 to a blip near the cavity
and its mirror images, we determine an expression for the field observables
near the cavity.

#####  7.2.1 The propagation of blips near a one-dimensional mirror

In free space, blips propagate along the boundaries of the light-cones as
illustrated in Fig. 1.1. In such a system the blips propagate away from the
source at the speed of light and never change their direction. Let us now
consider a single mirror placed at some point along the $x$ axis. For
convenience we shall choose this point to be the origin $x=0$. We shall name
the region to the left of the mirror Region I, and the region to the right of
the mirror Region II. In this chapter we shall treat all mirrors as classical
objects that interact locally with light. The mirrors we are describing are
perfect mirrors that reflect all incident light. As blips are the local energy
quanta of the EM field, it is the blips that interact locally with the mirror.
Therefore, when away from the mirror the blips evolve as if in a vacuum; when
in contact with the mirror, the blips are reflected by the mirror and their
direction of propagation is reversed ($s\mapsto-s$). The trajectory of a blip
initially approaching the mirror from the left along with its mirror image is
illustrated in Fig. 7.1.

Figure 7.1: The figure depicts the world-line of a single blip interacting
with a perfect mirror placed at the origin. The blip, initially approaching
from the left, propagates freely until it reaches the mirror at which point
the direction of propagation is reversed. While the incoming blip is
characterised by $s=1$, the outgoing blip is characterised by $s=-1$. A blip
initially approaching from the right would behave analogously, the reflection
implementing the transition from an $s=-1$ blip to an $s=1$ blip. The
translucent trajectory on the right-hand side of the mirror shows the mirror
image of the reflected blip.

Since the mirror is perfect and light therefore cannot pass through it, a blip
that is initially localised to the left of the mirror will remain to the left
of the mirror for all time. The reflection of light from the left-hand side of
the mirror can therefore be modelled entirely with the use of blips localised
in Region I. Since the light changes direction when it interacts with the
mirror, blips characterised by $s=1$ and $s=-1$ are required to give a full
account of this interaction. In particular, the reflection process would be
described by an interaction Hamiltonian of the kind studied in Ref. [110] that
would transform a blip characterised by a direction $s$ into another
characterised by $-s$. In this chapter, however, for simplicity we shall
describe the trajectory of the blip using the mirror image method of classical
electrodynamics.

Looking more closely at Fig. 7.1, one can see that the combination of the blip
being reflected from the left and its mirror image are equivalent to the
superposition of two blips propagating in free space, that is, as if the
mirror were not there. The two trajectories cross at the origin. Whereas the
blip propagating towards the mirror is defined by the operator
$a_{s\lambda}(x,t)$, its mirror image is defined by the operator
$a_{s\lambda}(-x,t)$. Denoting the operators for blips to the left and the
right of the mirror $a^{\text{I}}_{s\lambda}(x,t)$ and
$a^{\text{II}}_{s\lambda}(x,t)$ respectively, it is therefore possible to
express these operators conveniently as

$\displaystyle a^{\text{I}}_{s\lambda}(x,t)$ $\displaystyle=$
$\displaystyle\mathcal{X}^{\text{I}}\left(a_{s\lambda}(x,t)+a_{s\lambda}(-x,t)\right)$
$\displaystyle a^{\text{II}}_{s\lambda}(x,t)$ $\displaystyle=$
$\displaystyle\mathcal{X}^{\text{II}}\left(a_{s\lambda}(x,t)+a_{s\lambda}(-x,t)\right).$
(7.1)

In the above the $a_{s\lambda}(x,t)$ operators are the free-space blip
operators. Expressing the $a^{\text{I}}_{s\lambda}(x,t)$ and
$a^{\text{II}}_{s\lambda}(x,t)$ operators as a superposition of the free-space
blip operators is particularly useful as we already know the commutation
relations for the free operators. In Eq. (7.2.1), the operator
$\mathcal{X}^{\text{I}}$ tells us to neglect all blips located to the right of
the mirror for all times. This effectively removes the translucent trajectory
in Fig. 7.1 leaving only the physical trajectory. The operator
$\mathcal{X}^{\text{II}}$ tells us to neglect all blips located to the left of
the mirror for the same reason.

Note finally that the mirror image method is not needed to describe the
propagation of light near a double-sided mirror in the blip formalism [110].
We use the mirror image method only for convenience. However, whereas one
would usually need both parameters $s=\pm 1$ to describe the behaviour of
light at either side of the mirror, using the mirror image method, the entire
world-line of the blip near the mirror is described using only a single value
of $s$. Therefore, from now only we need only make use of one value of $s$.
Which value doesn’t matter so we shall just leave this parameter as $s$.

#####  7.2.2 The propagation of blips near a one-dimensional cavity

To construct a cavity system we place two double-sided mirrors at points along
the $x$ axis a width $D$ apart. For symmetry purposes we assume that the
leftmost mirror is placed at $x=-D/2$ whilst the rightmost is placed at
$x=D/2$. The two mirrors partition the real line into three regions. The
region to the left of the cavity spanning all $x<-D/2$ shall be denoted Region
I, the region between the mirrors spanning $-D/2<x<D/2$ shall be denoted
Region II, and the region to the right of the cavity spanning all $x>D/2$
shall be denoted Region III. As there are now two mirrors, there will be many
more mirror images to take into consideration in order to describe blips both
inside and outside the cavity. We now look at how to construct the appropriate
blip operators in terms of the free-space blip operators just like we did in
Eq. (7.2.1)

###### Blips outside the cavity

Consider a single blip propagating towards the cavity from the left. The blip
will propagate freely until it comes into contact with the leftmost mirror. As
these mirrors are perfect there is no chance of a blip penetrating the mirror.
Consequently all blips in Region I are unaffected by the mirror forming the
right-hand wall of the cavity. Therefore, the blips to the left of the cavity
propagate as if there were only one mirror, just as illustrated in Fig. 7.1.
Blips in region III behave analogously. We can therefore express the blips in
Regions I and III in a manner very similar to Eq. (7.2.1). Denoting blips in
Regions I and III $a^{\text{I}}_{s\lambda}(x,t)$ and
$a^{\text{III}}_{s\lambda}(x,t)$ respectively, and taking into account the new
position of the mirrors we find that

$\displaystyle a^{\text{I}}_{s\lambda}(x,t)$ $\displaystyle=$
$\displaystyle\mathcal{X}^{\text{I}}\left(a_{s\lambda}(x,t)+a_{s\lambda}(-D-x,t)\right)$
$\displaystyle a^{\text{III}}_{s\lambda}(x,t)$ $\displaystyle=$
$\displaystyle\mathcal{X}^{\text{III}}\left(a_{s\lambda}(x,t)+a_{s\lambda}(D-x,t)\right).$
(7.2)

As before the operators $\mathcal{X}^{\text{I}}$ and
$\mathcal{X}^{\text{III}}$ remove contributions that are outside their
designated regions.

###### Blips inside the cavity

Inside the cavity a single blip will bounce backwards and forwards between the
two mirrors for all time. This motion is illustrated in Fig. 7.2. This is true
for any size cavity as blips are infinitely well localised and therefore
guaranteed to fit inside the cavity. The trajectory of blips inside the cavity
differs therefore from that of those on the outside. As a blip inside the
cavity is reflected an infinite number of times there is an infinite number of
mirror images that must be taken into consideration. These can be imagined by
extending the world-lines of the blips in Fig. 7.2 through the cavity walls.
Taking these reflections into consideration we find that the blip operator for
a single blip placed inside the cavity is given by the expression

$\displaystyle
a^{\text{II}}_{s\lambda}(x,t)=\mathcal{X}^{\text{II}}\sum_{n=-\infty}^{\infty}\left(a_{s\lambda}(x+2nD,t)+a_{s\lambda}((2n-1)D-x,t)\right).$
(7.3)

The operation $\mathcal{X}^{\text{II}}$ disregards all blips located outside
Region II for all times. The infinite sum is representative of the infinite
number of reflections made by a blip inside the cavity.

Figure 7.2: The figure shows the world-line of a single blip placed inside a
cavity. A blip initially propagating to the left will meet the right-hand
mirror and be reflected to the left. At a later time the same blip will meet
the left-hand mirror and be reflected again causing it to propagate to the
right. This process continues indefinitely. Light initially inside the cavity
remains inside for all time, and light initially outside the cavity remains
outside for all time.

#####  7.2.3 The propagation of the EM field near a cavity

The Casimir effect arises due to changes in the zero-point energy of the EM
field as the cavity width is varied. The energy of the system is determined by
the fields, so in this section we shall investigate how the field observables
are altered by the presence of a cavity.

In Chapters 4 and 5, the bosonic blip operators are related to the field
observables by means of a regularisation operator $\mathcal{R}$. In both cases
the regularisation operator is highly non-local (see Eqs. (4.56) and (5.53)).
The field measured at a single point therefore depends on the distribution of
blips across all of space. We have seen, however, that when a mirror is
introduced to the system all blips are reflected, which prevents them from
reaching the other side. A mirror, therefore, will effectively shield an
observer on one side of the mirror from all blips present on the other. When
such an observer measures the field observables on their side of the mirror,
they ought not to be aware of any blips on the other side. For this reason,
all field observables measured on one side of a mirror cannot depend on blips
on the other. This is one consideration that must be taken into account. We
must also remember that the cavity walls are not dull but perfectly
reflecting. Therefore it is not only the blips on the correct side of the
mirror that contribute to the field observables, but all of their mirror
images too.

As the field observables defined in either Region I, II or III can only depend
on blips defined in that same region, and must also depend on all of their
mirror images, the field operators defined in either one of these regions is
simply the regularisation of the photon operators
$a^{\text{I},\text{II},\text{III}}_{s\lambda}(x,t)$ that we defined earlier.
Conveniently, as the regularisation operators in both one and three dimensions
are symmetric, the regularisation of the mirror images is equal to the mirror
images of the regularisation. Hence, the field observables near a cavity are
determined by taking the mirror images of the free field observables. Unlike
the blip operators that only become aware of the mirror when the two come into
contact, as the field observables are non-local, the fields at all locations
are changed quite significantly by the presence of the two mirrors. In
particular, by taking into account the mirror images of the fields both inside
and outside the cavity, the non-local field observables are folded over onto
themselves many times forming a concertina shape. This is illustrated for the
field observables both inside and outside the cavity in Fig. 7.3. In the
remainder of the section we shall determine explicit expressions for the field
observables near a one-dimensional cavity.

Figure 7.3: [173] a. The diagram shows the amplitude of the electric field
observable generated by a single blip shown by the grey dot. As the
regularisation operator ${\cal R}$ is non-local, local blip excitations
contribute to local electric and magnetic field expectation values everywhere
along the $x$ axis. b. Since a blip on one side of a highly reflecting mirror
cannot contribute to the field expectation value on the other side, its field
contribution must be reflected back on itself. This effect alters the electric
and magnetic field observables in the presence of a mirror. c. In the presence
of two highly-reflecting mirrors, blips outside the cavity cannot contribute
to field expectation values on the inside. Moreover, the field contributions
of blips on the inside need to be reflected as in the case of one mirror. Now,
however, the field contributions must be folded infinitely many times. d. By
comparing two cavities of different sizes we see that the behaviour of the
fields inside the cavity is dependent on the cavity width.

###### Outside the cavity

In the previous section we failed to mention that every time light is
reflected by a perfect mirror, the field observables gain a unitary phase
factor. In this section we choose a mirror that applies a factor of $-1$ and
$+1$ to the electric and magnetic fields respectively. As we calculate the
mirror images of the fields we must remember to include these additional phase
factors. Taking into account all mirror images, the field observables in
Region I are given by

$\mathbf{O}^{\text{I}}_{s}(x,t)=\mathcal{X}^{\text{I}}\left(\mathbf{O}_{s}(x,t)\mp\mathbf{O}_{s}(-D-x,t)\right).$
(7.4)

Here $\mathbf{O}_{s}=\mathbf{E}_{s},\mathbf{B}_{s}$ refers to the free field
observables defined in Eqs. (4.2.3) and (4.4.2), and the $\mp$ sign applies to
the electric and magnetic fields respectively.

In Eq. (7.4), the operation $\mathcal{X}^{\text{I}}$ restricts the Hilbert
space to blips characterised by $x$ in Region I only. This ensures that blips
outside of Region I cannot contribute in any way to the fields in Region I as
both are separated by a perfect mirror. After substituting into Eq. (7.4) the
expressions for the free field observables defined in Eq. (4.2.3) and using
the regularisation operator given in Eq. (4.56), one finds that

$\displaystyle\mathbf{E}^{\text{I}}_{s}(x,t)$ $\displaystyle=$
$\displaystyle-\int^{-D/2}_{-\infty}{\rm d}x^{\prime}\left({\hbar c\over
16\pi\varepsilon_{0}A}\right)^{1/2}\left[\left|x-x^{\prime}\right|^{-3/2}-\left|x+x^{\prime}+D\right|^{-3/2}\right]$
(7.5) $\displaystyle\hskip
62.59596pt\times\left[a_{s\mathsf{H}}(x^{\prime},t)\,\widehat{\boldsymbol{y}}+a_{s\mathsf{V}}(x^{\prime},t)\,\widehat{\boldsymbol{z}}\right]+\text{H.c.}$

and

$\displaystyle\mathbf{B}^{\text{I}}_{s}(x,t)$ $\displaystyle=$
$\displaystyle-\int^{-D/2}_{-\infty}{\rm d}x^{\prime}\;\frac{s}{c}\left({\hbar
c\over
16\pi\varepsilon_{0}A}\right)^{1/2}\left[\left|x-x^{\prime}\right|^{-3/2}+\left|x+x^{\prime}+D\right|^{-3/2}\right]$
(7.6) $\displaystyle\hskip
62.59596pt\times\left[a_{s\mathsf{H}}(x^{\prime},t)\,\widehat{\boldsymbol{z}}-a_{s\textsf{V}}(x^{\prime},t)\,\widehat{\boldsymbol{y}}\right]+\text{H.c.}$

respectively.

The field observables to the right of the cavity are similarly defined by
taking into account the mirror images of the field about the mirror on the
right. The field observables in Region III are given by

$\mathbf{O}^{\text{III}}_{s}(x,t)={\cal
X}^{\text{III}}\left(\mathbf{O}^{\text{free}}_{s}(x,t)\mp\mathbf{O}^{\text{free}}_{s}(D-x,t)\right).$
(7.7)

Here the operation $\mathcal{X}^{\text{III}}$ restricts the Hilbert space to
blip excitations that are to the right of the mirror. By again expanding out
the free field observables one finds that

$\displaystyle\mathbf{E}^{\text{III}}_{s}(x,t)$ $\displaystyle=$
$\displaystyle-\int^{\infty}_{D/2}{\rm d}x^{\prime}\left({\hbar c\over
16\pi\varepsilon_{0}A}\right)^{1/2}\left[\left|x-x^{\prime}\right|^{-3/2}-\left|x+x^{\prime}-D\right|^{-3/2}\right]$
(7.8) $\displaystyle\hskip
51.21504pt\times\left[a_{s\mathsf{H}}(x^{\prime},t)\,\widehat{\boldsymbol{y}}+a_{s\mathsf{V}}(x^{\prime},t)\,\widehat{\boldsymbol{z}}\right]+\text{H.c.}$

and

$\displaystyle\mathbf{B}^{\text{III}}_{s}(x,t)$ $\displaystyle=$
$\displaystyle-\int^{\infty}_{D/2}{\rm d}x^{\prime}\;\frac{s}{c}\left({\hbar
c\over
16\pi\varepsilon_{0}A}\right)^{1/2}\left[\left|x-x^{\prime}\right|^{-3/2}+\left|x+x^{\prime}-D\right|^{-3/2}\right]$
(7.9) $\displaystyle\hskip
51.21504pt\times\left[a_{s\mathsf{H}}(x^{\prime},t)\,\widehat{\boldsymbol{z}}-a_{s\mathsf{V}}(x^{\prime},t)\,\widehat{\boldsymbol{y}}\right]+\text{H.c.}$

respectively.

###### Inside the cavity

Inside the cavity we must take into consideration an infinite number of mirror
images that each contribute to a section of the zig-zagging trajectory
illustrated in Fig. 7.2. Analogous to Eq. (7.3), the expressions for the total
field observables inside the cavity are given by

$\mathbf{O}^{\text{II}}_{s}(x,t)=\mathcal{X}^{\text{II}}\sum_{n=-\infty}^{\infty}\left(\mathbf{O}_{s}(x+2nD,t)\mp\mathbf{O}_{s}((2n-1)D-x,t)\right)$
(7.10)

Here $\mathcal{X}^{\text{II}}$ reduces the Hilbert space to the Hilbert space
of blips located within Region II. By substituting in the expressions for the
free field observables, we arrive at the following expressions for the
electric and magnetic field observables:

$\displaystyle\mathbf{E}_{s}^{\text{II}}(x,t)$ $\displaystyle=$
$\displaystyle-\sum_{n=-\infty}^{\infty}\int^{D/2}_{-D/2}{\rm
d}x^{\prime}\left({\hbar c\over 16\pi\varepsilon_{0}A}\right)^{1/2}$ (7.11)
$\displaystyle\left[\left|x-x^{\prime}+2nD\right|^{-3/2}-\left|x+x^{\prime}+(2n-1)D\right|^{-3/2}\right]$
$\displaystyle\times\left[a_{s\mathsf{H}}(x^{\prime},t)\,\widehat{\boldsymbol{y}}+a_{s\mathsf{V}}(x^{\prime},t)\,\hat{\boldsymbol{z}}\right]+\text{H.c.}$

and

$\displaystyle\mathbf{B}_{s}^{\text{II}}(x,t)$ $\displaystyle=$
$\displaystyle-\sum_{n=-\infty}^{\infty}\int^{D/2}_{-D/2}{\rm
d}x^{\prime}\,{s\over c}\left({\hbar c\over
16\pi\varepsilon_{0}A}\right)^{1/2}$ (7.12)
$\displaystyle\left[\left|x-x^{\prime}+2nD\right|^{-3/2}+\left|x+x^{\prime}+(2n-1)D\right|^{-3/2}\right]$
$\displaystyle\times\left[a_{s\mathsf{H}}(x^{\prime},t)\,\widehat{\boldsymbol{z}}-a_{s\mathsf{V}}(x^{\prime},t)\,\widehat{\boldsymbol{y}}\right]+\text{H.c.}$

We now have a complete set of operators for the electric and magnetic field
observables inside and outside the cavity.

####  7.3 The Casimir Effect in one dimension

In this section we use the field observables derived in the previous section
to calculate the zero-point energy of the field near the cavity. We find that
the zero-point energy of the field inside the cavity is given by the usual
free-space expression plus a $D$-dependent term caused by the interference of
the field’s infinite tails. This leads to the usual expression for the Casimir
force in one dimension.

#####  7.3.1 The vacuum energy of the cavity

Now that we have determined the field observables in the presence of a cavity
we can calculate the total energy of the EM field inside and outside the
cavity. The purpose of this section is to determine whether the zero-point
energy of the field changes as the cavity width $D$ varies and, if so,
determine an expression for the accompanying Casimir force. As the field
observables now contain one or more mirror images, the expression for the
energy observable will become very complicated. Fortunately many of these
expressions disappear when we take the vacuum expectation value. Therefore in
this section we shall calculate the zero-point energy of the fields directly.
The first step is to calculate the zero-point energy of the EM field both
inside and outside the cavity.

###### Outside the cavity

The electric and magnetic fields to the left of the cavity are given by the
observables $\mathbf{E}_{s}^{\text{I}}(x,t)$ and
$\mathbf{B}_{s}^{\text{I}}(x,t)$ expressed in Eqs. (7.5) and (7.6). We only
use the field observables defined for one value of $s$ in this expression, as
these field observables describe the entire world-line of the blips. The
energy observable in Region I is obtained by substituting the above field
observables into the classical expression for the field energy given in Eq.
(2.21). As the fields defined in Region I only contain contributions from
blips also in Region I, this energy observable is indifferent to blips in
Regions II and III as we planned.

After taking the vacuum expectation value of the resulting energy observable
and using the blip commutation relation in Eq. (4.17), one finds that the
zero-point energy of the field to the left of the mirror is given by

$\displaystyle H^{\text{I}}_{\text{ZPE}}$ $\displaystyle=$
$\displaystyle\frac{\hbar
c}{8\pi}\int_{-\infty}^{-D/2}\text{d}x\int_{-\infty}^{-D/2}\text{d}x^{\prime}\;\left[|(x-x^{\prime})(x-x^{\prime})|^{-3/2}\right.$
(7.13) $\displaystyle\left.+|(x+x^{\prime}+D)(x+x^{\prime}+D)|^{-3/2}\right]$
$\displaystyle=$ $\displaystyle\frac{\hbar
c}{8\pi}\int_{-\infty}^{\infty}\text{d}x\int_{-\infty}^{-D/2}\text{d}x^{\prime}\;|(x-x^{\prime})(x-x^{\prime})|^{-3/2}.$

The final equality occurs after making the substitution
$x^{\prime}\mapsto-x^{\prime}-D$ in the second line. To simplify this
expression further we make use of the following identity

$\displaystyle\int_{-\infty}^{\infty}\text{d}x\,|(x^{\prime}-x)(x-x^{\prime\prime})|^{-3/2}$
$\displaystyle=$ $\displaystyle
4\int_{-\infty}^{\infty}\text{d}k\;|k|\,e^{ik(x^{\prime}-x^{\prime\prime})}$
(7.14) $\displaystyle=$
$\displaystyle-{8\over|x^{\prime}-x^{\prime\prime}|^{2}}.$

The final equality holds for $x^{\prime}\neq x^{\prime\prime}$ only. Using the
identity above we can show that Eq. (7.13) is equal to

$H^{\text{I}}_{\text{ZPE}}=-\frac{\hbar
c}{\pi}\int_{-\infty}^{-D/2}\text{d}x_{1}\;\left.\frac{1}{|x_{1}-x|^{2}}\right|_{x=x_{1}}.$
(7.15)

The above expression is properly defined as the Fourier transform given in the
first line of Eq. (7.14). This term is divergent and the factor under the
integral sign is independent of $D$. This term corresponds to the infinite
zero-point energy of the field in Region I.

The total vacuum energy of the fields to the right of the cavity is found in
the same way by calculating the expectation value of the energy observable
defined for the $\mathbf{E}^{\text{III}}(x,t)$ and
$\mathbf{B}^{\text{III}}(x,t)$ field observables given by Eqs. (7.8) and
(7.9). By an almost identical set of calculations one finds that

$H^{\text{III}}_{\text{ZPE}}=-\frac{\hbar
c}{\pi}\int_{D/2}^{\infty}\text{d}x_{1}\;\left.\frac{1}{|x_{1}-x|^{2}}\right|_{x=x_{1}}.$
(7.16)

The zero-point energy density to the right of the mirror is exactly the same
as the zero-point energy density to the left of the mirror. This would be
expected due to the symmetry of the system.

###### The vacuum energy inside the cavity

Inside the cavity the process is just the same. After calculating the energy
observable for the $\mathbf{E}_{s}^{\text{II}}(x,t)$ and
$\mathbf{B}_{s}^{\text{II}}(x,t)$ fields defined in Eqs. (7.11) and (7.12),
one finds that the total zero-point energy between the cavity walls is given
by

$\displaystyle H_{\text{ZPE}}^{\text{II}}$ $\displaystyle=$
$\displaystyle\frac{\hbar
c}{8\pi}\sum_{n,m=-\infty}^{\infty}\int_{-D/2}^{D/2}\text{d}x\int_{-D/2}^{D/2}\text{d}x^{\prime}$
(7.17)
$\displaystyle\times\left[\left|(x+x^{\prime}+(2n-1)D)(x+x^{\prime}+(2m-1)D)\right|^{-3/2}\right.$
$\displaystyle\left.+\left|(x-x^{\prime}+2nD)(x-x^{\prime}+2mD)\right|^{-3/2}\right]\,.$

This term can be simplified to a single line by a series of substitutions.
There are several steps to this substitution, so for clarity in this section
we set the full calculation apart in Appendix C. After performing the
mentioned substitutions we find that

$\displaystyle H_{\text{ZPE}}^{\text{II}}$ $\displaystyle=$
$\displaystyle\frac{\hbar
c}{8\pi}\sum_{n,m=-\infty}^{\infty}\int_{-D}^{D}\text{d}x\int_{-D/2}^{D/2}\text{d}x^{\prime}\,\left|(x-x^{\prime}+2nD)(x-x^{\prime}+2mD)\right|^{-3/2}$
(7.18) $\displaystyle=$ $\displaystyle\frac{\hbar
c}{8\pi}\sum_{n,m=-\infty}^{\infty}\int_{-D+2nD}^{D+2nD}\text{d}x\int_{-D/2}^{D/2}\text{d}x^{\prime}\,\left|(x-x^{\prime})(x-x^{\prime}+2(m-n)D)\right|^{-3/2}$
$\displaystyle=$ $\displaystyle\frac{\hbar
c}{8\pi}\sum_{m=-\infty}^{\infty}\int_{-\infty}^{\infty}\text{d}x\int_{-D/2}^{D/2}\text{d}x^{\prime}\,\left|(x-x^{\prime})(x-x^{\prime}+2mD)\right|^{-3/2}\,.$

The second line in Eq. (7.18) follows from the first by making the
substitution $x\mapsto x+2nD$. To make the transition from the second to the
third line we should first look at the expression $m-n$. As both $n$ and $m$
run between $-\infty$ and $\infty$, for each fixed value of $n$, $m-n$ also
takes all values between $-\infty$ and $\infty$. Therefore, it does no harm to
simply replace $m-n$ with $m$. At this point, the sum over all $n$ has the
sole effect of stretching the limits of the $x$ integral, which now covers the
entire real line. We are now in a position to apply the identity given in Eq.
(7.14) to Eq. (7.18). We then find that

$\displaystyle H_{\text{ZPE}}^{\text{III}}$ $\displaystyle=$
$\displaystyle-\frac{\hbar c}{4\pi
D}\sum_{m=-\infty}^{\infty}\frac{1}{m^{2}}\,.$ (7.19)

We shall examine this expression further in the next section.

#####  7.3.2 Divergent contributions to the total zero-point energy

The expression in Eq. (7.19) is the total zero-point energy of the quantised
EM field inside the cavity. As might be expected, this expression is
divergent. The divergent term, however is easily identified as the term for
which $m=0$. We can separate this divergent contribution from the remaining
terms which allows us to express the total zero-point energy in the form

$H_{\text{ZPE}}^{\text{II}}=\left.-\frac{\hbar c}{4\pi
Dm^{2}}\right|_{m=0}-\frac{\hbar c}{2\pi
D}\sum_{m=1}^{\infty}\frac{1}{m^{2}}.$ (7.20)

The second term in this expression is convergent and will be examined shortly.
For now we shall investigate more closely the divergent contribution to the
total zero-point energy inside the cavity. Let us look again at the final line
of Eq. (7.18). If we compare the $m=0$ contribution to this expression with
the final line of Eq. (7.13) we can see that the two are identical apart from
the limits of the integration over $x$. The divergent zero-point energy
density inside the cavity is therefore just the usual zero-point energy of the
free fields in the region between $x=-D/2$ and $x=D/2$. We can therefore
rewrite Eq. (7.20) in the form

$H_{\text{ZPE}}^{\text{II}}=H_{\text{ZPE}}^{\text{II}(\text{free})}-\frac{\hbar
c}{2\pi D}\sum_{m=1}^{\infty}\frac{1}{m^{2}}.$ (7.21)

A rough physical argument can be given as to why the $m=0$ contribution
corresponds to the zero-point energy of the free fields if we look again at
Eq. (7.18). The $m$ that appears in Eq. (7.20) and the final line of Eq.
(7.18) corresponds to the difference between $n$ and $m$ in the first line of
Eq. (7.18). As $n$ and $m$ here refer to the components of the fields that
have been reflected $2n$ and $2m$ times respectively, the $m=0$ contribution
to the zero-point energy of the field given in Eq. (7.20) is the contribution
to the total vacuum energy caused by the interference between components of
the fields that have been reflected an equal number of times. In part (d) of
Fig. 7.3 one can see that the effect of the mirrors is to fold the non-local
field amplitudes back in on themselves when they meet the cavity walls. If
this diagram were unfolded we would recover part (a) of the same figure. The
interference, then, between components of the field amplitudes that have been
folded an equal number of times corresponds exactly to the interference of the
free fields with themselves. The difference between the former and the latter
is only that the first set of fields have been scrunched up into a cavity.

The $m\neq 0$ contributions to Eq. (7.21), however, correspond to interference
between field amplitudes when one of the fields has been folded at least twice
more than the other; this is only possible inside a cavity. Moreover, the
greater the number of reflections that separate two contributions to the EM
field observables, the larger $m$ and the smaller their contribution to the
total zero-point energy. Next we examine more closely the remaining correction
to the free zero-point energy.

#####  7.3.3 The Casimir force

Using our expressions for the zero-point energy of the EM field inside and
outside the cavity we can now determine whether a Casimir force exists. To do
so we calculate how the total zero-point energy of the system changes as the
width of the cavity is varied. From Eqs. (7.15) and (7.16) we know that the
vacuum energy density in Regions I and III, which is given by the usual
expression for the free field zero-point energy density, is independent of the
cavity width and therefore does not contribute to an overall Casimir force. In
Region II the zero-point energy density is given by the usual cavity-
independent expression with a $D$-dependent correction. In the position
representation, as can be seen in Eq. (7.21), it is easy to separate the
$D$-independent free field contribution from the $D$-dependent correction.
Hence, the total Casimir force on the mirrors is given by the formula

$F_{\text{Casimir}}=-\frac{\partial E_{\text{Casimir}}}{\partial D}$ (7.22)

where $E_{\text{Casimir}}$ is the correction to the zero-point energy inside
the cavity. By substituting in our value for the Casimir energy
$E_{\text{Casimir}}$ in Eq. (7.21), we find that

$F_{\text{Casimir}}=-\frac{\hbar c}{2\pi
D^{2}}\sum_{m=1}^{\infty}\frac{1}{m^{2}}.$ (7.23)

This force is $D$-dependent and finite. The sum in Eq. (7.23) is the
convergent Basel sum equal to $\pi^{2}/6$. Hence we find that

$F_{\text{Casimir}}=-\frac{\hbar c\pi}{12D^{2}}.$ (7.24)

This is exactly the predicted Casimir force for photons with two possible
polarisations. This force is attractive and varies with the second negative
power of $D$ [197].

####  7.4 The Casimir effect in three dimensions

In this section we investigate the propagation of three-dimensional blips near
two infinite, flat and parallel mirrors. In a similar manner to Section 7.2,
we determine the appropriate expressions for the field observables near the
cavity, and then use these expressions to calculate the zero-point energy of
the system. The geometry of the cavity in this system is very similar to the
one-dimensional cavity described in Section 7.2.2. For this reason many of the
methods and results used in Sections 7.2 and 7.3 will be used again in this
section. We find that the zero-point energy of the field inside the cavity is
given by its free-space value with the addition of a $D$-dependent correction.
This result leads to the usual expression for the Casimir force in three
dimensions.

#####  7.4.1 The propagation of the EM field near a cavity

In three dimensions, the basic energy quanta of the EM field are the localised
$a_{s\lambda}(\mathbf{r},t)$ excitations. These excitations are completely
localised and propagate without obstruction until they come into contact with
the mirrors. At this point the blips will be reflected changing their
direction of propagation. To define this motion we shall again use the method
of mirror images. This also means that only one value of $s$ is required to
describe the entire trajectory of the blips near a cavity.

The field observables, on the other hand, are not localised, but depend upon
the configuration of blips across all space. In this section we must therefore
make sure that blips hidden at one side of a mirror cannot influence the
measurement of the fields at the other side. Moreover, the mirror images of
the blips in one particular region must also contribute to the field
observables in that region. As in Section 7.2.3, the resulting fields can be
determined by superposing the contributions from all the mirror images of the
field observables in each particular region. This will be our starting point
in this section.

###### Outside the cavity

Consider two infinite, flat and perfectly reflecting mirrors positioned
parallel to each other a distance $D$ apart. We shall work in an orthogonal
Cartesian system of coordinates such that the two mirrors lie in the $y$-$z$
plane. For symmetry purposes we shall assume that one mirror is positioned at
$x=-D/2$ and the other at $x=D/2$. It is convenient again to refer to the
regions to the left of the cavity, inside the cavity, and to the right of the
cavity as Regions I, II, and III respectively. As the mirrors are perfectly
reflecting, no blips can penetrate the mirrors. Consequently, blips initially
found in Regions I, II, or III will remain in those regions for all time. As
before we must remember that upon reflection the electric field observables
gain a phase factor $-1$ whereas the magnetic fields gain a phase factor $+1$.

In Region I the fields have only a single mirror image reflected about the
mirror placed at $x=-D/2$. Analogous to the field observables given in Eq.
(7.4) one finds that the field observables in Region I are given by

$\mathbf{O}^{\text{I}}_{s}(x,y,z,t)={\cal
X}^{\text{I}}\left(\mathbf{O}_{s}(x,y,z,t)\mp\mathbf{O}_{s}(-D-x,y,z,t)\right).$
(7.25)

In the above $\mathbf{O}_{s}=\mathbf{E}_{s},\mathbf{B}_{s}$ refer to the free
field observables defined in Eq. (5.4.1) and the operator
$\mathcal{X}^{\text{I}}$ restricts the Hilbert space to blips defined in
Region I only. Here also the negative sign applies to electric fields and the
positive sign to magnetic fields. After substituting in the expressions for
the free field observables (5.4.1) and the explicit form of the regularisation
operator in Eq. (5.53) one finds that

$\displaystyle\mathbf{E}^{\text{I}}_{s}(\mathbf{r},t)$ $\displaystyle=$
$\displaystyle-\int_{-\infty}^{-D/2}\text{d}x\int_{-\infty}^{\infty}\text{d}y\int_{-\infty}^{\infty}\text{d}z\;\left(\frac{9\hbar
c}{(4\pi)^{3}\varepsilon_{0}}\right)^{1/2}$ (7.26)
$\displaystyle\left[|\mathbf{r}-\mathbf{r}^{\prime}|^{-7/2}-|\mathbf{r}-\mathbf{r}^{\prime}_{-1}|^{-7/2}\right]\,\boldsymbol{a}_{s}(\mathbf{r}^{\prime})+\text{H.c.}$

and

$\displaystyle\mathbf{B}^{\text{I}}_{s}(\mathbf{r},t)$ $\displaystyle=$
$\displaystyle-\int_{-\infty}^{-D/2}\text{d}x\int_{-\infty}^{\infty}\text{d}y\int_{-\infty}^{\infty}\text{d}z\;\left(\frac{9\hbar
c}{(4\pi)^{3}\varepsilon_{0}}\right)^{1/2}$ (7.27)
$\displaystyle\left[|\mathbf{r}-\mathbf{r}^{\prime}|^{-7/2}+|\mathbf{r}-\mathbf{r}^{\prime}_{-1}|^{-7/2}\right]\,\boldsymbol{b}_{s}(\mathbf{r}^{\prime})+\text{H.c.}$

In the above
$\mathbf{r}^{\prime}=x^{\prime}\,\widehat{\boldsymbol{x}}+y^{\prime}\,\widehat{\boldsymbol{y}}+z^{\prime}\,\widehat{\boldsymbol{z}}$
is the position vector of a point to the left of the cavity. The vector
$\mathbf{r}^{\prime}_{-1}=(-x^{\prime}-D)\,\widehat{\boldsymbol{x}}+y^{\prime}\,\widehat{\boldsymbol{y}}+z^{\prime}\,\widehat{\boldsymbol{z}}$
is the position vector of the mirror image of $\mathbf{r}^{\prime}$ in the
left-hand mirror.

In Region III the fields are similarly defined once we have taken into account
the new position of the mirror. In this case one finds that

$\mathbf{O}^{\text{III}}_{s}(x,y,z,t)=\mathcal{X}^{\text{III}}\left(\mathbf{O}_{s\lambda}(x,y,z,t)\mp\mathbf{O}_{s}(D-x,y,z,t)\right).$
(7.28)

Here also the negative sign applies to the electric field and the positive
sign to the magnetic field, and the operator $\mathcal{X}^{\text{III}}$
restricts the Hilbert space to blips defined in Region III. After substituting
the free field observables into the above expression one finds that

$\displaystyle\mathbf{E}^{\text{III}}_{s}(\mathbf{r},t)$ $\displaystyle=$
$\displaystyle-\int_{D/2}^{\infty}\text{d}x\int_{-\infty}^{\infty}\text{d}y\int_{-\infty}^{\infty}\text{d}z\;\left(\frac{9\hbar
c}{(4\pi)^{3}\varepsilon_{0}}\right)^{1/2}$ (7.29)
$\displaystyle\left[|\mathbf{r}-\mathbf{r}^{\prime}|^{-7/2}-|\mathbf{r}-\mathbf{r}^{\prime}_{1}|^{-7/2}\right]\,\boldsymbol{a}_{s}(\mathbf{r}^{\prime})+\text{H.c.}$

and

$\displaystyle\mathbf{B}^{\text{III}}_{s}(\mathbf{r},t)$ $\displaystyle=$
$\displaystyle-\int_{D/2}^{\infty}\text{d}x\int_{-\infty}^{\infty}\text{d}y\int_{-\infty}^{\infty}\text{d}z\;\left(\frac{9\hbar
c}{(4\pi)^{3}\varepsilon_{0}}\right)^{1/2}$ (7.30)
$\displaystyle\left[|\mathbf{r}-\mathbf{r}^{\prime}|^{-7/2}+|\mathbf{r}-\mathbf{r}^{\prime}_{1}|^{-7/2}\right]\,\boldsymbol{b}_{s}(\mathbf{r}^{\prime})+\text{H.c.}$

In the above $\mathbf{r}^{\prime}$ is defined as before and
$\mathbf{r}^{\prime}_{1}=(-x^{\prime}+D)\,\widehat{\boldsymbol{x}}+y^{\prime}\,\widehat{\boldsymbol{y}}+z^{\prime}\,\widehat{\boldsymbol{z}}$
is now the position vector of the mirror image of $\mathbf{r}^{\prime}$ in the
right-hand mirror.

###### Inside the cavity

Finally, in region II the field observables are given by the expression

$\mathbf{O}^{\text{II}}_{s}(x,t)=\mathcal{X}^{\text{II}}\sum_{n=-\infty}^{\infty}\left(\mathbf{O}_{s}(x+2nD,y,z,t)\mp\mathbf{O}_{s}((2n-1)D-x,y,z,t)\right).$
(7.31)

The operator $\mathcal{X}^{\text{III}}$ restricts the Hilbert space to blips
defined in Region III only. The field observables inside the cavity contain
infinitely many image terms due to the infinitely many mirror images each blip
has. In this instance, after substituting in the free field observables one
finds that

$\displaystyle\mathbf{E}^{\text{II}}_{s}(\mathbf{r},t)$ $\displaystyle=$
$\displaystyle-\sum_{n=-\infty}^{\infty}\int_{-D/2}^{D/2}\text{d}x\int_{-\infty}^{\infty}\text{d}y\int_{-\infty}^{\infty}\text{d}z\;\left(\frac{9\hbar
c}{(4\pi)^{3}\varepsilon_{0}}\right)^{1/2}$ (7.32)
$\displaystyle\left[|\mathbf{r}-\mathbf{r}^{\prime}_{2n}|^{-7/2}-|\mathbf{r}-\mathbf{r}^{\prime}_{2n-1}|^{-7/2}\right]\,\boldsymbol{a}_{s}(\mathbf{r}^{\prime})+\text{H.c.}$

and

$\displaystyle\mathbf{B}^{\text{II}}_{s}(\mathbf{r},t)$ $\displaystyle=$
$\displaystyle-\sum_{n=-\infty}^{\infty}\int_{-D/2}^{D/2}\text{d}x\int_{-\infty}^{\infty}\text{d}y\int_{-\infty}^{\infty}\text{d}z\;\left(\frac{9\hbar
c}{(4\pi)^{3}\varepsilon_{0}}\right)^{1/2}$ (7.33)
$\displaystyle\left[|\mathbf{r}-\mathbf{r}^{\prime}_{2n}|^{-7/2}+|\mathbf{r}-\mathbf{r}^{\prime}_{2n-1}|^{-7/2}\right]\,\boldsymbol{b}_{s}(\mathbf{r}^{\prime})+\text{H.c.}$

In the above $\mathbf{r}^{\prime}$ is defined as before. The vector
$\mathbf{r}^{\prime}_{2n}=(x^{\prime}-2nD)\,\widehat{\boldsymbol{x}}+y^{\prime}\,\widehat{\boldsymbol{y}}+z^{\prime}\,\widehat{\boldsymbol{z}}$
is the position vector of the mirror image of a blip that has been reflected
$2|n|$ times. Positive $n$ applies when the first reflection is off the right-
hand mirror, and negative $n$ when the first reflection is off the left-hand
mirror. The vector
$\mathbf{r}^{\prime}_{2n-1}=(-x^{\prime}-(2n-1)D)\,\widehat{\boldsymbol{x}}+y^{\prime}\,\widehat{\boldsymbol{y}}+z^{\prime}\,\widehat{\boldsymbol{z}}$
is the position vector of the mirror image of a blip that has been reflected
an odd number of times. In this case, $n>0$ applies to blips that have been
reflected $2n-1$ times, initially off the left-hand mirror. Terms for which
$n\leq 0$ applies to blips that have been reflected $-(2n-1)$ times, the first
reflection being off the right-hand mirror.

#####  7.4.2 The vacuum energy of the cavity

We are now in a position to calculate the vacuum energy both inside and
outside the cavity. As before, the energy observable in each region is
determined by substituting the field observables into the classical expression
of energy for that region. We shall also calculate the vacuum expectation
value of the energy observable directly without writing out the full operator
itself.

###### Outside the cavity

After substituting the field observables in Region I (Eqs. (7.26) and (7.27))
into the expression for the total field energy to the left of the cavity and
taking the vacuum expectation value, one finds that

$\displaystyle H_{\text{ZPE}}^{\text{I}}$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{-D/2}\text{d}x\int_{-\infty}^{-D/2}\text{d}x^{\prime}\int_{-\infty}^{\infty}\text{d}y^{\prime}\int\text{d}y\int_{-\infty}^{\infty}\text{d}z^{\prime}\int\text{d}z\;\left(\frac{9\hbar
c}{2(4\pi)^{3}}\right)$ (7.34)
$\displaystyle\times\left[(|\mathbf{r}-\mathbf{r}^{\prime}||\mathbf{r}-\mathbf{r}^{\prime}|)^{-7/2}+(|\mathbf{r}-\mathbf{r}^{\prime}_{-1}||\mathbf{r}-\mathbf{r}^{\prime}_{-1}|)^{-7/2}\right]$
$\displaystyle=$
$\displaystyle\int_{-\infty}^{-D/2}\text{d}x\int\text{d}y\int\text{dz}\int_{\mathbb{R}^{3}}\text{d}^{3}\mathbf{r}^{\prime}\;\left(\frac{9\hbar
c}{2(4\pi)^{3}}\right)\,(|\mathbf{r}-\mathbf{r}^{\prime}||\mathbf{r}-\mathbf{r}^{\prime}|)^{-7/2}.$

In Eq. (7.34), the final equality holds by making the substitution
$x^{\prime}\mapsto-x^{\prime}-D$. In order to perform the integral over
$\mathbf{r}^{\prime}$ we make use of another identity. In particular we use
the identity

$\displaystyle\int_{\mathbb{R}^{3}}\text{d}^{3}\mathbf{r}\;(|\mathbf{r}-\mathbf{r}^{\prime}||\mathbf{r}-\mathbf{r}^{\prime\prime}|)^{-7/2}$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{3}}\text{d}^{3}\mathbf{k}\;\frac{16|\mathbf{k}|}{9}\,e^{i\mathbf{k}\cdot(\mathbf{r}^{\prime}-\mathbf{r}^{\prime\prime})}$
(7.35) $\displaystyle=$
$\displaystyle-\frac{128\pi}{9|\mathbf{r}^{\prime}-\mathbf{r}^{\prime\prime}|^{4}}.$

The final equality holds when
$\mathbf{r}^{\prime}\neq\mathbf{r}^{\prime\prime}$. By using Eq. (7.35) we can
see that the zero-point energy to the left of the cavity is given by

$H_{\text{ZPE}}^{\text{I}}=-\int_{-\infty}^{-D/2}\text{d}x\;\left.\frac{8\pi
A\hbar
c}{(2\pi)^{3}}\frac{1}{|\mathbf{r}-\mathbf{r}^{\prime}|^{4}}\right|_{\mathbf{r}=\mathbf{r}^{\prime}}.$
(7.36)

The area $A$ is the area inhabited by the field in the $y$-$z$ plane. This
term is divergent, as can be seen by substituting
$\mathbf{r}^{\prime}=\mathbf{r}^{\prime\prime}$ into the first line of Eq.
(7.35), and the term under the integral sign is independent of $D$. This term
corresponds to the infinite zero-point energy of the field to the left of the
cavity and is completely analogous to the one-dimensional result found in Eq.
(7.15).

In Region III the calculation follows almost identically. In this region we
find the similar result

$H_{\text{ZPE}}^{\text{I}}=-\int_{D/2}^{\infty}\text{d}x\;\left.\frac{8\pi
A\hbar
c}{(2\pi)^{3}}\frac{1}{|\mathbf{r}-\mathbf{r}^{\prime}|^{4}}\right|_{\mathbf{r}=\mathbf{r}^{\prime}}.$
(7.37)

As would be expected from symmetry arguments, the zero-point energy density of
the field to the left and the right of the cavity is the same.

###### Inside the cavity

Inside the cavity we calculate the zero-point energy in the same way. After
substituting the field observables given in Eqs. (7.32) and (7.33) into the
classical expression for the energy and calculating the vacuum expectation
value, we find that

$\displaystyle H_{\text{ZPE}}^{\text{II}}$ $\displaystyle=$
$\displaystyle\sum_{n,m=-\infty}^{\infty}\int_{-D/2}^{D/2}\text{d}x\int_{-D/2}^{D/2}\text{d}x^{\prime}\int_{-\infty}^{\infty}\text{d}y^{\prime}\int_{-\infty}^{\infty}\text{d}z\;\left(\frac{9A\hbar
c}{2(4\pi)^{3}}\right)$ (7.38)
$\displaystyle\times\left[(|\mathbf{r}-\mathbf{r}^{\prime}_{2n}||\mathbf{r}-\mathbf{r}^{\prime}_{2m}|)^{-7/2}+(|\mathbf{r}-\mathbf{r}^{\prime}_{2n-1}||\mathbf{r}-\mathbf{r}^{\prime}_{2m-1}|)^{-7/2}\right].$

In the above, $A$ has the same interpretation as before. The result above in
Eq. (7.38) is analogous to the expression for the zero-point energy inside a
one-dimensional cavity given in Eq. (7.17). The two terms are alike due to the
similar geometries of both cavities. To simplify Eq. (7.38) we can therefore
proceed exactly as before by making the substitutions described in Appendix C
and the paragraph beneath Eq. (7.18). The resulting expression for the zero-
point energy inside the three-dimensional cavity is then given by

$\displaystyle H_{\text{ZPE}}^{\text{II}}$ $\displaystyle=$
$\displaystyle\sum_{m=-\infty}^{\infty}\int_{-D/2}^{D/2}\text{d}x\int_{\mathbb{R}^{3}}\text{d}^{3}\mathbf{r}^{\prime}\;\left(\frac{9\hbar
cA}{2(4\pi)^{3}}\right)$ (7.39)
$\displaystyle\left[(|\mathbf{r}-\mathbf{r}^{\prime}||\mathbf{r}-\mathbf{r}^{\prime}_{2m}|)^{-7/2}\right].$

This expression is analogous to the final line of Eq. (7.18). To simplify this
expression further we again use the identity given in Eq. (7.35). As
$\mathbf{r}^{\prime}$ and $\mathbf{r}^{\prime}_{2m}$ differ only by a distance
$2mD$ along the $x$ axis, we find that

$H_{\text{ZPE}}^{\text{II}}=-\sum_{m=-\infty}^{\infty}\int_{-D/2}^{D/2}\text{d}x\;\frac{\hbar
cA}{\pi^{2}}\frac{1}{|2mD|^{4}}.$ (7.40)

#####  7.4.3 The Casimir force

The zero-point energy derived in Eq. (7.40) contains the addition of two
terms: one divergent term and one convergent term. The divergent term can be
immediately identified as the one for which $m=0$; let us look at this term
more closely. Compare the final line of Eq. (7.34) and the $m=0$ contribution
to Eq. (7.39). By noting that $\mathbf{r}^{\prime}_{2m}=\mathbf{r}^{\prime}$
when $m=0$, we can see that the energy density in both terms are equal. The
$m=0$ contribution to Eq. (7.40) is just the divergent zero-point energy of
the free fields. We may therefore rewrite Eq. (7.40) in the form

$H_{\text{ZPE}}^{\text{II}}=H_{\text{ZPE}}^{\text{II(free)}}-\frac{\hbar
cA}{8\pi^{2}D^{3}}\sum_{m=1}^{\infty}\frac{1}{m^{4}}.$ (7.41)

This expression is equal to the sum of the usual $D$-independent free field
contribution to the zero-point energy and an additional $D$-dependent
correction. As before, the negative correction to the zero-point energy is
caused by the interference between two components of the electric and magnetic
fields where one component has been reflected at least twice more than the
other.

To determine the total Casimir force on the mirrors we use the formula given
in Eq. (7.22). As the only $D$-dependent contribution to the total zero-point
energy density is the correction term to the zero-point energy inside the
cavity, the Casimir energy in this calculation is given by the second term on
the right-hand side of Eq. 7.41. This generates a force acting from the inside
of the cavity. Using this expression we find that the Casimir force per unit
area is given by

$F_{\text{Casimir}}=-\frac{1}{A}\frac{\partial E_{\text{Casimir}}}{\partial
D}=-\frac{3\hbar c}{8\pi^{2}D^{4}}\sum_{m=1}^{\infty}\frac{1}{m^{4}}.$ (7.42)

The infinite sum is well known and corresponds to the Riemann zeta function of
argument $4$. This sum is given by $\pi^{4}/90$. Hence, we find that the
Casimir force per unit area between two infinite parallel mirrors is given by

$F_{\text{Casimir}}=-\frac{\hbar c\pi^{2}}{240D^{4}}.$ (7.43)

This value is identical to the usual result found by other methods for light
with two distinct polarisations [192, 198, 199]. The resulting expression
describes an attractive force that pulls the two mirrors together and
increases with the negative fourth power of the mirror separation.

####  7.5 Discussion

The Casimir effect is usually calculated by examining changes to the standing
modes formed inside a perfect cavity. From this point of view only those modes
with a wavelength less than twice the width of the cavity are permitted inside
the cavity. Moreover, additional regularisation procedures must be applied in
order to wrestle the expression for the zero-point energy of the field into a
form where corrections caused by the cavity can be examined. In this chapter
we have shown that the usual well-known Casimir forces in one- and three-
dimensional cavities are an exact result that can be calculated by considering
the propagation of localised energy quanta both inside and outside the
cavities.

In both one and three dimensions the primary excitations of the EM field,
blips, are entirely localised. The blips fit perfectly inside cavities of any
width and move around as if in a vacuum until coming into direct contact with
the mirrors. From this point of view, light is treated as a collection of
small particles that interact with the cavity like classical point-like
particles. As blips are constructed by superposing monochromatic photons of
all frequencies, there are no restrictions on the frequency of light that can
be injected into the cavity, even if the cavity is exceptionally narrow. By
viewing the monochromatic photons as only a tool for constructing localised
wave packets we can see that more appropriate boundary conditions can be
applied just by specifying the trajectory of a blip.

Although the blip excitations are localised, the electric and magnetic field
observables have a non-local dependence on the configuration of blips, which
is implemented by the regularisation operator $\mathcal{R}$. As a result, the
electric and magnetic fields are altered by the presence of the cavity at
positions far removed from the mirrors themselves. In particular, as the two
mirrors are highly reflecting, a single blip inside the cavity contributes to
the electric and magnetic fields many times due to the many mirror images that
that blip has. These changes were implemented by taking the mirror images of
the field observables both inside and outside the cavity (see Sections 7.2.3
and 7.4.1). In this description, the Casimir effect arises due to interference
between the reflected components of the highly non-local field observables
that must be folded back into the cavity many times over.

This result also helps us to understand certain characteristics of the
resulting force. For example, looking again at Fig. 7.3, we can see that,
since the fields decrease the further one moves from the unregularised blip,
contributions that have been folded the most contribute the least to changes
of the field observables. This can be seen in Eqs. (7.23) and (7.42) where the
greater $m$ becomes the smaller the contribution to the overall force.
Moreover, by comparing parts (c) and (d) of Fig. 7.3, one can see that the
closer together the mirrors become, the more the reflected components
contribute to the field inside the cavity. Hence, as the mirrors are moved
closer the force between the mirrors increases.

The methods used in this chapter provide a particularly intuitive way of
deriving the Casimir effect. By piecing together the classical world-line of
localised particles inside a cavity we can calculate the zero-point energy of
the field in a way that easily distinguishes any divergent contributions
without the need for regularisation procedures or approximations [192], and
gives a more natural understanding of the physics at play. This approach is
therefore very well suited to investigations into Casimir forces in cavities
with different geometries, lossy or partially transparent mirrors, and even
inhomogeneous materials or curved space-times.

###  Chapter 8 Discussion

In this thesis we refer to single-photon wave packets as the set of
excitations associated with the normalised annihilation operators $a$
satisfying the commutation relation $[a,a^{\dagger}]=1$. It is usual to
construct these wave packets by superposing linear combinations of the
orthogonal and non-normalisable excitations associated with the monochromatic
plane wave solutions of Maxwell’s equations. These excitations are non-
localised and are characterised by a single wave vector $\mathbf{k}$ and
polarisation $\lambda=\mathsf{H},\mathsf{V}$, each carrying an energy
$\hbar|\mathbf{k}|c$. In this thesis we have taken an alternative approach and
constructed single-photon wave packets from a set non-normalisable bosonic
particles that have a well-defined position in space-time, so-called blips. In
the past it has been thought that localised excitations of the quantised EM
field do not exist [21]; however, here we overcome existing no-go theorems by
distinguishing between localised particles and the field observables
themselves.

In the classical theory of the free EM field in one dimension, localised wave
packets propagate in both directions at the speed of light without dispersion.
By considering a set of highly localised single-photon wave packets in one
dimension we have pointed out that an additional discrete parameter $s=\pm 1$
is needed in order to describe the unitary evolution of short light pulses
that behave in an analogous way. States characterised by different values of
$s$ must be orthogonal to one another in order to distinguish between the
different dynamics of left- and right-propagating photons. As we demonstrated
in Section 6.3.4, in standard quantisations of the EM field that neglect this
additional parameter, single-photon wave packets do not propagate causally but
spread out filling all of space immediately. When our quantisation scheme is
used, however, we find that single-photon wave packets propagate at the speed
of light. In three-dimensions, despite a much larger Hilbert space, light is
also only characterised by an additional parameter $s=\pm 1$. As demonstrated
in Chapter 5, in three dimensions the localised solutions of Maxwell’s
equations do not propagate in straight lines. The role of the parameter $s$ in
this case is to specify the overall motion of the system, for example, the
direction of a plane wave.

Following closely the quantisation scheme of Bennett et al. [131], we were
able to derive a set of field observables in terms of the orthogonal blip
operators. In order to ensure that the field observables possessed the correct
dimensions and transformation properties, the electric and magnetic field
observables were related to the blip operators by means of a non-local
regularisation operator $\mathcal{R}$. In this thesis, in contrast to many
other works, we differentiate between the basic excitations of the EM field,
which are completely localised, and the corresponding field observables, which
are spread out across all of space (see Figs. 4.2.b and 7.3.a). In this sense
our views are more in line with those of Fleming [12] who states that locally
orthogonal particle states also have some fundamental meaning. By
distinguishing between the particles and fields we were able to demonstrate
how the Casimir effect can be shown to arise entirely due to the non-local
characteristics of the field observables. More specifically, although the
localised blip excitations are unaffected by the mirror when the two are not
in contact, the non-local fields are necessarily reflected many times (see
Fig. 7.3.d) causing a large amount of interference inside the cavity. The
effect of this interference on the zero-point energy of the field varies with
the width of the cavity causing an attractive force between the mirrors.

In both one and three dimensions our quantisation differs from standard
approaches by introducing negative-frequency states. These states are often
disregarded as they do not usually possess a positive norm [200]; however, in
this thesis we have found that negative-frequency states are necessary for the
construction of wave packets corresponding to all possible solutions of
Maxwell’s equations. In quantum mechanics, the positive energy observable and
the dynamical Hamiltonian would usually be equal to each other. When both
positive and negative frequencies are taken into account, however, the
dynamical Hamiltonian cannot be bounded below.

To avoid this problem we no longer insist that the positive energy observable
and the generator of time translations be the same. In this way we avoid the
consequences of Hegerfeldt’s theorem [72] without the unnerving issue of
having an energy observable without a lower bound. Unlike the energy
observable, the dynamical Hamiltonian takes the form of an exchange operator.
The role of this operator is to continually recreate localised blips at new
positions at a rate equal to the energy expectation value of that state.
Without an energy observable to guide us, the exact form of the dynamical
Hamiltonian is determined from a constraint on the dynamics of the system. In
free space, light is restricted to the boundaries of the light-cones as
illustrated in Fig. 1.1, but similar constraints can be found for any other
trajectory.

The quantisation schemes presented in Chapters 4 and 5 not only provide
solutions to existing problems; for example, in Chapter 6 we found a possible
resolution to the Fermi problem and in Ref. [110] we constructed locally-
acting mirror Hamiltonians, but can also be used, as we have seen in Chapter
7, to predict experimentally verifiable results. In spite of these successes,
there are still important aspects of this quantisation that we do not yet
fully understand and that need to be examined further. For example, in
classical treatments, the momentum of a light pulse is given by its energy
multiplied by a unit vector oriented in the direction of propagation. Although
the energy of a blip is undetermined we appear to know something about its
direction, and therefore its momentum, whilst simultaneously having complete
knowledge of its position. In our case, when the position of a blip is known,
there is complete uncertainty in the wave vector $\mathbf{k}$ and therefore
complete uncertainty in the blip’s dynamics. Nevertheless, we’re not yet sure
if this can be related to the uncertainty of a measurable quantity and, if so,
what that quantity would be.

There is also the related question of, if our description of the quantised EM
field is approached from the point of view of canonical quantisation, what
would the canonical variables be and how would their commutation relation be
altered. This question has been asked in Refs. [146, 168, 169] that look at a
canonical quantisation approach to real fields containing negative-frequency
modes. In standard QED the canonical momentum is given by the electric field
observable which commutes non-trivially with the vector field
$\mathbf{A}(\mathbf{r},t)$ and therefore also $\mathbf{B}(\mathbf{r},t)$. In
our scheme the electric and magnetic fields are always related by a rotation
and therefore commute. This is an important result that would need to be
investigated and lends itself to be measured via uncertainties.

Other questions which could be addressed experimentally are, for instance, can
the negative-frequency photons be measured in a laboratory? As both positive-
and negative-frequency waves are required to localise a wave packet to any
length it is practically impossible to separate the two, and so a clever
experiment must be devised. The authors of Ref. [165] claim to have already
detected negative-frequency photons, but the real relation between what we
both call negative-frequency photons would need to be examined more carefully
before we can draw final conclusions. One possibility for detecting negative-
frequency photons could be to construct a transformation between the positive-
frequency and negative-frequency modes, and then probe this transformation by
means of an interference experiment.

In standard quantum electrodynamics a conversion from the positive-frequency
to negative-frequency modes is imposed by a non-unitary Bogoliubov
transformation between the annihilation and creation operators of the original
field operators. This results in an amplification effect such as the Unruh
effect for uniformly accelerating observers. In our case, however, it is
possible that we would require an anti-unitary transformation on the Hilbert
space (similar to time reversal) that converts between the positive- and
negative-frequency but positive-norm photon states. Materials with negative or
temporally oscillating refractive indices have been studied, for example, by
Pendry [201] and others [202, 203, 204], which reverse the phase evolution of
states, essentially inverting their frequencies. These experiments create the
possibility for realising the required transformation in a laboratory.

In this thesis we have also supposed that interactions take place by means of
a local exchange of blips. As it is the field observables that couple to
charged matter and currents (see Eqs. (2.4) and (2.7)), many other authors
hold the view that interactions take place by means of an exchange of field
excitations. An interesting extension to this work would therefore be to
couple blips to matter. This would not only increase the spectrum of problems
to which this theory could be applied, but also open an investigation into the
validity of the local blip approach to interactions by studying known effects.
It would also open the path to modelling experimental setups and photonic
devices which have not yet been accessible [195, 205]. This is a major
motivation for the work presented here.

Although there is still much to be understood and investigated, the work in
this thesis is beginning to open up new possibilities for studying complex or
non-intuitive systems. For example, this new formalism would also be very well
suited to studying gravitational systems in which the metric or curvature of
space-time varies from place to place. The Unruh effect is one effect that is
usually discussed in terms of frequency modes, which makes it difficult to
visualise the underlying physical processes. A blip description of this effect
is very likely to produce a more lucid explanation [206]. Casimir cosmology is
also an interesting field in gravitational physics [207, 208]. Regularisation
procedures, however, become increasingly difficult to manage in curved space-
times. As our approach has no need of regularisation procedures, this theory
may offer a satisfying resolution to this outstanding problem.

Our local theory of light also offers solutions to other long-standing
problems. For instance, the problem of time in quantum theory refers to the
impossibility of defining a time operator [209, 210, 211]. The Pauli objection
prohibits such an operator on the basis that energy is bounded from below
[212, 213, 214, 215] (and Refs. therein). As discussed in Ref. [215], this
theory evades this objection by defining a distinct dynamical Hamiltonian,
which is unbounded below, leading to the possibility of introducing time
observables or a quantum clock. This is perhaps not so surprising as our
Dirac-like equations of motion for the blips, Eqs. (4.12) and (5.55), closely
resemble the Wheeler-DeWitt equation of quantum gravity [216] which places a
constraint on the dynamics of a system.

In spite of remaining open questions, the results of this thesis and the way
in which our local theory appears to be so beautifully clearing the fog around
many other problems encourages me to believe that solutions will be found. By
looking at electromagnetic phenomena from the point of view of monochromatic
photons we only see half the picture; by introducing local photons we might in
the future not only be able to gain unique perspectives on a wide range of
existing problems, but create all the necessary tools for studying new ones.

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