Also at<EMAIL_ADDRESS>
Department of Informatics,
University of Sussex, Falmer, UK.

# The Quantum Vacuum of Spacetime with a Fundamental Length

Paul C.W. Davies<EMAIL_ADDRESS>Philip Tee  [<EMAIL_ADDRESS>The Beyond
Center for Fundamental Science, Arizona State University, Tempe AZ 85287, USA

###### Abstract

A quantum theory of gravity implies a fine-grained structure of spacetime,
which can be conveniently modeled as some form of ‘pixelation’ at the Planck
scale, with potentially observable consequences. In this work, we build upon
previous results to investigate the effect of pixelation on the quantum
vacuum, making use of the framework of Doubly Special Relativity (DSR). At the
center of the DSR approach is an observer dependent length scale, defining the
pixelation of spacetime. A key feature of quantum field theory in DSR is the
dispersive nature of the vacuum state and the associated appearance of
curvature in momentum space. As a result, the standard treatment of the
renormalized stress-energy-momentum tensor acquires correction terms. As an
illustration, we present here a calculation of the thermal vacuum and modified
Casimir effect, using both modified propagators and momentum measures. We
choose a consistent choice of momentum space metric that both generates the
modified dispersion relations we use and preserves the Lorentz invariant
character of the results obtained. Put together this constitutes a consistent
calculation framework we can apply to other more complex scenarios.

††preprint: APS/123-QED

##  I Introduction and Background

A familiar assumption is that spacetime is continuous, and the majority of
contemporary theories of physics are described using mathematics that assumes
continuity. However, even in classical physics the idea of a point particle
such as an electron inevitably leads to divergences, and in quantum fields
these problems become more serious. With electrodynamics (and the nuclear
forces), renormalization is successful at taming such divergences in a
systematic fashion, but attempts to directly quantize gravity lead to a
catastrophically divergent theory. Could it be that the fundamental assumption
of spacetime continuity needs to be reevaluated?

It is often speculated that the existence of a quantum theory of gravity (QG)
will naturally lead to the idea that geometry itself is quantized [1, 2] and
therefore ceases to be continuous. There have been attempts to formulate
theories of QG that do not require spacetime continuity such as Loop Quantum
Gravity [3], Combinatorial Gravity [4, 5], Causal Dynamical Triangulation [6]
and Quantum Geometry [7], but none have been universally accepted or
experimentally verified. If spacetime is discrete, there would have to exist
an observer-independent fundamental length scale usually expected to be the
Planck length $l_{p}=1.616\times 10^{-35}\text{m}$, although the precise value
could be different. In the absence of a full theory of quantum gravity (QG),
is it possible to investigate leading order effects such a fundamental length
scale would have on a quantized field? If we can arrive at a result which is
testable, or at least one that surfaces logical consistency issues, this may
throw light on the existence of a fine-grain structure or pixelation to
spacetime. The importance of a null result in such an investigation would be
to cast doubt on geometry being quantized at the highest energy scales, and by
implication perhaps bring into question whether gravity is in fact a quantum
theory. Given that experimental tests of QG are currently out of reach [8],
this provides an alternative route to establishing the existence of a theory
of QG. This is perhaps one of the most important open problems in theoretical
physics.

Doubly Special Relativity (DSR), provides such a framework in which to study
the effects of a fundamental length. The key merit of DSR is that it is able
to incorporate a fundamental length that is observer independent [9]. The
approach rests upon modifications to the underlying group structure of the
special theory of relativity [10, 11], by deforming the generators of the
Poincaré group. This has the effect of modifying energy-momentum dispersion
relations, but crucially also the underlying geometry of momentum space
$\mathcal{P}$. Such modifications are not unique to DSR, for example other
approaches to a quantum theory of gravity (QG), such as Hor̆ava-Lifshitz [12]
also propose that dispersion relations are modified. We do not intend to cover
the details of DSR here, but the key point is that we can take an approximate
approach to calculations that allow us to investigate the leading order
effects of modifications to dispersion relations.

In previous work [13, 14], we investigated the effect of modified dispersion
relations on accelerated particle detectors and gravitational deflection of
light. In the case of accelerated detectors, we identified that these
modifications to the dispersion relations can lead to transition probabilities
that are not positive definite when propagation is subluminal. This result
would appear to put into tension the Davies-Fulling-Unruh effect with the
existence of a fundamental length scale. Nevertheless, the dispersion
relations themselves are not the full story, as the use of them in isolation
causes the model to be in violation of Lorentz invariance. DSR restores
Lorentz invariance by affording momentum space a non-trivial curved geometry.
In our prior calculations we argued around this complication by asserting that
any adjustment to the modified propagators we derive must necessarily be
small, contributing extra powers in the expansion parameter we used
$\kappa\eta^{2}$, where $\eta$ is the inverse of the Planck Mass, and $\kappa$
a dimensionless parameter controlling the sign of any corrections and
intimately linked to whether propagation is subluminal (i.e. for values of
$\kappa<0$). In this work we explore this assertion more thoroughly by taking
the calculation back to basics and applying it to the most basic object in
quantum field theory, the quantum vacuum.

The effect of momentum curvature enters into the calculation by changing the
integration measure when we integrate over momenta. In the particle detector
example described above, we typically need position space propagators which we
obtain by a Fourier transform of the momentum space propagators suitably
deformed to be consistent with the modified dispersion relations. Some of the
earliest considerations of quantized spacetime dating back to Snyder’s
original proposal [15, 1], modelled Minkowski spacetime as a $3+1$ hyperbolic
surface embedded in a higher dimensional ($4+1$) space. This concept is
carried over into many interpretations of DSR, with the additional assumption
that momentum space is weakly de Sitter [16, 17]. As the momentum space now
has a non-trivial geometry any integration measure must include a factor of
$\sqrt{-g_{p}}$, where $g_{p}$ is the determinant of the momentum space metric
tensor.

The system we study here is the quantum vacuum of a massless scalar field,
both with and without a boundary. This is the simplest set up in which to test
further the calculational framework that exposed the pathologies in the
accelerated detector when considering spacetime with a fine-grained structure.
In the absence of a boundary we compute the energy density of a thermalized
vacuum, and in the presence of a static boundary we repeat the calculation for
the Casimir effect [18]. In both cases we are computing the energy density of
the vacuum, and we do so using two methods. The methods were chosen
specifically to rely upon the modified dispersion relations and the non-
trivial momentum measure independently. The first method is the technique of
“point splitting” to compute a renormalized stress-energy-momentum tensor.
This makes direct use of the propagator of the theory and therefore modified
dispersion relations. The second approach is more direct, computing the energy
density from an integral over field modes. This relies upon the density of
states for the thermal vacuum and the Hamiltonian for the Casimir effect, and
both involve an integral over momenta. The two approaches neatly separate the
measure from the propagator as the point splitting technique is local and does
not require a measure, whereas the integral over modes is global and does not
require knowledge of the propagator. If the two methods agree we can conclude
that the momentum measure and modified dispersion relations are internally
consistent, which will inform our reevaluation of the results obtained in
earlier work. As we describe in Section II, the momentum measure which is
consistent with the modified propagators we use here and in previous work, has
a first order correction of $\sqrt{-g}=1+\kappa\eta^{2}p^{2}$. The consequence
of this for our previous calculation is that to $\mathcal{O}(\kappa\eta^{2})$
we obtain a similar result, but the sign of the correction will be changed.
This has the consequence that the tension in the result for the Davies-
Fulling-Unruh effect on a fine-grained background is resolved when propagation
is subluminal. This is the first result of this work and confirms that the
inclusion of a modified momentum measure restores the Lorentz invariant
character to results obtained by using DSR to leading order.

The second set of results in the paper concern the application of these
methods to the vacuum energies of free thermal and bounded (Casimir)
spacetimes. When we perform the detailed calculations both methods yield
fundamentally consistent results for the energy density of both the free
vacuum and the Casimir effect. For the free vacuum we obtain,

$\displaystyle\expectationvalue{T_{tt}}$
$\displaystyle=\frac{\pi^{2}}{30\beta^{4}}+\frac{\kappa\eta^{2}\pi^{4}}{126\beta^{6}}\text{,}$
(1) $\displaystyle\expectationvalue{E}$
$\displaystyle=\frac{\pi^{2}}{30\beta^{4}}+\frac{8\kappa\eta^{2}\pi^{4}}{126\beta^{6}}\text{,}$
(2)

where the first result Eq. 1 is obtained using the modified propagator, and
the second Eq. 2 using the density of states. For the second result the
straight application of the density of states approach yields a correction
that is larger than that from point splitting, but we explore in Section III
how this can arise as a result of the over-counting of unphysical states. We
describe in more detail there how by the introduction of a cutoff to avoid
over-counting of unphysical states (i.e. wavelengths below the length scale of
spacetime) the correspondence can be made exact. For the Casimir effect [18]
we can also obtain the result from modified propagators by the use of ‘point-
slitting’ as first used by Brown and Maclay [19], to obtain the classical
result and a correction term,

$\expectationvalue{T_{\mu\nu}}=\frac{-\pi^{2}}{1440a^{4}}\left(1-\frac{5\kappa\eta^{2}\pi^{2}}{42a^{2}}\dots\right)\matrixquantity[1&0&0&0\\\
0&-1&0&0\\\ 0&0&-1&0\\\ 0&0&0&3\\\ ]\text{.}$ (3)

Approaching this calculation using the Hamiltonian for the field, and we
arrive at an almost identical result, with the exception that the correction
term $\frac{12\kappa\eta^{2}\pi^{2}}{42a^{2}}$ is again larger. In section
Section IV we explain how this slight difference arises from including non-
physical states in the calculation, and how their exclusion could sharpen the
correspondence.

The layout of this paper is as follows. We begin by an overview in Section II
of the calculation framework in terms of the modifications made to the
propagators and the momentum measure. We turn to the calculation of the free
space vacuum energy in Section III, and then the Casimir effect in Section IV.
As the computations for these results are lengthy we reserve the details for
the appendices to not clutter the arguments, and cover just the major points
of the calculations in the text. At all times we work in units where
$\hbar=c=k_{B}=G=1$, and work in a mostly minus metric signature.

We conclude in Section V, with a discussion of the results and a survey of
future directions.

##  II Modified Propagators and the Curvature of Momentum Space

Our approach is to model the presence of a finite scale by the use of modified
propagators inspired by DSR to recompute various well-known effects that arise
when studying quantum field theory in curved spacetimes. The modifications
extend the canonical energy-momentum dispersion relation to include terms that
are higher order in the momentum. This is the core proscription of both Doubly
Special Relativity [9] and also other approaches such as Hor̆ava-Lifshitz
theories of gravity (HLG) [12]). We focus on a modified dispersion relation of
the form,

$E^{2}=p^{2}+\kappa\eta^{2}p^{4}+m^{2}\text{,}$ (4)

where $\eta$ is a constant with dimension of inverse mass (usually assumed to
be the Planck mass), and $\kappa=\pm 1$ is a dimensionless constant that
controls the sign of the modification. In the case of other modified
dispersion relation theories such as HLG, the usual proscription sees higher
terms in $p$ such as $p^{6}$. The significance of $\kappa$ is that for
positive values the equations of motion indicate that propagation can be
superluminal, and only for $\kappa<0$ is the maximum speed of propagation $c$.
It is this ambiguity in the maximum speed of propagation that hints at the
non-Lorentzian nature of a theory that uses modified propagators and nothing
else. It is worth noting that the standard dispersion relation
$E^{2}=p^{2}+m^{2}$, is a simple consequence of the Minkowski norm of the four
vector $p_{\mu}p^{\mu}=m^{2}$ being evaluated in the rest frame of the
propagating particle. One arrives at the standard result assuming that
$p_{\mu}p^{\mu}=g_{p}^{\mu\nu}p_{\mu}p_{\nu}$, where $g_{p}^{\mu\nu}$ is the
metric of momentum space and is simply $\text{diag}(1,-1,-1,-1)$.

A Lorentz invariant DSR theory implies that momentum space is curved [17], and
as a result the use of modified dispersion relations must be done in a way
that respects the non-trivial nature of momentum space. The norm in a general
momentum space is now more complex than the simple $p_{\mu}p^{\mu}=m^{2}$
proscription, and relies upon the arc length of the interval in momentum space
along a geodesic. This intuitively computes the ‘length’ of the momentum four
vector in the more general momentum space, in the same way that for non-
trivial spacetimes we compute proper time. The full expression for this norm
$D(0,p^{\mu})$ is,

$D(0,p^{\mu})=m^{2}=\int\limits_{0}^{1}\sqrt{g^{\mu\nu}\dot{p}_{\mu}\dot{p}_{\nu}}~{}\differential
s\text{,}$ (5)

where we integrate along a geodesic in the momentum space described by a more
general interval $ds^{2}=g_{\mu\nu}\differential p^{\mu}\differential p^{\nu}$
(we have dropped the $p$ suffix to denote momentum space metric, and unless
otherwise stated metrics are for the momentum space $\mathcal{P}$). The
geodesic curve is parameterized by $s$ and in Eq. 5 the $\dot{p}_{\mu}$ should
be interpreted as differentiation with respect to $s$. We cover the details in
Appendix A, but our goal is to obtain a form for $g_{\mu\nu}$ that recovers
our modified propagator Eq. 4 from Eq. 5. We discover that the following
choice of non-diagonal metric for $n$-dimensional space,

$\begin{split}&g^{\mu\nu}=\begin{pmatrix}1&0\\\
0&-\cosh^{2}\left[\left(\alpha^{2}\right)^{\frac{1}{2}}p_{0}\right]\Omega_{ij}\\\
\end{pmatrix}\text{~{}with,~{}~{}}\\\
&\Omega_{ij}=\delta_{ij}+\frac{p_{i}p_{j}}{\alpha^{2}-p^{2}}\text{~{}~{},}i=1,\dots,n\text{,}\end{split}$
(6)

satisfies this constraint. With this choice of metric, we can then replace the
conventional Lorentz covariant momentum measure
$\int\differential^{4}p=\int\frac{\differential^{3}p}{(2\pi)^{3}E_{p}}$, where
$E_{p}$ is the on shell energy with $\int\sqrt{-g}\differential^{4}p$. We can
expand the metric determinant in powers of $\kappa\eta^{2}$, and to leading
order we obtain,

$\sqrt{-g}=1+\kappa\eta^{2}p^{2}\text{,}$ (7)

where $p^{2}$ here refers to the norm of the spatial momentum (i.e.
$p^{2}=\overrightarrow{\mathbf{p}}^{2}$), and going forward if we omit a Greek
index we assume $p$ is the spatial momentum.

In our prior investigations the importance of this measure is that we require
position space Green’s functions. Our starting point is to convert Eq. 4 into
an associated Feynman diagram [14],

prules (55,40) o1 o2 dashes_arrow, label=$p$o1,o2 o1
$=\frac{1}{p_{0}^{2}-p^{2}(1+\kappa\eta^{2}p^{2})-m^{2}+i\epsilon}$,o2

and position space propagator,

$G(t,x;t^{\prime},x^{\prime})=\int\limits_{-\infty}^{\infty}\frac{\differential^{n}p}{(2\pi)^{n}}\frac{e^{-i[p_{0}(t-t^{\prime})-\overrightarrow{\mathbf{p}}\dotproduct(\overrightarrow{\mathbf{x}}-\overrightarrow{\mathbf{x^{\prime}}})]}}{p_{0}^{2}-p^{2}(1+\kappa\eta^{2}p^{2})-m^{2}}\text{.}$
(8)

As obtained in [13] we have for the modified propagator for a massless scalar
field in position space,

$D^{+}(t,r)=\frac{-\theta(\sigma^{2})}{4\pi^{2}(\sigma^{2}-i\epsilon)}\left\\{1-\frac{\kappa\eta^{2}}{(\sigma^{2}-i\epsilon)}\right\\}\text{,}$
(9)

with the corresponding (retarded) space-like case,

$D^{-}(t,r)=\frac{\theta(-\sigma^{2})}{4\pi^{2}(-\sigma^{2}+i\epsilon)}\left\\{1+\frac{\kappa\eta^{2}}{(-\sigma^{2}+i\epsilon)}\right\\}\text{.}$
(10)

For economy of notation $\sigma^{2}=t^{2}-r^{2}$ the invariant interval, with
$\theta(x)$ being the normal sign function. Setting $\eta=0$ recovers the
standard result [20].

##  III The Free Space Vacuum in Pixelated Spacetime

###  III.1 Energy density from modified propagators

Our first approach to computing the energy density will make use of the
modified propagator described in Section II. The approach exploits the
approximate stress energy renormalization technique often referred to as point
splitting [21]. The details of the calculation are in Appendix B, but our
starting point for this calculation is the thermal Greens function for the
modified propagator of the massless scalar field in Cartesian coordinates,

$\displaystyle
D^{(1)}_{\beta}(t,\overrightarrow{\mathbf{x}};t^{\prime}\overrightarrow{\mathbf{x}}^{\prime})$
$\displaystyle=-\frac{1}{2\pi^{2}}\sum\limits_{m=-\infty}^{\infty}\left\\{\frac{1}{(t-t^{\prime}-im\beta)^{2}-(x-x^{\prime})^{2}-(y-y^{\prime})^{2}-(z-z^{\prime})^{2}}\right\\}$
(11)
$\displaystyle+\frac{\kappa\eta^{2}}{2\pi^{2}}\sum\limits_{m=-\infty}^{\infty}\left\\{\frac{1}{[(t-t^{\prime}-im\beta)^{2}-(x-x^{\prime})^{2}-(y-y^{\prime})^{2}-(z-z^{\prime})^{2}]^{2}}\right\\}\text{.}$
(12)

It should be understood that the summation will omit the infra-red divergence
for $m=0$.

The key to the method is to recognize that the propagator can also be written
as an expectation value of a field commutator. Specifically,
$D^{(1)}_{\beta}(t,\overrightarrow{\mathbf{x}};t^{\prime}\overrightarrow{\mathbf{x}}^{\prime})=\expectationvalue{\\{\phi(t,\overrightarrow{\mathbf{x}}),\phi(t^{\prime},\overrightarrow{\mathbf{x}}^{\prime})\\}}{\beta}$,
for a vacuum prepared as a mix of states at coolness (inverse temperature)
$\beta$, and
$D^{(1)}(t,\overrightarrow{\mathbf{x}};t^{\prime}\overrightarrow{\mathbf{x}}^{\prime})=\expectationvalue{\\{\phi(t,\overrightarrow{\mathbf{x}}),\phi(t^{\prime},\overrightarrow{\mathbf{x}}^{\prime})\\}}{0}$
for the zero temperature pure state vacuum. As the stress energy tensor is
generally expressed in terms of derivatives of the field $\phi$, one can
convert second derivatives of $\phi$ (and squares of first derivatives) to
derivatives of the propagator at two ‘split’ points that we then take the
limit of zero separation.

The specific expression for the conformal stress energy tensor is given by Eq.
41, and we can then exploit the infinite boundary conditions for the field of
$\phi=\partial\phi=0$ to simplify the expression to Eq. 42. As we are dealing
with flat space where $g_{\mu\nu}=\text{diag}(1,-1,-1,-1)$ the awkward cross
derivatives are not relevant, and we are left with the diagonal entries of the
stress tensor to evaluate,

$\expectationvalue{T_{\mu\mu}}{0}=\expectationvalue{(\partial_{\mu}\phi)^{2}-\frac{1}{4}g_{\mu\mu}\partial_{\nu}\phi~{}\partial^{\nu}\phi}{0}\text{.}$

Both derivatives are readily computed from the propagator and for the first
contribution Eq. 11 we obtain the expected standard result
$\expectationvalue{T_{tt}}=\frac{\pi^{2}}{30\beta^{4}}$. More care must be
taken with the correction Eq. 12, as the additional square in the denominator
will involve cross terms such as $(x-x^{\prime})^{2}(y-y^{\prime})^{2}$.
Fortunately when the derivatives involved in the point-splitting method are
applied, all of these are zero and do not contribute. In fact, the only
contribution comes from the time derivatives, and so we arrive at a correction
term of $\frac{\kappa\eta^{2}\pi^{4}}{126\beta^{6}}$. We can compute the other
diagonal components similarly as described in Appendix B, and we reproduce our
final result for the leading order thermal vacuum energy density from Eq. 46
here,

$\expectationvalue{T_{\mu\nu}}=\left(\frac{\pi^{2}}{30\beta^{4}}+\frac{\kappa\eta^{2}\pi^{4}}{126\beta^{6}}\right)\matrixquantity[1&0&0&0\\\
0&\frac{1}{3}&0&0\\\ 0&0&\frac{1}{3}&0\\\ 0&0&0&\frac{1}{3}\\\ ]\text{.}$ (13)

###  III.2 Energy density from modified measure

An alternative approach to the calculation involving point splitting proceeds
by using a density of states. From elementary considerations it will be
recalled that for modes of angular frequency $\omega$, the density of states
$\rho(\omega)$ can be written as,

$\rho(\omega)=\frac{\omega^{2}}{2\pi^{2}}\differential\omega\text{.}$ (14)

The standard derivation computes the number of states enclosed in an octant of
a sphere of radius $\omega$ in momentum space, in essence computing the volume
of momentum space $\mathcal{P}$ for the system. In the fundamental length
scale scenario we are considering, this must be modified to account for the
non-trivial geometry of $\mathcal{P}$. As discussed in Section II, this
amounts to inserting a factor of $\sqrt{-g}$ into the integral. Computing the
partition function for our scalar field at coolness $\beta$, allows us to
express the energy density as two integrals to $\mathcal{O}(\kappa\eta^{2})$
that we show in Eq. 51. We note here that the correction integral
$\frac{\kappa\eta^{2}}{4\pi^{2}}\int\limits_{0}^{\infty}\frac{\omega^{5}}{e^{\beta\omega}-1}\differential\omega$
has an infinite upper limit, which of course implies modes of zero wavelength.
Upon evaluation we have our result for the energy density Eq. 52, that we
restate here,

$\expectationvalue{E}=\frac{\pi^{2}}{30\beta^{4}}+\frac{8\kappa\eta^{2}\pi^{4}}{126\beta^{6}}\text{.}$
(15)

We see that the correction is eight times larger than the point splitting
result obtained in Eq. 13. A possible cause of this difference is the infinite
upper limit in the correction integral. We are implicitly assuming that
spacetime is discrete and so an infinite angular frequency $\omega$ as the
upper limit is not sensible and one would expect the integral to over-count
the contribution. If the upper limit did not extend to infinity the integral
will have a smaller numerical value. To progress, we introduce into the second
integral of Eq. 51 a cutoff, $\Lambda$ representing the maximum frequency
permissible with a wavelength at the scale of the pixelation of spacetime.
This is usually assumed to be the Planck frequency, but at this point is just
an arbitrary upper limit. When we evaluate the integral, we obtain a complex
expression involving Polylogarithms ([22] §9.5 and §9.55)
$\mathrm{Li}_{n}(e^{-\beta\Lambda})$ up to sixth order,

$\begin{split}&63\beta^{6}\int\limits_{0}^{\Lambda}\frac{\omega^{5}}{e^{\beta\omega}-1}\differential\omega=8\pi^{6}+63\beta^{5}\Lambda^{5}\log(1-e^{-\beta\Lambda})-315\beta^{4}\Lambda^{4}\mathrm{Li}_{2}(e^{-\beta\Lambda})\\\
&-1260\beta^{3}\Lambda^{3}\mathrm{Li}_{3}(e^{-\beta\Lambda})-3780\beta^{2}\Lambda^{2}\mathrm{Li}_{4}(e^{-\beta\Lambda})-7560\beta\Lambda\mathrm{Li}_{5}(e^{-\beta\Lambda})-7560\mathrm{Li}_{6}(e^{-\beta\Lambda})\text{.}\end{split}$

This expression can be expanded around $\beta$ for very high temperatures
(energies) or small $\beta$ and we obtain a $\Lambda$ dependent approximation,

$\begin{split}&\int\limits_{0}^{\Lambda}\frac{\omega^{5}}{e^{\beta\omega}-1}\differential\omega=\\\
&\frac{\pi^{6}}{63\beta^{6}}\left(\frac{63\Lambda^{5}\beta^{5}}{5\pi^{6}}-\frac{21\Lambda^{6}\beta^{6}}{4\pi^{6}}+\frac{3\Lambda^{7}\beta^{7}}{4\pi^{6}}+\mathcal{O}(\beta^{9})\right)\end{split}$

We propose a renormalization or re-scaling argument and choose a $\beta$
dependent value of $\Lambda$, such that the first term in the expansion is
simply $1$. On such choice would be
$\Lambda^{5}=\frac{5\pi^{6}}{63}(1+\beta^{-5})$, which would give for our
expansion,

$\int\limits_{0}^{\Lambda}\frac{\omega^{5}}{e^{\beta\omega}-1}\differential\omega=\frac{\pi^{6}}{63\beta^{6}}\left(1+\mathcal{O}(\beta^{n})\right)\text{,
with {\hbox{n\geq 1}}.}$

Inserting this back into our calculation Eq. 51, we have for our
“renormalized” energy density,

$\expectationvalue{E}=\frac{\pi^{2}}{30\beta^{4}}+\frac{\kappa\eta^{2}\pi^{4}}{126\beta^{6}}+\mathcal{O}(\beta)\text{.}$
(16)

This is only formally in agreement with the point-splitting results when we
take the limit $\beta\rightarrow 0$. However taking this limit is no longer be
consistent with the upper limit being $\Lambda$ we chose as in this limit the
energy scale is higher. This becomes a re-scaling argument similar to
renormalization. In fact, close inspection of the series expansion of the
Polylogarithms indicates that the choice of beta dependent $\Lambda$ would
have to be modified for each additional power of $\beta$. We have perhaps
motivated if not proven then principle that the excess value of the energy
density from the state density calculation is due to an unrealistic infinite
limit to the integrals involved.

##  IV The Casimir Effect in Pixelated Spacetime

In the previous section our spacetime was free of boundaries, and we were able
to obtain consistent results for the energy densities using either a modified
propagator or modified momentum measure. To push the method a little further,
we can complicate the calculation by the introduction of boundaries. The most
studied static system that exploits boundary conditions is the Casimir effect
[18], and there are two potential methods to compute the energy density as a
function of plate separation. We follow here the treatment of Brown and Maclay
[19], to compute the energy density from the field propagator, and then Fierz
[23] to compute using an integral over momenta.

The general set up is described in Fig. 1, where we have two perfect
conductors, infinite in the $x,y$ planes, separated by a small distance $a$.
Although we consider here a massless scalar field, we interpret conducting as
enforcing the field to satisfy Dirichlet boundary conditions on the plate
$\phi(x,y,0)=\phi(x,y,a)$ for all $x,y$.

$z$$y$$-4a$$-3a$$-2a$$-1a$$2a$$3a$$4a$$0$$\leftarrow
a\rightarrow$$\ast$$\ast$$\ast$$\ast$$\ast$$\ast$$\ast$ Figure 1: For our
Casimir system to have a value of $D^{(1)}=0$ on both plates we need an
infinite set of image sources placed at a distance of $z=\pm 2an$ for integer
$n$ for each plate. Placing the images at a spacing of $a$ locates a source on
each plate, and cause double placement of sources at each point along the $z$
axis. This doubling of the boundary terms translates into an extra factor of
$2$ on the numerator of the boundary term for the Hadamard function.

We set up the propagator in Cartesian coordinates as a pair of infinite image
sums to enforce these boundary conditions, which after subtraction of the free
space propagator is,

$\begin{split}D^{(1)}(t,\overrightarrow{\mathbf{x}};t^{\prime}\overrightarrow{\mathbf{x}}^{\prime})&=-\frac{1}{\pi^{2}}\sum\limits_{n=-\infty}^{\infty}\left\\{\frac{1}{(t-t^{\prime})^{2}-(x-x^{\prime})^{2}-(y-y^{\prime})^{2}-(z-z^{\prime}+2an)^{2}}\right\\}\\\
&+\frac{1}{\pi^{2}}\sum\limits_{n=-\infty}^{\infty}\left\\{\frac{\kappa\eta^{2}}{[(t-t^{\prime})^{2}-(x-x^{\prime})^{2}-(y-y^{\prime})^{2}-(z-z^{\prime}+2an)^{2}]^{2}}\right\\}\text{.}\end{split}$

From here we can follow the same calculation as in the previous section
Section III, with the details in Appendix C, and we restate the result Eq. 57
here,

$\expectationvalue{T_{\mu\nu}}=\frac{-\pi^{2}}{1440a^{4}}\left(1-\frac{5\kappa\eta^{2}\pi^{2}}{42a^{2}}\dots\right)\matrixquantity[1&0&0&0\\\
0&-1&0&0\\\ 0&0&-1&0\\\ 0&0&0&3\\\ ]\text{.}$ (17)

Turning to an integral over momentum space, the expected energy per unit area
[24] can be expressed as the following divergent integral,

$\frac{E(a)}{A}=\frac{1}{2}\sum\limits_{n=1}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\frac{\differential
p_{x}\differential
p_{y}}{(2\pi)^{2}}\sqrt{\left(\frac{n\pi}{a}\right)+p_{x}^{2}+p_{y}^{2}}\text{.}$

The standard approach is to insert a renormalization factor $\exp[-\alpha
E(n)]$, where $E(n)$ is the square root term in the integral and $\alpha$ is a
renormalization parameter such that when $\alpha=0$ one recovers the divergent
integral. Our tactic is to insert a factor of $\sqrt{-g}$ term into the
measure to account for the non-trivial momentum space geometry.

The details of the calculation are in Appendix C, and we reproduce the result
for $\expectationvalue{T_{tt}}$ from Eq. 62 here,

$\expectationvalue{T_{tt}}=\frac{-\pi^{2}}{1440a^{4}}\left(1-\frac{12\kappa\eta^{2}\pi^{2}}{42a^{2}}\right)\text{.}$
(18)

Once again we notice that the correction result is slightly larger than that
obtained by point-splitting. The measure in the integral also suffers from
infinite limits, implying modes of infinite momentum, and indeed the sum is
also an infinite sum involving modes of zero wavelength. We speculate that
this could be the source of the discrepancy.

The calculation of the renormalized integral is much more complex than the
free space version in the prior section, so following the same argument is
beyond reach at this point. We can however explore capping the sum over modes
to avoid contributions from modes of wavelength shorter than an arbitrary
pixelation length scale. If we introduce a regulator $\Lambda$ the upper limit
of the sum in the correction term Eq. 60, we will have a $\Lambda$ dependent
term to subtract from the integral. A simple computation shows that the
integrand would be smaller by,

$\frac{e^{-\frac{\alpha\pi}{a}\Lambda}[e^{\frac{\alpha\pi}{a}}(1+\Lambda)-\Lambda]}{(e^{\frac{\alpha\pi}{a}w}-1)^{2}}\text{.}$

As this is always positive (and of course tends to zero as
$\Lambda\rightarrow\infty$), we can see that our integral will over count the
contribution deriving from the curvature of momentum space associated with the
fundamental length scale.

##  V Discussion and Concluding Remarks

We set out in this work to evaluate the self-consistency of the DSR framework
in computing some well-known quantum field theory results. In our prior work
we arrived at the problematic result that for subluminal propagation
accelerated detectors do not have a positive definite probability of detecting
particles in the vacuum, albeit maintaining a thermal nature to the spectrum.
We remarked that our approach rested solely on the use of propagators modified
to agree with the deformed dispersion relations of DSR. On their own this is
not a Lorentz invariant calculation, but requires a modified momentum measure
in the production of position space propagators, which can be derived from the
metric of momentum space. This metric, in turn, must be consistent with the
modified propagators used.

In section Section II, we demonstrated that such a metric exists, and also
that to first order the metric would not affect our result for the accelerated
detectors. In essence, the selection of the metric modifies the norm of the
momentum four-vector to yield the modified dispersion relations we use in our
calculation. A different modification such as that proposed in HLG would
require a different metric. Taking the simplest quantum vacuum of a free space
thermal vacuum, and also one with boundaries we computed the energy density by
methods that require one but not both of a modified propagator and momentum
measure. We were able to demonstrate that the results obtained were
consistent. In addition, the form of the modified momentum measure if it were
to be used with the modified propagator, would not radically alter the
computations involved in the earlier results. Indeed, for integrals
encountered, when involving both the modified propagator and measure to
leading order we will be left with an overall
$(1+\kappa\eta^{2}p^{2})^{\frac{1}{2}}$ correction terms. We can then expand
this in the same manner and use it to compute leading order corrections. What
does change is the sign of the contribution, which will alter the values of
$\kappa$ that produce non-positive definite transition probabilities. The
natural consequence of this is that the previous Lorentz violating result
would hold, but with the pleasing result that propagation below the speed of
light produces positive definite probabilities, confirming that the momentum
measure restores Lorentz invariance to the calculations. Put together these
results build confidence in the rationality of using the DSR framework in more
complex systems to investigate the leading order effects of a fundamental
length. Perhaps more intriguingly, we have a complete set of tools to build
upon to analyze quantum field effects in general spacetimes. As a further
example, we could apply these results to curved spacetime examples such as the
Einstein universe [25]. Without providing the details, a simple application of
the methods used in this paper (specifically a modified momentum measure),
gives the following correction to the energy density $\rho$

$\rho=\frac{1}{480\pi^{2}a^{4}}-\frac{\kappa\eta^{2}}{1008\pi^{2}a^{6}}\text{.}$

Here $a$ is the constant scale factor of the Einstein metric $\differential
s^{2}=\differential t^{2}-a~{}\differential\Omega^{2}$. This is of course
positive when $\kappa<0$, that is for subluminal propagation.

Where can we go from here? As we stated earlier our original interest in the
Davies-Fulling-Unruh effect is that it is related to other phenomena such as
Hawking radiation. Although the combination of momentum measure with modified
propagator would seem to indicate that the Davies-Fulling-Unruh effect simple
acquires a small leading order contribution, it is not self-evident that this
is the case for the Hawking effect. The presence of thermally radiating black-
holes is at the origin of the Black-Hole information paradox, and should the
pixelation of spacetime cast doubt on that result then the existence of a QG
theory would seem to protect us from that paradox. In ongoing investigations,
we intend to test that hypothesis and apply the methods described in this
paper and in [13] to that question.

##  Appendix A Choice of momentum space metric

As has been remarked previously, the use of modified propagators as suggested
by DSR on their own, does not yield a theory that preserves Lorentz
symmetries. To restore Lorentz covariance requires other changes to the
theory, specifically the admission that momentum space is curved and not
trivially Minkowski [17]. The use of the standard Minkowski metric
$g_{\mu\nu}=\text{diag}(1,-1,-1,-1)$, when contracting the four momentum
$p^{\mu}$, yields the normal dispersion relations,

$g_{\mu\nu}p^{\mu}p^{\nu}=m^{2}=E^{2}-\overrightarrow{\mathbf{p}}^{2}\text{,}$

although in what follows when we write powers of $p$ we are referring to the
three-momentum $\overrightarrow{\mathbf{p}}$. This is different from our
proposed dispersion relation $E^{2}=m^{2}+p^{2}+\kappa\eta^{2}p^{4}$, and we
ask what choice of momentum space metric is consistent with this form.

To proceed, let us introduce some nomenclature, and we will follow closely the
analysis of [17] and private communication with one of the authors [26]. In
this discussion we will denote by $g^{\mu\nu}$ the non-trivial metric of
$n$-dimensional momentum space $\mathcal{P}$. We assume that $\mathcal{P}$ is
curved and denote by $C^{\mu\nu}_{\rho}$, the Christoffel coefficients of the
metric $g^{\mu\nu}$ of $\mathcal{P}$. The norm of the four-momentum is now
defined as the geodesic distance from the origin of an an arbitrary point in
momentum space $p^{\mu}=(p^{0},p^{i})$, where latin indices run from $i=1$ to
$i=n-1$. Using the metric of $\mathcal{P}$ we define the invariant interval
$ds$ in the usual way as,

$\differential s^{2}=g_{\mu\nu}\differential p^{\mu}\differential
p^{\nu}\text{,}$

which permits the definition of solutions to the geodesic equation in
$\mathcal{P}$,

$\ddot{p}_{\rho}+C^{\mu\nu}_{\rho}\dot{p}_{\mu}\dot{p}_{\nu}=0\text{,}$ (19)

where $\dot{p}$ denotes differentiation with respect to $s$. Once we have a
solution to this, our dispersion relation can be read off by computing the arc
length along a geodesic from the origin to $p^{\mu}$, as

$D(0,p^{\mu})=m^{2}=\int\limits_{0}^{1}\sqrt{g^{\mu\nu}\dot{p}_{\mu}\dot{p}_{\nu}}~{}\differential
s\text{.}$ (20)

A common choice for momentum space metric is to assert that it is de Sitter
like [27, 17, 16], with the de Sitter parameter controlling the degree to
which the geometry of momentum space $\mathcal{P}$ is curved. Usually $n$
dimensional de Sitter space is viewed as an $n$-dimensional hyperbolic slice
in an $n+1$ dimensional Minkowski space. For a $4$-dimensional hyperboloid, we
can define this as the surface,

$z_{A}z^{A}=z_{0}^{2}-z_{1}^{2}+z_{2}^{2}+z_{3}^{2}+z_{4}^{2}=-\frac{1}{\alpha^{2}}\text{,}$
(21)

where $A=0,1,2,3,4$ and the Minkowski metric in the $5$ dimensional space
$g_{AB}=(+,-,-,-,-)$. The radius of the hyperboloid $\alpha$ has dimensions of
length and in our notation we can associate it to $\eta$. However for later
convenience we actually set $\alpha^{2}=\frac{2\kappa\eta^{2}}{n}$, for an
$n$-dimensional space. The general $n$ dimensional hyperboloid has an
invariant interval,

$\differential s^{2}=dz_{0}^{2}-\sum\limits_{A=1}^{n}\differential
z_{A}^{2}\text{,}$ (22)

subject to the constraint $g_{AB}z^{A}z^{B}=-\alpha^{2}$. This constraint has
the effect of reducing the dimensionality of the space by creating a
constraint on one of the coordinates, which by convention we choose to be
$z_{n}$ leaving $n$ independent ones. For conventional use where $n=4$, we
have four independent coordinates and constrain $z_{4}$.

We introduce the following coordinates for $i=1,\dots,n$ (sometimes referred
to as a closed slicing),

$\displaystyle z_{0}$
$\displaystyle=\left(\alpha^{2}\right)^{-\frac{1}{2}}\sinh\left[\left(\alpha^{2}\right)^{\frac{1}{2}}p_{0}\right]\text{,}$
(23) $\displaystyle z_{i}$
$\displaystyle=\cosh\left[\left(\alpha^{2}\right)^{\frac{1}{2}}p_{0}\right]p_{i}\text{~{},with}$
(24)
$\displaystyle\sum\limits_{i=1}^{i=n}p_{i}^{2}=\left(\alpha^{2}\right)^{-1}\text{.}$
(25)

These coordinates parameterize the whole of de Sitter space and the $p_{i}$
coordinates parameterize an $n-1$ dimensional sphere of radius $\alpha^{-2}$,
noting the signs in this constraint are positive. It is easy to check that
with the constraint Eq. 25 these coordinates satisfy Eq. 21, and that in the
limit $\eta\rightarrow 0$, we recover Minkowski coordinates. Specifically if
we displace the origin of the sphere described by Eq. 25 to the point of
intersection of the sphere with the $p_{n}$ axis, i.e
$p_{1},\dots,p_{n-1}=0,p_{n}=\alpha^{-1}$, in the limit $\eta\rightarrow 0$ we
have

$\displaystyle z_{0}\xrightarrow{\eta\rightarrow 0}=p_{0}\text{,}$
$\displaystyle z_{1},\dots,z_{i-1}\xrightarrow{\eta\rightarrow
0}=p_{i}\text{,}$ $\displaystyle z_{4}\xrightarrow{\eta\rightarrow
0}=0\text{.}$

In these coordinates our invariant interval now becomes,

$\differential s^{2}=\differential
p_{0}^{2}-\cosh^{2}\left[\left(\alpha^{2}\right)^{\frac{1}{2}}p_{0}\right]\sum\limits_{i=1}^{i=n}\differential
p_{i}^{2}\text{,}$ (26)

subject to the constraint Eq. 21, and still involve the $n$ momentum
coordinates. In the limit $\alpha\rightarrow 0$, this reduces to the
‘Minkowski’ interval $ds^{2}=dp_{0}^{2}-\sum dp_{i}^{2}$.

As an aside, one can further re-parameterize the coordinates by introducing a
“conformal” time equivalent momentum coordinate,

$\tau=\left(\alpha^{2}\right)^{-\frac{1}{2}}\tan^{-1}\left[\sinh\left(\alpha^{2}\right)^{\frac{1}{2}}p_{0}\right]\text{,}$

that illustrates the fact that the interval is conformal to Minkowski as we
can write,

$\differential
s^{2}=\cosh^{2}\left[\left(\alpha^{2}\right)^{\frac{1}{2}}p_{0}\right]\left(\differential
p_{0}^{2}-\sum\limits_{i=1}^{i=n}\differential p_{i}^{2}\right)\text{.}$ (27)

The conformal factor
$\cosh^{2}\left[\left(\alpha^{2}\right)^{\frac{1}{2}}p_{0}\right]$ can be
converted to a function of $\tau$ to give
$C(\tau)=\sec^{2}\left[\left(\alpha^{2}\right)^{\frac{1}{2}}p_{0}\right]$. A
short calculation, following [21], can then compute from the conformal factor
the scalar curvature of the momentum space as,

$R=6\left(\alpha^{2}\right)\left\\{1+\sin^{2}\left[\left(\alpha^{2}\right)^{\frac{1}{2}}p_{0}\right]\right\\}\text{.}$
(28)

This result confirms our intuition that as our scale parameter
$\alpha\rightarrow 0$, the scalar curvature of the momentum space is zero,
consistent with it being Minkowski in this limit.

However, we wish to further constrain the embedding and eliminate the $p_{n}$
coordinate, using the constraint Eq. 25, re-written as,

$p_{n}=\sqrt{\alpha^{2}-\sum\limits_{i=1}^{n-1}p_{i}^{2}}=\sqrt{\alpha^{2}-p^{2}}\text{.}$

This has the consequence of creating a non-diagonal $n-1$ dimensional metric
because when we differentiate the constraint we find,

$\differential
p_{n}^{2}=\frac{1}{\alpha^{2}-p^{2}}\sum\limits_{i,j}^{n-1}p_{i}p_{j}\differential
p_{i}\differential p_{j}\text{.}$

After elimination of $p_{n}$ we have the following $n$ dimensional metric on
Cartesian momentum coordinates,

$\begin{split}&g^{\mu\nu}=\begin{pmatrix}1&0\\\
0&-\cosh^{2}\left[\left(\alpha^{2}\right)^{\frac{1}{2}}p_{0}\right]\Omega_{ij}\\\
\end{pmatrix}\text{,~{}~{}}\\\ &g_{\mu\nu}=\begin{pmatrix}1&0\\\
0&-\sech^{2}\left[\left(\alpha^{2}\right)^{\frac{1}{2}}p_{0}\right]\Omega^{-1}_{ij}\\\
\text{.}\end{pmatrix}\end{split}$ (29)

The cross terms $\Omega_{ij}$,$\Omega^{-1}_{ij}$ are defined as,

$\Omega_{ij}=\delta_{ij}+\frac{p_{i}p_{j}}{\alpha^{2}-p^{2}}\text{~{},~{}}\Omega^{-1}_{ij}=1-\alpha^{2}p_{i}p_{j}\text{.}$
(30)

In the above it should be understood that Eq. 29 describes an $n\times n$
metric, and for the $\Omega_{ij}$ terms $i,j=1,\dots,n-1$. To obtain the
leading order correction to the on-shell relation and metric determinant we
can expand Eq. 29 to $\mathcal{O}(()\kappa\eta^{2})$, to obtain,

$\begin{split}&g^{\mu\nu}=\begin{pmatrix}\\\\[-5.69054pt] 1&0\\\
0&-\delta_{ij}-\alpha^{2}[p_{0}^{2}\delta_{ij}+p_{i}p_{j}]\\\ \\\\[-5.69054pt]
\end{pmatrix}\text{,~{}~{}}\\\ &g_{\mu\nu}=\begin{pmatrix}\\\\[-5.69054pt]
1&0\\\ 0&-\delta_{ij}+\alpha^{2}(p_{0}^{2}\delta_{ij}+p_{i}p_{j})\\\
\\\\[-5.69054pt] \end{pmatrix}\text{.}\end{split}$ (31)

We compute the momentum space Christoffel symbols using the equivalent of the
position space formula,

$C^{\mu\nu}_{\rho}=\frac{1}{2}g_{\rho\sigma}(g^{\sigma\mu,\nu}+g^{\nu\sigma,\mu}-g^{\mu\nu,\sigma})\text{.}$

Applying this to Eq. 31 we find the only non-zero coefficients are,

$\begin{split}&C^{jk}_{0}=C^{0k}_{j}=C^{k0}_{j}=\alpha^{2}p_{0}\delta_{jk}\text{~{},~{}}\\\
&C^{kl}_{j}=\alpha^{2}p_{j}\delta_{ij}\text{,}\\\
&C^{0jk}=C^{j0k}=-C^{jk0}=-\alpha^{2}p_{0}\delta_{jk}\text{~{},~{}}\\\
&C^{jkl}=-\alpha^{2}\delta_{jk}p_{l}\text{.}\end{split}$ (32)

The geodesic equation can be solved, and following the analysis in [17], our
solution for $D(0,p^{\mu})$ is,

$D(0,p^{\mu})=m^{2}=p_{\mu}p^{\mu}+C^{\rho\mu\nu}p_{\rho}p_{\mu}p_{\nu}\text{.}$

Inserting our leading order Christoffel symbols Eq. 32, we have,

$m^{2}=p_{0}^{2}-p^{2}-\alpha^{2}(p_{0}^{2}+p^{2})p^{2}\text{.}$

For a massless particle, and noting a zero order we have $0=p_{0}^{2}-p^{2}$,
and that $E=p_{0}$, we can for the case of $n=4$, recover the on-shell
dispersion relations we have used in the text,

$E^{2}=p^{2}+\kappa\eta^{2}p^{4}\text{.}$

We conclude that our proposed metric Eq. 29 is consistent with the modified
dispersion relations used in the analysis in the form of our modified
propagators.

Turning to the integral measure, we can compute the leading order correction
to $\sqrt{-g}$ using Eq. 29 that we will use to perform our density of states
computations. We essentially have two approaches we can follow to compute
this, either by expanding directly expanding the determinant of Eq. 29, such
that,

$g=\det(g^{\mu\nu})=-\frac{\cosh^{2(n-1)}(\alpha^{2}p_{0})}{1-\alpha^{2}p^{2}}\text{,}$

or by exploiting the relationship between the determinant and the trace of a
matrix that is close to the identity. Expanding directly this expression for
the determinant, and taking the square root we obtain,

$\sqrt{-g}=1+\frac{\alpha^{2}}{2}((n-1)p_{0}^{2}+p^{2})=1+\frac{n\alpha^{2}}{2}p_{0}^{2}\text{,}$

for massless particles.

We can exploit the relationship between the trace and the determinant of an
$n\times n$ matrix $\mathbf{M}_{n}$ near the identity given by,

$det(\mathbf{I}_{n}+h\mathbf{M}_{n})=1+h\Tr(\mathbf{M}_{n})+\mathcal{O}(h^{2})\text{,}$
(33)

for small h, and the fact that $det(k\mathbf{M})=k^{n}det(\mathbf{M})$ to
compute $\sqrt{-g}$. The determinant, only depends upon the lower spatial part
as it is diagonal in the $p_{0}$ component. For the spatial part of the metric
we can write this as,

$g^{\mu\nu}=-1(\mathbf{1}+\alpha^{2}B^{\mu\nu})\text{,}$ (34)

where $\mathbf{1}_{n}$ is the $n\times n$ unit matrix and $B^{\mu\nu}$ is the
matrix,

$B^{\mu\nu}=\begin{pmatrix}(p_{0}^{2}+p_{1}^{2})&p_{1}p_{2}&p_{1}p_{3}\\\
p_{1}p_{2}&(p_{0}^{2}+p_{2}^{2})&p_{2}p_{3}\\\
p_{1}p_{3}&p_{2}p_{3}&(p_{0}^{2}+p_{3}^{2})\\\ \end{pmatrix}\text{.}$ (35)

Inserting this definition into Eq. 33 we have,

$\sqrt{-g}=\left[1+\alpha^{2}\\{(n-1)p_{0}^{2}+p^{2}\\}\right]^{\frac{1}{2}}\text{.}$
(36)

For a massless particle to zeroth order $p^{2}=p_{0}^{2}$ (higher order on-
shell terms will not contribute to this leading order estimate), and expanding
the square root we have,

$\sqrt{-g}=1+\frac{n\alpha^{2}}{2}p_{0}^{2}\text{,}$ (37)

in agreement with the result obtained by directly computing the determinant of
Eq. 29. Finally, setting $n=4$, and inserting our definition of $\alpha$, we
arrive at our final result,

$\sqrt{-g}=1+\kappa\eta^{2}p_{0}^{2}\text{~{}.}$ (38)

This approximate measure will be used in the text to compute energy densities
from state densities, and is fully consistent with our on-shell dispersion
relations to order $\mathcal{O}(\kappa\eta^{2})$.

##  Appendix B Calculating the Free Vacuum Energy Density

###  B.1 Point splitting and the modified propagator

Our starting point is the free space propagator for a massless scalar field
with correction term, from Eqs. 10 and 9, giving us for the Hadamard function,

$D^{(1)}(\sigma^{2})=D^{+}(\sigma^{2})+D^{-}(\sigma^{2})=-\frac{1}{2\pi^{2}}\left\\{\frac{1}{\sigma^{2}}-\frac{\kappa\eta^{2}}{\sigma^{4}}\right\\}\text{,}$
(39)

where $\sigma^{2}=\Delta
t^{2}-\Delta\overrightarrow{\mathbf{x}}^{2}-i\epsilon$ and $\sigma^{4}=\Delta
t^{4}-\Delta\overrightarrow{\mathbf{x}}^{4}-i\epsilon$. Technically, in
Cartesian coordinates the $\sigma^{4}$ terms contains awkward cross terms
$\Delta t^{2}\Delta\overrightarrow{\mathbf{x}}^{2}$, but elementary
calculation shows these do not contribute to the result. To compute the
thermal Green’s function $D^{(1)}_{\beta}$, we follow the normal procedure to
replace $\Delta t\rightarrow\Delta t-im\beta$, and then sum over all $m$, i.e.
$D^{(1)}_{\beta}=\sum\limits_{m=-\infty}^{\infty}D^{(1)}(\Delta
t-im\beta,\Delta\overrightarrow{\mathbf{x}})$, with $\beta=(k_{B}T)^{-1}$.

For the standard propagator in Cartesian coordinates we have,

$D^{(1)}_{\beta}(t,\overrightarrow{\mathbf{x}};t^{\prime}\overrightarrow{\mathbf{x}}^{\prime})=-\frac{1}{2\pi^{2}}\sum\limits_{m=-\infty}^{\infty}\left\\{\frac{1}{(t-t^{\prime}-im\beta)^{2}-(x-x^{\prime})^{2}-(y-y^{\prime})^{2}-(z-z^{\prime})^{2}}\right\\}\text{.}$
(40)

For a massless scalar field $\phi$, in conformal spacetime we have the
following expression for the stress-energy-momentum tensor $T_{\mu\nu}$ [21],

$T_{\mu\nu}=\frac{2}{3}\partial_{\mu}\phi\partial_{\nu}\phi-\frac{1}{6}g_{\mu\nu}g^{\rho\sigma}\partial_{\rho}\phi\partial_{\sigma}\phi-\frac{1}{3}\phi\partial_{\mu}\partial_{\nu}\phi+\frac{1}{12}\phi\square\phi\text{,}$
(41)

with $\xi=1/6$. Assuming flat space, we set
$g_{\mu\nu}=\text{diag}(1,-1,-1,-1)$. We seek to compute
$\expectationvalue{T_{\mu\nu}}{0}$, which simplifies Eq. 41 significantly as
this involved integrating over all space. We note the following identities,

$\displaystyle\partial_{\mu}(\phi\partial_{\nu}\phi)$
$\displaystyle=2(\partial_{\mu}\phi\partial_{\nu}\phi+\phi\partial_{\mu}\partial_{\nu}\phi)\text{,
and}$ $\displaystyle\partial_{\mu}(\phi\partial^{\mu}\phi)$
$\displaystyle=2(\partial_{\mu}\phi\partial^{\mu}\phi+\phi\square\phi)\text{.}$

When we take the integral, the left-hand side of both is a total derivative,
and so by Gauss’s theorem, and the fact that we assume at infinity that
$\partial\phi=\phi=0$, we have,

$\displaystyle\int_{\mathcal{M}}\partial_{\mu}(\phi\partial_{\nu}\phi)~{}\differential^{4}x$
$\displaystyle=\int_{\mathcal{\partial
M}}\phi\partial_{\nu}\phi~{}\differential^{4}x=0$
$\displaystyle\int_{\mathcal{M}}\partial_{\mu}(\phi\partial^{\mu}\phi)~{}\differential^{4}x$
$\displaystyle=\int_{\mathcal{\partial
M}}\phi\partial^{\mu}\phi~{}\differential^{4}x=0\text{.}$

Further as we are only considering diagonal entries for the stress-energy-
momentum tensor, we have the effective value for the expectation value,

$\expectationvalue{T_{\mu\nu}}{0}=\expectationvalue{\left(\partial_{\mu}\phi\partial_{\nu}\phi-\frac{1}{4}g_{\mu\nu}\partial_{\mu}\phi\partial^{\mu}\phi\right)}{0}\text{.}$
(42)

From here we note that as
$D^{(1)}(x,x^{\prime})=\expectationvalue{\\{\phi(x)\phi(x^{\prime})\\}}{0}$,
we can state the following identity,

$\expectationvalue{(\partial_{x}\phi)^{2}}{0}=\expectationvalue{\lim_{x^{\prime}\rightarrow
x}\partial_{x}\phi\partial_{x^{\prime}}\phi}{0}=\frac{1}{2}\lim_{x^{\prime}\rightarrow
x}\partial_{x}\partial_{x^{\prime}}D^{(1)}(x,x^{\prime})\text{.}$ (43)

For the standard thermal propagator including the leading
$-\frac{1}{2\pi^{2}}$, and intentionally not simplifying the fractions we
have,

$\displaystyle\lim_{t^{\prime}\rightarrow
t}\partial_{t}\partial_{t^{\prime}}D^{(1)}_{\beta}=\frac{6}{2\pi^{2}}\sum\limits_{-\infty}^{\infty}\frac{1}{(\beta
m)^{4}}$ $\displaystyle\left.\begin{aligned} &\lim_{x^{\prime}\rightarrow
x}\partial_{x}\partial_{x^{\prime}}D^{(1)}_{\beta}\\\
&\lim_{y^{\prime}\rightarrow
y}\partial_{y}\partial_{y^{\prime}}D^{(1)}_{\beta}\\\
&\lim_{z^{\prime}\rightarrow
z}\partial_{z}\partial_{z^{\prime}}D^{(1)}_{\beta}\end{aligned}\right\\}=\frac{2}{2\pi^{2}}\sum\limits_{-\infty}^{\infty}\frac{1}{(\beta
m)^{4}}\text{.}$

We can now compute the diagonal components of the stress-energy-momentum
tensor (all other terms are zero), by inserting these into Eq. 42. For
example, for the energy density we have
$T_{tt}=\frac{3}{4}(\partial_{t}\phi)^{2}+\frac{1}{4}\\{(\partial_{x}\phi)^{2}+(\partial_{y}\phi)^{2}+(\partial_{z}\phi)^{2}\\}$.
Omitting the infinite term at $m=0$, we obtain for our first term,

$\expectationvalue{T_{tt}}=\frac{6}{2\pi^{2}}\sum\limits_{1}^{\infty}\frac{1}{(\beta
m)^{4}}=\frac{\pi^{2}}{30\beta^{4}}\text{.}$ (44)

The remaining components are computed simply and each have the value of
$\frac{\pi^{2}}{90\beta^{4}}$.

For the correction term we have the following propagator,

$D^{{}^{\prime}(1)}_{\beta}(t,\overrightarrow{\mathbf{x}};t^{\prime}\overrightarrow{\mathbf{x}}^{\prime})=\frac{\kappa\eta^{2}}{2\pi^{2}}\sum\limits_{m=-\infty}^{\infty}\left\\{\frac{1}{[(t-t^{\prime}-im\beta)^{2}-(x-x^{\prime})^{2}-(y-y^{\prime})^{2}-(z-z^{\prime})^{2}]^{2}}\right\\}\text{.}$
(45)

As indicated earlier the extra two powers in the denominator lead to awkward
cross terms, but the point splitting derivatives simplify matters. Computation
of the derivatives demonstrate that all terms involving a spatial term gives a
value of $\lim_{x^{\prime}\rightarrow
x}\partial_{x}\partial_{x^{\prime}}D^{{}^{\prime}(1)}_{\beta}=0$. Accordingly,
we have,

$\displaystyle\lim_{t^{\prime}\rightarrow
t}\partial_{t}\partial_{t^{\prime}}D^{(1)}_{\beta}=\frac{20}{2\pi^{2}}\sum\limits_{-\infty}^{\infty}\frac{1}{(\beta
m)^{6}}$ $\displaystyle\left.\begin{aligned} &\lim_{x^{\prime}\rightarrow
x}\partial_{x}\partial_{x^{\prime}}D^{(1)}_{\beta}\\\
&\lim_{y^{\prime}\rightarrow
y}\partial_{y}\partial_{y^{\prime}}D^{(1)}_{\beta}\\\
&\lim_{z^{\prime}\rightarrow
z}\partial_{z}\partial_{z^{\prime}}D^{(1)}_{\beta}\end{aligned}\right\\}=0\text{.}$

Again we can use Eq. 42 to compute our diagonal components, and again we
perform the sum ommiting the $m=0$ divergent term. Bringing this all together
we arrive at our final result,

$\expectationvalue{T_{\mu\nu}}=\left(\frac{\pi^{2}}{30\beta^{4}}+\frac{\kappa\eta^{2}\pi^{4}}{126\beta^{6}}\right)\matrixquantity[1&0&0&0\\\
0&\frac{1}{3}&0&0\\\ 0&0&\frac{1}{3}&0\\\ 0&0&0&\frac{1}{3}\\\ ]\text{.}$ (46)

###  B.2 Density of states with modified momentum measure

If we assume that the vacuum state of the scalar massless field has the energy
spectrum, $E=n\omega$, with $\omega$ the angular frequency of the mode (recall
$\hbar=1$) and $n=0,1,2,3,\dots$, one can immediately write the partition
function for the canonical ensemble at inverse temperature $\beta$ as,

$Z=\sum\limits_{n=0}^{\infty}e^{-\beta
n\omega}=\frac{1}{1-e^{-\beta\omega}}\text{.}$ (47)

The standard textbook calculation uses the thermodynamic relation
$\expectationvalue{E}=-\partialderivative{\log Z}{\beta}$, to compute the
energy density by combining Eq. 47 with a density of states and integrating
over all possible $\omega$. However, this approach is inconsistent with the
assumption of minimal length, as the upper limit of the integral is
$\omega\rightarrow\infty$ that implies a zero wavelength. To accommodate an
upper limit of $\omega$, it would seem necessary to introduce a cut off.

In DSR the introduction of a minimal length, whilst maintaining Lorentz
covariance, is done by assuming that momentum space is not flat, as described
in Appendix A. This minimal length performs a similar function to the upper
momentum cut off by imposing a minimum wavelength. We seek to compare our
point splitting energy density with a calculation using a choice of metric
that is consistent with the modified dispersion relations. We analyzed this in
Appendix A, and inserting $\omega$ as our time-like component we have to
leading order for our momentum space metric determinant,

$\sqrt{-g}=1+\kappa\eta^{2}\omega^{2}\text{~{}.}$ (48)

For a covariant momentum measure, we need to replace $\int\differential^{4}p$
with $\int\sqrt{-g_{p}}\differential^{4}p$. As discussed in Section III, the
density of states is computed using the volume of momentum space and so should
include the $\sqrt{-g}$ term in the integral measure. Accordingly the modified
density of states is,

$\rho(\omega)=\frac{\omega^{2}\sqrt{-g}\differential\omega}{2\pi^{2}}\text{,}$
(49)

which gives for $\log Z$ on substitution,

$\log Z=\frac{1}{2\pi^{2}}\int\limits_{0}^{\infty}\omega^{2}\log
Z\sqrt{-g}\differential\omega\text{.}$ (50)

From here we expand the exponential from the metric determinant and we arrive
at two integrals at order $\mathcal{O}(\kappa\eta^{2})$,

$\begin{split}\expectationvalue{E}&=-\partialderivative{\log Z}{\beta}\\\
&=\frac{1}{2\pi^{2}}\left\\{\int\limits_{0}^{\infty}\frac{\omega^{3}}{e^{\beta\omega}-1}\differential\omega+\kappa\eta^{2}\int\limits_{0}^{\infty}\frac{\omega^{5}}{e^{\beta\omega}-1}\differential\omega\right\\}\end{split}$
(51)

These integrals are elementary, with the first evaluating to
$\frac{\pi^{4}}{15\beta^{4}}$ and the second to
$\frac{8\pi^{6}}{63\beta^{6}}$. Substituting in we arrive at our final result,

$\expectationvalue{E}=\frac{\pi^{2}}{30\beta^{4}}+\frac{8\kappa\eta^{2}\pi^{4}}{126\beta^{6}}\text{,}$
(52)

which is similar to Eq. 46, although larger by a factor of four. We discuss in
Section III how we can account for this difference from the over-counting of
unphysical states.

We can also compute the partition function by simply integrating the density
of states with the modified measure given by Eq. 49 to obtain,

$Z=\frac{\zeta(3)}{\pi^{2}\beta^{3}}+\frac{12\kappa\eta^{2}\zeta(5)}{\pi^{2}\beta^{5}}\text{,}$
(53)

where $\zeta(n)$ is the Riemann zeta function. For completeness we can also
use the relation $S=\partialderivative{T\log Z}{T}$ (in units where
$k_{B}=1$), to compute the entropy density, which when re-expressed in terms
of a derivative in $\beta$ is,

$S=-\beta^{2}\partialderivative{\beta}\left(\frac{\log
Z}{\beta}\right)\text{.}$

Using Eq. 50, and expanding in $\kappa\eta^{2}$, we have two integrals to
compute,

$\begin{split}\expectationvalue{S}&=-\partialderivative{\beta\log
Z}{\beta}=\\\
&\frac{1}{2\pi^{2}}\int\limits_{0}^{\infty}\left\\{\frac{\beta\omega^{3}}{e^{\beta\omega}-1}+\omega^{2}\log[1-e^{-\beta\omega}]\right\\}\differential\omega\\\
&+\frac{\kappa\eta^{2}}{2\pi^{2}}\int\limits_{0}^{\infty}\left\\{\frac{\beta\omega^{5}}{e^{\beta\omega}-1}+\omega^{4}\log[1-e^{-\beta\omega}]\right\\}\differential\omega\text{.}\end{split}$

These may be evaluated by elementary means and give us for the entropy
density,

$\expectationvalue{S}=\frac{2\pi^{2}}{45\beta^{3}}+\frac{16\kappa\eta^{2}\pi^{4}}{315\beta^{5}}\text{.}$
(54)

##  Appendix C Computing the Casimir Effect by Point Splitting

###  C.1 Casimir effect from point splitting

We follow the treatments of $\S 4.3$ of [21], and also [19] to compute the
energy momentum tensor and thus energy density associated with the vacuum of a
scalar field in the presence of two boundaries. The fundamental approach is
the same as in Appendix B, where we obtain an expression for the Green’s
function for the setup of two perfectly conducting plates separated by a
distance $a$, and then use the technique of point splitting to obtain an
approximate value for the expectation value of the stress-energy-momentum
tensor. The complication arises from the presence of two boundaries that due
to their perfectly conducting nature impose a Dirichlet boundary condition on
the Green’s function. We sketch the set-up in Fig. 1, and use the method of
images to obtain the full propagator $D^{(1)}$ for the scalar field in the
presence of this boundary. We note that to ensure the boundary condition is
met on both plates, we need to infinite series of images, one for each plate.
As the series of images stretches to infinity, we can write one image sum,
with an additional factor of $2$.

Using our modified propagator Eq. 39, the full expression for the propagator
including the image term is,

$\begin{split}D^{(1)}(t,\overrightarrow{\mathbf{x}};t^{\prime}\overrightarrow{\mathbf{x}}^{\prime})&=D^{(1)}_{0}(t,\overrightarrow{\mathbf{x}};t^{\prime}\overrightarrow{\mathbf{x}}^{\prime})-\frac{1}{\pi^{2}}\sum\limits_{n=-\infty}^{\infty}\left\\{\frac{1}{(t-t^{\prime})^{2}-(x-x^{\prime})^{2}-(y-y^{\prime})^{2}-(z-z^{\prime}+2an)^{2}}\right\\}\\\
&+\frac{1}{\pi^{2}}\sum\limits_{n=-\infty}^{\infty}\left\\{\frac{\kappa\eta^{2}}{[(t-t^{\prime})^{2}-(x-x^{\prime})^{2}-(y-y^{\prime})^{2}-(z-z^{\prime}+2an)^{2}]^{2}}\right\\}\text{.}\end{split}$
(55)

To compute the expectation value in this scenario, we subtract away the free
space propagator
$D^{(1)}_{0}(t,\overrightarrow{\mathbf{x}};t^{\prime}\overrightarrow{\mathbf{x}}^{\prime})$,
and utilize the point splitting technique to relate terms in derivatives of
the fields to propagators such as,

$\begin{split}\expectationvalue{(\partial_{\mu}\phi)^{2}}{0}&=\expectationvalue{\lim_{x_{\mu}^{\prime}\rightarrow
x_{\mu}}\partial_{x_{\mu}}\phi~{}\partial_{x_{\mu}^{\prime}}\phi}{0}\\\
&=\frac{1}{2}\lim_{x_{\mu}^{\prime}\rightarrow
x_{\mu}}\partial_{x_{\mu}}\partial_{x_{\mu}^{\prime}}D^{(1)}(x_{\mu},x_{\mu}^{\prime})\text{.}\end{split}$
(56)

We can then evaluate the expectation of the appropriate stress-energy-momentum
tensor, and in this treatment, we use the “new improved stress tensor” Eq. 41.

Dealing with the first (unmodified) term in the propagator, we note that Eq.
41 reduces to Eq. 42, and so we require the relevant derivatives of the
propagator, which are as follows,

$\displaystyle~{}\lim_{t^{\prime}\rightarrow
t}\partial_{t}\partial_{t^{\prime}}D^{(1)}=\frac{2}{\pi^{2}}\sum\limits_{-\infty}^{\infty}\frac{1}{(2an)^{4}}$
$\displaystyle\left.\begin{aligned} &\lim_{x^{\prime}\rightarrow
x}\partial_{x}\partial_{x^{\prime}}D^{(1)}\\\ &\lim_{y^{\prime}\rightarrow
y}\partial_{y}\partial_{y^{\prime}}D^{(1)}\end{aligned}\right\\}=-\frac{2}{\pi^{2}}\sum\limits_{-\infty}^{\infty}\frac{1}{(2an)^{4}}$
$\displaystyle~{}\lim_{z^{\prime}\rightarrow
z}\partial_{z}\partial_{z^{\prime}}D^{(1)}=-\frac{6}{\pi^{2}}\sum\limits_{-\infty}^{\infty}\frac{1}{(2an)^{4}}\text{.}$

For the infinite summations on the right-hand side, we omit the infinite value
at $n=0$, and note the standard result that
$\sum\limits_{-\infty}^{\infty}\frac{1}{(2an)^{4}}=\frac{\pi^{4}}{720a^{4}}$.
Substituting into Eq. 42 we arrive at the well known “book value” for
$\expectationvalue{T_{\mu\nu}}$,

$\expectationvalue{T_{\mu\nu}}=\frac{-\pi^{2}}{1440a^{4}}\matrixquantity[1&0&0&0\\\
0&-1&0&0\\\ 0&0&-1&0\\\ 0&0&0&3\\\ ]\text{.}$

Turning to the correction term, the calculation follows the same path as
above, but we have the extra two powers in the denominator to deal with.
However, the extra powers actually simplify matters, as any term involving
$(t-t^{\prime})^{2},(x-x^{\prime})^{2}$ or $(y-y^{\prime})^{2}$ are zero. This
simplifies the computation as the only surviving propagator term that
contributes to Eq. 42 is,

$\lim_{z^{\prime}\rightarrow
z}\partial_{z}\partial_{z^{\prime}}D^{(1)}=-\frac{20\kappa\eta^{2}}{\pi^{2}}\sum\limits_{-\infty}^{\infty}\frac{1}{(2an)^{6}}\text{.}$

Again, we perform the sum omitting the divergence at $n=0$, which gives the
result
$\sum\limits_{-\infty}^{\infty}\frac{1}{(2an)^{6}}=\frac{\pi^{6}}{30240a^{6}}$.
We can then substitute into Eq. 42 to obtain for the correction term,

$\expectationvalue{T_{\mu\nu}}=\frac{\kappa\eta^{2}\pi^{4}}{12096a^{6}}\matrixquantity[1&0&0&0\\\
0&-1&0&0\\\ 0&0&-1&0\\\ 0&0&0&3\\\ ]\text{.}$

Bringing this all together we arrive at our final result,

$\expectationvalue{T_{\mu\nu}}=\frac{-\pi^{2}}{1440a^{4}}\left(1-\frac{5\kappa\eta^{2}\pi^{2}}{42a^{2}}\dots\right)\matrixquantity[1&0&0&0\\\
0&-1&0&0\\\ 0&0&-1&0\\\ 0&0&0&3\\\ ]\text{.}$ (57)

###  C.2 Casimir effect from energy density

When analyzing the free space vacuum, we calculated the energy density from
both point splitting and density of states. With the Casimir effect our
alternative approach utilizing the modified momentum measure explored in
Appendix A starts with the regularized energy density. The treatment follows
the original calculation by Fierz [23], which uses as a starting point the
energy per unit area of a massless scalar field constrained by two infinite
area plates separated by a distance $a$. Using the same set up in Fig. 1, we
can write the energy per unit area as,

$\frac{E(a,\alpha)}{A}=\frac{1}{2}\sum\limits_{n=1}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\frac{\differential
p_{x}\differential
p_{y}}{(2\pi)^{2}}\sqrt{\left(\frac{n\pi}{a}\right)+p_{x}^{2}+p_{y}^{2}}~{}\exp\left\\{-\alpha\sqrt{\left(\frac{n\pi}{a}\right)+p_{x}^{2}+p_{y}^{2}}\right\\}\text{.}$
(58)

The energy is formally divergent and the exponential function is introduced to
regularize the integral. Setting $\alpha=0$ recovers the divergent energy per
unit area, and to arrive at the energy per unit volume
$\expectationvalue{T_{tt}}$ we will divide our result by $a$.

The introduction of the momentum measure proceeds by introducing the metric
determinant as a first order approximation from Eq. 38. For a massless scalar
field, we note that $p_{0}^{2}-\sum\limits_{i}p_{i}^{2}=0$, and that the terms
inside the square root in Eq. 58 are simply $\sum\limits_{i}p_{i}^{2}$, and so
we can re-write the momentum measure as,

$\sqrt{-g}=1+\kappa\eta^{2}\left[\left(\frac{n\pi}{a}\right)+p_{x}^{2}+p_{y}^{2}\right]\text{.}$

We can now rewrite the double integrals in polar coordinates, and make a
substitution by defining $z=(\frac{a}{n\pi})^{2}[p_{x}^{2}+p_{y}^{2}]$, such
that with $r^{2}=p_{x}^{2}+p_{y}^{2}$, $r\differential
r=\frac{1}{2}(\frac{n\pi}{a})^{2}\differential z$. With the modified measure
we are left with two integrals,

$\frac{E(a,\alpha)}{A}=\frac{1}{8\pi}\sum\limits_{1}^{\infty}\left(\frac{n\pi}{a}\right)^{3}\int\limits_{0}^{\infty}\sqrt{1+z}\exp\left\\{-\alpha\frac{n\pi}{a}\sqrt{1+z}\right\\}~{}\differential
z+\frac{\kappa\eta^{2}}{8\pi}\sum\limits_{1}^{\infty}\left(\frac{n\pi}{a}\right)^{5}\int\limits_{0}^{\infty}(1+z)^{3/2}\exp\left\\{-\alpha\frac{n\pi}{a}\sqrt{1+z}\right\\}~{}\differential
z$

A further substitution of $w=\sqrt{1+z}$ can be made, and we note that by
differentiating with respect to $\alpha$, we can further simplify to,

$\displaystyle\frac{E(a,\alpha)}{A}$
$\displaystyle=-\frac{1}{4\pi}\sum\limits_{1}^{\infty}\derivative[3]{\alpha}\int\limits_{0}^{\infty}\frac{\exp[-\alpha(\frac{n\pi}{a})w]}{w}~{}\differential
w$ (59)
$\displaystyle-\frac{\kappa\eta^{2}}{4\pi}\sum\limits_{1}^{\infty}\derivative[5]{\alpha}\int\limits_{0}^{\infty}\frac{\exp[-\alpha(\frac{n\pi}{a})w]}{w}~{}\differential
w$ (60)

In both of these integrals we can perform one differentiation by $\alpha$,
which then presents an elementary summation, which leaves the integrand in
both cases to be in the following form,

$\int\limits_{0}^{\infty}\frac{e^{\frac{\alpha\pi}{a}w}}{(e^{\frac{\alpha\pi}{a}w}-1)^{2}}~{}\differential
w=\left(\frac{\alpha\pi}{a}\right)^{-1}\frac{1}{e^{\frac{\alpha\pi}{a}}-1}$

We can now exploit the exponential generating identity for the Bernoulli
numbers ([22] §9.62),

$\frac{x}{e^{x}-1}=\sum\limits_{m=0}^{\infty}\frac{B_{m}}{m!}x^{m}\text{,}$

to arrive at the following result for the integrals,

$\frac{E(a,\alpha)}{A}=\frac{1}{4a}\derivative[2]{\alpha}\sum\limits_{m=0}^{\infty}\frac{B_{m}}{m!}\left(\frac{\alpha\pi}{a}\right)^{m-2}+\frac{\kappa\eta^{2}}{4a}\derivative[4]{\alpha}\sum\limits_{m=0}^{\infty}\frac{B_{m}}{m!}\left(\frac{\alpha\pi}{a}\right)^{m-2}\text{.}$
(61)

When we expand the series, we are left with some divergent quantities when we
set $\alpha=0$ that we discard for $m<2$. We can also see that the only
relevant terms after differentiating and setting $\alpha=0$ are $m=4$ for the
first terms, and $m=6$ for the second. These depend upon $B_{4}=-\frac{1}{30}$
and $B_{6}=\frac{1}{42}$. Performing the differentiation, substituting in
these values, and dividing by $a$, we arrive at our final result,

$\expectationvalue{T_{tt}}=\frac{-\pi^{2}}{1440a^{4}}\left(1-\frac{12\kappa\eta^{2}\pi^{2}}{42a^{2}}\right)\text{.}$
(62)

Comparing this result to Eq. 62, we notice that the correction term is
slightly larger than the point-splitting calculation, a result we address in
Section C.2.

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BMS symmetry in gravity: Front form versus Instant form

Sudarshan Ananth† and Sucheta Majumdar‡

$\dagger$ Indian Institute of Science Education and Research

Pune 411008, India

$\ddagger$ ENS de Lyon, CNRS, Laboratoire de physique,

UMR 5672, F-69342 Lyon, France

Abstract

In General Relativity, the allowed set of diffeomorphisms or gauge
transformations at asymptotic infinity forms the BMS group, an infinite-
dimensional extension of the Poincaré group. We focus on the structure of the
BMS group in two distinct forms of Hamiltonian dynamics - the instant and
front forms. Both similarities and differences in these two forms are examined
while emphasising the role of non-covariant approaches to symmetries in
gravity.

This article is dedicated to the memory of Lars Brink

Essay written for the

Gravity Research Foundation 2023 Awards for Essays on Gravitation

Symmetries, both spacetime and gauge, serve as powerful tools in formulating
consistent physical theories. Symmetries are essential signposts when
exploring theories like general relativity which lacks a consistent quantum
mechanical description. Gauge symmetries, which have to do with the
redundancies in our description of a physical theory, offer multiple
perspectives on the same theory. Asymptotic symmetries are a class of
spacetime symmetries that appear as one approaches infinity. One may move
along a space-like direction to approach spatial infinity, or along a null
direction - towards null infinity. At asymptotic infinity, the symmetry group
of gravity, known as the Bondi-van der Burg-Metzner-Sachs (BMS) group [1, 2],
is larger than the naïvely-expected Poincaré group. The BMS symmetry involves
an infinity of parameters, serving as a generalization of the finite-
dimensional Poincaré group. The BMS symmetry sets general relativity, where
the spacetime metric is dynamical, apart from special relativity where the
asymptotic symmetry group remains Poincaré. To describe the evolution of a
relativistic system in time, Dirac proposed three distinct forms of
Hamiltonian dynamics [3]. The forms vary essentially in their choice of the
time coordinate or ‘evolution parameter’. The light-cone form involves two
null coordinates $x^{\pm}=(t\pm x^{3})/\sqrt{2}$ along the light-cone and two
transverse spatial directions, resulting in a $(2+2)$ split of four-
dimensional spacetime. The dynamics in this frame, also called the front form,
is clearly in contrast with the standard description where one time coordinate
is singled out (the dynamics is then described on a constant time hypersurface
spanned by three spatial directions). Thus this approach prefers a $(3+1)$
split of spacetime and is also referred to as the instant form. Another
obvious candidate when choosing a time coordinate would be the proper time
referred to as the point form (from the study of the relativistic point
particle). This article focusses on the front and instant forms of dynamics,
since these forms have more relevance to the study of massless particles.
Starting with the structure of the Poincaré group in the instant and front
forms, we discuss how the enhancement to the infinite-dimensional BMS group
occurs in both these cases, for theories of gravity. A clear picture of the
symmetries in a system in one or more of these forms of Hamiltonian dynamics
is essential for the subsequent quantization procedures. Symmetries in the
Hamiltonian formulation are realized through canonical generators expressed in
terms of phase space variables, a set of physical fields $\\{\Phi\\}$ and
their conjugate momenta $\\{\pi_{\Phi}\\}$. The phase space is endowed with a
Poisson bracket $[\Phi(x),\pi_{\Phi}(y)]\sim\delta(x-y)$. A canonical
transformation $X$ in the phase space corresponds to a symmetry if it leaves
the Hamiltonian action $\mathcal{S}_{H}$ invariant

$(\Phi,\pi_{\Phi})\xrightarrow{\text{\quad
X\quad}}(\tilde{\Phi},\tilde{\pi}_{\Phi})\,\quad\text{such
that}\quad\delta_{X}\mathcal{S}_{H}=0\,.$

The transformation law of the fields may be derived from the canonical
generator $G[X]$ through the Poisson bracket

$\delta_{X}\Phi=[G[X],\Phi]\,,\quad\delta_{\xi}\pi_{\Phi}=[G[X],\pi_{\Phi}]\,.$

In the quantum theory, using the canonical quantization scheme, the Poisson
bracket is promoted to the commutator, $[\ ,\ ]_{PB}\rightarrow i\hbar[\ ,\
]$. The conserved quantities associated with the symmetries play the role of
physical observables in the quantum theory.

Quantization of gauge theories is, however, a particularly delicate issue
because the Hamiltonian description in these theories involve constraints. The
constraints imply that not all phase space variables are dynamical, which
reflects the gauge redundancy in the theory. In the instant form, such
constrained Hamiltonian systems can be studied using the Dirac-Bergmann
algorithm [4, 5] to classify the constraints and analyze their algebra in the
context of symmetries. In the front form, a key advantage is that the
constraints may be solved completely in order to eliminate the gauge degrees
of freedom. This allows us to obtain a Hamiltonian formulation solely in terms
of the physical fields of theory. Structure of Poincaré algebra

For quantum field theories in flat spacetime, the spacetime symmetry is
parameterized by $\xi^{\mu}=\omega^{\mu}{}_{\nu}x^{\nu}+a^{\mu}$,
corresponding to the Poincaré group. The constant parameters, $a^{\mu}$ and
$\omega^{\mu}{}_{\nu}$, label spacetime translations and Lorentz rotations
respectively. In terms of structure, there are two natural ways to divide up
this algebra.

1\. $P-M$ split: $G[\xi]=a^{\mu}P_{\mu}+\omega^{\mu\nu}M_{\mu\nu}$

The four spacetime translations are generated by $P^{\mu}$ while the
$M^{\mu\nu}$ produce the Lorentz transformations. The Poincaré algebra in this
split has the following structure

$[M,M]\sim M\,,\quad[P,M]\sim P\,,\quad[P,P]\sim 0\,.$ (1)

This split is ideal for working with theories formulated in a covariant
manner.

2\. $\mathcal{K}-\mathcal{D}$ split:
$G[\xi]=\xi^{\text{space}}\mathcal{K}+\xi^{\text{time}}\mathcal{D}$

The time component of $\xi^{\mu}$ corresponds to timelike diffeomorphisms,
namely time translation and Lorentz boosts. On the other hand, the spatial
components are associated with the transformations along a constant time
hypersurface that includes spatial translations and rotations. So the
kinematical part of the algebra $\mathcal{K}$ acts on a constant time surface,
while the dynamical part $\mathcal{D}$ maps the initial hypersurface to
another hypersurface at a later time. The algebra structure takes the form

$[\mathcal{K},\mathcal{K}]\sim\mathcal{K}\,,\quad[\mathcal{D},\mathcal{K}]\sim\mathcal{D}\,,\quad[\mathcal{D},\mathcal{D}]\sim
0\,.$ (2)

The dynamical generators collectively play the role of the Hamiltonian
operator in quantum field theory, unlike in quantum mechanics where the time
derivative $\partial_{t}$ alone suffices. In the instant form, we have the
time $x^{0}=t$ and three spatial coordinates $x^{i}$, labeled by a constant
time hypersurface, as depicted in figure 1.

$x^{0}=t$$x^{1}$$\\{x^{2,3}\\}$$\Sigma:t=0$ Figure 1: The $t=0$ spatial
surface in instant form labelled by coordinates $(x^{1},x^{2},x^{3})$

Thus, the Kinematical generators in instant form are the three spatial
translations $P^{i}$ and three Lorentz rotations $J^{i}$ . The dynamical
objects consist of the time translation generator $P^{0}$ and three Lorentz
boosts $B^{i}$, resulting in the four Hamiltonians

$\mathcal{K}\equiv\\{P^{i},J^{i}=\epsilon^{ijk}M_{jk}\\}\,,\quad\mathcal{D}\equiv\\{P^{0},B^{i}=M^{0i}\\}\,,\quad
i,j,\ldots=1,2,3\,.$

The front form involves two null coordinates $x^{\pm}$, of which either may be
chosen to be the time coordinate, and two transverse spatial coordinates.
Choosing $x^{+}$ to be light-cone time, the initial constant time hypersurface
is the $x^{+}=0$ surface. The coordinates on this three-dimensional
hypersurface are the other null coordinate $x^{-}$ and two transverse complex
coordinates, denoted by $x^{a}$ here, as shown in figure 2.

$x^{+}$$x^{-}$$\\{x^{a}\\}$$\Sigma:x^{+}=0$ Figure 2: The $x^{+}=0$ null
surface in front form labelled by coordinates $(x^{-},x,\bar{x})$

The kinematical part of the algebra in the front form involves the three
translation generators, $(P^{+},P^{a})$ and four rotation generators
$(J^{ab},J^{+a},J^{+-})$. The dynamical generators are $P^{-}$ and $J^{-a}$

$\mathcal{K}\equiv\\{P^{+},P^{a},J^{ab},J^{+a},J^{+-}\\}\,,\quad\mathcal{D}\equiv\\{P^{-},J^{-a}\\}\,,\quad
a,b,\ldots=x,\bar{x}\,.$

On the $x^{+}=0$ surface, the $J^{-+}$ is kinematical.

Three Hamiltonians, as opposed to four in the instant form, are easier to work
with, in deriving interacting Hamiltonians [6]. Unlike in the usual case,
Lorentz boosts in the light-cone Poincaré algebra commute with each other.
This highlights the underlying Carrollian structure of the light-cone Poincaré
group [7]. From Poincaré to BMS Diffeomorphism invariance that lies at the
heart of general relativity allows for a change of coordinates
$x^{\mu}\rightarrow x^{\mu}+\xi^{\mu}(x)$ with an arbitrary local parameter
$\xi^{\mu}$. This gauge freedom in gravity is associated with the Hamiltonian
and momentum constraints in the Dirac-ADM formulation of gravity. As alluded
to before, the asymptotic symmetry group of gravity is the BMS group, which
comprises Lorentz transformations and infinitely many supertranslations. The
study of asymptotic symmetries is very sensitive to the choice of coordinate
system, gauge and boundary conditions imposed on the fields. This is because
the asymptotic symmetries are nothing but residual gauge transformations or
diffeomorphisms that do not vanish at the boundary of spacetime. There are
multiple approaches to the study of asymptotic symmetries, including the
following three key ones.

  * •

Bondi Aproach: This approach employs a special set of coordinates, known as
the Bondi coordinates, which in four dimensions involve one null time
coordinate, one radial coordinate and two spatial coordinates. The BMS
symmetry in gravity was originally discovered at null infinity in this setup
[1, 2]. For asymptotically flat spacetimes, the BMS symmetry arises at null
infinity as the set of allowed diffeomorphisms that preserve the form of the
metric at the asymptotic boundary. The BMS group at null infinity has been
studied extensively and further enhanced to include superrotations,
Diff($\mathbb{S}^{2}$) and other larger groups.

  * •

Carrollian approach: The Carroll group has been formally known to exist in the
literature as the ‘ultrarelativistic’ $c\rightarrow$ limit of the Poincaré
group for decades [7], but had not been associated with any physical setup
until recently. In 2014, the conformal extension of the Carroll group was
shown to be isomorphic to the BMS group [8]. This led to a rebirth of the
Carroll group as the spacetime symmetry of null hypersurfaces, that are formed
by a plane wave propagating with the velocity of light. This also resulted in
new efforts to formulate ultrarelativistic theories with the Carroll group as
the underlying symmetry as opposed to relativistic theories with Lorentz
symmetry.

  * •

Hamiltonian approach: The BMS symmetry can also be realised using Hamiltonian
methods in one of more forms of relativistic dynamics discussed above. This
serves as a vital step forward from a formal field-theoretic standpoint as it
offers a consistent canonical description of BMS symmetry in terms of phase
space variables, crucial for investigating implications for the quantum
theory. We now discuss the canonical realization of BMS in the instant and
front forms of Hamiltonian dynamics.

BMS symmetry in instant form In the (3+1) Dirac-ADM formulation [9, 10] with
standard boundary conditions associated with well-defined canonical charges,
the asymptotic symmetry group was found to be just Poincaré and not BMS. This
canonical realization of the Poincaré algebra in the phase space of gravity in
the instant form can be found in [11]. Attempts to recover the BMS group at
spatial infinity were met with failure, as the required choice of boundary
conditions would lead to ill-defined symmetry generators and conserved
charges. This gap in our knowledge was bridged recently, where the BMS group
was obtained at spatial infinity by virtue of meticulously relaxing the
boundary conditions while respecting the notion of canonical conserved
charges. The enhancement of the Poincaré group to BMS occurs in the
translation part of the algebra. As a result of the relaxed boundary
conditions introduced in [12], the spacetime translations $P^{\mu}$ in
Poincaré labelled by the four constant parameter $a^{\mu}$ are replaced by an
infinite set of supertranslations $\mathcal{ST}$ labelled by an angle-
dependent or local parameter $\tau(\theta,\phi)$. Here, the angles $\theta$
and $\phi$ may be identified with the coordinates of the sphere at infinite
radial distance $r$ as in the spherical polar coordinates
$x^{i}=(r,\theta,\phi)$. The Lorentz part $M^{\mu\nu}$ of the algebra,
however, remains unchanged. Thus, the map from the ten-dimensional Poincaré
group to the infinite-dimensional BMS group in the instant form can be
schematically written as

$M^{\mu\nu}\rightarrow M^{\mu\nu}\,,\quad P^{\mu}\rightarrow\mathcal{ST}\,.$

The BMS algbera has the same form as in 1 with $P^{\mu}$ replaced with the
supertranslations $\mathcal{ST}$. The local parameter $\tau(\theta,\phi)$ can
be decomposed into even and odd modes in a basis of spherical harmonics
$Y_{lm}(\theta,\phi)$

$\tau(\theta,\phi)=T_{0,0}Y_{0,0}+\sum_{m=-1}^{1}T_{1,m}Y_{1,m}+\sum_{m=-2}^{2}T_{2,m}Y_{2,m}+\cdots\,.$

The Poincaré subgroup of BMS can be obtained by the restricting the parameter
to lowest four harmonics, where $T_{0,0}$ corresponds to the time translation
and $T_{1,m}$ to the three spatial translations. The structure of the BMS
group at spatial infinity, thus, matches with the original interpretation of
BMS at null infinity, wherein angle-dependent supertranslations replace the
four ordinary translations.

* * *

## The need for a unifying framework

There has been significant progress recently, in the study of asymptotic
symmetries in the context of modern scattering amplitude techniques, such as
soft theorems, double copy and twistor theory (see for example [16]). BMS
symmetries have been shown to constrain the gravitational scattering problem.
But the two sides of this connection, namely asymptotic symmetries and
scattering amplitudes, are conventionally studied in different setups.
Asymptotic symmetries are typically examined as a boundary-value problem of
classical solutions at null or spatial infinity while scattering amplitudes
are investigated using on-shell approaches that often involve working in a
physical gauge, such as the light-cone gauge [17, 18]. It is then natural to
ask whether we can identify one framework that will offer a unified
description of both asymptotic symmetries and scattering amplitudes. A light-
cone realization of the BMS algebra constitutes a first step in that
direction, one platform from which we may examine asymptotic symmetries from
an action principle, which is also conducive for deriving on-shell physics.
Instead of working with the boundary data available at asymptotic infinity, we
can formulate an initial-value problem in the bulk spacetime, where we study
the evolution of the initial data on a null hypersurface.

BMS symmetry in front form In the front form, the gauge constraints can be
solved to eliminate unphysical degrees of freedom. In the case of gravity, the
light-cone action is written entirely in terms of two physical modes and
contains only first-order time derivatives (an inherently Hamiltonian system).
This (2+2) Hamiltonian formulation of gravity in the front from was first
presented in [13] and the corresponding light-cone Poincaré algebra was
studied in [6]. The light-cone BMS symmetry is again a local extension of the
light-cone Poincaré algebra, but the difference from the instant form is that
the enhancement occurs in a different part of the algebra [14]. In light-cone
BMS, the time component of the diffeomorphisms $\xi^{+}$ gets enhanced from a
global to a local parameter, $\xi^{+}=T(x,\bar{x})$. Therefore, the extension
from Poincaré to BMS algebra affects the dynamical part of the algebra.
Instead of three dynamical tranformations $P^{-}$, $J^{-a}$ we end up with
infinitely many ‘supertranslations’ denoted by $\mathcal{D}(T)$. The
kinematical part of the algebra remains the same and hence, the map from
light-cone Poincaré to light-cone BMS is schematically

$\mathcal{K}\rightarrow\mathcal{K}\,,\quad\mathcal{D}\rightarrow\mathcal{D}(T)\,,$

in contrast to the instant form case. The BMS algbera, then, has the same form
as in 2 with $\mathcal{D}$ replaced by the supertranslations $\mathcal{D}(T)$.
The local parameter $T(x,\bar{x})$ may be decomposed in a complex coordinate
basis $(x,\bar{x})$

$T(x,\bar{x})=c_{0,0}+c_{1,0}x+c_{0,1}\bar{x}+c_{1,1}x\,\bar{x}+\cdots+c_{m,n}x^{m}\bar{x}^{n}\,.$

The Poincaré subgroup of the light-cone BMS is then the restriction to the
lowest three modes that are at most linear in coordinates, with $c_{0,0}$
corresponding to time translations and $c_{1,0}$, $c_{0,1}$ corresponding to
the two boosts. Clearly, from a group theory perspective, the BMS structure
here is very different from the instant form. Instead of the normal subgroup
of Poincaré $P^{\mu}$, the enhancement in this case occurs in the Abelian
ideal $\mathcal{D}$. In recent work, the BMS algebra was also obtained as the
residual gauge symmetry in light-cone gravity without any explicit reference
to asymptotic limits [15]. This result resonates with the idea that the BMS
symmetry may be more than just an asymptotic symmetry of the theory [8].

Figure 3: Light-cone BMS as a bridge between Hamiltonian and on-shell methods

In the light-cone formulation, one can eliminate all unphysical gauge degrees
of freedom from the theory, a feature that has proved to be fruitful in many
instances. Therefore, it serves as a natural starting point for such a
unifying framework because (i) one can reformulate the asymptotic symmetries
as an initial-value problem in the front form and (ii) the transition to on-
shell methods in a physical gauge is quite straightforward. As a matter of
fact, the light-front is a natural basis for scattering amplitudes with spinor
helicity structures emerging naturally [19]. It is therefore worth exploring
the potential links between the asymptotic symmetries in the front form and
celestial or flat space holography constituting an effort to establish a
duality between the S-matrix of four-dimensional quantum gravity and
correlation functions in a two-dimensional “celestial” CFT, akin to the idea
of AdS/CFT duality.

Acknowledgments The work of SA is partially supported by a MATRICS grant -
MTR/2020/000073 - of SERB. The work of SM is supported by the LABEX Lyon
Institute of Origins (ANR-10-LABX-0066) Lyon within the Plan France 2030 of
the French government operated by the National Research Agency (ANR).

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