Physics Prof. H. F. Dowker To My Parents

# Stress-Energy in the Conical Vacuum and its Implications for Topology Change

Eric Jones

(September 18, 2015)

###### Abstract

This dissertation presents a semiclassical analysis of conical topology change
in $1+1$ spacetime dimensions wherein, to lowest order, the ambient spacetime
is classical and fixed while the scalar field coupled to it is quantized. The
vacuum expectation value of the scalar field stress-energy tensor is
calculated via two different approaches. The first of these involves the
explicit determination of the so called Sorkin-Johnston state on the cone and
an original regularization scheme, while the latter employs the conformal
vacuum and the more conventional point-splitting renormalization. It is found
that conical topology change seems not to suffer from the same pathologies
that trousers-type topology change does. This provides tentative agreement
with conjectures due to Sorkin and Borde, which attempt to classify topology
changing spacetimes with respect to their Morse critical points and in
particular, that the cone and yarmulke in $1+1$ dimensions lack critical
points of unit Morse index.

###### Acknowledgements.

I would foremost like to thank Fay Dowker, whose depth of knowledge, ready
support, and contagious sense of wonder about the universe have been elemental
to the completion of this project. For guidance and many fruitful discussions,
much gratitude is due Ian Jubb. The work of Michel Buck and initial
conversations with him facilitated an easy apprehension of the background on
this subject. And it will be evident that much of the foundation for this
analysis has been laid by Rafael Sorkin. To these two and all of the other
authors whose work I have drawn on, I extend my thanks. Immeasurable love and
thanks to my Dad, Darrell, who got this whole ball rolling when he first lent
me his copy of “A Brief History of Time,” my Mom, Leah, who has always
inspired me to find beauty in the world, my brother, Trevor, who has been a
stalwart intellectual companion over the years, and the rest of my friends and
family for their encouragement and support. I thank Matthias Beuerle, Iacopo
Russo, Marlene Weber, Nick Duch, Colin Price, and Manuel Costa Campos for
their friendship, and Meera Patel, John Ronayne, Azin Khan, and Eduardo García
for lighting up the computational suite with their colorful dispositions.

###### Contents

  1. 0 Introduction
    1. 1 The Zero
    2. 2 Gravitational Topology Change
      1. 1 Geons and the Quantum Foam
      2. 2 String Theory
      3. 3 Low Dimensions
  2. 1 Preliminaries
    1. 1 Free Scalar Quantum Field Theory in Flat Spacetime
    2. 2 Scalar Quantum Field Theory in Curved Spacetime
    3. 3 Causal Set Theory
    4. 4 Green’s Functions and the Sorkin-Johnston State
    5. 5 The Bogolyubov Method
    6. 6 The Discrete Sorkin-Johnston State
  3. 2 Quantum Field Theory on the Lorentzian Cone
    1. 1 The Sorkin-Johnston State on the Lorentzian Cone
    2. 2 Another Route To Stress-Energy
  4. 3 Conclusion
    1. 1 Beyond the Zero

##  Chapter 0 Introduction

###  1 The Zero

The existence, and corresponding conception or perception, of nothingness is a
notion that has beleaguered natural philosophers back into antiquity. Even the
semantic contradiction inherent in using “existence” and “nothingness” in the
same breath highlights the crux of the issue as posed by Parmenides of Elea in
the fifth century BCE and paraphrased by Bertrand Russel [1]: “You say there
is the void; therefore the void is not nothing; therefore it is not the void.”
Already, another term has been introduced to try and clarify things. Indeed,
atomists such as Democritus postulated that the void must exist to allow atoms
to move through it and to occupy the volumes between them, but this position
was retrograded by Aristotle and finally Newton who delineated between an
“eternal space” upon which physical matter played out its dynamics and the
philosophical concept of the void, or absolute nothingness.

Furthermore, we now know through the paradigm of general relativity, that this
eternal space is anything but. Rather, it is dynamical and ever changing,
rendering its candidacy for playing the role of void substantially more
suspect. And finally, when one considers the ground state of quantum
electrodynamics, with its roiling sea of virtual particles, continuously
materializing and dematerializing, the idea that physical nothingness could be
identified with metaphysical nothingness becomes outright untenable.

However, we might still run some thought experiment wherein all of the
physical fields of the Standard Model are banished and we are left with pure
spacetime and perhaps a scalar field coupled to it acting as a sort of
indicator field. Is there some modest sense in which we can recover the
concept of void as it relates to this system? The physical situation to which
we refer of course is the vacuum state, and the theoretical framework we will
use is that of semiclassical gravity. As it turns out, for a general classical
spacetime, the answer to this question is still not straightforward. And when
one tries to incorporate some proposed features of quantum gravitational
spacetimes into the analysis, things become even further nettled with
uncertainty.

###  2 Gravitational Topology Change

####  1 Geons and the Quantum Foam

The great triumph of Einstein’s relativity was the discovery not only that
spacetime is a geometric object, but that it is also a dynamical one. However,
for the purpose of being able to describe a complete classical physics, matter
still has to be introduced by hand through parameters such as mass and charge.
In 1955, John Archibald Wheeler proposed the concept of the geon, or
“gravitational electromagnetic entity” [2]. The hope was that these self-
cohering geometric structures could be used in lieu of the comparatively ad
hoc “bodies” of classical physics in order to provide: “gravitation without
gravitation, electromagnetism without electromagnetism, charge without charge,
and mass without mass” [3]. In order to achieve charge without charge, for
example, electromagnetic field lines are confined to pass through a wormhole
into two disparate regions of spacetime, giving the macroscopic appearance of
source charges. It soon became apparent though, that if a non-trivial
topological structure such as this were to exist in a spacetime, physicists
might also need a topological dynamics in order to complement the
geometrodynamics.

In order to see why this might be, consider another concept due to Wheeler,
that of the quantum foam. Classical spacetime is described as a smooth
manifold endowed with a Lorentzian metric, $(M,g_{\mu\nu})$. Once the matter
content of the physical system under consideration is specified, the dynamics
are determined through the Einstein or Einstein-Maxwell field equations. So
long as we should like to retain a Lorentzian metric everywhere, the spatial
topology of the spacetime must be fixed in the absence of closed timelike
curves [4]. However, suppose we concentrate our attention towards a very small
patch of spacetime over a very short period of time at around the Planck
length, $\ell_{p}\sim 1.6\times 10^{-33}$ cm and time, $t_{p}\sim 5.4\times
10^{-44}$ s. On this scale, quantum gravitational effects are thought to
dominate the dynamics of spacetime. In particular, if we interpret the energy-
time uncertainty relation, $\Delta E\Delta t\sim\hbar$ as allowing the
temporary existence of virtual particles with arbitrarily high energies over
commensurately short timescales, then at the Planck scale, existence of these
virtual particles should cause violent fluctuations to the stress-energy
tensor, which in turn ought to result in correspondingly large fluctuations in
the metric. Wheeler argued that such agitation to the continuum manifold could
cause topology change at the quantum level [5]. This is colloquially referred
to as the quantum foam.

While seemingly disallowed classically, topology change can be admitted by
relaxing the requirement that $M$ have an everywhere Lorentzian metric, and
allowing the metric to degenerate at a finite set of points [6]. Moreover,
topological processes could occur as virtual processes at the quantum level,
entering into the gravitational amplitude as a phase in the path integral
$Z=\int_{geometries}e^{\frac{i}{\hbar}S_{geometries}}$ [7]. Just as the non-
relativistic and relativistic path integrals allow particle paths and field
configurations which do not obey Hamilton’s principle, to contribute phases to
the amplitude, it is not unthinkable that a similar situation might be in view
for classically forbidden topological contributions in gravity.

Finally, we can also ask how one could go about formulating a consistent
theory of quantum geons, and in fact, this was one of the initial motivations
for Wheeler’s introduction of the classical geon. The modern term for the
relevant physical entity is the topological geon, and it has been shown that
in the absence of spatial topology change, topological geons as fundamental
particles suffer from violations of the spin-statistics theorem [8].

####  2 String Theory

Perhaps the contemporary theory of quantum gravity which makes most ready use
of topology change is string theory. This can be seen from the very outset
within the context of bosonic string theory. In the simplest scenario, let
$(M,\delta_{\mu\nu})$ be 26 dimensional Euclidean space after a Wick rotation
from $(M,\eta_{\mu\nu})$, and $(\Sigma_{g},\gamma_{\alpha\beta})$ a Riemann
surface of genus $g$ embedded therein with coordinates,
$X^{\mu}(\xi^{0},\xi^{1})$. The string action is given by [9]:

$S[X,\gamma,g]:=\frac{1}{4\pi\alpha^{\prime}}\int
d^{2}\xi\sqrt{\gamma}\gamma^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}\delta_{\mu\nu}+\lambda\chi_{g}.$
(1)

The first term is the Polyakov action with Regge slope $\alpha^{\prime}$,
while $\chi_{g}=\frac{1}{4\pi}\int d^{2}\xi\sqrt{\gamma}\mathcal{R}=2-2g$ is
the Euler characteristic, which is a topological invariant on $\Sigma_{g}$,
and $\mathcal{R}$ is the scalar curvature on $\Sigma_{g}$. The naive vacuum
amplitude of the theory would read:

$Z=\sum_{g=0}^{\infty}\int\mathcal{D}X\mathcal{D}\gamma e^{-S[X,\gamma,g]}.$
(2)

It turns out that the correct partition function is:

$Z=\sum_{g=0}^{\infty}\int_{\mathcal{E}\times
M_{g}}\frac{\mathcal{D}X\mathcal{D}\gamma}{V(Diff\ast
Weyl)}e^{-S[X,\gamma,g]},$ (3)

owing to the fact that the functional integration measure needs to be
normalized to take into account the gauge invariance of $S[X,\gamma,g]$ under
both diffeomorphisms, $f:\Sigma_{g}\rightarrow\Sigma_{g}$ and Weyl rescalings,
$\gamma_{\alpha\beta}\rightarrow e^{\phi}\gamma_{\alpha\beta}$. If
$Diff(\Sigma_{g})\ast Weyl(\Sigma_{g})$ is the semi-direct product space of
the groups of these transformations on $\Sigma_{g}$, then $V(Diff\ast Weyl)$
is the infinite volume of this space. So, each term in (3) integrates over all
possible metric configurations on, and embeddings of, a given Riemann surface
modulo geometries equivalent under the gauge group.

Crucially, the vacuum amplitude also sums over Riemann surfaces of every genus
with $e^{-2\lambda g}$ playing the role of perturbative expansion
parameter111Schematically, the expansion looks like: (jelly doughnut) $+$
(glazed doughnut) $+$ (double-handed doughnut)$+\cdot\cdot\,\cdot$. When one
moves towards incorporating external legs, one simply stitches ingoing and
outgoing strings to these topologies. According to bosonic string theory then,
topology change in the 26 dimensional vacuum is a fairly pedestrian
occurrence. From the point of view of world sheet conformal field theory, the
free Polyakov action describes the field theory of 26 scalar fields on a 2
dimensional spacetime. Upon quantization, one of these degrees of freedom
always remains as the scalar dilaton under $SO(24)$ [10].

####  3 Low Dimensions

The fundamental closed string vertex then looks rather similar to 26 uncoupled
massless scalar fields propagating on the so called “trouser spacetime,” which
allows an $S^{1}\rightarrow S^{1}\times S^{1}$ spatial topology change. The
trousers fall into the category of topological cobordisms222Mathematically,
given two $n$ dimensional manifolds M and N, with distinct topologies, a
cobordism is an $n+1$ dimensional compact manifold W such that $\partial
W=M\sqcup N$ [11]. mentioned above whose metric can be chosen to degenerate at
a single point. In this instance, the degeneracy occurs at the “crotch” as
viewed with the trousers hanging upside down and time moving upwards. Various
analyses of the trouser spacetime have showed that calculation of the stress-
energy tensor of a massless scalar field yields an unrenormalizable divergence
at the crotch, which propagates into the future lightcones therefrom [12]
[13].

While this may seem to work against topology change on the quantum level as
well, it is interesting to note two points. First, in the case of the trousers
the metric is “causally discontinuous” [14]. That is, the forward lightcone of
every point on the trousers is unique, except at the crotch where it
bifurcates in two with one branch leading up each leg. There is a conjecture
related to this fact due to Sorkin, which attributes such infinite energy
bursts to causal discontinuities in a spacetime as opposed to just
irregularities in the metric. The second observation is that the trousers
cobordism, as a $1+1$ dimensional submanifold of a $3+1$ dimensional product
spacetime with $2$ flat codimensions, is argued to be exponentially suppressed
in any sum-over-histories formulation of quantum gravity by an extra factor of
$e^{-\frac{A}{4G}}$ over its typical phase suppression due to its distance
from stationarity in the Einstein-Hilbert action. $G$ is Newton’s constant and
$A$ is the area subtended by the flat $2$ codimensional submanifold [15].

There is another type of cobordism in two dimensions however, the yarmulke.
Along with its time-reverse: a structure we will refer to as the cone, and the
trousers, these three cobordisms constitute all of the irreducible $1+1$
dimensional cobordisms. In contrast to the trousers, the cone not only is
causally continuous, but according to Louko and Sorkin, its amplitude in the
SOH is expected to be exponentially enhanced by $e^{\frac{A}{4G}}$ in the same
$3+1$ product manifold scenario described in the preceding paragraph [15]. The
aim of this thesis then will be to determine the behavior of a massless scalar
field on the Lorentzian cone and to contextualize the results with respect to
Sorkin’s conjecture.

##  Chapter 1 Preliminaries

Here we present the mathematical tools we require in order to go about
evaluating conical-type topology change. We quickly review canonical
quantization of a scalar field in ordinary Minkowski spacetime before moving
on to address the procedure in curved spacetime. The treatment given to these
subjects closely follows Birrell and Davies [16]. Then, upon motivating the
use of the Sorkin-Johnston (SJ) state as a preferred vacuum state, it will be
defined first in terms of a theory’s Green’s functions and then via Bogolyubov
transformations from an arbitrary, orthogonal and complete Klein-Gordon
solution set. This will position us to address in the next chapter SJ state
construction on spacetimes that are less than amenable to its original
definition via Green’s functions.

###  1 Free Scalar Quantum Field Theory in Flat Spacetime

Early attempts in the history of quantum theory to construct a
relativistically covariant, phenomenologically relevant single particle wave
equation with a well defined probabilistic interpretation were quickly riddled
with inconsistencies. Among the myriad difficulties encountered were artifacts
such as the existence of infinite “seas” of negative energy states, non-
positive definite probability densities, and the inability to cope, in
general, with systems with variable number of particles [17]. As such, the
portrait of the dynamics of fundamental particles through the lens of the
probabilistic wavefunction was quickly abandoned in favor of quantization of
the classical field itself rather than merely its coordinates. The view we
take then is that of “Second Quantization.”

Let $\phi(x)\in\mathbb{R}$ be a spinless massive scalar particle with mass
parameter, $m$, such as might describe the post-symmetry breaking Standard
Model Higgs degree of freedom [18], which satisfies the Klein-Gordon field
equation:

$(\partial_{\mu}\partial^{\mu}+m^{2})\phi(x)=0.$ (1)

The field equation can be derived as a result of demanding that the action:

$S=\int d^{n}x\mathcal{L}=\int
d^{n}x\frac{1}{2}(\partial_{\mu}\phi(x)\partial^{\mu}\phi(x)-m^{2}\phi^{2}(x))$
(2)

be invariant under infinitesimal deformations of the field:

$\delta S=0.$ (3)

That the Minkowski metric in $n$ dimensions,

$ds^{2}=\eta_{\mu\nu}dx^{\mu}dx^{\nu}=dt^{2}-\sum_{i=1}^{n-1}(dx^{i})^{2}$ (4)

has a timelike Killing vector field, $\partial_{t}\eta_{\mu\nu}=0$, indicates
a particular class of solutions to the Klein-Gordon equation:

$u_{\textbf{k}}(x)=N_{\textbf{k}}e^{-ik_{\mu}x^{\mu}}=N_{\textbf{k}}e^{-i(\omega_{\textbf{k}}t-\textbf{k}\cdot\textbf{x})},$
(5)

where $\omega_{\textbf{k}}>0$ must satisfy
$\omega_{\textbf{k}}=\sqrt{\textbf{k}^{2}+m^{2}}$. These are known as the
positive frequency modes, and are taken together with the negative frequency
solutions:

$\bar{u}_{\textbf{k}}(x)=\bar{N}_{\textbf{k}}e^{ik_{\mu}x^{\mu}}=\bar{N}_{\textbf{k}}e^{i(\omega_{\textbf{k}}t-\textbf{k}\cdot\textbf{x})}.$
(6)

Of course, Minkowski space possesses the full Poincaré isometry group. So,
there is nothing particularly special about $\partial_{t}$ other than the fact
that it allows us to conveniently and intuitively foliate spacetime into
hypersurfaces, $\Sigma_{t}$, normal to it and define a bilinear form thereon,
which is also invariant in time:

$\left(f,g\right)_{KG}=i\int_{\Sigma_{t}}d^{n-1}x\bar{f}\,\overset{\leftrightarrow}{\partial_{t}}\,g=i\int_{\Sigma_{t}}d^{n-1}x\\{\bar{f}(\partial_{t}g)-(\partial_{t}\bar{f})g\\}.$
(7)

This also results in the relativistic normalization constraint:

$N_{\textbf{k}}=\frac{1}{\sqrt{2\omega_{\textbf{k}}(2\pi)^{n-1}}},$ (8)

which enforces the following on the solution set
$\\{u_{\textbf{k}},\bar{u}_{\textbf{k}}\\}$:

$\left(u_{\textbf{k}},u_{\textbf{k'}}\right)_{KG}=\delta^{(n-1)}(\textbf{k}-\textbf{k'})$
(9)
$\left(\bar{u}_{\textbf{k}},\bar{u}_{\textbf{k'}}\right)_{KG}=-\delta^{(n-1)}(\textbf{k}-\textbf{k'})$
(10) $\left(u_{\textbf{k}},\bar{u}_{\textbf{k'}}\right)_{KG}=0.$ (11)

We can then expand any field configuration in terms of linear combinations of
these modes, in the classical case, with complex coefficients $a_{\textbf{k}}$
and $\bar{a}_{\textbf{k}}$:

$\phi(x)=\int
d^{n-1}k(a_{\textbf{k}}u_{\textbf{k}}(x)+\bar{a}_{\textbf{k}}\bar{u}_{\textbf{k}}(x)).$
(12)

In order to make the transition to the quantum theory, we elevate the status
of our field to an operator valued distribution (this will be defined more
precisely in section 2.4), which operates on the Fock space of multiparticle
states111More accurately, the irreducible, symmetric Fock representation of
the Hilbert space, $\mathcal{H}$., and impose equal time commutation
relations. With the scalar Lagrangian density (2), the conjugate momentum to
$\phi$ is:

$\pi(t,\textbf{x})=\frac{\partial\mathcal{L}}{\partial(\partial_{t}\phi)}=\dot{\phi}(t,\textbf{x}).$
(13)

The equal time commutation relations are then:

$\left[\phi(t,\textbf{x}),\pi(t,\textbf{y})\right]=i\delta^{(n-1)}(\textbf{x}-\textbf{y})$
(14) $\left[\phi(t,\textbf{x}),\phi(t,\textbf{y})\right]=0$ (15)
$\left[\pi(t,\textbf{x}),\pi(t,\textbf{y})\right]=0.$ (16)

These are completely equivalent to the following commutation relations imposed
upon the complex coefficients, $a_{\textbf{k}}$ and $\bar{a}_{\textbf{k}}$,
now elevated to Fock space annihilation and creation operators,
$a_{\textbf{k}}$ and $a^{\dagger}_{\textbf{k}}$,

$\left[a_{\textbf{k}},a^{\dagger}_{\textbf{k'}}\right]=\delta^{(n-1)}(\textbf{k}-\textbf{k}^{\prime})$
(17) $\left[a_{\textbf{k}},a_{\textbf{k'}}\right]=0$ (18)
$\left[a^{\dagger}_{\textbf{k}},a^{\dagger}_{\textbf{k'}}\right]=0,$ (19)

with the vacuum state being defined as that, which is annihilated by every
$a_{\textbf{k}}$:

$a_{\textbf{k}}\left|0\right>=0.$ (20)

Because it bears heavily upon what follows, it will be useful to examine the
Poincaré invariance of the vacuum. Let:
$\tilde{x}^{\mu}=\Lambda^{\mu}_{\nu}\,x^{\nu}+a^{\mu}$ be a Poincaré
transformation on the coordinate system corresponding to the vacuum state
above, with $\Lambda^{\mu}_{\nu}\in SO(1,3)$ and $a^{\mu}$ a spacetime
translation. Note first that:

$a_{\textbf{k}}\left|0\right>=0\implies\int
d^{n-1}k\frac{1}{\sqrt{2\omega_{\textbf{k}}(2\pi)^{n-1}}}e^{-ik_{\mu}x^{\mu}}a_{\textbf{k}}\left|0\right>=0.$
(21)

Now, set ${a^{\mu}=0}$ and consider just Lorentz transformations. Then, under
$x\rightarrow\tilde{x}$,

$\left|0\right>\rightarrow\left|\tilde{0}\right>,$ (22)

and

$k_{\mu}x^{\mu}\rightarrow\tilde{k}_{\mu}\tilde{x}^{\mu}=k_{\alpha}\Lambda^{\alpha}_{\mu}\Lambda^{\mu}_{\nu}x^{\nu}=k_{\alpha}\delta^{\alpha}_{\nu}x^{\nu}=k_{\mu}x^{\mu},$
(23)

while,

$\int\frac{d^{n-1}k}{2\omega_{\textbf{k}}(2\pi)^{n-1}}$ (24)

and

$\sqrt{{2\omega_{\textbf{k}}(2\pi)^{n-1}}}a_{\textbf{k}}$ (25)

are the Lorentz invariant integration measure and annihilation operator,
respectively [19]. Therefore, $a_{\textbf{k}}\left|\tilde{0}\right>=0,$ which
by definition means that under Lorentz transformations,
$\left|0\right>=\left|\tilde{0}\right>.$ Meanwhile, allowing $a^{\mu}$ to be
nonzero while fixing $\Lambda^{\mu}_{\nu}=\delta^{\mu}_{\nu},$ and noting that
wave vectors transform trivially under translations implies that these also
are a symmetry of the vacuum.

###  2 Scalar Quantum Field Theory in Curved Spacetime

We now generalize the flat, Minkowski metric, $\eta_{\mu\nu}$ to $g_{\mu\nu}$,
the components of the symmetric, nondegenarate, bilinear form $g(\cdot,\cdot)$
on an n-dimensional, smooth Lorentzian manifold M, with signature
$(+-\cdot\cdot\cdot\,-)$. Writing $g_{\mu\nu}$ as an $n\times n$ symmetric
matrix, let: $g(x)=det(g_{\mu\nu}(x))$ in a coordinate chart, $\\{x^{\mu}\\}$.
Then the Lagrangian density for the real scalar field coupled to gravity is:

$\mathcal{L}=\frac{1}{2}\sqrt{-g(x)}\\{g^{\mu\nu}(x)\partial_{\mu}\phi(x)\partial_{\nu}\phi(x)-\left(m^{2}+\xi
R(x)\right)\phi^{2}(x)\\},$ (26)

where $R(x)$ is the Ricci scalar and $\xi$ is the dimensionless, numerical
coupling of the scalar field to gravity. $\xi=0$ is called minimal coupling
while $\xi=\frac{1}{4}[(n-2)/(n-1)]$ corresponds to conformal coupling.
Minimizing the corresponding action with respect to variations in $\phi$,

$\delta S=\delta\int d^{n}x\mathcal{L}=0,$ (27)

yields the field equation:

$\left(\Box+m^{2}+\xi R(x)\right)\phi(x)=0,$ (28)

where,

$\Box\phi(x)=g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\phi(x)=\frac{1}{\sqrt{-g}}\partial_{\mu}\left[\sqrt{-g}g^{\mu\nu}\partial_{\nu}\phi(x)\right].$
(29)

In order to perform a well defined quantization of the field theory on
$\left(M,g_{\mu\nu}\right),$ we specify that the spacetime be globally
hyperbolic. This ensures that $M$ can be foliated into spacelike Cauchy
hypersurfaces, $\Sigma$, along a timelike coordinate, and that once a set of
conditions for $\phi$ is initialized upon one of the $\Sigma$, its value is
determined uniquely on all of $\left(M,g_{\mu\nu}\right)$ through (28).

The generalization of the Klein-Gordon inner product is

$\left(f,g\right)_{KG}=i\int_{\Sigma}d\Sigma\,n^{\mu}\bar{f}\,\,\overset{\leftrightarrow}{\nabla}_{\mu}\,g$
(30)

where

$d\Sigma=d^{n-1}x\sqrt{-g_{\Sigma}(x)}$ (31)

is the volume form integration measure on the hypersurface $\Sigma$, $n^{\mu}$
are the components of the future-directed, timelike normal vector, orthogonal
to $\Sigma$, and $g_{\Sigma}(x)$ is the determinant of the induced metric on
$\Sigma$. The Klein-Gordon inner product is invariant across timelike
translations. That is, it is independent of the $\Sigma$ upon which it is
calculated. As such, by existence and uniqueness theorems for hyperbolic
differential equations on $C^{\infty},$ globally hyperbolic manifolds, a
complete solution set to the field equation exists, $\\{u_{i},\bar{u}_{i}\\}$,
and once its elements are found to satisfy:

$\left(u_{i},u_{j}\right)_{KG}=\delta_{ij}$ (32)
$\left(\bar{u}_{i},\bar{u}_{j}\right)_{KG}=-\delta_{ij}$ (33)
$\left(u_{i},\bar{u}_{j}\right)_{KG}=0,$ (34)

on any one of the Cauchy hypersurfaces, they will remain so orthonormalized
forever. Here, the Roman subscripts are generalized labels to delineate
between different modes. For example, $i$ plays the role of the wavenumber $k$
in the case of Minkowski spacetime, or takes on discrete values if spatial
sections of the spacetime are compact. Once such a set of modes is found, we
can expand the quantized field by analogy with the Minkowski case:

$\phi(x)=\sum_{i}(a_{i}u_{i}(x)+{a}^{\dagger}_{i}\bar{u}_{i}(x)),$ (35)

where,

$\left[a_{i},a^{\dagger}_{j}\right]=\delta_{ij}$ (36)
$\left[a_{i},a_{j}\right]=0$ (37)
$\left[a^{\dagger}_{i},a^{\dagger}_{j}\right]=0,$ (38)

and define a vacuum by:

$a_{i}\left|0\right>=0\;\forall\,i.$ (39)

The operative word above is “a.” In the absence of a high degree of symmetry
of the spacetime222The most desirable situation is when the spacetime admits a
global timelike Killing vector field, $\kappa=\frac{d}{dt}$, which commutes
with the Klein-Gordon operator, $\left(\Box+m^{2}+\xi R(x)\right)$. In this
case, a solution set can be found, which is positive frequency with respect to
the Killing vector, $\kappa\,u_{k}(x)=-i\omega_{k}u_{k}(x)$ and the
corresponding vacuum will be unique., the solution set
$\\{u_{i},\bar{u}_{i}\\}$ will not be unique, and the number of corresponding
vacua will be as multitudinous as the different sets available with which to
decompose $\phi$ [20]. The fundamental idea in general relativity, that
physics occurs on $M$ rather than in the coordinate charts, finds its
expression in semiclassical gravity by rendering the vacuum generally
meaningless. In particular, if we let $\\{v_{i},\bar{v}_{i}\\}$ be another
Klein-Gordon orthonormal solution set, then the field can be expanded as:

$\phi(x)=\sum_{i}(b_{i}v_{i}(x)+{b}^{\dagger}_{i}\bar{v}_{i}(x)),$ (40)

where now the vacuum is defined by:

$b_{i}\left|\tilde{0}\right>=0\;\forall\,i,$ (41)

and the two sets are related by a Bogolyubov transformation:

$v_{i}(x)=\sum_{j}\left(A_{ij}u_{j}(x)+B_{ij}\bar{u}_{j}(x)\right).$ (42)

With the Klein-Gordon orthonormalization conditions, the Bogolyubov
coefficients can be obtained by:

$A_{ij}=\left(v_{i},u_{j}\right)_{KG}\;\;\;\;\;B_{ij}=-\left(v_{i},\bar{u}_{j}\right)_{KG}$
(43)

Equating then, the two field expansions,

$\phi(x)=\sum_{i}(a_{i}u_{i}(x)+{a}^{\dagger}_{i}\bar{u}_{i}(x))=\sum_{i}(b_{i}v_{i}(x)+{b}^{\dagger}_{i}\bar{v}_{i}(x))$
(44)
$=\sum_{ij}\left((b_{i}A_{ij}+b^{\dagger}_{i}\bar{B}_{ij})u_{j}(x)+(b_{i}B_{ij}+b^{\dagger}_{i}\bar{A}_{ij})\bar{u}_{j}(x)\right),$

and taking the Klein-Gordon inner products again, once with $u_{k}(x)$ and
once with $\bar{u}_{k}(x),$ yields the transformation properties of the
operator coefficients:

$a_{k}=\sum_{j}\left(b_{j}A_{jk}+b^{\dagger}_{j}\bar{B}_{jk}\right)$ (45)
$a^{\dagger}_{k}=\sum_{j}\left(b_{j}B_{jk}+b^{\dagger}_{j}\bar{A}_{jk}\right).$
(46)

That the vacuum is a generally variant quantity can then be seen by
calculating:

$\left<\tilde{0}\right|a^{\dagger}_{k}a_{k}\left|\tilde{0}\right>=\left<\tilde{0}\right|\sum_{i}\left(b_{i}B_{ik}+b^{\dagger}_{i}\bar{A}_{ik}\right)\sum_{j}\left(b_{j}A_{jk}+b^{\dagger}_{j}\bar{B}_{jk}\right)\left|\tilde{0}\right>=\sum_{i}\left|B_{ik}\right|^{2},$
(47)

which of course does not vanish unless each $B_{ik}$ does. This non-uniqueness
of the vacuum has led to concrete physical predictions such the Unruh effect
[21], Hawking radiation [22], and the production of Gaussian-distributed
random perturbations within the theoretical framework of cosmic inflation
[23]. However, the lack of even a preferred ground state, to say nothing of a
unique one, presents a serious ontological quandary to anyone attempting to
retain the familiar notion of quantal excitations as particles. It would seem
pertinent then to try and identify either some criterion by which to recognize
such a preferred vacuum or methods by which to construct one. In the next
section, we will present the fundamental ideas of causal set theory with a
view towards using the closely associated Sorkin-Johnston state as a natural
option.

For completeness, we include the remaining transformation properties of the
field modes, operator coefficients, and Bogolyubov coefficients as will be
relevant for our discussion later on. The inverse transformation from the
$\\{v_{i},\bar{v}_{i}\\}$ to the $\\{u_{i},\bar{u}_{i}\\}$ is:

$u_{i}(x)=\sum_{j}\left(\bar{A}_{ji}v_{j}(x)-B_{ji}\bar{v}_{j}(x)\right),$
(48)

while for the operator coefficients, it is

$b_{i}=\sum_{j}\left(\bar{A}_{ij}a_{j}-\bar{B}_{ij}a^{\dagger}_{j}\right).$
(49)

Finally, preservation of (32)-(34) under the transformation requires that:

$\sum_{k}\left(A_{ik}\bar{A}_{jk}-B_{ik}\bar{B}_{jk}\right)=\delta_{ij},$ (50)
$\sum_{k}\left(A_{ik}B_{jk}-B_{ik}A_{jk}\right)=0.$ (51)

###  3 Causal Set Theory

Causal set theory is a framework that proposes to rectify the three
“infinities” of modern theoretical physics, namely, the ultraviolet
divergences arising in relativistic quantum field theory333The most egregious
example here, of course, is the perturbative non-renormalizability of
interactions involving the graviton. [24] [19], the curvature singularities of
General Relativity [25], and the unbounded result when the Bekenstein-Hawking
entropy is calculated directly by way of quantum gravitational degrees of
freedom [26]. The basic idea that spacetime should be in some way discrete is
common to many of the approaches aimed at successfully quantizing gravity
[24]. The causal sets program shares this view and adopts discreteness along
with causality as its fundamental kinematic axioms [27].

In fact, given only the causal relations between the elements of a set of
discrete points in a given manifold, all of the topological information and
nearly all of the metrical information of the host manifold can be recovered,
in addition to its dimension and differential structure [28] [29]. Then, the
only piece of information missing is an overall conformal factor of the
metric, which can be naturally fixed by requiring that the number of causal
set elements (at some embedding density) corresponds to the volume of the
continuum spacetime region under consideration. Ongoing formulations have then
typically involved a sum over histories or quantum measure theoretic approach
to the dynamics on the space of all causal sets [30] [31]. The eventual goal
is to discard the concept of a manifold entirely in favor of the causal set as
the most fundamental structure. Interestingly, even without a full quantum
dynamics in place, the effects of this type of discrete geometry have already
been shown to be phenomenologically prescient as regards, most notably, the
minscule value of the cosmological constant [32]. For our purposes however, we
will employ the causal set as a Lorentz invariant discretization of spacetime,
upon which it is convenient, for example, to perform semiclassical
calculations numerically. To make this all concrete, we introduce a
definition.

A causal set, denoted by $\left(\mathcal{C},\prec\right),$ is a set
$\mathcal{C}$ endowed with a locally finite partial order relation $\prec$,
which satisfies the following properties:

1.) Transitivity: $x\prec y$ and $y\prec z\implies x\prec z$
$\forall\,\,x,y,z\in\mathcal{C}$

2.) Irreflexivity: $x\nprec x$ $\forall\,\,x\in\mathcal{C}$

3.) Local Finiteness: $\left|\\{y\in\mathcal{C}\,|\,x\prec y\prec
z\\}\right|<\infty$ $\forall\,\,x,y,z\in\mathcal{C}.$

To borrow the nomenclature of global causal analysis, we say that “$x$ is in
the causal past of $y$” if $x\prec y$. In addition, we write $x\preceq y$ in
the case that $x\prec y$ or $x=y$. It is illustrative to consider these
conditions within the context of embedding in a continuum manifold (while
keeping in mind that we should then refocus on the causal set). Transitivity
formalizes the intuitive idea that if we can establish timelike or null
worldlines from $x$ to $y$ and from $y$ to $z$, then we ought also to be able
to establish one from $x$ to $z$. Meanwhile, irreflexivity prohibits the
existence of the discrete equivalent of closed timelike curves of the form
$x\prec y\prec z\prec\cdot\cdot\cdot\prec w\prec x.$ In other words, $x$
cannot be in its own causal past. Finally, the condition that the causal set
be locally finite is how one specifies that its elements are discrete. That
is, if we look at the region marked out by the intersection of the past
lightcone of $z$ and the future lightcone of $x$, the cardinality of the
causal subset comprised of elements therein should be finite.

Now, given an n-dimensional Lorentzian manifold $(M,g_{\mu\nu}),$ we can
construct a causal set $\left(\mathcal{C},\prec\right)$ whose partial order
relation bears the geometrical information of the parent manifold by way of a
process called sprinkling. While discretization of a manifold might seem to be
most easily carried out by simply laying down a rectangular, or other regular,
lattice whose elements obey the causal precedence relations of the manifold,
specialization to Minkowski space and consideration of a boost along some axis
quickly shows that this will not be Lorentz invariant444Inspect for example
the effect of length contraction and time dilation on a spacetime square of
side length $\ell$.. However, distribution of points into Minkowski space by
way of a Poisson process does respect Lorentz symmetry.

Putting aside for a moment the dire problem of quantum gravity, consider some
stochastic process such as children dropping their ice cream cones at the
beach, which occurs along a homogenous time function $\tau$ with an average
rate of $\omega.$ The probability that there will be $n$ instances of this
happening in some time interval $\Delta\tau$ can then be modeled by the
Poisson distribution [33]:

$P(n)=\frac{e^{-\omega\Delta\tau}(\omega\Delta\tau)^{n}}{n!}.$ (52)

The generalization to the probability of $n$ elements being counted within a
spacetime volume $V$ given an average point embedding density of $\rho$
is555In accordance with the idea that distance measures will likely be
meaningless below roughly the Planck length ($\ell_{p}=1.6\times 10^{-33}$ cm)
in any final quantum theory of gravity, it is typically taken to be the case
that $\rho\sim\ell_{p}^{-n}$ [30].:

$P(n)=\frac{e^{-\rho V}(\rho V)^{n}}{n!}.$ (53)

Since the expectation value of the distribution, $\left<N\right>=\rho V$ is
clearly Lorentz invariant, so too is $P(n)$, as will be the embedding
predicated thereon, but in a statistical sense. This is what is meant by a
sprinkling- a discretization of the manifold whose elements obey the counting
statistics of (53). If in the limit $\rho\rightarrow\infty$, the original
manifold $M$ is recovered, the embedded causal set $\mathcal{C}$ will be
called a faithful embedding. In practice, the statistical Lorentz invariance
means that in order to derive meaningful physical quantities, one can use a
garden variety random number generator computationally to discretize $M$, but
will need to average results over a number of such discretizations.

Once such a causal set has been thus obtained, a natural next step is to see
how scalar quantum field theory might be affected by this discreteness. While
still not fully quantum gravitational, this sort of analysis could lead to the
extension of the semiclassical causal set phenomenology666While tangential for
this work, the heuristic prediction of $\Lambda$ is interesting. It goes as
follows. Assume $\left<\Lambda\right>=0$ and what actually drives acceleration
is its quantum fluctuation, $\Delta\Lambda.$ Identify $\Lambda$ and $V$ as
conjugate quantities as viewed by unimodular gravity in the Einstein-Hilbert
action, $-\Lambda\int d^{4}x\sqrt{-g}=-\Lambda V$. They then satisfy an
uncertainty relation: $\Delta\Lambda\Delta V\sim 1$. In natural units,
$\hbar=c=G=1\implies\ell_{p}=1\implies\rho=1$ in (53). But
$N\sim\left<N\right>\pm\Delta N\sim V\pm\sqrt{V},$ using the mean and standard
deviation of the Poisson distribution. Then, $V\sim
N\pm\sqrt{V}\sim\left<V\right>\pm\Delta V\implies\Delta V=\sqrt{V}$, so that
$\Delta\Lambda\sim V^{-\frac{1}{2}}$. Taking $V\sim H^{-4}$, where $H\sim
10^{-61}$ is the Hubble constant in natural units, $\Delta\Lambda\sim
10^{-122}$. of the type found in [34] [35]. Discrete, Lorentz invariant
d’Alembertian operators have been found for flat spacetimes of arbitrary
dimension $n$ [36]. So, we could in principle use these to find sets of
discrete eigenmodes of the Klein-Gordon field. We will follow a different
route to determine these modes based on the field’s discrete retarded
propagator on the causal set. To do this, we will need the $p\times p$ causal
adjacency matrix $C$ whose entries are defined by:

$C_{ij}:=\left\\{\begin{tabular}[]{cc}$1$&if $x_{i}\prec x_{j}$\\\
$0$&otherwise,\end{tabular}\right.$ (54)

where $x_{i},x_{j}\in\mathcal{C}$ and $p$ is the cardinality of $\mathcal{C}$.
The causal matrix encodes all of the binary order relations within the set.
Later, we’ll see how this relates to the field’s discrete retarded propagator,
but first, we review some aspects of Green’s functions in the continuum
theory.

###  4 Green’s Functions and the Sorkin-Johnston State

In our discussion of canonical quantization of the scalar field in curved
spacetime (Section 2.2), recall that we stipulated commutation relations for
the creation and annihilation operators in the expansion of the field,
(36)-(38). However, nothing was said about the corresponding form of the field
commutators as we had for Minkowski space. It turns out that they can be
stated in fully covariant form [37]. In order to do this, calculate the vacuum
expectation value (VEV) of the field commutator in some mode set:

$\left<0\right|\left[\phi(x),\phi(y)\right]\left|0\right>=\sum_{i}\left(u_{i}(x)\bar{u}_{i}(y)-\bar{u}_{i}(x)u_{i}(y)\right).$
(55)

Then, defining the Pauli-Jordan Function, $\Delta(x,y)$, as

$i\Delta(x,y):=\sum_{i}\left(u_{i}(x)\bar{u}_{i}(y)-\bar{u}_{i}(x)u_{i}(y)\right),$
(56)

we can write

$\left[\phi(x),\phi(y)\right]=i\Delta(x,y)\,\mathbb{I},$ (57)

where $\mathbb{I}$ is the Fock space identity operator. From now on, we will
drop $\mathbb{I}$ under the understanding that its existence is implicit when
dealing with Fock states. Since the left hand side of (57) bears no indication
of the coordinate basis with which the field is decomposed, $i\Delta(x,y)$ had
better not either. Luckily, so long as (50) and (51) are satisfied, such as
between two bases, which are correctly orthonormalized under
$(\cdot,\cdot)_{KG},$ the Pauli-Jordan function indeed is also basis
independent [38].

Moreover, we now make precise the sense in which $\phi$ is an operator valued
distribution in the general case. Given some smooth functions of compact
support on M, $f,h\in C^{\infty}_{0}(M),$ the rigorous expression of (57) is:

$\left[\phi(f),\phi(h)\right]=i\int_{M}d^{n}x\,d^{n}y\sqrt{-g(x)}\sqrt{-g(y)}f(x)\Delta(x,y)h(y).$
(58)

Additionally, expressions such as: $\phi(x)\left|\psi\right>$ should formally
read

$\phi(f)\left|\psi\right>=\int_{M}d^{n}x\sqrt{-g(x)}\phi(x)f(x)\left|\psi\right>,$
(59)

for all $f$. The requirement that field operators be defined as distributional
addresses ultraviolet divergence issues arising from the ill-definition of a
field at a point by “smearing” with $f$ and ensures that after application to
Fock space kets, the states are still normalizable [39]. In practice,
equations such as (59) are most relevant in axiomatic quantum field theory
(see for example [40]), which will not be the approach most closely followed
here. More importantly for us, equation (58) suggests the use of the Pauli-
Jordan function as a bilinear integral operator. As such, define:

$\left(i\Delta f\right)(x):=i\int_{M}d^{n}y\sqrt{-g(y)}\Delta(x,y)f(y).$ (60)

As can be seen from its definition in (56), $i\Delta(x,y)$ enjoys two
important properties as an operator, namely:

1.) Skew-symmetry: $i\Delta(y,x)=-i\Delta(x,y)$

2.) Hermiticity: $(i\Delta(x,y))^{\dagger}=i\Delta(x,y).$

The second property means that if:

$\left<f,h\right>=\int_{M}d^{n}x\sqrt{-g(x)}\,\bar{f}(x)h(x)$ (61)

is the $L^{2}(M)$ inner product, then $i\Delta(x,y)$ is self-adjoint on
$L^{2}(M)$. That is:

$\left<f,i\Delta h\right>=\left<i\Delta f,h\right>,$ (62)

and by the spectral theorem, $i\Delta(x,y)$ can be diagonalized. This object
then becomes the linchpin for the Sorkin-Johnston proposal for a preferred
vacuum state [41] [42].

In the canonical formulation of quantum field theory, Wick’s Theorem allows us
to write any time-ordered product of fields as a polynomial in Wightman two-
point functions [19]. In a given Klein-Gordon solution set
$\\{u_{i},\bar{u}_{i}\\}$, the Wightman function can be written as:

$W(x,y)=\left<0\right|\phi(x)\phi(y)\left|0\right>=\sum_{i}u_{i}(x)\bar{u}_{i}(y).$
(63)

Notice that, in contrast to the Pauli-Jordan function, the Wightman function
is not in general invariant under Klein-Gordon norm preserving Bogolyubov
transformations. So long as the vacuum state can be taken to be Gaussian,
specifying which Wightman function to use in the Wick polynomials is exactly
equivalent to choosing a mode set in which to expand $\phi$. Now, since
$i\Delta(x,y)$ is diagonalizable and Hermitian, it has a set of $L^{2}(M)$
orthonormal eigenfunctions, $\\{u_{a}\\}$ with real eigenvalues,
$\\{\lambda_{a}\\}$ such that:

$(i\Delta u_{a})(x)=\lambda_{a}u_{a}(x).$ (64)

Taking the complex conjugate of (64), and noting that $\Delta(x,y)$ is a real
valued kernel and $\lambda_{a}\in\mathbb{R}\,\,\forall\,\,a$,

$(i\Delta\bar{u}_{a})(x)=-\lambda_{a}\bar{u}_{a}(x).$ (65)

Evidently, the eigenvalues come in positive-negative pairs. We can now
decompose $i\Delta(x,y)$ in the $\\{u_{a},\bar{u}_{a}\\}$ modes:

$i\Delta(x,y)=\sum_{a}\lambda_{a}\left(u_{a}(x)\bar{u}_{a}(y)-\bar{u}_{a}(x)u_{a}(y)\right).$
(66)

The first and second terms are referred to respectively as the positive and
negative parts of the decomposition. If we now define

$u^{SJ}_{a}(x):=\sqrt{\lambda_{a}}u_{a}(x),$ (67)

then the Sorkin-Johnston vacuum is obtained by expanding our field in terms of
these modes,

$\phi(x)=\sum_{b}\left(a_{b}u^{SJ}_{b}(x)+a^{\dagger}_{b}\bar{u}^{SJ}_{b}(x)\right),$
(68)

and using the corresponding coefficients as the correct Fock space creation
and annihilation operators:

$a_{b}\left|SJ\right>=0\,\,\forall\,\,b.$ (69)

The Wightman two-point function in this vacuum is:

$\displaystyle W^{SJ}(x,y)$
$\displaystyle=\left<SJ\right|\phi(x)\phi(y)\left|SJ\right>$ (70)
$\displaystyle=\sum_{a}u^{SJ}_{a}(x)\bar{u}^{SJ}_{a}(y)$
$\displaystyle=\sum_{a}\lambda_{a}u_{a}(x)\bar{u}_{a}(y)$
$\displaystyle=Pos\left[i\Delta(x,y)\right].$

In adopting this two-point function, we specify our full quantum field theory
through Wick’s Theorem. If one considers that $i\Delta(x,y)$ encodes
everything that is “quantum” in a theory via the commutator, and does so in a
fully covariant way, thereby satisfying also the relativistic notion that
coordinates are unimportant, it seems reasonable to think that imprinting this
information upon the vacuum state via diagonalization in the expansion basis
should select the most physically relevant of the vacua available.

We would like to check that after convolution with the SJ Wightman function,
the projection of a test function $f$ onto itself is positive-semidefinite. In
other words, $W^{SJ}$ should be positive as should any reliable two point
function. This follows from the fact that using the SJ modes is just another
way of decomposing the field:

$\displaystyle\left<f,W^{SJ}f\right>$
$\displaystyle=\int_{M}\int_{M}d^{n}xd^{n}y\sqrt{-g(x)}\sqrt{-g(y)}\bar{f}(x)W^{SJ}(x,y)f(y)$
(71)
$\displaystyle=\int_{M}\int_{M}d^{n}xd^{n}y\sqrt{-g(x)}\sqrt{-g(y)}\bar{f}(x)\left<SJ\right|\phi(x)\phi(y)\left|SJ\right>f(y)$
$\displaystyle=\left<SJ\right|\int_{M}\int_{M}d^{n}xd^{n}y\sqrt{-g(x)}\sqrt{-g(y)}\bar{f}(x)\bar{\phi}(x)\phi(y)f(y)\left|SJ\right>$
$\displaystyle=\left<\phi(f)|\phi(f)\right>,$

where the reality of $\phi$ has been used. So long as the fields satisfy the
Wightman axioms, this Hilbert inner product is positive-semidefinite. Hence,
so is $W^{SJ}$.

$\left<f,W^{SJ}f\right>\geq 0.$ (72)

It is also easy to see that we can recover the commutator via:

$\displaystyle W^{SJ}(x,y)-W^{SJ}(y,x)$
$\displaystyle=W^{SJ}(x,y)-\bar{W}^{SJ}(x,y)$ (73)
$\displaystyle=i\Delta(x,y)$ $\displaystyle=\left[\phi(x),\phi(y)\right],$

and that the two point function satisfies the Klein-Gordon equation in both
arguments:

$\left(\Box+m^{2}+\xi R\right)W^{SJ}=0.$ (74)

That $W^{SJ}$ specifies a unique Wightman function is given by (72) and (73)
along with a criterion which the author of [13] has termed “orthogonal
supports”:

$W^{SJ}\bar{W}^{SJ}:=\int_{M}d^{n}z\sqrt{-g(z)}W^{SJ}(x,z)\bar{W}^{SJ}(z,y)=0.$
(75)

This can be shown for finite matrix operators quickly. Let $W_{1}$ and $W_{2}$
be two such Wightman functions. Then, equation (73) implies
$W_{1}-\bar{W}_{1}=W_{2}-\bar{W}_{2}$. Squaring both sides gives
$(W_{1}-\bar{W}_{1})^{2}=(W_{2}-\bar{W}_{2})^{2}$, which implies
$(W_{1}+\bar{W}_{1})^{2}=(W_{2}+\bar{W}_{2})^{2}$ by (75). But by (72) we can
take the unique square root of both sides to get
$W_{1}+\bar{W}_{1}=W_{2}+\bar{W}_{2}$. Together with (73) again this proves
that $W_{1}=W_{2}$.

Along with the Pauli-Jordan and Wightman functions, we can also define the
retarded and advanced Green’s functions of the theory:

$G_{R}(x,y):=\chi(y\prec x)\Delta(x,y)$ (76) $G_{A}(x,y):=-\chi(x\prec
y)\Delta(x,y),$ (77)

$\chi(s)$ being 1 if its argument is logically true and 0 otherwise.
$\Delta(x,y)$ can then be recovered by:

$\Delta(x,y)=G_{R}(x,y)-G_{A}(x,y)$ (78)

by using the property that $\chi(y\prec x)+\chi(x\prec y)=1$ on the domain
over which the commutator (57) is non-vanishing. Applying (28) to (57) implies
that:

$\left(\Box+m^{2}+\xi R\right)\Delta(x,y)=0$ (79)

in both arguments, and the retarded and advanced propagators obey the Green’s
function equations:

$\left(\Box+m^{2}+\xi
R\right)G_{R,A}(x,y)=-\frac{\delta^{(n)}(x-y)}{\sqrt{-g}}$ (80)

in both arguments as well. Note that we could have begun with (80) and run the
argument in reverse in order to construct $i\Delta(x,y)$ through (78). This
actually furnishes another way to find the Pauli-Jordan kernel, and has been
put to good effect to find the SJ state on the causal diamond in [43].

A final Green’s function, which typically finds employment in curved spacetime
is Hadamard’s elementary function:

$\displaystyle G^{(1)}(x,y)$
$\displaystyle:=\left<0\right|\\{\phi(x),\phi(y)\\}\left|0\right>$ (81)
$\displaystyle=W(x,y)+\bar{W}(x,y)$
$\displaystyle=\sum_{i}\left(u_{i}(x)\bar{u}_{i}(y)+\bar{u}_{i}(x)u_{i}(y)\right).$

Since the Pauli-Jordan function is state-independent, the Hadamard function
must carry all of the state-dependence that the Wightman function does, as can
be seen by the relation:

$W(x,y)=\frac{i}{2}\Delta(x,y)+\frac{1}{2}G^{(1)}(x,y).$ (82)

It should be noted also that although these Green’s functions have been
discussed within the context of the SJ formalism, (76)-(82) are valid outside
the formalism as well.

###  5 The Bogolyubov Method

In principle, we are now equipped to find the SJ state on any spacetime with
which it is compatible888The spacetime will need to be globally hyperbolic and
bounded. Global hyperbolocity has been addressed in section 2.2. The
boundedness requirement arises from the fact that $i\Delta$ is an integral
operator and is hence highly non-local and potentially divergent. In order to
find SJ modes on unbounded spacetimes, integral cutoffs can typically be
introduced and then their limits taken to infinity, although there are
exceptions where this procedure is ill-defined and the two point function
shows sensitivity to the limiting method [44]., be it flat, curved, or
topologically unconventional. In practice however, the geometry of the
spacetime may be such that implementing either the mode sum in (56) or finding
the retarded and advanced propagators through (80) is prohibitively difficult.
In cases such as these it would be advantageous to have an alternate method to
find the Sorkin-Johnston modes. Happily, the SJ modes, just like any other
orthonormal Klein-Gordon solution set, are also related to each other solution
set by a Bogolyubov transformation, and following [38] [44], we can exploit
this fact to find them.

Let $\\{u_{a},\bar{u}_{a}\\}$ be the desired mode set in which $i\Delta$ is
diagonal and $\\{v_{i},\bar{v}_{i}\\}$ some other arbitrary set, which solves
(28) and satisfies (32)-(34). Then,

$\displaystyle(i\Delta u_{a})(x)$ $\displaystyle=\int
d^{n}y\sqrt{-g(y)}\sum_{i}\left(v_{i}(x)\bar{v}_{i}(y)-\bar{v}_{i}(x)v_{i}(y)\right)u_{a}(y)$
(83)
$\displaystyle=\sum_{i}\left(\left<v_{i},u_{a}\right>v_{i}(x)-\left<\bar{v}_{i},u_{a}\right>\bar{v}_{i}(x)\right)$
$\displaystyle=\lambda_{a}u_{a}(x)$

In order to bring this into the form (42), make the definitions
$A_{ai}:=\frac{\left<v_{i},u_{a}\right>}{\lambda_{a}}$ and
$B_{ai}:=-\frac{\left<\bar{v}_{i},u_{a}\right>}{\lambda_{a}}$. Then,

$u_{a}(x)=\sum_{i}\left(A_{ai}v_{i}(x)+B_{ai}\bar{v}_{i}(x)\right).$ (84)

Taking the $L^{2}(M)$ inner product of (84), once with $v_{j}(x)$ and once
with $\bar{v}_{j}(x)$ yields the two relations for the Bogolyubov
coefficients:

$A_{aj}=\frac{1}{\lambda_{a}}\sum_{i}\left(A_{ai}\left<v_{j},v_{i}\right>+B_{ai}\left<v_{j},\bar{v}_{i}\right>\right)$
(85)
$B_{aj}=-\frac{1}{\lambda_{a}}\sum_{i}\left(A_{ai}\left<\bar{v}_{j},v_{i}\right>+B_{ai}\left<\bar{v}_{j},\bar{v}_{i}\right>\right).$
(86)

These, along with (50) and (51) place a set of four constraints on the
coefficients, which when satisfied, will determine the modes
$\\{u_{a},\bar{u}_{a}\\}$ in terms of $\\{v_{i},\bar{v}_{i}\\}$. In general,
this is no less trivial than the previously discussed solution methods for
$i\Delta$. Let us assume that whatever solutions we find for the
$\\{v_{i},\bar{v}_{i}\\}$ set are orthogonal under the $L^{2}(M)$ inner
product:

$\left<v_{i},v_{j}\right>=\left<v_{i},v_{i}\right>\delta_{ij}$ (87)
$\left<v_{i},\bar{v}_{j}\right>=\left<v_{i},\bar{v}_{-i}\right>\delta_{i,-j}.$
(88)

Like with $i\Delta$, the integrals involved in the $L^{2}(M)$ inner products
may not be finite. In this case, it will similarly be necessary to introduce
integral limit regulators and then take them to infinity later on. This
postulate greatly simplifies (85) and (86):

$\displaystyle A_{aj}$
$\displaystyle=\frac{1}{\lambda_{a}}\sum_{i}\left(A_{ai}\left<v_{j},v_{j}\right>\delta_{ji}+B_{ai}\left<v_{j},\bar{v}_{-j}\right>\delta_{j,-i}\right)$
(89)
$\displaystyle=\frac{1}{\lambda_{a}}\left(A_{aj}\left<v_{j},v_{j}\right>+B_{a-j}\left<v_{j},\bar{v}_{-j}\right>\right)$
$\displaystyle B_{aj}$
$\displaystyle=-\frac{1}{\lambda_{a}}\sum_{i}\left(A_{ai}\left<\bar{v}_{j},v_{-j}\right>\delta_{j,-i}+B_{ai}\left<\bar{v}_{j},\bar{v}_{j}\right>\delta_{ji}\right)$
(90)
$\displaystyle=-\frac{1}{\lambda_{a}}\left(A_{a-j}\left<\bar{v}_{j},v_{-j}\right>+B_{aj}\left<\bar{v}_{j},\bar{v}_{j}\right>\right).$

Another simplification comes about, and the consistency of all four constraint
equations holds, if we write $A_{aj}=A_{a}\delta_{aj}$ and
$B_{aj}=B_{a}\delta_{a,-j}$. This says that whatever linear combination of
$\\{v_{i},\bar{v}_{i}\\}$ constitutes the $\\{u_{a},\bar{u}_{a}\\}$ modes, it
will only ever mix modes of the same frequency, up to a sign. The set of
equations, which now needs to be solved is:

$A_{a}=\frac{1}{\lambda_{a}}\left(A_{a}\left<v_{a},v_{a}\right>+B_{a}\left<v_{a},\bar{v}_{-a}\right>\right)$
(91)

$B_{a}=-\frac{1}{\lambda_{a}}\left(A_{a}\left<\bar{v}_{-a},v_{a}\right>+B_{a}\left<\bar{v}_{-a},\bar{v}_{-a}\right>\right)$
(92)

$\left|A_{a}\right|^{2}-\left|B_{a}\right|^{2}=\frac{1}{\lambda_{a}}$ (93)

$A_{a}B_{-a}-B_{a}A_{-a}=0.$ (94)

The modification to (50) in (93) is so that the transformation to the SJ
modes, rather than the Pauli-Jordan eigenfunctions, will be unitary. By
rearranging and dividing equations (91) and (92) and noting that
$\left<\bar{v}_{-a},\bar{v}_{-a}\right>=\left<v_{-a},v_{-a}\right>$, one finds
a quadratic equation whose solution renders the set of eigenvalues:

$\lambda_{a}=\frac{1}{2}\left[\left<v_{a},v_{a}\right>-\left<v_{-a},v_{-a}\right>\pm\sqrt{\left(\left<v_{a},v_{a}\right>+\left<v_{-a},v_{-a}\right>\right)^{2}-4\left|\left<v_{a},\bar{v}_{-a}\right>\right|^{2}}\,\right].$
(95)

With a view towards the Lorentzian cone under consideration999See equation
(11). in Chapter 3, we will take
$\left<v_{a},v_{a}\right>=\left<v_{-a},v_{-a}\right>$ so that:

$\lambda_{a}=\lambda_{-a}=\pm\sqrt{\left|\left<v_{a},v_{a}\right>\right|^{2}-\left|\left<v_{a},\bar{v}_{-a}\right>\right|^{2}},$
(96)

where now the signs in the eigenvalue solutions just duplicate the parity with
which they appear in the solution set. In the expressions below, we take them
to be positive for convenience.

Under these conditions, we can now finally state what the Bogolyubov
coefficients are [38]:

$A_{a}=\left[\frac{\lambda_{a}+\left<v_{a},v_{a}\right>}{2\lambda_{a}^{2}}\right]^{\frac{1}{2}}$
(97)
$B_{a}=-\left[\frac{-\lambda_{a}+\left<v_{a},v_{a}\right>}{2\lambda_{a}^{2}}\right]^{\frac{1}{2}}e^{-iArg\left(\left<v_{a},\bar{v}_{-a}\right>\right)}.$
(98)

They can be re-written slightly as

$A_{a}=\frac{1}{\sqrt{2\lambda_{a}}}\left[1+\frac{1}{\sqrt{1-\frac{|\left<v_{a},\bar{v}_{-a}\right>|^{2}}{\left|\left<v_{a},v_{a}\right>\right|^{2}}}}\right]^{\frac{1}{2}}$
(99)

and

$B_{a}=-\frac{1}{\sqrt{2\lambda_{a}}}\left[-1+\frac{1}{\sqrt{1-\frac{|\left<v_{a},\bar{v}_{-a}\right>|^{2}}{\left|\left<v_{a},v_{a}\right>\right|^{2}}}}\right]^{\frac{1}{2}}e^{-iArg\left(\left<v_{a},\bar{v}_{-a}\right>\right)},$
(100)

with the Sorkin-Johnston modes now given by:

$u_{a}^{SJ}=\sqrt{\lambda_{a}}\left(A_{a}v_{a}+B_{a}\bar{v}_{-a}\right).$
(101)

The validity of these solutions can be checked by substituting (97) and (98)
into (91)-(94). Notice that this method obviates the need to find $i\Delta$
even though the SJ prescription relies principally upon it in its definition.
This will be particularly important for the construction of the SJ state on
the Lorentzian cone in Chapter 3.

###  6 The Discrete Sorkin-Johnston State

The continuum framework is now complete. However, it would be remiss not to
discuss how the Sorkin-Johnston formalism is applied to the discrete setting
since the origination of the concept was motivated thereby. Here we give up
the relative generality of the preceding sections and confine our attention to
a causal set $\left(C,\prec\right)$ obtained by a $p$ element sprinkling into
a $1+1$ dimensional spacetime, $\left(M,g_{\mu\nu}\right)$. The discrete
retarded propagator has been found in this instance [45]:

$R:=\frac{1}{2}C\left(\mathbb{I}+\frac{m^{2}}{2\rho}C\right)^{-1},$ (102)

where $C$ is the causal matrix defined in (54), $m$ is the mass parameter of
the field, $\rho$ is the sprinkling density, and $\mathbb{I}$ is the $p\times
p$ identity matrix. The Pauli-Jordan function is found by analogy to the
continuum case where $G_{A}(x,y)=G_{R}(y,x),$

$\Delta:=R-R^{T},$ (103)

and $R^{T}$ denotes transposition. As before, $i\Delta$ is Hermitian,
$(i\Delta)^{\dagger}=(i\Delta)$ and skew-symmetric,
$(i\Delta)_{ji}=-(i\Delta)_{ij}$. Hence there is a set of $p$ dimensional
eigenvectors now $\\{u_{a},\bar{u}_{a}\\}$ such that:

$(i\Delta)u_{a}=\lambda_{a}u_{a}$ (104)
$(i\Delta)\bar{u}_{a}=-\lambda_{a}\bar{u}_{a},$ (105)

where for example (104) explicitly reads:

$\sum_{j}(i\Delta)_{ij}(u_{a})_{j}=\lambda_{a}(u_{a})_{i}.$ (106)

Spectral decomposition proceeds similarly,

$i\Delta=\sum_{a}\lambda_{a}\left(u_{a}u_{a}^{\dagger}-\bar{u}_{a}\bar{u}_{a}^{\dagger}\right),$
(107)

and the Wightman function is given by:

$W^{SJ}:=Pos[i\Delta]=\sum_{a}\lambda_{a}u_{a}u_{a}^{\dagger}.$ (108)

The discrete Wightman function too will satisfy positivity, the discrete field
equation, and orthogonal support.

##  Chapter 2 Quantum Field Theory on the Lorentzian Cone

This chapter concerns the explicit construction of the massless scalar Sorkin-
Johnston state on the Lorentzian cone. This geometry can be viewed simply as a
toy cosmological model. However, similar structures do arise by way of
embedding in models of cosmic string inflation [46]. It is therefore important
to analyze the effect of such topological defects on extant fields. By
implementing a series of coordinate transformations, one is able to able to
“unfurl” the cone into a flat spacetime, which can alternately be viewed as
(i.) a subspace of the Milne universe with the spacelike coordinate
periodically identified [16], (ii.) an analytic extension of Rindler space
into the forward lightcone of the origin, again with the spacelike coordinate
periodically identified [47], or (iii.) a subspace of Misner space restricted
solely to within the forward lightcone of the origin, the periodicity of which
is inbuilt to its construction [48]. Mathematically, Misner spacetime’s cover
is the quotient space of Minkowski spacetime under boost isometries with fixed
rapidity parameter [49]. The full covering space suffers from pathologies such
as chronology horizons and closed timelike curves, and it is generally
acknowledged that existence of closed timelike curves leads to
unrenormalizable divergences in the stress-energy tensor of quantum fields at
the chronology horizons [50]. These divergences are thought to enforce the
chronology protection conjecture by causing the spacetime to be unstable. By
requiring that our subspace be within the forward lightcone of the origin, we
avoid these pathologies since the closed curves in the spacetime can only be
spacelike. It is not obvious then that this region of Misner space should be
unstable as the others are. This is what we aim to determine.

###  1 The Sorkin-Johnston State on the Lorentzian Cone

We begin with the cosmological metric on the cone

$ds^{2}=dt^{2}-(at)^{2}dx^{2},$ (1)

which describes a $1+1$ dimensional cosmology with $S^{0}$ creation event at
$t=0$ and whose $S^{1}$ spatial sections grow in time with constant rate, $a$,
and have periodicity under the identification $x\sim x+L$. Under the
coordinate transformation $t=a^{-1}e^{a\eta}$ with $t\geq 0$ (and hence
$-\infty<\eta<\infty$), the metric takes on its form in conformal coordinates:

$ds^{2}=e^{2a\eta}(d\eta^{2}-dx^{2}).$ (2)

That this spacetime is diffeomorphic to the restriction of $1+1$ flat
Minkowski space to within the forward lightcone of the origin is illuminated
by a third transformation:

$y^{0}=a^{-1}e^{a\eta}\cosh(ax)$ (3) $y^{1}=a^{-1}e^{a\eta}\sinh(ax),$ (4)

with $0<y^{0}<\infty$, $-\infty<y^{1}<\infty$, and $|y^{1}|<y^{0}.$

For the purposes of constructing the SJ state and subsequently finding its
stress-energy tensor VEV, we’ll use the conformal version of the metric. In
these coordinates the massless, minimally coupled $(\xi=0)$ scalar field
equation is:

$\Box\phi=e^{-2ax}\left(\frac{\partial^{2}}{\partial\eta^{2}}-\frac{\partial^{2}}{\partial
x^{2}}\right)\phi=0,$ (5)

which being conformally flat,

$e^{2ax}\Box\phi=\left(\frac{\partial^{2}}{\partial\eta^{2}}-\frac{\partial^{2}}{\partial
x^{2}}\right)\phi=0,$ (6)

means that we can select the plane waves,

$u_{k}(\eta,x)=\frac{1}{\sqrt{2|k|L}}e^{-i(|k|\eta-kx)}$ (7)

with periodic boundary condition, $u_{k}(\eta,x+L)=u_{k}(\eta,x)$, enforced by

$k=k_{n}=\frac{2\pi n}{L},$ (8)

as the properly Klein-Gordon normalized, positive frequency mode set with
which to construct the SJ state. Henceforth, we will often drop the subscript
on $k_{n}$ under the understanding that it is restricted to the discrete set
above. Indeed, in addition to

$(u_{k},u_{k^{\prime}})_{KG}=i\delta_{kk^{\prime}},$ (9)

the set $\\{u_{k},\bar{u}_{k}\\}$ also satisfies the $L^{2}(M)$ inner
products, defined in (61):

$\left<u_{k},u_{k^{\prime}}\right>=\left<u_{k},u_{k}\right>\delta_{kk^{\prime}}$
(10)
$\left<u_{k},\bar{u}_{k^{\prime}}\right>=\left<u_{k},\bar{u}_{-k}\right>\delta_{k-k^{\prime}},$
(11)

where:

$\left<u_{k},u_{k}\right>=\frac{1}{2|k|L}\int_{-\frac{L}{2}}^{\frac{L}{2}}dx\int_{-\infty}^{\Lambda}d\eta\,e^{2a\eta}=\frac{e^{2a\Lambda}}{4|k|a}$
(12)
$\left<u_{k},\bar{u}_{-k}\right>=\frac{1}{2|k|L}\int_{-\frac{L}{2}}^{\frac{L}{2}}dx\int_{-\infty}^{\Lambda}d\eta\,e^{2(a+i|k|)\eta}=\frac{e^{2(a+i|k|)\Lambda}}{4|k|(a+i|k|)},$
(13)

$\Lambda$ being an integral cutoff in the $\eta$ coordinate, which can be kept
finite or, depending upon the reader’s proclivity for ice cream, taken to
infinity at the conclusion of the calculations we will perform in this
section. As mentioned previously, this limiting procedure is not always well
defined or unambiguously unique, but for our purposes, it will do. In
addition, the eigenvalues are computed using (96):

$\lambda_{k}=\frac{e^{2a\Lambda}}{4a\sqrt{a^{2}+\left|k\right|^{2}}}.$ (14)

These quantities in hand, we can proceed to construct the SJ modes via the
Bogolyubov method introduced in Section 2.5:

$u_{k}^{SJ}=\sqrt{\lambda_{k}}\left(A_{k}u_{k}+B_{k}\bar{u}_{-k}\right),$ (15)

where

$A_{k}=\frac{1}{\sqrt{2\lambda_{k}}}\left[1+\frac{1}{\sqrt{1-\frac{|\left<u_{k},\bar{u}_{-k}\right>|^{2}}{\left|\left<u_{k},u_{k}\right>\right|^{2}}}}\right]^{\frac{1}{2}}$
(16)

and

$B_{k}=-\frac{1}{\sqrt{2\lambda_{k}}}\left[-1+\frac{1}{\sqrt{1-\frac{|\left<u_{k},\bar{u}_{-k}\right>|^{2}}{\left|\left<u_{k},u_{k}\right>\right|^{2}}}}\right]^{\frac{1}{2}}e^{-iArg\left(\left<u_{k},\bar{u}_{-k}\right>\right)}.$
(17)

Noting that $\left<u_{k},u_{k}\right>=\left<u_{-k},u_{-k}\right>$ and using
our $L^{2}(M)$ expressions from above, we find these coefficients to be:

$A_{k}=\frac{1}{\sqrt{2\lambda_{k}}}\left[1+\frac{\sqrt{a^{2}+|k|^{2}}}{|k|}\right]^{\frac{1}{2}},$
(18)

and

$B_{k}=-\frac{1}{\sqrt{2\lambda_{k}}}\left[-1+\frac{\sqrt{a^{2}+|k|^{2}}}{|k|}\right]^{\frac{1}{2}}\left(\frac{a+i|k|}{\sqrt{a^{2}+|k|^{2}}}\right)e^{-2i|k|\Lambda}.$
(19)

Now, in order to evaluate the stress energy tensor, we expand our field in
terms of these modes:

$\phi(\eta,x)=\sum_{k=-\infty}^{\infty}(a_{k}u_{k}^{SJ}+a_{k}^{\dagger}\bar{u}_{k}^{SJ}),$
(20) $a_{k}\left|SJ\right>=0.$ (21)

The diagonal terms in the stress-energy tensor are:

$T_{\eta\eta}=T_{xx}=\frac{1}{2}\left[\left(\frac{\partial\phi}{\partial\eta}\right)^{2}+\left(\frac{\partial\phi}{\partial
x}\right)^{2}\right],$ (22)

while the off diagonal terms are zero. In terms of the SJ modes, the VEV

$\left<SJ\right|T_{\eta\eta}\left|SJ\right>=\frac{1}{2}\left[\left<SJ\right|\phi,_{\eta}\phi,_{\eta}\left|SJ\right>+\left<SJ\right|\phi,_{x}\phi,_{x}\left|SJ\right>\right],$
(23)

reduces to:

$\left<SJ\right|T_{\eta\eta}\left|SJ\right>=\frac{1}{2}\sum_{k=-\infty}^{\infty}\left[(\partial_{\eta}u_{k}^{SJ})(\partial_{\eta}\bar{u}_{k}^{SJ})+(\partial_{x}u_{k}^{SJ})(\partial_{x}\bar{u}_{k}^{SJ})\right].$
(24)

The intermediate result is fairly simple:

$\left<SJ\right|T_{\eta\eta}\left|SJ\right>=\frac{\pi\beta}{L^{2}}+\frac{2\pi}{L^{2}}\sum_{n=1}^{\infty}\sqrt{\beta^{2}+n^{2}},$
(25)

where $\beta=\frac{La}{2\pi}$. Notice that the sum is now on the natural
numbers $\mathbb{N}$. Also note in particular that our integral regulator
$\Lambda$ has dropped out of the calculation. Now, this is clearly a divergent
sum as it stands, and moreover, it is not solvable in closed analytic form.
However, we can follow a regularization procedure analogous to the one used
for an infinite cylinder by introducing a regulator on each term and seeing
where it takes us.

$\left(\left<SJ\right|T_{\eta\eta}\left|SJ\right>-\frac{a}{2L}\right)\rightarrow\frac{2\pi}{L^{2}}\sum_{n=1}^{\infty}\sqrt{\beta^{2}+n^{2}}x^{n},$
(26)

where $0<x=e^{-\epsilon}<1$, $\epsilon=\frac{\delta 2\pi}{L}$, and the
regulation parameter, $\delta$, will be taken to zero at the conclusion of the
calculation. As $n$ becomes large, terms are exponentially suppressed. This
motivates postulating that $\beta$ (or equivalently L) is large, and expanding
the square root in a power series about $\beta=\infty$, since the presence of
the regulator means any terms where $n$ is nearly as large as $\beta$ will be
exponentially suppressed and thus contribute little to the sum. To wit:

$\sqrt{\beta^{2}+n^{2}}=\sum_{m=0}^{\infty}\frac{C_{m}n^{2m}}{\beta^{2m-1}}.$
(27)

The $C_{m}$ are just numerical constants. For example, the first few terms
are:

$=\beta+\frac{n^{2}}{2\beta}-\frac{n^{4}}{8\beta^{3}}+\frac{n^{6}}{16\beta^{5}}+...$
(28)

In fact, the expansion above converges for any $\beta$ away from the origin.
So, as long as the conformal circumference of our cone remains non-zero, we
shall feel justified in its validity.

What is now proposed is to calculate $\left<T_{\eta\eta}\right>^{SJ}$ term for
term in $m$ and to isolate and subtract the divergent part attributable to the
SJ state in the Milne forward wedge, namely
$\left<T_{\eta\eta}\right>^{SJ}_{L\rightarrow\infty}$. We will be left with
the desired vacuum energy density of the cone:

$\left<T_{\eta\eta}\right>^{SJ}_{Reg}=\left<T_{\eta\eta}\right>^{SJ}-\left<T_{\eta\eta}\right>^{SJ}_{L\rightarrow\infty}.$
(29)

Consider then, the contribution from the $m^{th}$ term in the sum to
$\left<T_{\eta\eta}\right>^{SJ}$:

$\frac{2\pi
C_{m}}{L^{2}\beta^{2m-1}}\sum_{n=1}^{\infty}n^{2m}x^{n}=\frac{(2\pi)^{2m}C_{m}}{L^{2m+1}a^{2m-1}}\sum_{n=1}^{\infty}n^{2m}x^{m}$
(30) $=\frac{(2\pi)^{2m}C_{m}}{L^{2m+1}a^{2m-1}}Li_{-2m}(x)$ (31)
$=\frac{(2\pi)^{2m}C_{m}}{L^{2m+1}a^{2m-1}}Li_{-2m}(e^{-\epsilon}),$ (32)

where $Li_{-2m}(x)$ is a polylogarithm. We can then expand this about
$\epsilon=0$ since we will eventually be taking our regulator to zero [51]:

$=\frac{(2\pi)^{2m}C_{m}}{L^{2m+1}a^{2m-1}}\left(\frac{\Gamma(1+2m)}{\epsilon^{2m+1}}+\sum_{l=0}^{\infty}\frac{\zeta(-(2m+l))}{l!}(-\epsilon)^{l}\right),$
(33)

where $\Gamma(1+2m)$ is the usual gamma function and $\zeta(-(2m+l))$ is the
Riemann zeta function. Showing explicitly the first few terms:

$=\frac{(2\pi)^{2m}C_{m}}{L^{2m+1}a^{2m-1}}\left(\frac{L^{2m+1}\Gamma(1+2m)}{(\delta
2\pi)^{2m+1}}+\zeta(-2m)-\frac{\zeta(-(2m+1))(\delta
2\pi)}{L}+O\left(\frac{\delta^{2}}{L^{2}}\right)\right)$ (34)

$=\frac{(2\pi)^{2m}C_{m}}{a^{2m-1}}\left(\frac{\Gamma(1+2m)}{(\delta
2\pi)^{2m+1}}+\frac{\zeta(-2m)}{L^{2m+1}}-\frac{\zeta(-(2m+1))(\delta
2\pi)}{L^{2m+2}}+O\left(\frac{\delta^{2}}{L^{2m+3}}\right)\right).$ (35)

Here, we can see that the only divergent term in the limit $\delta\rightarrow
0$ is the first one. Since this evidently corresponds to the Milne vacuum
energy, we remove this term. Now we can safely take the limit and we are left
with:

$=\frac{(2\pi)^{2m}C_{m}}{a^{2m-1}}\left(\frac{\zeta(-2m)}{L^{2m+1}}\right).$
(36)

The reader might now be ecstatic upon noticing that the Riemann zeta function
vanishes for any negative, even integer. Hence, the only contribution to the
renormalized vacuum energy comes from the $m=0$ term, which with $C_{0}=1$ and
$\zeta(0)=-\frac{1}{2}$ takes the neat value of:

$\left<T_{\eta\eta}\right>^{SJ}_{Reg}=\frac{a}{2L}-\frac{a}{2L}=0.$ (37)

###  2 Another Route To Stress-Energy

One criticism, which has been leveled against the use of the Sorkin-Johnston
state is that it fails to be Hadamard in a general spacetime [52]. A full
discussion of the merits or even necessity of using states, which satisfy the
Hadamard condition is outside the scope of this dissertation, but a typical
counterargument runs as follows. The Hadamard condition requires that a
state’s symmetric two point function ought to have the same singularity
structure as the corresponding two point function in Minkowski space. The
postulate is motivated by the equivalence principle in that the coincidence
limit of the two points under consideration should render the propagator as it
would look in a locally inertial frame. However, considering that nearly all
of quantum gravitational research deals with how physicists need to alter the
fundamental kinematical structure underlying spacetime away from the continuum
that begets its singularities, the thought that the Hadamard condition will
ultimately have any deep physical meaning is tenuous.

Nevertheless, in the case that the maneuvers in the preceding section will
strike anyone as doubtful, it will behoove us to consider alternate means of
obtaining the renormalized stress-energy tensor. Topically, this will require
the use of the Hadamard ansatz and its related regularization scheme. As such,
we will proceed independently of the Sorkin-Johnston prescription and
calculate directly with the conformal modes
$\\{u^{C}_{k_{n}},\bar{u}^{C}_{k_{n}}\\}$ in equations (7)111The placement of
the $C$ superscript is new, but only to emphasize that these are different
than the SJ modes. and (8) rather than
$\\{u^{SJ}_{k_{n}},\bar{u}^{SJ}_{k_{n}}\\}$ in order to calculate, via
Hadamard regularization, the renormalized stress-energy tensor in the
conformal vacuum. Begin by restating the definition of Hadamard’s elementary
function in terms of Wightman functions:

$G^{(1)}(X,Y)=W(X,Y)+\bar{W}(X,Y),$ (38)

where $X$ and $Y$ are used to indicate a general coordinate system. As a mode
sum in conformal coordinates, the Wightman function reads,

$\displaystyle W^{C}(\eta,x;\eta^{\prime},x^{\prime})$
$\displaystyle=\sum_{n=-\infty}^{\infty}u^{C}_{k_{n}}(\eta,x)\bar{u}^{C}_{k_{n}}(\eta^{\prime},x^{\prime})$
(39)
$\displaystyle=\sum_{n=-\infty}^{\infty}\frac{1}{2|k_{n}|L}e^{-i|k_{n}|\eta}e^{ik_{n}x}e^{i|k_{n}|\eta^{\prime}}e^{-ik_{n}x^{\prime}}$
$\displaystyle=\sum_{n=-\infty}^{\infty}\frac{1}{2|k_{n}|L}e^{-i|k_{n}|(\eta-\eta^{\prime})}e^{ik_{n}(x-x^{\prime})}$
$\displaystyle=\frac{1}{2L}\frac{L}{2\pi}\sum_{n=-\infty}^{\infty}\frac{1}{|n|}e^{-i|k_{n}|\Delta\eta}e^{ik_{n}\Delta
x}$
$\displaystyle=\frac{1}{4\pi}\left(\sum_{n=1}^{\infty}\frac{\alpha^{n}}{n}+\sum_{n=1}^{\infty}\frac{\beta^{n}}{n}+\mathcal{N}_{0}\right).$

In the step moving to the last line, we have split the sum into its negative
and positive integer domains and a divergent, but functionally constant
$(n=0)$ term, $\mathcal{N}_{0}$. The sum on the negative integers was then
reoriented by accounting for signs in the complex exponentials, and the
definitions,

$\alpha:=e^{-i\frac{2\pi}{L}(\Delta\eta+\Delta x)}$ (40)
$\beta:=e^{-i\frac{2\pi}{L}(\Delta\eta-\Delta x)},$ (41)

were made. The sums for the Wightman function using these definitions are not
strictly convergent since the complex exponentials do not lie entirely within
the radius of convergence for the Mercator expansion of the complex logarithm,
namely:

$-\ln(1-z)=\sum_{n=1}^{\infty}\frac{z^{n}}{n},$ (42)

which is convergent so long as $|z|\leq 1$ and $z\neq 1$. In order to get the
sums in (39) to converge, insert a regulator $e^{-\epsilon}$ next to both
$\alpha$ and $\beta$ in each term and then take $\epsilon\in\mathbb{R}$ to
zero at the calculation’s conclusion. With this mechanism, both sums converge
to the principle branch of the complex logarithm:

$W^{C}(\eta,x;\eta^{\prime},x^{\prime})=\frac{1}{4\pi}\left(-\ln(1-\alpha
e^{-\epsilon})-\ln(1-\beta e^{-\epsilon})+\mathcal{N}_{0}\right).$ (43)

The Hadamard elementary function is then

$\displaystyle G^{(1)C}(\eta,x;\eta^{\prime},x^{\prime})$
$\displaystyle=-\frac{1}{4\pi}\left(\ln\left[(1-\alpha e^{-\epsilon})(1-\beta
e^{-\epsilon})(1-\bar{\alpha}e^{-\epsilon})(1-\bar{\beta}e^{-\epsilon})\right]-2\mathcal{N}_{0}\right)$
(44)
$\displaystyle=-\frac{1}{4\pi}\left(\ln\left[(2-\Re[\alpha]e^{-\epsilon}-O(\epsilon))(2-\Re[\beta]e^{-\epsilon}-O(\epsilon))\right]\right).$

At this point we will consider it safe to take $\epsilon$ to zero since the
argument of the logarithm is entirely real. After a little bit of
trigonometric manipulation, the result is:

$G^{(1)C}(\eta,x;\eta^{\prime},x^{\prime})=-\frac{1}{4\pi}\left(\ln\left[16\sin^{2}(\frac{\pi}{L}(\Delta\eta+\Delta
x))\sin^{2}(\frac{\pi}{L}(\Delta\eta-\Delta
x))\right]-2\mathcal{N}_{0}\right).$ (45)

Note that this calculation has been nearly identical to the treatment given to
the infinite cylinder in section 4.2 of [16]. Elements of that treatment will
find further use in the remainder of the calculation.

Now, a function is Hadamard if in the limit $X\rightarrow Y$, it can be put
into the form of the Hadamard ansatz. In two dimensions, this is [53]:

$G^{(1)}(X,Y)=v(X,Y)\ln\sigma(X,Y)+w(X,Y),$ (46)

where $\sigma(X,Y)$ is one half the square of the geodesic distance between
$X$ and $Y$. Here also, $v(X,Y)$ and $w(X,Y)$ are nonsingular, smooth
functions. If the Hadamard function is initially unknown, it’s coincidence
form can ostensibly be found by expanding $v(X,Y)$ and $w(X,Y)$ as:

$v(X,Y)=\sum_{n=0}^{\infty}v_{n}(X,Y)\sigma^{n}$ (47)
$w(X,Y)=\sum_{n=0}^{\infty}w_{n}(X,Y)\sigma^{n},$ (48)

operating on the Hadamard ansatz with the Klein-Gordon operator, and solving
for the expansion coefficients recursively using the fact that $G^{(1)}$
solves the field equation. Since we have an explicit expression for $G^{(1)C}$
in conformal coordinates, we simply take the limit
$(\eta,x)\rightarrow(\eta^{\prime},x^{\prime}),$ or equivalently,
$\left(\Delta\eta,\Delta x\right)\rightarrow 0.$ We can then expand the sin
functions in the logarithm about a zero argument. This gives:

$\displaystyle G^{(1)C}(\eta,x;\eta^{\prime},x^{\prime})$
$\displaystyle\cong-\frac{1}{4\pi}\left(\ln\left[16\left(\frac{\pi}{L}\right)^{4}(\Delta\eta+\Delta
x)^{2}(\Delta\eta-\Delta x)^{2}\right]-2\mathcal{N}_{0}\right)$ (49)
$\displaystyle=-\frac{1}{4\pi}\left(2\ln\left[\frac{\left(\Delta\eta^{2}-\Delta
x^{2}\right)}{L^{2}}\right]+4\ln(2\pi)-2\mathcal{N}_{0}\right).$

This is beginning to look familiar. Note that there is an $L^{2}$ in the
denominator of the first logarithm in order to keep the argument
dimensionless. For any reasonable yarmulke or cone, $L$ will be nonzero and as
such will not ruin the singularity structure. in order to get this into its
correct form, add and subtract $\frac{1}{2\pi}\ln[e^{2a\eta^{\prime}}]$ to
(49). It will then read:

$G^{(1)}(\eta,x;\eta^{\prime},x^{\prime})=-\frac{1}{4\pi}\left(2\ln\left[\frac{e^{2a\eta^{\prime}}\left(\Delta\eta^{2}-\Delta
x^{2}\right)}{L^{2}}\right]+4\ln(2\pi)-2\mathcal{N}_{0}-2\ln[e^{2a\eta^{\prime}}]\right).$
(50)

Finally, notice that in the true coincidence limit, the geodesic distance
squared is just the familiar expression for the line element in our spacetime:

$\displaystyle\lim_{(\Delta\eta,\Delta x)\rightarrow
0}\sigma^{2}(\Delta\eta,\Delta x)$ $\displaystyle=ds^{2}$ (51)
$\displaystyle=e^{2a\eta^{\prime}}(d\eta^{\prime 2}-dx^{\prime 2})$
$\displaystyle=e^{2a\eta^{\prime}}(\Delta\eta^{2}-\Delta x^{2}).$

To lowest order in $n$, one can then identify (50) with (46) and the smooth
functions as:

$v_{0}(\eta,x;\eta^{\prime},x^{\prime})=-\frac{1}{\pi}$ (52)
$w_{0}(\eta,x;\eta^{\prime},x^{\prime})=-\frac{1}{4\pi}(4\ln(2\pi)-\mathcal{N}_{0}-2\ln[e^{2a\eta^{\prime}}]).$
(53)

Generally, the way Hadamard renormalization then proceeds is by
differentiating in the point splitting limit and removing the previously
logarithmically singular term in order to get the stress-energy tensor.

$\left<T_{\mu\nu}\right>_{Reg}=\frac{1}{2}\lim_{X\rightarrow
Y}\mathcal{D}_{\mu(X)\nu(Y)}G_{Reg}^{(1)}(X,Y),$ (54)

where $G_{Reg}^{(1)}(X,Y)=w(X,Y)$. The bilocal differential operator is given
in minimally coupled, massless case by [54]:

$\mathcal{D}_{\mu(\eta,x)\nu(\eta^{\prime},x^{\prime})}=\frac{1}{2}\\{g^{\mu^{\prime}}_{\mu}\nabla_{\mu^{\prime}}\nabla_{\nu}+g^{\nu^{\prime}}_{\nu}\nabla_{\mu}\nabla_{\nu^{\prime}}-g_{\mu\nu}g^{\beta^{\prime}}_{\beta}\nabla_{\beta^{\prime}}\nabla^{\beta}\\},$
(55)

where $g^{\beta^{\prime}}_{\beta}$ is the bivector of parallel transport,
which when $X\rightarrow Y$, limits as
$g^{\beta^{\prime}}_{\beta}\rightarrow\delta^{\beta^{\prime}}_{\beta}$. With
the understanding that $\mathcal{D}$ will be operating on the biscalar
Hadamard function, (55) reads:

$\mathcal{D}_{\mu\nu}=\frac{1}{2}\\{g^{\mu^{\prime}}_{\mu}\partial_{\mu^{\prime}}\partial_{\nu}+g^{\nu^{\prime}}_{\nu}\partial_{\mu}\partial_{\nu^{\prime}}-g_{\mu\nu}g^{\beta^{\prime}}_{\beta}\partial_{\beta^{\prime}}\partial^{\beta}\\},$
(56)

and the diagonal components are:

$\mathcal{D}^{C}_{\eta\eta}=\mathcal{D}^{C}_{xx}=\frac{1}{2}\\{g^{\eta^{\prime}}_{\eta}\partial_{\eta^{\prime}}\partial_{\eta}+g^{x^{\prime}}_{x}\partial_{x^{\prime}}\partial_{x}\\},$
(57)

with the off-diagonal components again zero. Applying this to (45) and
following [16] in that $\partial_{\mu}(\mathcal{N}_{0})=0$, we get:

$\mathcal{D}^{C}_{\mu\nu}G^{(1)C}=-\frac{\pi}{2L^{2}}\left(\csc^{2}\left(\frac{\pi}{L}(\Delta\eta+\Delta
x)\right)+\csc^{2}\left(\frac{\pi}{L}(\Delta\eta-\Delta
x)\right)\right)(g^{\eta^{\prime}}_{\eta}+g^{x^{\prime}}_{x}).$ (58)

Upon expanding both squared cosecant functions,

$\csc^{2}\left(\frac{\pi}{L}(\Delta\eta\pm\Delta
x)\right)=\frac{L^{2}}{(\pi(\Delta\eta\pm\Delta
x))^{2}}+\frac{1}{3}+O((\Delta\eta\pm\Delta x)^{2}),$ (59)

we observe that the divergent term in the stress-tensor corresponds to the
logarithmic term in the coincident Hadamard function. Once these are removed,
we can take the coincidence limit. Terms of all higher orders vanish, and we
are left with:

$\left<T_{\eta\eta}\right>^{C}_{Reg}=-\frac{\pi}{6L^{2}},$ (60)

which is a slightly different result than that calculated in the previous
section222Note that if we had used the pure conformal modes instead of the SJ
modes in the renormalization method of 3.1, the result would have been
identically $-\frac{\pi}{6L^{2}}$. This speaks to the robustness of the
different regularization methods on the same vacuum.. We turn to this
presently.

##  Chapter 3 Conclusion

###  1 Beyond the Zero

With two contending expressions now available for the vacuum energy of the
Lorentzian cone, a more general discussion of what might be expected for
conical topology change is in order. The first issue to address is the
discrepancy between (37) and (60). While the classical stress-energy tensor is
by construction a covariant quantity, the renormalized expectation value of
the quantal stress-energy tensor is generally not [55]. Moreover, it is not
always the case that different regularization schemes give the same result.
This, in fact, was part of the original impetus to introduce Hadamard
regularization and Wald’s axiomatic renormalization [56]. This was the
formalism most closely followed in section 3.2 and the resulting Casimir
energy is concordant with the idea that such a topology induces a non-zero
vacuum energy.

On the other hand, although Wald’s axioms and their corresponding uniqueness
theorem are motivated by physically sensible requirements, such as covariant
conservation of $\left<T_{\mu\nu}\right>_{Reg}$ and respect for causality, it
is not strictly necessary to take up all of these axioms in every instance.
For example, the second axiom typically reads [56]:

“In the case of Minkowski spacetime, $T_{\mu\nu}$ is given by normal ordering,
$T_{\mu\nu}=:T_{\mu\nu}:$.”

However, this axiom is unsuitable for use with topologies of the form
$\mathbb{R}\times S^{(n-1)}$ since Minkowski space does not have this
topology. For the case of the temporally infinite cylinder, the substitute
prescription is to renormalize against the infinite term when the
circumference of the cylinder, $L$, is taken to infinity. This is ostensibly
the correct prescription for the cylinder, because it becomes effectively
$1+1$ dimensional Minkowski space when the limit is taken. But what then is to
be done when the limiting case of a topologically non-trivial spacetime is not
Minkowski space, such as for the cone or yarmulke?

Here, the limiting open spacetime is the $1+1$ Milne universe, which itself
has ambiguities as regards its vacua and thus its vacuum energy [16]. It has
both an adiabatic vacuum, which one can write as a linear superposition of
Minkowski plane waves, and in which no particles are detected. The other
vacuum is the conformal vacuum wherein the energy density has the functional
dependence $\left<T_{\eta\eta}\right>^{Mil}_{Reg}=-\frac{1}{24\pi t^{2}}$
where $t$ is the time coordinate introduced in (1). The question as to which
vacuum we are limiting to is not straightforward. The optimistic resolution
would be to say that the null vacuum energy found in the cone with the Sorkin-
Johnston modes (as a linear combination of conformal modes) somehow reflects
the adiabatic, particle-less vacuum of Milne, while the Casimir energy found
with the conformal modes corresponds to the conformal vacuum in Milne with the
singularity at creation being somehow stabilized by the topological structure
of the conical cobordism. However, these suggestions are by no means sure.

Perhaps the most striking aspect of these results is the distinct lack of any
geometrically singular expressions for the stress-energy expectation on the
cone. Although one might expect the cone to inherit the creation singularity
from the Milne universe, this at first glance does not seem to be the case.
Furthermore, there looks to be no pathological energy burst such as is
witnessed in the trousers. One however could reason that since the Milne
conformal vacuum, where there is no Casimir energy, should be the limiting
case of the Misner conformal vacuum, we should add
$\left<T_{\eta\eta}\right>^{Mil}_{Reg}=-\frac{1}{24\pi t^{2}}$ to (60). This
seems rather ad hoc, but plausible. Even still, this divergence decays to the
causal future of the singularity, making it of a distinctly different nature
than the trouser singularity. Conceptually, an observer at any point to the
causal future of the trouser crotch will be besieged by an infinite bath of
radiation, while an observer on the cone will witness either an ever cooling
universe111Of course semiclassically, the $\left<T_{\eta\eta}\right>\propto
t^{-2}$case would have a fairly significant back-reaction on the spacetime.
Moreover, at times close to $t\sim t_{p}$, the Planck time, the creation event
should actually be described by quantum gravity proper. This analysis assumes
some temporal cutoff at around $t_{p}$ then and that the deleterious effects
of a decaying energy burst will not be so severe on the $S^{1}$ topology as
would a $\delta^{2}$ type divergence., one with homogenous negative pressure,
or one devoid of any detectable matter at all. As a side note, only the first
two instances are consistent with the common appearance of a trace anomaly in
the stress-energy VEV of a conformally invariant field theory [56].

This brings us back to Sorkin’s conjecture. The causal continuity of the cone
and yarmulke can be seen by placing an arbitrary point in the unfurled
spacetime of either and considering the future and past lightcones emanating
therefrom. As the point is imagined to be pulled closer and closer to the
singularity, one lightcone diminishes smoothly to zero while the other
transitions smoothly to illuminate the entire yarmulke or cone. At no point
does the volume of the future or past lightcone change abruptly. In this
light222Pun intended., our results that the cone displays no irreparable
divergences may be viewed as some measure of corroboration for the Sorkin
conjecture. Put differently, there is another conjecture due to Sorkin and
Borde that in $n$ dimensions, only Morse metrics on topological cobordisms
with index $1$ or $n-1$ critical points are causally discontinuous [57]. Since
the yarmulke and cone have Morse indices $\lambda=2$ and $\lambda=0$
respectively with only one critical point each, we can rephrase our result.
The mild or non-singularity of $\phi$ on the cone is consistent with with the
fact that the Morse index of its critical point $t=0$ is neither $1$ nor
$n-1=2-1=1$, which correspond to the trousers. This signifies roughly that
conical cobordisms at the quantum level are at least stable in terms of
intrinsic geometry in the semiclassical limit. It would be interesting in
future work to examine whether coupling to the Ricci scalar $(\xi\neq 0)$ or
allowing the field to be massive remains consistent with these observations.

One corollary of this analysis is that we can now revisit the question posed
at the outset of this dissertation. Notwithstanding the ill-definition of the
semiclassical vacuum, we have actually been able to address a notion of
nothingness one level deeper. While we initially described the cone as an
$S^{0}\rightarrow S^{1}$ cobordism, it can in fact be viewed as an
$\\{\,\\}\rightarrow S^{1}$ cobordism333Note that this does not cohere with
the definition of cobordism in terms of manifold topology, but we could make a
similar definition using causal sets. In addition, all $S^{n}$ are cobordant
in this sense with $\\{\,\\}$. [57]. Mathematically, the empty set is perhaps
the closest thing we can define that approaches the concept of nothing. But if
$S^{1}$ is cobordant with it, and the $S^{1}$ vacuum energy is the negative
Casimir energy found in section 3.2, then assuming that the scalar Hamiltonian
VEV in $\\{\,\\}$ is zero, the existence of such structures is actually
preferred and nothingness will be unstable. Then, we might rest easy at night
knowing that the once dark void is little more than a gently lilting sea of
cones and yarmulkes.

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