# Classical Simulations of Quantum Field Theory in Curved Spacetime I:
Fermionic Hawking-Hartle Vacua from a Staggered Lattice Scheme

Adam G. M. Lewis  Perimeter Institute for Theoretical Physics, 31 Caroline St.
N., Waterloo, Ontario, Canada, N2L 2Y5<EMAIL_ADDRESS>Guifré
Vidal  Perimeter Institute for Theoretical Physics, 31 Caroline St. N.,
Waterloo, Ontario, Canada, N2L 2Y5  X, The Moonshot Factory, Mountain View, CA
94043

###### Abstract

We numerically compute renormalized expectation values of quadratic operators
in a quantum field theory (QFT) of free Dirac fermions in curved two-
dimensional (Lorentzian) spacetime. First, we use a staggered-fermion
discretization to generate a sequence of lattice theories yielding the desired
QFT in the continuum limit. Numerically-computed lattice correlators are then
used to approximate, through extrapolation, those in the continuum. Finally,
we use so-called point-splitting regularization and Hadamard renormalization
to remove divergences, and thus obtain finite, renormalized expectation values
of quadratic operators in the continuum. As illustrative applications, we show
how to recover the Unruh effect in flat spacetime and how to compute
renormalized expectation values in the Hawking-Hartle vacuum of a
Schwarzschild black hole and in the Bunch-Davies vacuum of an expanding
universe described by de Sitter spacetime. Although here we address a non-
interacting QFT using free fermion techniques, the framework described in this
paper lays the groundwork for a series of subsequent studies involving
simulation of interacting QFTs in curved spacetime by tensor network
techniques.

##  1 Introduction

While by now it is generally expected that black holes will radiate and
evaporate [27, 52, 12] in the presence of a quantum field, the details remain
murky. The specific interplay between gravity and quantum fields during the
early Universe’s presumed inflationary [24, 48, 36, 2] phase is similarly
unclear. This is not (necessarily) because we lack a quantum theory of
gravity: one expects that at least some of these effects might already be
acceptably described in some sort of semiclassical regime. For example, one
might consider the semiclassical field equations

$G_{\mu\nu}=\frac{1}{8\pi}\langle T^{\mathrm{ren}}_{\mu\nu}\rangle$ (1)

that relate the classical Einstein tensor $G_{\mu\nu}$ to the expectation
value $\langle T^{\mathrm{ren}}_{\mu\nu}\rangle$ of a quantum field’s
suitably-renormalized stress-energy tensor $T_{\mu\nu}$. Given this regime
indeed exists, the real problem is that (1) is hard to solve. On the left hand
side, we have the fully non-linear Einstein equations, while on the right sits
an out-of-equilibrium, perhaps strongly-interacting, quantum field theory in
curved spacetime. Even taken separately, such problems are typically
intractable.

But much further progress can be made with either _numerically_. Since initial
breakthroughs in 2005, [46, 45] fully nonlinear simulations of general
relativity are now routine. The quantum field theory is a bit trickier, due to
the inability of classical hardware to represent highly entangled
wavefunctions. In two spacetime dimensions, however, a tensor network ansatz
known as the matrix product state [37, 58, 59, 40, 47, 54, 55, 17, 60, 42, 49,
50] can efficiently simulate low-energy states of local Hamiltonians, as well
as sufficiently short real-time evolutions of them.

This is the first in a series of studies attacking (1) by combining numerical
relativity with tensor network approaches. The first step is to write down the
Hamiltonian evolving the quantum field forward in some time coordinate. This
Hamiltonian is then mapped to the continuum limit of a sequence of lattice
Hamiltonians. On each lattice, interesting states such as the ground state and
other low energy states, as well as thermal states and their real-time
evolutions, will be simulated with matrix product state techniques. From that
sequence we extract $\langle T^{\mathrm{ren}}_{\mu\nu}\rangle$, the
(renormalized) continuum limit of the analogous lattice expectation value. The
expectation value $\langle T^{\mathrm{ren}}_{\mu\nu}\rangle$ can then serve as
the source term in a numerical relativity simulation.

A necessary first step towards the use of tensor networks such as matrix
product states is a careful translation of the continuum QFT problem into one
on the lattice. This paper documents that undertaking: the mapping between
expectation values in a QFT on a _fixed_ curved spacetime [9, 57, 41] and the
aforementioned renormalized sequence of lattice expectation values.

We study Dirac fermions in a curved, two-dimensional Lorentzian spacetime.
Dirac fermions in two spacetime dimensions are among the easiest fields to
study on the lattice, because they map straightforwardly to an approximation
by staggered lattice fermions [32, 51, 7] which in turn can mapped exactly to
a quantum spin chain [30]. The quantum spin chain can then be simulated by
matrix product state techniques without further ado, as we will demonstrate in
[35]. In the current derivations, however, in order to not further complicate
an already rather involved discussion, we consider a non-interacting Dirac
field, which can be solved without resorting to matrix products states, using
the free-fermion formalism. We note this part of the program is comparable to
that proposed in [61] and, more closely, [10], which studied lattice
Hamiltonians equivalent to the one we eventually derive.

In general spacetimes it is not always clear what states ought to be of
physical interest. In the literature one often proceeds by making mode
expansions of field operators, whose forms are somehow determined by the
problem at hand. These mode expansions might not necessarily relate
straightforwardly to the Hamiltonian formulation we use. To make comparisons
with results based on them, we consider a class of states called “Hawking-
Hartle vacua” [26, 28, 22, 53, 33, 31, 29]. They are defined in spacetimes
containing static observers, bounded by a so-called “Killing horizon”. In the
Hawking-Hartle vacuum, that horizon radiates, but is at thermal equilibrium
with its environment. It can be viewed as a thermal state of the Hamiltonian
defined by static observers outside the horizon, at a certain “Unruh”
temperature. We identify it (in the continuum limit) with thermal states of
the analogous lattice Hamiltonian, at the relevant metric’s Unruh temperature.

In a quantum field theory, the expectation value $\langle T_{\mu\nu}\rangle$
of the stress-energy tensor $T_{\mu\nu}$ in the Hawking-Hartle vacuum is
formally infinite even for free fields, and to extract a meaningful finite
number it needs to be renormalized. This same problem occurs for any operator
with quadratic terms in the fields. One possible solution, formulated directly
in coordinate space and thus appropriate to curved spacetime, is Hadamard
renormalization [25, 16, 39, 1, 15, 19, 5, 34]. Here, the quadratic operator
is replaced with the coincidence (zero-separation) limit of some correlation
function, with any divergent terms subtracted before the limit is taken. In
fact, the lattice two point functions have the same leading order form as
their continuum limits in a short-distance expansion. Thus, substituting them
for the continuum correlation functions within the coincidence limit yields
the same result.

For illustrative purposes we present four applications in the continuum using
lattice data. By order of complexity, they are: (i) computation of ground
state expectation values as measured by inertial Minkowski observers; (ii)
computation of thermal expectation values as measured by accelerated Minkowski
observers (and thus reproduction of the Unruh effect [52]); (iii) computation
of thermal expectation values as measured by static observers in the Hawking-
Hartle vaccum of Schwarzschild black hole spacetime; and, finally, (iv)
computation of thermal expectation values as measured by static observers in
the Hawking-Hartle vacuum (often called the Bunch-Davies vacuum) of an
expanding universe described by de Sitter spacetime [11, 4, 14, 23, 18, 21].

This paper is divided into four main sections. The first is this introduction.
The second develops the continuum problem to be numerically solved: it defines
the free Dirac field in a curved two-dimensional spacetime, explicates its
canonical quantization, and details how point-splitting regularization and
Hadamard renormalization allow quadratic expectation values to be defined. It
also introduces examples of spacetimes with a bifurcate Killing horizon and
defines the Hawking-Hartle vacuum. The third section outlines our lattice
approximation: it introduces the lattice and then reviews the staggered
fermion technique to map between its Hamiltonian and that of the continuous
field theory, as well as the free fermion formalism to find correlation
functions of non-interacting fermions on the lattice. Finally, the fourth
section uses the above method to find Hadamard-renormalized expectation values
in the four example cases (i)-(iv) previously listed.

##  2 Continuum Problem

To obtain $\langle T^{\mathrm{ren}}_{\mu\nu}\rangle$ in (1), we must
understand how to compute renormalized expectation values of quadratic
operators. The goal of this paper is to explain how to do so for so-called
“Hawking-Hartle vacuum”, describing a horizon in thermal equilibrium, of a
theory of free Dirac fermions in two spacetime dimensions. In this section, we
define our terms and outline this problem in detail.

###  2.1 Free Dirac Fields in Curved Spacetime

We work in a two-dimensional Lorentzian manifold
$(\mathcal{M},g_{\mu\nu}(t,x))$. Every two-dimensional spacetime is
conformally flat [13], and thus can be covered by finite-size patches in which
the metric $g_{\mu\nu}(t,x)$ differs from the Minkowski metric
$\eta_{\mu\nu}\equiv\mathrm{diag}\begin{pmatrix}-1,1\end{pmatrix}$ by an
overall function $\Omega^{2}(t,x)$, called the conformal (Weyl, scaling)
factor. We will always adopt coordinates in which this is manifestly true,

$g_{\mu\nu}(t,x)=\Omega^{2}(t,x)\eta_{\mu\nu}.$ (2)

The strategy to be outlined will not qualitatively depend upon this choice,
though the specific equations we present will.

The manifold will shelter a “Dirac” field of two-component spinors
$\psi_{A}(t,x)$, whose spinor components are indexed with capital-Latin
letters. In order to simplify the notation, we will usually suppress these.
When spinor indices are suppressed $(\psi_{A}(t,x)\to\psi(t,x)$ and
$\psi^{{\dagger}A}(t,x)\to\psi^{\dagger}(t,x))$, $\psi(t,x)$ and
$\psi^{\dagger}(t,x)$ are to be formally treated, apart from their
transformation properties, respectively as two-component column and row
vectors.

The spinors form a separate spinor representation of the Lorentz group at each
point in $\mathcal{M}$. In curved spacetime [20, 41], along with the usual
“flat spacetime” gamma matrices $\gamma^{\mu}$ we need “curved spacetime”
equivalents $\tilde{\gamma}^{\mu}(t,x)$, respectively defined by their
anticommutation relations,

$\displaystyle\\{\gamma\indices{{}^{\mu}_{A}{}^{B}},\gamma\indices{{}^{\nu}_{B}{}^{A}}$
$\displaystyle=2\eta^{\mu\nu},$ (3a)
$\displaystyle\\{\tilde{\gamma}\indices{{}^{\mu}_{A}{}^{B}}(t,x),\tilde{\gamma}\indices{{}^{\nu}_{B}{}^{A}}(t,x)\\}$
$\displaystyle=2g^{\mu\nu}(t,x).$ (3b)

When spinor indices are suppressed
$(\gamma\indices{{}^{\mu}_{A}{}^{B}}\to\gamma{{}^{\mu}}$,
$\tilde{\gamma}\indices{{}^{\mu}_{A}{}^{B}}\to\tilde{\gamma}{{}^{\mu}})$,
$\gamma{{}^{\mu}}$ and $\tilde{\gamma}{{}^{\mu}}$ interact with the Dirac
spinors as $2\times 2$ matrices. In the manifestly conformal chart (2), we
will choose

$\tilde{\gamma}^{\mu}(t,x)=\Omega^{-1}(t,x)\gamma^{\mu},$ (3d)

so that (3a) imply (3b). It is convenient to define the Dirac adjoint,

$\bar{\psi}(t,x)\equiv i\psi^{\dagger}(t,x)\gamma^{0},$ (3e)

along with the inaccurately-named “fifth” gamma matrix,

$\gamma^{5}\equiv i\gamma^{0}\gamma^{1}.$ (3f)

We will distinguish between two derivative operators. The first is the partial
derivative, denoted interchangeably by either the usual $\partial_{\mu}$ or a
comma,

$\partial_{\mu}T(t,x)\equiv T_{,\mu}(t,x).$ (3g)

The partial derivative of a tensor generically does not transform as a tensor.
To remedy this, one introduces the covariant derivative, denoted
interchangeably by either $\nabla_{\mu}$ or a semicolon,

$\nabla_{\mu}T(t,x)\equiv T_{;\mu}(t,x).$ (3h)

The covariant derivative satisfies the axioms of a derivative operator [56],
notably including linearity and the product rule. It differs from the partial
derivative by the subtraction of “connection” terms, such that the full
expression transforms covariantly. The specific connection depends on the
transformation properties of the operand. Details are available in e.g. [56,
20], but in the case of the spinor fields we have

$\displaystyle\bar{\psi}_{;\mu}(t,x)$
$\displaystyle=\bar{\psi}_{,\mu}(t,x)-\bar{\psi}(t,x)C_{\mu}(t,x),$
$\displaystyle
C_{0}(t,x)=-\frac{1}{2}\frac{\Omega_{,1}(t,x)}{\Omega(t,x)}\gamma^{0}\gamma^{1}$
(3ia) $\displaystyle\psi_{;\mu}(t,x)$
$\displaystyle=\psi_{,\mu}(t,x)+C_{\mu}(t,x)\psi(t,x),$ $\displaystyle
C_{1}(t,x)=-\frac{1}{2}\frac{\Omega_{,0}(t,x)}{\Omega(t,x)}\gamma^{0}\gamma^{1},$
(3ib)

where the expressions in the right column (but not the left) are obviously
specific to the manifestly conformal chart (2). We can work out
$\tilde{\gamma}\indices{{}^{\mu}_{;\nu}}(t,x)=0$ [20], which in fact holds
generally.

We will sometimes use the “arrow” notation

$\displaystyle\bar{\psi}(t,x)\overset{\leftrightarrow}{\nabla}_{\mu}\psi(t,x)=\bar{\psi}(t,x)\psi_{;\mu}(t,x)-\bar{\psi}_{;\mu}(t,x)\psi(t,x),$
(3ja)
$\displaystyle\bar{\psi}(t,x)\overset{\leftrightarrow}{\partial}_{\mu}\psi(t,x)=\bar{\psi}(t,x)\psi_{,\mu}(t,x)-\bar{\psi}_{,\mu}(t,x)\psi(t,x).$
(3jb)

Applying (3i) we see

$\bar{\psi}(t,x)\tilde{\gamma}^{\mu}(t,x)\overset{\leftrightarrow}{\nabla}_{\mu}\psi(t,x)=\frac{1}{\Omega(t,x)}\left(\bar{\psi}(t,x)\gamma^{\mu}\overset{\leftrightarrow}{\partial}_{\mu}\psi(t,x)\right),$
(3k)

which is specific to 2 dimensions.

Dynamics will be set by the free Dirac action $S$ and its corresponding
Lagrangian density $\mathcal{L}(t,x)$,

$\displaystyle S[\bar{\psi}(t,x),\psi(t,x)]$ $\displaystyle=-\int
dtdx\mathcal{L}(t,x),$ (3l) $\displaystyle\mathcal{L}(t,x)$
$\displaystyle=\sqrt{-g(t,x)}\bar{\psi}(t,x)\left(\frac{1}{2}\tilde{\gamma}^{\mu}(t,x)\overset{\leftrightarrow}{\nabla}_{\mu}-m\right)\psi(t,x),$
(3m)

where $g(t,x)$ is the determinant of the spacetime metric. In the manifestly
conformal frame (2) we have

$\sqrt{-g(t,x)}=\Omega^{2}(t,x).$ (3n)

In flat spacetime, one more commonly uses a Lagrangian density containing
$\gamma^{\mu}\partial_{\mu}$, as opposed to the
$\frac{1}{2}\gamma^{\mu}\overset{\leftrightarrow}{\partial}_{\mu}$ that
appears in ours. The two Lagrangian densities differ only by a total
derivative, but the present choice (3m) is more convenient for our purposes
due to (3k), which lets us write

$\mathcal{L}(t,x)=\Omega(t,x)\bar{\psi}(t,x)\left(\frac{1}{2}\gamma^{\mu}\overset{\leftrightarrow}{\partial}_{\mu}-\Omega(t,x)m\right)\psi(t,x).$
(3o)

The canonically conjugate momentum to $\psi(t,x)$, $\pi(t,x)$, is defined by

$\displaystyle\pi(t,x)$
$\displaystyle\equiv\frac{\partial\mathcal{L}}{\partial\left(\psi_{,0}(t,x)\right)}=\frac{1}{2}\sqrt{-g(t,x)}\bar{\psi}(t,x)\tilde{\gamma}^{0}(t,x),$
(3pa) $\displaystyle=-\frac{i}{2}\Omega(t,x)\psi^{\dagger}(t,x).$ (3pb)

Time evolution (Lie transport) of the fields between successive constant-$t$
hypersurfaces of the spacetime is generated by the Hamiltonian,

$\displaystyle H_{\psi}(t,x)$
$\displaystyle\equiv\int~{}dx~{}\left(\pi\psi_{,0}+(\pi\psi_{,0})^{\dagger}-\mathcal{L}(t,x)\right),$
(3qa)
$\displaystyle=\int~{}dx~{}\Omega(t,x)\bar{\psi}(t,x)\left(-\frac{1}{2}\gamma^{1}\overset{\leftrightarrow}{\partial_{1}}+\Omega(t,x)m\right)\psi(t,x),$
(3qb)
$\displaystyle=\int~{}dx~{}\Omega(t,x)\psi^{\dagger}(t,x)\left(-\frac{1}{2}\gamma^{5}\overset{\leftrightarrow}{\partial_{1}}+i\Omega(t,x)m\gamma^{0}\right)\psi(t,x)$
(3qc)

The stress-energy tensor $T_{\mu\nu}(t,x)$ of the Dirac field is [20]

$T_{\mu\nu}(t,x)\equiv\frac{-2}{\sqrt{-g(t,x)}}\frac{\delta S}{\delta
g^{\mu\nu}(t,x)}=\frac{1}{2}\bar{\psi}(t,x)\left(\tilde{\gamma}_{(\mu}(t,x)\overset{\leftrightarrow}{\nabla}_{\nu)}\right)\psi(t,x).$
(3r)

A field’s stress-energy tensor measures its response to an infinitesimal
diffeomorphism, in the sense that

$Q(t,x)=\int d\Sigma^{\mu}(t,x)T_{\mu\nu}(t,x)\xi^{\nu}(t,x),$ (3s)

where $d\Sigma^{\mu}(t,x)$ is the integration measure on a hypersurface
$\Sigma$, generates Lie transport of the portion of that field from $\Sigma$
along $\xi^{\nu}(t,x)$. The Hamiltonian (3q) can also be derived from (3s) by
fixing $\Sigma$ to be constant-$t$ hypersurfaces with spatial coordinate $x$
and choosing $\xi^{\nu}(t,x)$ to point along lines of constant $x$ (see
Appendix B). Unlike (3q), however, (3s) could also be used to propagate the
fields along a different foliation than the constant-$t$ one, an issue we will
not explore further here.

###  2.2 The Quantum Field Theory

The theory can now be quantized e.g. canonically, by introducing the canonical
anticommutation relations [41]

$\displaystyle\\{\psi_{A}(t,x),\pi^{B}(t,x^{\prime})\\}=i\delta(x^{\prime}-x)\delta\indices{{}_{A}^{B}},$
(3ta)
$\displaystyle\\{\psi_{A}(t,x),\psi_{B}(t,x^{\prime})\\}=\\{\pi^{A}(t,x),\pi^{B}(t,x^{\prime})\\}=0,$
(3tb)

which require $\psi(t,x)$ to be understood as a field of operators. Inserting
(3pb) into (3t), we find

$\displaystyle\\{\psi_{A}(t,x),\psi^{{\dagger}B}(t,x^{\prime})\\}$
$\displaystyle=\frac{\delta(x^{\prime}-x)\delta\indices{{}_{A}^{B}}}{2\sqrt{\Omega(t,x^{\prime})\Omega(t,x)}},$
(3ua)
$\displaystyle\\{\psi^{{\dagger}A}(t,x),\psi^{{\dagger}B}(t,x^{\prime})\\}$
$\displaystyle=0.$ (3ub)

Note that the Dirac delta in (3ua) makes the powers on the $\Omega(t,x)$ and
$\Omega(t,x^{\prime})$ arbitrary, provided they sum to $-1$. We have chosen
the present form to ease discretization later.

We take as given a Hilbert space of quantum states $\ket{\Psi(t)}$, such that
the so-called Hadamard function (which is actually a 2$\times$2 matrix
containing 4 functions, see Appendix C)

$G\indices{{}^{(1)}_{A}{}^{B^{\prime}}}(t,x,t^{\prime},x^{\prime})\equiv\frac{1}{2}\left\langle\left[\psi_{A}(t,x),\bar{\psi}^{B^{\prime}}(t^{\prime},x^{\prime})\right]\right\rangle,$
(3v)

is finite, provided the primed and unprimed points not be lightlike separated:
notably, provided they are not the same point. The primed index in (3v)
transforms as a spinor with respect to Lorentz transformations at
$(t^{\prime},x^{\prime})$, rather than at $(t,x)$ as does the unprimed index,
but these indices will again usually be suppressed,
$G\indices{{}^{(1)}_{A}{}^{B^{\prime}}}(t,x,t^{\prime},x^{\prime})\mapsto
G^{(1)}(t,x,t^{\prime},x^{\prime})$. The superscript (1) is a decorator often
used in the literature to distinguish $G^{(1)}(t,x,t^{\prime},x^{\prime})$
from other two-point functions such as the Feynman propagator (which will not
appear here).

We will most often be interested in the _equal-time_ Hadamard function

$G^{(1)}(t,x,x^{\prime})\equiv G^{(1)}(t,x,t,x^{\prime}).$ (3w)

As will be discussed below, the Hadamard function can be used to regulate and
renormalize “quadratic” operators involving quadratic terms in the fields.
Many interesting operators, including the stress-energy tensor (3r), fall in
this class. However, the stress-energy tensor must be further manicured even
after renormalization in order for its expectation value to meet, for example,
the divergence-free condition $\langle T_{\mu\nu}(t,x)\rangle^{;\mu}=0$. This
poses a significant, though surmountable [38, 34], complication that will not
be addressed here. In this study we instead use as prototypes the condensate
$\langle C_{I}(t,x)\rangle$, the pseudo-scalar $\langle C_{5}(t,x)\rangle$,
and the current $\langle j_{\mu}(t,x)\rangle$,

$\displaystyle C_{I}(t,x)$ $\displaystyle\equiv m\bar{\psi}(t,x)\psi(t,x),$
$\displaystyle\langle
C_{I}(t,x)\rangle=m\langle\bar{\psi}(t,x)\psi(t,x)\rangle,$ (3xa)
$\displaystyle C_{5}(t,x)$
$\displaystyle\equiv\bar{\psi}(t,x)\gamma^{5}\psi(t,x),$ $\displaystyle\langle
C_{5}(t,x)\rangle=\langle\bar{\psi}(t,x)\gamma^{5}\psi(t,x)\rangle,$ (3xb)
$\displaystyle j_{\mu}(t,x)$
$\displaystyle\equiv\bar{\psi}(t,x)\tilde{\gamma}_{\mu}(t,x)\psi(t,x),$
$\displaystyle\langle
j_{\mu}(t,x)\rangle=\left\langle\bar{\psi}(t,x)\tilde{\gamma}_{\mu}(t,x)\psi(t,x)\right\rangle.$
(3xc)

Together, they span the four possible quadratic expressions of the form
$\psi^{{\dagger}A}\psi_{B}$ that can be obtained with components of the Dirac
spinors (see Appendix C).

###  2.3 Point-splitting regularization

The expectation values on the right column of (3x) are formally infinite and
thus ill-defined. We will replace them with well-defined quantities via so-
called “point-splitting” [25, 16, 39, 1, 15, 19, 5, 34]. Thus, we regulate our
quadratic operators by explicitly treating them as
$(t^{\prime},x^{\prime})\to(t,x)$ “coincidence” limits of traces of the
Hadamard function (see Figure 1). For example, in flat spacetime, we would
write

$\displaystyle\langle C_{I}(t,x)\rangle$ $\displaystyle\mapsto$
$\displaystyle\lim_{(t^{\prime},x^{\prime})\to(t,x)}-m\Tr
G^{(1)}(t,x,t^{\prime},x^{\prime}),$ (3y) $\displaystyle\langle
C_{5}(t,x)\rangle$ $\displaystyle\mapsto$
$\displaystyle\lim_{(t^{\prime},x^{\prime})\to(t,x)}-\Tr\gamma^{5}G^{(1)}(t,x,t^{\prime},x^{\prime}),$
(3z) $\displaystyle\langle j_{\mu}(t,x)\rangle$ $\displaystyle\mapsto$
$\displaystyle\lim_{(t^{\prime},x^{\prime})\to(t,x)}-\Tr\gamma_{\mu}G^{(1)}(t,x,t^{\prime},x^{\prime}),$
(3aa)

where the trace is over the suppressed spinor indices, and the limit is along
the unique geodesic connecting $(t^{\prime},x^{\prime})$ to $(t,x)$.

In a curved spacetime, the point-split expressions will be more complicated,
because as e.g. $\bar{\psi}(t^{\prime},x^{\prime})$ is parallel-dragged to
$(t,x)$, it will be deformed and rotated by the spacetime curvature. This is
expressed symbolically by introducing the “spin parallel propagator”
$\mathcal{J}\indices{{}_{B^{\prime}}^{A}}(t,x,t^{\prime},x^{\prime})\mapsto\mathcal{J}(t,x,t^{\prime},x^{\prime})$,
defined by

$\displaystyle\mathcal{J}_{;\mu}(t,x,t^{\prime},x^{\prime})\sigma^{;\mu}(t,x,t^{\prime},x^{\prime})$
$\displaystyle=0,$ (3aba)
$\displaystyle\lim_{(t^{\prime},x^{\prime})\to(t,x)}\mathcal{J}(t,x,t^{\prime},x^{\prime})$
$\displaystyle=\mathbbm{1},$ (3abb)

where Synge’s world function $\sigma(t,x,t^{\prime},x^{\prime})$ is one-half
the squared geodesic distance from $(t^{\prime},x^{\prime})$ to $(t,x)$, while
$\mathbbm{1}\indices{{}_{B^{\prime}}^{A}}\mapsto\mathbbm{1}$ is the identity
over bispinor indices. As always, the limit is understood to be along the
coincident geodesic. These are, in fact, the parallel transport equations, so
that multiplication of $\bar{\psi}(t^{\prime},x^{\prime})$ by
$\mathcal{J}(t,x,t^{\prime},x^{\prime})$ correctly parallel-drags the
suppressed spinor indices from $(t^{\prime},x^{\prime})$ to $(t,x)$.

Figure 1: Schematic of point-splitting regularization at equal times. The
manifold $\mathcal{M}$ is foliated by equal-time surfaces $\Sigma_{t}$.
Instances of quadratic expectation values such as
$\langle\bar{\psi}(t,x)\psi(t,x)\rangle$ are replaced with coincidence ($r\to
0,\;r\equiv x^{\prime}-x$) limits of two-point functions such as
$\langle\bar{\psi}(t,x^{\prime})\psi(t,x)\rangle$, expressed in terms of the
equal-time Hadamard function $G^{(1)}(t,x,x^{\prime})$.

To point split, we now treat the formal expectation values in (3x) as the
replacements

$\displaystyle\langle C_{I}(t,x)\rangle$
$\displaystyle\mapsto\lim_{(t^{\prime},x^{\prime})\to(t,x)}\mathbbm{C}_{I}(t,x,t^{\prime},x^{\prime}),$
$\displaystyle\mathbbm{C}_{I}(t,x,t^{\prime},x^{\prime})\equiv-m\Tr\mathcal{J}(t,x,t^{\prime},x^{\prime})G^{(1)}(t,x,t^{\prime},x^{\prime}),$
(3aca) $\displaystyle\langle C_{5}(t,x)\rangle$
$\displaystyle\mapsto\lim_{(t^{\prime},x^{\prime})\to(t,x)}\mathbbm{C}_{5}(t,x,t^{\prime},x^{\prime}),$
$\displaystyle\mathbbm{C}_{5}(t,x,t^{\prime},x^{\prime})\equiv-\Tr\mathcal{J}(t,x,t^{\prime},x^{\prime})\gamma^{5}G^{(1)}(t,x,t^{\prime},x^{\prime}),$
(3acb) $\displaystyle\langle j_{\mu}(t,x)\rangle$
$\displaystyle\mapsto\lim_{(t^{\prime},x^{\prime})\to(t,x)}\mathbbm{j}_{\mu}(t,x,t^{\prime}x^{\prime}),$
$\displaystyle\mathbbm{j}_{\mu}(t,x,t^{\prime},x^{\prime})\equiv-\Tr\mathcal{J}(t,x,t^{\prime},x^{\prime})\tilde{\gamma}_{\mu}(t,x)G^{(1)}(t,x,t^{\prime},x^{\prime}).$
(3acc)

Notice that, had the spin parallel propagator
$\mathcal{J}\indices{{}_{B^{\prime}}^{A}}(t,x,t^{\prime},x^{\prime})$ not been
included in these expressions, the respectively unprimed and primed spinor
indices of $G\indices{{}^{(1)}_{A}{}^{B^{\prime}}}(t,x,t^{\prime},x^{\prime})$
would transform at different points, and thus could not be traced over to
yield a scalar.

To get expressions we can compare to lattice data, we need to specialize each
of (3ac) to the manifestly conformal metric (2) with $t=t^{\prime}$. We will
achieve this by expanding $\mathcal{J}(t,x,t,x^{\prime})$ in the spatial
coordinate separation $r\equiv x^{\prime}-x$. As we will see in (3ag), the
strongest divergences in (3ac) are proportional to $1/r$. Thus recovery of the
coincidence limit requires this expansion to linear order in $r$. In Appendix
A we find

$\mathcal{J}(t,x,t,x^{\prime})=\mathbbm{1}\left(1-\frac{1}{2}\frac{\Omega_{,0}(t,x)}{\Omega(t,x)}\gamma^{0}\gamma^{1}r\right)+\cdots.$
(3ad)

Writing
$\mathbbm{C}_{I}(t,x,x^{\prime})\equiv\mathbbm{C}_{I}(t,x,t,x^{\prime})$,
$\mathbbm{C}_{5}(t,x,x^{\prime})\equiv\mathbbm{C}_{5}(t,x,t,x^{\prime})$ and
$\mathbbm{j}_{\mu}(t,x,x^{\prime})\equiv\mathbbm{j}_{\mu}(t,x,t,x^{\prime})$,
we thus have

$\displaystyle\mathbbm{C}_{I}(t,x,x^{\prime})$
$\displaystyle=-m\mathrm{Tr}\left(G^{(1)}(t,x,x^{\prime})\right)-r~{}\frac{im\Omega_{,0}(t,x)}{2\Omega(t,x)}\mathrm{Tr}\left(\gamma^{5}G^{(1)}(t,x,x^{\prime})\right)+\cdots,$
(3aea) $\displaystyle\mathbbm{C}_{5}(t,x,x^{\prime})$
$\displaystyle=-\mathrm{Tr}\left(\gamma^{5}G^{(1)}(t,x,x^{\prime})\right)+r~{}\frac{i\Omega_{,0}(t,x)}{2\Omega(t,x)}\mathrm{Tr}\left(G^{(1)}(t,x,x^{\prime})\right)+\cdots,$
(3aeb) $\displaystyle\mathbbm{j}_{0}(t,x,x^{\prime})$
$\displaystyle=\frac{-1}{\Omega(t,x)}\mathrm{Tr}\left(\gamma_{0}G^{(1)}(t,x,x^{\prime})\right)-r~{}\frac{\Omega_{,0}(t,x)}{2\Omega^{2}(t,x)}\mathrm{Tr}\left(\gamma_{1}G^{(1)}(t,x,x^{\prime})\right)+\cdots,$
(3aec) $\displaystyle\mathbbm{j}_{1}(t,x,x^{\prime})$
$\displaystyle=\frac{-1}{\Omega(t,x)}\mathrm{Tr}\left(\gamma_{1}G^{(1)}(t,x,x^{\prime})\right)+r~{}\frac{\Omega_{,0}(t,x)}{2\Omega^{2}(t,x)}\mathrm{Tr}\left(\gamma_{0}G^{(1)}(t,x,x^{\prime})\right)+\cdots.$
(3aed)

###  2.4 Hadamard renormalization

The coincidence limits of these expressions (3ae) are still undefined, because
via $G^{(1)}(t,x,x^{\prime})$ they contain terms that diverge as
$x^{\prime}\to x$. However, given certain smoothness assumptions, bundled
together as the requirement that physical quantum states be “Hadamard”, said
divergences are completely determined by the local geometry and the mass. In
particular, they are independent of the specific choice of “Hadamard” quantum
state.

The locally determined contributions to (3ae), including all the divergences,
made by any Hadamard state can then be computed as the first step of a
procedure known as Hadamard renormalization. The next step is to subtract the
locally-determined terms from each of (3ae) before the coincidence limits are
taken, thus yielding finite answers.

We label the locally-determined terms with the superscript “loc”, e.g.
$\mathbbm{j}^{\mathrm{loc}}_{\mu}(t,x,x^{\prime})$. Subtracting each local
contribution from its corresponding bare two-point function within the limit
then yields a convergent remainder - which we label e.g.
$\mathbbm{j}^{\mathrm{ren}}_{\mu}(t,x,x^{\prime})$ \- with a finite
coincidence limit. Next, we identify each of (3x) with the relevant
coincidence limit:

$\displaystyle\langle C_{I}^{\mathrm{ren}}(t,x)\rangle$
$\displaystyle\equiv\lim_{x^{\prime}\to
x}\mathbbm{C}^{\mathrm{ren}}_{I}(t,x,x^{\prime})=\lim_{x^{\prime}\to
x}\left(\mathbbm{C}_{I}(t,x,x^{\prime})-\mathbbm{C}^{\mathrm{loc}}_{I}(t,x,x^{\prime})\right),$
(3afa) $\displaystyle\langle C_{5}^{\mathrm{ren}}(t,x)\rangle$
$\displaystyle\equiv\lim_{x^{\prime}\to
x}\mathbbm{C}^{\mathrm{ren}}_{5}(t,x,x^{\prime})=\lim_{x^{\prime}\to
x}\left(\mathbbm{C}_{5}(t,x,x^{\prime})-\mathbbm{C}^{\mathrm{loc}}_{5}(t,x,x^{\prime})\right),$
(3afb) $\displaystyle\langle j_{\mu}^{\mathrm{ren}}(t,x)\rangle$
$\displaystyle\equiv\lim_{x^{\prime}\to
x}\mathbbm{j}^{\mathrm{ren}}_{\mu}(t,x,x^{\prime})=\lim_{x^{\prime}\to
x}\left(\mathbbm{j}_{\mu}(t,x,x^{\prime})-\mathbbm{j}^{\mathrm{loc}}_{\mu}(t,x,x^{\prime})\right).$
(3afc)

The locally-determined terms were computed in [34]. They are

$\displaystyle\mathbbm{C}^{\mathrm{loc}}_{I}(t,x,x^{\prime})$
$\displaystyle=\frac{m^{2}}{2\pi}\ln\left(\mu\frac{1}{2}\Omega^{2}(t,x)r^{2}\right),$
(3aga) $\displaystyle\mathbbm{C}^{\mathrm{loc}}_{5}(t,x,x^{\prime})$
$\displaystyle=0,$ (3agb)
$\displaystyle\mathbbm{j}^{\mathrm{loc}}_{0}(t,x,x^{\prime})$
$\displaystyle=\frac{\Omega_{,0}(t,x)}{2\pi\Omega(t,x)},$ (3agc)
$\displaystyle\mathbbm{j}^{\mathrm{loc}}_{1}(t,x,x^{\prime})$
$\displaystyle=-\frac{1}{\pi r}+\frac{\Omega_{,1}(t,x)}{2\pi\Omega(t,x)},$
(3agd)

where $\mu$ is an undetermined dimensionful parameter that must be “measured”.
We will choose

$\mu=\mu_{0}\equiv\frac{1}{2}m^{2}e^{2\gamma_{E}}$ (3ah)

where $\gamma_{E}=0.5772\ldots$ is the Euler-Mascheroni constant. This
arbitrary choice cancels a constant term that otherwise appears in a short-
distance expansion of $\mathbbm{C_{I}}(t,x,x^{\prime})$ (see Appendix D), and
is thus implicitly made by the standard practice of normal ordering. If a
different choice $\mu^{\prime}$ were made, our results for the condensate
would suffer the replacement

$\langle C_{I}^{\mathrm{ren}}(t,x)\rangle\mapsto\langle
C_{I}^{\mathrm{ren}}(t,x)\rangle-\frac{m^{2}}{2\pi}\ln\left(\frac{\mu^{\prime}}{\mu_{0}}\frac{1}{2}\Omega^{2}(t,x)\right).$
(3ai)

Our results are otherwise fixed by the Hadamard renormalization procedure. It
is perhaps worth emphasizing that not all of the terms in (3ag) are divergent.
Apart from the choice of $\mu$, however, the finite terms are not arbitrary,
as e.g. the expected behaviour of the Hawking-Hartle two-point functions would
not obtain without them.

###  2.5 Hawking-Hartle Vacua

To specify a calculation we must identify a quantum state. In Minkowski
spacetime, one “by default” considers the Minkowski vacuum, selected for
example by the demand that $G^{(1)}(t,x,t^{\prime},x^{\prime})$ be Poincaré
invariant. But there is no general such prescription yielding an interesting
state in an arbitrary spacetime. States must be identified in a manner
suggested by the problem at hand. Our target in this study will be the
Hawking-Hartle vacuum, which describes a (Killing) horizon in thermal
equilibrium with its environment.

In spacetimes containing such Killing horizons, the Hawking-Hartle vacuum is,
in a free theory, the unique state that is both stationary and possessed of a
Hadamard function with no divergences in $x$ independently of $x^{\prime}$.
That such a state appears thermal to certain observers is the essential reason
for the Unruh effect, the inflationary power spectrum, and black hole
radiance, arguably the “poster children” of quantum field theory in curved
spacetime. Given access to a direct simulation of the Hawking-Hartle vacuum,
these effects can correspondingly all be studied, perturbed, and prodded in
various ways. One could, for example, excite the state, and then study e.g.
real time evolution of the von Neumann entropy [35].

In the specific spacetimes we study, the Hawking-Hartle vacuum is often
referred to by a specific name. In Minkowski spacetime it is the _Minkowski
vacuum_. In the prototypical case of Schwarzschild spacetime, it is the
_Hawking-Hartle vacuum_ , or sometimes the _Kruskal vacuum_. In de Sitter, it
is the _Bunch-Davies_ or _Euclidean_ vacuum. These and similar comparisons are
tabulated in Table 1.

Spacetime | Global Chart | Hawking-Hartle Vacuum | Static Chart | Static Vacuum
---|---|---|---|---
Minkowski | Minkowski | Minkowski vacuum | Rindler (Lass) | Rindler vacuum
Schwarzschild | Kruskal | Hawking-Hartle/Kruskal vacuum | Schwarzschild | Boulware vacuum
de Sitter | Global | Bunch-Davies/Euclidean vacuum | Static | Static vacuum
Table 1: Names commonly used in the literature for the global chart, static
chart, Hawking-Hartle vacuum, and static vacuum in the three bifurcate Killing
horizon spacetimes we work with.

####  2.5.1 Bifurcate Killing Horizons

The Hawking-Hartle vacuum is defined in terms of geometric objects called
“bifurcate Killing horizons”. Before reviewing the Hawking-Hartle vacuum
itself, we will illustrate the concept of a bifurcate Killing horizon, via
concrete examples in Minkowski, two-dimensional Schwarzschild, and de Sitter
spacetime.

In all three spacetimes, we will first identify _global observers_ at constant
$X$ in the _global coordinates_ $(T,X)$. Next we will identify _static
observers_ at constant $x$ in the _static coordinates_ $(t,x)$. We will use
these same labels for the global and static coordinates in all three
spacetimes. The corresponding Weyl factors (2) will be written
$\Omega_{\mathrm{global}}^{i}(T,X)$ and $\Omega_{\mathrm{static}}^{i}(x)$,
where the superscript i labels the spacetime. That is, we will write
$\Omega_{\mathrm{global}}^{\mathrm{Mink.}}(T,X)$,
$\Omega_{\mathrm{global}}^{\mathrm{Schw.}}(T,X)$, and
$\Omega_{\mathrm{global}}^{\mathrm{dS}}(T,X)$ for global observers and
$\Omega_{\mathrm{static}}^{\mathrm{Mink.}}(x)$,
$\Omega_{\mathrm{static}}^{\mathrm{Schw.}}(x)$, and
$\Omega_{\mathrm{static}}^{\mathrm{dS}}(x)$ for local observers. Notice the
static Weyl factors are, by definition, independent of the static time $t$.

Bifurcate Killing horizon spacetimes are defined by the presence of a
lightlike _Killing horizon_ beyond which the static timelike coordinate
becomes spacelike, so that the static observers are confined to an exterior
_static patch_. The Killing horizon is “bifurcate” because of the reflection-
like symmetry relating the top and right patches to the bottom and left ones.
They meet at the so-called _bifurcation point_ , where a constant $t$
hypersurface in either static wedge intersects the horizon.

The important geometric features are thus the global observers, the static
observers, their confinement by the horizon to the static patch, and the
bifurcation point. Now, we will highlight these in concrete examples, starting
with Minkowski spacetime.

Minkowski.— The global chart $(T,X)$ on Minkowski spacetime is given by

$\Omega_{\mathrm{global}}^{\mathrm{Mink.}}(T,X)=1,$ (3an)

called “inertial” or “Minkowski” coordinates. Figure 2 depicts Minkowski
spacetime aligned with the $T,X$ global coordinates. The Minkowski metric is
_static_ because it does not depend on $T$. This is a special feature of
Minkowski spacetime; i.e. in other examples the global coordinates will not be
static. Vectors parallel to $T$ translations form a symmetry of the spacetime,
and are thus timelike _Killing vectors_.

Figure 2: Minkowski spacetime, drawn so that inertial (global) coordinates
form straight lines. One constant global-space-$X$ surface and a few constant
global-times-$T$ observers are depicted as straight horizontal and vertical
lines. A particular Killing horizon pair is shown as the diagonal X-shape: it
divides Minkowksi spacetime into left and right static wedges, and into top
and bottom interior wedges. These horizons are irrelevant to the global
observers, but confine the static observers, whose static time $t$ runs
hyperbolically in the diagram, to one or the other static wedge. A few static
observers are depicted in the right wedge, along with a pair of constant
static-space-$x$ hypersurfaces.

Minkowski spacetime has another set of timelike Killing vectors, parallel to
boosts. Observers parallel to these are called _Rindler observers_. They are
at rest in, for example, the _Lass chart_

$\Omega^{\mathrm{Mink.}}_{\mathrm{static}}(x)=e^{\alpha x},$ (3ao)

which is also static because it does not depend on $t$.

The Lass coordinate $x$ ranges from $-\infty$ to $\infty$, but only covers the
right static wedge in the diagram 2, where the Killing vector is future-
pointing and timelike. In the left static wedge, it is past-pointing and
timelike. In the top or bottom wedge, it is spacelike. At the boundary, the
Killing horizon, it is lightlike.

The Killing horizon causally disconnects a particular group of timelike
Killing observers from half of Minkowski spacetime. Which group depends on
which point of Minkowski spacetime is identified with the origin, and upon the
choice of acceleration parameter $\alpha$. This parameter defines the rate at
which accelerations diverge as observers closer to the horizons are
considered, and can be viewed as characterizing the “strength” of the horizon.
This can be quantified by defining the _surface gravity_ $\kappa$,

$k^{a}k^{b}_{;a}=\kappa k^{b},$ (3ap)

where $k^{a}$ is the Killing vector, and the expression is to evaluated on the
horizon. In Minkowski spacetime we have

$\kappa_{\mathrm{Mink.}}=\alpha.$ (3aq)

Schwarzschild.— Now consider the Schwarzschild black hole, described by the
familiar metric

$ds^{2}=-\left(1-\frac{2M}{\xi}\right)dt^{2}+\left(1-\frac{2M}{\mathcal{\xi}}\right)^{-1}d\mathcal{\xi}^{2}.$
(3ar)

We use the term “Schwarzschild” black hole to describe the 2D manifold
partially covered by (3ar), not the 4D solution to the Einstein equations.
Since this line element is independent of $t$, the constant-$\xi$
“Schwarzschild” observers are timelike Killing observers. The spatial
coordinate $\xi$ ranges from $2M$ (the event and Killing horizon) to $\infty$.

Figure 3: Schwarzschild spacetime, drawn so that Kruskal (global) coordinates
form straight lines. As in Minkowski spacetime, one constant global-space-$X$
surface and a few constant global-times-$T$ observers are depicted as straight
horizontal and vertical lines. In this case the spacetime is divided into four
wedges by the event horizon: these wedges are the antiuniverse and universe,
analogous to Minkowski spacetime’s left and right static wedges, along with
the black and white holes, analogous to Minkowski spacetime’s top and bottom
interior wedges. A few static “Schwarzschild” observers are depicted in the
universe, along with a pair of constant static-space-$x$ hypersurfaces. Unlike
Minkowski spacetime, Schwarzschild spacetime exhibits a singularity at $x=0$:
curves parameterized by $t$ simply terminate here.

An analogous chart to the Lass metric (3ao) is found by defining the tortoise
coordinate $x$

$x\equiv\xi+2M\ln\left(\frac{\xi-2M}{2M}\right)$ (3as)

to find

$\Omega_{\mathrm{static}}^{\mathrm{Schw.}}(x)=\left(1-\frac{2M}{\xi}\right),$
(3at)

which is again static. These coordinates cover the same region as the
Schwarzschild coordinates, but have infinite range. The horizon is at
$x\to-\infty$.

The region of Schwarzschild spacetime outside the $\xi=2M$ Killing horizon is
thus analogous to the static patch of Minkowski spacetime. This can be made
obvious by adopting the “Kruskal” chart, which covers all of Schwarzschild
spacetime just as the inertial chart covered all of Minkowski. The Kruskal
chart is defined by

$\displaystyle T^{2}-X^{2}$
$\displaystyle=\left(1-\frac{\xi}{2M}\right)e^{\xi/2M},$ (3aua)
$\displaystyle\Omega_{\mathrm{global}}^{\mathrm{Schw.}}(T,X)$
$\displaystyle=\sqrt{\frac{32M^{3}}{\xi}e^{-\xi/2M}}.$ (3aub)

The const-$\xi$ “Schwarzschild” observers in the patches are Killing, but they
are not geodesic. Their local acceleration diverges towards the horizon at a
rate set by the Schwarzschild mass $M$, which is in this way analogous to
$\alpha$ in (3ao). However, while $\alpha$ parameterized a family of charts
upon the same manifold, $M$ parameterizes a family of physically inequivalent
manifolds. The surface gravity is

$\kappa_{\mathrm{Schw.}}=\frac{1}{4M}.$ (3av)

De Sitter.— Our final example will be the exponentially expanding de Sitter
universe, often labelled $dS_{2}$ in the two-dimensional case. This spacetime
is most straightforwardly discussed in terms of its embedding in higher
dimensional flat spacetime, as in Figure 4. The global coordinates $(T,X)$
with line element

$ds^{2}=-dT^{2}+\alpha_{dS}^{2}\cosh^{2}(T/\alpha_{dS})dX^{2}$ (3aw)

cover the full hyperboloid, and may be viewed as projections of the higher-
dimensional Minkowski coordinates upon it. Constant-$T$ slices are horizontal
rings, which contract until $T=0$ and then expand. The “de Sitter radius”
$\alpha_{dS}$ controls the rate of expansion.

Figure 4: $dS_{2}$ visualized as an embedding in Minkowski spacetime. The
horizontal rings are the constant-global-time $T$ surfaces. Constant
global-$X$ geodesic observers, which are the zero-energy geodesics, are
constantly boosted with respect to the external flat coordinates, and thus in
the embedding diagram follow hyperbolic trajectories like the dotted line. Due
to the spacetime expansion, all observers asymptote to such a zero-energy
geodesic, which themselves asymptote to lightlike lines, their cosmological
horizons. The lightcone of any point on the $T=0$ ring when the expansion
reverses forms a bifurcate Killing horizon. The corresponding timelike Killing
observers, defining static coordinates $(t,x)$, are not zero energy geodesics,
but asymptote to them, as do all worldlines in $dS_{2}$. Thus, each geodesic’s
cosmological horizon is also a Killing horizon, confining some set of static
observers to some static patch, like the one shaded in the diagram. These
static patches also wrap around behind the hyperboloid, and thus form
spatially compact diamonds, enclosed on either side.

Constant-$X$ observers follow timelike geodesics, which appear hyperbolic and
thus approach lightlike lines in the embedding diagram, their cosmological
horizons. These horizons segment $dS_{2}$ into causally disconnected patches.
In fact, they are Killing horizons, and the geodesic observers are Killing
observers, though this is not obvious from (3aw).

We can construct static coordinates bounded by a given Killing horizon as

$\Omega^{\mathrm{dS}}_{\mathrm{static}}(x)=\mathrm{Sech}\frac{x}{\alpha_{dS}},$
(3ax)

which cover half the circumference of the central ring. These observers are
accelerated relative to the higher-dimensional Minkowski coordinates. The
surface gravity is

$\kappa_{\mathrm{dS}}=\frac{1}{\alpha_{dS}}.$ (3ay)

Figure 5: Time evolution operators in a bifurcate Killing horizon spacetime.
That generated by the global Hamiltonian $H_{g}(T)$ implements forward
translations in the global time $T$, for example from $\Sigma_{g}$ to
$\Sigma_{0}$. This translation is always timelike, but it cannot in general be
used to define a stationary quantum state because $H_{g}(T)$ generically
depends on the global time. The static Hamiltonian $H_{s}$ generates forward
translations in the static time $t$, for example from the right half of
$\Sigma_{0}$ to $\Sigma_{s}$. Since $H_{s}$ is independent of $t$ this
translation can be used to define a stationary quantum state, but since the
translation is only timelike within the static wedges, that state will
generically be singular on the horizon. A nonsingular state that is still
stationary with respect to $H_{s}$ can nevertheless be defined as a Euclidean
path integral on the Euclidean manifold found by analytically continuing from
e.g. one half of $\Sigma_{0}$ to the other. This Euclidean manifold is
depicted as the half-cone in the inset. The edge of that half-cone overlaps
with $\Sigma_{0}$ in the Lorentzian spacetime, and the path integral is from
one edge to the other along the Euclidean time coordinate $\tau$. To fix the
singularity at the tip of the cone, $\tau$ must be made periodic with period
$\beta=\frac{2\pi}{\kappa}$, resulting in a thermal state of $H_{s}$ at
temperature (3az).

####  2.5.2 Hawking-Hartle Vacua

Now let us try to construct a physically reasonable “ground” state in a
spacetime with a bifurcate Killing horizon. We will make two demands. First,
we would like the equal-time Hadamard function to be independent of the static
time $t$, so that we can interpret the state as “equilibrated”. Second, we
would like the state not to diverge in any special way at the horizon, since
this would be very surprising to the global observers who can cross it.

These stipulations pick out, uniquely [29] in a free theory, the _Hawking-
Hartle vacuum_. To reiterate, the Hawking-Hartle vacuum is the unique state on
a bifurcate Killing horizon which is both smooth everywhere and static to the
static observers.

In Figure 5 we see the time evolution operators generated by two Hamiltonians
on the bifurcate Killing horizon spacetime, the static Hamiltonian $H_{s}$ and
the global Hamiltonian $H_{g}(T)$. These respectively evolve the field forward
in the static time $t$ and the global time $T$. The first, $H_{s}$, evolves
the field forward along the worldlines of the static observers - e.g. from the
right half of $\Sigma_{0}$ to $\Sigma_{s}$ \- and is therefore independent of
$t$.

The “ground state” of $H_{s}$, called the _static vacuum_ , is thus also
independent of $t$, our first criterion. However, $H_{s}$ generates a
lightlike rather than timelike translation at the horizon. Because of this,
its Hadamard function diverges at the horizon independently of the distance
between its arguments. It thus violates our second criterion.

Though unphysical, the static vacuum comes up often enough to have earned its
own special names in each of Minkowski (“Rindler vacuum”), Schwarzschild
(“Boulware vacuum”) and de Sitter (“static vacuum”).

The global Hamiltonian $H_{g}(T)$ evolves the field forward in the global time
$T$, e.g. from $\Sigma_{g}$ to $\Sigma_{0}$, which is well behaved at the
horizon. The “ground state” of $H_{g}(T)$ at a particular time is thus well-
behaved there. However, except in Minkowski spacetime, $H_{g}(T)$ will depend
on $T$, and through it, on $t$. This violates our first criterion. Though such
a state is potentially physical, it is not “equilibrated” in any useful sense.

Let us now construct the Hawking-Hartle vacuum, which fulfills both criteria.
Consider a hypersurface such as $\Sigma_{0}$ in Figure 5, formed by the union
of constant-time hypersurfaces in the right and left static wedges. A
stationary state of $H_{s}$ prepared everywhere on that hypersurface would be
smooth. The difficulties in preparing one stem entirely from the bifurcation
point $\mathcal{O}$.

We will resolve them by analytically continuing past that point. That is, we
will make the substitution $t\to\tau=-it$ in the static chart. Because that
chart is by definition time-independent, this only changes the signature of
the metric from Lorentzian to Euclidean. For example, the hyperbolic
trajectories of the static observers in the Lorentzian manifold become circles
in the Euclidean one, as depicted in the inset of Figure 5. The “time”
evolution operator $e^{-\tau H_{s}}$ advances along those circles.

We can follow those circles to move from the part of $\Sigma_{0}$ in the right
static patch to that in the left, without needing to cross the horizon. Of
course the Euclidean circles are not physical trajectories, but path integrals
along them can be used to define quantum states. These will be stationary with
respect to the static Hamiltonian $H_{s}$, the generator of motion along the
circles.

One part of the horizon does lie upon the Euclidean cone: the bifurcation
point $\mathcal{O}$. This can be shown to be a literal conical singularity by
going to polar coordinates in its vicinity. So long as that singularity
persists, the quantum state defined by the path integral on the cone will be
singular at the horizon.

Like any conical singularity, however, the singularity can be smoothed by
choosing a certain periodicity $\beta_{U}$ of the relevant angular coordinate,
$\tau$,

$\beta_{U}=\frac{2\pi}{\kappa},$ (3az)

the inverse Unruh temperature [52]. States $\rho$ defined as path integrals on
closed Euclidean manifolds are thermal with respect to the Hamiltonian
generating motion in the direction of the path integral,

$\rho\equiv\frac{e^{-\beta_{U}H_{s}}}{\Tr\left[e^{-\beta_{U}H_{s}}\right]}$
(3ba)

Thus the Hawking-Hartle vacuum describes the thermal equilibrium state of the
horizon. Despite the apparent complexity of the above discussion, preparing it
from a Hamiltonian formulation is simple: just find a thermal state of $H_{s}$
at the relevant Unruh temperature.

###  2.6 Summary

Everything needed for a well-defined calculation is now at hand. The steps to
be followed are:

  1. 1.

Choose a spacetime with a bifurcate Killing horizon.

  2. 2.

Find the Hamiltonian of its static observers by specializing (3q) to the
relevant Weyl factor $\Omega(x)$.

  3. 3.

Prepare the Hawking-Hartle vacuum as a thermal state of this Hamiltonian at
the appropriate Unruh temperature (3az).

  4. 4.

Compute the equal-time Hadamard function of that state via (3v).

  5. 5.

Compute renormalized quadratic expectation values via (3af).

We will next show how to approximately follow this procedure using data
obtained by numerically simulating a sequence of lattice theories.

##  3 Staggered Fermion Discretization

One possible strategy to perform numerical simulations is to map the continuum
quantum field theory onto a sequence of lattice models. In this section we
outline a means to do this, based on a slight adaptation of the well-known
“staggered fermion” discretization [32, 51, 7].

###  3.1 Staggered Fermions

We will numerically simulate the quantum field theory by extrapolating data
from a sequence of lattice theories. Thus, consider a one-dimensional lattice
whose sites, labelled by $n$, lie at points $x_{n}=na$ separated by constant
lattice spacing $a\equiv x_{n+1}-x_{n}$. The lattice may be finite or
infinite; when finite, it will have an even number of sites $N$. We use a
staggered-fermion discretization [32, 51, 7] to relate fermionic operators
$\phi_{n}$ on this lattice to components of the two-component continuum
spinors $\psi_{A}(t_{0},x)$ at some reference time $t_{0}$

$\displaystyle\\{\phi^{\dagger}_{n},\phi_{n^{\prime}}\\}$
$\displaystyle=\delta_{n,n^{\prime}},$ (3bba)
$\displaystyle\\{\phi_{n},\phi_{n^{\prime}}\\}$
$\displaystyle=\\{\phi^{\dagger}_{n},\phi^{\dagger}_{n^{\prime}}\\}=0.$ (3bbb)

As we move from site to site, these operators are identified with alternating
components of the continuum spinors,

$\displaystyle\psi_{0}(t_{0},x_{n})$
$\displaystyle\mapsto\frac{1}{\sqrt{2a\Omega(t_{0},x_{n})}}\phi_{n}\;\;(n\;\;\mathrm{even}),$
(3bca) $\displaystyle\psi_{1}(t_{0},x_{n})$
$\displaystyle\mapsto\frac{1}{\sqrt{2a\Omega(t_{0},x_{n})}}\phi_{n}\;\;(n\;\;\mathrm{odd}).$
(3bcb)

The normalization $\frac{1}{\sqrt{2a\Omega(t_{0},x_{n})}}$ recovers the
canonical anticommutation relations (3t) and (3u) from the lattice
anticommutation relations (3bb) in the continuum limit $a\to 0$. In addition,
it rescales $\phi_{n}$ to be dimensionless: one of $a$ or
$\Omega(t_{0},x_{n})$ has units of length, depending on whether in
$ds^{2}=g_{\mu\nu}(t,x)dx^{\mu}dx^{\nu}$ the coordinates $dx^{\mu}$ or the
metric components do.

Since this prescription involves individual components of the continuum
spinors, we need to commit to a representation of the gamma matrices in order
to use it. Our choice will be

$\displaystyle\gamma^{0}=-i\sigma^{z},$ $\displaystyle\gamma^{1}=\sigma^{y},$
$\displaystyle\gamma^{5}\equiv i\gamma^{0}\gamma^{1}=-i\sigma^{x}.$ (3bd)

Quadratic operators are transcribed onto the lattice as in [51]. Those which
couple differing components of $\psi(t,x)$ are mapped to nearest-neighbour
couplings such as $\phi^{\dagger}_{n+1}\phi_{n}$. Those that couple like
components are instead mapped to on-site couplings such as
$\phi^{\dagger}_{n}\phi_{n}$, with oscillating signs introduced while moving
from site to site as required by (3bd). Noting our slightly different
conventions from [51], we have

$\displaystyle\psi^{\dagger}(t_{0},x_{n})\gamma^{0}\psi(t_{0},x_{n})$
$\displaystyle\mapsto\frac{i}{2a\Omega(t,x)}\phi^{\dagger}_{n}\phi_{n}\;\;(n\;\;\mathrm{even}),$
(3bea) $\displaystyle\psi^{\dagger}(t_{0},x_{n})\gamma^{0}\psi(t_{0},x_{n})$
$\displaystyle\mapsto-\frac{i}{2a\Omega(t,x)}\phi^{\dagger}_{n}\phi_{n}\;\;(n\;\;\mathrm{odd}).$
(3beb)

Per [51] and our differing normalization of $\phi_{n}$, the derivative term in
the Hamiltonian (3q) maps to

$\frac{1}{2}\psi^{\dagger}(t_{0},x_{n})\overset{\leftrightarrow}{\partial_{1}}\gamma^{5}\psi(t_{0},x_{n})\mapsto-\frac{i}{2a^{2}\Omega(t,x)}\left(\phi^{\dagger}_{n}\phi_{n+1}-\phi^{\dagger}_{n+1}\phi_{n}\right)$
(3bf)

Comparing (3be) and (3bf) to the continuum Hamiltonian (3q) yields

$H_{\phi}(t)=\sum^{N}_{n}\left[\frac{i}{2a}\left(\phi_{n}^{\dagger}\phi_{n+1}-\phi^{\dagger}_{n+1}\phi_{n}\right)+(-1)^{n}\Omega(t,x_{n})m\phi^{\dagger}_{n}\phi_{n}\right],$
(3bg)

where $dx\mapsto a$ was adopted.

Thus regulated, such quadratic operators are well-defined on the lattice, but
their continuum limits $a\to 0$ will diverge. We extract defined quadratic
expectation values by way of the point-split definitions made in Section 2.2.
First, we map correlation functions onto the lattice, by first expanding their
spinor dependence into components using the representation (3bd), and then
applying (3bc), as illustrated in Appendix C. Starting from

$\langle\psi^{\dagger}_{i}(t_{0},x^{\prime})\psi_{j}(t_{0},x)\rangle=\lim_{a\to
0}\begin{cases}\left(2a\sqrt{\Omega(t,x)\Omega(t,x^{\prime})}\right)^{-1}\left\langle\phi^{\dagger}_{n}\phi_{n^{\prime}}\right\rangle,&\;n,i\;\mathrm{same}\;\mathrm{parity};\;n^{\prime},j\;\mathrm{same}\;\mathrm{parity}\\\
0&\mathrm{otherwise}\end{cases}$ (3bh)

we make the correspondences

$\displaystyle\Tr G^{(1)}(t,x,x^{\prime})$ $\displaystyle\mapsto\lim_{a\to
0}K_{I}(t,x,x^{\prime},a),$ $\displaystyle
K_{I}(t,x,x^{\prime},a)\equiv\frac{-(-1)^{n}}{2a}\left(\frac{\langle\phi^{\dagger}_{n^{\prime}}\phi_{n}\rangle}{\sqrt{\Omega(t,x^{\prime})\Omega(t,x)}}-\frac{\langle\phi^{\dagger}_{n^{\prime}+1}\phi_{n+1}\rangle}{\sqrt{\Omega(t,x^{\prime}+a)\Omega(t,x+a)}}\right),$
(3bia) $\displaystyle\Tr\gamma^{5}G^{(1)}(t,x,x^{\prime})$
$\displaystyle\mapsto\lim_{a\to 0}K_{5}(t,x,x^{\prime},a)$ $\displaystyle
K_{5}(t,x,x^{\prime},a)\equiv
i\frac{(-1)^{n}}{2a}\left(\frac{\langle\phi^{\dagger}_{n^{\prime}+1}\phi_{n}\rangle}{\sqrt{\Omega(t,x^{\prime}+a)\Omega(t,x)}}-\frac{\langle\phi^{\dagger}_{n^{\prime}}\phi_{n+1}\rangle}{\sqrt{\Omega(t,x^{\prime})\Omega(t,x+a)}}\right),$
(3bib) $\displaystyle\Tr\tilde{\gamma}_{0}(t,x)G^{(1)}(t,x,x^{\prime})$
$\displaystyle\mapsto\lim_{a\to 0}K_{0}(t,x,x^{\prime},a),$ $\displaystyle
K_{0}(t,x,x^{\prime},a)\equiv\frac{i}{2a}\left(\sqrt{\frac{\Omega(t,x)}{\Omega(t,x^{\prime})}}\langle\phi^{\dagger}_{n^{\prime}}\phi_{n}\rangle+\sqrt{\frac{\Omega(t,x^{\prime}+a)}{\Omega(t,x+a)}}\langle\phi^{\dagger}_{n^{\prime}+1}\phi_{n+1}\rangle\right),$
(3bic) $\displaystyle\Tr\tilde{\gamma}_{1}(t,x)G^{(1)}(t,x,x^{\prime})$
$\displaystyle\mapsto\lim_{a\to 0}K_{1}(t,x,x^{\prime},a),$ $\displaystyle
K_{1}(t,x,x^{\prime},a)\equiv-\frac{i}{2a}\left(\sqrt{\frac{\Omega(t,x)}{\Omega(t,x^{\prime}+a)}}\langle\phi^{\dagger}_{n^{\prime}+1}\phi_{n}\rangle+\sqrt{\frac{\Omega(t,x+a)}{\Omega(t,x^{\prime}+2a)}}\langle\phi^{\dagger}_{n^{\prime}+2}\phi_{n+1}\rangle\right).$
(3bid)

The correlation functions in (3bid) have been chosen so that the total number
of lattice units between sites is the same in both terms. This choice is made
so that the coincidence ($x^{\prime}\to x$) limit is correctly renormalized by
the locally determined terms (3ag). It is not necessary to do this in the case
of (3bib), because the coincidence limit is well-defined in that case. Notice
also that the coordinate dependencies of the $\Omega(t,x)$ prefactors are
located at the same point of the lattice correlator they are proportional to.
This is how, for example, the derivative term in (3agd) gets reproduced on the
lattice.

As alluded to, the expressions (3bi) are finite in the continuum ($a\to 0$)
limit, but the continuum quantities they approximate diverge in the subsequent
coincidence ($x^{\prime}\to x$) limit. We can renormalize the latter
divergence by subtracting the locally determined terms (3ag) after the
continuum limit is taken. In fact, we find empirically that the continuum and
coincidence limits can be renormalized simultaneously, by defining

$\displaystyle K_{I}^{\mathrm{ren}}(t,x,\Delta n,a)$ $\displaystyle\equiv
K_{I}(t,x,x+a\Delta n,a)-\frac{m}{2\pi}\ln\frac{1}{2}\mu\Omega^{2}(x)(a\Delta
n)^{2},$ (3bja) $\displaystyle K_{5}^{\mathrm{ren}}(t,x,\Delta n,a)$
$\displaystyle\equiv K_{5}(t,x,x+a\Delta n,a),$ (3bjb) $\displaystyle
K_{0}^{\mathrm{ren}}(t,x,\Delta n,a)$ $\displaystyle\equiv K_{0}(t,x,x+a\Delta
n,a)-\frac{\Omega_{,0}(t,x)}{2\pi\Omega(t,x)},$ (3bjc) $\displaystyle
K_{1}^{\mathrm{ren}}(t,x,\Delta n,a)$ $\displaystyle\equiv K_{1}(t,x,x+a\Delta
n,a)+\frac{1}{\pi(a(\Delta n+1))}-\frac{\Omega_{,1}(t,x)}{2\pi\Omega(t,x)}.$
(3bjd)

The limit $a\to 0$ with $\Delta n$ held fixed now also brings $x$ and
$x^{\prime}$ together. The remaining terms may depend on $\Delta n$, but do
not diverge in the relevant limit $\Delta n\to\infty$. The limiting process is
illustrated in Figure 6. In practice the $\Delta n$ dependence seems to be
quite weak.

Figure 6: Extracting the coincidence limit from lattice correlators. The
lattice correlator between sites at $x$ and $x+\Delta na$, with $\Delta n$ the
number of lattice sites separating the two operators and $a$ the lattice
spacing, is identified with the continuum correlator
$\langle\psi^{\dagger}_{i}(t,x+\Delta na)\psi_{0}(t,x)\rangle$, $i=\Delta
n\;\mathrm{mod}\;2$. Holding $\Delta n$ constant and comparing correlators on
successively refined lattices, we get functions of $r=\Delta na$ with the same
divergent terms in $r$ as their continuum analogues. Subtracting those
divergent terms (in practice by fitting them, up to a constant, to the
numerical data) leaves a remainder whose difference from the Hadamard
renormalized quadratic expectation value
$\langle\psi^{\dagger}_{i}(t,x)\psi_{0}(t,x)\rangle^{\mathrm{ren}}$ falls off
with $\Delta n$.

We can now make lattice correspondences to the Hadamard renormalized quadratic
operators by combining (3bj) in the $a\to 0$ and $\Delta n\to\infty$ limits
with (3af) and (3ae), to find

$\displaystyle\langle C_{I}^{\mathrm{ren}}\rangle(t,x)$
$\displaystyle=\lim_{\Delta n\to\infty}m\lim_{a\to
0}K_{I}^{\mathrm{ren}}(t,x,\Delta n,a),$ (3bka) $\displaystyle\langle
C_{5}^{\mathrm{ren}}\rangle(t,x)$ $\displaystyle=\lim_{\Delta
n\to\infty}\lim_{a\to 0}K_{5}^{\mathrm{ren}}(t,x,\Delta n,a),$ (3bkb)
$\displaystyle\langle j_{0}^{\mathrm{ren}}\rangle(t,x)$
$\displaystyle=\lim_{\Delta n\to\infty}\lim_{a\to
0}\left(K_{0}^{\mathrm{ren}}(t,x,\Delta n,a)+\frac{a}{2}(\Delta
n+1)\frac{\Omega_{,0}(t,x)}{\Omega(t,x)}K_{1}^{\mathrm{ren}}(t,x,\Delta
n,a)\right),$ (3bkc) $\displaystyle\langle j_{1}^{\mathrm{ren}}\rangle(t,x)$
$\displaystyle=\lim_{\Delta n\to\infty}\lim_{a\to
0}\left(K_{1}^{\mathrm{ren}}(t,x,\Delta n,a)-\frac{a}{2}(\Delta
n+1)\frac{\Omega_{,0}(t,x)}{\Omega(t,x)}K_{0}^{\mathrm{ren}}(t,x,\Delta
n,a)\right).$ (3bkd)

These expressions are now both well-defined and straightforward to compute
numerically. Indeed, we could now make a Jordan-Wigner transformation [30] to
exactly map the lattice Hamiltonian (3bg) and lattice Hadamard functions (3bi)
to expressions in terms of Pauli matrices while retaining their local
character. This enables simulations of the ground state of (3bg) and other
related states of interest using matrix product state techniques, as we
explore in Ref. [35] in the context of simulating interacting QFTs.

###  3.2 The Free Fermion Method

At present, however, since here we work with a quadratic fermion Hamiltonian
$H_{\phi}(t)$, we can exploit the free fermion formalism to completely and
efficiently characterize its (spontaneous) ground state, as well as thermal
states and more general Gaussian states, in terms of the equal-time two-point
correlation matrix

$C(t,n,n^{\prime})=\frac{1}{a}\langle\phi^{\dagger}_{n}\phi_{n^{\prime}}\rangle=\frac{1}{a}\Tr(\phi^{\dagger}_{n}\phi_{n^{\prime}}\rho(t)),$
(3bl)

where the possible time dependence comes from the Gaussian state $\rho(t)$
under consideration.

For instance, let us briefly recall how to characterize the instantaneous
ground state and thermal states of a more generic quadratic Hamiltonian
$H(t)\equiv\sum_{n,k}A_{nn^{\prime}}(t)\phi_{n}^{\dagger}\phi_{n^{\prime}}$
(with $A_{nn^{\prime}}(t)=A_{n^{\prime}n}(t)^{*}$). We proceed by first
identifying a unitary transformation $O(t)$ that diagonalizes
$A_{nn^{\prime}}$, that is $O(t)^{\dagger}A(t)O(t)=D(t)$ where $D(t)$ is a
diagonal matrix with single-particle energies $\lambda_{p}(t)$ in its
diagonal, that is $D_{pp^{\prime}}(t)=\delta_{pp^{\prime}}\lambda_{p}(t)$. The
unitary $O(t)$ in turn defines a unitary $U(t)$ acting on the Hilbert space of
the lattice such that it diagonalizes $H(t)$,

$U^{\dagger}(t)H(t)U(t)=\sum_{p}\lambda_{p}(t)c^{\dagger}_{p}(t)c_{p}(t),$
(3bm)

where $c_{p}(t)$ are a new set of fermionic operators. Then the two point
correlator of the Gaussian thermal state with inverse temperature $\beta$,

$\rho_{\beta}(t)\equiv\frac{1}{Z_{\beta}(t)}e^{-\beta
H(t)},~{}~{}~{}Z_{\beta}(t)\equiv\Tr e^{-\beta H(t)},$ (3bn)

is given by

$\left\langle
c^{\dagger}_{p}(t)c_{p^{\prime}}(t)\right\rangle=\delta_{p,p^{\prime}}\frac{e^{-\beta\lambda_{p}(t)}}{1+e^{-\beta\lambda_{p}(t)}},$
(3bo)

which implies

$C(t;n,n^{\prime})=\frac{1}{a}\langle\phi^{\dagger}_{n}\phi_{n^{\prime}}\rangle=\frac{1}{a}\sum_{p,p^{\prime}}O_{np}(t)\left\langle
c^{\dagger}_{p}(t)c_{p^{\prime}}(t)\right\rangle
O_{p^{\prime}n^{\prime}}^{\dagger}(t)=\frac{1}{a}\sum_{p}O_{np}(t)O_{n^{\prime}p}^{*}(t)\frac{e^{-\beta\lambda_{p}(t)}}{1+e^{-\beta\lambda_{p}(t)}}.$
(3bp)

In our applications below, we will consider the case of a time independent
Hamiltonian $H_{\phi}$. Correspondingly, the above expressions simplify to

$\displaystyle\rho_{\beta}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{Z_{\beta}}e^{-\beta
H_{\phi}},~{}~{}~{}Z_{\beta}\equiv\Tr e^{-\beta H_{\phi}},$ (3bq)
$\displaystyle C(n,n^{\prime})$ $\displaystyle=$
$\displaystyle\frac{1}{a}\langle\phi^{\dagger}_{n}\phi_{n^{\prime}}\rangle=\frac{1}{a}\sum_{p}O_{np}O_{n^{\prime}p}^{*}\frac{e^{-\beta\lambda_{p}}}{1+e^{-\beta\lambda_{p}}}.$
(3br)

We can now compare (3br) with e.g. (3bi) to estimate traces of
$G^{(1)}(x,x^{\prime})$. In the simplest case of $\Omega(t,x)=1$, zero mass
$m=0$, zero temperature ($\beta=\infty$) and infinite lattice length
$N\to\infty$, we can easily compute the two point lattice correlator exactly
to obtain

$C(n,n^{\prime})=\frac{i}{2\pi r}\left(-1+(-1)^{\Delta n}\right),$ (3bs)

where $r\equiv a\Delta n$. Comparing with (3bi) and then (3ae), we see that
the Hadamard divergences (3ag) are indeed obtained also on the lattice by
sending $a$ to 0 with $\Delta n$ held constant. More generally, we find
empirically that this is also true at finite mass $m>0$, finite temperature
($0<\beta<\infty$), and finite lattice length, as well as for space-dependent
$\Omega(x)$ (given that the normalizations (3bc) are taken into account).

##  4 Applications

In this section we apply the formalism described above to compute, in the
Hawking-Hartle vacua of the two-dimensional Minkowski, Schwarzschild, and de
Sitter spacetimes, expectation values of the Hadamard-renormalized quadratic
operators defined in (3af), namely those of the condensate $\langle
C_{I}^{\mathrm{ren}}(x)\rangle$, the current $\langle
j_{i}^{\mathrm{ren}}(x)$, and the pseudoscalar $\langle
C_{5}^{\mathrm{ren}}(x)\rangle$. Note that since all of the metrics we work in
will be manifestly static, our notation here differs from that preceding by
the suppression of temporal dependencies, e.g. $\langle
C_{I}^{\mathrm{ren}}(t,x)\rangle\to\langle C_{I}^{\mathrm{ren}}(x)\rangle$.

We will compute the Hawking-Hartle vacuum in four different coordinate charts.
First, we will find it in Minkowski spacetime, both as the $\beta\to\infty$
state set by the inertial observers
($\Omega^{\mathrm{Mink.}}_{\mathrm{global}}=1$) and as the
$\beta_{U}=\frac{2\pi}{\alpha}$ state of the Rindler observers,
$\Omega^{\mathrm{Mink.}}_{\mathrm{static}}(x)=e^{\alpha x}$. We will find
equivalent results in both cases despite the quite different lattice
Hamiltonians, and in particular will find smooth behaviour when approaching
the horizon in the Rindler chart, but only when $\beta=\beta_{U}$.

Next we will find it in Schwarzschild spacetime, (3at). We will find that at
the “Hawking” temperature $\beta_{H}=8\pi M$ results approach those of
Minkowski far from the black hole, and yield smooth behaviour approaching it.
Finally, we will find the “Bunch-Davies” vacuum in de Sitter spacetime,
$\Omega^{\mathrm{dS}}_{\mathrm{static}}(x)=\mathrm{Sech}(x/\alpha_{dS})$, and
again will show smooth behaviour approaching the horizon at the Unruh
temperature $\beta_{U}=2\pi\alpha_{dS}$.

###  4.1 Minkowski: Minkowski Vacuum

####  4.1.1 Inertial Frame

Figure 7: Approach to the coincidence limit of the renormalized point-split
operators (3ae) in the ground state within the inertial Minkowski chart,
computed on a finite and discrete lattice via the discretizations (3bj),
plotted against the inertial coordinate $X$. We use $m=0.6$. Darkening colours
indicate simultaneously decreasing point-separation $r$ and lattice spacing
$a$. Red lines marked with crosses show the results of the infinite size
($L\to\infty$) and coincidence $r\to 0$ extrapolations, themselves detailed in
the insets to the top left and bottom right plot.
$\mathcal{J}_{0}^{\mathrm{ren}}(X,X+r)$ (bottom left) and
$\mathcal{C}_{5}^{\mathrm{ren}}(X,X+r)$ (top right) vanish essentially to
machine precision even at finite separation and lattice size.
$\mathcal{J}_{1}^{\mathrm{ren}}(X,X+r)$ (bottom right) and
$\mathbb{C}_{I}^{\mathrm{ren}}(X,X+r)$ (top left) are nonzero on the lattice
at finite separation, but vanish in the continuum/coincidence limit.

We first discuss the somewhat trivial case of Minkowski spacetime in Minkowski
coordinates, $\Omega^{\mathrm{Mink.}}_{\mathrm{global}}=1$. The results in
this case (Figure 7) provide a baseline to compare our other results against.
Here, we have computed each of (3ae), by identifying each with its lattice
counterpart in (3bj). The superscript ren indicates that the locally-
determined terms (3ag) have been subtracted.

The coloured lines with circular markers show results at finite point-
separation, computed on a finite lattice with periodic boundary conditions,
after extrapolation to infinite $L$ as detailed in the top inset of the top
left panel. We have chosen $m=0.6$ here. The dependence on $X$ is artificial
due to the periodic boundary conditions, but we maintain it nevertheless for
easier comparison with subsequent results.

Since the locally determined Hadamard terms have been subtracted, the $r\to 0$
extrapolation depicted in the bottom inset of the top left plot in Figure 7
yields an estimate of the associated quadratic expectation value in the field
theory, $\eqref{eq:QuadraticRenormalized}$. We have not depicted the $\Delta
n\to\infty$ limit since it has no visible effect. The results here and
throughout are at $\Delta n=2$.

$\langle C_{5}\rangle$ and $\langle J_{0}\rangle$ already vanish on the
lattice. $\langle J_{1}\rangle$, which measures the current density of
fermions in the continuum limit, also appears to approach zero, though slowly;
the actual extrapolation in this case yields about $10^{-4}$, but can be made
smaller by extrapolating from smaller values of $r$. The condensate $\langle
C_{I}\rangle$ vanishes similarly.

####  4.1.2 Unruh Effect

Figure 8: Hadamard-renormalized point-split expectation values (3ae) computed,
once again in Minkowski spacetime with $m=0.6$, but this time using the Lass
chart (3ao). At $\beta=\beta_{U}$ this should, in the continuum limit, be the
same state as shown in Figure (7); indeed we qualitatively recover those
results, in the continuum limit. The top-left plot of $\langle
C_{I}^{\mathrm{ren}}(x)\rangle$ shows extrapolated results at a few choices of
$\beta_{U}$. The Hawking-Hartle vacuum at $\beta=\beta_{U}$ is the unique
state which does not diverge towards the horizon. It also recovers the
inertial values computed in Appendix D, shown as dashed black lines.

Thermal states defined by the accelerated Lass chart
$\Omega^{\mathrm{Mink.}}_{\mathrm{static}}(x)=e^{\alpha x}$ have the same two-
point functions as zero-temperature states defined by the inertial Minkowski
observers: this is the famous “Unruh effect”. We reproduce this on the lattice
by making the same calculations and extrapolations using that metric as we did
for the inertial observers. The results are depicted as Figure 8, where we
indeed qualitatively recover the same straight lines as we did in Figure 7.
Note that in this case the lattice is not translationally invariant, as
reflected in the very different curves away from the continuum limit.

The most interesting example is the condensate $\langle
C^{\mathrm{ren}}_{I}(x)\rangle$, depicted in the top left plot, since it does
not limit to zero. In this case we have plotted results both at the Unruh
temperature $\beta_{U}$ and at temperatures differing from the latter by
factors of 0.75 and 1.25, which thus do not yield the Hawking-Hartle vacuum.
These, indeed, visibly diverge as the horizon is approached with
$x\to-\infty$, while the $\beta_{U}$ curve remains smooth.

###  4.2 Schwarzschild: Hawking-Hartle Vacuum

Figure 9:  Hadamard-renormalized point-split expectation values (3ae) computed
in Schwarzschild spacetime at the Hawking temperaure $\beta_{H}=8\pi M$. All
operators except $\langle C_{I}^{\mathrm{ren}}\rangle(x)$ fall off to zero in
the continuum/coincidence limit. The top-left plot of $\langle
C_{I}^{\mathrm{ren}}\rangle(x)$ again shows results both at and slightly away
from the Hawking increasing temperature $\beta_{H}$, with brightening colours
indicating increasing temperature (and thus decreasing $\beta$). Approaching
the black hole, the results away from $\beta_{H}$ diverge, but those at
$\beta_{H}$ converge to a finite horizon value.

Now we repeat the calculation in Schwarzschild spacetime in tortoise
coordinates (3at). The Hawking-Hartle vacuum is now no longer the Minkowski
vacuum, although it approaches the latter far from the black hole
$x\to\infty$. Again, all operators but the condensate vanish in the continuum
limit, albeit nontrivially (the small bump in $\langle J_{0}(x)\rangle$ around
$x=0$ shrinks with the smallest value of $r$ used in the extrapolation). Note
the full current is conserved, $\langle J_{\mu}(x)\rangle^{;\mu}=0$. For a
time-independent metric the stipulation that time derivatives ought to vanish
as they must in the Hawking-Hartle vacuum, $\langle J_{0}(x)\rangle^{,0}$,
implies $\langle J_{1}(x)\rangle=0$ as well.

The top-left plot again shows the condensate at temperatures at and near the
Hawking temperature $\beta_{H}=8\pi M$. Far from the black hole, results
approach the inertial Minkowski results at this temperature. Close to it, they
approach the Rindler results for a horizon with the same surface gravity. A
$x$ around 0, the Hawking-Hartle vacuum smoothly interpolates between these
asymptotes.

###  4.3 de Sitter: Bunch-Davies Vacuum

Figure 10: Hadamard-renormalized point-split expectation values (3ae) computed
in de Sitter spacetime at the Unruh temperaure $\beta_{U}=2\pi\alpha_{dS}$.
Here we take $\alpha_{dS}=\frac{1}{2\pi}$. Once again, all operators except
$\langle C_{I}^{\mathrm{ren}}\rangle(x)$ fall off to zero, and once again, the
top-left plot of the latter shows results both at and slightly away from the
Unruh temperature $\beta_{U}$. Here we used $\delta n=2$. Notice the slight
asymmetry about $x=0$ of the unconverged data, due to $r$ always being added
to $x$ (rather than subtracted).

We finally turn to the exponentially expanding de Sitter universe, whose
static patch is covered by
$\Omega^{\mathrm{dS}}_{\mathrm{static}}(x)=\mathrm{Sech}{\frac{x}{\alpha}}$.
The Hawking-Hartle vacuum in this case is called the “Bunch-Davies vacuum”.
Again, everything except the condensate vanishes in the field theory limit,
whereas the condensate smoothly approaches the horizon only at the Unruh
temperature.

##  5 Conclusion

Let us review before parting ways. We have illustrated, by simple computations
using free Dirac fermions, a method to estimate two-point functions and
quadratic expectation values of quantum expectation values upon curved 1+1
dimensional backgrounds. To get finite expectation values we made use of
Hadamard renormalization, which perhaps unexpectedly can be applied directly
to lattice data, provided the latter be computed holding the lattice length
and separation constant. We used these results to demonstrate the Unruh
effect, and to compute expectation values in de Sitter spacetime.

Our eventual goal is to study time-dependent processes for interacting
theories in 3+1, which presents three separate avenues for future study.
Moving to 3+1 requires us to consider more complicated lattices and to use a
different set of Hadamard terms. The numerical study of time-dependent
processes is methodologically similar to that of finite temperature states,
but doing so in curved spacetime raises the issue of what evolution equation
to use. If the Schr odinger equation is construed as a projection of the
Wheeler-de Witt equation upon a spacetime foliation, it picks up decohering
nonlinear terms which are not present if it is instead taken as a first
principle. A numerical study would allow the effects of these terms to be
analyzed in detail.

The presence of interactions raises two issues. In this case the free fermion
numerical method is no longer available, so we must resort to more
complicated, though well-understood, tensor network techniques, which are
especially efficient in one spatial dimension. A more serious conceptual
difficulty arises when attempting to take the continuum limit. In an
interacting theory the appropriate operator weights that terms in the
Hamiltonian must incur to yield lines of constant physics are difficult to
predict a priori. We mean to study this issue in a future paper.

##  6 Acknowledgments

A. G. M. Lewis is supported by the Tensor Network Initiative at Perimeter
Institute. Research at Perimeter Institute is supported by the Government of
Canada through the Department of Innovation, Science and Economic Development
Canada and by the Province of Ontario through the Ministry of Research,
Innovation and Science. G. Vidal is a CIFAR fellow in the Quantum Information
Science Program. X is formerly known as Google[x] and is part of the Alphabet
family of companies, which includes Google, Verily, Waymo, and others
(www.x.company).

##  Appendix A Coordinate expansion of spin parallel propagator

In this Appendix we find (3ad) by expanding the spin parallel propagator
$\mathcal{J}(t,x,t^{\prime},x^{\prime})$ defined by (3ab) to linear order in
$r\equiv x^{\prime}-x$ with $t^{\prime}=t$ in the manifestly conformal metric
(2). Writing the Ansatz

$\mathcal{J}(t,x,t,x^{\prime})=\mathbbm{1}+j(t,x)r+\ldots,$ (3bt)

this task amounts to finding $j(t,x)$. The partial derivatives of (3bt) in a
coordinate basis are

$\displaystyle\mathcal{J}_{,0}(t,x,t,x^{\prime})$
$\displaystyle=j_{,0}(t,x)r,$ (3bua)
$\displaystyle\mathcal{J}_{,1}(t,x,t,x^{\prime})$
$\displaystyle=j_{,1}(t,x)r-j(t,x).$ (3bub)

In this Appendix we will adopt the standard practice of denoting coincidence
limits with square brackets, e.g.

$\left[\mathcal{J}(t,x,t^{\prime},x^{\prime})\right]=\lim_{(t^{\prime},x^{\prime})\to(t,x)}\mathcal{J}(t,x,t^{\prime},x^{\prime}),$
(3bv)

where as always the limits are understood to be along the coincident geodesic.
Applied to (3bu) we find

$\left[\mathcal{J}_{,\mu}(t,x,t,x^{\prime})\right]=-\delta_{\mu,1}j(t,x).$
(3bw)

On the other hand, using the spin covariant derivatives listed in (3i), we
find

$\left[\mathcal{J}_{;1}(t,x,t^{\prime},x^{\prime})\right]=\left[\mathcal{J}_{,1}(t,x,t^{\prime},x^{\prime})\right]-\frac{1}{2}\frac{\Omega_{,0}(t,x)}{\Omega(t,x)}\gamma^{0}\gamma^{1}.$
(3bx)

Taking the covariant derivative of the definition (3ab), we next find

$\mathcal{J}_{;\mu\nu}(t,x,t^{\prime},x^{\prime})\sigma^{;\mu}(t,x,t^{\prime},x^{\prime})+\mathcal{J}_{;\mu}(t,x,t^{\prime},x^{\prime})\sigma\indices{{}^{;\mu}_{\nu}}(t,x,t^{\prime},x^{\prime})=0$
(3by)

Coincidence limits of $\sigma(t,x,t^{\prime},x^{\prime})$ and its derivatives
are documented in many sources, e.g. [1, 15, 44]. The ones we need are

$\displaystyle\left[\sigma^{;\mu}(t,x,t^{\prime},x^{\prime})\right]$
$\displaystyle=0,$ (3bza)
$\displaystyle\left[\sigma\indices{{}^{;\mu}_{\nu}}(t,x,t^{\prime},x^{\prime})\right]$
$\displaystyle=g\indices{{}^{;\mu}_{\nu}}(t,x).$ (3bzb)

In light of (3bz), taking the coincidence limit of both sides of (3by) yields

$\left[\mathcal{J}_{;\nu}(t,x,t^{\prime},x^{\prime})\right]=0.$ (3ca)

Combining (3ca), (3bw), and (3by) we find

$j(t,x)=-\frac{1}{2}\frac{\Omega_{,0}(t,x)}{\Omega(t,x)}\gamma^{0}\gamma^{1}\mathbbm{1}$
(3cb)

which, inserted back into (3bt), yields the desired result (3ad).

##  Appendix B ADM derivation of Hamiltonian

In this Appendix we compute (3q) from (3s) and (3r). We thus consider a
foliation of spacetime by hypersurfaces $\Sigma_{t}$, upon each of which the
spacetime coordinate $t$ is held constant. The hypersurfaces will be charted
by the same spatial coordinate $x$ as the full spacetime, see e.g. [6, 3, 8].
The integration measure on each hypersurface is

$d\Sigma^{\mu}=-dx\sqrt{h(t,x)}n^{\mu}(t,x),$ (3cc)

where $\nu^{\mu}$ is the unit ($\nu^{\mu}\nu_{\mu}=-1$) normal vector to the
foliation, and $h(t,x)$ is the determinant of the intrinsic metric
$h_{\mu\nu}(t,x)$ to the hypersurfaces,

$h_{\mu\nu}(t,x)\equiv g_{\mu\nu}(t,x)+n_{\mu}(t,x)n_{\nu}(t,x).$ (3cd)

We wish to generate Lie transport of the Dirac field in the direction of
constant $x$, and thus choose $\xi^{\mu}(t,x)$ to be the constant coordinate
vector, which points along lines of constant $x$ and is normalized by

$\xi^{\mu}(t,x)t_{;\mu}=1.$ (3ce)

Now define the lapse function $\alpha(t,x)$ and shift vector
$\beta^{\mu}(t,x)$ by

$\xi^{\mu}(t,x)=\alpha(t,x)n^{\mu}(t,x)+\beta^{\mu}(t,x).$ (3cf)

We have

$\displaystyle n^{\mu}=(1/\alpha,-\beta^{i}/\alpha),$ $\displaystyle
n_{\mu}=(-\alpha,0),$ (3cg)
$g_{\mu\nu}(t,x)=\begin{pmatrix}-\alpha^{2}(t,x)+\beta_{k}(t,x)\beta^{k}(t,x)&&\beta_{i}(t,x)\\\
\beta_{j}(t,x)&&h_{ij}(t,x)\end{pmatrix}$ (3ch)

where indices $i,j,\ldots$ run over space only. In the manifestly conformal
chart (2) we can read off

$\displaystyle\beta^{\mu}(t,x)$ $\displaystyle=0,$ $\displaystyle
n^{\mu}(t,x)=(\Omega^{-1}(t,x),\;0),$ (3ci) $\displaystyle\alpha(t,x)$
$\displaystyle=\Omega(t,x),$ $\displaystyle\xi^{\mu}(t,x)=(1,\;0).$ (3cj)

Noting

$\sqrt{h(t,x)}=\Omega(t,x)$ (3ck)

we find from (3s)

$H(t)=-\int dxT_{00}(t,x)$ (3cl)

and from (3r)

$T_{00}(t,x)=\frac{1}{2}\bar{\psi}(t,x)\tilde{\gamma}_{0}\overset{\leftrightarrow}{\nabla}_{0}\psi(t,x).$
(3cm)

In a sense this completes the derivation, but the time derivatives are
inconvenient for us. To eliminate them, we need to use the equations of motion
(Dirac equations) for $\psi(t,x)$ and $\bar{\psi}(t,x)$, which themselves are
found by varying the action (3l) with respect to the fields. In the presently
most convenient form, they are

$\displaystyle\psi_{;0}(t,x)$
$\displaystyle=\gamma^{0}\gamma^{1}\psi_{;1}(t,x)-m\tilde{\gamma}^{0}(t,x)\psi(t,x)$
(3cna) $\displaystyle\bar{\psi}_{;0}(t,x)$
$\displaystyle=-\bar{\psi}_{;1}(t,x)\gamma^{0}\gamma^{1}+m\bar{\psi}(t,x)\tilde{\gamma}_{0}(t,x)$
(3cnb)

Applying (3cn) to (3cm) yields

$T_{00}=\Omega(t,x)\bar{\psi}(t,x)\left(\frac{1}{2}\gamma^{1}\overset{\leftrightarrow}{\nabla}_{1}-\Omega(t,x)m\right)\psi(t,x),$
(3co)

which, in light of (3cl), indeed recovers (3q).

##  Appendix C Spinor Components

For the sake of simplicity, in the main text we have mostly suppressed the
spinor indices to lighten an otherwise already heavy notation. In this
Appendix we will rewrite a few key expressions explicitly in terms of spinor
components, given the important role the two spinor components play in the
staggered lattice discretization. Recall that we use the gamma matrix
representation $\gamma^{0}=-i\sigma^{z}=-\gamma_{0}$,
$\gamma^{1}=\sigma^{y}=\gamma_{1}$ and $\gamma^{5}\equiv
i\gamma^{0}\gamma^{1}=-i\sigma^{x}$.

In two spacetime dimensions the Dirac field $\psi(t,x)$ is a two-component
column spinor

$\psi(t,x)=\left(\begin{array}[]{c}\psi_{0}(t,x)\\\
\psi_{1}(t,x)\end{array}\right)$ (3cr)

whereas its adjoint conjugate $\psi^{\dagger}(t,x)$ and Dirac conjugate
$\bar{\psi}(t,x)$ correspond to the two-component row spinors

$\psi^{\dagger}(t,x)=\left(\begin{array}[]{cc}\psi^{\dagger}_{0}(t,x)&\psi^{\dagger}_{1}(t,x)\end{array}\right),~{}~{}~{}~{}\bar{\psi}(t,x)\equiv
i\psi^{\dagger}(t,x)\gamma^{0}=\left(\begin{array}[]{cc}\psi^{\dagger}_{0}(t,x)&-\psi^{\dagger}_{1}(t,x)\end{array}\right).$
(3cu)

Then the Hadamard function $G^{(1)}(t,x,t^{\prime},x^{\prime})$ is a $2\times
2$ matrix given by

$\displaystyle G^{(1)}(t,x,t^{\prime},x^{\prime})$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}\left\langle\left[\left(\begin{array}[]{c}\psi_{0}(t,x)\\\
\psi_{1}(t,x)\end{array}\right),\left(\begin{array}[]{cc}\psi^{\dagger}_{0}(t^{\prime},x^{\prime})&-\psi^{\dagger}_{1}(t^{\prime},x^{\prime})\end{array}\right)\right]\right\rangle$
(3cy) $\displaystyle=$
$\displaystyle\frac{1}{2}\left\langle\left(\begin{array}[]{cc}\left[\psi_{0}(t,x),\psi^{\dagger}_{0}(t^{\prime},x^{\prime})\right]&\left[\psi_{0}(t,x),-\psi^{\dagger}_{1}(t^{\prime},x^{\prime})\right]\\\
&\\\
\left[\psi_{1}(t,x),\psi^{\dagger}_{0}(t^{\prime},x^{\prime})\right]&\left[\psi_{1}(t,x),-\psi^{\dagger}_{1}(t^{\prime},x^{\prime})\right]\end{array}\right)\right\rangle.$
(3dc)

It is useful to compute the following traces involving
$G^{(1)}(t,x,t^{\prime},x^{\prime})$ and $\gamma^{5}=-i\sigma^{x}$,
$\gamma_{0}=i\sigma^{z}$ and $\gamma_{1}=\sigma^{y}$,

$\displaystyle\Tr\left(G^{(1)}(t,x,t^{\prime},x^{\prime})\right)$
$\displaystyle=\frac{-1}{2}\left(\langle[\psi^{\dagger}_{0}(t,x^{\prime}),\psi_{0}(t,x)]\rangle-\langle[\psi^{\dagger}_{1}(t,x^{\prime}),\psi_{1}(t,x)]\rangle\right),$
(3dda)
$\displaystyle\Tr\left(\gamma^{5}G^{(1)}(t,x,t^{\prime},x^{\prime})\right)$
$\displaystyle=\frac{i}{2}\left(\langle[\psi^{\dagger}_{0}(t^{\prime},x^{\prime}),\psi_{1}(t,x)]\rangle-\langle[\psi^{\dagger}_{1}(t^{\prime},x^{\prime}),\psi_{0}(t,x)]\rangle\right),$
(3ddb)
$\displaystyle\Tr\left(\gamma_{0}G^{(1)}(t,x,t^{\prime},x^{\prime})\right)$
$\displaystyle=\frac{-i}{2}\left(\langle[\psi^{\dagger}_{0}(t^{\prime},x^{\prime}),\psi_{0}(t,x)]\rangle+\langle[\psi^{\dagger}_{1}(t^{\prime},x^{\prime}),\psi_{1}(t,x)]\rangle\right),$
(3ddc)
$\displaystyle\Tr\left(\gamma_{1}G^{(1)}(t,x,t^{\prime},x^{\prime})\right)$
$\displaystyle=\frac{i}{2}\left(\langle[\psi^{\dagger}_{0}(t^{\prime},x^{\prime}),\psi_{1}(t,x)]\rangle+\langle[\psi^{\dagger}_{1}(t^{\prime},x^{\prime}),\psi_{0}(t,x)]\rangle\right).$
(3ddd)

In order to reproduce these calculations, it is helpful to note

$\mathrm{Tr}\left(\mathcal{O}\psi\bar{\psi}\right)=-\bar{\psi}\mathcal{O}\psi=-i\psi^{\dagger}\gamma^{0}\mathcal{O}\psi=-\psi^{\dagger}\sigma^{z}\mathcal{O}\psi,$
(3de)

which readily implies

$\displaystyle\mathrm{Tr}\left(\psi\bar{\psi}\right)$
$\displaystyle=-\bar{\psi}\psi=-\psi^{\dagger}\sigma^{z}\psi=\left(\begin{array}[]{cc}\psi^{\dagger}_{0}&\psi^{\dagger}_{1}\end{array}\right)\left(\begin{array}[]{cc}-1&0\\\
0&1\end{array}\right)\left(\begin{array}[]{c}\psi_{0}\\\
\psi_{1}\end{array}\right)$ (3dff)
$\displaystyle=-\psi^{\dagger}_{0}\psi_{0}+\psi^{\dagger}_{1}\psi_{1},$ (3dfg)
$\displaystyle\mathrm{Tr}\left(\gamma^{5}\psi\bar{\psi}\right)$
$\displaystyle=-\bar{\psi}\gamma^{5}\psi=-\psi^{\dagger}\sigma^{z}\gamma^{5}\psi=\psi^{\dagger}(-\sigma^{y})\psi=\left(\begin{array}[]{cc}\psi^{\dagger}_{0}&\psi^{\dagger}_{1}\end{array}\right)\left(\begin{array}[]{cc}0&i\\\
-i&0\end{array}\right)\left(\begin{array}[]{c}\psi_{0}\\\
\psi_{1}\end{array}\right)$ (3dfm)
$\displaystyle=i\psi^{\dagger}_{0}\psi_{1}-i\psi^{\dagger}_{1}\psi_{0},$
(3dfn) $\displaystyle\mathrm{Tr}\left(\gamma_{0}\psi\bar{\psi}\right)$
$\displaystyle=-\bar{\psi}\gamma_{0}\psi=-\psi^{\dagger}\sigma^{z}\gamma_{0}\psi=-\psi^{\dagger}(-iI)\psi=\left(\begin{array}[]{cc}\psi^{\dagger}_{0}&\psi^{\dagger}_{1}\end{array}\right)\left(\begin{array}[]{cc}-i&0\\\
0&-i\end{array}\right)\left(\begin{array}[]{c}\psi_{0}\\\
\psi_{1}\end{array}\right)$ (3dft)
$\displaystyle=-i\psi^{\dagger}_{0}\psi_{0}-i\psi^{\dagger}_{1}\psi_{1},$
(3dfu) $\displaystyle\mathrm{Tr}\left(\gamma_{1}\psi\bar{\psi}\right)$
$\displaystyle=-\bar{\psi}\gamma_{1}\psi=-\psi^{\dagger}\sigma^{z}\gamma_{1}\psi=-\psi^{\dagger}(i\sigma^{x})\psi=\left(\begin{array}[]{cc}\psi^{\dagger}_{0}&\psi^{\dagger}_{1}\end{array}\right)\left(\begin{array}[]{cc}0&i\\\
i&0\end{array}\right)\left(\begin{array}[]{c}\psi_{0}\\\
\psi_{1}\end{array}\right)$ (3dfaa)
$\displaystyle=i\psi^{\dagger}_{0}\psi_{1}+i\psi^{\dagger}_{1}\psi_{0}.$
(3dfab)

The condensate, pseudo-scalar, and current operators can be similarly
evaluated and read

$\displaystyle C_{I}(t,x)$ $\displaystyle\equiv
m\bar{\psi}(t,x)\psi(t,x)=m\left(\psi^{\dagger}_{0}(t,x)\psi_{0}(t,x)-\psi^{\dagger}_{1}(t,x)\psi_{1}(t,x)\right),$
(3dga) $\displaystyle C_{5}(t,x)$
$\displaystyle\equiv\bar{\psi}(t,x)\gamma^{5}\psi(t,x)=-i\psi^{\dagger}_{0}(t,x)\psi_{1}(t,x)+i\psi^{\dagger}_{1}(t,x)\psi_{0}(t,x),$
(3dgb) $\displaystyle j_{0}(t,x)$
$\displaystyle\equiv\bar{\psi}(t,x)\tilde{\gamma}_{0}(t,x)\psi(t,x)=i\Omega(t,x)\left(\psi^{\dagger}_{0}(t,x)\psi_{0}(t,x)+\psi^{\dagger}_{1}(t,x)\psi_{1}(t,x)\right),$
(3dgc) $\displaystyle j_{1}(t,x)$
$\displaystyle\equiv\bar{\psi}(t,x)\tilde{\gamma}_{1}(t,x)\psi(t,x)=-i\Omega(t,x)\left(\psi^{\dagger}_{0}(t,x)\psi_{1}(t,x)+\psi^{\dagger}_{1}(t,x)\psi_{0}(t,x)\right).$
(3dgd)

##  Appendix D Minkowski-Spacetime Calculation of Hadamard-Renormalized
Condensate

In this Appendix we calculate the Hadamard renormalized condensate $\langle
C^{\mathrm{ren}}_{I}\rangle$ defined by (3af), in the inertial vacuum of
Minkowski spacetime. The equal-time Hadamard function can be found in e.g.
[43],

$G^{(1)}(t,x,x^{\prime})=-\left(i\gamma^{\mu}\partial_{\mu}+m\mathbbm{1}\right)\int\frac{dp}{2\pi}\frac{1}{2\sqrt{p^{2}+m^{2}}}e^{-ipr},$
(3dh)

where $r=x^{\prime}-x$. Suppressing the artificial $t$ dependence, the point-
split condensate defined by (3ae) is

$\displaystyle\mathbbm{C}_{I}(x,x^{\prime})$
$\displaystyle=m\mathrm{Tr}\left(i\gamma^{\mu}\partial_{\mu}+m\mathbbm{1}\right)\int\frac{dp}{2\pi}\frac{1}{2\sqrt{p^{2}+m^{2}}}e^{-ipr}$
(3di)
$\displaystyle=\int\frac{dp}{2\pi}\frac{m^{2}}{\sqrt{p^{2}+m^{2}}}e^{-ipr}.$
(3dj)

Now the modified Bessel function of the first kind, $K_{0}(\omega)$, may be
expressed as

$K_{0}(\omega)=\frac{1}{2}\int db\frac{e^{i\omega b}}{\sqrt{b^{2}+1}},$ (3dk)

so that

$\mathbbm{C}_{I}(x,x^{\prime})=-\frac{m^{2}}{\pi}K_{0}(mr)=\frac{m^{2}}{2\pi}\ln\mu_{0}\frac{1}{2}r^{2}+\mathcal{O}(r^{2})$
(3dl)

where $\mu_{0}=\frac{1}{2}m^{2}e^{2\gamma_{E}}$ as in (3ah), and $\gamma_{E}$
is the Euler-Mascheroni constant. In the Minkowski metric, the locally
determined term in (3ag) is

$\mathbbm{C}^{\mathrm{loc}}_{I}(x,x^{\prime})=\frac{m^{2}}{2\pi}\ln\mu\frac{1}{2}r^{2},$
(3dm)

and so choosing $\mu=\mu_{0}$ we have

$\langle C_{I}^{\mathrm{ren}}\rangle=\lim_{r\to
0}\left(\mathbbm{C}_{I}(x,x^{\prime})-\mathbbm{C}^{\mathrm{loc}}_{I}(x,x^{\prime})\right)=0.$
(3dn)

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