11institutetext: Yasaman K. Yazdi 22institutetext: Imperial College London,
Blackett Laboratory, SW7 2AZ, London, UK,
22email<EMAIL_ADDRESS>
# Entanglement Entropy and Causal Set Theory

Yasaman K. Yazdi

###### Abstract

We review a formulation of the entanglement entropy of a quantum scalar field
in terms of its spacetime two-point correlation functions. We discuss
applications of this formulation to studying entanglement entropy in various
settings in causal set theory. These settings include sprinklings of causal
diamonds in various dimensions in flat spacetime, de Sitter spacetime,
massless and massive theories, multiple disjoint regions, and nonlocal quantum
field theories.

###### Keywords:

Entanglement Entropy, Algebraic Quantum Field Theory, Covariant
Regularization, Correlation Functions, Black Hole Thermodynamics, Green
Functions, Spacetime Domains, Nonlocality, Poisson Fluctuations

##  1 Introduction

Entanglement entropy is a useful measure of our limited access to quantum
fields’ degrees of freedom. This limited access can occur for example in the
presence of an event horizon, where correlations between field values at
points in the interior and exterior of the event horizon, $\langle
0|\phi(x_{int})\phi(x_{ext})|0\rangle$, are no longer available.

Entanglement entropy is one of the most important concepts in quantum gravity.
It naturally combines both quantum (entanglement) and gravitational (area
laws) properties. It was originally inspired by the search for a fundamental
understanding of black hole entropy: we know that black holes classically have
the Bekenstein-Hawking entropy bekenstein ; hawking  associated to them, which
scales as the spatial area of the event horizon, but we do not know what the
fundamental or microscopic origin of this entropy is (e.g. in the statistical
mechanical sense of what the microstates leading to this entropy are). It is
one of the important tasks of quantum gravity to provide insight into this.
Entanglement entropy, as first shown in Sorkin:1984kjy , also generically
scales as the area of the boundary of the entangling region (which in a black
hole spacetime is the area of the event horizon). Hence, it is a promising
direction to investigate this question. Ultimately, we expect entanglement
entropy to contribute to black hole entropy; the open question is whether or
not it will be the dominant contribution.

Let us now review some general aspects of entanglement entropy. Entanglement
entropy is conventionally defined using a density matrix $\rho$, which is
initially pure, meaning that we have full information about the quantum system
and the von Newumann entropy vanishes:

$S=-\text{Tr}\rho_{\text{pure}}\ln\rho_{\text{pure}}=0.$ (1)

Subsequently, we trace out of the density matrix the parts of the system that
we do not have information about. Traditionally this tracing out is done on a
spatial hypersurface $\Sigma$, as in Fig. 1, which is divided into two
complementary subregions, region $A$ and region $B$, one of which represents
the degrees of freedom we do have access to and the other the degrees of
freedom that we do not have access to. After we trace out the degrees of
freedom in one of the subregions, for example those in $B$111We would get the
same answer if we instead traced out the degrees of freedom in $A$ to get
$\rho_{B}$ and computed $S_{B}$. We refer to this as complementarity below.,
we get a reduced density matrix

$\rho_{A}=\text{Tr}_{B}\rho,$ (2)

and the entropy of entanglement between region $A$ and $B$ is defined as
Sorkin:1984kjy

$S_{A}=-\text{Tr}\rho_{A}\ln\rho_{A}\ .$ (3)

Figure 1: A spatial hypersurface $\Sigma$ divided into two complementary
subregions $A$ and $B$.

It is crucial that the theory one is working with has a UV cutoff with respect
to which we count how many degrees of freedom we do or do not have access to.
Without a UV cutoff, we would get an infinite answer for the entanglement
entropy in (3).

Since the original work Sorkin:1984kjy  in the context of black hole entropy,
entanglement entropy has found many additional useful applications in other
areas of physics such as quantum information Wootters:1997id  and condensed
matter physics Laflorencie:2015eck . Depending on the specific application in
mind, different techniques may be used to evaluate the entropy in (3). This
choice of technique is often motivated by physical and computational
considerations.

In causal sets, there is no analogue of field data on a spatial hypersurface
(i.e. a Cauchy surface). Fig. 2 illustrates the reason for this, which is
essentially that there is no guarantee that there will not be relations that
will not make an imprint on a subset of unrelated elements. Therefore we
cannot work on a hypersuface as in Fig. 1 and must use an intrinsically
spacetime approach to compute the entanglement entropy. Fortunately, a
spacetime definition of entanglement entropy, in terms of the spacetime two-
point correlation function, exists and can be used in causal set calculations.
We review this formulation in Section 2. While we are led to a spacetime
formulation of entanglement entropy in causal set theory out of necessity,
there are in fact numerous attractive features of working with a spacetime
definition of entanglement entropy. For example, quantum fields themselves are
really spacetime quantities (their domain is spacetime). Therefore, it is more
natural to study them in spacetime. Furthermore, quantum fields are highly
singular and may not always admit meaningful restrictions to spatial
hypersurfaces. We devote Section 3 to supporting these statements with some
concrete examples. Finally, in Section 4 we discuss several calculations of
entanglement entropy, in settings including sprinklings into flat spacetime
and de Sitter spacetime, before ending in Section 5 with a discussion of some
subtleties of and the future directions for this work.

Figure 2: The pink elements form a maximal antichain, which is the largest set
of unrelated elements such that every other element is to the past or future
of at least one of the elements of this set. This is the analogue of a spatial
hypersurface in a causal set. In order for it to be a viable analogue of a
Cauchy surface, we must be able to deduce all causal relations using relations
involving the antichain elements. However, this is not possible, as
illustrated by the counterexample above: the causal relation represented by
the dashed line has no imprint on the antichain, hence we do not have an
analogue of a Cauchy hypersurface in a causal set.

##  2 Entanglement Entropy from Spacetime Two-point Correlation Functions

A quantum field theory with local operators is typically fully determined by
the set of all its n-point correlation functions. If we consider a scalar
field $\phi$, these are $\langle 0|\phi(x_{1})\phi(x_{2})|0\rangle$,… $\langle
0|\phi(x_{1})\phi(x_{2})\dots\phi(x_{n})|0\rangle$, where
$\\{x_{1},x_{2},...\\}$ represent points in spacetime. In a Gaussian quantum
field theory, things are much simpler because we only need to know the two-
point correlation function $\langle 0|\phi(x_{1})\phi(x_{2})|0\rangle$. In
this case, all the higher n-point correlation functions are derivable from the
two-point function through Wick’s theorem. For the remainder of this chapter,
except where explicitly mentioned otherwise, we will restrict the discussion
to Gaussian scalar field theories.

###  2.1 Entropy from Correlation Functions

Since we are focusing on a Gaussian scalar field theory, as mentioned above,
everything (including the entanglement entropy (3)) must be expressible in
terms of the two-point correlation function $\langle
0|\phi(x)\phi(x^{\prime})|0\rangle$. The definition of entropy introduced in
Sorkin:2012sn , which we review in this subsection, does precisely that.
Specifically, the entropy is given by the sum

$S=\sum_{\lambda}\lambda\ln|\lambda|$ (4)

over the solutions $\lambda$ to the generalized eigenvalue problem

$W\,v=i\lambda\;\Delta\,v,\ $ (5)

while excluding components in the kernel of $\Delta$

$\Delta v\neq 0~{}.$ (6)

$W$ in (5) is the spacetime two-point correlation function or Wightman
function, $W(x,x^{\prime})=\langle 0|\phi(x)\phi(x^{\prime})|0\rangle$, and
$i\Delta$ is the Pauli-Jordan function or spacetime commutator of the field,
$i\Delta(x,x^{\prime})=[\phi(x),\phi(x^{\prime})]$. In a Gaussian theory,
$i\Delta$ is a c-number. We can obtain $\Delta$ using the retarded Green
function, $G_{R}$, as
$\Delta(x,x^{\prime})=G_{R}(x,x^{\prime})-G_{R}(x^{\prime},x)$. If we already
have a state and therefore a $W$ to work with, we can also obtain it through
the imaginary or anti-symmetric part of $W$, i.e.
$i\Delta(x,x^{\prime})=W(x,x^{\prime})-W(x^{\prime},x)=2~{}\text{Im}(W(x,x^{\prime}))$.

More precisely, we must start with a global state (and its corresponding $W$)
which is initially pure. In terms of the eigenvalue equation (5) and entropy
(4), purity would mean solutions $\lambda$ that are $1$’s and $0$’s. In the
next subsection we review one choice of pure state, the Sorkin-Johnston state,
that can be defined in causal set theory. With a pure state at hand, we
subsequently exclude parts of the quantum system that we don’t have access to
by restricting the elements $x$ and $x’$ in $W(x,x^{\prime})$ and
$i\Delta(x,x^{\prime})$ to lie in the _spacetime_ subregion that we do have
access to, before solving (5). Then we can interpret the resulting entropy
from substituting the solutions into (4) as the entanglement entropy between
the spacetime region which is the domain of $x$ and $x^{\prime}$ and its
causal complement. Note that the causal complement will not in general be the
complementary spacetime domain (see e.g. Fig. 3).

Figure 3: Causally complementary spacetime subregions $A$ and $B$ in a causal
set.

It is in this way, owing to $W(x,x^{\prime})$ and $i\Delta(x,x^{\prime})$
being spacetime functions, that this formulation of entanglement entropy is a
spacetime approach. Additionally, due to its spacetime nature it allows one to
use a spacetime UV cutoff such as the discreteness scale of a causal set. This
can give our counting of degrees of freedom a covariance and universality that
is not present in the more spatial formulations.

We can in some cases compare the results obtained from this method to the
results from the conventional spatial methods. We can do this for example when
the regions of spacetime that we consider are domains of dependence of Cauchy
surfaces. Fig. 3 illustrates an example of this, since the causal diamonds are
domains of dependence of the $1$D intervals connecting the left and right
spatial corners (e.g. their diameters).

This entropy formulation is derived in Sorkin:2012sn . An alternative
derivation of it using the replica trick can be found in Chen:2020ild . Quite
surprisingly, this formula can actually be used beyond Gaussian theories. In
Chen:2020ild  it was shown that up to first order in perturbation theory, the
entanglement entropy of even non-Gaussian or interacting theories, is captured
by the two-point correlation function via (5) and (4). The only difference in
the interacting theory case is that the $W$ that enters (5) is the interacting
one and while $i\Delta$ would still be the antisymmetric part of $W$, it is no
longer the commutator.

###  2.2 The Sorkin-Johnston (SJ) Vacuum

The Sorkin-Johnston (SJ) Wightman function sj1 ; sj2  is defined in the same
algebraic and spacetime spirit as the entropy formulation we have just
reviewed. It uniquely picks out a vacuum state in any globally hyperbolic
spacetime.

To define the SJ state, we require the spacetime commutator function
$i\Delta$. As we saw above, we can express $i\Delta$ in terms of the retarded
Green
function222$(\Box+m^{2})G_{R}(x,x^{\prime})=-\frac{\delta^{D}(x-x^{\prime})}{\sqrt[]{-g}}$,
where $G_{R}(x,x^{\prime})$ is only non-zero if $x^{\prime}\prec x$. as

$\Delta(x,x^{\prime})=G_{R}(x,x^{\prime})-G_{R}(x^{\prime},x).$ (7)

$i\Delta$ is anti-symmetric and Hermitian. We can then rewrite it as an
expansion in terms of its positive and negative eigenvalues and their
respective eigenfunctions as

$i\Delta=\sum_{i}\tilde{\lambda}_{i}u_{i}u^{\dagger}_{i}-\tilde{\lambda}_{i}v_{i}v^{\dagger}_{i},$
(8)

where $u_{i}$ and $v_{i}$ are the normalized positive and negative
eigenvectors respectively, and $\tilde{\lambda_{i}}>0$. The SJ Wightman
function is defined by restricting to the positive eigenspace of $i\Delta$

$W_{SJ}:=\text{Pos}(i\Delta)=\sum_{i}\tilde{\lambda}_{i}u_{i}u^{\dagger}_{i}.$
(9)

When we define W in this way, the only solutions to the generalized eigenvalue
equation (5) are $\lambda=0$ or $\lambda=1$, and this makes the entropy (4)
vanish, as required if the state is pure. It is also worth mentioning that in
static spacetimes, the SJ vacuum is the same one that is picked out by the
timelike and hypersurface-orthogonal Killing vector Aslanbeigi , hence the SJ
state is known to reflect the symmetries of the background geometry if there
are any. In all of the entanglement entropy applications that we will review
in Section 4, the SJ state is used as the pure state.

Both the entanglement entropy formulation and SJ state prescription can also
be used in continuum spacetimes. Before moving on to their applications in
causal set theory, we will in the next section motivate why it is desirable to
work with spacetime formulations, in general, when studying quantum field
theories.

##  3 Quantum Field Operators are Distributions in Spacetime

At least as early as a 1933 work by Bohr and Rosenfeld bohr , it has been
recognized that quantum field operators are only well-defined as averages over
finite regions of spacetime rather than at individual spacetime points. This
averaging is achieved through smearing the quantum field with smooth, real-
valued functions with compact support

$\phi(f)=\int\phi(x)f(x)\,dV.$ (10)

See also geroch  for a modern review of the convergence issues that arise if
quantum fields are defined at points in spacetime rather than as
distributions. Less well-known is the fact that generic nonlinear operators in
free quantum field theories are not well-defined if they are smeared with
functions with support only on spatial hypersurfaces rather than spacetime
regions streater ; haag . They become well-defined only after some smearing in
a duration of time as well. Furthermore, based on perturbation theory, then,
it is expected that operators in generic interacting quantum field theories
are also not well-defined on spatial hypersurfaces, since they would contain
the ill-defined nonlinear terms from the free theory.

Below we study this behaviour for two generic nonlinear operators ($\phi^{2}$
and $T_{ab}$) in free scalar field theory in Minkowski spacetime.

###  3.1 Variance of a Free Scalar Field $\phi^{2}$ Operator

Consider the usual quantum scalar field operator in $3+1$ spacetime dimensions

$\phi(x)=\int\frac{d^{3}p}{(2\pi)^{3}\sqrt{2E_{\vec{p}}}}\left(a_{\vec{p}}\,e^{ip\cdot
x}+a^{\dagger}_{\vec{p}}\,e^{-ip\cdot x}\right),$ (11)

where $E_{\vec{p}}=\sqrt{|\vec{p}|^{2}+m^{2}}$ and $x=(t,\vec{x})$.

Let us now examine the normal-ordered $\phi^{2}$ operator, and smear it with a
test function $f(x)=f_{1}(t)f_{2}(\vec{x})$ with compact support on a time
$t^{\prime}=\text{const}$ slice, i.e. $f_{1}(t)=\delta(t-t^{\prime})$,

$:\phi^{2}(t^{\prime},f):\,=\int
dt\,d^{3}x\,f_{2}(\vec{x})\delta(t-t^{\prime})\int\frac{d^{3}p\,d^{3}k}{2(2\pi)^{6}\sqrt{E_{\vec{p}}E_{\vec{k}}}}\left(a_{\vec{p}}\,a_{\vec{k}}e^{i(p+k)\cdot
x}+\dots\right).\,\,\,\,$ (12)

We then square the result and compute its expectation value in the Minkowski
vacuum,

$\displaystyle\langle$ $\displaystyle
0|:\phi^{2}(t^{\prime},f_{2})::\phi^{2}(t^{\prime},f_{2}):|0\rangle=\int\frac{d^{3}p\,d^{3}k|\tilde{f}_{2}(\vec{k}+\vec{p})|^{2}}{2(2\pi)^{12}E_{\vec{p}}E_{\vec{k}}},$
(13)

where $\tilde{f}_{2}$ is the Fourier transform of $f_{2}$. The integral can be
seen to be linearly divergent by simply counting the powers of $\vec{p}$ and
$\vec{k}$, and assuming that $f_{2}$ is square-integrable. However, (13)
converges if the smearing is done over an extent in time as well. To see this,
let us for simplicity assume that the smearing function is separable, i.e.,
$f(x)=f_{1}(t)f_{2}(\vec{x})$ as before (where $f_{1}$ is no longer a delta
function) and both $f_{1}$ and $f_{2}$ are square-integrable. Then

$\displaystyle\langle 0|:\phi^{2}(f)::\phi^{2}(f):|0\rangle$
$\displaystyle\propto\int\frac{d^{3}p\,d^{3}k\,|\tilde{f}_{2}(\vec{p}+\vec{k})|^{2}\,|\tilde{f}_{1}(p^{0}+k^{0})|^{2}}{p^{0}k^{0}}$
(14)
$\displaystyle\leq\int\frac{d^{3}k\,I(\vec{k})\tilde{F}_{1}(k^{0})^{2}}{\sqrt{|\vec{k}|^{2}+m^{2}}},$
(15)

with $\tilde{F}_{1}(k^{0})$ defined as the absolute maximum of $\tilde{f}_{1}$
with respect to $p^{0}$ at a given $k^{0}$, and $I(\vec{k})$ being the
$p$-integral of the remaining $p$-dependent variables. Since it can be shown
that $I(\vec{k})\sim\frac{1}{|\vec{k}|}$ for large $|\vec{k}|$, and
$\tilde{F}_{1}$ is square-integrable, we see by counting powers of $k$ that
the integral is convergent.

Expectation values such as the one we have considered above are ubiquitous in
operator product expansions (OPEs) and appear in generic correlation
functions. Similar divergences occur for other nonlinear operators such as
$\phi^{n}$ for $n\geq 2$ and the stress energy tensor $T_{ab}$ (as we show
below), and suggest that quantum field theories generally need to be
considered in a spacetime region.

###  3.2 Variance of Scalar Field Energy Density $T_{00}$

The simple construction above shows that certain operator expectation values
are divergent when smeared only on a spatial hypersurface and convergent when
smeared in spacetime. Let us next consider the analogous calculation for a
physically more interesting quantity, namely, the scalar field’s stress-energy
tensor

$:T_{ab}:=:\partial_{a}\phi\partial_{b}\phi:-\frac{1}{2}\eta_{ab}:\partial_{c}\phi\,\partial^{c}\phi:-\frac{m^{2}}{2}\eta_{ab}:\phi^{2}:$
(16)

We will focus our attention on the energy density component $:T_{00}:$, and
show that its variance diverges when smeared on a hypersurface, but can be
made convergent with an appropriate spacetime smearing. Substituting in the
definition of $\phi$ and smearing with a square integrable test function
$f(x)=\delta(t-t^{\prime})f_{2}(\vec{x})$ defined on a timeslice, the
expectation value of the variance is given by

$\displaystyle\langle 0|$
$\displaystyle:T_{00}(f):^{2}\,|0\rangle=\int\frac{d^{3}p\,d^{3}k}{8(2\pi)^{12}}\left(\frac{E_{\vec{p}}E_{\vec{k}}+\vec{p}\cdot\vec{k}-m^{2}}{\sqrt{E_{\vec{p}}}\sqrt{E_{\vec{k}}}}\right)^{2}|\tilde{f}_{2}(\vec{p}+\vec{k})|^{2}.$
(17)

We then see in the large $|\vec{p}\,|$ and $|\vec{k}\,|$ limit that the
leading order contribution to the integral is

$\displaystyle\langle 0|$ $\displaystyle:T_{00}(f):^{2}\,|0\rangle\sim\int
d^{3}p\,d^{3}k\,|\vec{p}\,|\,|\vec{k}\,|\left(1+\cos\theta\right)^{2}|\tilde{f}_{2}(\vec{p}+\vec{k})|^{2},$
(18)

$\theta$ being the angle between $\vec{p}$ and $\vec{k}$ in the dot product
above. Since this integral has higher powers of $\vec{p}$ and $\vec{k}$ than
the integral in (13) which we already showed diverges, it follows trivially
that (18) diverges as well.

In order for a time smearing to make (18) converge, more care needs to be
taken than in (14) where generic square integrable smearing functions suffice.
Here we have in effect four extra factors of $\vec{p}$ and $\vec{k}$, so we
need to ensure that the Fourier transforms of our smearing functions decay
fast enough near infinity to counteract this. For this reason and for
simplicity of analysis, we will use Gaussian smearing functions.333While
technically a Gaussian does not have compact support, its rapid decay
restricts meaningful values to a local enough region. In any case, one could
also work with “bump” functions which have compact support, but nonetheless
have Fourier transforms that decay faster than any power law, as demonstrated
in bump . We again write the spacetime smearing function as the separable
function $f(t,\vec{x})=f_{1}(t)\,f_{2}(\vec{x})$, where

$\displaystyle f_{1}(t)=e^{-a\,t^{2}}$
$\displaystyle\implies|\tilde{f}_{1}(p^{0}+k^{0})|^{2}\propto
e^{-\frac{\left(p^{0}+k^{0}\right)^{2}}{2\,a}},$ (19) $\displaystyle
f_{2}(\vec{x})=e^{-b\,\vec{x}\,^{2}}$
$\displaystyle\implies|\tilde{f}_{2}(\vec{p}+\vec{k})|^{2}\propto
e^{-\frac{\left(\vec{p}+\vec{k}\right)^{2}}{2\,b}}.$ (20)

The variance of the spacetime smeared energy density then at large
$|\vec{p}\,|$ and $|\vec{k}\,|$ has leading order contribution

$\displaystyle\langle 0|\,\,:T_{00}(f):^{2}\,|0\rangle$ $\displaystyle\sim\int
d^{3}p\,d^{3}k\,|\vec{p}\,|\,|\vec{k}\,|\left(1+\cos\theta\right)^{2}\,|\tilde{f}_{1}(p^{0}+k^{0})|^{2}\,|\tilde{f}_{2}(\vec{p}+\vec{k})|^{2}$
(21) $\displaystyle\lesssim\int
d^{3}k\,|\vec{k}\,|\,\tilde{f}_{1}(k^{0})|^{2}\,I(\vec{k}),$ (22)

where the inequality follows from $\tilde{f}_{1}(v+w)<\tilde{f}_{1}(v)$ for
all positive $v$, $w$ when $\tilde{f}_{1}$ is a Gaussian, and we have simply
grouped the remaining terms into the $p$ integral $I$. One can then
asymptotically evaluate $I$ as

$\displaystyle I(\vec{k})$ $\displaystyle\propto\int
d^{3}p\,|\vec{p}\,|\left(1+\cos\theta_{p}\right)^{2}e^{-\frac{\left(\vec{p}+\vec{k}\right)^{2}}{2\,b}}$
(23)
$\displaystyle\sim\mathcal{O}(|\vec{k}\,|^{4}\log(|\vec{k}\,|))\,+\,\mathcal{O}(|\vec{k}\,|)\,+\,\mathcal{O}(|\vec{k}\,|^{2}e^{-\frac{\vec{k}\,^{2}}{2\,b}}),$
(24)

in the large $|\vec{k}\,|$ limit. Replacing the leading-order term back into
(22), the upper bound on the variance of the energy density becomes asymptotic
to the $|\vec{k}\,|$-integral

$\displaystyle\langle 0|\,\,:T_{00}(f):^{2}\,|0\rangle$
$\displaystyle\lesssim\int^{\infty}_{0}d|\vec{k}\,|\,|\vec{k}\,|^{7}\log(|\vec{k}\,|)\,e^{-\frac{|\vec{k}\,|\,^{2}}{2\,a}}.$
(25)

The decaying Gaussian will dominate asymptotically, so this integral is
finite.

Therefore, as anticipated, $\langle 0|:T_{ab}::T_{ab}:|0\rangle$ is well-
defined only as a distribution in spacetime. The smearing in the above
discussion can be considered as a model for making a measurement. We would
certainly not expect the energy in a bounded region to have an infinite
variance, but we see that a finite variance is only obtained if the bounded
region is in spacetime rather than in space alone. Hence we can conclude that
the quantum stress tensor is only meaningful as a distribution in _spacetime_.
See also ford  for evidence that a well-defined probability distribution for
the quantum stress tensor is only obtained for averages in time or spacetime.

##  4 Applications

Having established the importance of entanglement entropy in quantum gravity,
as well as the importance of treating quantum fields in a spacetime domain,
let us now explore what work on entanglement entropy in causal set theory has
been done.

As a reminder, the calculation of entanglement entropy schematically amounts
to

$G_{R}\rightarrow i\Delta\rightarrow W_{SJ}\rightarrow S.$ (26)

However, note that the scheme above does not tell one how to obtain the
retarded Green function, $G_{R}$, which is the starting point. In fact, in
general there exists no known expression for $G_{R}$ in terms of quantities
intrinsic to the causal set (such as the link or causal matrices). The
examples below represent some important and interesting cases where an
expression for $G_{R}$ is known.

###  4.1 Causal Diamond in $1+1$D Flat Spacetime

The most studied and well-understood setting for entanglement entropy in a
causal set, is the causal diamond in $1+1$D. This setting also benefits from
numerous analytic results from the continuum being known and available for
comparison.

We will mainly focus on the massless theory, for simplicity. The retarded
Green function in $1+1$D Minkowski spacetime, for a massless scalar field
theory is

$G_{R}(x,x^{\prime})=\frac{1}{2}\theta(t-t^{\prime})\theta(\tau^{2}),$ (27)

where $\tau=\sqrt{|t-t|^{2}-|\vec{x}-\vec{x^{\prime}}|^{2}}$ is the proper
time between the two points and $\theta$ is the Heaviside step function. In
other words, $G_{R}$ is only nonzero if $x^{\prime}$ causally precedes $x$ and
when it does, $G_{R}$ takes the value $\frac{1}{2}$. With $G_{R}$ so directly
related to the causal structure, we have an exact analogue of it in the causal
set in terms of the causal matrix $C$. The causal matrix similarly has a
nonzero value of $1$ for each entry $C_{x^{\prime}x}$ where $x^{\prime}$
causally precedes $x$ or $x^{\prime}\prec x$. Therefore, all we have to do is
multiply the causal matrix by the constant $\frac{1}{2}$ to get the analogue
of the retarded Green function in the causal set:

$G_{R}=\frac{1}{2}C$ (28)

Next, we form $i\Delta$ from $\frac{i}{2}$ times the antisymmetric part of the
causal matrix444When we write $C_{xy}$, we mean the entry corresponding to
elements $x$ and $y$ in a point basis representation of the the matrix $C$.

$i\Delta_{xx^{\prime}}=\frac{i}{2}(C_{x^{\prime}x}-C_{xx^{\prime}}).$ (29)

From here it is a simple algebraic exercise to diagonalize (29) and restrict
to its positive eigenspace to define $W_{SJ}$. The SJ state in the $1+1$D
causal diamond has been extensively studied Afshordi_2012 . It resembles the
standard Minkowski Wightman function (with an IR cutoff) away from the
boundaries of the diamond. Near the left and right corners it resembles the
Minkowski Wightman function with a static mirror at these corners.

Figure 4: The lower plot shows the SJ Wightman function in a causal set causal
diamond, away from the boundaries. The SJ state is computed in the larger blue
diamond at the top. The values of $W_{SJ}$ versus proper time have been shown
for elements in the inner Green subdiamond. Agreement is seen with the
Wightman function associated with the IR-regulated Minkowski vacuum state (red
curve).

In Sorkin_2018 ; Yazdi:2017pbo  it was shown that if we go ahead and compute
the entanglement entropy according to the steps in Section 2.1 , we obtain an
unexpected answer: instead of the usual spatial area law scaling of the
entanglement entropy with the UV cutoff, we obtain a _spacetime volume_
scaling. Since we have $N\propto V$ in a causal set, a volume scaling means
that the entanglement entropy scales linearly with $N$ instead of the expected
$N^{D-2}$ scaling in $D>2$ or logarithmic scaling when $D=2$.

More specifically, to study the scaling of the entanglement entropy with the
UV cutoff, which in length dimensions is $\rho^{-1/D}$ (where $\rho$ is now
the sprinkling density), we fix the geometry into which the sprinkling is
performed (e.g. the blue diamond in Fig. 4), as well as the entangling
subregion (e.g. the green subdiamond in Fig. 4). We then vary the number of
elements $N$ that we sprinkle, thereby varying $\rho$ and the UV cutoff. Fig.
5 shows an example of the volume law scaling obtained, for the diamond setup
shown in Fig. 4.

Figure 5: The entanglement entropy grows linearly with $N$, demonstrating a
spacetime volume law scaling. The points represent entropy values from the
calculation of (4) in the setup of the upper plot in Fig.4. The line is a
linear fit to the data. In this example, the ratio of the side lengths of the
two diamonds was $\frac{1}{4}$.

This result is peculiar to the causal set calculation, as the continuum
analogue of the same calculation (performed in Saravani_2014 ), showed no sign
of this behaviour. A closer look at the eigenvalues of $i\Delta$, with the
help of insight from analytic results from the continuum, reveals the source
of the extra entropy. In the continuum diamond with side length $2\ell$, the
eigenfunctions of $i\Delta$ with nonzero eigenvalues are SJthesis

$\displaystyle f_{k}(u,v)$ $\displaystyle=e^{-iku}-e^{-ikv},$
$\displaystyle\textrm{with }k=\frac{n\pi}{{\ell}},\;n=\pm 1,\pm 2,\ldots$ (30)
$\displaystyle g_{k}(u,v)$ $\displaystyle=e^{-iku}+e^{-ikv}-2\cos(k{\ell}),$
$\displaystyle\textrm{with }k|\tan(k{\ell})=2k{\ell}\wedge k\in\mathbb{R}\,$
(31)

The eigenfunctions above have been expressed in terms of lightcone coordinates
$u=\frac{t+x}{\sqrt{2}}$ and $v=\frac{t-x}{\sqrt{2}}$. The eigenvalues are
${\tilde{\lambda}}_{k}={\ell}/k$, with eigenvalues from both sets of
eigenfunctions $f_{k}$ and $g_{k}$ approaching
${\tilde{\lambda}}_{k}=\frac{\ell^{2}}{n\pi}$ in the large $k$ limit. The
eigenfunctions (30) and (31) span the solutions of the Klein Gordon
equation.555$\text{Ker}(\Box+m^{2})=\overline{\text{Im}(\Delta)}$. Therefore,
if we were to consider a finite number of eigenfunctions up to some $k_{max}$,
we can think of $k_{max}$ as a cutoff. In other words, with a finite set of
eigenfunctions up to $k_{max}$, we would only be able to expand solutions
(e.g. initial data) up to this maximum wavenumber. Turning this argument
around, if there were reason to believe that solutions beyond some UV cutoff
$k_{max}$ ought not to be supported in a setting, we would expect to obtain a
finite number $\sim n_{max}$ of eigenvalues and eigenfunctions corresponding
to $k_{max}=\frac{n_{max}\pi}{\ell}$. This is precisely the scenario we are
faced with in causal sets. In our $1+1$D diamond causal set, the discreteness
length is $\frac{1}{\sqrt{\rho}}=\sqrt{\frac{V}{N}}=\frac{2\ell}{\sqrt{N}}$.
Hence we do not expect to be able to meaningfully describe wavelengths shorter
than this. Converting this wavelength to a wavenumber we get
$\frac{2\ell}{\sqrt{N}}=\frac{2\pi}{k_{max}}\implies
k_{max}=\frac{\pi\sqrt{N}}{\ell}\implies n_{max}\approx 2\sqrt{N}$, where the
factor of $2$ in $2\sqrt{N}$ comes from the fact that we have two sets of
eigenfunctions $f_{k}$ and $g_{k}$ that will each have this $k_{max}$ and
contribute $\sqrt{N}$.

However, a look at the eigenvalues and eigenfunctions of $i\Delta$ in the
causal set reveals that we in fact end up with many more nonzero eigenvalues
than $n_{max}$. An example of the spectrum of $i\Delta$ is shown in Fig. 6.
The values of the positive eigenvalues are shown on a log-log scale, where
they have been ordered from largest to smallest, and each $i^{th}$ eigenvalue
in this ordering is paired with $i$ on the horizontal axis. Circled are the
extra eigenvalues that are beyond the expected $n_{max}$. Interestingly, these
extra eigenvalues behave qualitatively differently from the rest: they do not
follow a power law like the larger eigenvalues.

Figure 6: The positive eigenvalues of $i\Delta$ for a sample sprinkling into a
causal diamond.

When the contributions to the entanglement entropy from these extra
eigenvalues and their corresponding eigenvectors are removed, we recover the
expected spatial area law scaling of the entanglement entropy with respect to
the UV cutoff. This removal is referred to as a _truncation_ , and it must be
implemented at two stages of the calculation: (1) A first truncation when
$W_{SJ}=\text{Pos}(i\Delta)$ is constructed, and (2) a second truncation when
the generalized eigenvalue equation (5) is solved. These truncations can also
be regarded as projections down to the subspace where
$k_{max}=\frac{\pi\sqrt{N_{\diamond}}}{\ell_{\diamond}}$, where the diamond
subscript indicates that in the first truncation which is in the global
diamond, $N$ and $\ell$ are the total number of elements and size of this
diamond, whereas in the second truncation which occurs in the smaller
subdiamond, $N$ and $\ell$ are the number of elements in and size of the
subdiamond. When this so-called “double truncation” is performed, we obtain
the result shown in Fig. 7

Figure 7: The entanglement entropy versus the UV cutoff, following a
logarithmic scaling with a coefficient consistent with $\frac{1}{3}$. This is
the conventional result according to the expectation of a spatial area law.

Note that, as mentioned earlier, a logarithmic scaling with respect to the UV
cutoff is the expected “area law” result in $1+1$D Chandran_2016 , and this is
what is obtained after the double truncation. A coefficient of $\frac{1}{3}$
to the logarithmic scaling is also an expected666A coefficient of
$\frac{1}{3}$ is expected in the case of two spatial boundaries (such as in
the configuration of Fig. 4). The case of one spatial boundary, where a
coefficient of $\frac{1}{6}$ is expected, was also studied in Duffy_2022  and
agreement with a logarithmic scaling and the $\frac{1}{6}$ coefficient was
confirmed. universal constant Calabrese_2009  that the causal set results
agree with.

These results in the causal diamond were extended to the massive scalar field
theory in keseman . In the massive theory, because mass is an intermediate
scale lying between the UV (discreteness scale) and IR (diamond size) scales,
the same double truncation procedure of the massless theory can be used. This
is extremely useful as we lack analytic results in the massive theory to
otherwise give us some guidance towards the nature of the eigenvalues. In
keseman  scalings of the entanglement entropy with both the UV cutoff and mass
were studied, and in both cases the expected result of logarithmic scaling
with a coefficient of $\frac{1}{3}$ was obtained. In the same work, the
entanglement entropy results were also extended to Rényi entropies Renyi . The
Rényi entropy of order $q$, in terms of the quantities we are working with in
our formulation, is given by Sorkin_2018 ; keseman

$S^{(q)}=\frac{-1}{1-q}\sum_{\lambda}\ln(\lambda^{q}-(\lambda-1)^{q}).$ (32)

The solutions to (5) come in pairs of $\lambda$ and $1-\lambda$ and
$|\lambda|\geq 1$. Each term in the sum (32) represents the contribution from
one such pair. Similarly, other measures of entropy such as Tsallis entropy
Tsallis:1987eu  can also be calculated and studied in this manner.

###  4.2 Disjoint Causal Diamond regions

Another setting in which entanglement entropy in causal set theory has been
studied, is that of disjoint causal diamonds within a larger global causal
diamond in $1+1$D Duffy_2022 . An example setup with two disjoint subdiamonds
is shown in Fig. 8. Entanglement entropy of disjoint regions has been studied
in several places in the literature PhysRevB.81.060411 ; Ryu_2006 ; Arias_2018
and the calculations tend to be quite involved and difficult. In contrast, the
calculation using (5) and (4) for the disjoint diamonds is very similar to the
calculation for the single diamond, which is now well-understood and
relatively easy to do. Therefore, this is an example where there are
calculational advantages, in addition to physical ones, to working with the
spacetime formulation of Section 2.1 in a causal set.

Figure 8: Two smaller causal diamonds within a larger causal diamond.

In Duffy_2022  explicit calculations were done for the case of two and three
disjoint subdiamonds; the entanglement entropy scalings in both of these cases
were shown to be consistent with the logarithmic scalings expected. While
scalings with respect to the UV cutoff were the focus of Duffy_2022 , there
are several other scales in the problem (e.g. the sizes of the subdiamonds,
the separation(s) between the diamonds, the distance away from the boundary of
the global diamond, etc.) whose relation to the entanglement entropy would be
interesting to investigate.

The mutual information for a two-subdiamond setup was also studied in
Duffy_2022 . The mutual information in this case is the difference between the
entanglement entropy associated with the union of the two subdiamonds and the
sum of the entropies of the individual subdiamonds. Specifically, the relation
between the mutual information and the separation distance between two
diamonds was studied. The results demonstrated the expected qualitative
behavior that the mutual information asymptotically vanishes as the separation
between the subdiamonds grows and diverges as the separation goes to zero.

###  4.3 De Sitter Spacetime

Cosmological event horizons, just like black hole event horizons, also have a
classical entropy, the Gibbons-Hawking entropy gh , associated with them that
scales as their spatial area. Hence applications of entanglement entropy to
cosmological spacetimes are of particular interest. De Sitter spacetime offers
an especially convenient setting to study the entanglement entropy. This is
partly due to its maximal symmetry, which makes sprinkling into it
considerably easier in comparison to sprinkling into more generic curved
spacetimes. Of course, any sprinkling into de Sitter spacetime would not
represent the full global de Sitter spacetime, as that has infinite volume and
would therefore require an infinite number of elements, which is
computationally not feasible. Instead, sprinklings into finite slabs of global
de Sitter spacetime are used, and it is ensured that any results obtained are
stable under making the volume of the slab larger and larger.

Another attractive feature of working with de Sitter spacetime is that an
expression for the retarded Green function, in terms of causal set quantities,
is known in this context nx . This gives us the starting point in (26). The SJ
Wightman function in causal sets sprinkled into de Sitter spacetime was
studied in sjds . The entanglement entropy in causal set sprinklings of de
Sitter slabs, using the formalism reviewed in this chapter, was studied in
eeds . In particular the entanglement entropy associated to the subregion
within the horizon of a static observer at the North pole was considered. This
subregion and its causal complement are shown in the conformal diagram in Fig.
9.

Figure 9: Entangled wedges corresponding to the horizon of a static observer
at the North Pole (Green) and its causal complement (Purple). The dashed lines
represent the boundaries of the slab, $-T_{max}<T<T_{max}$, in de Sitter
spacetime $ds^{2}=\frac{1}{\cos^{2}T}\left(-dT^{2}+d\Omega_{d-1}^{2}\right)$.

Similar to the case of the causal diamond, the spectrum of $i\Delta$ has two
characteristic regimes: a branch of eigenvalues that are larger in magnitude
and follow a power law, and a branch of more numerous small but nonzero
eigenvalues that do not follow a power law. Without truncating away this
second branch, once again a spacetime volume scaling is obtained. A spatial
area scaling is recovered only after implementing a double truncation.
However, choosing a precise double truncation scheme is more subtle in this
case, as we lack analytic results in de Sitter spacetime to guide us. In other
words, we do not know exactly how the eigenvalues of $i\Delta$ relate to
something like a $k_{max}$ in de Sitter space. In the absence of analytic
results to guide us, one can estimate the transition between the power law
regime and non-power law regime using a number of different strategies. Some
of these strategies were studied in eeds  and shown to yield spatial area
laws. However, in that work it was found challenging to hone in on a unique
prescription for the truncations, as many different choices yielded area laws.
On the other hand, complementarity of the entanglement entropy was found to be
a property that was difficult to preserve, since in the second truncation it
was unclear which (non-unique) truncation in a subregion ought to be paired
with which (non-unique) counterpart truncation in the causally complementary
subregion. We will return to this point of finding a general truncation scheme
in the concluding section of this chapter.

###  4.4 Nonlocal Quantum Field Theory

Ordinarily, the retarded Green function which is the starting point of (26)
would be obtained via the Klein Gordon equation in the continuum. In causal
set theory, we do not have a local field equation that is the analogue of the
Klein Gordon equation. Instead, we must obtain $G_{R}$ through other means.
There is no general recipe in causal set theory for deriving $G_{R}$.
Expressions for it are known in a few distinct cases with help from the
continuum analogues of these Green function and/or dimensional analysis.

While a local analogue of the d’Alembertian $\Box$ and therefore the Klein
Gordon equation does not exist in causal set theory, a _nonlocal_ analogue of
it does sorkin_box ; Benincasa_2010 ; Dowker_2013 ; Aslanbeigi_2014 . In fact
a whole family $\Box_{k}$ of nonlocal d’Alembertians exists, with each member
distinguished by a nonlocality scale $\ell_{k}$. For example in $1+1$D,
$\Box_{k}\phi$ at the element $x\in\mathcal{C}$ is defined to be sorkin_box

$\Box_{k}\phi(x)=\frac{4\epsilon}{\ell_{\rho}^{2}}\left(\frac{1}{2}\phi(x)+\epsilon\sum_{y\prec
x}f\left(n(x,y),\epsilon\right)\phi(y)\right),$ (33)

where $\ell_{\rho}$ is the discreteness scale,
$\epsilon\equiv\ell_{\rho}^{2}\ell_{k}$, $n(x,y)$ is the number of elements in
the causal diamond between $x$ and $y$, and

$f(n,\epsilon)=(1-\epsilon)^{n}\left(1-\frac{2\epsilon
n}{1-\epsilon}+\frac{\epsilon^{2}n(n-1)}{2(1-\epsilon)^{2}}\right).$ (34)

The nonlocality of (33) is evident in the fact that in order to know the
action of the d’Alembertian on the field at the point $x$, we must consider a
sum of quantities involving a set of other elements $y$ to the past of $x$.
Therefore this nonlocality has a causal, or more specifically retarded,
nature.

In the infinite density limit ($\ell_{\rho}\rightarrow 0$), the mean of
$\Box_{k}$ over all sprinklings into a spacetime reduces to the usual
continuum d’Alembertian plus a term containing the Ricci scalar curvature
Benincasa_2010 :

$\displaystyle{\lim_{\ell_{\rho}\rightarrow
0}}\,\bar{\Box}_{k}\,\phi(x)=(\Box-\frac{1}{2}R(x))\,\phi(x).$ (35)

With these nonlocal retarded causal set d’Alembertians at hand, we can now
invert them to obtain their corresponding retarded Green functions $G_{R,k}$,
for use in (26). This was done in Belenchia_2018  for nested causal diamonds
in $1+1$, $2+1$, and $3+1$-dimensional Minkowski spacetime. Once again, as in
the local calculations, only after the use a double truncation, the expected
spatial area scalings were obtained.

##  5 Discussion and Outlook

Entanglement entropy in causal set theory is a powerful way to covariantly and
unambiguously count the quantum field degrees of freedom one has access to in
settings such as spacetimes with event horizons. One must know the scalar
field retarded Green function in the spacetime of interest in order to carry
out the entanglement entropy calculations, as well as a double truncation rule
in order to project out the irrelevant degrees of freedom in the causal set.

An expression for the retarded Green function in causal sets approximated by
generic curved spacetimes, in terms of quantities intrinsic to the causal set,
is not at present known. Such an expression is known in a few cases, such as
the Minkowski and de Sitter examples reviewed above. Nonlocal versions of
these Green functions, as discussed in Section 4.4, can be computed more
generally. Alternatively, viewing the causal set as a Lorentzian and covariant
discretization of the continuum, we can also simply take the Green functions
and/or correlator expressions from the continuum and restrict them to the
causal set elements. Thereby we would be regulating any coincidence limit
divergences that may be present, by imposing the minimum distance set by the
causal set discreteness scale.

As mentioned, it is also necessary to have a prescription for the double
truncation in order to meaningfully study entanglement entropy in causal set
theory. This prescription is well understood in the massless theory in causal
diamonds in $1+1$D flat spacetime. As shown in Duffy_2022  and keseman , the
same prescription can be used for the case of multiple disjoint causal
diamonds as well as the massive scalar field theory. More generally, the same
prescription can be used in any $1+1$D scalar field theory (e.g. the nonlocal
theory or in curved spacetimes), as long as the intermediate scale is far from
the discreteness scale. This is because the truncations concern the deep UV
regime of the theory, which has the same character in all theories where the
other length scales are far from the discreteness scale.

In eeds  some generalizations of the $1+1$D causal diamond double truncation
scheme were studied and applied to calculations in causal diamonds in $3+1$D
Minkowski spacetime as well as slabs in de Sitter spacetime. One strategy was
to keep $\alpha N^{\frac{D-1}{D}}$ of the largest eigenvalues and their
corresponding eigenfunctions, where $\alpha$ is a constant that is a free
parameter (several choices for $\alpha$ were investigated). This counting is
motivated by the expectation that the number of independent degrees of freedom
are the number that would lie within some approximate Cauchy surface-like
submanifold (e.g. a thickened antichain). The number of elements in such a
submanfiold would be proportional to its volume, which is approximately
$V^{\frac{D-1}{D}}$. This strategy succeeds in yielding area laws, but it does
not produce a unique prescription (many choices of $\alpha$ are possible) and
complementarity is difficult to achieve. Another strategy was to try to
estimate the transition between the power law to non-power law regime in the
spectrum, through estimating when the approximate linear trend in Fig. 6 ends
and begins to curve. This strategy sometimes succeeds in producing an area law
but it too suffers from an ambiguity in how and how sensitively to define the
transition from power law (line on the log-log scale) to non-power law (curve
on the log-log scale). There are some other possible truncation schemes that
would merit future investigation. For example, one ansatz could be that in the
power law regime, each $n^{th}$ (positive) eigenvalue $\tilde{\lambda}_{n}$ of
$i\Delta$ (in any dimension), when sorted from largest to smallest, is
proportional to $\frac{1}{n^{p}}$, where the proportionality constant is given
unambiguously by the value of the largest eigenvalue (where $n=1$) and $p$ can
be approximated from the spectrum. For example we know that $p=1$ in the
causal diamond in $1+1$D and $p=\frac{1}{2}$ in the causal diamond in $2+1$D.
We can then choose $n_{max}$ to be $N^{\frac{D-1}{D}}$ and estimate the
magnitude of the smallest eigenvalue in the power law regime to then be
$\tilde{\lambda}_{min}=\frac{\tilde{\lambda}_{max}}{N^{\frac{p(D-1)}{D}}}$.

More analytic results for the eigenvalues of $i\Delta$ in the continuum would
also aid our understanding of the eigenvalues in the causal set and better
inform our truncation schemes. It is, however, quite difficult to analytically
solve for the eigenfunctions of integral operators.

Another perspective is that there should be no truncations, and that all
nonzero eigenvalues and eigenfunctions must contribute to the entropy
Mathur:2022ivs . However, even in this case we must face the question of how
small of an eigenvalue we can really expect to exist in the causal set
calculations. Remember that we have the condition (6) that $i\Delta v\neq 0$.
Due to the numerical nature of the calculations, there is always some degree
of numerical error, and we must identify a threshold beyond which to set the
values to zero. While doing so, we must also assess whether we are throwing
away anything physical due to the numerical error. Therefore, a better
understanding of the truncated contributions and what solutions can be
meaningfully supported on the causal set is needed.

In keseman  some insight was gained into the nature of the truncated
contributions. Motivated by the observation that the untruncated contributions
had continuumlike analogues whereas the truncated ones did not, as well as the
observation that the truncated eigenfunctions had many sharp variations at the
discreteness scale, it was investigated whether the truncated contributions
may be fluctuations particular to a given sprinkling. Namely, it was
investigated whether these contributions were random fluctuations that behaved
differently from one sprinkling to the next, or whether they had features
which persisted over an ensemble of different sprinklings. There are different
prescriptions one can use to investigate this; one particular scheme,
involving fixing a coarser sub-causal set in order to use it to take averages,
was used in keseman . Indeed, evidence was found in favor of this conjecture
that the truncated contributions are fluctuation-like, and the scheme studied
in keseman  indicated a transition point to the fluctuation-like regime that
was consistent with the double truncation in the causal diamond in $1+1$D.
This is promising both as insight into the nature of the truncated
contributions, as well as practically in order to use it to inform a double
truncation scheme in more general settings. The transition to fluctuation-like
behaviour can thus potentially be used in general to distinguish between
contributions we must keep and ones we must not.

There are many other applications of the entanglement entropy formulation
reviewed in this chapter that would be interesting to explore in causal set
theory. For example, up to first order in perturbation theory, the
entanglement entropy for interacting scalar field theories such as those
introduced in Sorkin_2011 ; emma ; kasia , can be studied. There is also an
analogue of (5) for Fermionic field theories, except with the anti-commutator
appearing instead of the commutator. Currently, there is no known construction
of a Fermionic field theory on a causal set, in terms of quantities intrinsic
to the causal set. When such a construction is available, the entanglement
entropies of Fermionic field theories could also be studied.

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# Relation between Fresnel transform of input light field and Radon transform
of Wigner function of the field

Hong-yi Fan and Li-yun Hu∗
Department of Physics, Shanghai Jiao Tong University,
Shanghai, 200030, China
*Corresponding author<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We prove a new theorem about the relationship between optical field Wigner
function’s Radon transform and optical Fresnel transform of the field, i.e.,
when an input field $\psi\left(x^{\prime}\right)$ propagates through an
optical $\left[D\left(-B\right)\left(-C\right)A\right]$ system, the energy
density of the output field is equal to the Radon transform of the Wigner
function of the input field, where the Radon transform parameters are $D,B.$
We prove this theorem in both spatial-domain and frequency-domain.

In optical communication theory every signal or image can be uniquely and
indirectly described by a Wigner distribution function (WDF) [1, 2, 3]. The
WDF (or named Wigner transform) of an optical signal field
$\psi\left(x^{\prime}\right)$ is defined as

$W_{\psi}(\nu^{\prime},x^{\prime})=\int_{-\infty}^{+\infty}\frac{du}{2\pi}e^{i\nu^{\prime}u}\psi^{\ast}\left(x^{\prime}+\frac{u}{2}\right)\psi\left(x^{\prime}-\frac{u}{2}\right),$
(1)

$W_{\psi}(\nu^{\prime},x^{\prime})$ involves both spatial distribution
information and space-frequency distribution information of the signal. $\nu$
is named space frequency. $W_{\psi}(\nu^{\prime},x^{\prime})$ is said to be
bilinear in the signal because the signal enters twice in its definition. The
WDF undergoes certain variations if something happens to the signal. For
examples, passage through a lens corresponds to a vertical shearing of the
WDF, propagation in free space means a horizontal shearing of the WDF [4].
However, the WDF preserves space and space frequency marginal properties of
any signal,

$\displaystyle\int_{-\infty}^{+\infty}d\nu^{\prime}W_{\psi}(\nu^{\prime},x^{\prime})$
$\displaystyle=$ $\displaystyle|\psi\left(x^{\prime}\right)|^{2},$ (2)
$\displaystyle\int_{-\infty}^{+\infty}dx^{\prime}W_{\psi}(\nu^{\prime},x^{\prime})$
$\displaystyle=$
$\displaystyle|\tilde{\psi}\left(\nu^{\prime}\right)|^{2},\text{ }$ (3)

where
$\tilde{\psi}\left(\nu\right)=\int_{-\infty}^{+\infty}\frac{dx}{\sqrt{2\pi}}\psi\left(x\right)e^{ix\nu}.$
If one wants to reconstruct the Wigner function by using various probability
distribution, obviously the position density
$|\psi\left(x^{\prime}\right)|^{2}$ and the space-frequency density
$|\tilde{\psi}\left(\nu^{\prime}\right)|^{2}$ were not enough, so the Radon
transform [5, 6] of the Wigner function is introduced [7],

$R\left(x\right)\equiv\iint\limits_{-\infty}^{\infty}dx^{\prime}d\nu^{\prime}\delta\left(x-Dx^{\prime}+B\nu^{\prime}\right)W_{\psi}(\nu^{\prime},x^{\prime}),$
(4)

$R\left(x\right)$ is also a probability distribution along an infinitely thin
phase space strip denoted by the real parameters $D,B$. The inverse relation
of (4) is the foundation of optical tomographic imaging techniques (the
techniques derive two-dimensional data from a three-dimensional object to
obtain a slice image of the internal structure and thus have the ability to
peer inside the object noninvasively.)

On the other hand, an optical system can be analyzed by using either
diffraction theory or ray optics. It is worth setting up a connection between
ray transfer matrix and diffraction theory, i.e., the diffraction integration
written in terms of ray transfer matrix—named Collins’ diffraction integral
formula [8, 9, 10] which describes the propagation of a general beam
$\psi\left(x^{\prime}\right)$ through an $\left(ABCD\right)$ optical paraxial
system,

$\phi\left(x\right)=\int_{-\infty}^{\infty}\mathcal{K}\left(x,x^{\prime}\right)\psi\left(x^{\prime}\right)dx^{\prime},$
(5)

where $\phi\left(x\right)$ is the output field, $AD-BC=1,$ and
$\mathcal{K}\left(x,x^{\prime}\right)$ is the integral kernel

$\mathcal{K}\left(x,x^{\prime}\right)=\frac{1}{\sqrt{2\pi
iB}}\exp\left[\frac{i}{2B}\left(Ax^{\prime
2}-2x^{\prime}x+Dx^{2}\right)\right].$ (6)

If the energy in the initial beam is normalized,
$\int_{-\infty}^{+\infty}\left|\psi\left(x^{\prime}\right)\right|^{2}dx^{\prime}=1,$
then the output beam is normalized too,
$\int_{-\infty}^{+\infty}\left|\phi\left(x\right)\right|^{2}dx=1.$ Clearly, if
the $\left[ABCD\right]$ system is changed to
$\left[D\left(-B\right)\left(-C\right)A\right]$ system, then Eq. (5) should
read

$\phi\left(x\right)=\int_{-\infty}^{\infty}\mathcal{\vec{K}}\left(x,x^{\prime}\right)\psi\left(x^{\prime}\right)dx^{\prime},$
(7)

where the integral kernel $\mathcal{\vec{K}}\left(x,x^{\prime}\right)$ is

$\mathcal{\vec{K}}\left(x,x^{\prime}\right)=\frac{1}{\sqrt{-2\pi
iB}}\exp\left[\frac{-i}{2B}\left(Dx^{\prime
2}-2x^{\prime}x+Ax^{2}\right)\right].$ (8)

In this Letter, we shall reveal the following theorem:

When an input field $\psi\left(x^{\prime}\right)$ propagates through an
optical $\left[D\left(-B\right)\left(-C\right)A\right]$ system, the energy
density of the output field $\phi\left(x\right)$ is equal to the Radon
transform of the Wigner function of the input field, where the Radon transform
parameters are $D,B$. The proof is demonstrated as follows.

With the use of Dirac $\delta$-function, we can re-express the Wigner function
of input field $\psi\left(x^{\prime}\right)$ in Eq. (1) as

$\displaystyle W_{\psi}(\nu^{\prime},x^{\prime})$ $\displaystyle=$
$\displaystyle\iint\limits_{-\infty}^{\infty}dx^{\prime\prime}dx^{\prime\prime\prime}\int_{-\infty}^{+\infty}\frac{du}{2\pi}e^{i\nu^{\prime}u}\psi^{\ast}\left(x^{\prime\prime}\right)\psi\left(x^{\prime\prime\prime}\right)\delta\left(x^{\prime\prime}-x^{\prime}-\frac{u}{2}\right)\delta\left(x^{\prime}-\frac{u}{2}-x^{\prime\prime\prime}\right)$
(9) $\displaystyle=$
$\displaystyle\frac{1}{\pi}\iint\limits_{-\infty}^{\infty}dx^{\prime\prime}dx^{\prime\prime\prime}\psi^{\ast}\left(x^{\prime\prime}\right)\psi\left(x^{\prime\prime\prime}\right)e^{i2\nu^{\prime}\left(x^{\prime\prime}-x^{\prime}\right)}\delta\left(2x^{\prime}-x^{\prime\prime}-x^{\prime\prime\prime}\right).$

Substituting (9) into (4) we rewrite the Radon transform of
$W_{\psi}(\nu^{\prime},x^{\prime})$ as

$\displaystyle R\left(x\right)$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}\iint\limits_{-\infty}^{\infty}dx^{\prime}d\nu^{\prime}\delta\left(x-Dx^{\prime}+B\nu^{\prime}\right)\iint\limits_{-\infty}^{\infty}dx^{\prime\prime}dx^{\prime\prime\prime}\psi^{\ast}\left(x^{\prime\prime}\right)\psi\left(x^{\prime\prime\prime}\right)e^{i2\nu^{\prime}\left(x^{\prime\prime}-x^{\prime}\right)}\delta\left(2x^{\prime}-x^{\prime\prime}-x^{\prime\prime\prime}\right)$
(10) $\displaystyle=$ $\displaystyle\frac{1}{2\pi
B}\iint\limits_{-\infty}^{\infty}dx^{\prime\prime}dx^{\prime\prime\prime}\psi^{\ast}\left(x^{\prime\prime}\right)\psi\left(x^{\prime\prime\prime}\right)\exp\left\\{\frac{i}{2B}[D\left(x^{\prime\prime
2}-x^{\prime\prime\prime
2}\right)-2x\left(x^{\prime\prime}-x^{\prime\prime\prime}\right)\right\\}.$

On the other hand, when the beam $\psi\left(x^{\prime}\right)$ propagates
through the $\left[D\left(-B\right)\left(-C\right)A\right]$ optical system,
according to the Fresnel integration (7)-(8), we have

$\displaystyle\left|\phi\left(x\right)\right|^{2}$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}dx^{\prime}\mathcal{\vec{K}}\left(x,x^{\prime}\right)\psi\left(x^{\prime}\right)\int_{-\infty}^{\infty}dx^{\prime\prime}\psi^{\ast}\left(x^{\prime\prime}\right)\mathcal{\vec{K}}^{\ast}\left(x,x^{\prime\prime}\right)$
(11) $\displaystyle=$ $\displaystyle\frac{1}{2\pi
B}\int_{-\infty}^{\infty}dx^{\prime}\exp\left[\frac{-i}{2B}\left(Dx^{\prime
2}-2x^{\prime}x+Ax^{2}\right)\right]\psi\left(x^{\prime}\right)$
$\displaystyle\times\int_{-\infty}^{\infty}dx^{\prime\prime}\exp\left[\frac{i}{2B}\left(Dx^{\prime\prime
2}-2x^{\prime\prime}x+Ax^{2}\right)\right]\psi^{\ast}\left(x^{\prime\prime}\right)$
$\displaystyle=$ $\displaystyle\frac{1}{2\pi
B}\iint\limits_{-\infty}^{\infty}dx^{\prime}dx^{\prime\prime}\psi\left(x^{\prime}\right)\psi^{\ast}\left(x^{\prime\prime}\right)\exp\left\\{\frac{i}{2B}\left[D\left(x^{\prime\prime
2}-x^{\prime
2}\right)+2x\left(x^{\prime}-x^{\prime\prime}\right)\right]\right\\},$

which is the same as $R\left(x\right)$ in (10). So combining (4), (10)-(11),
(7) and (8) we reach the conclusion

$\left|\frac{1}{\sqrt{-2\pi
iB}}\int_{-\infty}^{\infty}\exp\left[\frac{-i}{2B}\left(Dx^{\prime
2}-2x^{\prime}x+Ax^{2}\right)\right]\psi\left(x^{\prime}\right)dx^{\prime}\right|^{2}=\iint\limits_{-\infty}^{\infty}dx^{\prime}d\nu^{\prime}\delta\left(x-Dx^{\prime}+B\nu^{\prime}\right)W_{\psi}(\nu^{\prime},x^{\prime}),$
(12)

where $DA-BC=1$. The physical meaning of Eq. (12) is: when an input field
propagates through an optical $\left[D\left(-B\right)\left(-C\right)A\right]$
system, the energy density of the output field is equal to the Radon transform
of the Wigner function of the input field. So far as our knowledge is
concerned, this conclusion seems new.

Eq. (8) is the relationship between the input amplitude and output one in
spatial-domain. Now we turn the above discussion to the case of space-
frequency domain.

For a $\left[D\left(-B\right)\left(-C\right)A\right]$ optical system, the
Collins’ diffraction integral formula in space-frequency (angle spectrum [11])
domain is [12]

$\widetilde{\phi}\left(v\right)=\int_{-\infty}^{\infty}\widetilde{\mathcal{K}}\left({\nu},\nu^{\prime}\right)\widetilde{\psi}\left({\nu}^{\prime}\right)d{\nu}^{\prime},$
(13)

where the kernel is

$\widetilde{\mathcal{K}}\left({\nu},{\nu}^{\prime}\right)=\frac{1}{\sqrt{2\pi
iC}}\exp\left[\frac{i}{2C}\left(D{\nu}^{2}-2{\nu}^{\prime}{\nu}+A{\nu}^{\prime
2}\right)\right].$ (14)

On the other hand, the Wigner function expressed in terms of the space-
frequency field is

$W_{\tilde{\psi}}(\nu^{\prime},x^{\prime})=\int_{-\infty}^{+\infty}\frac{ds}{2\pi}e^{-ix^{\prime}s}\tilde{\psi}^{\ast}\left(\nu^{\prime}+\frac{s}{2}\right)\tilde{\psi}\left(\nu^{\prime}-\frac{s}{2}\right).$
(15)

In space-frequency domain the Radon transform is along an infinitely thin
phase space strip denoted by the real parameters $A,C$,

$R\left(\nu\right)\equiv\iint\limits_{-\infty}^{\infty}dx^{\prime}dp^{\prime}\delta\left({\nu}-A{\nu}^{\prime}+Cx^{\prime}\right)W_{\tilde{\psi}}({\nu}^{\prime},x^{\prime})$
(16)

rather than the parameters $D,B$ of the Radon transform (4) in spatial-domain.
We want to examine if the above conclusion still holds in the space-frequency
domain. For this propose, we rewrite (15) as

$\displaystyle W_{\tilde{\psi}}(\nu^{\prime},x^{\prime})$ $\displaystyle=$
$\displaystyle\iint\limits_{-\infty}^{\infty}d{\nu}^{\prime\prime}d{\nu}^{\prime\prime\prime}\int_{-\infty}^{+\infty}\frac{ds}{2\pi}e^{-ix^{\prime}s}\tilde{\psi}^{\ast}\left({\nu}^{\prime\prime}\right)\tilde{\psi}\left({\nu}^{\prime\prime\prime}\right)\delta\left(\nu^{\prime\prime}-\nu^{\prime}-\frac{s}{2}\right)\delta\left(\nu^{\prime}-\frac{s}{2}-\nu^{\prime\prime\prime}\right)$
(17) $\displaystyle=$
$\displaystyle\iint\limits_{-\infty}^{\infty}d{\nu}^{\prime\prime}d{\nu}^{\prime\prime\prime}\tilde{\psi}^{\ast}\left({\nu}^{\prime\prime}\right)\tilde{\psi}\left({\nu}^{\prime\prime\prime}\right)e^{-i2x^{\prime}\left({\nu}^{\prime\prime}-{\nu}^{\prime}\right)}\delta\left(2{\nu}^{\prime}-{\nu}^{\prime\prime}-{\nu}^{\prime\prime\prime}\right)$

Then we substitute (17) into (16), the result is

$\displaystyle R\left(\nu\right)$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}\iint\limits_{-\infty}^{\infty}dx^{\prime}d{\nu}^{\prime}\delta\left({\nu}-A{\nu}^{\prime}+Cx^{\prime}\right)\iint\limits_{-\infty}^{\infty}d{\nu}^{\prime\prime}d{\nu}^{\prime\prime\prime}\tilde{\psi}^{\ast}\left({\nu}^{\prime\prime}\right)\tilde{\psi}\left({\nu}^{\prime\prime\prime}\right)e^{-i2x^{\prime}\left({\nu}^{\prime\prime}-{\nu}^{\prime}\right)}\delta\left(2{\nu}^{\prime}-{\nu}^{\prime\prime}-{\nu}^{\prime\prime\prime}\right)$
(18) $\displaystyle=$ $\displaystyle\frac{1}{2\pi
C}\iint\limits_{-\infty}^{\infty}d{\nu}^{\prime\prime}d{\nu}^{\prime\prime\prime}\tilde{\psi}^{\ast}\left({\nu}^{\prime\prime}\right)\tilde{\psi}\left({\nu}^{\prime\prime\prime}\right)\exp\left\\{-\frac{i}{2C}\left[A\left({\nu}^{\prime\prime
2}-{\nu}^{\prime\prime\prime
2}\right)-2{\nu}\left({\nu}^{\prime\prime}-{\nu}^{\prime\prime\prime}\right)\right]\right\\}.$

On the other hand, from (13) we calculate

$\displaystyle|\tilde{\phi}\left({\nu}\right)|^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi
C}\int_{-\infty}^{+\infty}\tilde{\psi}\left({\nu}^{\prime}\right)\exp\left[\frac{i}{2C}\left(D{\nu}^{2}-2{\nu}^{\prime}{\nu}+A{\nu}^{\prime
2}\right)\right]d{\nu}^{\prime}$ (19)
$\displaystyle\times\int_{-\infty}^{+\infty}\tilde{\psi}^{\ast}\left({\nu}^{\prime\prime}\right)\exp\left[\frac{-i}{2C}\left(D{\nu}^{2}-2{\nu}^{\prime\prime}{\nu}+A{\nu}^{\prime\prime
2}\right)\right]d{\nu}^{\prime\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi
C}\int\int_{-\infty}^{+\infty}d{\nu}^{\prime}d{\nu}^{\prime\prime}\tilde{\psi}\left({\nu}^{\prime}\right)\tilde{\psi}^{\ast}\left({\nu}^{\prime\prime}\right)\exp\left[-\frac{i}{2C}\left(A\left({\nu}^{\prime\prime
2}-{\nu}^{\prime
2}\right)-2{\nu}\left({\nu}^{\prime\prime}-{\nu}^{\prime}\right)\right)\right].$

Comparing Eq. (19) with Eq. (18) and using (13)-(14) we see

$\left|\frac{1}{\sqrt{2\pi
iC}}\int\exp\left[\frac{i}{2C}\left(D{\nu}^{2}-2{\nu}^{\prime}{\nu}+A{\nu}^{\prime
2}\right)\right]\tilde{\psi}\left({\nu}^{\prime}\right)d{\nu}^{\prime}\right|^{2}=\iint\limits_{-\infty}^{\infty}dx^{\prime}dp^{\prime}\delta\left({\nu}-A{\nu}^{\prime}+Cx^{\prime}\right)W_{\tilde{\psi}}({\nu}^{\prime},x^{\prime}),$
(20)

this is the theorem expressed in space-frequency domain.

We now take an example to confirm the theorem. When the input field is
described by a Gaussian function (Gaussian chirplet)

$\psi_{0}\left(x^{\prime}\right)\equiv\sqrt[4]{\frac{\epsilon}{\pi}}\exp\left\\{-\left(\epsilon-i\beta\right)\frac{x^{\prime
2}}{2}\right\\},\text{ }\epsilon>0,\text{\ }$ (21)

which depicts a Gaussian windowed linear chirp signal [13]. Using Eq.(1) we
obtain its Wigner function

$\displaystyle W_{\psi_{0}}(\nu^{\prime},x^{\prime})$ $\displaystyle=$
$\displaystyle e^{-\epsilon
x^{\prime}{}^{2}}\sqrt{\frac{\epsilon}{\pi}}\int_{-\infty}^{+\infty}\frac{du}{2\pi}e^{-\frac{1}{4}\epsilon
u^{2}+i\left(\nu^{\prime}-\beta x^{\prime}\right)u}$ (22) $\displaystyle=$
$\displaystyle\frac{1}{\pi}\exp\left\\{-\left[\epsilon
x^{\prime}{}^{2}+\frac{1}{\epsilon}\left(\nu^{\prime}-\beta
x^{\prime}\right)^{2}\right]\right\\},$

which shows that the energy of Gaussian chirplet is concentrated at
$\nu^{\prime}=\beta x^{\prime}.$ According to Eq. (4), the Radon transform of
(22) with the parameters $D,B$ is

$\displaystyle\iint\limits_{-\infty}^{\infty}dx^{\prime}d\nu^{\prime}\delta\left(x-Dx^{\prime}+B\nu^{\prime}\right)W_{\psi_{0}}(\nu^{\prime},x^{\prime})$
(23) $\displaystyle=$ $\displaystyle\frac{1}{\pi
D}\int_{-\infty}^{+\infty}d\nu^{\prime}\exp\left\\{-\frac{\epsilon}{D^{2}}\left(x+B\nu^{\prime}\right){}^{2}-\frac{1}{\epsilon
D^{2}}\left[\left(D-\beta B\right)\nu^{\prime}-\beta x\right]^{2}\right\\}$
$\displaystyle=$
$\displaystyle\sqrt{\frac{\epsilon/\pi}{\left(D-B\beta\right)^{2}+B^{2}\epsilon^{2}}}\exp\left[\frac{-\epsilon
x^{2}}{\left(D-B\beta\right)^{2}+B^{2}\epsilon^{2}}\right]\equiv
R_{0}\left(x\right).$

On the other hand, according to (7) and (8), the Fresnel transform of input
field $W_{\psi_{0}}(\nu^{\prime},x^{\prime})$ through a
$\left[D\left(-B\right)\left(-C\right)A\right]$ optical system is given by

$\displaystyle\frac{1}{\sqrt{-2\pi
iB}}\sqrt[4]{\frac{\epsilon}{\pi}}\int_{-\infty}^{\infty}dx^{\prime}\exp\left[\frac{-i}{2B}\left(Dx^{\prime
2}-2x^{\prime}x+Ax^{2}\right)-\left(\epsilon-i\beta\right)\frac{x^{\prime
2}}{2}\right]$ (24) $\displaystyle=$
$\displaystyle\sqrt[4]{\frac{\epsilon}{\pi}}\frac{1}{\sqrt{-i}}e^{-\frac{iA}{2B}x^{2}}\sqrt{\frac{1}{\epsilon
B-i\left(\beta B-D\right)}}\exp\left[\frac{-x^{2}}{2B\left(B\epsilon+iD-i\beta
B\right)}\right]\left.\equiv\right.\phi_{0}\left(x\right).$

Comparing (24) with (23) and noticing

$\frac{1}{2B\left(B\epsilon+iD-i\beta B\right)}+\frac{1}{2B\left(B\epsilon-
iD+i\beta
B\right)}=\allowbreak\frac{\epsilon}{\left(D-B\beta\right)^{2}+B^{2}\epsilon^{2}}$
(25)

we see

$\left|\phi_{0}\left(x\right)\right|^{2}=R_{0}\left(x\right),$ (26)

as expected.

In summary, we have derived a new theorem governing the connection between
optical field’s Fresnel transform and its Wigner function’s Radon
transformation, since both the Wigner function and the Fresnel transform are
widely used in optical propagation, we hope this theorem would have new
applications in the analysis of optical communication and optical tomography.
For the application of Wigner function in deriving the quantum-mechanical
photocount formula, we refer to [14].

This work was supported by the National Natural Science Foundation of China
under grant 10775097.

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