Locally detecting UV cutoffs on a sphere with particle detectors

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Locally detecting UV cutoﬀs on a sphere with particle detectors

Ahmed Shalabi,1, ∗Laura J. Henderson,2, 1, †and Robert B. Mann1, 3, 4

1Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1

2Centre for Engineered Quantum Systems, School of Mathematics and Physics,

The University of Queensland, St. Lucia, Queensland 4072, Australia

3Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1

4Perimeter Institute, 31 Caroline St. N. Waterloo Ontario, N2L 2Y5, Canada

The potential breakdown of the notion of a metric at high energy scales could imply the existence

of a fundamental minimal length scale below which distances cannot be resolved. One approach to

realizing this minimum length scale is construct a quantum ﬁeld theory with a bandlimit on the

ﬁeld. We report on an investigation of the eﬀects of imposing a bandlimit on a ﬁeld on a curved and

compact spacetime and how best to detect such a bandlimit if it exists. To achieve this operationally,

we couple two Gaussian-smeared UDW detectors to a scalar ﬁeld on a S2×Rspherical spacetime

through delta-switching. The bandlimit is implemented through a cut-oﬀ of the allowable angular

momentum modes of the ﬁeld. We observe that a number of features of single detector response in

the spherical case are similar to those in ﬂat spacetime, including the dependence on the geometry

of the detector, and that smaller detectors couple more strongly to the ﬁeld, leading to an optimal

size for bandlimit detection. We ﬁnd that in ﬂat spacetime squeezed detectors are more senstive to

the bandlimit provided they are larger than the optimal size; however, in spherical spacetime the

bandlimit itself determines if squeezing improves the sensitivity. We also explore setups with two

detectors, noting that in the spherical case, due to its compact nature, there is a lack of dissipation

of any perturbation to the ﬁeld, which results in locally excited signals being refocused at the poles.

Quite strikingly, this feature can be exploited to signiﬁcantly improve bandlimit detection via ﬁeld

mediated signalling. Moreover, we ﬁnd that squeezing on a sphere introduces extra anisotropies

that could be exploited to amplify or weaken the response of the second detector.

I. INTRODUCTION

The theories of General relativity and quantum ﬁeld

theory describe all fundamental interactions in nature.

Yet they are based on entirely diﬀerent mathematical

structures and are empirically applicable over very dif-

ferent energy and length scales. While semi classical de-

scriptions of quantum ﬁeld theory on curved spacetime

exist, a fundamental step towards their uniﬁcation into

some higher energy theory of quantum gravity will en-

tail understanding what happens at short distance scales.

Fluctuations of quantum ﬁelds might potentially break

down the notion of a general relativistic metric upon ap-

proaching the Planck scale. As such, it is believed that

there is a ﬁnite minimum length beneath which distances

cannot be resolved [1].

There are several consistent high energy theories of

quantum gravity, each with its own treatment of space-

time. Generally, there are two overarching approaches

to dealing with the nature of spacetime in theories of

quantum gravity [2]. One is to model spacetime as some

sort of discrete structure. This approach is conducive to

quantization and would naturally entail some sort of ul-

traviolet (UV) cutoﬀ. However it comes at the price of

a loss of local Lorentzian symmetries. The other broad

approach, based on continuous structures, does not suﬀer

∗ashalabi@uwaterloo.ca

†l7henderson@uwaterloo.ca

from these issues. However understanding the notion of

metric breakdown remains an open problem [3].

In an attempt to reconcile these issues a hybrid pro-

posal [4] treats spacetime as both continuous and dis-

crete, analogous to the way that Shannon’s sampling

theorem [5] regards information as both continuous and

discrete. More concretely, consider a continuous signal

modelled by a function f(t). Shannon’s sampling theo-

rem states that if f(t)is bandlimited i.e. contains fre-

quencies in a ﬁnite interval (−Λ,Λ), then taking a dis-

crete set of samples {f(tn)}n=∞

n=−∞ is enough to recon-

struct the signal via the Shannon sampling formula for

all times, provided the samples are taken at intervals

tn+1 −tn= 1/(2Λ). This was generalized to physical

ﬁelds on Lorentzian manifolds, establishing how this form

of bandlimitation on the momentum modes of a ﬁeld is

equivalent to a UV cutoﬀ [6]. It is important to note

that, unlike quantizing a ﬁeld on discrete lattice, ban-

dlimitation of a QFT preserves local Euclidean symme-

tries. Furthermore, although this form of bandlimitation

is not Lorentz invariant, it can be generalized to a fully

covariant cutoﬀ [7,8].

With all of this established, it is of utmost importance

to study UV cutoﬀs at the intersection between quan-

tum ﬁeld theory and general relativity – in other words,

imposing a cutoﬀ on a quantum ﬁeld on a curved back-

ground. To this end, we here study this question on

an S2×Rspacetime and, for comparison, its (2 + 1)-

dimensional Minkowski counterpart. We do so for sev-

eral reasons. First, a quantized scalar ﬁeld on an S2×R

background has well deﬁned angular momentum modes.

Moreover, this spacetime is compact and bounded, so

arXiv:2210.11503v1 [gr-qc] 20 Oct 2022

a quantized scalar ﬁeld would have a countably inﬁnite

number of modes if no UV cutoﬀ existed. Furthermore,

AdS3is conformal to S2×R, making our results straight-

forwardly transferrable to that context. AdS spacetimes

have been studied extensively in the context of holo-

graphic duality and the AdS/CFT correspondence. In

addition, the ﬁeld correlation functions on AdS3are re-

lated to those in BTZ spacetimes via image sums [9].

The most straightforward way to probe quantum ﬁelds

locally is through particle detectors. First proposed by

Unruh [10], the detector is modelled as a two-level system

that (linearly) couples to the ﬁeld. As such, it serves as a

local probe of the ﬁeld, providing both a concrete notion

of locality and an operational deﬁnition of a particle. In

other words, "A particle is what a particle detector de-

tects" [11].

Particle detectors probe and study the semi classi-

cal regime of quantum ﬁeld theory on curved spacetime.

They sample the ﬂuctuations and (if more than one de-

tector is employed) correlations of a quantum ﬁeld by

coupling to its momentum modes. By smearing a parti-

cle detector over a region of spacetime, we can probe the

quantum ﬁeld in question via local interactions over that

region. Since the ﬁeld degrees of freedom and the spatial

proﬁle of the detector (which quantiﬁes where the detec-

tor couples to the ﬁeld) enter the model at the same level,

an investigation of how they interact can yield a better

operational understanding of the ﬁnite spatial volume of

the discrete degrees of freedom. Such a study was re-

cently carried out in (3 + 1)-dimensional ﬂat spacetime

[12]. Here we take the next natural step by considering

this problem in S2×R, with appropriate comparison to

(2 + 1)-dimensional ﬂat spacetime.

There are several models for the ﬁeld-detector cou-

pling. These include non linear scalar ﬁeld coupling

[13], ﬁeld derivative coupling [14,15], fermionic couplings

[16,17] and delocalized matter [18,19]. However a sim-

ple linear coupling [20] is an appropriate approximation

to the full light-matter interaction if angular momentum

exchange is negligible [21,22]. We shall only consider

this coupling in our investigation.

Particle detector models have found many applications

in the study of quantum information in both ﬂat and

curved spacetimes. These include studies of the entan-

glement structure of quantum ﬁelds using the entangle-

ment harvesting protocol [20,23], the Unruh eﬀect [24],

Hawking radiation [25], probing the geometry [26,27]

and topology [28] of spacetime, providing a measurement

framework for quantum ﬁeld theory [29], and other appli-

cations like communication protocols [30,31] and ther-

modynamics [32].

We consider here the question of how to best detect

the presence of a cutoﬀ using particle detectors in both

ﬂat and curved spacetimes. We utilize particle detec-

tors by coupling them to vacuum states of quantum

ﬁelds in each of (2 + 1)-dimensional Minkowski space-

time and on S2×Ras a prototypical (2 + 1)-dimensional

curved spacetime. We will implement the UV cutoﬀs via

a hard/conventional bandlimitation on the momentum

modes of the scalar ﬁeld. We will take the ﬁeld-detector

coupling duration to be the shortest length scale in the

problem by modelling it as a δ-function. This delta cou-

pling has several advantages. It removes the need for

time ordering and allows a full non-perturbative determi-

nation of the ﬁnal detector-ﬁeld joint state [33]. After the

ﬁeld is traced out, the ﬁnal state of the detector carries

information about the geometry of the underlying space-

time [27]. Furthermore, as discussed earlier, although

the conventional bandlimit is not covariant, we expect

our results to be similar to a full covariant generalization

since the duration of the coupling we employ is smaller

than any other length scales in the problem. Finally, we

will use two detectors, switching on one before the other

to study the impact of ﬁeld mediated signalling on the

detection of the bandlimit.

The rest of the paper is organized as follows. In sec-

tion II we present the basic formalism of the UDW model

in the context of δ-switching and bandlimited quantum

ﬁelds for both the ﬂat and spherical cases we consider,

and in section III we describe the spatial proﬁles of the

detectors. We then present our results for bandlimit de-

tection using a single detector in section IV and for two

detectors in section V A. We present our conclusions in

section VI along with a discussion of directions for fur-

ther work. A set of appendices contains technical details

pertinent to our investigation.

II. THE UDW DETECTOR MODEL AND

DIRAC δSWITCHING

The Unruh-DeWitt (UDW) detector [10,34,35] is a 2-

level system whose ground and excited states are respec-

tively given by |giDand |eiD, separated by an energy

gap ΩD. We shall consider two such detectors A and B

linearly coupled to a massless scalar ﬁeld such that the

initial joint detector-ﬁeld state is given by

ˆρi=|giA Ahg|⊗|giB Bhg|⊗|0iφ φh0|(1)

or in other words, the ﬁeld is in the vacuum state and

the detectors are in their ground states. The interaction

detector-ﬁeld Hamiltonian is

HI,AB (t) = ˆ

HI,A(t) + ˆ

HI,B (t)(2)

in the interaction picture, where ˆ

HI,D (t)is given by

HI,D (t) = λDχD(t)eiΩDtˆσ+

D+ e−iΩDtˆσ−

D

⊗ZdnxFD[x−xD]ˆ

φ(x, t)(3)

with D∈ {A, B}, where λDis the ﬁeld-detector coupling

constant and χD(t)is the switching function that controls

the duration of the ﬁeld-detector interaction. The oper-

ators ˆσ+

D:=|eiD Dhg|and ˆσ−

D:=|giD Dhe|are the SU(2)

ladder operators acting on the Hilbert space of detector

D. We have introduced a spacial proﬁle FD(x−xD)for

each detector, centred around its position xD. We inter-

pret this as describing the size and shape of the detector

[36,37].

The time evolution of the full detector-ﬁeld system is

U=Texp −iZ∞

−∞

dtˆ

HI,AB(t)(4)

where Tis the time ordering operator. The ﬁnal state of

the two detector-ﬁeld system is given by

ˆρf=ˆ

Uˆρiˆ

U†(5)

in turn yielding the reduced density matrix describing

the ﬁnal state of the two detector system

ˆρAB := Trφˆρf(6)

obtained by tracing out the Hilbert space of the ﬁeld.

The general approach from here would be to solve for

the matrix elements of ˆρAB perturbatively. However it is

possible to solve the two detector density matrix exactly

[33,38] by using the switching function

χD(t) = ηDδ(t−TD)(7)

where TDis the time at which the interaction takes place.

We shall brieﬂy review this ‘δ-switching’ formalism with-

out imposing the original constraint [33] of working in

Minkowski space.

We will assume without loss of generality that detec-

tor A switches before B (TA≤TB). Applying the δ-

switching allows us to write the time evolution operator

(4) as

Uδ= exp ˆ

HI,B (TB)exp ˆ

HI,A(TA)(8)

or alternatively as

Uδ= exp ˆµB(TB)⊗ˆ

YBexp ˆµA(TA)⊗ˆ

YA(9)

where the operator

ˆµD(t)=eiΩDτD(t)ˆσ+

D+ e−iΩDτD(t)ˆσ−

D(10)

describes the evolution of the detector and

YD:=−iλDηDZdnxFD(x−xD)ˆ

φ(x, TD)(11)

which is the smeared ﬁeld operator.

By expanding the Taylor series of the exponential and

noting that ˆµD(t)2=, we can write the time evolution

operator ˆ

Uδas

Uδ=A⊗B⊗cosh( ˆ

YB) + A⊗ˆµB(TB)⊗sinh( ˆ

YB)

×A⊗B⊗cosh( ˆ

YA) + ˆµA(TA)⊗B⊗sinh( ˆ

YA)

(12)

Moreover, we can rewrite ˆ

Uδusing the complex ex-

ponential form of the hyperbolic trigonometric functions

utilizing the following deﬁnition. Let j, k ∈ {1,−1}and

write

X(j,k)=1

4(eˆ

YB+je−ˆ

YB)(eˆ

YA+ke−ˆ

YA)(13)

which gives

Uδ=A⊗B⊗ˆ

X(1,1) + ˆµA(TA)⊗ˆµB(TB)⊗ˆ

X(−1,−1)

+ ˆµA(TA)⊗B⊗ˆ

X(1,−1) +A⊗ˆµB(TB)⊗ˆ

X(−1,1)

(14)

from (12). The two detector subsystem evolves to the

ﬁnal state

ˆρAB = Trφ[ˆ

Uδˆρiˆ

U†

δ](15)

which in the |aiA⊗ |biBbasis for a, b ∈ {g, e}contains

cross terms of the form h0|ˆ

X(j,k)ˆ

X(l,m)|0i. This moti-

vates the following deﬁnition:

fjklm :=D0ˆ

X†

(j,k)ˆ

X(l,m)0E(16)

where j, k, l, m ∈ {1,−1}. Using the Baker-Campbell-

Hausdorﬀ (BCH) formula eXeY=eZwith Zgiven by

Z=X+Y+1

2[X, Y ]+ 1

12X, [X, Y ]−1

12Y, [X, Y ]+···

(17)

and using arguments similar to those in the Minkowski

space case [33], we rewrite the fjklm matrix elements as

fjklm =1

16 [(1 + j` +km +jk`m) + [(1 + j`)(k+m)]fA

+(`+jkm)e2iΘ+ (j+k`m)e−2iΘfB

+(jk +`m)eω+ (jm +k`)e−ωfAfB(18)

where

fD=h0|exp(2 ˆ

YD)|0i(19)

and the quantities Θand ωare deﬁned as:

Θ := −iD0hYA,ˆ

YBi0E=D0ˆ

Θ0E

ω:= 2 D0nˆ

YA,ˆ

YBo0E=h0|ˆω|0i

(20)

which are respectively the vacuum expectation values of

the smeared ﬁeld commutator and anti-commutator. For

two regions in a spacetime, these operators encode the

signalling through the ﬁeld and the correlations in the

ﬁeld between the two regions.

The smeared ﬁeld commutator Θis non zero when two

smeared detectors are in casual contact, in which case

communication between them is possible. It is zero when

the detectors are spacelike separated. It is important to

note that for two detectors that are initially separable

there is no dependence of the ﬁnal state on ωafter the

interaction. This is to be expected since ωencodes the

amount of correlations between two detectors – if they are

uncorrelated initially then they cannot harvest entangle-

ment from the ﬁeld via delta coupling [33,39]. The role

played by the anti-commutator ωin detector correlations

with delta coupling was recently studied [39]. We will an-

alyze in section IV A the relevance of Θin signalling.

The ﬁnal reduced density matrix ˆρAB can then be writ-

ten in terms of the fjk`m matrix elements as

ˆρAB =





ρ11 0 0 ρ14

0ρ22 ρ23 0

0ρ∗

23 ρ33 0

ρ∗

14 0 0 ρ44





(21)

where the non zero ρij matrix elements are given by

ρ11 =1

41 + fA+fBcos(2Θ) + fAfBcosh(ω)(22a)

ρ14 =1

4e−i(ΩATA+ΩBTB)fBi sin(2Θ) + fAsinh(ω)

(22b)

ρ22 =1

41 + fA−fBcos(2Θ) −fAfBcosh(ω)(22c)

ρ23 =−1

4e−i(ΩATA−ΩBTB)fBi sin(2Θ) + fAsinh(ω)

(22d)

ρ33 =1

41−fA+fBcos(2Θ) −fAfBcosh(ω)(22e)

ρ44 =1

41−fA−fBcos(2Θ) + fAfBcosh(ω)(22f)

We can also trace out the detectors individually to obtain

the following density operators

ˆρA= TrB[ˆρAB ] = 1

21 + fA0

0 1 −fA(23)

and

ˆρB= TrA[ˆρAB ] = 1

21 + fBcos(2Θ) 0

0 1 −fBcos(2Θ)

(24)

Note that the dynamics of detector B is modiﬁed by

the commutator of the ﬁeld. This is a consequence of the

fact that detector B interacts with an evolved state of the

ﬁeld subsequent to its interaction with the ﬁrst detector.

Finally, we can read oﬀ the transition probabilities for

the ﬁrst and second detector to be

PA=1

2(1 −fA)and PB=1

21−fBcos(2Θ)(25)

The dynamics and the response of δ-coupled detectors

can be extended to those on any curved spacetime by

quantizing the scalar ﬁeld on the background spacetime.

The Klein-Gordon equation in curved spacetime is

p|g|

∂

∂xµgµν p|g|∂

∂xν!ˆ

φ(x, t)=0 (26)

We can solve the Klein-Gordon equation by assuming the

following ansatz for the scalar ﬁeld operator

φ(x, t) = X

khuk(x, t)ˆak+uk(x, t)∗ˆa†

ki(27)

where the functions uk(x, t)are solutions to the Klein-

Gordon equation, and ˆak,ˆa†

kare the raising and lowering

operators of the scalar ﬁeld. If the spacetime is globally

hyperbolic, then a set of modes uk(x, t)exists. If we can

quantize the scalar ﬁeld on some background spacetime,

the task would be to derive the smeared ﬁeld operator ˆ

and from it the expressions for fDand Θthat deﬁne the

response of the two detectors to the coupling. Moreover,

we can particularize those expressions to the shape and

localization of the detectors on the background geometry.

A. Flat spacetime

Here we brieﬂy summarize the recent (3 + 1)-

dimensional ﬂat space analysis [12] in a (2 + 1)-

dimensional context for ease of comparison with the

spherical case. We decompose the scalar ﬁeld into plane-

wave modes as

φ(x, t) = 1

(2π)n/2Zd2k

p2|k|hei(|k|t−k·x)ˆa†

k+H.c.i

(28)

where ˆa†

k,ˆakare creation and annihilation operators that

obey the canonical commutation relations

hˆak,ˆa†

k0i=δ(2)(k−k0).(29)

After writing down the ﬁeld, we can now calculate the

density matrix describing the joint state of the two de-

tector system (21). In the case of (2 + 1)-dimensional

Minkowski space, the smeared ﬁeld operator is

YD=−iλDηDZd2k

p2|k|e

FD(k)e−i(|k|TD−k·xD)ˆak+H.c.

(30)

where

FD(k) = 1

2πZd2xFD(x)eik·x(31)

is the Fourier transform of the spacial proﬁle.

It is straightforward to calculate the matrix element

functions (19) and (20) as [33]

fD= exp −λ2

Dη2

DZ|k|<Λ

d2k

|k|e

FD(k)

Θ = −iλAλBηAηB

2Z|k|<Λ

d2k

|k|e

F∗

A(k)e

FB(k)e−i|k|(TB−TA)eik·(xB−xA)−H.c.

ω=λAλBηAηBZ|k|<Λ

d2k

|k|e

F∗

A(k)e

FB(k)e−i|k|(TB−TA)eik·(xB−xA)+H.c.(32)

Expanding the ﬁeld in the plane-wave modes of Eq.

(28), we are able to easily introduce a hard momentum

cutoﬀ by removing modes where |k|>Λ[12]. We note

that this cutoﬀ is not Lorentz invariant, but expect that

results will be similar to the case of the full covariant

cutoﬀ, since the switching time of the detector is shorter

than any other scale in the problem [7,8].

B. Spherical Spacetime

To generalize δ-switching to the spherical spacetime

we will need to quantize the scalar ﬁeld ˆ

φand derive

from it the smeared ﬁeld operator ˆ

YDfor each detector.

We shall then obtain expressions for fDthat deﬁne the

transition probability PDfor each detector. We shall also

need an expression for the commutator Θof the smeared

ﬁeld operators to obtain the transition probability of the

second detector after it has interacted with the evolved

state of the ﬁeld.

Quantizing a conformally coupled scalar ﬁeld on R×S2

was done in [40]. From the metric

ds2=−dt2+dθ2+ sin2(θ)dϕ2(33)

where −∞ <t<∞,0< θ < π and 0<ϕ<2π, the

Klein-Gordon equation then becomes

ψ−1

8Rψ = 0 (34)

whose solutions are given by

ψ`m =1

√2`+ 1e−i(`+1

2)tY`m(θ, ϕ)(35)

where

Y`m(θ, ϕ)=(−1)mN`mPm

`(cos θ)eimϕ (36)

are the spherical harmonics basis functions, Pm

`are As-

sociated Legendre Polynomials and

N`m ≡s(2`+ 1)

4π

(`−m)!

(`+m)! (37)

This allows us to expand the scalar ﬁeld ˆ

φin the modes

ψ`m as follows

φ=X

`,m

ψ`mˆa`m +ψ∗

`mˆa†

`m (38)

where ˆa`m and ˆa†

`m are creation and annihilation opera-

tors such that ˆa†

`m |0i=|`, mi, and where

[ˆa†

ij ,ˆa`m] = δi`δjm (39)

That is, ˆa`m and ˆa†

`m raise and lower the angular mo-

mentum of the scalar ﬁeld. We can expand any function

on S2in terms of spherical harmonics as:

h(θ, ϕ) = X

l,m

hlmYlm(θ, ϕ)(40)

and using the orthogonality condition

Z Z dθdϕsin(θ)Y`m(θ, ϕ)Y∗

pq(θ, ϕ) = δmqδpl (41)

we can compute the coeﬃcients

hlm =Z Z dθdϕsin(θ)h(θ, ϕ)Y∗

`m(θ, ϕ).(42)

in (40).

Expanding the smeared ﬁeld operator in terms of the

scalar ﬁeld modes we get

YD=−iλDηDZdΩFD(θ−θD)

X

`,m

ψ`mˆa`m +ψ`mˆa†

`m

,

(43)

where RdΩ :=RRdθdϕsin(θ)for a proﬁle F(θ)centered

at θD= (θD, ϕD). Expressing the spacial localization

function in the spherical harmonics basis, with FD=

Pp,q fpqYpq we obtain

YD=−iλDηDZdΩ X

p,q

fpqYpq(θ, ϕ)X

l,m ψ`mˆa`m +ψ`mˆa†

lm

(44)

for the smeared ﬁeld operator. Writing ˆ

YD:= ˆy+ ˆy∗we

have

ˆy∗=−iλDηDZdΩ X

p,q

fpqYpq(θ, ϕ)X

l,m

ψ`mˆa†

=−iλDηDX

l,m X

p,q

ei(`+1

2)TDfpq

p(2`+ 1) ZdΩYpq(θ, ϕ)Y∗

lmˆa†

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摘要：

LocallydetectingUVcutosonaspherewithparticledetectorsAhmedShalabi,1,LauraJ.Henderson,2,1,yandRobertB.Mann1,3,41DepartmentofPhysicsandAstronomy,UniversityofWaterloo,Waterloo,Ontario,Canada,N2L3G12CentreforEngineeredQuantumSystems,SchoolofMathematicsandPhysics,TheUniversityofQueensland,St.Lucia,Quee...

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