Physical significance of generalized boundary conditions an Unruh-DeWitt detector viewpoint on AdS2 S2

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Physical signiﬁcance of generalized boundary conditions:

an Unruh-DeWitt detector viewpoint on AdS2×S2

Lissa de Souza Campos1†?, Claudio Dappiaggi2†?, Luca Sinibaldi3†?

†Dipartimento di Fisica, Universit`a degli Studi di Pavia, Via Bassi, 6, 27100 Pavia, Italy

?Istituto Nazionale di Fisica Nucleare – Sezione di Pavia, Via Bassi 6, 27100 Pavia, Italy

Abstract

On AdS2×S2, we construct the two-point correlation functions for the ground and thermal states of a real

Klein-Gordon ﬁeld admitting generalized (γ, v)-boundary conditions. We follow the prescription recently

outlined in [1] for two diﬀerent choices of secondary solutions. For each of them, we obtain a family of

admissible boundary conditions parametrized by γ∈0,π

2. We study how they aﬀect the response of a

static Unruh-DeWitt detector. The latter not only perceives variations of γ, but also distinguishes between

the two families of secondary solutions in a qualitatively diﬀerent, and rather bizarre, fashion. Our results

highlight once more the existence of a freedom in choosing boundary conditions at a timelike boundary which

is greater than expected and with a notable associated physical signiﬁcance.

Keywords: Generalized Robin boundary conditions, Unruh-DeWitt detector, Klein-Gordon ﬁeld, AdS2×S2

1. Introduction

Since the introduction of Unruh-DeWitt particle

detectors in quantum ﬁeld theory [2, 3], they have

been employed to probe a plethora of features of quan-

tum ﬁelds and of spacetimes, stretching from pick-

ing out properties of quantum states [4] to identify-

ing topological aspects of underlying backgrounds [5].

On a globally-hyperbolic spacetime with a timelike

boundary, they are particularly sensible to the choice

of an underlying boundary condition for the matter

ﬁelds: phenomena such as the divergence of physical

observables [6, 7] and the anti-Hawking eﬀect [8, 9]

depend crucially on the boundary condition set. This

is consonant with the fact that diﬀerent boundary

conditions are associated to diﬀerent quantum states,

to diﬀerent dynamics, to diﬀerent physics [10].

In this arena we have recently shown that within

free, scalar quantum ﬁeld theories on static, curved

spacetimes with a timelike boundary one may take

into account generalized (γ, v)-Robin boundary con-

ditions in the construction of physically meaningful

states [1]. These arise whenever the underlying dy-

namics can be reduced thanks to the background

isometries to a second order diﬀerential equation on

the half line and they are characterized by two data:

a ﬁxed parameter γ∈Rand a so-called secondary

solution v. For each admissible pair (γ, v), one can

construct a two-point function both for the ground

state and also for arbitrary thermal states. All these

correlation functions are physically sensible since, per

construction, they enjoy the local Hadamard prop-

erty, see [11]. In [1], we have discussed the example

of a wave propagating on the two-dimensional half-

Minkowski spacetime showing that an Unruh-DeWitt

detector can discern not only the value of γ, but also

the choice of secondary solution v. Here, we elab-

orate on this idea and we provide, in full detail, a

full-ﬂedged example that demonstrates how the hid-

den freedom in the mode expansion of the scalar ﬁeld,

which is associated to the choice of v, inﬂuences the

rate of transition of an Unruh-DeWitt particle detec-

tor.

In this work, we consider a real, free, scalar ﬁeld

Ψ with mass m0on M≡AdS2×S2spacetime. This

background approximates the near horizon geometry

of an extremal black hole with unit charge and it is

known as the Bertotti-Robinson solution of Einstein-

Maxwell equations, see e.g. [12] but also [13, 14]. For

t∈Rand r∈(0,∞), its line-element can be written

ds2=1

r2−dt2+dr2+r2dS2(θ, ϕ),(1)

where dS2(θ, ϕ) is the standard line-element on the

unit 2-sphere. Observe that the manifold possesses

1lissa.desouzacampos@unipv.it

2claudio.dappiaggi@unipv.it

3luca.sinibaldi01@universitadipavia.it

Preprint submitted to ... October 6, 2022

arXiv:2210.02395v1 [hep-th] 5 Oct 2022

a timelike conformal boundary at r= 0. On top of

Mwe consider a real scalar ﬁeld Ψ : AdS2×S2→R

whose dynamics is ruled by the Klein-Gordon equa-

tion

PΨ := (−m2

0)Ψ = 0,(2)

where is the D’Alembert wave operator built out

of Equation (1), while m0is the mass parameter.

In Section 2, we recollect the main ingredients nec-

essary to construct the two-point functions for ground

and thermal states for Ψ. The procedure follows the

prescription outlined in [1] and it generalizes the re-

sults obtained in [15] by considering two diﬀerent fam-

ilies of generalized (γ, v)-boundary conditions at the

conformal boundary, dubbed (γ, vκ) with κ= 1,2.

The analysis for κ= 2 coincides with that of [15],

while for κ= 1 it yields novel two-point functions.

Nonetheless, since also in this case the overall analy-

sis is structurally identical to that for κ= 2, we do

not dwell into many details and we limit ourselves to

sketching the main steps of the construction.

Subsequently, in Section 3 we obtain an explicit

expression for the transition rate of a static Unruh-

DeWitt detector interacting either with the ground

or with a thermal state of a Klein-Gordon ﬁeld. We

scrutinize the response of the detector to grasp if and

how it is aﬀected by the choice of boundary condi-

tion. In particular, we focus on the consequences of

the freedom in selecting a secondary solution in the

construction of Section 2. In Section 4, we summarize

the main results of this work.

2. Dynamics and Boundary Conditions

Working with the coordinates of Equation (1), a

real Klein-Gordon ﬁeld Ψ that solves Equation (2)

can be decomposed as

Ψ(t, r, θ, ϕ) = e−iωtψ(r)Ym

`(θ, ϕ),

where Y`(θ, ϕ) are the spherical harmonics on the 2-

dimensional unit sphere. Setting λ=−`(`+ 1), the

function ψ(r) solves the radial equation

Lψ(r) :=d2ψ(r)

dr2+ω2+λ−m2

r2ψ(r) = 0.(3)

2.1. The radial equation

In the following we assume that m0≥0 and we in-

troduce the auxiliary quantities

ν:= 1

2q1−4λ+ 4m2

0and p:= √ω2,(4)

subject to the constraint ν > 0,∀`≥0. Equation

(3) is a Sturm-Liouville problem with eigenvalue ω2,

which admits a basis of solutions written in terms of

Bessel functions of ﬁrst and second kind:

y1(r) = √rJν(pr),(5a)

y2(r) = √rYν(pr).(5b)

Following [16], it is convenient to work on the

Hilbert space L2((0,∞), dr) and allow pto take a

priori any complex value. Hence, according to Weyl’s

endpoint classiﬁcation r→ ∞ is a limit point, see also

[15] for a succinct summary of the nomenclature. As a

matter of fact, assuming that Im p6= 0, the most gen-

eral solution of Equation (3) lying in L2((c, ∞), dr),

∀c∈(0,∞) can be written in terms of Hankel func-

tions of ﬁrst and second kind:

ψ∞(r) = √r[H(1)

ν(pr)Θ(Im p) + H(2)

ν(pr)Θ(−Im p)] .

On the other hand, it turns out that r= 0 is a

limit circle point if ν∈(0,1), since both y1, y2∈

L2((0, c0), dr), ∀c0∈(0,∞), as one can infer from the

following asymptotic expansion close to the origin:

|y1(r)|2r→0

∼r1+2ν,

|y2(r)|2r→0

∼r1−2ν.

Observe that

ν∈[0,1) ⇐⇒ `= 0 and m2

0∈−1

4,3

4.(6)

Still following the theory of Sturm-Liouville, this en-

tails that only the `= 0 mode calls for a boundary

condition at r= 0, provided that the mass lies in the

range individuated in Equation (6). When ν≥1, no

boundary condition needs to be imposed at both ends

and since this case is not of interest in this work, we

shall not consider it further.

2.2. Generalized (γ, v)-boundary conditions

At the limit circle endpoint, r= 0, we impose

generalized (γ, v)-boundary conditions as introduced

in [1]. These identify self-adjoint extensions of the

radial operator Lin L2((0,∞), dr) and they are fully

characterized in terms of a parameter γ∈R, of the

principal solution, u, and of a secondary solution, v,

which we introduce in the following. The principal

solution at r= 0 reads

u:= √rJν(pr) = y1(r).

This is unambiguously identiﬁed by the condition

that, for any solution vof Equation (3) such that

v6=λu,λ∈C, then lim

r→0

v= 0. On the contrary

choosing a secondary solution at r= 0, linearly in-

dependent from u, is fully arbitrary. In the following

we consider two possible choices vκ,κ= 1,2:

v1:= p√rYν(pr) (7a)

v2:= −p2ν√rJ−ν(pr).(7b)

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摘要：

Physicalsignicanceofgeneralizedboundaryconditions:anUnruh-DeWittdetectorviewpointonAdS2S2LissadeSouzaCampos1y?,ClaudioDappiaggi2y?,LucaSinibaldi3y?yDipartimentodiFisica,UniversitadegliStudidiPavia,ViaBassi,6,27100Pavia,Italy?IstitutoNazionalediFisicaNucleare{SezionediPavia,ViaBassi6,27100Pavia,It...

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Physical significance of generalized boundary conditions an Unruh-DeWitt detector viewpoint on AdS2 S2

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