KEK-TH-2449 Baby universes in 2d and 4d theories of quantum gravity Yuta HamadaHikaru Kawaiand Kiyoharu Kawana

2025-05-06 0 0 564.68KB 24 页 10玖币

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KEK-TH-2449

Baby universes in 2d and 4d theories of quantum gravity

Yuta Hamada,∗Hikaru Kawai,†and Kiyoharu Kawana,‡

∗Theory Center, IPNS, KEK, 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan,

∗Graduate University for Advanced Studies (Sokendai),

1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan,

∗Department of Physics, Harvard University, Cambridge, MA 02138 USA,

†Department of Physics and Center for Theoretical Physics,

National Taiwan University, Taipei 106, Taiwan, R.O.C.

†Physics Division, National Center for Theoretical Sciences,

Taipei 106, Taiwan, R.O.C.

‡Center for Gravitational Physics and Quantum Information,

Yukawa Institute for Theoretical Physics,

Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan

Abstract

The validity of the Coleman mechanism, which automatically tunes the fundamental

constants, is examined in two-dimensional and four-dimensional quantum gravity theories.

First, we consider two-dimensional Euclidean quantum gravity on orientable closed man-

ifolds coupled to conformal matter of central charge c≤1. The proper time Hamiltonian

of this system is known to be written as a ﬁeld theory of noncritical strings, which can also

be viewed as a third quantization in two dimensions. By directly counting the number of

random surfaces with various topologies, we ﬁnd that the contribution of the baby uni-

verses is too small to realize the Coleman mechanism. Next, we consider four-dimensional

Lorentzian gravity. Based on the diﬀerence between the creation of the mother universe

from nothing and the annihilation of the mother universe into nothing, we introduce a

non-Hermitian eﬀective Hamiltonian for the multiverse. We show that Coleman’s idea

is satisﬁed in this model and that the cosmological constant is tuned to be nearly zero.

Potential implications for phenomenology are also discussed.

∗E-mail: yhamada@post.kek.jp

†E-mail: hikarukawai@phys.ntu.edu.tw

‡E-mail: kiyoharukawana@gmail.com

arXiv:2210.05134v2 [hep-th] 23 Dec 2022

Contents

1 Introduction 3

2 2d Euclidean quantum gravity with various topologies 3

2.1 Formulations in Continuum theory . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.1 Hamiltonian Formalism: Non-critical string ﬁeld theory . . . . . . . . 4

2.1.2 Path Integral Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 The absence of the Coleman mechanism . . . . . . . . . . . . . . . . . . . . . 8

2.3 Modiﬁcation of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Coleman mechanism in Lorentzian multiverse 12

3.1 LorentzianModel.................................. 12

3.2 Fine-tuning of the cosmological constant . . . . . . . . . . . . . . . . . . . . . 16

3.3 Maximum entropy principle and maximum matter principle . . . . . . . . . . 16

4 Conclusion 18

A Random Walk 19

1 Introduction

The smallness of the cosmological constant is one of the great mysteries of particle physics.

In the late 1980’s, Coleman proposed a solution [1, 2] based on the eﬀects of Euclidean

wormholes [3] (see also Refs. [4, 5] for reviews). After summing the wormholes, the low

energy theory is described by an ensemble average of various coupling constants, including the

cosmological constant. However, Coleman’s original proposal has problems such as [6, 7, 8].

These problems seem to stem from the pathology of the 4d Euclidean gravity associated

with the conformal mode. To overcome this problem, a Lorentzian formulation of Coleman’s

mechanism was proposed and studied [9, 10, 11, 12, 13].

On the other hand, signiﬁcant progress has recently been made toward resolving the

information paradox of the black hole [14, 15]. At least in two dimensions, the replica

wormhole [16, 17] plays an important role in reproducing the unitary page curve of the black

hole entropy (see Refs. [18, 19] for reviews).

Given the importance of wormhole, it is interesting to revisit the Coleman’s mechanism

in two dimensions. Indeed, 2d Euclidean quantum gravity on closed manifolds coupled to

a matter ﬁeld with central charge c≤1 is well-deﬁned. Its proper time Hamiltonian can

be regarded as a kind of ﬁeld theory of noncritical strings. Thus, the validity of Coleman’s

proposal can be clearly discussed.1

In this paper, we ﬁrst show that the sum of topologies in 2d Euclidean gravity does not

lead to an automatic tuning of the cosmological constant by explicitly counting the number

of random surfaces.2We argue that this is true for a wide range of modiﬁcations of the 2d

Euclidean gravity based on the matrix model. Next, we consider 4d Lorentzian gravity and

introduce an eﬀective Hamiltonian of the multiverse consisting of the creation and annihilation

operators of the mother and baby universes. The Hamiltonian is non-Hermitian due to the

diﬀerence between the creation of the mother universe from nothing and the annihilation of

the mother universe into nothing. In this model, the Coleman mechanism is realized and the

eﬀective cosmological constant is tuned to almost zero.

The paper is organized as follows. In Section 2, we ﬁrst outline the path integral formula-

tion of 2d Euclidean gravity (non-critical strings). In particular, we introduce the Hamiltonian

formulation for 2d gravity coupled to (2, q) minimal matter.3We then show that the eﬀect of

the microscopic baby universes is too small compared to the macroscopic topology changes

to realize the Coleman mechanism. We then consider modiﬁcations of 2d gravity based on

the matrix model, and discuss that the Coleman mechanism works in Lorentzian gravity.

In Section 3, we consider Lorentzian gravity. The processes of creation and annihilation of

the mother and baby universes are investigated, and a non-Hermitian eﬀective Hamiltonian

describing a Lorentzian multiverse is introduced. We show that Coleman’s idea is satisﬁed in

this model. We also discuss the potential implications for phenomenology.

2 2d Euclidean quantum gravity with various topologies

In this section, we examine the possibility of obtaining ensemble averages for the coupling

constants from the sum of topologies in two-dimensional gravity. The Euclidean 2d gravity

1The analysis of wormholes in the worldsheet theory of critical strings has been done in [20]. See also

Ref. [21] for recent study of the Liouville theory coupled to c= 1 matter using the matrix quantum mechanics.

2The ﬂuctuation of the cosmological constant in 2d gravity is also considered in [22].

3It includes the Jackiw-Teitelboim (JT) gravity as a limit q→ ∞ [23, 24].

V(P;D)

S(P;D)

Figure 1: Illustration of V(P;D) and S(P;D). Here V(P;D) is the set of points whose

geodesic distance from Pis less than or equal to D. The boundary of V(P;D) is denoted by

S(P;D). In the ﬁgure, S(P;D) consists of three loops.

coupled to a matter ﬁeld with central charge c≤1 is well-deﬁned without suﬀering from the

problem of the conformal mode.4It can be deﬁned either by continuum theory [25, 26, 27] or

by dynamical triangulation [28, 29, 30, 31, 32]. In particular, all topologies can be summed

using the matrix model [33, 34, 35]. (See also Ref. [36, 37, 38, 39] for reviews. )

As is well known, 4d Euclidean gravity has diﬃculties due to instability of the conformal

mode. Also, whether a microscopic wormhole is more important than a macroscopic topology

change depends on the dimension of spacetime. Nevertheless, the 2d Euclidean wormhole is

a good clue to investigate the 4d Lorentzian multiverse, as we will see in the next section.

2.1 Formulations in Continuum theory

In this subsection, we introduce two formalisms of 2d Euclidean gravity.

2.1.1 Hamiltonian Formalism: Non-critical string ﬁeld theory

In Refs. [40, 41], a Hamiltonian formalism was proposed for 2d Euclidean gravity, in which

the geodesic distance is considered as time (See Ref. [42] for the Hamiltonian in dynamical

triangulation). This theory can be regarded as a string ﬁeld theory, since the Hamiltonian

describes the creation and annihilation of universes of spatial dimension 1. It can also be

viewed as a two-dimensional third quantization theory [43]. This formalism is convenient to

generalize to Lorentzian spacetime, which we will explore in Section 3.

Let us consider 2d spacetime, and take an arbitrary point P(See Fig. 1 for illustration).

Then, the set of points V(P;D) is deﬁned as

V(P;D) = {Q∈(spacetime)|d(P, Q)≤D}(1)

where d(P, Q) is the geodesic distance between Pand Q. Let S(P;D) be the boundary of

V(P;D).

Next, we introduce operators ψ†(`) and ψ(`), which create and annihilate loops (1d spaces)

of length `, respectively. They satisfy the relation,

[ψ(`), ψ†(`0)] = δ(`−`0),(2)

4This is related to the fact that the number of degrees of freedom is negative in 2d gravity.

and the vacuum (the absence of space) is deﬁned as

ψ(`)|0i=h0|ψ†(`)=0.(3)

For simplicity, we take the (2, q) minimal model as the matter ﬁeld.5In that case, there is no

need to introduce any extra degrees of freedom other than l.6Then, the state with kloops

of length `1,··· , `kcan be written as

|`1,··· , `ki=ψ†(`1)···ψ†(`k)|0i,(4)

and the state of the boundary surface S(P;D) is represented by their superposition:

|S(P;D)i=∞

k=0 Z∞

d`1···Z∞

d`kck(`1,··· , `k)|`1,··· , `ki.(5)

We can deﬁne Hamiltonian [42, 40], which describes the inﬁnitesimal translation of the

proper time D.

dD |S(P;D)i=−HEuclid|S(P;D)i,(6)

where HEuclid is given by

HEuclid =Z∞

d`1d`2ψ†(`1)ψ†(`2)ψ(`1+`2) + Z∞

d`1d`2ψ†(`1+`2)ψ(`1)ψ(`2)

+Z∞

d` ρ(`)ψ(`).(7)

The source function ρ(`) is7

ρ(`) = (λ δ(`) for (2,1) topological gravity (c=−2) [45]

λ δ(`) + δ00(`) for (2,3) pure gravity (c= 0) [40] .(8)

This function is related to the disk amplitude:

˜ρ(ζ) = ∂

∂ζ (˜

D(ζ))2,(9)

where ˜

D(ζ) is the Laplace transformation of the disk amplitude D(l).

The function ρcorresponding to the JT gravity [23, 24] can be obtained as follows. Its

action is given by

SJT =−S0

2π1

2ZM

√gR +Z∂M

√hK−1

2ZM

√gΦ(R+ 2) + Z∂M

√hΦ(K−1),(10)

where S0is a constant, Kis the boundary extrinsic curvature, and Φ is the dilaton. We

consider the case where Mhas the disk topology. From the variation of Φ, we obtain that

5The (2, q) minimal model has c=−3q+13−12

q. For example, q= 1,3,∞gives c=−2,0,−∞,respectively.

6Non-critical string ﬁeld theory for the minimal unitary series (p, p + 1) (p= 2,3,· · · ) is given in Ref. [44].

7To be precise, the source term is deﬁned through the Laplace transformation.

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KEK-TH-2449Babyuniversesin2dand4dtheoriesofquantumgravityYutaHamada,*HikaruKawai,andKiyoharuKawana,TheoryCenter,IPNS,KEK,1-1Oho,Tsukuba,Ibaraki305-0801,Japan,GraduateUniversityforAdvancedStudies(Sokendai),1-1Oho,Tsukuba,Ibaraki305-0801,Japan,DepartmentofPhysics,HarvardUniversity,Cambridge,MA021...

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KEK-TH-2449 Baby universes in 2d and 4d theories of quantum gravity Yuta HamadaHikaru Kawaiand Kiyoharu Kawana

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