Comments on Lorentzian topology change in JT gravity Mykhaylo Usatyuk Center for Theoretical Physics and Department of Physics University of California Berkeley

2025-04-27 0 0 492.23KB 26 页 10玖币

侵权投诉

Comments on Lorentzian topology change in JT gravity

Mykhaylo Usatyuk

Center for Theoretical Physics and Department of Physics, University of California, Berkeley,

CA 94720, USA

musatyuk@berkeley.edu

Abstract

We propose a deﬁnition for the Lorentzian Jackiw-Teitelboim (JT) gravity path integral that includes

Lorentzian topology changing conﬁgurations. The construction is inspired by the bosonic string genus

expansion on singular Lorentzian worldsheets, with geometries known as lightcone diagrams playing

a prominent role. The Lorentzian path integral is deﬁned through a suitable analytic continuation

of the Euclidean path integral, and includes metrics that become degenerate at isolated points

allowing for Lorentzian topology changing transitions. We discuss the relation between Euclidean

JT amplitudes and the proposed Lorentzian amplitudes.

arXiv:2210.04906v1 [hep-th] 10 Oct 2022

Contents

1 Introduction 1

2 Lightcone diagrams 4

2.1 Euclidean and Lorentzian lightcone diagrams . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Negative curvature lightcone diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 JT gravity on lightcone diagrams 12

3.1 LorentzianJTpathintegral.................................. 15

4 Discussion 18

A Lorentzian pair of pants 20

B Details on punctured Riemann surfaces and measures 22

1 Introduction

Euclidean wormholes have played an important role in recent developments in quantum gravity. The

study of two dimensional Jackiw-Teitelboim (JT) gravity [1] has been central to these developments.

Topology changing Euclidean conﬁgurations have been found to be important in the calculation of a

unitary page curve [2,3], the spectral form factor [1,4,5], and a variety of other interesting eﬀects,

see [6–8] for a few selected examples.

However, so far a satisfactory Lorentzian explanation for Euclidean wormhole calculations has been

lacking. The standard procedure is to calculate a Euclidean amplitude and then analytically continue

it to obtain a Lorentzian result. The role that Euclidean wormholes play in the Lorentzian theory is

not apparent in this analytic continuation. Similarly, in the standard canonical quantization treatment

of Lorentzian JT gravity [9] the topology of the Lorentzian spacetime is ﬁxed, and it’s unclear how the

Euclidean path integral formulation of the theory [1] can be related to the Lorentzian formulation.

In this work we formulate a Lorentzian theory of JT gravity that includes Lorentzian topology

changing conﬁgurations. The theory is deﬁned through a special analytic continuation of the standard

Euclidean path integral. Euclidean wormholes are turned into topology changing Lorentzian conﬁgu-

rations with degenerate points in the metric, see ﬁgure 1.

Figure 1: Lorentzian topology changing transition where two spatial circles evolves into three circles. The

metric is Lorentzian everywhere except the splitting points (blue circles) where it is degenerate. At the

splitting points the topology of spatial slices changes. The spacetime is an analytic continuation of a genus

two Euclidean geometry with ﬁve circular boundaries.

Our proposal is inspired by a formulation of bosonic string theory on degenerate Lorentzian world-

sheets known as the interacting string picture [10,11]. In [12,13] it was argued that the interacting

string picture is equivalent to the standard Euclidean path integral formulation to all orders in the genus

expansion. This is accomplished by gauge-ﬁxing the Euclidean path integral to lightcone diagrams [12],

which are a special class of metrics that can be analytically continued to Lorentzian signature where

they become the topology changing geometries of the interacting string picture.

To deﬁne the Lorentzian JT path integral we closely follow the construction of the bosonic string

genus expansion with singular Euclidean/Lorentzian worldsheets [12–14]. We start with the standard

Euclidean JT path integral and with a suitable gauge choice and analytic continuation we end up with

a Lorentzian path integral over degenerate metrics. We now brieﬂy explain this construction, leaving

technical details to the main sections.

Summary of results

We begin with the two dimensional Euclidean gravity path integral with speciﬁed boundary condi-

tions. In this paper we consider boundary conditions given by ngeodesic circles of given lengths

b“ pb1,¨¨¨ , bnq. For simplicity we assume the bulk geometry has ﬁxed genus gand is fully connected.

The path integral is computed by

Z“żDgµν DΦ

Vol e´IJTrg,Φs“(1.1)

The integral over Euclidean metrics can be split into an integral over the Weyl factor ωand over the

moduli ˆg, where all metrics can be represented as g“e2ωˆg. The moduli space of ˆgis the moduli

space of punctured Riemann surfaces Mg,n of genus gwith npunctures. It was shown by Giddings and

Wolpert [12] that this Moduli space has a representative metric ˆgthat is ﬂat with curvature singularities

at isolated points, this is known as a Euclidean lightcone metric. Choosing the Euclidean lightcone

metric as our representative metric ˆggives us a path integral over singular Euclidean geometries

Z“żmoduli

dpmeasureqżDωDΦe´IJTre2ωˆg,Φs“(1.2)

In the above ﬁgure, the circles (blue) corresponds to points where the metric ˆgbecomes degenerate

det ˆg“0. It is at these points that the topology of spatial slices changes. The integral over the moduli

is partially over all possible locations of the degenerate points. The Euclidean lightcone diagram has a

globally deﬁned euclidean time τin terms of which the metric is ﬂat everywhere except the degenerate

points. The boundary conditions are now speciﬁed at τ“ ˘8, and we must make a choice to send

boundary conditions either to the future or the past. In the above ﬁgure the boundary of length b1has

been sent to the past while the boundaries of length b2, b3have been sent to the future.

To turn the above Euclidean path integral into a Lorentzian path integral, we will analytically

continue the Euclidean lightcone geometries. Since the time τis globally deﬁned, we can analytically

continue τÑiτ to get a Lorentzian signature metric ˆg, known as a Lorentzian lightcone diagram.

We now have a path integral over degenerate Lorentzian metrics that incorporate topology changing

transitions

ZL“żmoduli

dpmeasureqżDωDΦeiIJTre2ωˆg,Φs.(1.3)

Our deﬁnition for the Lorentzian JT path integral will be the above integral over Lorentzian lightcone

diagrams. The role of the Weyl factor ωwill be to give the geometry constant negative curvature away

from the degenerate points.

In the above procedure we started with the usual Euclidean path integral and through a gauge

choice and a suitable analytic continuation we ended up with a Lorentzian path integral. It might

then be expected that the amplitudes computed with this Lorentzian path integral should agree with

the corresponding Euclidean amplitudes. Indeed, this is the argument of D’Hoker and Giddings [13]

that the interacting string picture [11] is equal to the Euclidean path integral to all orders in the

genus expansion1. In the case of JT gravity there are some additional subtleties that arise due to the

degenerate points, and we return to this question in section 3and in the discussion. We now brieﬂy

summarize the rest of the paper.

In Section 2we review the basic aspects of Euclidean and Lorentzian lightcone diagrams. We also

review the work of Giddings and Wolpert [12] where it was shown that Euclidean lightcone diagrams

give a single cover of the moduli space of punctured Riemann surfaces. Lastly, we discuss the problem

of ﬁnding a Weyl factor to turn a Euclidean/Lorentzian lightcone metric into a constant negative

curvature geometry with degenerate points.

In Section 3we ﬁll in the technical details of the path integral over lightcone diagrams. We explain

how boundary conditions are implemented, and we discuss the integration measure over the moduli

space of lightcone diagrams. The inclusion of degenerate points introduces certain ambiguities into the

path integral, and we discuss how these ambiguities modify the relation of the Euclidean amplitudes

to the Lorentzian amplitudes.

In the Appendices we construct the Lorentzian pair of pants with constant negative curvature, and

we include additional details on punctured Riemann surfaces and the integration measure.

2 Lightcone diagrams

In this section we review aspects of Euclidean and Lorentzian lightcone diagrams, their connection to

punctured Riemann surfaces, and how to construct constant negative curvature analogues of lightcone

diagrams.

2.1 Euclidean and Lorentzian lightcone diagrams

A Euclidean/Lorentzian lightcone diagram is a two dimensional geometry with degenerate metric g

built out of ﬂat Euclidean/Lorentzian cylinders joined together at singular points, see ﬁgure 2. The

two basic properties of a lightcone diagram are the number of asymptotic cylinders nand the genus

g. The number of cylinders running oﬀ to inﬁnity is given by ně2 with a ﬁxed number extending to

past or future inﬁnity2. In the next section we will see that in/out states are speciﬁed by introducing

boundary conditions on the asymptotic cylinders. The genus gand number of boundaries ndetermine

the number of times the diagram splits apart and joins together in the interior of the geometry. At the

splitting and joining points the metric is degenerate, and there are 2g´2`nsuch degenerate points.

We will also call these interaction/singular points, and denote where they occur by the subscript zI.

1Giddings and D’Hoker [13] stopped short of analytically continuing the Euclidean lightcone diagrams to Lorentzian

signature. Thus they argued that the analytically continued interacting string picture was equivalent to the usual Euclidean

path integral.

2At least one cylinder must run to the future and one to the past.

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

CommentsonLorentziantopologychangeinJTgravityMykhayloUsatyukCenterforTheoreticalPhysicsandDepartmentofPhysics,UniversityofCalifornia,Berkeley,CA94720,USAmusatyuk@berkeley.eduAbstractWeproposeadenitionfortheLorentzianJackiw-Teitelboim(JT)gravitypathintegralthatincludesLorentziantopologychangingcong...

展开>> 收起<<

Comments on Lorentzian topology change in JT gravity Mykhaylo Usatyuk Center for Theoretical Physics and Department of Physics University of California Berkeley.pdf

共26页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Comments on Lorentzian topology change in JT gravity Mykhaylo Usatyuk Center for Theoretical Physics and Department of Physics University of California Berkeley

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: