de Sitter space extremal surfaces and time-entanglement K. Narayan

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de Sitter space, extremal surfaces

and “time-entanglement”

K. Narayan

Chennai Mathematical Institute,

H1 SIPCOT IT Park, Siruseri 603103, India.

Abstract

We reﬁne previous investigations on de Sitter space and extremal surfaces anchored

at the future boundary I+. Since such surfaces do not return, they require extra

data or boundary conditions in the past (interior). In entirely Lorentzian de Sitter

spacetime, this leads to future-past timelike surfaces stretching between I±. Apart

from an overall −ifactor (relative to spacelike surfaces in AdS) their areas are real and

positive. With a no-boundary type boundary condition, the top half of these timelike

surfaces joins with a spacelike part on the hemisphere giving a complex-valued area.

Motivated by these, we describe two aspects of “time-entanglement” in simple toy

models in quantum mechanics. One is based on a future-past thermoﬁeld double type

state entangling timelike separated states, which leads to entirely positive structures.

Another is based on the time evolution operator and reduced transition amplitudes,

which leads to complex-valued entropy.

arXiv:2210.12963v4 [hep-th] 21 Apr 2023

Contents

1 Introduction and summary 1

2dS extremal surfaces from I+, boundary conditions 2

2.1 Lorentzian dS ................................................................ 3

2.2 dS no-boundarysurfaces .......................................................... 6

2.3 2-dim CFT, timelike subsystems, complex EE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 “Time-entanglement” in quantum mechanics 9

3.1 Afuture-pastthermoﬁelddoublestate................................................... 9

3.2 Time-evolution and reduced transition amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4 Discussion: dS surfaces, time contours, rotations 12

1 Introduction and summary

It is of great interest to understand holography for de Sitter space (see the review [1]). In de

Sitter (and cosmology more generally) perhaps the natural asymptotics are in the far future

or the far past: this thinking leads to dS/CF T [2, 3, 4] (and [5] in the higher spin context),

which associates a hypothetical non-unitary dual Euclidean CFT at the future boundary

I+, with several dramatic diﬀerences from AdS [6, 7, 8]. A particularly fascinating question

is whether de Sitter entropy [9] can be understood as some sort of entanglement entropy.

It is then natural to ask if the extensive investigations of holographic entanglement in AdS

[10, 11, 12] can be generalized to de Sitter space.

One possible generalization of the Ryu-Takayanagi formulation to de Sitter space is to

consider the bulk analog of setting up entanglement entropy in the dual Euclidean CF T on

the future boundary [13]. We restrict to some boundary Euclidean time slice as a crutch,

deﬁne subregions on these slices, and look for extremal surfaces anchored at I+dipping

into the holographic (time) direction. Analysing this extremization interestingly shows that

surfaces anchored at I+do not return to I+,i.e. there is no I+→I+turning point, so there

are no spacelike surfaces connecting points on I+. There exist analytic continuations of RT

surfaces in AdS which lead to complex extremal surfaces [13, 14, 15, 16]. In [17, 18], entirely

timelike future-past extremal surfaces were studied, stretching from I+to I−.

In this note, we develop this further, stitching together an overall perspective which

hopefully adds value to the understanding of these studies. The absence of I+→I+returns

for surfaces implies that surfaces starting at I+continue inward, to the past: this suggests

that they require extra data or boundary conditions in the interior, or far past to be well-

deﬁned. One obvious possibility for an entirely Lorentzian de Sitter space (sec. 2.1) is that

the surfaces then end at the past boundary I−. Analysing this in detail leads to future-

past surfaces stated above [17, 18]. These are timelike extremal surfaces stretching between

subregions at I+and equivalent ones at I−: they are akin to rotated analogs of the Hartman-

Maldacena surfaces [19] in the eternal AdS black hole. Being entirely timelike, their area

has an overall −ifactor, relative to the familiar spacelike extremal surfaces in AdS (this

overall −iwas discarded in [17, 18]; see below). Since we obtain codim-2 surfaces (when

they exist), their area scales as de Sitter entropy.

Another possibility for the interior boundary conditions arises from modifying de Sitter

from being entirely Lorentzian in accord with the Hartle-Hawking no-boundary prescription,

i.e. to cut dS in the middle and remove the bottom half, replacing it with a hemisphere

(sec. 2.2). Now we join the top timelike part of the extremal surfaces above with regularity

at the mid-slice to a spatial extremal surface that goes around the hemisphere (thus turning

around): see [20, 21] for dS3. This spacelike part has real area so that the total area is

complex-valued. The top part of the surface (in the Lorentzian de Sitter) is the same as in

the entirely timelike surfaces above: this reﬂects consistency of the future-past surfaces with

Hartle-Hawking boundary conditions. The ﬁnite real part of the area of the no-boundary

surfaces arises from the hemisphere and is precisely half de Sitter entropy for any dimension

when the subregion at I+becomes maximal. In sec. 4, we give some comments on these

future-past and no-boundary surface areas in terms of time contours, and argue that they

can be regarded as space-time rotations from timelike subregions in AdS-like spaces.

Imaginary values also arise in studies of quantum extremal surfaces in de Sitter with re-

gard to the future boundary [22, 23], stemming from timelike-separations (sec. 2.3). Complex-

valued entanglement entropy was also found quite explicitly in studies of ghost-like theories,

including simple toy quantum-mechanical models of “ghost-spins”, e.g. [24, 25].

For entirely Lorentzian dS, the entirely timelike future-past surfaces are akin to entirely

timelike geodesics for ordinary particles moving in time. Removing the overall −iin their

pure imaginary areas (relative to real spacelike surface areas) is akin to calling the length

of timelike geodesics as “time” rather than “−i·space”. Overall this suggests that the areas

of these dS extremal surfaces with timelike components encode some new object, “time-

entanglement”, distinct from usual spatial entanglement. In sec. 3, we describe two aspects

of this in ordinary quantum mechanics, which incorporate this entry of late and early time

boundary conditions. One is based on a future-past thermoﬁeld-double state [17] (see also

[26, 27]) which leads to entirely positive structures despite the timelike separation. The other

involves the time-evolution operator and “reduced transition amplitudes”, giving complex-

valued entropy. As we were preparing this, the work [28] appeared with partial overlap.

2dS extremal surfaces from I+, boundary conditions

The simplest place to see the absence of I+→I+turning points [13] is in the Poincare

slicing with planar foliations, so

ds2

d+1 =R2

τ2(−dτ2+dy2

i) = R2

τ2(−dτ2+dw2+dx2

i).(1)

Here we have singled out w∈yias boundary Euclidean time, without loss of generality.

Taking the w=const slice, we consider at I+a strip-shaped subregion (the natural sub-

regions consistent with planar symmetries), with width along x∈xiand extremal surfaces

anchored from one boundary interface of the strip. This leads to the area functional and

extremization,

SdS =−iRd−1

dS Vd−2

4Gd+1 Zdτ

τd−1p1−(∂τx)2→(∂τx)2=B2τ2d−2

1 + B2τ2d−2.(2)

where B2is some constant. The fact that there is a minus sign relative to the extremization

equation in AdS is the reﬂection of the absence of turning points back to I+. We see that

(∂τx)21 near the boundary τ∼0 and remains bounded with (∂τx)2<1 throughout, for

any real B2>0. (The surfaces with B2<0 are equivalent to analytic continuations from

AdS RT surfaces [13, 14, 15, 16].) We will return to this later.

The absence of I+→I+return implies that the surfaces march on inward: this suggests

they end at I−if we focus on entirely Lorentzian de Sitter space. These lead to future-past

extremal surfaces, timelike codim-2 surfaces stretching from I+to I−. We describe this now,

ﬁrst in part reviewing the studies in [17, 18]. Alternatively we could modify Lorentzian dS

in accord with the Hartle-Hawking no-boundary prescription replacing the bottom half of

dS by a hemisphere, and then impose a no-boundary type boundary condition on extremal

surfaces. We will discuss these now.

2.1 Lorentzian dS

Static coordinates: These coordinates exhibit static patches exhibiting time translation

symmetry, but allowing analytic extensions to the entire de Sitter space. We have

ds2=−(1 −r2

l2)dt2+dr2

1−r2

+r2dΩ2

d−1.(3)

In the Northern/Southern diamond regions N/S, the static patches, tis time enjoying trans-

lation symmetry. Event horizons for observers in N/S are at r=l: the area of these

cosmological horizons is de Sitter entropy. Towards studying the future boundary, we use

τ=l

r, w =t

l, to recast as ds2=l2

τ2−dτ2

1−τ2+ (1 −τ2)dw2+dΩ2

d−1: now τis bulk time,

with τ= 0 the future/past boundary and the future/past universes described by 0 ≤τ < 1.

In this case the boundary at I+is R×Sd−1. We can take the boundary Euclidean time slice

as any Sd−1equatorial plane or as the w=const slice.

Taking the boundary Euclidean time slice as some Sd−1equatorial plane, we deﬁne a

subregion as ∆w×Sd−2∈I+and an equivalent one at I−. Then we obtain the area functional

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deSitterspace,extremalsurfacesandime-entanglement"K.NarayanChennaiMathematicalInstitute,H1SIPCOTITPark,Siruseri603103,India.AbstractWerenepreviousinvestigationsondeSitterspaceandextremalsurfacesanchoredatthefutureboundaryI+.Sincesuchsurfacesdonotreturn,theyrequireextradataorboundaryconditionsinthep...

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