Unimodular Hartle-Hawking wave packets and their probability interpretation Bruno Alexandre and Jo ao Magueijo Theoretical Physics Group The Blackett Laboratory Imperial College

2025-05-06 0 0 811.27KB 17 页 10玖币

侵权投诉

Unimodular Hartle-Hawking wave packets and their probability interpretation

Bruno Alexandre and Jo˜ao Magueijo∗

Theoretical Physics Group, The Blackett Laboratory, Imperial College,

Prince Consort Rd., London, SW7 2BZ, United Kingdom

(Dated: )

We re-examine the Hartle-Hawking wave function from the point of view of a quantum theory

which starts from the connection representation and allows for oﬀ-shell non-constancy of Λ (as in

unimodular theory), with a concomitant dual relational time variable. By translating its structures

to the metric representation we ﬁnd a non-trivial inner product rendering wave packets of Hartle-

Hawking waves normalizable and the time evolution unitary; however, the implied probability mea-

sure diﬀers signiﬁcantly from the naive |ψ|2. In contrast with the (monochromatic) Hartle-Hawking

wave function, these packets form travelling waves with a probability peak describing de Sitter space,

except near the bounce, where the incident and reﬂected waves interfere, transiently recreating the

usual standing wave. Away from the bounce the packets get sharper both in metric and connection

space, an apparent contradiction with Heisenberg’s principle allowed by the fact that the metric is

not Hermitian, even though its eigenvalues are real. Near the bounce, the evanescent wave not only

penetrates into the classically forbidden region but also extends into the a2<0 Euclidean domain.

We work out the propagators for this theory and relate them to the standard ones. The a= 0 point

(aka the “nothing”) is unremarkable, and in any case a wave function peaked therein is typically

non-normalizable and/or implies a nonsensical probability for Λ (which the Universe would pre-

serve forever). Within this theory it makes more sense to adopt a Gaussian state in an appropriate

function of Λ, and use the probability associated with the evanescent wave present near the time

of the bounce as a measure of the likelihood of creation of a pair of time-symmetric semiclassical

Universes.

I. INTRODUCTION

The Hartle-Hawking wave function of the Universe was

one of the ﬁrst proposals of a concrete framework for

quantum creation of the Universe out of nothing [1]. Its

interpretation and derivation has aroused much interest

(e.g. [2–4]), with a revival in recent years (e.g. [5–7]).

Even back in the 1980s, when the pioneering work of

Hartle and Hawking was done, an almost orthogonal ap-

proach to quantum gravity was developed, making the

connection, rather than the metric, the central character

of the theory (e.g. [8]). One of its earliest solutions was

the Chern-Simons-Kodama state [9, 11], but in the face

of its problems (e.g. [10]) this was superseded by the loop

representation, with sporadic backtracking [12–18].

The metric- and connection-driven approaches then led

separate lives. It was not until recently that it was real-

ized that the Chern-Simons-Kodama state is in fact the

Fourier dual of the Hartle-Hawking wave function un-

der the most minimal assumption: that the connection

is real [19]. The point of this paper is to explore what

can be learnt from this duality regarding the initial con-

ditions of the Universe.

The metric and connection appear as duals in the

quantum theory and the choice of representation in quan-

tum cosmology is not innocuous. It can lead to inequiv-

alent theories: diﬀerent natural ranges of variation for

the variables and diﬀerent natural inner products and

probability interpretations, for example. It is only in the

∗magueijo@ic.ac.uk

most standard setting, where the inner product is trivial

and ﬁxed a priori and the ranges for the variables are a

given on physical grounds, that one may appeal to the

Stone-von Neumann theorem and claim unitary equiva-

lence between the representations. In contrast, in quan-

tum cosmology the choice of representation may shed new

light on the old issue of “boundary conditions”. The rea-

son why the position representation is usually favoured

in standard Quantum Mechanics is that it is physically

clearer for deﬁning boundary conditions. But in quantum

gravity/cosmology it is far from obvious which represen-

tation should receive primacy in this respect. In this pa-

per we reassess the metric driven no-boundary proposal

and the Hartle-Hawking wave function from the point of

view of a theory which starts from the connection repre-

sentation.

In order to do this, an extra ingredient is needed. The

physical interpretation of the Chern-Simons-Kodama

state is improved by a minimal extension of Einstein’s

gravity: “unimodular” gravity [28] in its fully diﬀeomor-

phism formulation [30]. This happens for two reasons.

First, the unimodular extension introduces a physical

time variable (unimodular or 4-volume time [35]), so that

the waves now move “in physical time”. Second, it in-

troduces a natural (unitary) inner product with respect

to which normalizable wave packets, superposing states

with diﬀerent Λ, may be built [21, 22]. From the point

of view of unimodular theory, ﬁxed-Λ wave functions

are just the “spatial” factors of monochromatic partial

waves1. Pathologies found in the ﬁxed-Λ theory (inﬁnite

1“Spatial” here is used not in the sense of x, which is trivial

arXiv:2210.02179v2 [hep-th] 16 Feb 2023

norms, lack of a time variable) are cured for the wave

packets built from these partial waves, and physical be-

havior is found. In particular the peak of the proba-

bility (induced by the inner product) follows the classi-

cal trajectory in unimodular time in the semi-classical

regime [22]. Quantum deviations around this trajectory

give predictability to the theory (see [23, 24] for exam-

ples).

The narrative line of this paper is as follows. We ﬁrst

review results on the connection-driven (Section II) and

unimodular (Section III) quantum theories, as well as the

metric-connection duality exposed in [19] (Section IV).

In Section V we then translate into the metric formalism

the construction of unimodular wave packets selecting

the connection contour associated with Hartle-Hawking

waves, paying particular attention to the inner product

they acquire, the implied probability measure and the

structure of the Hilbert space.

Surprises are found at once. In Section VI we explicitly

construct packets of Hartle-Hawking waves for a Gaus-

sian amplitude in φ= 3/Λ. These packets are dramati-

cally diﬀerent from the usual standing waves: they form

well separated travelling waves even without the need of

Vilenkin boundary conditions. Their peaks follow the

semiclassical limit and get sharper as the Universe gets

larger, in an apparent contradiction with the Heisenberg

principle which we resolve by inspection of the eﬀects of

our unusual inner product.

In an attempt to make contact with creation out of

nothing and the no-boundary proposal, we evaluate the

unimodular propagators in Sections VII and VIII. The

mathematics is straightforward but serious problems are

identiﬁed in Section IX when trying to force the theory

into creation out of a= 0. Brieﬂy, such an initial condi-

tion is naturally non-normalizable, and would in any case

imply a non-viable distribution for Λ (which the Universe

would have to live with for ever). Having started from

the connection representation, the dual is also a2, not a,

with the whole real line naturally appearing in the theory.

All of this points to the creation of a pair of

time-symmetrical semiclassical Universes out of the full

evanescent wave resent around the bounce, with a Gaus-

sian state, as argued in Section X. In a concluding section

we summarise the take-home messages of this paper.

II. SUMMARY OF CONNECTION-BASED

RESULTS

The roots of connection led approaches, such as the

Ashtekar formalism, are in the Einstein-Cartan (EC) for-

malism [8]. We will only need the reduction to minisu-

perspace (MSS) in this paper, but start by presenting the

in minisuperspace, but in the sense of dependent on the non-

time variables, here the metric or the connection, as well as the

“frequency” conjugate to the time variable, here Λ itself. This

will be clearer later, cf. (6) vs (15), or (31) vs (34).

full theory because this will illuminate some peculiarities.

The EC action subject to a 3 + 1 split takes the form:

SEC =1

16πG Zdt d3x[2 ˙

aEa

i−(NH +NaHa+NiGi)],

where Ki

ais the extrinsic curvature connection (from

which the Ashtekar connection can be built by a canoni-

cal transformation [8]), Ea

iis the densitized inverse triad,

and the last three terms are the Hamiltonian, Diﬀeomor-

phism and Gauss constraints, enforced by corresponding

Lagrange multipliers. Quantization derives from imple-

mentations of:

[Ki

a(x), Eb

j(y)] = il2

Pδb

aδi

jδ(x−y) (1)

where lP=√8πG~is the reduced Planck length.

Adding Λ, in MSS this action becomes (e.g. [19, 25]):

S0=3Vc

8πG Zdt˙

ba2−Na −(b2+k) + Λ

3a2,(2)

where ais the expansion factor, bis the only MSS con-

nection variable (an oﬀ-shell version of the Hubble pa-

rameter, since b= ˙aon-shell, if there is no torsion), k

is the normalized spatial curvature (assumed k= 1, as

usual), Nis the lapse function and Vc=Rd3xis the co-

moving volume of the region under study, assumed ﬁnite

throughout this paper (in the quantum cosmology clas-

sical literature one usually chooses k= 1 and Vc= 2π2;

see [45] for a discussion of the criteria for the choice of

Vc). Hence, (1) becomes:

hˆ

b, ˆ

a2i=il2

3Vc≡ih,(3)

so that in the brepresentation:

ˆa2=−il2

3Vc

∂

∂b =−ih∂

∂b ,(4)

leading (with suitable ordering) to the WDW equation:

−(b2+k)−ihΛ

∂

∂b ψs= 0,(5)

where we introduce the subscript sto solutions of the

WDW equation for later conveninece. This is solved by

the Chern-Simons-Kodama (CSK) state reduced to MSS:

ψs(b, φ) = ψCS (b, φ) = Nbexp i

hφX(b),(6)

where Nbis a normalization factor2,φ= 3/Λ and

X(b) = LCS =b3

3+kb (7)

2Irrelevant in much of the original work on the CSK state, but

essential here, as we will see.

is the MSS reduction of the Chern-Simons functional.

It is known [19] that the CSK state is the Fourier dual

of both the Hartle-Hawking (HH) and Vilenkin (V)wave

functions, depending on the choice of contour (and sign

of Λ, as we will see). Indeed the literature on the HH

and V state uses the CSK state unwittingly (see [4] for

example). Although not strictly needed in this paper we

assume that bcovers the whole real line (so that we have

a HH dual for Λ >0). Dropping this assumption will

be investigated elsewhere: there are some technical dif-

ferences.

As announced in the introduction, the representation

from which one starts matters. By starting from the con-

nection we have made the following choices which are not

innocuous when re-examined from the metric viewpoint:

•The natural variable is φ= 3/Λ and not Λ. Clas-

sically this amounts to a canonical transformation:

Λ→φ(Λ) and T→Tφ=T/φ0(Λ). The quantum

mechanical theories that follow are not equivalent,

as we shall presently see.

•Starting from the metric we are led to the pair

{a, pa}(possibly with a > 0), whereas starting from

the connection the natural pair is {b, a2}. This

is because the conjugate of the connection is the

densitized inverse triad Ei

a, and in MSS this is

a2. Again they are canonically related, but lead

to quantum theories naturally based on diﬀerent

assumptions. Instead of a > 0, in the brepresen-

tation a2should cover the whole real line including

the negative-Euclidean section. This is because b

generates translations in a2, and remains most nat-

urally Hermitian if left to act unencumbered.

III. REVIEW OF THE UNIMODULAR

CHERN-SIMONS STATE

We use the Henneaux and Teitelboim formulation of

“unimodular” gravity [30], where full diﬀeomorphism in-

variance is preserved (so that “unimodular” is actually

a misnomer). In this formulation one adds to S0a new

term:

S0→S=S0−3

8πG Zd4x φ ∂µTµ(8)

(the pre-factor is chosen for later convenience). Here Tµ

is a density, so that the added term is diﬀeomorphism

invariant without the need of a √−gfactor in the volume

element or of the connection in the covariant derivative.

Since the metric and connection do not appear in the

new term, the Einstein equations and other standard ﬁeld

equations are left unchanged. The only new equations of

motion are:

δS

δT µ= 0 =⇒∂µφ=∂µΛ = 0 (9)

δS

δΛ= 0 =⇒∂µTµ∝√−g(10)

i.e. on-shell-only constancy for Λ (the deﬁning charac-

teristic of unimodular theories [28, 30–34]) and the fact

that T0is proportional to a prime candidate for relational

time: 4-volume time [28–30, 35, 36].

Reduction to MSS gives:

S0→S=S0+3Vc

8πG Zdtx ˙

φ T (11)

(where we identify T≡T0), so classically nothing

changes except that we gain a canonical pair enforcing

the constancy of Λ as an equation of motion, and a “time”

variable:

T=Na3

φ2=NΛ2

9a3.(12)

However, the quantum mechanics is very diﬀerent, since:

[φ, T ] = ih,(13)

that is φand Tare quantum complementaries. Hence, we

can choose either the φ(i.e. Λ) representation, leading

to the WDW equation (5) for ψs(b, φ), or its dual time

representation, leading to a Schrodinger equation:

−ih1

b2+k

∂

∂b −ih∂

∂T ψ(b, T )=0,(14)

for a wave function depending on time Tinstead. From

the unimodular perspective [20] the CSK state is just

the spatial factor, ψs=ψCS , of a monochromatic wave

(with ﬁxed Λ) moving in unimodular time Tconjugate

to φ. The general solution to the Hamiltonian constraint

is the superposition:

ψ(b, T ) = Z∞

−∞

dφA(φ) exp −i

hφT ψs(b, φ),

=Z∞

−∞

dφ

√2πhA(φ) exp i

hφ(X(b)−T),(15)

for some amplitude function A(φ). We have chosen nor-

malization N2

b=|ψs|2= 1/(2πh) so that the inversion

formula is symmetric:

A(φ) = ZdX ψ(b, T )e−i

hφ(X−T)

√2πh.(16)

For a Gaussian amplitude centered on φ0leading to a

probability with variance σφwe ﬁnd wave packets:

ψ(b, T ) = e−i

hφ0(X−T)

(2πσ2

T)1

exp −(X−T)2

4σ2

T(17)

with

σT=h

2σφ

(18)

saturating the Heisenberg uncertainty relation following

from (13).

Within the unimodular perspective, the natural inner

product between two states is given by:

hψ1|ψ2i=Zdφ A?

1(φ)A2(φ),(19)

and this product is automatically conserved with respect

to time T, i.e. unitarity is enforced, since it is deﬁned

in terms of T-independent amplitudes. By virtue of Par-

seval’s theorem (with the assumption that bis real) this

product is equivalent:

hψ1|ψ2i=ZdXψ?

1(b, T )ψ2(b, T ).(20)

Hence the probability in terms of bis:

P(b) = |ψ(b, T )|2dX

db =|ψ(b, T )|2(b2+k) (21)

where we note the measure factor3. Note that this could

have been guessed directly from the conserved current:

jT=jX=|ψ|2(22)

associated with the Schrodinger equation (14) written as:

∂

∂X +∂

∂T ψ= 0,(23)

(∂aja= 0, for a=T, X). Note also that one can bypass

expansion (15) to ﬁnd the the general solution:

ψ(b, T ) = F(T−X),(24)

where Fcan be any function. Hence the waves written

in terms of Xare non-dispersive. This allows us to guess

the propagator directly.

Given that we have unitarity, it is reasonable to deﬁne

physical states as any state derived from an amplitude

A(φ) such that:

hψ|ψi=Zdφ |A(φ)|2= 1,(25)

that is any state with norm 1. Hence a delta function

is not acceptable since the integral of the square of a

delta function is not 1, supporting the view aired pre-

viously that ﬁxed Λ “monochromatic” waves (whether

HH or V, or CSK states) are not physical, because non-

normalizable. The Gaussian states are normalizable, and

as we will see, lead to a sound semi-classical limit. How-

ever, we stress that not all A(φ) lead to a semiclassical

limit, so there is no a priori reason why perfectly physical

states should be semi-classical at all. For example, a nor-

malized uniform distribution in φ(conﬁned within a ﬁnite

3This inner product is exact and should not be confused with its

approximate semi-classical cousins required in more complicated

situations (such as multi-ﬂuids or minority clocks [22, 23, 26]).

range) is never semi-classical. We should not be surprised

that semi-classicality is a matter of choice/selection of

state, rather than an imposition from mathematical con-

sistency, or physical Hilbert space. Fully and endemic

quantum behaviour is perfectly physical.

We close this review by specifying why the canonical

transformations Λ →φ(Λ) and T→Tφ=T /φ0(Λ), lead

to theories which are all classically equivalent (and in fact

have the same semiclassical limit), but their quantum

mechanics is diﬀerent. Their solutions (15) are diﬀerent:

a Gaussian in Λ is not a Gaussian in a generic φ(Λ);

the frequency ΛTis not invariant under the canonical

transformation. The natural unimodular inner product

(19) is also not invariant [22, 23]. Although all these

quantum theories are diﬀerent, for a generic φchosen

within reason, their border with the semi-classical limit

is the same, as we will comment in more detail later.

IV. THE MONOCHROMATIC METRIC DUALS

OF THE CSK STATE

Our ﬁrst purpose is to translate the constructions re-

viewed in the last two Sections to the metric represen-

tation. We start with the metric dual of the CSK state,

extending [19] to suit our purposes. We assume that both

band a2are real and unconstrained, so that:

ψs(a2, φ) = Z∞

−∞

√2πhe−iba2

hψs(b, φ) (26)

ψs(b, φ) = Z∞

−∞

da2

√2πheiba2

hψs(a2, φ) (27)

(in [19] we did not deﬁne (27); note the symmetrical con-

vention adopted here in contrast with [19]). Note that

(25) implies boundary conditions both as a2→ ∞ and

a2→ ∞. We will often assume that A(φ) is peaked at

a positive φand exponentially suppressed at φ < 0, but

in some parts of this paper a general φwill be required.

It was shown in [19] that the CSK state is the Fourier

dual of both the HH and V wave functions, depending on

the choice of contour and that imposing the reality of b

selects the HH wave function if Λ >0 (which for simplic-

ity and deﬁniteness was assumed throughout [19]). We

will relax this assumption here, and not only conﬁrm the

results of [19] for Λ >0, but also show that the reality

of bselects the V wave function if Λ <0.

This is straightforward to show using the integral rep-

resentation of the Airy-like functions:

f(−z) = 1

2πZ∞

−∞

eit3

3−ztdt, (28)

noting that it maps onto (26) with [19]:

z=−φ

h2/3k−a2

φ,(29)

t=φ

h1/3

b. (30)

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：

UnimodularHartle-HawkingwavepacketsandtheirprobabilityinterpretationBrunoAlexandreandJo~aoMagueijoTheoreticalPhysicsGroup,TheBlackettLaboratory,ImperialCollege,PrinceConsortRd.,London,SW72BZ,UnitedKingdom(Dated:)Were-examinetheHartle-Hawkingwavefunctionfromthepointofviewofaquantumtheorywhichstartsf...

展开>> 收起<<

Unimodular Hartle-Hawking wave packets and their probability interpretation Bruno Alexandre and Jo ao Magueijo Theoretical Physics Group The Blackett Laboratory Imperial College.pdf

共17页,预览4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！

Unimodular Hartle-Hawking wave packets and their probability interpretation Bruno Alexandre and Jo ao Magueijo Theoretical Physics Group The Blackett Laboratory Imperial College

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: