Issues in Quantum-Geometric Propagation

Home , Wigner rotation

CERN{TH/97{167

ISSUES IN QUANTUM{GEOMETRIC PROPAGATION

M.A. CLAYTON

Abstract. A discussion of relativistic quantum{geometric mechanics on phase space and its gener-

alisation to the propagation of free, massive, quantum{geometric scalar elds on curved spacetimes

is given. It is shown that in an arbitrary co ordinate system and frame of reference in a at spacetime,

the resulting propagator is necessarily the same as derived in the standard Minkowski co ordinates

up to a Lorentz b o ost acting on the momentum content of the eld, which is therefore seen to play

the role of Bogolub ov transformations in this formalism. These results are explicitly demonstrated

in the context of a Milne universe.

1. Introduction

That a fundamental scale should exist in the `ultimate' theory of nature seems bynow to b e fairly

well accepted, and it is therefore of interest to examine a program that incorp orates such a scale into

physics from the outset [28 , 29 , 31 ]. That the program involves a mo di cation of non{relativistic

quantum mechanics (without disturbing its agreement with exp eriment) is overshadowed by the

improved measurement{theoretic scenario that emerges, as well as the extension to a relativistic

quantum mechanical picture in b oth at and curved spacetimes; leading to a quantum gravitational

scenario. The twofold purp ose of this manuscript is to giveanoverview of the basic ideas and results

of scalar Quantum{Geometric (QG) eld theory on a curved spacetime, as well as to clarify QG

eld propagation in at spacetime using a Milne universe for illustrative purp oses. Even in sucha

simple spacetime mo del the conventional eld quantisation is not without it's ambiguities [4 , 35 ].

That the conventional treatment of lo cal quantum elds, or the resulting formal `p erturbative'

series, is not adequate follows from attempts to construct the simplest scalar quantum eld theory.

These results indicate that, at b est, the current p erturbative treatment is not in accord with the

these rigorous results [16 , Section 15.1.6] and it is felt that ultimately the mo del will turn out

to be trivial [18 , Section 21.6]. Despite this, it is by now also well{established that the formal

p erturbation theory that results from considering such mo dels must have some physical validity,

due to the high accuracy of its agreement with exp erimental results [22 , 13 ]. One of the remarkable

results from QG eld theory is that the series that results from considering the analogous QG

eld theory mo dels formally repro duces the conventional series term by term in the sharp p oint

limit (i.e., when the fundamental length scale is taken to zero). Thus although the question of

triviality of such mo dels has not b een rigorously answered, the formal agreement of QG mo dels

with exp eriment is established, even at such high accuracies.

Furthermore, the naive extension of lo cal quantum elds to curved spacetimes provides a \grossly

inadequate foundation for the theory [11 ]" as it requires an (approximate) timelike Killing vector

for its formulation [25 ]. Although QG eld propagation is free of these criticisms [30 ], as we shall see

there are still issues to b e resolved. However it is interesting that the consistent incorp oration of a

quantum analogue of the strong equivalence principle into the formalism implies that the violation

of lo cal energy{momentum that leads to ex{nihilo particle pro duction in non{inertial frames of

reference in the conventional eld theory cannot o ccur.

The plan of the manuscript is straightforward; we b egin by giving an overview of Minkowski

space relativistic sto chastic quantum mechanics and conclude the general discussion with the ge-

ometrization of this program and some results from generic at spacetime mo dels. Following this,

we make some of these results more concrete by considering a Milne universe.

Date : July 18, 1997.

PACS: 04.62, 04.20, 02.40.-k, 03.70.+k. 1

ISSUES IN QUANTUM{GEOMETRIC PROPAGATION 2

2. Background

Here we will describ e that part of relativistic QG mechanics that is necessary background to

consider the evolution of free QG scalar elds in a classical, globally hyp erb olic spacetime; further

details on interacting quantum elds and gauge elds has app eared elsewhere [31 ]. Although we

shall discuss relativistic QG elds exclusively, The construction pro ceeds in an analogous manner

in the case of non-relativistic elds [1 ], and may be reached through a group contraction of the

Poincare group of the relativistic QG formalism to the Galilei group of the non-relativistic QG

formalism [28 ].

Lo osely sp eaking, the wave function (x) of the conventional (lo cal) theory which represents the

amplitude to nd the system lo calised ab out a p oint x, is replaced in the QG formalism byawave

function (q; p) which contains information ab out the spatial lo calisation as well as the velo cityof

the system. However whereas the lo cal wave function of the former (and its extension to a lo cal eld

op erator) is intended to allow for arbitrarily sharply lo calised p osition measurements of the state,

the outcome of measurements on the latter state may only be interpreted in a sto chastic sense.

That is, instead of the practically unrealizable limit of measurements of the eld at a p oint [6 ], the

outcome of the sharp est p ossible measurement of the p osition and momentum of a QG eld would

indicate that the state is sto chastically lo calised ab out q with a (sto chastic) average momentum p.

The distribution of p ossible values ab out this outcome is related to the fundamental scale through a

resolution generator; a necessary bypro duct of the reduction of the Poincare group action on phase

space, also leading to a conserved probability current with a p ositive{de nite timelike comp onent.

As we shall see, emphasis is placed on a lo cal vacuum which represents (roughly sp eaking) a

micro-detector at sto chastic rest, sto chastically lo calised ab out the spacetime p oint in question.

Due to the fact that the formalism do es not rely on the existence of a global vacuum for its

de nition, the geometrization of the at spacetime theory in Section 2.2 survives the passage to

curved spacetimes.

2.1. Relativistic Quantum{Geometric Mechanics. To b egin we must intro duce some nota-

4 + 8

tion. The action of the Poincare group ISO (1; 3) on the phase space manifold = M V R

4 4

of pairs (q; p) with q 2 M = (R ;) from Minkowski space ( = diag (1; 1; 1; 1)) and

+ 4 2 2 0

p 2 V = fp 2 R j p = m ;p > 0g from the p ositive mass hyp erb oloid is

(;b) ? (q; p)=(q+b; p); 2 SO(1; 3); b 2 R ; (1a)

with Poincare group pro duct and inverse given by

1 1 1

( ;b ) ? ( ;b )=( ; b +b ); (;b) =( ; b): (1b)

2 2 1 1 2 1 2 1 2

Here we will use the convention that, for example, q represents the vector comp onents q (A;B;::: 2

A B

f0;1;2;3g), q the pro duct q (so we will not indicate the contraction explicitly), and the

A B

pro duct of twovectors is written as p q := p q . From the de nition of the Lorentz matrices

AB A B CD 1 1 A AC D

= wehave that p q = pq where ( ) = .

C D BD C

+ 4 + 4

De ning the pro duct = V M V of a maximal spacelike hyp ersurface in M

m m

and the forward mass hyp erb oloid, one intro duces the Hilb ert space of square{integrable functions

on these surfaces L ( ) with inner pro duct

h j i = (q; p) (q; p)d (q; p); (2)

1 2 2 m

where the Poincareinvariant measure is given by [28 ]

d (q; p):=2pd (q )d (p); (3a)

m m

d p~

2 2 4

; (3b) m )d p= d (p)=(p

0 2 2

p = p~ +m

A 4

and d (q ) is the volume element of the hyp ersurface 2 M . Note that d is normalised so

0 3 3

that on a spatial hyp er-plane de ned by q = 0 it reduces to d ~qd p~.

ISSUES IN QUANTUM{GEOMETRIC PROPAGATION 3

The induced spin zero representation of ISO (1; 3) on

1 1 1

U (;b):(q; p) ! U (;b) (q; p)= (;b) ? (q; p) = (q b); p ; (4)

was found (in contrast to the usual representations on momentum or con guration space [5 , Chapter

7]) to b e highly reducible [29 , Section 3.4]. Each irreducible sub-representation de nes a subspace

of L ( ) of functions of the form

Z Z

p k

iq k

~ ~

(q; p)= e ~ (k )d (k )= ~ (k )(k )d (k ); (5)

m m

q;p

+ +

V V

m m

de ning the Hilb ert space PL ( ). Here wehave assumed that the irreducible sub-representations

are characterised by a real, rotationally invariant resolution generator ~, which in this work will b e

chosen as

lk +

e ~(k )= ; k 2 V ; (6)

l;m

which is a minimum uncertainty state, and emerges as the unique ground state wave function of the

recipro cally invariant quantum metric op erator [28 , Section 4.5]. (Note that all of the relations given

here may either b e viewed in Planck units [38 , App endix F] where all quantities are dimensionless,

or in units where ~ = c = 1 so that l and m retain the dimensions of a length. In the former

case it is still useful to retain the parameter l in order to examine the sharp p oint limit l ! 0.)

Considering the in nitesimal form of the Poincare transformation in (4) : (;b) (1;0) + (; ),

A B A A A

M and = P ,we nd that the generators fM ;P g of the Lie algebra where =

B A A B

iso (1; 3) acting on PL ( )as

A B A A B A

1 1

~ ~

M (q; p)=i P + M (q; p) (7) (q; p)= P +

B A A B A A

2 2

satisfy the Lie algebra of ISO(1; 3)

[P ;P ]=0; [M ;P ]= P P ; (8a)

A B AB C AC B BC A

[M ;M ]= M M M + M ; (8b)

AB CD AC BD AD BC BC AD BD AC

and are de ned in terms of the eight op erators [31 , Section 4.4]

~ ~

P := i@ ; Q := q i@ ; (9a)

A A

A A A

q p

~ ~ ~ ~ ~ ~

P = iP ; M = iM =: i(Q P Q P ): (9b)

A A AB AB A B B A

~ ~

Furthermore, the op erators P and Q satisfy the relativistic commutation relations

A A

~ ~ ~ ~ ~ ~

[Q ; P ]=i ; [Q ; Q ]=[P ;P ]=0; (10)

A B AB A B A B

whichis related to the relativistic generalisation of the Heisenb erg uncertainty relations [31 , Sec-

tion 3.8].

The collection of

s s

3 3

m m

ik q k p ik

~ (k ):= U( ;q)~ (k)= e = e ; (q; p) 2 ; (11)

q;p

p=m

~ ~

Z Z

l;m l;m

( is the b o ost to velo city p=m, and we have intro duced the complex variables := q i p)

p=m

generated by the action of the Poincare group on the resolution generator, allows the de nition of

2 2

the pro jection of 2 L ( )onto PL ( ) via the kernel

m m

Z Z

ik ( )

K (q ;p ;q ;p ):= d (k )e ; (12) d (k)~ (k )~ (k )=

2 2 1 1 m m q ;p

q ;p 1 1

2 2

+ +

V V

l;m

m m

0 0 0 0

(q ;p )= K(q;p ;q; p)(q; p)d (q; p): (13)

ISSUES IN QUANTUM{GEOMETRIC PROPAGATION 4

Cho osing the normalisation constant Z as [31 ]

l;m

K (2lm)

4 5

Z =(2) m ; (14)

l;m

2lm

~ ~

implies that h j i = h j i, and the kernel in (12) acts as a repro ducing kernel on PL ( ) [29 ,

1 2 1 2 m

Section 3.6]

K m ( )

1 2

K ( ; )=K ( ; )= ; (15) K( ;)K(; )d ()=

2 1 1 2 2 1 m

Z m ( )

l;m

where K (z ) is the mo di ed Bessel function of the rst kind. (Note that the di erence between

the normalisation factors app earing here and those in [31 ] is due to the fact that we are employing

momentum variables exclusively, as opp osed to velo city variables.) Renormalising the kernel and

taking the l ! 0 limit results (in the distributional sense) in the p ositive frequency Pauli{Jordan

function [5 , App endix F] of the Klein{Gordon wave op erator [32 ]

l;m

0 + 0 0

K ( ; ) ! K (q q ):= exp ik (q q ) d (k ): (16) lim

3 3 3

l !0

(2 ) m (2 )

Notice that for small arguments K (x) 2=x , and therefore an in nite renormalisation has b een

p erformed in order to achieve this limit.

The alternative form of the inner pro duct holding only on elements of PL ( ) (the normalisa-

tion constant is given by Z = m K (2lm)=K (2lm) [31 , page 105])

m 2 1

h j i = i (q; p) @ (q; p)d (q )d (p); (17a)

1 2 2 m

[ (q; p)] (q; p); (17b) [ (q; p)] @ (q; p):= (q; p)@ (q; p) @

A A A

2 2 2

q q q

1 1 1

implies the existence of the system of covariance [29 , Section 3.4] made up of p ositive op erator{

valued measures

A 4

E (B )=i j in @ h j d qd (p) (18a)

q;p q;p m

(B is a Borel set), which satisfy the Poincarecovariance relations

U (;b)E(B)U(;b) = E(B + b): (18b)

This in turn secures the interpretation of the integration of such measures over a Borel set B

j(P)(q; p)j d (q; p); (19) P (B )=hjE(B)i =

as the outcome of a simultaneous measurement of sto chastic p osition and momentum. The existence

of the Poincarecovariant, conserved, probability current

A 2

J (q ):= j(q; p)j d (p); (20a)

A A 1 A

U (;b):J (q)!(J ) (q b) ; @ J (q )=0; (20b)

with p ositive{de nite probability density J related to the measures (18a) , not only secures the

probabilistic interpretation of the theory, but also guarantees that the integration over a hyp er-

surface in the inner pro duct (2) can be taken over any spatial hyp ersurface and not just spatial

hyp er-planes. Since this will b e of imp ortance later on, it is useful to review the arguments leading

to this result, which is a generalisation of that app earing in [33 , pages 57-8].

First generalise the currentby considering the pro duct of anytwo elements of PL ( )

(q; p) (q; p)d (p); (21) J (q ):=

1 m

2 21

V m

ISSUES IN QUANTUM{GEOMETRIC PROPAGATION 5

which due to the Lorentz invariance of the momentum measure, transforms as

A 1

J (q ) ! (q )J (q ) q b(q ) (22)

under a lo cal Poincare transformation (q );b(q) . Using the momentum space representation and

taking the partial derivativeof (21) gives

Z Z Z

2im

d (k ) d (k ) d (p) @ J (q )=

m 2 m 1 m

+ + +

V V V

l;m

m m m

(23)

p(k +k )

iq (k k )

1 2

2 1

~ ~

; (k ) (k )p (k k )e e

2 1 1 1 2

which (once again using invariance of d (p)) vanishes when one b o osts to the rest frame of

(k + k ). (Note that this probability current is conserved for any choice of resolution generator

1 2

~(k ) provided that it is a real function of k [28 , Section 2.8].) Integrating this over a region R M

and making use of Gauss' law gives

Z I

4 A

d q@ J (q ) 0= J (q ) d (q ); (24)

R @R

1 2 1

which, when sp ecialised to the region b etween twohyp ersurfaces and (corresp onding to

and resp ectively) that coincide at spatial in nity, gives

Z Z

d (q; p) (q; p) (q; p)= d (q; p) (q; p) (q; p): (25)

m 2 m 2

1 1

1 2

m m

For later reference it is useful to consider the form of an integral over an arbitrary hyp ersurface

lab elled by the constantvalue of a function of the q variables T (q )=T . The vector comp onents

T 0

A AB CD

of the normal to the surface is then de ned by n (q ):= @ [T (q )]= @ [T (q)]@ [T (q )], and

B C D

q q q

the surface measure may b e written as

Z Z

d (q; p) ! 2p n(q) T (q) T d qd (p): (26a)

m 0 m

Intro ducing a new co ordinate system (T; ~x) in which the time variable is constant on , the

gradientofq with resp ect to the spatial variables @ [q (x)] (the pullback of the emb edding map [24 ])

A B

gives the induced surface metric (x):= @ [q (x)]@ [q (x)]. In this co ordinate system the

i j

ij AB

x x

surface measure is given by

Z Z

2p n(x) 2pd (x) d (p); (26b) (x) d ~xd (p)=

m m

+ +

V V

t t

m m

0 0

where the second form has b een generalised to include the use of an arbitrary Lorentz frame; we

nd that the spatial measure has taken on the standard form [38 , App endix B].

In the case where the 's are replaced by the kernel (12) and we are therefore considering the

repro ducing prop erty (15) , we nd a stronger result. Considering anytwo spatial hyp ersurfaces

and with a surface at spatial in nity C with (outward{p ointing) normalr ^, the surface integral

in (24) then implies

d (q; p)K ( ; q; p)K (q; p; )

m 2 1

Z Z

= d (q; p)K ( ; q; p)K (q; p; )+ 2prd^ (q)d (p)K ( ; q; p)K (q; p; ): (27)

m 2 1 m 2 1

The fact that for large spacelike separations dominated by the spatial interval ~q , m ( )

mj~q jil (p + p ) ^q and using the asymptotic b ehaviour of the mo di ed Bessel function of the

2 1

second kind [19 , Equation 8.451 #6, page 963] K (z ) =(2z )e as z ! 1, we nd that as

2mj~qj 3

the surface C is taken to spatial in nity, it's contribution b ehaves as dt d e =(m j~qj), which

vanishes in the limit. Therefore for establishing the repro ducing prop erty of the kernel (12) , any

maximal spacelike hyp ersurface at all will suce. Note however that if p oints on the surface C

ISSUES IN QUANTUM{GEOMETRIC PROPAGATION 6

1 2

have light-like (or timelike) separation from p oints on b oth of and , then we cannot conclude

m m

that its contribution vanishes in the limit since the kernel (15) will no longer vanish as C is taken

to spatial in nity.

Before concluding this brief description of relativistic QG mechanics, we should make a few com-

ments on interacting elds. Intro ducing a relativistic p otential (e.g.,avector p otential in the case

of the scattering of charged scalars o a background electro-magnetic eld) involves the pro jection

of the p otential function onto PL ( ), with the resulting dynamics given by an iterated kernel

very similar to that which will app ear when we consider a curved background spacetime (65) .

However, since the phenomena of pair creation in strong elds is an exp erimental fact, one intro-

duces many{body, second{quantised QG eld theory, in which the QG eld op erators are bona

de densely de ned op erators on Fock space, in contrast to the conventional construction of lo cal

eld theory. Due to this fact, there are no divergences in the resulting p erturbation series for the

scattering matrix, and therefore no in nite renormalizations to b e p erformed. Furthermore, in the

non-relativistic limit the role of the resolution generator in such measurements are interpretable

as that of a con dence function, and the outcome of a measurement as a sto chastic or average

value only [1 ]. Then taking the sharp p oint limit l ! 0 leads identically to ortho dox quantum

mechanics [29 , Section 2.6]. Taking the sharp p oint limit rst however, leads to the formal p er-

turbative series of lo cal quantum eld theory in any particular mo del, but only after an in nite

renormalisation is p erformed [31 , Section 5.8]. Thus agreement of QG eld theory mo dels with the

remarkably accurate predictions of the formally de ned p erturbative series of conventional lo cal

quantum eld theory mo dels [13 , 22 ] is guaranteed at the formal level for small (but non{zero)

choice of the fundamental scale l , which one exp ects on fundamental grounds should b e of Planck

length order [10 ].

2.2. Geometrization. The rst step toward the generalisation of this structure to a curved space-

time manifold M is to note that one need not ever consider the underlying momentum space struc-

ture in (5) at all, instead working completely within [28 , Section (2.2.7)]. Intro ducing the phase

space wave function of the rest frame leads to the lo cal wave functions

Z Z

ik q

~ (k)~(k)d (k); (28) e ~(k p=m)~(k)d (k)= ( ):=

m m m

+ +

V V

m m

and generating the wave functions of the b o osted and translated frames through the representation

of ISO (1; 3) on the rest frame, leads to

0 0

( ):= U( ;q ) ()=K(; ): (29)

p =m

Considering the action of an arbitrary group element on one of these frames, we nd [15 ]

U (;b) ()= U(;b)U( ;q ) ()

p =m

(30a)

0 0

= U (;b) ? ( ;q ) ()= U( ; q + b) ( );

p =m p =m

and using the fact that = uniquely, where is the Wigner rotation and

R R

p=m p=m p=m

is the b o ost to the velo cityp=m,we nd that

0 0

U (;b) ( )= U( ; q + b) ( )= ( );

(30b)

q ;p

p =m (;b)?

where wehave used the fact that the rest frame is rotationally invariant U ( ; 0) = . Abstracting

away the phase space functional dep endence and considering as a basis in PL ( ), we de ne

the set of quantum frames [29 , Section 3.7]

4 +

p; q 2 M ;p 2 V Q := j = q i

(31)

4 +

j = U( ;q) ;q 2 M ;p 2 V ;

p=m

with the group action from (30)

U (g ) = ; g 2 ISO (1; 3): (32)

ISSUES IN QUANTUM{GEOMETRIC PROPAGATION 7

The kernel (12) then plays the role of the overlap amplitude of these quantum frames

= K ( ; ) (33)

(and indeed may be considered as the lo cal quantum metric in a curved spacetime [29 , Section

0 2

5.2] [31 , Section 4.5]). The fact that K ( ; ) repro duces on PL ( ) indicates that the op erator

de ned by

P := j id ( )h j; (34)

2 2

acts either as continuous resolution of the identityonPL ( ), or a pro jection from L ( )onto

m m

PL ( ). Therefore a scalar state may b e decomp osed on Q as

= d ( )( ) ; (35a)

where

Z Z

0 0 0 0 0 0

( ):=h ji= ( )( )d ( )= K(; )( )d ( ): (35b)

m m

(The structure describ ed is actually a sp eci c example of a more general pro cedure [2 , 3 ].)

Since the frames in Q carry a representation of the Poincare group, in order to relate the Hilb ert

space structure of PL ( ) to a curved spacetime M the ane generalisation [23 ] P M of the

Lorentz frame bundle LM over M is employed. This bundle consists of all frames ab ove each

x 2 M of the form u = (e ; a), where we will use a b oldface letter to denote the vector itself

a = a e 2 T M (and continue to use, for example, a to denote the vector comp onents) and

the linear frame is de ned with resp ect to a particular co ordinate frame via e = E @ . The

A A

dual (coframe) basis is then given by = dx E , where the vierb eins satisfy the following

A A A

orthogonality relations E E = , and E E = . The structure group of P M is the

B A

Poincare group ISO(1; 3) with right action on these frames de ned by

0 0 B A A

u? (;b)=(e ;a)?(;b)=(e ;a)= e ;(a +b )e : (36)

A B A A

In practice we will actually employ P M, constructed from Lorentz frames with future{p ointing

timelike and right{handed spatial basis vectors, which has ISO (1; 3) as the structure group.

Implicit in this construction is the existence of a metric on M which has b een used to reduce

the ane frame bundle AM to P M. With this metric comes the unique, metric compatible, Levi{

Civita connection, which is uniquely extendible to a connection on P M [23 , Section I I I.3]. In a given

section (dep endence on which will not be indicated) this connection determines the in nitesimal

parallel transp ort, or covariant derivative, op erator which acts on a tensor T asso ciated to P M

through a tensor representation of the Poincare group with generators M and P satisfying (8) ,

B A

along a path in M with tangentvector X 2 T M as [29 , Section 2.6]

A B A A A

! [X]M + [X]P T; X 2 T M; (37) r T = X e [T ]+![X]T = X e [T]+

B A A X A A

where ! is the (Poincare) Lie algebra{valued, Levi{Civita, connection one form acting in the

representation of T , with action on a frame describ ed by [14 ]

B C A A

! [X ]:(e ;a)! X e ; X + r [a] e ; (38a)

A C X A

A B A C A

: (38b) r [a] := X e [a ]+a

X B

app earing in (38) are those of the torsion{free, Levi{Civita connection written The comp onents

in terms of the structure constants [e ;e ]=C e of the Lorentz frame (e ; 0) as

A B AB C A

A A A A A

= ! [e ]= (C + C + C ); (38c)

C B BC BC CB

A A A AD E

C := E e [E ] e [E ] ; C := C ; (38d)

BC B C C B BC CE DB

ISSUES IN QUANTUM{GEOMETRIC PROPAGATION 8

and satisfy the metric compatibility condition ! = 0. The action of ! on the frame is an

(AB )

in nitesimal Poincare transformation of the form

! [X ]:(e ;a)!(e ;a)? ![X];X + r [a] : (39)

A A X

Since parallel transp ort plays a fundamental role in QG eld propagation in curved spacetimes,

it is worthwhile at this p oint to review the computation of the results of parallel transp ort in a

Poincare frame bundle. A frame that is parallel transp orted along a curve with tangent X 2 T M

is one that do es not change along , and may be written in a section s = (e ; a) as (e ; a) ?g .

A A

Writing as a parameterisation of ,wehave @ [(e ;a) ?g ] = 0 which gives [27 , Section 10.1.4]

(e ; a) ? ! [X ];X + r [a] ?g +(e ;a) ?X[g ]=0; (40a)

A X A

and so the transition element satis es

@ [g ]= ![X];X + r [a] ?g : (40b)

It is convenient to transfer this action to the comp onents of a vector in the Mobius represen-

tation [21 ] where V 2 T M is written as a column vector in R and the in nitesimal Poincare

transformation as a 5 5 matrix, which acts on V as

(; ) ?X ! ; = ![X]; = X + r [a] : (41)

0 0 1

The restriction ! = 0 ensures that this may be lo oked up on as an in nitesimal Poincare

(AB )

transformation with generators that satisfy the Lie algebra of the Poincare group.

In this representation the exp onential map (of iso (1; 3) into ISO(1; 3)) may be split up into a

Lorentz b o ost and a translation by a straightforward induction argument

n 1

(; ) = exp(); (exp () 1) ; (42) exp(; )=

n=0

where exp() is the exp onential map of so (1; 3) to SO(1; 3). Also by induction, the pro duct of N

such exp onential maps may b e re{represented as

1 1 N m+1

Y Y X Y

exp( ; )= exp( ); exp( ) exp( ) 1 : (43)

m m

k k k k

m=1

k =N k=N k =N

Note that the pro duct is ordered: g := g g g g , and that (exp() 1) :=

n N N 1 2 1

n=N

=(n + 1)! .

n=0

If we consider parallel transp ort along a curve :[ ; ] R ! M, one nds [27 ] that the nite

ISO (1; 3) parallel transp ort map that is generated by the in nitesimal connection may b e written

g ( ; ) = P exp ( ); () d ; (44)

where P is the path ordering op erator (larger values of the parameter app earing to the left).

Computing this `path integral' by splitting up the time interval up into N equal pieces [31 ]: =

( )=N = , = , = , = + n and writing

i n n1 i 0 N n i

f f

g ( ; ) = P exp ( ); () d exp ( ); ( ) ; (45)

n+1 n n n

noting that the evolution along is comp osable: g ( ; ) = g ( ; )g ( ; ), and

n n2 n n1 n1 n2

taking the N !1 limit in which the approximation b ecomes exact, the transition element is given

exp ( ); ( ) : (46) g ( ; )= lim

n n N 0

N !1

n=N 1

ISSUES IN QUANTUM{GEOMETRIC PROPAGATION 9

Using the ab ove results for the representation of the Lie algebra, this is equal to (in the N !1

limit)

Z Z Z

f f f

0 0

g ( ; )= P exp ( )d ; d P exp ( )d ( ) : (47)

N 0

i i

In practice these results will b e applied to geo desics that are forward{p ointing in the chosen time

co ordinate, so that the path ordering may be replaced by time ordering; however in general note

that the P op erator disapp ears in cases where [( );( )] = 0.

It will b e useful to display the b ehaviour of this transition element under a change of Poincare

frame. Writing the transition element corresp onding to parallel transp ort along a path b eginning

0 0 s 0

at x and ending at x in a section s as g (x ;x), if the section is related to another choice by

s (x)=s(x)?g(x), from the structure of (40) we see that the transition elements are related by [27 ]

s 0 0 s 0 1

g (x ;x)=g(x)?g (x ;x) ?g(x) : (48a)

In particular, the relationship b etween an arbitrary Poincare section s = e ; a and the related

Lorentz section s := (e ; 0) is

0 A

s 0 0 s 0

g (x ;x)= 1;a(x) ?g (x ;x) ? 1;a(x) ; (48b)

where wehave used s = s ? (1; a).

2.3. Quantum Propagation. With the Poincare frame bundle P M and the set of quantum

frames Q (31) , wehave what may b e considered as the classical and quantum counterparts of the

same physical entity, namely, the frames of reference on which prop erties of the physical system are

measured. Since b oth sets of frames transform under a representation of the Poincare group, the

problem of howto relate the two sets is resolved by de ning the principle quantum frame bundle

over M as Q M = P M Q, whichisthe G{pro duct [23 , page 54] of the Poincare frame

0 0

ISO (1;3)

bundle with Q. Atypical elementofQ Mis given by the asso ciated quantum frame

u(x)

= u(x); ; (49)

which is a pair consisting of a Poincare frame u(x) and a quantum frame . The bre bundle

structure of Q M is completed by de ning the pro jection map back to the underlying manifold

u(x)

= x 2 M; (50)

and the lo cal trivialisation map onto the typical bre which is isomorphic to PL ( )

u(x)

u(x) u(x)

= u(x); = : (51)

Equivalence classes are identi ed via the left action

1 1

g?(u; )= u? g ;U(g) ? = u? g ; ; (52a)

from which it is clear that

1 1

u?g u u?g u 1 1

= U (g ) ; =( ) U(g ): (52b)

This pro duct `solders' the quantum frames Q to the Poincare frames P M of M, and parallel

transp ort as de ned by the Levi-Civita connection in P M is then transferable to the frames in Q.

Given a section s(x)= e (x);a(x) of P M, the asso ciated quantum frames are given by

A 0

s(x)

= ; (53)

which on accountof (52) satis es

s?g

s?g 1 s 1 s 1 s

=( ) =( ) U(g) =( ) = : (54)

For simplicitywe will adopt the notation

E E D D

s(x) s(x) s(x) s(x)

s(x) 0 s(x)

= h j i = K ( ; ) (55) :=

0 0

ISSUES IN QUANTUM{GEOMETRIC PROPAGATION 10

for the overlap amplitude (33) of quantum frames in a given bre of QM. Similarly, the continuous

resolution of the identity (34) b ecomes an identity in each bre of Q M

E D

s(x) s(x) s(x)

P = d ( ) : (56)

Then, if the outcome of parallel transp ort along a path connecting two p oints x; x 2 M in the

0 0

section s(x) as computed in section 2.2 is the map s(x) ! s(x ) ?g (x;x), then this induces the

change of quantum frame

0 0

s(x)?g (x ;x)

s(x) s(x) s(x)

! (x ;x) = = ; (57)

g (x;x)?

from whichwe de ne the propagator for parallel transp ort along in the section s as the amplitude

D E D E

0 0 0

s(x) s(x) s(x ) s(x )

0 0 s 0 0 0

(x ; ;x; ):= K (x;x) = = K ; g (x ;x) ? ; (58)

0 0

(x ;x)

g (x ;x)?

0 1 0

and from (12) we also know that K ( ;g ? )=K(g ? ;). An imp ortant consequence of (58)

is the b ehaviour of the propagator under changes of Poincare gauge. From

s?g

0 0 s 0 0 0

K (x ; ;x; )=K x ;g(x ) ? ;x; g (x) ? ; (59a)

(x;x)

(x ;x)

we deduce that

0 s?g 0 0 0 s 0

K ; g (x ;x) ? =K g(x )? ;g (x ;x) ?g(x)?

(59b)

0 0 1 s 0

=K ;g(x ) ?g (x ;x) ?g(x) ? ;

where g is the transition element from parallel transp ort in the Poincare section s. Clearly (59b)

expresses the consistency of the action of the Poincare group on the Poincare frames (48) and the

propagator. An imp ortant sp ecial case of this that will arise later is when we take s = (e ; 0)

L A

and g = 1;a(x) , and we are therefore relating the amplitude in a Poincare gauge to the related

Lorentz frame.

Notice that the construction of Q M has attached the entire Hilb ert space PL ( ) to each

0 m

p oint x 2 M, and the repro ducing kernel (55) and therefore the resolution of the identity (56) act

within each bre of Q M. Although the entire set of states in each bre is not easily physically

interpretable (unless spacetime is at, in which case each bre may be identi ed [32 ] as will be

discussed further b elow), since the formalism dep ends on parallel transp ort and the lo cal overlap

amplitudes (58) it is necessary to have the entire space repro duced lo cally. The states that are

interpretable in a straightforward manner are those which are sto chastically lo calised at the p oint

of contact of the Poincare frame with M, and are therefore describ ed in a section s = e ; a(x)

by the collection [30 , 31, 32 ]

p; p 2 V ; (60) (x):=a(x)i

where recall that a non{zero a(x) indicates that the section is a translation of the related Lorentz

section s (x) := (e ; 0) by a(x), and so transforming (60) back to the related Lorentz frame

L A

results in the collection of quantum frames of all p ossible momenta sto chastically lo calised at the

p. origin of the frame (x)=i

The assignment (60) is unique in any section of P M, and given a change of frame

0 0 0

s(x)= e (x);a(x) = e (x);a(x) ? (x);b(x) =: s (x) ? (x);b(x) ; (61a)

the action is transferred to the quantum frame

0 0

l l

^ ^

(x) ! (x);b(x) ? (x)=(x)(a(x) i p)+b(x)=a(x)i p; (61b)

m m

where wehave identi ed the comp onents of the momentum in the primed frame p =(x)p as well

as the comp onents of the translation vector a (x)=(x)a(x)b(x). This recognises the fact that

due to the nature of the construction of the bundle Q M, in general may be considered as a

vector in the complexi ed tangent space T M

A A

l l

p e (x)= qi p e (x)a(x): (62) := q a(x) i

A A

m m

ISSUES IN QUANTUM{GEOMETRIC PROPAGATION 11

The transformation prop erty (61) implies that the form of the propagator for parallel transp ort in

a section s is invariant under an arbitrary change of frame

s?g

s 0 0 0 0 0

^ ^ ^ ^

K x ; (x ); x; (x) = K x ;g(x ) ? (x ); x; g (x) ? (x) ; (63)

(x ;x)

and guarantees the consistency of the choice of lo cal frame with p oincare invariance, where the

frame is transp orted as b efore, and the outcome of parallel transp ort along a path between

frames lo calised at x and x is de ned by

D E D E

0 0 0

s(x ) s(x) s(x ) s(x )

s 0 0 0

^ ^

K x ; (x ); x; (x) := (x ;x) =

(x ;x)

0 0 0

^ ^ ^ ^

(x ) (x) (x ) g (x ;x)? (x)

(64)

0 0

^ ^

=K (x ); g (x ;x) ? (x) :

The picture that emerges from this is that the Poincare frame (e (x); a(x)) related to the classical

spacetime manifold is replaced by the set of quantum states (x) which represents the set of states

of all p ossible momenta which are sto chastically lo calised at x. Combined with the exp onential

map that generates Riemann normal co ordinates in a suitably small neighb ourho o d of a p oint

(limited by the magnitude of the lo cal curvature [26 , Sections 8.6 and 11.6]), this also allows an

interpretation of the quantum frame wave function ( ) which is expressed in terms of at, tangent

space co ordinates, as the wave function of an extended state in spacetime in accordance with the

discussion in Section 2.1. It is in this sense that QG propagation is said to b e geometrically lo cal

(i.e., o ccurring within individual bres above M) but not top ologically lo cal (i.e., not interpretable

as arbitrarily precisely lo calizable quantum states in M).

Although the propagator for parallel transp ort (58) or (64) provides an appropriate description

of quantum propagation in the case of Minkowski spacetime (and, as we will see, in any frame of

reference whatso ever), due to the path dep endence of parallel propagation in curved spacetimes,

in general it do es not provide a repro ducing kernel. The recti cation of this situation is to con-

struct the quantum{geometric propagator [31 , Section 4.6] by iterating the propagator (58) over all

broken, forward{p ointing, geo desic paths b etween the initial and nal p oints, in analogy with the

computation of the transition element in Section 2.2.

Begin by considering a foliation of the spacetime into spacelikehyp ersurfaces lab elled by the

value of a time parameter t. Given an initial p oint x 2 and a nal p oint x 2 , a sequence

i t t

of hyp ersurfaces is de ned by the nite time intervals as b efore: t =(t t )=N =t t ,

t i n n1

t := t , t := t , and using the propagators for parallel transp ort along geo desics (x ;x ) with

0 i N n n1

endp oints x 2 , the quantum{geometric propagator is de ned as the limit [32 ]

n t

^ ^

K x ; (x ); x ; (x ) :=

i i

f f

^ ^

d (x ;p )K x ; (x ); x ; (x ) lim

m n n n+1 n+1 n n

(x ;x )

n+1

N !1

n=N 1

^ ^

K x ; (x ); x ; (x ) ; (65a)

1 1 0 0

(x ;x )

1 0

where by construction, this satis es the curved spacetime analogue of (15) (where t >t>t )

2 1

s s

^ ^ ^ ^

x ; (x ); x ; (x ) K x ;(x ); x ; (x ) =K

1 1 2 2 2 2 1 1

(65b)

s s

^ ^ ^ ^

= d(x; p)K x ; (x ); x; (x) K x; (x); x ; (x ) :

2 2 1 1

The measure app earing in equation (65) is de ned by d (x; p):=2pd (x)d (p), and although

m m

at rst sight the resulting construction would app ear dep end on the gauge to a greater degree than

do es (63) , the fact that it is not may b e seen by making use of (61) . The repro ducing prop ertyof

the QG{propagator emb o dies the spirit of the `principle of path indep endence' as formulated by

Teitelb oim [36 , 25 ] and extended by the present author [8, 7 ], although the extent to which results

dep end on the choice of foliation is not clear at this time.

Even though wehave b een primarily describing parallel transp ort and QG propagation in Q M,

it is easily adapted to a scalar eld by constructing the Klein{Gordon quantum bundle E M = 0

ISSUES IN QUANTUM{GEOMETRIC PROPAGATION 12

Q M PL ( ). A section of E M would represent an assignment of scalar eld mo des

0 m 0

ISO (1;3)

ab ove each p ointin M expanded in terms of the quantum frames as

s(x)

= ( ) d ( ); (66)

x x m

which is just (35) expressed in each bre of Q M. Alternatively, making use of the identi cation of

the sto chastically lo calised frames, a complete description of initial data for a scalar eld is given

s(x)

= x; (x) d (x; p): (67)

(x)

Parallel transp ort is transferred to the lo cal comp onents by considering the amplitudes to nd the

scalar eld in those mo des that are sto chastically lo calised ab out the nal p oint

s(x )

! = x ;(x ) (x; x ) d (x ;p) (x; x ):

0 0 0 m 0 0

(x )

(68)

s(x)

^ ^ ^

! x ;(x ) K x; (x); x ; (x ) d (x ;p);

0 0 0 0 m 0

(x)

where in the second form the result of parallel transp ort has b een pro jected backonto the sto chas-

tically lo calised mo des. The iteration of this parallel transp ort op eration over all forward{p ointing

geo desic arcs repro duces (65) , and the time evolution of the initial data is given by replacing the

propagator for parallel transp ort with the quantum{geometric propagator

s(x)

^ ^ ^

x ; (x ) K x; (x); x ; (x ) = d (x ;p); (69)

0 0 0 0 m 0

(x)

generating a section of E M from the initial data (67) .

It is worthwhile at this stage to make a few comments on the integrals app earing in (65) . In

Minkowski spacetime, the repro ducing prop erty (27) may b e rewritten in the following way

Z Z

00 0 00 0

^ ^

d ( )K ( ;)K(; )= d (x; p)K ;g (x; x ) ? (x ) K g (x; x ) ? (x ); ;

m m 0 0 0 0

(70)

where we have used the fact that the outcome of parallel transp ort in this case just gives the

separation vector between the two p oints (this will be discussed in more detail b elow), and so

x may be chosen arbitrarily as all dep endence on it will cancel identically. In (70) we recognise

the form of the surface integral in (26) , and in fact have a sp ecial case of the integrals app earing

in (65) . Thus we nd that what is happ ening (op erationally) in (65) is that the outcome of parallel

transp ort g (x; x ) ? (x ) is (at least for spacetimes with small enough curvature) a map from

0 0

the hyp ersurface that spans R , and may therefore b e considered to b e a surface integral in the

same sense as (70) . (Note that the outcome of the integration in curved spacetimes will dep end in

general on x in a nontrivial way, so the kernel is not exp ected to repro duce identically. Physically

the iteration may be viewed in analogy to a multi{slit exp eriment: a quantum state, initially

sto chastically lo calised on the Cauchy surface will propagate indiscriminantly along all p ossible

broken geo desic paths to a nal state lo calised on .

It is useful to see how this construction pro ceeds over a at (Minkowski) spacetime, however

considering an arbitrary co ordinate chart (assumed for simplicity to b e global), frame of reference

and family of hyp ersurfaces on whichto construct the quantum geometric propagator. To b egin,

note [15 ] that any section of the Poincare frame bundle may be written as an arbitrary Poincare

transformation of the parallel propagation of a Lorentz frame at O (taken as the origin of co or-

O O

dinates) as s(x) := s (x) ?g (x), where s (x) := (e (x); 0) and e (x) is the chosen frame at

O O O

A A

O parallelly transp orted to all p oints of M, expressed in terms of some arbitrary co ordinate sys-

tem. (This can easily b e generalised to the parallelly propagated frame s (x):= e (x);X (x) ,

k A

ISSUES IN QUANTUM{GEOMETRIC PROPAGATION 13

where X is the Minkowski co ordinate vector of the p oint x 2 M again expressed in some arbi-

trary co ordinates). If the arbitrary section is written as s(x) = e (x); a(x) , then the choice of

Poincare transformation may b e written as g (x)= (x);0 ? 1;a(x) , and so (59a) relates the

O O

propagator in s to that in s

s 0 0 0 s 0

^ ^ ^ ^

K x ; (x ); x; (x) =K (x ); g (x ;x) ? (x)

0 0 1 s 0

^ ^

(71a)

=K (x ); g (x ) ?g (x ;x) ?g (x)?(x)

O O

0 0 s 0

l l

=K i (x )p ;g (x)p) ; (x ;x) ? (i

O O

m m

where we have used the fact that by de nition (x)= 1;a(x) ?(i p). Since we know that

s 0 0

the outcome of parallel transp ort in s is given by g (x ;x)= 1;X (x)X (x ) where X (x)

O M M M

is co ordinate comp onents of the p oint x in the Minkowski co ordinate system related to O , the

propagator in a generic frame s may b e written as

s s

^ ^ ^ ^ ~ ~

x ; (x ); x ; (x ) = K x ; (x ); x ; (x ) = K (x ); (x ) ; (71b) K

2 2 2 1 1 1 2 2 2 1 1 1 2 2 1 1

(x ):=X (x )i (x )p : (71c)

n n M n O n n

This result has some very intriguing features; not only is the spatial dep endence identical to

that of the sto chastic quantum mechanics result (12) (and so equivalent to the hyp er-plane results

quoted in [32 , 9, 15 ]), but the only di erence b etween the results in a general frame and those in a

hyp er-plane slicing is the presence of the Lorentz factors acting on the momenta at the endp oints.

This is not surprising since one should b e able to view the propagator in a at spacetime as a b o ost

back to the Minkowski frame, a trivial parallel transp ort that just induces a translation factor, and

a b o ost back to the general frame at the endp oint of propagation. Indeed, rightaway we see that

what replaces the Bogolub ov transformations of the conventional lo cal quantum eld theory on a

curved background [11 ], is a physically reasonable shift of the momentum sp ectrum of the state.

As a sp eci c example, consider an initial state that is at rest at O in the Minkowski frame. The

amplitude to nd the state sto chastically lo calised ab out a p oint x with sto chastic momentum p

on a later hyp ersurface is evidently given by

^ ~

x; (x) = K (x); il n ; (72)

where n = (1; 0) and x 2 . Clearly only the momentum sp ectrum is distorted by the relative

motion of the initial state and observer.

Lest one think that the QG propagator from (65) will b e somewhat more complicated than (71b) ,

note rst of all that since we are dealing with a globally at spacetime, the propagators for parallel

transp ort are path indep endent and so one would not exp ect there to b e a more general propagator

built from it that would give a di erent result. However, explicitly considering the integral

l l

~ ~

d (x; p)K (x ); X (x) i (x)p K X (x) i (x)p; (x ) ; (73a)

m 2 M O M O 1

m m

and transforming the momentum variables as p ! (x) p we nd

Z Z

l l

~ ~

2p (x)d (x) d (p)K (x ); X (x) i p K X (x) i p; (x ) : (73b)

O m 2 M M 1

m m

Now we see that the argument X (x) i p is precisely the Minkowski co ordinates evaluated in

(x), and furthermore the b o osted measure (x)d (x) is just the parallelly propagated frame e

the surface measure evaluated in these same hyp er-plane frames (the surface measure transforms

as a vector under changes of frame [38 , App endix B]). By assumption the chosen co ordinates x

constitutes a global chart (at least in some op en region containing ), and therefore x ! X (x)is

smo oth and invertible. The measure may therefore b e written in terms of X , the result of which

is an integral over written in terms of standard Minkowski co ordinates. This is a sp ecial case

of (27) , which may be used to complete the pro of that the propagator (71b) repro duces, and is

therefore equivalent to the quantum{geometric propagator K de ned by (65) .

ISSUES IN QUANTUM{GEOMETRIC PROPAGATION 14

3. Propagation in a Milne Spacetime

Wenow turn to a sp eci c illustration of the results of the last section, for simplicitycho osing to

work in a Milne universe which has constant time slices that are partial Cauchy surfaces (timelike

geo desics are complete to the future). This simple case gives further insight into the problems

of the conventional lo cal quantum eld theory, since there exists in the literature inequivalent

quantisations of a scalar eld on hyp erb oloids of Minkowski space [20 , 12 , 34 ]. At present, there are

authors who claim that the hyp erb oloids are inappropriate for imp osing the canonical commutation

relations and de ning an inner pro duct [35 ], whereas others [4] claim that one must make an

appropriate choice of orthonormal mo de expansion. In either case one must app eal to the b ehaviour

of the propagator or orthonormal mo de functions in the extension of the Milne universe to the

Rindler wedge, even though physics in this region of spacetime should (in a truly lo cal theory) have

no e ect on the causal evolution of a system in the Milne universe. This is merely a re ection of

the nonlo cal nature of the vacuum of conventional lo cal quantum eld theory [17 ].

In the case of QG propagation similar issues arise. Although we are dealing with a region of

spacetime which is at, since the hyp ersurfaces are asymptotically null the propagator do es not

repro duce on them. Indeed as we shall see, one cannot p erform the sum in (65) in this case since

the integrals are not convergent. This presents no conceptual problem since this form of quantum

propagation is fundamentally nonlo cal, however it do es emphasise the role of strictly spacelike

hyp ersurfaces; indeed it is not clear that one could formulate QG propagation of initial data given

on a null or partially null hyp ersurface.

3.1. The Milne Universe. The Milne universe that we will consider here is the region of Min-

kowski space (we will write the conventional Minkowski co ordinates as (t ;x ;~y)) that is in the

M M

causal future of the origin in the t x plane, while the remaining two spatial directions will b e

M M

covered by the at co ordinate vector ~y . Sp eci cally, with the (t; x; ~y ) co ordinate domain t>0 and

2 1 2

g = diag (1; t ; 1; 1); g = diag (1; t ; 1; 1); (74a)

g=t or E := det(E )=t, dep ending on whether one works in the co ordinate frame so that

or the Lorentz frame f@ ;e ;@ g with dual coframe basis (dt; ;d~y) where

t x

x 1

:= t dx; e := t @ : (74b)

x x

Since the metric has the form of a at Euclidean metric in the y z plane, we will employ the

vector symb ol in that sector to simplify the notation. Making use of the co ordinate transformation

between the adopted Milne co ordinates and the standard Minkowski co ordinates

2 2

t = t cosh(x); x = t sinh(x); t = ; x = tanh (x =t ); (75) x t

M M M M

M M

shows that the metric (74) covers the t > 0, t < x < t , and 1 < ~y < 1 region

M M M M

of Minkowski space. The chosen Lorentz frame may be related to the at Minkowski co ordinate

frame by

3 2

cosh (x) sinh(x) 0

5 4

@ e @ @ @ @

:= (@ = )(x) ; (76)

sinh(x) cosh (x) 0

t x t x

~y ~y

M M

~ ~

0 0 1

B 0

, and implies that the Milne = e which is the usual Lorentz frame transformation e ! e

A B A

frames are related to the standard Minkowski frames via a b o ost along the x^ axis by x, which

has b een de ned as the Lorentz b o ost (x) .

3.2. Geo desics. It is a straightforward calculation to show that the only non{zero comp onents of

the Levi{Civita connection one{form (38c) are

t t x 1 x x x x

! := = dx = t = ! := ; (77)

x t

xx xt

ISSUES IN QUANTUM{GEOMETRIC PROPAGATION 15

the geo desics of which are of course straight lines. However, since they will not app ear so in Milne

co ordinates, and to demonstrate how the calculation of the propagator pro ceeds, we will compute

the geo desics explicitly.

A t x A

The tangent to a geo desic will b e written as u =(u ;u ;~u):=E d [x ] where d indicates the

t x

derivative with resp ect to the geo desic parameter , so that u =d [t], u = td [x], and ~u =d [~y],

and the geo desic equations are

t x 2 x t x

d [u ]+(u ) =t =0; d [u ]+u u =t =0; d [~u]=0: (78)

This, combined with the normalisation g(u; u)= , results in

x t

(u =t) + S; u =: u =t; ~u =: ~u ; (79) u =

0 0 0

where S := (~u ) + , the + and signs refer to forward and backward{p ointing geo desics

resp ectively, and

+1 for timelike geo desics

= : (80)

0 for null geo desics

1 for spacelike geo desics

Using u , the geo desic equations may b e reparameterised in terms of the time co ordinate as

u ~u

0 0

p p

d x = ; d ~y = : (81)

t t

2 2 2

t (u =t) + S (u =t) + S

0 0

Integrating these gives the geo desics

p t

(u =t) + S for S 6=0

= ;

1 0

for S =0

2ju j

p t

(82)

(u =t) + S for S 6=0 ln ju j=t +

0 0

x x = ;

1 0

ju j

ln 1=t for S =0

~y ~y =~u ( ):

1 0 0 1 0

We have included the full set of solutions of the geo desic equations in order to discuss the

following imp ortant issue. If we consider an initially future{p ointing spacelike geo desic emanating

from x (in the ~y = 0 plane) on the surface lab elled by t = t (which requires that ju j > t ),

0 0 0 0

then (79) tells us that these geo desics continue to b e forward{p ointing until t := ju j. At this

max 0

t x

p oint, u = 0 and ju j = 1, and we see that the geo desic is tangentto , after which the geo desic

max

then b ecomes past{p ointing. Using (82) we nd the maximum spatial distance that the geo desic

may `travel' b efore encountering some later surface is found by cho osing ju j = t , resulting

t 0 1

t =t + in x = ln (t =t) 1 . In order to connect p oints on to p oints on with

1 max 1 t t

0 1

x> x , one has to consider geo desics that pass through as a forward{p ointing geo desic,

max t

continue on until it tangents at for some ju j >t at which p oint it b ecomes past{p ointing,

0 1

ju j

and then continues on to reach for a second time. (This is straightforward to see if one considers

the surfaces as hyp erb oloids in Minkowski space, and draws straight lines b etween arbitrary p oints

on nearbyhyp ersurfaces.)

A similar scenario arose in the consideration of parallel propagation in a Rob ertson{Walker

spacetime [9], however in that case the geo desic could b e smo othly matched (at x ) to a geo desic

max

that remained within the spatial hyp ersurface, and could therefore be used to connect all p oints

with greater separation. Here we will assume that some choice of path has b een made to join such

p oints in the construction of the QG{propagator (65) ; since the spacetime is at, all choices yield

the same transition element. This is justi ed post facto by the agreement of the resulting propagator

with that derived using the standard Minkowski space hyp er-planes as the foliation of spacetime.

However in a curved spacetime where parallel transp ort is path{dep endent, the QG{propagator

ISSUES IN QUANTUM{GEOMETRIC PROPAGATION 16

may have to be constructed with some care as to how to make such a choice. In principle, this

should follow from taking the semi-classical limit of quantum{geometric gravitational propagation

given in [31 , Chapter 8], although it is p ossible that a study of di eomorphism invariance of a

semi-classical mo del may shed some light on this issue.

3.3. Parallel Transp ort. Since we are interested in hyp ersurfaces of constant t, the geo desics

may b e reparameterised by t (as in (81) ) and all integrals in may b e converted to integrals in t.

Using (77) , the matrices app earing in (41) (and the remainder of that section) are given by

A A B B A A A

1 1

~ ~ ~ ~

! ! (u)M (u)M + (u)P = +(u +r [a] )P

B B A A A u A

2 2

(83)

1 A A x 1

~ ~

M +(u +r [a] )P : =u t

0 u A

Considering any geo desic with endp oints relating the value of the time co ordinate to the value of

the x co ordinate as for example x := x(t ), from (47) one nds that the exp onential determining

1 1

the Lorentz transformation has as it's argument

Z Z

2 2

dx u

d = dt = (x x ); (84a)

2 1

t dt

1 1

and so the Lorentz b o ost is given by

(t ;t ) := exp d K =(x )(x ) ; (84b)

2 1 1 2 1

where (x) was de ned in (76) , and the hyp erb olic angle addition formulae: cosh (a + b) =

cosh(a) cosh (b) + sinh(a) sinh(b) and sinh(a + b) = cosh(a) sinh(b) + sinh(a) cosh (b)have b een used.

If one works in the Lorentz section s =(e ;0), the translational contribution is given by

0 A

Z Z

2 2

dt x(t) u(t)=u (t) d ( ;)u()=(x ) b(t ;t )=

2 2 2 1

1 1

3 2 3 2

(85)

cosh x(t) sinh x(t) 0

5 4 5 4

t@ x

: dt = (x )

sinh x(t) cosh x(t) 0

~ ~

@ ~y

0 0 1

The contribution to the y z plane is straightforward to integrate, and the remainder is the two

comp onentvector in the t x plane

cosh x(t) + t@ [x] sinh x(t)

dt : (86)

sinh x(t) + t@ [x] cosh x(t)

The second term in each of these is of the form t@ [sinh or cosh] x(t) and may be integrated by

parts, cancelling o the rst term and leaving the surface contributions, giving the total contribution

to the translation

2 3

t cosh(x ) t cosh(x )

2 2 1 1

4 5

t sinh(x ) t sinh(x )

b(t ;t )=(x ) =(x )(x ) b(x ) b(x ); (87a)

2 2 1 1

2 1 2 2 1 1 2

(~y ~y )

2 1

where wehave de ned the vector

b(x):=(t; 0;~y): (87b)

A A

@ Note that b(x):=b e is the Minkowski co ordinate vector X := x translated to this frame.

A M

Using these results the Poincare transformation (46) is given by

g (t ;t )= (x ;x );b(x ;x ) = (x ); b(x ) ? (x ); b(x ) : (88)

2 1 2 1 2 1 2 2 1 1

Cho osing instead the Poincare frame s := (e ; a) with

a = (t t cosh(x x );t sinh(x x );~y ~y )=(x) b b(x); (89)

0 0 0 0 0 0

we nd that in this case b 0. Here the vector a(x) may be seen to be the translation of

A A

the Minkowski co ordinate vector X (X ) := (x (x ) )@ to the Lorentz frame, where

M M 0 M

M M

ISSUES IN QUANTUM{GEOMETRIC PROPAGATION 17

(t ) = t cosh(x ), (x ) = t sinh(x ), := (x ) and b := b(x ). Using (48) these two

M 0 0 0 M 0 0 0 0 0 0 0

transition elements may b e related by

(x ;x ) ? 1;a(x ) : (90) g (x ;x )= 1;a(x ) ?g

2 1 1 2 1 2

In the case at hand, we have that g is a Lorentz b o ost in the Poincare section s chosen with a

as in (89) , and so transforming back to the Lorentz frame s do es not a ect the Lorentz contribu-

tion (84b) , only intro ducing the translational factor

b(t ;t )=a(t )(t ;t )a(t );

2 1 2 2 1 1

1 1 1

=(x ) b b(x ) (x )(x ) (x ) b b(x )

(91)

2 0 2 2 1 1 0 1

0 0

=(x )(x ) b(x ) b(x );

2 1 1 2

which is the Lorentz frame result (87) .

3.4. The Quantum{Geometric Propagator. The computation of the parallel transp ort ele-

ments allows one to write down the propagator for parallel transp ort corresp onding to (64) for each

of the choices of section s and s , which turn out to give an identical (up to a constant factor that

either cancels or is removed bycho osing (x ) = 0) result

M 0

1 1

l l

^ ^

(x ) p ; X (x ) i (x ) p ; K x ; (x ); x ; (x ) =K X (x ) i

(92)

2 2 M 1 1 1 2 2 1 1 M 2

m m

(x ;x )

2 1

where we have identi ed X (x) = (x) b(x) as the comp onents of the Minkowski co ordinate

vector (note that b(x) := x but X (x) = x , that is, the rst is the actual vector translated

M M M

to the new frame whereas the second is the comp onents of the vector transformed). Using the

0 3

measure d (x; p)=2p t d xd (p), the integral analogous to (73a) is

m 0 m

Z Z

0 2 1 1

l l

~ ~

d (p)K (x ); X (x) i 2p t dx d ~y (x) p K X (x) i (x) p; (x ) (93)

m 2 M 0 M 1

m m

(x ) p . Following the argument at the end where we have identi ed (x ) := X (x ) i

n n n M n

of Section 2.3, we transform the momentum variables (this time p ! (x)p) resulting in p =

p n(x) ! p (x) n(x) where n(x) := (1; 0) is the normal to the surface evaluated in the

frame e (x). The vector (x) n(x)may b e written as n (x )=(t =t; x =t; 0 ), which are the

A M M M M

A AB

vector comp onents of the normal to (n := @ [t]{which in this case is already normalised)

evaluated in the Minkowski frame asso ciated with the co ordinates t ;x ;~y. The surface integral

M M

R R

2 2

may also be rewritten as tdxd ~y (t t )tdtdxd ~y, which is translated to Minkowski

2 2

t dt dx d ~y . Thus we have transformed the integral x co ordinates to give t

0 M M

M M

into one over a hyp erb oloid in Minkowski space, the surface contribution to which, as we noted

following (27) , do es not vanish, and the kernel (92) do es not repro duce (in fact the integral (93) is

divergent).

4. Discussion

We have shown how the construction of the free QG propagator on a classical background

spacetime pro ceeds, and furthermore shown that it fails for the Milne universe that we considered.

The problem is that wehave b een using hyp ersurfaces that are only partial Cauchy surfaces of the

spacetime, which are not sucient due to the fundamentally non-lo cal nature of QG propagation.

If one had a truly lo cal theory, then one would exp ect that the evolution of any system to the future

of a Milne hyp ersurface would b e completely determined by data on this surface. Indeed, one may

consider a system which has radiated away prior to t =0 (so that the Milne sector is at). In

terms of any foliation that consists of global Cauchy surfaces, the quantum{geometric propagator

will not b e exactly equivalent to that of a at spacetime, even for p oints that are well within the

at region of spacetime. It is clear that the construction in Section 3 cannot see this, and the

lack of complete information is re ected in the fact that the propagator sum (65) is ill{de ned. In

ISSUES IN QUANTUM{GEOMETRIC PROPAGATION 18

general the foliation of spacetime must consist of global Cauchy surfaces (and therefore spacetime

must b e globally hyp erb olic) in order to derive a consistent result.

Although the actual construction of a quantum{geometric propagator in a curved spacetime has

not b een carried out as yet, the knowledge of its general form in a at spacetime already has some

remarkable features. Not only is the at spacetime propagator (71b) well{de ned for any choice

of co ordinates, Poincare frame of reference, or foliation of spacetime into spacelike hyp ersurfaces,

but the results of propagation viewed by di erent observers is re ected by physically reasonable

b o osts of the observed momentum sp ectrum of a state which takes the place of the Bogolub ov

transformations of conventional eld theory.

Thus we are naturally lead to consider the observer dep endence of conventional quantum eld

theory on curved spacetime. In particular, the ex{nihilo particle pro duction in Rindler spacetimes

and it's interpretation as the thermal radiation of a detector [37 ] may b e assessed in the context of

a theory that is consistently extendible in a straightforward manner to curved spacetimes. Further-

more, the preliminary results [32 , 9] lead to the consideration of particle pro duction in an expanding

universe, as well as a study of the p ossibility of macroscopic acausal e ects due to the presence of

strong curvatures.

It is also interesting to lo ok for nontrivial e ects of a consistent quantum eld theory on a curved

background in cosmological scenarios by considering the back-reaction of the presence of the scalar

eld on the classical spacetime through the lo cal average quantum stress{energy hT i. Although

the ultimate formulation of Quantum{Geometric Gravity [32 ] is exp ected to be di eomorphism

invariant, this typ e of (ultimately approximate) mo del may not b e. It would therefore b e interesting

to determine how this lack of di eomorphism invariance manifests itself, and therefore determine

the limitations of this typ e of semi{classical mo del.

Acknowledgements

The author thanks the Natural Sciences and Engineering Research Council of Canada for a

p ostdo ctoral fellowship.

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CERN{Theory Division, CH{1211 Geneva 23, Switzerland

E-mail address : [email protected]