Concept of a Holonomy and Some of its Properties

1.9 Introduction to Loop Quantum Gravity

1.9.3 Concept of a Holonomy and Some of its Properties

The connectiona, tells us how a spinorD.1 2/>is parallel transported along a curve, parametrized by variables. As we have done in (6.1.5) in Chap. 6 of Vol. I, we have the similar equation.dxa=dsD Pa.s//

d ds

.s/

D iPa.s/ a

.s/

.s/

; (1.9.10)

in matrix form, which is easily integrated78fromstos, corresponding to two points on the curve, to give

..s// D P exp

i Z s

s

dsPa.s/ a

.s/ .s/

; (1.9.11)

whereP means path-ordering, exactly as is done for time-ordering. That is, in particular,

PΠa

.s1/ b

.s2/ Db..s2// a

.s1/

if s2>s1; along the curve, and so on. The object

h.s;s/ D P exp

i Z s

s

dsPa.s/ a

.s/ ; (1.9.12)

is referred to as a holonomy.79The holonomy is the basic variable used in the “Loop Quantum Gravity” formulation.

For two curves introduced via the equation

.s/D 8ˆ ˆ<

ˆˆ :

1.s/; s<s;

2.s/; s<s;

(1.9.13)

joining at the points, we obviously have

h.s;s/ D h2.s;s/h1.s;s/: (1.9.14) Under an SU.2/gauge transformation, via a unitary matrixV.x/locally, we have the well known gauge transformation80

a.x/ ! aV.x/ D V.x/

a.x/ @a

i

V1.x/: (1.9.15) In view of the above transformation law, we investigate the gauge transformation of h.s;s/. To this end, we consider a limiting cases of a holonomy which is a trivial one involving zero integration, where one doesn’t move from the starting point and

78See Problem1.19.

79Some authors define a holonomy as the trace of the expression in (1.9.12).

80See, e.g., Chap. 6, (6.1.10) of Vol. I [43].

is given by

h.s;s/ˇˇsDs D 1; (1.9.16) Given an SU.2/- transformation viaV.x.s//, define

Q.s/ D V.x.s//h.s;s/: (1.9.17) Upon taking the derivative d=dsof the latter equation, using (1.9.10), and the facts that.d=ds/V.x.s//D Pa.s/ @aV.x.s//,.@aV/V1D V@aV1, we obtain

d

dsQ.s/ D Pa.s/h

@aV.x.s// C iV x.s/

a

x.s/ i h.s;s/ D Pa.s/h

@aV.x.s// C iV x.s/

a.x.s//i V1

x.s/

V x.s/

h.s;s/

Di Pa.s/h V

x.s/ a.x/1 i @a

V1

x.s/i Q.s/;

Di Pa.s/ V x.s/

Q.s/; (1.9.18)

where in writing the last line, we have used the gauge transformation rule fora.x/

in (1.9.15), and have also used, in the process, the definition ofQ.s/in (1.9.17) all over again. The solution of (1.9.18) is given by

Q.s/ D hV.s;s/Q.s/; (1.9.19) where

hV.s;s/ D P exp

i Z s

s

dsPa.s/ aV

.s/ : (1.9.20)

The last two equations give from (1.9.16), (1.9.17), the explicit gauge transforma- tion ofh.s;s/,

hV.s;s/ D V.x.s//h.s;s/V1 x.s/

: (1.9.21)

Hence TrŒh.s;s/ , referred to as Wilson loop,81corresponding to aclosed loop, is invariant under anSU.2/transformationgiven in (1.9.15).

The two fieldsAjbandEiaare canonical conjugate variables, and the Hamiltonian of general relativity may be expressed in terms of the fieldsEia,jb.82We will not

81See also Sect. 6.12. of Vol. I [43].

82See, e.g., [50,62].

need the corresponding explicit expression of the Hamiltonian here, however. In particular, the Poisson brackets are given by.ia !Aia/

fAia;EjbgPBDıij ıabı3.x;x0/; fia;EjbgPBD! ıijıabı3.x;x0/; (1.9.22) wherex;x0are in˙, and„, c, have been set equal to one,83and!is taken as

! D 8 ˇGN; (1.9.23)

where GNis Newton’s constant, and ˇ, a numerical, is a free parameter of loop quantum gravity, referred to as a Barbero-Immirzi constant.84 Its value is usually chosen by making a comparison with physically established results.0 < ˇ < 1/

such as in studies of the entropy of black holes, via the Bekestein-Hawking Entropy Formula. The latter expression is given Appendix E of this chapter.

A transition to a quantum formulation may be achieved by a functional represen- tation of the densitized triadEiaas follows

Ebia.x/ D .i/ •

•Aia.x/; (1.9.24)

consistent with (1.9.22), now as a commutator, withAia.x/acting as a multiplicative operator.

Consider the application of the flux in (1.9.9), now as an operator, to a holonomy with the underlying curvecrossing the surfaceS, at a given point, where on one side of the surface the curve is denoted by1, and on the other side it is denoted by2, as shown in Fig.1.8a: going, say, in the direction from2 to1. In view of application to the Riemann sum in (1.9.8), we may consider the surfaceS to be arbitrary small so that only one line crosses it. To the above end, we consider variation of a holonomyh.s;s/ about some points:s < s < s. From (1.9.12)–

(1.9.14), we may first write. D=2/

•h.s;s/Dh

s;sC s 2

f:gh ss

2 ;s

; (1.9.25)

f:gD1Ci Z sC2s

s2s ds1Pa.s1/ iia.x.s1//C.i/2 2

Z sC2s s2s ds2

Z sC2s

s2s ds1Pa.s1/Pb.s2/ h

ij.s1s2/Cji.s2s1/ ia

x.s1/ jb

x.s2/ i C :

(1.9.26)

83In this section, the variablesx;x0; : : :are understood to correspond to three components.

84Barbero [11], Immirzi [33].

Fig. 1.8 (a) Acurve, with parts denoted by1; 2, as it pierces an area element—see (1.9.28).

(b) Acurvepiercing an area several times—see (1.9.30).

From the above two equations, we obtain by functional differentiation .i/ •

•Aia.x/Ajb.x.s//D iıjiıabı3

xx.s/

; in the process, withs!0,

EbiŒSh.s;s/ D 8 ˇGN

Z

S

d1d2 Z sC2s

s2s ds1"abc

@xa

@1

@xb

@2

@xc

@s1 ı3 x

1; 2/x.s1/

h1.s;s/ i h2.s;s/: (1.9.27) We recognize"abc.@xa=@1/.@xb=@2/.@xc=@s1/as the Jacobian of transformation of variables of integration leading to integrations over the variablesx1;x2;x3, thus obtaining the simple expression

bEiŒSh.s;s/ D 8 ˇGNh1.s;s/i h2.s;s/; (1.9.28) since we have considered the intersection of the curve at a single point as shown in Fig.1.8a.

Similarly, referring to the second term in (1.9.26), we have by taking the limit s!0

EbiŒSEbiŒSh.s;s/D.8 ˇGN/2iih.s;s/D.8 ˇGN/21 2

1

2C1 h.s;s/;

(1.9.29) where we have, in the process, used the elementary property:.s1s2/C.s2 s1/D1,iiD3I=4, and (1.9.14).

As a further generalization, consider a curve with no parts of it touching, which crosses a surfaceSat several points as shown in Fig.1.8b, then clearly by partitioningSinto very fine cells so that at most one intersection occurs per cell, by using (1.9.28) for each cell involving an intersection, and summing over these cells, we obtain

EbiŒSh D 8 ˇGNX

p

ıphp1i hp2; (1.9.30)

as a sum over all points of intersection, with theıptaking the values˙1.

Before closing this subsection, we consider higher spin representations of SU.2/

as well. To this end, irreducible representations of this group are labeled by the spin jtaking values inf1=2; 1; 3=2; 2; : : :g. The generators of the group will be denoted byTi.j/;iD1; 2; 3;satisfying the quantization conditionTi.j/Ti.j/Dj.jC1/I, with ŒTi.j/;Tk.j/Di"i k lTl.j/,Ti1=2D i=2 i.

LetRjŒA;hD hhjRji, denote the spinjirreducible matrix representation of the holonomy of the connectioniorAi, along a curve. For spin1=2, the so-called fundamental representation,R1=2ŒA;hDh. The action of the operatorEbŒSon RjŒA;his the same as the one given in (1.9.28), for the curve intersecting the surface at a given point, obtained by simply replacingibyTi.j/, i.e.,

EbiŒSRj A;h

D 8 ˇGNRj A;h1

Ti.j/Rj A;h2

; (1.9.31)

and similarly for the case involving several intersections.85

Now we are ready to define the concept of spin networks, construct space itself and derive formally a major result of loop quantum gravity.