Quantum Regge calculus

Volume 104B, number 1

PHYSICS LETTERS

13 August 1981

QUANTUM REGGE CALCULUS M. RO(2EK Department of Applied Mathematics and TheoreticalPhysics, University of Cambridge, Cambridge, England and Ruth M. WILLIAMS Girton College, University o f Cambridge, Cambridge, England and Department of Applied Mathematics and TheoreticalPhysics, Universityof Cambridge, Cambridge, England Received 7 May 1981

We consider the quantization of Regge's discrete description of gravity using functional methods• We show that in the weak field limit the standard continuum theory emerges.

Regge calculus provides a geometric and coordinate-independent description of gravity in which spacetime has a discrete rather than continuous structure [1]. A Regge skeleton is specified by a discrete set of link lengths, which are the dynamical variables of the theory, and a connection matrix which provides the topological or kinematical substructure. The approach has been applied to a number of classical problems [2], and has been generalized to include matter couplings [3,4]. The initial value problem is understood at least in principle [5], though it has not yet resulted in a practical technique in the field of numerical relativity. Discrete approximations to gauge theories on fixed flat lattices (lattice gauge theories) have given insight into aspects of the quantization of gauge theories, (e.g., confinement), by providing a manifestly gauge-invariant and non-perturbative regularization scheme. Attempts to copy this approach for gravity have not been geometrical: they are defined on passive flat lattices which in no way reflect the geometry of spacetime [6'] * 1. It is therefore tempting to try to formulate a quantum theory of gravity using Regge calculus .2 . To quantize Regge calculus, one must choose amongst several options: we use functional rather than canonical operator methods, and we integrate over skeletons with a fixed number of points and a fixed connection matrix, but with varying link lengths.This effectively restricts tire path integral to a fixed topology. Beyond this one must also choose a measure which insures that adjacent link lengths satisfy the triangle inequalities and their higher dimensional analogues, understand gauge invariance, and deal with the unboundedness of the gravitational action (which survives in Regge calculus). We discuss these issues, and give the derivation of the results presented below in a longer paper currently being prepared [9]. As a first step toward understanding quantum Regge calculus we consider the simplest possible problem: the free "field theory" of small fluctuations about a flat euclidean background. We find that the free propagator in a de Donder-like gauge is precisely the discrete version of the usual continuum theory [10]. In Regge calculus [1 ], the curvature in a four-dimensional space is restricted to two-dimensional subspaces, the ,1 An alternative approach by Macrae [7] is expressed not in terms of the intrinsic geometry of the four-dimensional spacetime, but in terms of its embedding in flat ten-dimensional space. t 2 Martellini [8 ] follows a course completely orthogonal to ours by treating the deficit angles as independent "coupling constants' unrelated to the link lengths. In our opinion, such a theory does not describe gravitation. 0 0 3 1 - 9 1 6 3 / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 02.50 © North-Holland Publishing Company

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flat triangular hinges of the 4-simplices into which the space is decomposed. A measure of the curvature at a hinge is given by the deficit angle e between the blocks that meet there; the Regge action S is given by S =

~

hinges

(1)

A(l)e(l),

where A is the area of the hinge. The Regge calculus approximation to Einstein's equations is the set of nonlinear difference equations which is found by varying the action with respect to the link lengths l. Regge has shown [1] that the term involving the variation of the deficit angle vanishes, and so the first variation of the action is: hinges

"

-~7 51 e .

(2)

Thus for a variation about fiat space, where the deficit angles vanish, the second variation of the action is:

+2s= 2

hinges hnks ~ - 5 l

2-

links ~I 5l

.

(3)

It is this expression we must compute to find the free propagator. To proceed, we must choose a lattice for our flat background space. We use a lattice which in n dimensions is generated by the translation of 2 n - 1 lattice vectors. These are just the numbers 1,2,..., 2 n - 1 written in base 2 and interpreted as vectors from the origin to the corners of the unit n-cube; for two (see ref. [4] ) and three dimensions, the lattice vectors, the unit n-cube, and a sample n-simplex are shown in fig. 1. Note that the lattices consist of n-cubes subdivided into n! identical n-simplices. The lattice vectors in four dimensions are listed in table 1. In the dynamical lattice the connection matrix remains the same, but the link lengths l vary. Note that this lattice has 2 4 - 1 = 15 components per point, whereas the continuum theory is described by a metric tensor with only 10 components. This discrepancy can be understood by noting that in addition to curvature on the hinges between adjacent hypercubes, there can be curvature between the simplices that make up a single hypercube. We will see below that these extra modes are eliminated by the dynamics of the system. We now allow each link length l i to fluctuate by a small perturbation 5 i I} =(1 + 5 i ) l i ~ 5 l i = 5 i ' l i ,

(4)

and compute (3). This is a quadratic expression in the 5i's which can be written as 52S = ~

li +M+8,

(5)

where 6 is an infinite-dimensional vector with 15 components per lattice point, and M is the infinite-dimensional matrix whose inverse is the free propagator. As in the continuum theory,M cannot actually be inverted because it has zero eigenvalues. The zero modes span an infinite-dimensional subspace of the 6's with five independent parameters per lattice point; four correspond to translations of the points which leave the space flat, and are just the usual gauge transformations, whereas the fifth corresponds to fluctuations of the hyperbody diagonal, and serves to eliminate one of the extraneous "internal" degrees of freedom described above. The matrix M consists of 15 X 15 dimensional matrices associated with (infinity) 4 × (infinity) 4 pairs of lattice points. It is sparse in that it only couples lengths in the same or in adjacent hypercubes. As a first step to finding the propagator, we expand 6 in periodic eigenmodes, that is, modes such that 5;(ku) ^ (i = 1,..., 15) at a point k steps in the ~ direction ~ = 1 , 2 , 4 , 8) from 5}0) is related by 5} k~) = (co~)ks}0),whe're ¢o~ = exp (27ri/n~) and n~ is the period in the t£ direction. Acting on periodic modes, M reduces to a block diagonal matrix with 15 × 15 dimensional blocks M w given in table 2; schematically, M w has the form

M+ 32

l0

I '+'+1 ° ' Lo

I0

Volume 104B, number 1

PHYSICS LETTERS

13 August 1981

I.

"

d)

a.[

a) l" / / / / / / / / / / / / a .

.."/~, .'/

i I' •.'57 .);/

.lr.

I-/~

2.

l., "//'] /

/

/

/

tl.

e)

/

l

//

/

.,'/

.~V

/ /

¢) [¢

f"

/

/"

//

/

/

/ ~"

f ft

/ /

i- 1/

...

.7 .' /

;¢," .. /

/

Table 1 Lattice vectors in four dimensions.

."//

J Eb~ES

Ed des (l = 1)

Face diagonals (l = ,J'2)

Body diagonals (l = x/3)

Hyperbody diagonal (l = 2)

1 (0,0,0,1) 2 (0,0,1,0) 4 (0,1,0,0) 8 (1,0,0,0)

3 (0,0,1,1) 5 (0,1,0,1) 6 (0,1,1,0) 9 (1,0,0,1) 10 (1,0,1,0) 12 (1,1,0,0)

7 (0,1,1,1) 11 (1,0,1,1) 13 (1,1,0,1) 14 (1,1,1,0)

15 (1,1,1,1)

FACE D~AGON~,LS ...................................

I~ODy DIAGONALS"

Fig. 1. (a) Lattice vectors in two dimensions. Vector coordinates: 1. (0,1), 2. (1,0), 3. (1,1). (b) N-cube in two dimensions. (c) N-simplex in two dimensions~ (d) Lattice vectors in three dimensions. Vector coordinates: 1. (0,0,1), 2. (0,1,0), 3. (0,1,1), 4. (1,0,0), 5. (1,0,1), 6. (1,1,0), 7. (1,1,1). (e)Ncu be in three dimensions. (f) N-simp lex in three dimensions.

where A 10 is a 10 × 10 dimensional matrix, B is a 4 × 10 dimensional matrix, and 14 is the 4 × 4 dimensional identity matrix. M w has four zero modes corresponding to periodic translations of the points of the lattice, given in table 3, and a fifth zero mode corresponding to periodic fluctuations of 815 ; it can be easily block-diagonalized by a (nonunitary) unimodular similarity transformation M ~ = S+Mw S, where

~0

I o I1/

I 0

10

. , Note that this• transformation completely decouples 5 7, , 811, (~,13 and 8 ,14 [8 i, = ( Sp -_)17 8Iii ].. from the , rest of the system, and that they enter the action without derivatives (co's) simply as: 18 (16712 + I&Pl1i2 + 1613 j2 + i814 ]2). Thus, at this level, they do not contribute to the dynamics at all: they are constrained to vanish. If one applies S -1 to the four translation zero modes, one discovers that (5}, etc., are gauge invariant fluctuations: they are pre-

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Volume 104B, number 1

PHYSICS LETTERS

13 August 1981

cisely the deficits of four " i n t e r n a l " triangles in the h y p e r c u b e ( " i n t e r n a l " triangles are shared only by 4-simplices within the same hypercube). The problem is thus reduced to the study of the 10 X 10 dimensonial m a t r i x L = A 10 - ~8 BB+. We p o s t p o n e the derivation of our results to ref. [9], and merely present them. The most convenient variables for our purposes are the discrete analogues of the trace-reversed metric fluctuations huv = huv Table 2 The small fluctuation operator Moo acting on periodic modes. Moo is written in terms of blocks whose dimension is indicated and a, ..., e are given, with rows and columns labeled ; Ct2 = exp (-27ri/n/2).

-2b _

4

~

6

2b +

4d

C÷

-6e +

18 1

0

0

0

0

Mo9 = 4 1 -

6

4

1 F6 C02(C04 + C08)

t

1 ,--dimension 8

,- column roy

2

4

tOI(W4 + ~O8) + ~.~2(c~4+ ~ 8)

Wl(tO2 + ~O8) + ~4(c~2+ ~ 8 )

L+ c~8(~2 ~ 4 )+

6

C02(COl + COS) + C~4 (C~1 + C~8)

CO2 (OJ I +CO 4 ) + C~8(C01 + ~ 4 ) |

c°4(c°I+C°2L I + ~ 8(,~1 + co2~

+ c~ 1(CO4 + C~8)

[

1 2

a= CO4 (cO2 + COs) + ¢.~I(t~ + ~S)

C04(c01+ C08) + ¢~2((-~I + (~8)

6

tO8(CO2 + L04) + ~1(¢~)2 + t.~4)

~o8(~ol + ~2) + ~2(~1 + 54)

to8(~1

3

5

+ ~2)

+ c~4(t~l + ¢-~2)

6

9

col + c~2c~4 +2¢58 b=

2+2~t

w2+

co4 +G1 t~2

2+ 2~1

2 + 2~ 2

11

13

14

-

+- column row ~.

~ 2 + ~ 4 I ~ + L ~ 8 ~4 + ¢.~8 ¢~20,. 1 C~I + C~4 C~l L ~ _ 0 ~4 + ~8

~i+~2

34

I ~1 + t~4

_ t~1+t~2¢58

l c°4 + ~1c°8

c08 + co 1co2] c08 + COlco4 ¢~8 + C~2C~4 / 2 + 2 ~ 1 7

10

co2 + c~1c~8 2 + 2~ 8

2+ 2¢~ 4

~t~4

6

+~4

1 2

8

i 8 12

,-- column row

coi+¢B4¢581 1 ~o2 + ~ 4 ~ 8 / 2

c04+c~2c~8~_+__2638 I 2+2~2

4

I 2+2~4

.~4 ] 8

Volume 104B, number 1

PHYSICS LETTERS

13 August 1981

Table 2 (continued)

5

3

~ column row

092 + 5 4

~ 4 + O92

4

091

~4+ 5 1

092 + 5 1

4

o98 + o92

098 + 5 4

0

~ 8 + 091

0

co 8

0

~8 + ~1

0

54

092 + 098

6

1+51

1+58

0

5

0

0

1+58

6

4

o91+ t~2

o91 + 5 4

9

i+5 2

1+ 5 4

0

9

o92 + 5 1

4

oa2 + 5 4

10

1+51

0

1 + 5~

10

12

J 12

4

8uvha; ; t h e s e are f o u n d b y c o m b i n i n g a u n i t a r y ( b u t n o n u n i m o d u l a r ) formation to produce): \0

3

1+ 5 2 0

+~8

1

=T+LT= [ 8 - - ( ] ~ + ] ~ ) 1

I1+54

5

_~

14 +._column row

13

II

7

3

4

d=

12

10

6

with a unimodular (but nonunitary trans-

I 16 / - C + C '

where -1

z= ~,~i l

~=~

-t 1

-1

-1

-1 -1

1 -1

-1 -1

' 1

Table 3 Translation zero modes. The four independent parameters at each point are x I, x 2, x4, and x 8 . "81

~1 -- co I

"li 64

68

'

1

83 [ 1 ½ ( 1 - w i t ° 2 ) (55 I ~ (1 -- o91o94) 86 kl 89 = "~ (1 610 , 612

Wl~d8)

~1

i (1 -- C°1O92094 ) ]~ (1 -- 031 092098)

813

I ~(1 -- 091~4098)

18'4[

L~lsj

L~- (1 - 091w2o94ws)

0

0

0

1 - w2

0

0

1 - 094

0 0

0 -12(1 - co I co2)

0 0

1- ~8 0

0 1(1 0

1 (1 - ¢Olo94) ~1 (1 -- 092094) 0

0 0 1(1-

C02 O94)

½ (1 - ¢o2w s) 0 I (1 -- 091O92094) 1 (1 -- ¢01O92098) 0 1 ( 1 - ~o2w4098) I (1 -- 091 o92 094o98)

0 1 ~ (1 1 (1 -0 1 ] (1 -1(1k (1 --

¢04098) t~l O9"2094) Wlo94ev8) co2o94w 8) 091092 094o98)

o91098) 1 ~ (1 - o92 098) 1 ('1 - ~4098) 0 1 ( 1 - - ¢0 1 092 O98 ) 1 ( 1 - - COl 094c08) 1 ( 1 - O92094O98) 1 ( 1 - - cO1092 O94098 )

X X2 d e f R × x4 fi 8

x1 x2

I

X4 X8

35

Volume 104B, number 1 6oI --1 0 C= 0 0

0 602 -- 1 0 0

(~4~-].(1~4

PHYSICS LETTERS 0 0 604 - 1 0

0 0 0 608 -- 1

"16/'

7=--

0

1

1

1

0

0

1

0 1

1 0

1 1

0 ' 0 /

ll g24 =diag(601,602,604,608),

1 -c~ 4 0 1 --c~ 1 0

0 1-ff~4

1 - c~8 0

0 1-o3 8

0 0

1 -

0

0

1 - c~ 8

1 - c~ 1

1 -co 2

1-~4

c~ 2

0

'\

)

'

11001 '/'

.@~ t

T = \ O 7- ] 16 /, \ ~ 6 9 ' 1

1 - ~2 1 -c~ 1 0 0

13 August 1981

11

1 o o/

~6 =diag(601602,601604,602604,601608,602608,604608)"

(8)

We observe that for 60~ = exp (ik~a) ~ 1 + ik~ a + O((kga)2), where " a " is a typical lattice spacing and kfi is the m o m e n t u m k~ = 2zr/an;~, L becomes (up to a conventional normalization factor of - ½) identical to L s y M in ref. [ 10] when expressed in terms of huu" The de Donder gauge is obtained by adding a gauge breaking term C+C to L and including the appropriate ghost term Eij~i(O/3xj)(Ch)igj where gi and gj are four (per lattice site) antighost and ghost fields, respectively, The variation of the gauge fixing condition is

(8/Sxj)(Ch)i = (8/Sx/)(CT-lT)i = (CT-1R)i]

= [8 - (]~ + ~)]1 4 ,

where R is given in table 3. This corresponds to a ghost term f d4x x / ~ u E 3 gu in the continuum case (which is the correct result). Finally, we must fix the gauge for the fifth zero mode; rather than fix the hyperbody'diagonal 815 itself, it is more convenient to fix a linear combination of 615 and other link lengths which is invariant under the four translation zero modes, e.g., 515 = ~15 - 1 (8 7 + 511 + 513 + 514 + 60867 + 6°4611 + 6°2613 + 6°1614) +1

[601(512 +~10 + 5 6 ) + 602(512 +59 + 5 5 ) + 604(~10 +69 + 5 3 ) + 608(56 +55 + 6 3 ) ] .

(9)

Such a 8'15 Still undergoes an arbitrary co-independent shift under the fifth zero mode fluctuation, and is invariant under the translation modes fixed above ; it can be constrained without introducing any propagating ghosts. It is a combination of internal deficits which is linearly independent of the four corresponding to 8~, etc., above. Thus we have seen that at least to this level, perturbative Regge calculus is equivalent to Einstein's theory. It should be emphasized that this equivalence is not trivial: it involved an intricate decoupling of 5 extra modes. Of course, we do not propose that this is a useful way to do weak-field perturbation theory;just as in conventional lattice gauge theories the continuum theory is a far more efficient way to do that. The hope which motivated our study is that having established the connection in this regime, one can now go on to use the framework for methods which are not available in the continuum theory. In particular, if the problem of the unboundedness of the action can be solved, then Monte Carlo calculations should become practical. It might also be possible to develop some strong-coupling expansion, although the non-compact domain of integration of the link lengths makes usual methods fail. More speculative ideas are reserved for ref. [9]. We would like to thank S.W. Hawking, B. Whiting, J. Saxl and S. Austin for encouragement and/or helpful discussions. One of us (R.M.W.) expresses her gratitude for the hospitality of the Theory Division at CERN, where some of this work was done.

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Volume 104B, number 1

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References [1] T. Regge, Nuovo Cimento 19 (1961) 558. [2] J.A. Wheeler, in: Relativity, groups and topology, eds. C. DeWitt and B. DeWitt (Gordon and Breach, New York, 1964); C .-Y. Wong, J. Math. Phys. 12 (1971) 70; P.A. Collins and R.M. Williams, Phys. Rev. D5 (1972) 1908; R.M. Williams and G.F.R. Ellis, Regge calculus and observations: I. Formalism and applications to radial motion and circular orbits, to be published in Gen. Rel. Grav. [3] P.A. Collins and R.M. Williams, Phys. Rev. D7 (1973) 965 ; P.A. Collins and R.M. Williams, Phys. Rev. D10 (1974) 3537. [4] R. Sorkin, J. Math. Phys. 16 (1975) 2432. [5] R. Sorkin, Ways. Rev. D12 (1975) 385. [6] L. Smolin, Nucl. Phys. B146 (1979) 333; A. Das, M. Kaku and P.K. Townsend, Phys. Lett. B81 (1979) 11; M. Kaku, in: Superspace and supergravity, eds. S.W. Hawking and M. Ro~ek (Cambridge U.P., Cambridge, 1981). [7] K.I. Macrae, Phys. Rev. D23 (1981) 900. [8] M. Martellini, preprint (1980). [9] M. Ro~ek and R.M. Williams, in preparation. [ 10] M. Veltman, in: Methods in field theory, eds. R. Balian and J. Zinn-Justin (North-Holland, Amsterdam, 1976).

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