Measures in the Space of Metrics and Diifeomorphisms

In order to define an invariant measure for the h integration, we shall start from the definition of the metric in the space of metrics h, and then use the “metric tensor” in this functional space for finding the volume element. The only local expression for the “distance” \\Sh\\ between the metrics /i(t) and h(z) + has the form:

\\Shf = dT(Sh(r)fh-^^\x) (9.23)

It is quite obvious that (9.23) is invariant under reparametrizations.

Let us notice now that any metric h (T ) can be made r-independent by means of a properly chosen gauge transformation / (t). Due to (9.9) this implies that any h can be represented as:

(9.24) with some T and / . Our strategy will be to pass from the integration over S>h(T) to integration over the new variables T and / . According to general rules:

Q>h{x) = dT^f(x) X (Jacobian) (9.25) If we manage to find the Jacobian, our problem will be solved, since instead of ^h{x)/^f{x) we shall obtain a properly defined integral over dT. The easiest way for solving this problem is to substitute the decomposition (9.24) into (9.23) so as to find the “distance” in terms of the new coordinates.

The computation is greatly simplified if we use a general geometrical formula, which we derive now in its n-dimensional form (needed for future applications). Suppose, that we have two metric tensors Qabi^) and h^biO^ where are coordinates of some n-dimensional Riemannian manifold. Let these two metrics be connected by the coordinate transformation (or diffeomorphism) ^ /(^):

g = (9.26a)

or, explicitly:

,,, d r n ) d m ) , , , , , , ,

dabiO = KdifiO) {9.26b)

We should like to prove the following relation, connecting small variations of g, h, and / :

with

co^{i) = s r { f - \ o )

(9.27)

(9.28)

Here by we mean the standard covariant derivative computed in the metric /i, and the symbol / “ ^ means the inverse function. This formula (9.27) can be checked by direct computations, but it can be easily understood without that by the following consideration, based on the group properties of diffeomorphism. We can consider the transforma­

tion / + ¿/ as the transformation / , followed by the infinitesimal transformation 1 + co. Hence it is suificient to compute the change of the metric tensor under the infinitesimal transformation co, which is given by the expression in brackets in (9.27). Let us also notice, that the analogous formula for gauge fields has the form:

CO = V ^ c o = c ^ c o +

(9.29)

(here / (x) is the field of unitary matrices, performing gauge transforma­

tions).

It is straightforward now to use these general geometrical formulas in our special case. Substitution of (9.24) into (9.23) gives:

\ \ S h f = [ T S T + w f

dtl r-^ (T ó r + c¿[/])"

dt J (ST)^

+ T

-1

' Í ^(9.30)

Here d>(f) = dw/d/, and the relation w(l) = (u(0) = 0 has been used (which follows from the conditions / ( I ) = 1 and /(O) = 0).

As the next step let us determine the measure and the metric in the space of diffeomorphisms. Just as in the case of ordinary groups there is only one metric invariant under left and right multiplication simultaneously. It is given by the following general formula:

«“(0 = W ‘(0)

(9.31) This formula is a consequence of two facts. Let us consider first “right multiplication”, namely the change:

/ ( 0 - / ( a ( 0 )

(9.32) Then we have:

/ - ‘( 0 - a " ‘( / ‘ ‘(i)) d m ^ s m o )

« ( 0 - ¿ /( a ( a " ‘( / ‘ ‘(0)) = o)(0

(9.33)

So, the form a>(^) is invariant under “right multiplication” of a diffeomorphism. If we look at “left multiplication”, we have:

Sp’if) .

W i O )

-

co%0-W )

■d i r ^ f d n r ' )

(9.34)

Therefore co“ transforms as a standard covariant vector under the left multiplication. If we transform the metric simultaneously, the distance (9.31) will be invariant. It is obviously the only local expression with these two properties. This expression generalizes the well known double invariant Killing metrics for finite dimensional groups. In the

finite dimensional case, if / is a group element then the Killing metric is given by:

\\Sff = Tv{cD^l co = ¿ /./- ^ This metric is invariant under the change:

(9.35)

(9.36) (where dots mean the usual matrix multiplication.)

Diffeomorphisms, while forming an infinite dimensional group, be­

have very similar to the finite dimensional cases. The only crucial difference is that in order to define the analogue of the trace in (9.35) we have to use the metric of the manifold as seen from (9.31).

Certainly, our manipulations with metrics in the functional spaces make sense only if some invariant regularization and renormalization is performed. This will be the case in all our future applications.

Returning to our problem we see that : w \ f ) d f

(9.37) CO = df

After rescaling t = T - f , a> ^ Te our expressions can be written as 2 ^{_ ( S T f}■ +

i

(9.38) W\\

I dts\t)

From (9.38) we deduce that Sihix) = 3>h(r) m o

(9.39)

and hence we have found the desired Jacobian. Now it is time to regularize and compute the determinant in (9.39). Its infiniteness reflects the formal nature of our manipulations with infinite dimen­

sional measures. However, we know that the ultraviolet divergences, according to (9.15), have to be cut off by splitting the interval into equal small pieces, since our metric tensor on the interval [0, T] is equal to 1.

There are many equivalent and more convenient regularizations. We shall use the following expression:

-T72 =_dr

fd iZ e- ^(9.40)

(where are the eigenvalues for —d^/dt^ which are equal to: = For the n for which the contribution to (9.40) is just log(l/2„8^). Harmonics with nnT~^ > give negligible contribution to (9.40). But nn/T is just the wave vector corresponding to the eigen­

function \l/„ of — d^di^:

- sininntIT) (9.41)

Therefore, the definition (9.40) accounts correctly for the eigenmodes which vary slowly in the time intervals ^ a, but cuts off higher modes.

Hence s plays the role of a lattice spacing. A little later we shall see that after renormalization we can take the limit £ 0, and that the concrete form of the cut-off is irrelevant.

To do the computation, we represent the sum in (9.40) in the form:

X exp(-7i^n^TlT^) = ^ Y. n^nh/T^)

-n = l ^ n = - 00

dn exp( — — t + 0(exp( —c/t))

- ^ + 0 ( e x p ( - c /T ) )

2yJ{nx) 2 ^(9.42)

(where the exponential smallness of the correction follows from the Poisson summation formula). We obtain:

-logdetRi - ,1 '

di

QO

" I ^t I,

exp(-7T^M^T/r^)

r dx ®

= — Z exp(-7t^/i^x) J ^ n= I

(£IT)2

J n = l e x p ( — TT^/I^x)

-I-dx

^ e x p ( — TT^M^x)

(e/T)2

(e/T)2 0

1 1

2 2 ^ { n x )

+ ^dx— ^ e x p ( — Tc^M^x) (9.43)

Now, the last two terms in the last formula are 7-independent constants. Therefore:

, d^\ T T

- l o g d e t - = — / lo g

-' d f7 ,,o « V ’' « (9.44)

If we substitute this expression into (9.39) we obtain:

| | = c o n s t.e x p (-T /2 a V .)^ .r‘/^

= const • exp( — T/2e^n) dT (9.45) The divergences are condensed into the constant factor in front and the term exp(—T/2e^n).

Now we are ready to compute (9.6). Combining (9.45), (9.22) and (9.7) we obtain:

ou

G(x, x') = const. 1 d r exp( —(mo — const./£)T

X T exp( —(x — x'Y /eT ) (9.46)

00

= const. J dT exp( —|I£T) exp( —(x— x 'Y ls T )

= const.

i

d >

+ Hexp(i/>(jc - Ac')),

ii = £ ‘(mo - mo.cr) = s ‘ mo - ^const

All our derivations make sense only if relevant values of T in the integral (9.46) are much larger than the cut-off (or “lattice spacing”) £.

Therefore, the integral (9.6) has a continuum limit only if we adjust Mq

to be very close to the critical value, or, in other words we have to take the limit £ -► 0 for the cut-off simultaneously with mo(£) ^ Wo,cr- Then the terms like e^^® will be compensated and we obtain the continuum theory. This result is quite easy to understand from the lattice point of view. On a lattice we have:

G (x,x')= X N ^(x,x')e“"’«^ (9.47)

where A^(x, x') is the number of paths of the length L connecting points X and x'. We know that for large L, N^(x, x') ^ (c)^ where c depends on the type of the lattice. We see, that for Mq > log c, the relevant paths in (9.47) have a length of the order of the lattice spacing and no continuum limit is possible. But as we approach ytiq -► Wo,cr = 1^8 G then a typical L ~ (mo — ^Wo,cr) ” ^ ^ 1 and the theory becomes continuous and lattice independent. Our description refers precisely to this limit.

We see from (9.46) that the correlation length or the physical mass have a nontrivial critical exponent:

“ phys = yl P l i n t o - W o , „ ) ‘ (9.48)

The procedure of renormalization in this case consists of expressing everything in terms of the physical mass and eliminating the irrelevant

constant factor in front of (9.46). After that we obtain a finite amplitude in the limit a -► 0.

In view of the future generalizations we have to develop the same formalism as above for closed paths.

9.3 Closed Paths

In this case there are some technical differences in the integration over the metric field. Let us begin with the integration over x(t), provided that the Lagrange multiplier is replaced by its average, which we shall choose to be ^ by a choice of scale. We have:

jTCMt)]

= i

9x(x) exp(

-x (0 ) = -x(T )

d t

T= h^'\T)dx

(9.49)

First of all we have to remove the trivial divergence in (9.49) connected with the fact that the integrand has translation invariance -►

and so we have to fix some point of our loop. Such fixing is most conveniently performed by inserting the relation

del Y ^{(5(x(t) — c}) d i I = 1 (9.50)

and omitting the j dc = K After that we have (with the choice c = 0):

I T

= I ¿ (x(t) ) e x p ^ - (9.51)

The ¿-function in (9.51) will exclude the dangerous zero mode of the operator —d^/dt^. If we expand

■«(t) = ^ « 0 + Z ^ (9 .5 2 )

then

. ^ . 4n^n^

dr = /l„ =

n^O ^

^x(t) = dflo n d«„

n# 0 From this we derive:

I

^ J^I2 det' -dr

-SI/2

(9.53)

(where det'(x) is defined as the product of the nonzero eigenvalues).

Using the representation of the preceding section we get:

00

- lo g D e t'^ - ^ ^ = L e\p(-4n^n^s/T^

ds ^ exp( —47r^n^s/T^) — 1

00

= j* — exp( —47T^n^x) — 1

(e/T)2

= J — exp( —47c^n^x) — 1

dx

— j Y, exp( — 47T^n^x) — 1

dx f 1

(£/T)2

_____

X [27(7TX)- 1 -I- 0 ( e x p ( - c / x ) ) l 4- const

T T

= 2 log

-Syjn e ^(9.54)

This gives the result:

jT[^(t)] = 7^/2 QxpicTls)

= exp(cT/e) (9.55)

Now we have to find the measure Here again we must treat zero modes carefully. Since our parametrization space is a circle it now admits an isometry—a transformation which does not change the metric. This is just a translation t -> t -h a and it must be excluded from our gauge group. This can be done in the same way as in the case of the X integration, by inserting the relation:

da dr

S ( f ( x ) - a ) = l (9.56)

As a result the measure in T will be the same as for open paths and we get the following answer for the number of closed paths of length T:

d T

dN(T) = — exp( - c T/ £) (9.57)

Of course, this formula could have been anticipated from (9.46). If we set X = x' we get the integrand in this formula to be exp(cT/e).

The extra 1/T in (9.57) follows from the fact that we should count paths with different starting points x as one path. Since for a path of the length T we have T different possibilities for a choice of the starting point we obtain a 1/T factor.

Different physical quantities can be expressed in terms of the amplitudes for a path to pass through a prescribed set of points {xj.

These amplitudes are obtained as expectation values of the following type:

= ( n dtj dix(Xj) - X j)

OD

dT ^jc(t) expl dxj d(x(xj) - Xj)

(9.58)

In order to compute these integrals it is convenient to go to the momentum representation:

00 T T

F{q r dT

1» • • •» Qn) — ^{^x(t) expl dr}

(9.59) The integrals in (9.59) are Gaussian. According to the general rules, to find them we have first to solve the classical equations:

■id = - ' I 9j ¿(i - ty)

J

and then to compute the classical action. We have:

(9.60)

(9.61) where ^ (t|t') is a Green function for the operator —d^/dt^. The substitution of (9.61) into the classical action gives:

1 r

Sci = xi, dt - i X q jX jT j)

(i,J>

(9.62) The Green function on a circle contains a zero mode contribution:

(9.63) (here X„ = 4n^n^lT^, and C is the zero mode contribution which is arbitrary). However if momentum is conserved (Lj qj = 0) this arbitrar­

iness disappears from (9.62):

cZ?,?y = q ^ i ; ) = 0 (9.64)

The necessity of momentum conservation can be seen also from (9.60) by integrating it over t. Then periodicity of requires = 0.

We obtained the following expression:

00 T

F(.qi,-.,qN) = dTY~^/2 Q-m^T/l j n * ,

(9.65)

In order to elucidate its meaning, we shall consider the case of the two point function with = —9i = 9- Let us compute, first of all, the

^-function. We have:

_ exp(i^yl„(t — t')) 4 n ^ n ^

^ (T |0 = Z = T2

-df^{T\x') = SÌT - t' ) - \ / T (9.66) (where the second terms in the r.h.s. follows from the constraint n # 0 in the definition of the ^-function). Since the ^-function on a circle depends on s = |i — t'| we can trivially solve (9.66), finding the result:

5 ( 7 - 5)

^ (t|t') = --- --- 1- const, for 0 < 5 < 7 (9.67) 27

Therefore:

m = ^{r d 7 ^ ^}

Y ~ ^12 Q-m^T/2 rdr J 7

Q-q^siT-s)l2T

0 0

T

dT Q-rn^TU ds Q

-=H

_{0 0}

1

~^/2 Q-m^TI2 Q-q^x(l -x)T/2

dx

+ q^x{{ - x )] ^(9.68)

(here we have introduced a new variable x = s/T = |ti — T2I/L). It is quite interesting that the amplitude (9.67) can be interpreted in terms of ordinary Feynman diagrams. Let us consider the expression:

F(q) =

q - k d^k

dx

H- m^){{q - kY m^) d^k

{xk^ -H (1 - x)(q - ky -h m^y

dx

(m^ -f x(l - x)q^y ^(9.69)

w h ere w e h a v e u sed th e sta n d a rd r e p r esen ta tio n :

169

1 AB

dx ( ^ x -I- B(1 — x))^

The derivation we have given serves as a check that all normaliza­

tions, measures etc. in the path integrals were correctly chosen because at the end it produced the standard answer for the amplitude (9.69). In the same way it can be shown that defined by (9.59) reduces to the standard diagram:

. . - b e "

^(9.70)

In this section we have developed an unusual formalism which deals with path integrals but in the end all the answers have appeared to be much more easily described by the standard Feynman graphs. How­

ever, there are serious reasons for working with geometrical integrals directly. They lie in the fact that as we go from paths to surfaces and higher dimensional objects, geometrical functional integrals remain our only existing tool. In the next section we are going to show how this tool works in the case of string theory.

Im Dokument Gauge Fields and Strings (Seite 168-180)