Analogies Between Gauge and Chiral Fields. Loop Dynamics
7.2 Quantum Theory of Loops
It appears more practical to use a set of equations slightly different from (7.29). It exploits the fact that, according to the transformation law (7.2), only (p(C) = Tr 'F(C) is a gauge-invariant quantity. Therefore the only quantities which make sense in quantum theory are:
M^(C)=<(P(C)>; W2(C,C) = W i C M C ) y
etc. Using formulas (7.24), (7.25) we derive:
Sx/s)Sx/s') ^ j dx'‘j^x,(s)x,(s')
+ ¿(s - s')x,(s){^Tr P^V^f^,(x(s)) exp^ dx"
Let us now introduce a local derivative;
(7.33)
(7.34)
def,.
= lim d f
5x„(s + tl2)6xJs - t/2) (7.35)
which picks up ¿-like terms in the second functional derivative. If the quantum theory is regularized, then the first term in (7.34) does not contain singularities as s ^ s'. From this we infer that:
= ( r r ( ^ P i V , F , M s ) ) exp(^ j A , x , (7.36)
For classical fields the r.h.s. of (7.36) would be zero. In the quantum case it is finite and calculable. To find it we consider the following (non
invariant) functional integral:
tr A, dx^ (7.37)
and change variables by A öA^. There are two effects coming from this change, which must cancel each other. The first is the variation of the action, proportional to while the second is the variation of the phase factor. The cancellation condition gives the identities:
- ^ V ,F ; ,( z m x , x ) ) = ( \ ) dy,^(z - yX'Fix, y)A^'F(y, x)>
0
1 \ % (7.38)
p Tr(V,F,,(z)'F(x, x)) \ = (j) d y ^ z - y)<Tr X^^(x, x)>
c
Here the index a labels the generators A" of our Lie algebra and y) is a piece of the phase factor for the part of the contour C connecting x and y(x, yeC). For the SU(N) group, (7.38) is further simplified through the use of the identity:
Z Kß^ö — ^OLÖ^ßy ~
1
^Oiß^ (7.39)We have:
- ^ T r ( V ,F ,,( z ) 'F ( x ,x ) )
^0
= d) - y)
X <Tr >P(x, y) Tr x)> - - <Tr '¥(x, y m y , x)> [. (7.40)
ANALOGIES BETWEEN GAUGE AND CHIRAL FIELDS
Substitution of (7.40) into (7.36) gives
121
dx^{s)
= - el S(x(s) - W(C)W(C) - ~ W ( Q C) 1 dy^ (7.41)
which is the first equation in the Schwinger-like chain.
This equation requires some explanations. First of all, C, C and C are defined as following. The loop starts and terminates at the point X = x(0). If this loop does not have a self-intersection at the point x{s) then the r.h.s. of (7.41) will be zero because of the ¿-function. If it does, then the point y splits the loop onto two closed contours C and C:
CO
(7.42)A very important point, not to be forgotten, is that eq. (7.41) was derived in an unrenormalized but regularized version of the theory. So the ¿-function in (7.41) must be somehow smeared, and £, entering in (7.35) must be taken much less than the smearing length. The equation itself corresponds to a particular cut-off of the gauge theory. It may be untrue for a different cut-off. All this is quite unpleasant. It would be much nicer to have an equation for the finite, renormalized W{C).
Unfortunately this equation is not known, and we are unable at present to remove the scaffolding (the regularization) from our con
struction.
Nevertheless, equation (7.41) is meaningful. It reproduces perturba
tion theory and in the large N limit, presents a closed equation which sums up all planar Feynman diagrams (see the next chapter). Here we shall show how the first order of perturbation theory for W(C) arises from equation (7.41).
For this, let us consider the following ansatz for W(C):
m c ) = 1 + ( t f r ,,( x i, x^) dxï dxl
-f ^2» ^3) dxï dx^ dx^ + (7.43)
This ansatz is true for any W{C) which can be represented as an average of the loop factor (7.1), with:
• (^1. X2, X 3 , ...) = <Tr A^(x^)AXx2)A^{x^). . .> (7.44)
In fact, it can be shown that this ansatz is true for more or less any functional W{C\ being an analogue of the Taylor expansion in loop space. To see how the perturbation theory arises let us find d^W2{C)/dx\s) where W2 is the second term in (7.43). We have:
6W f
= J
ds, ds2x / s2){x ^ (s ,)a ,„ r;i/ x ,(s ,), X jiS jM s - s,) + - s)r2p(x(Si), x(S2»}= J d S jX ^ iS jX ^ i.^ r ^ p iX iC s ), X2(S2))
- 5 i ,A r ,,( x ,{ s ) ,x ( s 2 ) ) (7.45)
Next we have to take the second derivative <5/<5x^(s') and to pick out the terms containing 6{s — s'). That gives:
3x^(s)Sx^(s') - s' )Us)x, (s2) ds2 (7.46)
-h terms, which do not contain S(s — s').
Thus:
d^W2
dx^(s) = j e ? r 2 , ( x ( s ) , X ( 0 ) - 5 , , 2 5 i . , r , , ( x ( s ) , X ( 0 ) X 2 ( S ) X , ( 0 di (7.47)
We must substitute this result into the r.h.s. of (7.41) replacing W(C) in the l.h.s. by 1. This gives the equation for T^^:
x') - 5i,2^i,„r,p(x, x')
= ^ x') (7.48)
Here </> is an arbitrary function which is needed to make (7.48) solvable, and which it is possible to add, since
The solution of (7.48) has the form:
^0 ^Ap aiip(x, x')
5x, (7.49)
where is arbitrary. This arbitrariness does not appear in H^2 ( 0 itself which has the form:
^0 dx^ dx;
e
(7.50)The formula (7.50) is precisely the first nontrivial perturbative contribu
tion to W(C).
In order to obtain higher orders this contribution has to be substi
tuted into the r.h.s. of (7.41). After some complicated combinatorics (which can be found in A. A. Migdal (1977)) one finds, order by order, the standard Feynman diagrams contributing to W(C). It is remarkable that no a priori gauge fixing is needed, since (7.41) is an equation for gauge invariant quantities. Ghost diagrams in the higher orders appear automatically in the process of iteration (A. A. Migdal (1977)).
So we conclude that in spite of some dubious operations performed in deriving (7.41), namely separating from the complete derivative S^/Sx^(s)dx^(s') the part containing ¿(s — s') only, which we called d^jdx^{s\ we did not lose any information. It appears that in the frame of the general ansatz (7.43), knowledge of d^!dx^{s) is sufficient for the reconstruction of W{C). Again, we have to warn the reader that this is true only in an unrenormalized but regularized version of the theory in which the coefficients in (7.43) do not have singularities at coincident points. Renormalized quantities, while being finite, are singular in these cases and that makes our definition of 5^/5x^(s) inoperative. I believe that there should exist some kind of renormalized equations but they have not yet been found.
What is the use of loop equations, like (7.41)? Their main purpose is to provide us with a description of gauge fields in terms of their natural elementary excitations. We have seen in Chapter 3, that in the confine
ment phase those excitations are closed strings. In general these strings interact with each other. In some cases, in particular in the large N limit, they must become free, as will be shown in Chapter 8.
The loop equation (7.41) is aimed at choosing from among possible free string theories the one describing or being described by the gauge fields. This task does not yet have a final solution, although consider
able progress has been achieved.
Our next step will be to consider the large N approximation and to prove that in this limit particles in chiral theories and strings in gauge theories become free.