ANALOGIES BETWEEN GAUGE AND CHIRAL FIELDS
W ith the a id o f th ese fo rm u la s w e o b ta in :
¿‘F(C) ^
P F„„(x(s)) exp
¿n
ds
117
(7.25)
where
is the Yang-Mills field strength. We see that the continuum
counterpart of the quantity (7.20) is given by:
c) = / y - > - H C )
SxJs)
= X J P exp j di F^,(x(s)) P expl I x ,d t (7.26)
By its definition it satisfies:
x^(sW^(s, 0 = 0
C) S^Js, C)
'^v(^', C)] = 0
dx^(s )
dx^{s )
(7.27)
The second relation expresses the fact that a gauge field having nonzero
field-strength in ordinary space, in the loop space has zero curvature
being thus a chiral field.
If we take a functional divergence of (7.26) we obtain:
C)
, /
A^x„ di
X
+ [A^, J ) P expl - j A^x, x„dt X,
(7.28)
From this relation we conclude that the Yang-Mills equations in terms
of phase factors have the form:
S r
A
T ' M = 0
(7.29)
3x^(s) \Sx^(s)
To be sure (7.29) is just a continuum version of the lattice equations.
(7.18) and (7.21).
These formulas show a remarkable correspondence with equations of
motion for chiral fields. Indeed, on the lattice the action
s = - -*2 Z T^idx'yx+i)
^0 X.8
(7.30)