224
GAUGE FIELDS AND STRINGS
We can avoid the nonlocal form (9.322a) at the price of introducing an
extra field
with which (9.322a) can be replaced by
L =
^(t) di
^5 di +
di
(9.322b)
(biAs = -a )
Integrating out ij/^ returns us to (9.322a).
Now, the functional integral for a supersymmetric path isf
^ exp
(9.323)
For a free particle, which we are considering, this integral is easily
computed, using the following gauge fixing:
¿ = 0, e = r, x = o, x = ^
(9.324)
(where T and 6 should be integrated over, since gauge freedom is not
sufficient to eliminate them completely). For an open path from (x^, ij/^)
to (xj,, ij/^) (the quantity must not change since it satisfies a first order
differential equation in contrast with x^) we have in the momentum
representation (changing (1/T)x^ -► ip^):
G(p) =
■ i
(9.325)
If we change
-► and ^As 7 s we obtain the propagator for Dirac
particles.
The formulas for closed loops, similar to (9.65), also can be written.
We shall not give a detailed derivation here. The answer replacing
t This form was found by Brink, Di Vecchia, How (1975) and anticipated by Bezezin
and Mazinov (1974).