228
GAUGE FIELDS AND STRINGS
amplitude for the propagation of the ^-dimensional fermion trapped
on the path:
S!(p exp
(py.cpx, di
(9.337)
This is almost (9.335). Presumably the quartic term is needed for proper
regularization. But how come that the supersymmetric expression
(9.335), describing bosons and fermions together, is obtained from the
purely fermionic path integral? The answer to this question is simple.
When we treat the functional integral (9.337) we find that it describes a
second quantized Dirac particle, living on the path. In order to
reproduce fermionic results in (9.336) we implicitly assumed that the
occupation number for this particle is equal to one. If we remove this
constraint and integrate over all possible (^-fields, we add to this sector
another one, with the occupation number zero. In this other case we
have no particle on the path and thus describe a boson.
There are obvious gaps in our derivation, but I believe they can be
filled without damage to the general ideas outlined above. This would
be a theme for an interesting investigation.
9.12 Fermionic Strings
We shall now describe the string analogue of a Dirac particle. Just as in
the latter case we had a field
distributed on the trajectory, which
eventually played the role of the Dirac matrices
in the case of strings
we have to consider a field
living on the world sheet, which should
be a supersymmetric partner of the field x^(^. As before, it is convenient
to start from the superspace, which now must have two fermionic
directions. If we describe ^-space by complex variables z and z, each of
them has its Fermi-partner 9 and 9. The supersymmetry transformation
is given by:
Sz = —e9, ÒZ— —e9
SO = e,
SO = £
(9.338)
We see that we have a direct product of one dimensional supergroups.
The corresponding covariant derivatives take the form:
dz’
e
-e
= — = “I" 9 —
80
8z
(9.339)
GAUGE FIELDS AND STRINGS
amplitude for the propagation of the ^-dimensional fermion trapped
on the path:
S!(p exp
(py.cpx, di
(9.337)
This is almost (9.335). Presumably the quartic term is needed for proper
regularization. But how come that the supersymmetric expression
(9.335), describing bosons and fermions together, is obtained from the
purely fermionic path integral? The answer to this question is simple.
When we treat the functional integral (9.337) we find that it describes a
second quantized Dirac particle, living on the path. In order to
reproduce fermionic results in (9.336) we implicitly assumed that the
occupation number for this particle is equal to one. If we remove this
constraint and integrate over all possible (^-fields, we add to this sector
another one, with the occupation number zero. In this other case we
have no particle on the path and thus describe a boson.
There are obvious gaps in our derivation, but I believe they can be
filled without damage to the general ideas outlined above. This would
be a theme for an interesting investigation.
9.12 Fermionic Strings
We shall now describe the string analogue of a Dirac particle. Just as in
the latter case we had a field
distributed on the trajectory, which
eventually played the role of the Dirac matrices
in the case of strings
we have to consider a field
living on the world sheet, which should
be a supersymmetric partner of the field x^(^. As before, it is convenient
to start from the superspace, which now must have two fermionic
directions. If we describe ^-space by complex variables z and z, each of
them has its Fermi-partner 9 and 9. The supersymmetry transformation
is given by:
Sz = —e9, ÒZ— —e9
SO = e,
SO = £
(9.338)
We see that we have a direct product of one dimensional supergroups.
The corresponding covariant derivatives take the form:
dz’
e
-e
= — = “I" 9 —
80
8z
(9.339)