JHEP05(2018)138
Let us consider a free Dirac fermion ψ and view the equations of motion
S
0
=
Z
¯
ψ(iγ · ∂
x
)ψ =⇒ iγ
a
∂
a
ψ = 0 (2.4)
as a physical state condition in the underlying quantum constrained system. The con-
straints are first class and read
− ∂
x
· ∂
x
, iγ · ∂
x
, [iγ · ∂
x
, iγ · ∂
x
] = −2∂
x
· ∂
x
, (2.5)
where the brackets denote super-commutator, with the Grassmann degree being the stan-
dard Z
2
-grading in the Clifford algebra of γ-matrices, so that in the above relations it is the
anti-commutator. The constraints are operators acting in the space of “wave functions”
with values in the Dirac spinor representation. Here and below we exploit the analogy
between free fields and first-quantized particle models.
Now suppose we would like to add at least infinitesimal sources for all operators (2.1),
i.e. couplings of the type
S
1
=
X
s,q
Z
J
a
1
...a
s
,c
1
...c
q
(h
a
1
...a
s
,c
1
...c
q
+ ∂
a
1
ξ
a
2
...a
s
,c
1
...c
q
+ perm − traces) , (2.6)
where we also indicated the desired gauge invariance. Here, the sources h
s,q
have the same
algebraic properties as J
s,q
, i.e. they are traceless and have the Young symmetry depicted
in (2.1). Since most of J
s,q
are conserved tensors (save for (2.2)), the sources h
s,q
enjoy a
gauge symmetry.
At this stage the sources in (2.6) are infinitesimal. If they are not, the currents J
s,q
are not conserved: the invariance of the effective action W [h] can be restored upon adding
appropriate counterterms
¯
ψ(h . . . h)ψ to S
1
and correcting the gauge symmetry δh by non-
linear terms. This also results in nonlinear deformation of the conservation condition for
the currents. As we explained in the Introduction, an elegant way to introduce sources
so that the total action S
0
+ S
1
remains gauge invariant even for non-infinitesimal sources
was proposed in [26] in the case of scalar field (see also [29] for a recent discussion of
curved backgrounds and nonlinear gauge symmetries). Now we are going to apply the
same strategy to the Dirac fermion.
The crucial step is to realize that all derivatives of
¯
ψ that are hidden in J
s,q
can be
integrated by parts and the coupling can be written in the following suggestive form
S
1
= i
Z
¯
ψ f(γ, ∂
x
|x)ψ , (2.7)
where f is a function of γ and ∂, whose Taylor coefficient can depend on x:
f(γ, ∂
x
|x) =
X
s,q
f
a
1
...a
s
|c
1
...c
q
(x)γ
c
1
...c
q
∂
a
1
. . . .∂
a
s
. (2.8)
Here f
s,q
are not irreducible tensors anymore. These coefficients are related to the source
h
s,q
, but it makes sense not to work at this stage with the operators that are quasi-
primary and to extend the base of local bilinear operators to the one covered by generating
– 6 –
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