Suppose given a Dirac field
ψ
{\displaystyle \psi }
representation
ρ
{\displaystyle \rho }
compact Lie group G ; and we have a background connection form of taking values in the Lie algebra
g
.
{\displaystyle {\mathfrak {g}}\,.}
Dirac operator (in Feynman slash notation ) is
D
/
=
d
e
f
∂
/
+
i
A
/
{\displaystyle D\!\!\!\!/\ {\stackrel {\mathrm {def} }{=}}\ \partial \!\!\!/+iA\!\!\!/}
and the fermionic action is given by
∫
d
d
x
ψ
¯
i
D
/
ψ
{\displaystyle \int d^{d}x\,{\overline {\psi }}iD\!\!\!\!/\psi }
The partition function is
Z
[
A
]
=
∫
D
ψ
¯
D
ψ
e
−
∫
d
d
x
ψ
¯
i
D
/
ψ
.
{\displaystyle Z[A]=\int {\mathcal {D}}{\overline {\psi }}{\mathcal {D}}\psi \,e^{-\int d^{d}x\,{\overline {\psi }}iD\!\!\!/\,\psi }.}
The axial symmetry transformation goes as
ψ
→
e
i
γ
d
+
1
α
(
x
)
ψ
{\displaystyle \psi \to e^{i\gamma _{d+1}\alpha (x)}\psi \,}
ψ
¯
→
ψ
¯
e
i
γ
d
+
1
α
(
x
)
{\displaystyle {\overline {\psi }}\to {\overline {\psi }}e^{i\gamma _{d+1}\alpha (x)}}
S
→
S
+
∫
d
d
x
α
(
x
)
∂
μ
(
ψ
¯
γ
μ
γ
d
+
1
ψ
)
{\displaystyle S\to S+\int d^{d}x\,\alpha (x)\partial _{\mu }\left({\overline {\psi }}\gamma ^{\mu }\gamma _{d+1}\psi \right)}
Classically, this implies that the chiral current,
j
d
+
1
μ
≡
ψ
¯
γ
μ
γ
d
+
1
ψ
{\displaystyle j_{d+1}^{\mu }\equiv {\overline {\psi }}\gamma ^{\mu }\gamma _{d+1}\psi }
0
=
∂
μ
j
d
+
1
μ
{\displaystyle 0=\partial _{\mu }j_{d+1}^{\mu }}
Quantum mechanically, the chiral current is not conserved: Jackiw discovered this due to the non-vanishing of a triangle diagram. Fujikawa reinterpreted this as a change in the partition function measure under a chiral transformation. To calculate a change in the measure under a chiral transformation, first consider the Dirac fermions in a basis of eigenvectors of the Dirac operator :
ψ
=
∑
i
ψ
i
a
i
,
{\displaystyle \psi =\sum \limits _{i}\psi _{i}a^{i},}
ψ
¯
=
∑
i
ψ
i
b
i
,
{\displaystyle {\overline {\psi }}=\sum \limits _{i}\psi _{i}b^{i},}
where
{
a
i
,
b
i
}
{\displaystyle \{a^{i},b^{i}\}}
Grassmann valued coefficients, and
{
ψ
i
}
{\displaystyle \{\psi _{i}\}}
Dirac operator :
D
/
ψ
i
=
−
λ
i
ψ
i
.
{\displaystyle D\!\!\!\!/\psi _{i}=-\lambda _{i}\psi _{i}.}
The eigenfunctions are taken to be orthonormal with respect to integration in d-dimensional space,
δ
i
j
=
∫
d
d
x
(
2
π
)
d
ψ
†
j
(
x
)
ψ
i
(
x
)
.
{\displaystyle \delta _{i}^{j}=\int {\frac {d^{d}x}{(2\pi )^{d}}}\psi ^{\dagger j}(x)\psi _{i}(x).}
The measure of the path integral is then defined to be:
D
ψ
D
ψ
¯
=
∏
i
d
a
i
d
b
i
{\displaystyle {\mathcal {D}}\psi {\mathcal {D}}{\overline {\psi }}=\prod \limits _{i}da^{i}db^{i}}
Under an infinitesimal chiral transformation, write
ψ
→
ψ
′
=
(
1
+
i
α
γ
d
+
1
)
ψ
=
∑
i
ψ
i
a
′
i
,
{\displaystyle \psi \to \psi ^{\prime }=(1+i\alpha \gamma _{d+1})\psi =\sum \limits _{i}\psi _{i}a^{\prime i},}
ψ
¯
→
ψ
¯
′
=
ψ
¯
(
1
+
i
α
γ
d
+
1
)
=
∑
i
ψ
i
b
′
i
.
{\displaystyle {\overline {\psi }}\to {\overline {\psi }}^{\prime }={\overline {\psi }}(1+i\alpha \gamma _{d+1})=\sum \limits _{i}\psi _{i}b^{\prime i}.}
The Jacobian of the transformation can now be calculated, using the orthonormality of the eigenvectors
C
j
i
≡
(
δ
a
δ
a
′
)
j
i
=
∫
d
d
x
ψ
†
i
(
x
)
[
1
−
i
α
(
x
)
γ
d
+
1
]
ψ
j
(
x
)
=
δ
j
i
−
i
∫
d
d
x
α
(
x
)
ψ
†
i
(
x
)
γ
d
+
1
ψ
j
(
x
)
.
{\displaystyle C_{j}^{i}\equiv \left({\frac {\delta a}{\delta a^{\prime }}}\right)_{j}^{i}=\int d^{d}x\,\psi ^{\dagger i}(x)[1-i\alpha (x)\gamma _{d+1}]\psi _{j}(x)=\delta _{j}^{i}\,-i\int d^{d}x\,\alpha (x)\psi ^{\dagger i}(x)\gamma _{d+1}\psi _{j}(x).}
The transformation of the coefficients
{
b
i
}
{\displaystyle \{b_{i}\}}
D
ψ
D
ψ
¯
=
∏
i
d
a
i
d
b
i
=
∏
i
d
a
′
i
d
b
′
i
det
−
2
(
C
j
i
)
,
{\displaystyle {\mathcal {D}}\psi {\mathcal {D}}{\overline {\psi }}=\prod \limits _{i}da^{i}db^{i}=\prod \limits _{i}da^{\prime i}db^{\prime i}{\det }^{-2}(C_{j}^{i}),}
where the Jacobian is the reciprocal of the determinant because the integration variables are Grassmannian, and the 2 appears because the a's and b's contribute equally. We can calculate the determinant by standard techniques:
det
−
2
(
C
j
i
)
=
exp
[
−
2
t
r
ln
(
δ
j
i
−
i
∫
d
d
x
α
(
x
)
ψ
†
i
(
x
)
γ
d
+
1
ψ
j
(
x
)
)
]
=
exp
[
2
i
∫
d
d
x
α
(
x
)
ψ
†
i
(
x
)
γ
d
+
1
ψ
i
(
x
)
]
{\displaystyle {\begin{aligned}{\det }^{-2}(C_{j}^{i})&=\exp \left[-2{\rm {tr}}\ln(\delta _{j}^{i}-i\int d^{d}x\,\alpha (x)\psi ^{\dagger i}(x)\gamma _{d+1}\psi _{j}(x))\right]\\&=\exp \left[2i\int d^{d}x\,\alpha (x)\psi ^{\dagger i}(x)\gamma _{d+1}\psi _{i}(x)\right]\end{aligned}}}
to first order in α(x).
Specialising to the case where α is a constant, the Jacobian must be regularised because the integral is ill-defined as written. Fujikawa employed heat-kernel regularization , such that
−
2
t
r
ln
C
j
i
=
2
i
lim
M
→
∞
α
∫
d
d
x
ψ
†
i
(
x
)
γ
d
+
1
e
−
λ
i
2
/
M
2
ψ
i
(
x
)
=
2
i
lim
M
→
∞
α
∫
d
d
x
ψ
†
i
(
x
)
γ
d
+
1
e
D
/
2
/
M
2
ψ
i
(
x
)
{\displaystyle {\begin{aligned}-2{\rm {tr}}\ln C_{j}^{i}&=2i\lim \limits _{M\to \infty }\alpha \int d^{d}x\,\psi ^{\dagger i}(x)\gamma _{d+1}e^{-\lambda _{i}^{2}/M^{2}}\psi _{i}(x)\\&=2i\lim \limits _{M\to \infty }\alpha \int d^{d}x\,\psi ^{\dagger i}(x)\gamma _{d+1}e^{{D\!\!\!/\,}^{2}/M^{2}}\psi _{i}(x)\end{aligned}}}
(
D
/
2
{\displaystyle {D\!\!\!\!/}^{2}}
D
2
+
1
4
[
γ
μ
,
γ
ν
]
F
μ
ν
{\displaystyle D^{2}+{\tfrac {1}{4}}[\gamma ^{\mu },\gamma ^{\nu }]F_{\mu \nu }}
=
2
i
lim
M
→
∞
α
∫
d
d
x
∫
d
d
k
(
2
π
)
d
∫
d
d
k
′
(
2
π
)
d
ψ
†
i
(
k
′
)
e
i
k
′
x
γ
d
+
1
e
−
k
2
/
M
2
+
1
/
(
4
M
2
)
[
γ
μ
,
γ
ν
]
F
μ
ν
e
−
i
k
x
ψ
i
(
k
)
{\displaystyle =2i\lim \limits _{M\to \infty }\alpha \int d^{d}x\int {\frac {d^{d}k}{(2\pi )^{d}}}\int {\frac {d^{d}k^{\prime }}{(2\pi )^{d}}}\psi ^{\dagger i}(k^{\prime })e^{ik^{\prime }x}\gamma _{d+1}e^{-k^{2}/M^{2}+1/(4M^{2})[\gamma ^{\mu },\gamma ^{\nu }]F_{\mu \nu }}e^{-ikx}\psi _{i}(k)}
=
−
−
2
α
(
2
π
)
d
/
2
(
d
2
)
!
(
F
)
d
/
2
,
{\displaystyle =-{\frac {-2\alpha }{(2\pi )^{d/2}({\frac {d}{2}})!}}(F)^{d/2},}
after applying the completeness relation for the eigenvectors, performing the trace over γ-matrices, and taking the limit in M. The result is expressed in terms of the field strength 2-form,
F
≡
1
2
F
μ
ν
d
x
μ
∧
d
x
ν
.
{\displaystyle F\equiv {\tfrac {1}{2}}F_{\mu \nu }\,dx^{\mu }\wedge dx^{\nu }\,.}
This result is equivalent to
(
d
2
)
t
h
{\displaystyle ({\tfrac {d}{2}})^{\rm {th}}}
Chern class of the
g
{\displaystyle {\mathfrak {g}}}
chiral anomaly , responsible for the non-conservation of the chiral current.
![{\displaystyle Z[A]=\int {\mathcal {D}}{\overline {\psi }}{\mathcal {D}}\psi \,e^{-\int d^{d}x\,{\overline {\psi }}iD\!\!\!/\,\psi }.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/39689035711b3ff0b0547c3de59c52a2251defd5)
![{\displaystyle C_{j}^{i}\equiv \left({\frac {\delta a}{\delta a^{\prime }}}\right)_{j}^{i}=\int d^{d}x\,\psi ^{\dagger i}(x)[1-i\alpha (x)\gamma _{d+1}]\psi _{j}(x)=\delta _{j}^{i}\,-i\int d^{d}x\,\alpha (x)\psi ^{\dagger i}(x)\gamma _{d+1}\psi _{j}(x).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0ff4a9b91e03ec9e3a18e5f57eaadd03b10b1ec)
![{\displaystyle {\begin{aligned}{\det }^{-2}(C_{j}^{i})&=\exp \left[-2{\rm {tr}}\ln(\delta _{j}^{i}-i\int d^{d}x\,\alpha (x)\psi ^{\dagger i}(x)\gamma _{d+1}\psi _{j}(x))\right]\\&=\exp \left[2i\int d^{d}x\,\alpha (x)\psi ^{\dagger i}(x)\gamma _{d+1}\psi _{i}(x)\right]\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/26a547accf4438f355faec9182c227ad3d642ad3)
![{\displaystyle D^{2}+{\tfrac {1}{4}}[\gamma ^{\mu },\gamma ^{\nu }]F_{\mu \nu }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3b53ceb460bc4993a4bfd9e4e103daf9e2726f7)
![{\displaystyle =2i\lim \limits _{M\to \infty }\alpha \int d^{d}x\int {\frac {d^{d}k}{(2\pi )^{d}}}\int {\frac {d^{d}k^{\prime }}{(2\pi )^{d}}}\psi ^{\dagger i}(k^{\prime })e^{ik^{\prime }x}\gamma _{d+1}e^{-k^{2}/M^{2}+1/(4M^{2})[\gamma ^{\mu },\gamma ^{\nu }]F_{\mu \nu }}e^{-ikx}\psi _{i}(k)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7950abb404ba18c112133787793ba0fe6d8c870)
