I'm interested to know under what conditions γμDμ is a hermitian operator.
I am studying the Fujikawa method of anomalies and I see that many sources have different answers for this. Some claim that γμDμ is hermitian or possibly anti hermitian in Euclidean space, or that iγμDμ is hermitian in Minkowski space. I need the hermiticiy condition so that I can expand the Dirac spinors Ψ(x) and ˉΨ(x) in a basis of orthonormal eigenvectors of the Dirac operator to perform the path integral.
Another question I have is this expansion. I would like something of the form
Ψ(x)=∑nψn(x)an
ˉΨ(x)=∑n¯ψn(x)ˉbn
where an and ˉbn are elements of a Grassmann algebra. But how is ˉψn defined? Some references define ˉψn(x) = ψ(x)†, but I guess it depends on how you are defining your inner product.
Thanks, I've attached some references below.
Fujikawa 1) https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.42.1195
Fujikawa 2) https://journals.aps.org/prd/abstract/10.1103/PhysRevD.21.2848
TASI Anomalies https://arxiv.org/abs/hep-th/0509097
Answer
The (operator-valued) matrix D̸=γ⋅D is anti-hermitian with respect to the inner product ⟨v,u⟩:=∫Rdˉv(x)u(x) dx
Indeed, write D̸=⧸∂−iA̸
It is clear that the second factor satisfies ¯iA̸=−i¯A̸=−iA̸
On the other hand, the first factor in (3) satisfies ⟨v,⧸∂u⟩=∫Rdˉv(x)⧸∂u(x) dx=−∫Rdˉv(x)←⧸∂u(x) dx=−∫Rd¯(¯⧸∂v)(x)u(x) dx=−⟨⧸∂v,u⟩
From this we learn that D̸ satisfies ¯D̸=−D̸, which is equivalent to the statement that D̸ is anti-hermitian with respect to ⟨⋅,⋅⟩, as claimed.
Note that this notion of anti-hermiticity guarantees that iD̸ is diagonalisable, with real eigenvalues and orthogonal eigenvectors, by the usual reasons (spectral theorem, etc.): iD̸vi=λivi⇒{λi∈R⟨vi,vj⟩∝δij
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