Thursday, 13 October 2016

quantum field theory - Hermiticity of Dirac Operator gammamuDmu and Expansion in eigenmodes


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 =γD is anti-hermitian with respect to the inner product v,u:=Rdˉv(x)u(x) dx

where ˉv(x):=v(x)γ0


Indeed, write =i


It is clear that the second factor satisfies ¯i=i¯=i

where we have used the fact that γμ satisfies γ0γμγ0=γμ
and that Aμ is hermitian.


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

where in the first equality we integrated by parts and in the last one we used (5).


From this we learn that satisfies ¯=, which is equivalent to the statement that is anti-hermitian with respect to ,, as claimed.



Note that this notion of anti-hermiticity guarantees that i is diagonalisable, with real eigenvalues and orthogonal eigenvectors, by the usual reasons (spectral theorem, etc.): ivi=λivi{λiRvi,vjδij


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