I'm trying to define a Particle-Hole symmetry operator in the CAR Algebra in a general way.I am finding very confusing to understand weather it should be treated as a linear or antilinear operator when acting on the CAR algebra.

In particular I would like to define the operation consistently on an element$\psi^\dagger(f)$ and$\psi(f)$, where f is an element on the Hilbert space, for then applying it to the usual definition one uses in BdG formalism for topological superconductors.

In my mind there are two ways to define it properly through an homomorphism$P$, either:$P[\psi^\dagger(f)]=\psi(U_Pf)$ or$P[\psi^\dagger(f)]=\psi(U_PKf)$ . Where$U_P$ is a unitary operator and$K$ is some complex conjugate operator on the Hilbert space.From what I understand then one would obtain, respectively, an antilinear operator on the Fock space (without K) and a linear operator in the Fock space (with K).(This is because$\psi(af)=a^*\psi(f)$ and$\psi^\dagger(af)=a\psi^\dagger(f)$)

What should be the correct choice for the definition? Am I doing something else wrong?

Thank you for the help.

• 1
  $\begingroup$The C operation is antilinear in the single-particle (wavefunction) space and linear on the many-particle Fock space. Seephysics.stackexchange.com/q/273626$\endgroup$

  mike stone

  – mike stone

  2025-10-18 12:12:30 +00:00

  CommentedOct 18 at 12:12

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