- #1
Trifis
- 167
- 1
Hi everyone!
I have two questions that arose during the path integral quantization of theories involving fermions.
First of all, when we prove the equivalence between the path integral formalism and the canonical quantization we make use of the eigenvalue defining equation: [itex] \hat{φ}(x)|Φ>=φ(x)|Φ> [/itex] in order to get rid of the operators and start working with numbers.
My first question is: What happens when we try to find the eigenvalue of a fermionic field? To my understanding, the eigenvalues of creation/annihilation operators for fermions, somehow, have to be decomposable into normal functions, i.e. numbers, and Grassman numbers.
And then a more general consideration came to mind: Does the anticommutation property of fermionic operators follow solely from the Lorentz group representation theory? I know that Dirac spinor part of the fermionic field transforms as the (1/2,0)⊕(0,1/2) representation of the Lorentz group but what about the creation/annihilation operators? As far as I can tell, anticommutation relations are not imposed on the spinors but on the operators! In other words, is the usage of Grassman numbers justified by the spacetime symmetry or there is some other kind of fundamental axiom?
I have two questions that arose during the path integral quantization of theories involving fermions.
First of all, when we prove the equivalence between the path integral formalism and the canonical quantization we make use of the eigenvalue defining equation: [itex] \hat{φ}(x)|Φ>=φ(x)|Φ> [/itex] in order to get rid of the operators and start working with numbers.
My first question is: What happens when we try to find the eigenvalue of a fermionic field? To my understanding, the eigenvalues of creation/annihilation operators for fermions, somehow, have to be decomposable into normal functions, i.e. numbers, and Grassman numbers.
And then a more general consideration came to mind: Does the anticommutation property of fermionic operators follow solely from the Lorentz group representation theory? I know that Dirac spinor part of the fermionic field transforms as the (1/2,0)⊕(0,1/2) representation of the Lorentz group but what about the creation/annihilation operators? As far as I can tell, anticommutation relations are not imposed on the spinors but on the operators! In other words, is the usage of Grassman numbers justified by the spacetime symmetry or there is some other kind of fundamental axiom?