giving a Lie algebra representation on the fermionic state space. We get the
same formulas for operatorsN̂(equation 38.3),X̂(equation 38.5),P̂(equation
38.11),̂L(equation 38.12) andŜ(equation 38.14) , but with anticommuting
field operators. These give unitary representations on the multi-particle state
space of the Lie algebras ofU(1),U(n),R^3 translations andSO(3) rotations re-
spectively. For the free particle, these operators commute with the Hamiltonian
and act as symmetries on the state space.
38.5 For further reading
The material of this chapter is often developed in conventional quantum field
theory texts in the context of relativistic rather than non-relativistic quantum
field theory. Symmetry generators are also more often derived via Lagrangian
methods (Noether’s theorem) rather than the Hamiltonian methods used here.
For an example of a detailed physics textbook discussion relatively close to this
one, getting quadratic operators based on group actions on the space of solutions
to field equations, see [35].