37.5 Fermion fields

37.5 Fermion fields

Most everything discussed in this chapter and in chapter 36 applies with little change to the case of fermionic quantum fields using fermionic instead of bosonic oscillators, and changing commutators to anticommutators for the annihilation and creation operator. This gives fermionic fields that satisfy anticommutation relations

$$[\hat{\Psi }(x),{\hat{\Psi }}^{†}(x^{\prime }){]}_{+}=\delta (x-x^{\prime })$$

and states that in the occupation number representation have $n_p=0,$ 1, while also having a description in terms of antisymmetric tensor products, or polynomials in anticommuting coordinates. Field operators will in this case generate an infinite dimensional Cliford algebra. Elements of this Cliford algebra act on states by an infinite dimensional version of the construction of spinors in terms of fermionic oscillators described in chapter 31.

For applications to physical systems in three dimensional space, it is often the fermionic version that is relevant, with the systems of interest for instance describing arbitrary numbers of electrons, which are fermionic particles so need to be described by anticommuting fields. The quantum field theory of nonrelativistic free electrons is the quantum theory one gets by taking as singleparticle phase space ${\mathcal{H}}_1$ the space of solutions of the two-component Pauli-Schr¨odinger equation 34.3 described in section 34.2 and then quantizing using the fermionic version of Bargmann-Fock quantization. The fermionic Poisson bracket is determined by the inner product on this ${\mathcal{H}}_1$ discussed in section 34.3.

More explicitly, this is a theory of two quantum fields ${\hat{\Psi }}_1(\mathbf{x}),{\hat{\Psi }}_2(\mathbf{x})$ satisfying the anticommutation relations

$$[{\hat{\Psi }}_j(\mathbf{x}),{\hat{\Psi }}_k^{†}({\mathbf{x}}^{\prime }){]}_{+}={\delta }_{jk}{\delta }^3(\mathbf{x}-{\mathbf{x}}^{\prime })$$

These are related by Fourier transform

$${\hat{\Psi }}_j(\mathbf{x})=\frac{1}{(2\pi {)}^{\frac{3}{2}}}{\int }_{{\mathbf{R}}^3}{e}^{i\mathbf{p}\cdot \mathbf{x}}{a}_j(\mathbf{p})d^3\mathbf{p}$$ $${\hat{\Psi }}_j^{†}(\mathbf{x})=\frac{1}{(2\pi {)}^{\frac{3}{2}}}{\int }_{{\mathbf{R}}^3}{e}^{-i\mathbf{p}\cdot \mathbf{x}}{a}_j^{†}(\mathbf{p})d^3\mathbf{p}$$

to annihilation and creation operators satisfying

$$[a_j(\mathbf{p}),a_k^{†}({\mathbf{p}}^{\prime }){]}_{+}={\delta }_{jk}{\delta }^3(\mathbf{p}-{\mathbf{p}}^{\prime })$$

The theory of non-relativistic electrons is something diferent than simply two copies of a single fermionic field, since it describes spin $\frac{1}{2}$ particles, not pairs of spin 0 particles. In section 38.3.3 we will see how the group $\stackrel{\sim }{E}(3)$ acts on the theory, giving angular momentum observables corresponding to spin $\begin{array}{l}\frac{1}{2}\end{array}$ rather than two copies of spin 0.

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