38.4 Fermionic fields

38.4 Fermionic fields

Concept links · terms present in this machine draft; source roles are unverified: complex inner product · orthogonal group · unitary group · Lie algebra · Lie algebra representation · unitary representation · symmetry group · group action

It is an experimentally observed fact that elementary particles with spin $\frac{1}{2}$ behave as fermions and are described by fermionic fields. In non-relativistic quantum field theory, such spin $\frac{1}{2}$ elementary particles could in principle be bosons, described by bosonic fields as in section 38.3.3. There is however a “spin-statistics theorem” in relativistic quantum field theory that says that spin $\begin{array}{l}\frac{1}{2}\end{array}$ fields must be quantized with anticommutators. This provides an explanation of the observed correlation of values of the spin and of the particle statistics, due to the fact that the non-relativistic theories describing fundamental particles should be low-energy limits of relativistic theories.

The discussion of the symplectic and unitary group actions on ${\mathcal{H}}_1$ of section 38.1 has a straightforward analog in the case of a single-particle state space ${\mathcal{H}}_1$ with Hermitian inner product describing fermions, rather than bosons. The analog of the infinite dimensional symplectic group action (preserving the imaginary part of the Hermitian inner product) of the bosonic case is an infinite dimensional orthogonal group action (preserving the real part of the Hermitian inner product) in the fermionic case. The multi-particle state space will be an infinite dimensional version of the spinor representation for this orthogonal group. As in the bosonic case, there will be an infinite dimensional unitary group preserving the full Hermitian inner product, and the groups of symmetries we will be interested in will be subgroups of this group.

In section 31.3 we saw in finite dimensions how unitary group actions on a fermionic phase space gave a unitary representation on the fermionic oscillator state space, by the same method of annihilation and creation operators as in the bosonic case (changing commutators to anticommutators). Applying this to the infinite dimensional case of the single-particle space ${\mathcal{H}}_1$ of solutions to the free particle Schr¨odinger equation is done by taking

$$\begin{array}{rl}{\theta }_j& \to \Psi (x)\text{or}A(p)\\ {\overline{\theta }}_j& \to \overline{\Psi }(x)\text{or}\overline{A}(p)\\ \{{\theta }_j,{\overline{\theta }}_k{\}}_{+}& ={\delta }_{jk}\to \{\Psi (x),\overline{\Psi }(x^{\prime }){\}}_{+}=\delta (x-x^{\prime })\text{or}\{A(p),\overline{A}(p^{\prime }){\}}_{+}=\delta (p-p^{\prime })\end{array}$$

Quantization generalizes the construction of the spinor representation from sections 31.3 and 31.4 to the ${\mathcal{H}}_1$ case, taking

$$\begin{array}{rl}{a}_{Fj}& \to \hat{\Psi }(x)\text{or}a(p)\\ a_{Fj}^{†}& \to {\hat{\Psi }}^{†}(x)\text{or}{a}^{†}(p)\\ {}_{+}& ={\delta }_{jk}\to [\hat{\Psi }(x),{\hat{\Psi }}^{†}(x^{\prime }){]}_{+}=\delta (x-x^{\prime })\text{or}[a(p),a^{†}(p^{\prime }){]}_{+}=\delta (p-p^{\prime })\end{array}$$

Quadratic combinations of the ${\theta }_j,{\overline{\theta }}_j$ give the Lie algebra of orthogonal transformations of the phase space $M$. We will again be interested in the generalization to $M={\mathcal{H}}_1$ , but for very specific quadratic combinations, corresponding to certain finite dimensional Lie algebras of unitary transformations of ${\mathcal{H}}_1$ . Quantization will take these to quadratic combinations of fermionic field operators, giving a Lie algebra representation on the fermionic state space. We get the same formulas for operators $\hat{N}$ (equation 38.3), $\hat{X}$ (equation 38.5), $\hat{\mathbf{P}}$ (equation 38.11), Lb (equation 38.12) and $\hat{\mathbf{S}}$ (equation 38.14) , but with anticommuting field operators. These give unitary representations on the multi-particle state space of the Lie algebras of $\mathcal{U}(1),\tilde{\mathcal{U}}(n),{\mathbf{R}}^3$ translations and SO(3) rotations respectively. For the free particle, these operators commute with the Hamiltonian and act as symmetries on the state space.

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