Fermionic Fock space

The fermionic Fock space of $n$ modes is the exterior algebra ${\Lambda }^{\bullet }{\mathbb{C}}^n$. Creation acts by exterior multiplication and annihilation by contraction.

An orthogonal projection $P$ splits a one-particle Hilbert space into empty and filled modes. Take completed exterior Fock spaces of empty particles and of holes in the filled modes, with the graded tensor product enforcing the canonical anticommutation relations between both factors. The vacuum is killed by particle annihilation in $PH$ and by particle creation in $(I-P)H$. Only finite particle-hole excitations are needed for a dense basis. Normal ordering subtracts the filled-vacuum expectation from bilinear observables.

In charge $q$, let ${\Omega }_q$ fill all integer modes below $q$. Its rotation energy is $q(q-1)/2$. The negative Heisenberg current algebra of a fermion on the circle modes create mutually commuting oscillators $b_n^{\ast }=J_{-n}/\sqrt{n}$ with $[b_n,b_m^{\ast }]={\delta }_{nm}$. Their normalized occupation monomials applied to ${\Omega }_q$ are orthonormal. The Jacobi triple product identifies the fermionic charge-energy generating function with ${\sum }_q{u}^q{t}^{q(q-1)/2}{\prod }_{n>0}(1-t^n{)}^{-1}$, proving completeness in each finite-dimensional energy space. Thus the rotation Hamiltonian is $Q(Q-1)/2+{\sum }_{n>0}n{b}_n^{\ast }{b}_n$.

For integer modes filled at indices below zero, set $E_{rs}=c_r^{†}{c}_s-{\delta }_{rs}{1}_{r<0}$ and $J_n={\sum }_r{E}_{r,r+n}$ on the finite-energy domain. The canonical anticommutation relations give a central term ${\delta }_{st}{\delta }_{ur}(1_{r<0}-1_{s<0})$ in $[E_{rs},E_{tu}]$. Summing across the filled-empty boundary gives $[J_n,J_m]=n{\delta }_{n+m,0}I$, with $J_n^{\ast }=J_{-n}$. The current's level is the coefficient of this central term.

A Fourier mode of the normal-ordered fermion density is a fermionic current. On the finite-energy domain its defining sum has finitely many nonzero actions on each occupation vector. The modes satisfy $J_n^{\ast }=J_{-n}$ and $[J_n,J_m]=n{\delta }_{n+m,0}I$. The zero mode is charge; positive modes annihilate charge ground states and negative modes create the oscillators in the fermion-boson correspondence on the circle.

For the irreducible representation of the canonical anticommutation relations determined by $P$, a unitary one-particle map $u$ is unitarily implementable precisely when $P-uP{u}^{\ast }$ is Hilbert-Schmidt, equivalently when $[P,u]$ is Hilbert-Schmidt. The implementer is unique up to a scalar phase. The equivalent representation criterion for two polarizations is that their projections differ by a Hilbert-Schmidt operator. This yields a continuous projective unitary representation of the restricted unitary group.

A fermionic operator is odd under fermion parity: if $P=(-1{)}^N$ is the parity operator, then $PAP=-A$. A fermionic creation operator or fermionic annihilation operator is an example. Interchanging odd insertions in a time-ordered product contributes the corresponding fermionic sign. Odd parity alone does not imply that every pair of such operators has zero anticommutator.

A Clifford vacuum is a state annihilated by all the chosen fermionic annihilation operators. In a massless supermultiplet it labels the starting helicity, and applying fermionic raising and lowering operators constructs the remaining states.

Fermionic creation and annihilation operators satisfy $\{a_i,a_j^{†}\}={\delta }_{ij}$ and vanishing equal-type anticommutators. These relations make each mode either empty or singly occupied.

For constant internal matrices acting on Dirac fields, the equal-time canonical anticommutation relations givewith the corresponding spatial delta functions for fields. The quartic terms cancel after fermion reordering. Integrating proves the displayed algebra for well-defined global Noether charges, with normal ordering and a regulator that preserves the nonanomalous internal symmetry. Orthogonal chiral projectors make the left/right mixed commutators zero. This reduces current-charge closure to a finite matrix calculation.

For normalized triplets $(E^{+},n,e{)}_L$ and $(E^{+},N,e{)}_R$, let the raising matrix have ones in entries $(1,2)$ and $(2,3)$. Its commutator with its adjoint is $\mathrm{diag}(1,0,-1)$. Neutral mixing in $n=c_{\alpha }\nu +s_{\alpha }N$ leaves this result unchanged because $c_{\alpha }^2+s_{\alpha }^2=1$. With current normalization using $1\mp {\gamma }^5=2P_{L,R}$, fermionic bilinear charge algebra gives the displayed proportionality to the leptonic electric charge $Q_{\ell }=e\int :(E^{+†}{E}^{+}-e^{†}e):$. Other current normalizations change the numerical prefactor, not closure. Equal-time algebra does not itself imply conservation of a fermionic weak charge in a symmetry-broken theory.

For mode normalization $1/\sqrt{2E_{\mathbf{p}}}$, particle and antiparticle operators satisfy $\{a_{\mathbf{p}}^s,a_{\mathbf{q}}^{r†}\}=\{b_{\mathbf{p}}^s,b_{\mathbf{q}}^{r†}\}=(2\pi {)}^3{\delta }^{sr}{\delta }^3(\mathbf{p}-\mathbf{q})$. The Dirac spinor completeness relations imply ${\sum }_s[u_s(\mathbf{p})u_s^{†}(\mathbf{p})+v_s(-\mathbf{p})v_s^{†}(-\mathbf{p})]=2E_{\mathbf{p}}I$. Inserting this into the mode expansion of a Dirac field gives the displayed local anticommutator. The reversed momentum in the antiparticle completeness term is essential.

The fermion number operator counts occupied fermionic modes. Its parity $(-1{)}^F$ is positive on even exterior degree and negative on odd exterior degree.

A supertrace is the trace weighted by fermion parity. Supersymmetry makes paired positive-energy states cancel in a supertrace.

An equivariant supertrace inserts a symmetry $f$ as well as fermion parity, $\mathrm{Tr}((-1{)}^{F}fA)$. In a path integral, $f$ produces boundary conditions twisted by its action.