Fermionic Fock space

The fermionic Fock space of

n

modes is the exterior algebra

Λ^{∙} C^{n}

. Creation acts by exterior multiplication and annihilation by contraction.

Table of contents

Polarized fermionic Fock space ( $F_{P} = Λ (P H) \otimes Λ (\overline{(I - P) H})$ )

 0  0

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

P H

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.

Fermion-boson correspondence on the circle ( $F_{P} ≅ ⨁_{q \in Z} F_{boson}$ )

 0  0

In charge

q

, let

Ω_{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}^{*} = J_{- n} / n

with

[b_{n}, b_{m}^{*}] = δ_{nm}

. Their normalized occupation monomials applied to

Ω_{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}^{*} b_{n}

Heisenberg current algebra of a fermion on the circle ( $[J_{n}, J_{m}] = n δ_{n + m, 0} I$ )

 0  0

For integer modes filled at indices below zero, set

E_{rs} = c_{r}^{†} c_{s} - δ_{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

δ_{s t} δ_{u r} (1_{r < 0} - 1_{s < 0})

[E_{rs}, E_{t u}]

. Summing across the filled-empty boundary gives

[J_{n}, J_{m}] = n δ_{n + m, 0} I

, with

J_{n}^{*} = J_{- n}

. The current's level is the coefficient of this central term.

Fermionic current on the circle ( $J_{n} = \sum_{r} : c_{r}^{*} c_{r + n} :$ )

 0  0

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}^{*} = J_{- n}

and

[J_{n}, J_{m}] = n δ_{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.

Segal quantization criterion for fermions ( $u implementable ⟺ [P, u] \in S_{2}$ )

 0  0

For the irreducible representation of the canonical anticommutation relations determined by

P

, a unitary one-particle map

u

is unitarily implementable precisely when

P - u P u^{*}

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.

Fermionic operator ( $P A P = - A$ )

 0  0

A fermionic operator is odd under fermion parity: if

P = (- 1)^{N}

is the parity operator, then

P A P = - 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.

Clifford vacuum ( $∣ λ ⟩$ )

 0  0

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.

Canonical anticommutation relations (CAR)

 0  0

Fermionic creation and annihilation operators satisfy

{a_{i}, a_{j}^{†}} = δ_{ij}

and vanishing equal-type anticommutators. These relations make each mode either empty or singly occupied.

Fermionic bilinear charge algebra ( $[Q_{A}, Q_{B}] = Q_{[A, B]}, Q_{A} = \int d^{3} x : ψ^{†} A ψ :$ )

 0  0

For constant internal matrices acting on Dirac fields, the equal-time canonical anticommutation relations give

[ψ_{i}^{†} ψ_{j}, ψ_{k}^{†} ψ_{l}] = δ_{jk} ψ_{i}^{†} ψ_{l} - δ_{i l} ψ_{k}^{†} ψ_{j}

(1)

with 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.

Local fermion density commutator with the Dirac Hamiltonian ( $[ψ^{†} ψ, H] = - i \nabla \cdot (\overset{ˉ}{ψ} γ ψ)$ )

 0  0

The equal-time canonical anticommutation relations imply

[n (x), ψ_{b}^{†} (y) ψ_{c} (z)] = [δ^{(3)} (x - y) - δ^{(3)} (x - z)] ψ_{b}^{†} (y) ψ_{c} (z), n = ψ^{†} ψ .

(1)

Apply

- i α_{b c}^{k} \partial_{z^{k}} + m β_{b c}

, set

z = y

, and integrate over

y

. The mass term cancels, leaving

i \int d^{3} y ψ^{†} α^{k} ψ \partial_{y^{k}} δ^{(3)} (x - y)

. Integration by parts proves the title formula as a distribution identity. The Heisenberg equation of motion then yields

\partial_{t} n = - \nabla \cdot j

for the physical Dirac current

j^{k} = \overset{ˉ}{ψ} γ^{k} ψ

. The factor

- i

is required because the commutator of Hermitian operators is anti-Hermitian.

Triplet charged-current closure on electromagnetic charge ( $[T^{+}, T^{-}] = 4 Q_{ℓ} / e$ )

 0  0

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

diag (1, 0, - 1)

. Neutral mixing in

n = c_{α} ν + s_{α} N

leaves this result unchanged because

c_{α}^{2} + s_{α}^{2} = 1

. With current normalization using

1 \mp γ^{5} = 2 P_{L, R}

, fermionic bilinear charge algebra gives the displayed proportionality to the leptonic electric charge

Q_{ℓ} = 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.

Equal-time canonical anticommutator of a Dirac field ( ${ψ_{α} (t, x), ψ_{β}^{†} (t, y)} = δ_{α β} δ^{3} (x - y)$ )

 0  0

For mode normalization

1/ 2 E_{p}

, particle and antiparticle operators satisfy

{a_{p}^{s}, a_{q}^{r †}} = {b_{p}^{s}, b_{q}^{r †}} = (2 π)^{3} δ^{sr} δ^{3} (p - q)

. The Dirac spinor completeness relations imply

\sum_{s} [u_{s} (p) u_{s}^{†} (p) + v_{s} (- p) v_{s}^{†} (- p)] = 2 E_{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.

Fermion number operator ( $F$ )

 0  0

The fermion number operator counts occupied fermionic modes. Its parity

(- 1)^{F}

is positive on even exterior degree and negative on odd exterior degree.

Supertrace ( $Str A = Tr ((- 1)^{F} A)$ )

 1  0

A supertrace is a trace weighted by the grading of a Hilbert space. For fermion parity

(- 1)^{F}

, it is

Str A = Tr [(- 1)^{F} A]

: even states contribute with a plus sign and odd states with a minus sign.

In supersymmetric quantum mechanics, paired positive-energy states have opposite fermion parity and cancel in the supertrace of

e^{- β H}

. Unpaired zero modes then contribute to the Witten index. This cancellation requires a well-defined trace or a consistent regulator; a continuum reaching zero energy can invalidate a naive term-by-term argument. The supertrace is therefore useful for counting protected differences between sectors, while an ordinary thermal trace adds both sectors and counts a different quantity.