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In second quantization, expand the annihilating quantized field in a complete orthonormal basis of one-particle states, $\hat{\psi }(x)={\sum }_{\alpha }{\phi }_{\alpha }(x)a_{\alpha }$. At equal times, the canonical commutation relation for bosons and the canonical anticommutation relations for fermions are, respectively,andHere $\delta$ is a Dirac delta distribution. Multiplying by the basis wave functions and integrating gives $[a_{\alpha },a_{\beta }^{†}]={\delta }_{\alpha \beta }$ for bosons, or $\{a_{\alpha },a_{\beta }^{†}\}={\delta }_{\alpha \beta }$ for fermions; the two-annihilator and two-creator brackets vanish in the corresponding convention.

The mode number operator and total particle number operator areThe bosonic occupation numbers are $0,1,2,\dots$. For fermions, $a_{\alpha }^2=0$ and the canonical anticommutation relations imply $n_{\alpha }^2=n_{\alpha }$, so the fermionic occupation numbers are only $0$ and $1$, as required by the Pauli exclusion principle.

Keeping only two modes later is a projection: the projected field bracket is ${\sum }_{j=1}^2{\phi }_j(x){\phi }_j^{\ast }(x^{\prime })$, rather than the full Dirac delta distribution.

For independent bosonic modes, the canonical commutation relation isFor independent fermionic modes, the canonical anticommutation relations arewhere $[A,B]=AB-BA$ and $\{A,B\}=AB+BA$. The creation operator is the adjoint of the annihilation operator. In either case the mode's number operator iswith $a_i=b_i$ or $c_i$ as appropriate. It measures the mode's occupation number. Its possible eigenvalues areFor fermions, $c_i^2=0$ implies $n_i^2=c_i^{†}(1-c_i^{†}{c}_i)c_i=n_i$, so the number operator is a projection. This expresses the exclusion of double occupation of one fermionic mode. The total particle number is $N={\sum }_i{n}_i$.

For bosonic creation operators and bosonic annihilation operators, the canonical commutation relations areFor fermionic annihilation operators and their adjoints, the canonical anticommutation relations areThe respective number operators are $n_i=a_i^{†}{a}_i$ and $n_i=f_i^{†}{f}_i$. Their allowed eigenvalues are $\boxed{0,1,2,\dots \text{ for bosons};0,1\text{ for fermions}}$. In the fermionic case $n_i^2=n_i$, so its eigenvalues must satisfy $n^2=n$; this is the single-mode Pauli exclusion principle.

There is a normalization mismatch in the printed definitions: ${\sigma }^x\pm i{\sigma }^y$ have nonzero entries of magnitude two, whereas a single fermionic creation or annihilation operator has magnitude one. To obtain the intended Jordan–Wigner transformation, use normalized spin ladders $s_j^{\pm }=({\sigma }_j^x\pm i{\sigma }_j^y)/2$. The given fermion expressions represent these $s_j^{\pm }$. The two-site Jordan-Wigner normalization and signs then fix every factor unambiguously.

Put $n_j=f_j^{†}{f}_j$ and $P_1=e^{i\pi n_1}=1-2n_1$. The canonical anticommutation relations give $P_1^2=I$, $P_1{f}_1=-f_1{P}_1$, and $[P_1,f_2]=[P_1,f_2^{†}]=0$. HenceandThese are the spin commutation relations, with distinct sites commuting. If one retains the unscaled symbols ${\sigma }_j^{\pm }=2s_j^{\pm }$, the first and last displayed commutators instead equal $-4{\sigma }_j^z$; the off-site commutators remain zero. Setting the unscaled ${\sigma }_j^{-}$ equal to $f_j$ would contradict the original Pauli matrices.

Using the consistently normalized Jordan–Wigner transformation from part (c), write ${\sigma }_1^x{\sigma }_2^x+{\sigma }_1^y{\sigma }_2^y=2(s_1^{+}{s}_2^{-}+s_1^{-}{s}_2^{+})$. Since $f_1^{†}{P}_1=f_1^{†}$ and $f_1{P}_1=-f_1$, the parity string givesThereforeThe orthogonal transformation $b=(f_1+f_2)/\sqrt{2}$, $c=(f_1-f_2)/\sqrt{2}$ preserves the canonical anticommutation relations and diagonalizes this bilinear Hamiltonian:The Fock vacuum is $|\downarrow \downarrow ⟩$, with energy $2h$. Occupying $b$ alone gives the symmetric opposite-spin state with energy $-2J$; occupying $c$ alone gives the antisymmetric state with energy $2J$; occupying both gives $|\uparrow \uparrow ⟩$, up to a global phase, with energy $-2h$. This reproduces the entire spectrum in part (b).

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.

Second quantization describes identical-particle quantum states in Fock space using creation operators and annihilation operators. Relative to an orthonormal basis of one-particle states, a one-body operator becomes ${\sum }_{\alpha \beta }{h}_{\alpha \beta }{a}_{\alpha }^{†}{a}_{\beta }$. The canonical commutation relation implements bosonic statistics, while the canonical anticommutation relations implement fermionic statistics. Expanding the quantized field in a complete basis gives a delta-function equal-time bracket; restricting the expansion to finitely many modes replaces that delta function by the corresponding projection operator kernel.

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.

In a unit-spacing periodic spinless chain, self-paired momenta are $k=0$ and, when available, $k=\pi$. An anomalous pair of the same creation operator vanishes by the canonical anticommutation relations. Such a mode is a single normal number level, filled or empty according to its energy sign, rather than a two-distinct-mode Bogoliubov transformation. This excludes the paired-angle construction at those special momenta.

With occupied fermion sites representing up-spins, use $s_1^{-}=f_1$, $s_2^{-}=P_1{f}_2$ and $P_1=(-1{)}^{n_1}=1-2n_1$. The canonical anticommutation relations imply $P_1^2=I$, $P_1{f}_1=-f_1{P}_1$ and $[P_1,f_2]=0$. Consequently the two sites commute as spin operators, while $[s_j^{-},s_j^{+}]=1-2n_j=-{\sigma }_j^z$. This normalization is essential: the unscaled combinations ${\sigma }_j^x\pm i{\sigma }_j^y$ are twice $s_j^{\pm }$.

The two-spin XY Hamiltonian becomesIn particular, $f_1^{†}{P}_1=f_1^{†}$ and $f_1{P}_1=-f_1$ fix the hopping signs. The modes $b=(f_1+f_2)/\sqrt{2}$ and $c=(f_1-f_2)/\sqrt{2}$ preserve the canonical anticommutation relations and give $H=2h+(-2h-2J)b^{†}b+(-2h+2J)c^{†}c$.

A fermionic mode with canonical anticommutation relations $\{b,b^{†}\}=1$ has the symmetrically ordered Hamiltonian operator $H=\frac{E}{2}(b^{†}b-b{b}^{†})=E(b^{†}b-1/2)$. Its empty-state zero-point contribution is $-E/2$. A quantum harmonic oscillator with bosonic canonical commutation relations contributes $+E/2$ instead. For a Dirac field, sum over both particle and antiparticle modes and over spin: there are four independent oscillators per spatial momentum in four spacetime dimensions. Comparing bosonic and fermionic vacuum energies requires a common regularization in quantum field theory, matching degrees of freedom and a stated vacuum-energy subtraction convention.