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A Bogoliubov--de Gennes Hamiltonian is the matrix representation of a quadratic fermion Hamiltonian in a Nambu spinor combining annihilation operators and creation operators. It treats ordinary one-particle motion and fermionic pairing in one eigenvalue problem.

For a column $c$ of $N$ fermionic annihilation operators, let $h=h^{†}$ and $\Delta =-{\Delta }^T$. With $\Psi =(c,c^{†}{}^T{)}^T$, defineThe canonical anticommutation relations give the identityThe factor $1/2$ compensates for the full particle-hole doubling; the trace constant follows when the hole block is normal-ordered.

A two-by-two opposite-spin pairing blockhas eigenvalues $E_{\pm }=\pm \sqrt{{\xi }^2+|\Delta {|}^2}$, as follows from its characteristic polynomial. Pairing opens a spectral gap at $\xi =0$. Negative and positive partners in a fully doubled representation are related by Particle-hole symmetry of a Bogoliubov--de Gennes Hamiltonian; they are not independent species of physical quasiparticles. A reduced block and a full doubled basis therefore require consistent counting conventions.

For $S_n^{+}=P_n{c}_n$ with $P_n={\prod }_{m<n}(1-2c_m^{†}{c}_m)$, neighbouring Jordan–Wigner transformation strings differ by only the local parity factor. The canonical anticommutation relations give $(c_n+c_n^{†})(1-2c_n^{†}{c}_n)=c_n^{†}-c_n$, proving the displayed local spin identity. The closing bond of a periodic chain instead retains total fermion parity; cancellation on bulk bonds does not justify discarding that global sector.

A real two-mode Bogoliubov transformation preserving the full canonical anticommutation relations. The self anticommutators become $\{a,a^{†}\}=\{b,b^{†}\}=u^2+v^2$, while $\{a,b\}=-uv+uv=0$ and every other cross anticommutator vanishes. Thus its matrix is an orthogonal matrix. For the two-orbital fermionic pairing Hamiltonian, let $E=\sqrt{{\epsilon }^2+{\Delta }^2}$, take $u=\sqrt{(E+\epsilon )/(2E)}$ and $v=\sqrt{(E-\epsilon )/(2E)}$, and obtain $H=E(c^{†}c+d^{†}d)+\epsilon -E$. Inverting gives $c=ua-v{b}^{†}$, $d=v{a}^{†}+ub$.

A linear change of creation operators and annihilation operators is canonical when it preserves their defining brackets. For finitely many bosonic modes, the canonical commutation relations require $U{U}^{†}-V{V}^{†}=I$ and $U{V}^T-V{U}^T=0$. For fermions, the canonical anticommutation relations instead require $U{U}^{†}+V{V}^{†}=I$ and $U{V}^T+V{U}^T=0$. A Bogoliubov transformation uses such mixing to diagonalize a quadratic Hamiltonian or relate different vacuum descriptions. Infinite-mode unitary implementation has further conditions, as in bosonic mode mixing implementability.

A Dirac field is a quantum field transforming as a Dirac spinor. Its free Lagrangian density is $\mathcal{L}=i\bar{\psi }{\gamma }^{\mu }{\partial }_{\mu }\psi -m\bar{\psi }\psi$, giving the Dirac equation. Unlike a single-particle wave function, a quantized Dirac field can create and remove particles and antiparticles.

Its mode expansion of a Dirac field contains fermionic annihilation operators for particles and fermionic creation operators for antiparticles, with canonical anticommutation relations enforcing Fermi-Dirac statistics. For the electron field, these excitations are electrons and Positrons. A global phase symmetry gives the conserved Dirac current; coupling that current to the electromagnetic field produces the matter interaction in quantum electrodynamics.

Exclusion constrains complete one-particle states, not a collection of little spatial boxes. In particular, two opposite-spin electrons in the same spatial orbital haveThe spatial factor is symmetric; the spin-one-half singlet state supplies the required antisymmetry. At ${\mathbf{r}}_1={\mathbf{r}}_2$, this wavefunction need not vanish for opposite spin labels. For equal spin labels it does vanish. In a continuous position description, an exact coincident point has zero measure anyway; the useful comparison concerns joint probability densities or detection in finite regions. Equal positions do not imply equal complete quantum states.

The fermionic occupation numbers $n_{\alpha }=0,1$ refer to a chosen orthonormal basis of one-particle states. A quantum superposition need not have definite occupation numbers in that basis, and an expectation value $⟨n_{\alpha }⟩$ can be fractional. For example, one electron in $(u+v)/\sqrt{2}$ has mean occupation number $1/2$ in each of two orthogonal spin orbitals, without splitting into two particles. The canonical anticommutation relations give $(a_{\alpha }^{†}{)}^2=0$ and make $a_{\alpha }^{†}{a}_{\alpha }$ a projector, so its expectation value stays between zero and one in every normalized quantum state.

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 two odd fermionic operators, time ordering contributes a minus sign whenever the chronological rearrangement exchanges them. For several fields the sign is the sign of a permutation restricted to the odd fields. This convention makes the free Dirac propagator a Green function of the Dirac equation with its contact term fixed by the canonical anticommutation relations.

For $d$ fermionic creation operators with the canonical anticommutation relations, the Hilbert space $L^2({\mathbb{R}}^d)\otimes {\Lambda }^{\bullet }{\mathbb{C}}^d$ splits into ${\mathcal{H}}_n=L^2({\mathbb{R}}^d)\otimes {\Lambda }^n{\mathbb{C}}^d$, $0\le n\le d$. The fermion number operator $N={\sum }_a{\psi }_a^{†}{\psi }_a$ has value $n$ on ${\mathcal{H}}_n$; its internal multiplicity is $\left(\genfrac{}{}{0ex}{}{d}{n}\right)$. An odd supercharge $Q=q+q^{†}$, with $q$ raising $n$ and $q^2=0$, gives $H=\{q,q^{†}\}$ and preserves these sectors.

On an eigenspace with $E>0$, $Q/\sqrt{E}$ is an odd involution pairing bosonic and fermionic states. More precisely,gives an orthogonal decomposition of each state into the images of $q$ and $q^{†}$: their inner product vanishes because $(q^{†}{)}^2=0$. If $u\in \mathrm{im}{q}^{†}$, then $q^{†}u=0$, $q^{†}qu=Eu$ and $\Vert qu{\Vert }^2=E\Vert u{\Vert }^2$. Thus $q/\sqrt{E}$ pairs such a state in degree $n$ with one in degree $n+1$, and $q^{†}/\sqrt{E}$ is the inverse. States with zero energy are killed by both operators and may remain unpaired. The pairing applies to positive spectral subspaces as well; discreteness is not implied.

For a real smooth function $\chi$ on Euclidean space, put $A_a={\partial }_a-{\partial }_a\chi$ and $q={\sum }_a{\psi }_a^{†}{A}_a$. The adjoint operator for the usual $L^2$ inner product is $A_a^{†}=-{\partial }_a-{\partial }_a\chi$. Since $[A_a,A_b]=0$, the canonical anticommutation relations give $q^2=0$. Expanding $H=\{q,q^{†}\}$ and using $[A_a,A_b^{†}]=-2{\partial }_a{\partial }_b\chi$ givesIn the empty fermionic Fock space sector the last term vanishes; in the filled sector ${\psi }_a^{†}{\psi }_b={\delta }_{ab}$. Consequently the two scalar Hamiltonian operators are $H_0=-\Delta +|\nabla \chi {|}^2+\Delta \chi$ and $H_d=-\Delta +|\nabla \chi {|}^2-\Delta \chi$. On the compatible self-adjoint operator domain the Hamiltonian operator is nonnegative because $⟨v,Hv⟩=\Vert qv{\Vert }^2+\Vert q^{†}v{\Vert }^2$.

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.

For a paired non-self-conjugate momentum, rotate the Ising-chain Nambu spinor by an angle with $\cos \theta =(g-\cos k)/r_k$, $\sin \theta =-\sin k/r_k$ and $r_k=\sqrt{1+g^2-2g\cos k}$. Opposite angles at $k,-k$ preserve the canonical anticommutation relations. The result is $H={\sum }_k{ϵ}_k({\gamma }_k^{†}{\gamma }_k-1/2)$. Self-paired momenta have no pairing and are handled directly as number levels. Periodic spin-chain sectors additionally constrain fermion parity and allowed mode quantization.

Use dimensionless spin components, equivalently units $\hslash =1$, in the following formulas.

The Jordan–Wigner transformation represents a one-dimensional spin one-half chain by fermions with a parity string preceding each site. With $P_n={\prod }_{m<n}(1-2n_m)$, one has $S_n^{+}=P_n{c}_n$ and $S_n^z=1/2-n_n$. The string converts the off-site canonical anticommutation relations into commuting spin operators. Adjacent exchange terms become quadratic fermion expressions, while a periodic end bond depends on global fermion parity.

The equal-time canonical anticommutation relations implyApply $-i{\alpha }_{bc}^k{\partial }_{z^k}+m{\beta }_{bc}$, set $\mathbf{z}=\mathbf{y}$, and integrate over $\mathbf{y}$. The mass term cancels, leaving $i\int d^{3}y{\psi }^{†}{\alpha }^k\psi {\partial }_{y^k}{\delta }^{(3)}(\mathbf{x}-\mathbf{y})$. Integration by parts proves the title formula as a distribution identity. The Heisenberg equation of motion then yields ${\partial }_{t}n=-\nabla \cdot \mathbf{j}$ for the physical Dirac current $j^k=\bar{\psi }{\gamma }^k\psi$. The factor $-i$ is required because the commutator of Hermitian operators is anti-Hermitian.

A Majorana fermion operator is a self-adjoint operator in a fermionic algebra satisfying $\{{\gamma }_i,{\gamma }_j\}=2{\delta }_{ij}I$. A single complex fermionic mode $c$ can be written using two such operators:The canonical anticommutation relations verify their Hermiticity and normalization, and the occupation is $c^{†}c=(I+i{\gamma }_1{\gamma }_2)/2$.

This algebraic rewriting is always possible for a complex mode. A spatially separated Majorana zero mode requires additional physical properties, including localization and near-zero commutator with the Hamiltonian. The existence of two Majorana operators in a formula alone therefore does not establish protected boundary states or a topological phase.

Let $S=\bar{q}q$, $P_a=i\bar{q}{\gamma }_5{\tau }_{a}q$ and $Q_{5,a}=\int q^{†}{\gamma }_5{\tau }_{a}q/2d^{3}x$. The canonical anticommutation relations imply $[Q_{5,a},q]=-{\gamma }_5{\tau }_{a}q/2$ and $[Q_{5,a},\bar{q}]=-\bar{q}{\gamma }_5{\tau }_a/2$. The Leibniz rule for the commutator, together with $\{{\tau }_a,{\tau }_b\}=2{\delta }_{ab}$ and ${\gamma }_5^2=1$, gives the displayed transformations. These are equal-time operator relations; common nonzero quark masses remove conservation of the charges without changing their canonical action on the fields.

For a spinless Bogoliubov transformation $c_p=\cos {\theta }_p{\alpha }_p+i\sin {\theta }_p{\alpha }_{-p}^{†}$, consistency of the doubled Nambu spinor requires $\cos {\theta }_{-p}=\cos {\theta }_p$ and $\sin {\theta }_{-p}=-\sin {\theta }_p$. Thus choose ${\theta }_{-p}=-{\theta }_p$ modulo $2\pi$. The cross anticommutator from the first-row formulas is $\{c_p,c_{-p}\}=i\sin ({\theta }_p+{\theta }_{-p})$; a nontrivial even angle would violate the canonical anticommutation relations. A self-paired zero mode must instead have vanishing mixing in this convention. Real-angle unitarity within each two-by-two block alone does not enforce consistency between duplicated momentum labels.

An operator algebra is an algebra of linear operators, with multiplication given by composition. In quantum examples it can be generated by creation and annihilation operators subject to the canonical commutation relations or canonical anticommutation relations. Its analytic realization requires specifying a common domain for unbounded operators.

Start from electrons moving in the periodic equilibrium ionic potential. Displace ion $l$ by ${\mathbf{u}}_l$ and expand its contribution as $V(\mathbf{r}-{\mathbf{R}}_l-{\mathbf{u}}_l)\simeq V(\mathbf{r}-{\mathbf{R}}_l)-{\mathbf{u}}_l\cdot \nabla V(\mathbf{r}-{\mathbf{R}}_l)$. Quantizing the harmonic normal modes expresses ${\mathbf{u}}_l$ as a sum of creation and annihilation operators with displacement scale $\sqrt{\hslash /(2{\rho }_{\mathrm{ion}}V{\omega }_{\mathbf{q}})}$ for a single simple branch. Expanding the electronic field in bloch states then gives a band-projected Fröhlich HamiltonianHere $c,c^{†}$ obey canonical anticommutation relations and remove/create an electron with crystal momentum $\mathbf{k}$ and spin $\sigma$; $b,b^{†}$ are boson operators for lattice modes. The matrix element $M_{\mathbf{q}}$ contains the derivative of the ionic potential, phonon polarization, displacement normalization and electronic wavefunction overlaps. With phases chosen appropriately, $M_{-\mathbf{q}}=M_{\mathbf{q}}^{\ast }$ ensures a Hermitian Hamiltonian. The displayed one-band, one-branch notation suppresses additional branch and band labels. “Fröhlich Hamiltonian” here means the linear electron-phonon model used in the question; it is not a requirement to use specifically the polar-optical-phonon coupling of a polar semiconductor. A harmonic lattice, small displacements and a weak enough coupling to organize perturbation theory are the underlying approximations.

The Fröhlich canonical transformation removes real linear phonon operators from the electron-only effective problem to first order. Choose an anti-Hermitian $S$ with $[S,H_0]=-V_{e\text{-}ph}$. For $A_{\mathbf{kq}\sigma }=c_{\mathbf{k}+\mathbf{q},\sigma }^{†}{c}_{\mathbf{k}\sigma }$ and ${\Delta }_{\mathbf{kq}}={ϵ}_{\mathbf{k}+\mathbf{q}}-{ϵ}_{\mathbf{k}}$, one possible generator isIts Hermitian conjugate is its negative after relabelling momenta. The Baker--Campbell--Hausdorff formula givesThe commutator describes virtual emission and absorption of a phonon. normal ordering yields an electron-electron term as well as one-electron energy renormalizations and phonon terms. Retain the former, absorb single-particle shifts into ${ϵ}_{\mathbf{k}}$ and eliminate the explicit phonon sector. For elastic pair scattering the two virtual time orderings have the combined amplitudeUsing the usual factor one-half to count unordered pairs gives the coefficient in the printed Hamiltonian, with $\Delta ={ϵ}_{\mathbf{k}}-{ϵ}_{\mathbf{k}-\mathbf{q}}$. The two outgoing momenta are $\mathbf{k}-\mathbf{q}$ and ${\mathbf{k}}^{\prime }+\mathbf{q}$, so total momentum and the spins are conserved. A fully off-shell effective operator can instead use symmetrized denominators for both external legs; the displayed expression is the usual elastic-scattering reduction. Eliminating nearly resonant real phonons requires more care than this second-order construction.

For $|\Delta |<{\Omega }_{\mathbf{q}}$ the denominator is negative: virtual phonon exchange is attractive in this low-energy channel. Physically, an electron changes the ionic density and a second electron can gain energy by responding to the delayed distortion. This is a retarded interaction, not an unscreened instantaneous attraction at every energy transfer. At large transfer the sign changes, and screened Coulomb repulsion must also be included when deciding the net low-energy pairing strength. The connection between lattice dynamics and superconductivity is supported by the isotope effect, but the mathematical attraction alone does not determine a measured transition temperature.^[ref]

For the pair calculation, make the model restrictions explicit. The filled Fermi sea is inert, and $|0⟩$ denotes that sea with no added particles, not an empty electronic vacuum. Project the interaction onto the previously empty shell $0\le {\xi }_{\mathbf{k}}\le D$, where ${\xi }_{\mathbf{k}}={\hslash }^2{k}^2/(2m)-E_F$ and $D=\hslash {\omega }_D$. Measure the added-pair energy relative to $2E_F$; this is the negative $E$ in the final formula. If absolute one-electron energies are retained in the preceding eigenvalue equation, its eigenvalue must instead be $E_{\mathrm{abs}}=2E_F+E$. A shell-restricted state would not in general be an exact eigenstate of the completely unprojected, all-momentum contact Hamiltonian.

Write $g={\gamma }^2>0$ and $|\mathbf{k}⟩=c_{-\mathbf{k}\downarrow }^{†}{c}_{\mathbf{k}\uparrow }^{†}|0⟩$. Within the zero-total-momentum pair subspace the two opposite-spin terms cancel the interaction's factor one-half, while same-spin s-wave terms cancel by fermionic antisymmetry. Thus the projected Hamiltonian acts asFor $|\Psi ⟩={\sum }_{⟨\mathbf{k}⟩}{\phi }_{\mathbf{k}}|\mathbf{k}⟩$ and $C={\sum }_{⟨\mathbf{k}⟩}{\phi }_{\mathbf{k}}$, matching each basis coefficient givesThis is the stated coefficient formula in the consistent Fermi-level energy convention. Its even dependence on $\mathbf{k}$ produces an even orbital state with a spin singlet. The constant $C$ is fixed up to an overall phase by normalization; for a bound state, $|C{|}^{-2}=g^2{\sum }_{⟨\mathbf{k}⟩}(E-2{\xi }_{\mathbf{k}}{)}^{-2}$. It cannot be set to zero for the nontrivial bound solution.

Put $E=-B$ with $B>0$ and impose the sum defining $C$:Here $\rho (E_F)$ counts orbital states for one spin species, as appropriate to the unrestricted $\mathbf{k}$ sum in this pair basis, and is approximately constant over the thin shell. If a two-spin total density of states is used, it must be divided by two in this integral. State-counting and the volume convention in $g$ and $\rho$ must be consistent. With $\xi =g\rho (E_F)$ as the paper's dimensionless coupling, inversion yieldsThe first approximation is the constant-density-of-states, restricted-shell model; the final exponential additionally assumes weak attraction. The binding energy is nonanalytic at zero coupling, so a finite-order power expansion in $g$ cannot produce it.

Cooper's problem asks whether two electrons added above an inert Fermi sea bind despite Pauli exclusion of lower states. An arbitrarily weak net attraction binds a zero-momentum singlet below the free-pair threshold, because the near-Fermi-surface integral diverges logarithmically as $B\downarrow 0$. Its wavefunction draws on many time-reversed momentum states and becomes dominated by a narrow region near the Fermi surface at weak coupling; the corresponding pair is spatially extended. This establishes a Cooper instability of the unpaired reference state.^[ref] The full superconducting ground state needs a coherent many-electron treatment in which overlapping pairs and the reorganized sea are included. BCS theory supplies that step; the two-electron binding scale alone is neither a calculation of the many-body gap nor a derivation of zero resistance.^[ref]

For real angles write $u_p=\cos {\theta }_p$ and $v_p=\sin {\theta }_p$. For one independent pair $p>0$, the transformation and its conjugate readUsing the canonical anticommutation relations for the $\alpha$ modes givesThe remaining anticommutators vanish for independent momentum pairs. The displayed two-by-two matrix is unitary and its inverse is obtained by changing $i$ to $-i$.

When the same formula is indexed over both signs of momentum, its first row at $-p$ must agree with the conjugate of its second row at $p$. Thus a consistent global convention requiresThis is the odd-momentum fermionic Bogoliubov angle convention. If one checks only the first-row formulas, their cross anticommutator is $i\sin ({\theta }_p+{\theta }_{-p})$; its vanishing gives the condition modulo $\pi$, with possible signs absorbable in the $\alpha$ modes. Requiring both rows to describe the same operators fixes the convention above. Merely checking a separate unitary matrix at each momentum misses this redundancy: choosing ${\theta }_p={\theta }_{-p}=\pi /4$ in both first rows would give $\{c_p,c_{-p}\}=i$.

At $p=0$, the annihilation operator and its conjugate belong to one mode. The full displayed formula is consistent only with $v_0=0$; choose ${\theta }_0=0$ and ${\alpha }_0=c_0$. This mode is not a two-mode pair.