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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 each complex fermion define two Majorana fermion operatorsThey obey $c_r^{†}=c_r$, $\{c_r,c_s\}=2{\delta }_{rs}$, and $a_j=(c_{2j-1}+i{c}_{2j})/2$. Substituting these inverse relations expands every hopping and pairing monomial as a bilinear in the $c_r$. Diagonal terms $c_r^2=1$ contribute only a constant.

For $r\ne s$, $c_r{c}_s$ is anti-Hermitian. Hermiticity of the original quadratic fermion Hamiltonian therefore makes its coefficient purely imaginary, so it can be written $i{A}_{rs}$ with $A_{rs}$ real. Sincethe symmetric part again changes only the constant. Absorbing conventional factors into $A$ gives

The inverse definitions areSubstitution into the hopping and pairing terms expresses the Hamiltonian as a bilinear in the Majorana fermion operators. Hermiticity makes its off-diagonal coefficients purely imaginary in the Majorana bilinear, whileremoves the symmetric part; diagonal terms are constants because $c_j^2=1$. Therefore, up to that additive constant,whereConversely every real antisymmetric $A$ makes this expression Hermitian, so this is the general Majorana form of a quadratic fermion Hamiltonian.

A quadratic fermion Hamiltonian is bilinear in creation and annihilation operators. In a Majorana basis it is determined, up to a constant, by a real antisymmetric matrix.