Useful stuff

Commutator properties

$[A,B+C]=[A,B]+[A,C]$
$[A+B,C]=[A,C]+[B,C]$
$[A,BC]=B[A,C]+[A,B]C$
$[AB,C]=A[B,C]+[A,C]B$

Densities of Slater determinants

The above decomposition is unique if:

1. $\chi$ has only ph and hp matrix elements in the basis in which ${\rho }_0$ is diagonal:

   $${\rho }_0\chi {\rho }_0={\sigma }_0\chi {\sigma }_0=0$$

   (${\sigma }_0=1-{\rho }_0$ projects onto particle states.)

2. The eigenvalues of $\chi$ obey:

   $$-\frac{\pi }{4}\le {\chi }_{\mu }<\frac{\pi }{4}$$

   We are not proving these theorems 🤫.

Some rules for calculating with Slater determinant densities

An arbitrary matrix A has the following "parts" in a basis in which $\rho$ is diagonal:

The three following statements are equivalent:

If two matrices $A$, $B$ obey $B=[A,\rho ]$, then:

If $A^{pp}=A^{hh}=0$, then $A=[B,\rho ]$. For Hermitian matrices $A$, $B$, with vanishing pp and hh elements, we can define these useful vectors:

and find:

Densities of HF-BCS and HFB states

zzz

Wick's theorem

Say we want

Particle-Hole picture

For many-body systems, it is useful to use basis states that are labeled as deviations/excitations from some reference many body Slater determinant (vs states with N creation operators acting on the vacuum):

Occupied single particle states below the Fermi level are hole states, and unoccupied single particle states above it are particle states.

Quasiparticle formalism

Very useful for Hartree-Fock and beyond. We start with a simple redefinition of the creation and annihilation operators, like so (see Notation and more):

Normal ordering w.r.t. $|\varphi ⟩$

We will use curly braces instead of $\mathrm{N}[\dots ]$ to distinguish from normal ordering from respect to the vacuum:

Everything else (fundamental contractions, etc) remains the same when swapping in the $b$ operators.