Bogoliubov transformation

Consider the canonical commutation relation for bosonic creation and annihilation operators in the harmonic basis

${\displaystyle \left[{\hat {a}},{\hat {a}}^{\dagger }\right]=1.}$

Define a new pair of operators

${\displaystyle {\hat {b}}=u{\hat {a}}+v{\hat {a}}^{\dagger },}$

${\displaystyle {\hat {b}}^{\dagger }=u^{*}{\hat {a}}^{\dagger }+v^{*}{\hat {a}},}$

for complex numbers u and v, where the latter is the Hermitian conjugate of the first.

The Bogoliubov transformation is the canonical transformation mapping the operators ${\displaystyle {\hat {a}}}$ and ${\displaystyle {\hat {a}}^{\dagger }}$ to ${\displaystyle {\hat {b}}}$ and ${\displaystyle {\hat {b}}^{\dagger }}$. To find the conditions on the constants u and v such that the transformation is canonical, the commutator is evaluated, namely,

${\displaystyle \left[{\hat {b}},{\hat {b}}^{\dagger }\right]=\left[u{\hat {a}}+v{\hat {a}}^{\dagger },u^{*}{\hat {a}}^{\dagger }+v^{*}{\hat {a}}\right]=\cdots =\left(|u|^{2}-|v|^{2}\right)\left[{\hat {a}},{\hat {a}}^{\dagger }\right].}$

It is then evident that ${\displaystyle |u|^{2}-|v|^{2}=1}$ is the condition for which the transformation is canonical.

Since the form of this condition is suggestive of the hyperbolic identity

${\displaystyle \cosh ^{2}x-\sinh ^{2}x=1,}$

the constants u and v can be readily parametrized as

${\displaystyle u=e^{i\theta _{1}}\cosh r,}$

${\displaystyle v=e^{i\theta _{2}}\sinh r.}$

This is interpreted as a linear symplectic transformation of the phase space. By comparing to the Bloch–Messiah decomposition, the two angles ${\displaystyle \theta _{1}}$ and ${\displaystyle \theta _{2}}$ correspond to the orthogonal symplectic transformations (i.e., rotations) and the squeezing factor ${\displaystyle r}$ corresponds to the diagonal transformation.

For the anticommutation relations

${\displaystyle \left\{{\hat {a}},{\hat {a}}\right\}=0,\left\{{\hat {a}},{\hat {a}}^{\dagger }\right\}=1,}$

the Bogoliubov transformation is constrained by ${\displaystyle uv=0,|u|^{2}+|v|^{2}=1}$. Therefore, the only non-trivial possibility is ${\displaystyle u=0,|v|=1,}$ corresponding to particle–antiparticle interchange (or particle–hole interchange in many-body systems) with the possible inclusion of a phase shift. Thus, for a single particle, the transformation can only be implemented (1) for a Dirac fermion, where particle and antiparticle are distinct (as opposed to a Majorana fermion or chiral fermion), or (2) for multi-fermionic systems, in which there is more than one type of fermion.

The Hilbert space under consideration is equipped with these operators, and henceforth describes a higher-dimensional quantum harmonic oscillator (usually an infinite-dimensional one).

The ground state of the corresponding Hamiltonian is annihilated by all the annihilation operators:

${\displaystyle \forall i\qquad a_{i}|0\rangle =0.}$

All excited states are obtained as linear combinations of the ground state excited by some creation operators:

${\displaystyle \prod _{k=1}^{n}a_{i_{k}}^{\dagger }|0\rangle .}$

One may redefine the creation and the annihilation operators by a linear redefinition:

${\displaystyle a'_{i}=\sum _{j}(u_{ij}a_{j}+v_{ij}a_{j}^{\dagger }),}$

where the coefficients ${\displaystyle u_{ij},v_{ij}}$ must satisfy certain rules to guarantee that the annihilation operators and the creation operators ${\displaystyle a_{i}^{\prime \dagger }}$, defined by the Hermitian conjugate equation, have the same commutators for bosons and anticommutators for fermions.

The equation above defines the Bogoliubov transformation of the operators.

The ground state annihilated by all ${\displaystyle a'_{i}}$ is different from the original ground state ${\displaystyle |0\rangle }$, and they can be viewed as the Bogoliubov transformations of one another using the operator–state correspondence. They can also be defined as squeezed coherent states. BCS wave function is an example of squeezed coherent state of fermions.[10]

Because Bogoliubov transformations are linear recombination of operators, it is more convenient and insightful to write them in terms of matrix transformations. If a pair of annihilators ${\displaystyle (a,b)}$ transform as

${\displaystyle {\begin{pmatrix}\alpha \\\beta \end{pmatrix}}=U{\begin{pmatrix}a\\b\end{pmatrix}}}$

where ${\displaystyle U}$is a ${\displaystyle 2\times 2}$ matrix. Then naturally

${\displaystyle {\begin{pmatrix}\alpha ^{\dagger }\\\beta ^{\dagger }\end{pmatrix}}=U^{*}{\begin{pmatrix}a^{\dagger }\\b^{\dagger }\end{pmatrix}}}$

For Fermion operators, the requirement of commutation relations reflects in two requirements for the form of matrix ${\displaystyle U}$

${\displaystyle U={\begin{pmatrix}u&v\\-v^{*}&u^{*}\end{pmatrix}}}$

and

${\displaystyle |u|^{2}+|v|^{2}=1}$

For Boson operators, the commutation relations require

${\displaystyle U={\begin{pmatrix}u&v\\v^{*}&u^{*}\end{pmatrix}}}$

and

${\displaystyle |u|^{2}-|v|^{2}=1}$

These conditions can be written uniformly as

${\displaystyle U\Gamma _{\pm }U^{\dagger }=\Gamma _{\pm }}$

where

${\displaystyle \Gamma _{\pm }={\begin{pmatrix}1&0\\0&\pm 1\end{pmatrix}}}$

where ${\displaystyle \Gamma _{\pm }}$ applies to Fermions and Bosons, respectively.

Diagonalizing a quadratic Hamiltonian using matrix description

Bogoliubov transformation lets us diagonalize a quadratic Hamiltonian

${\displaystyle {\hat {H}}={\begin{pmatrix}a^{\dagger }&b^{\dagger }\end{pmatrix}}H{\begin{pmatrix}a\\b\end{pmatrix}}}$

by just diagonalizing the matrix ${\displaystyle \Gamma _{\pm }H}$. In the notations above, it is important to distinguish the operator ${\displaystyle {\hat {H}}}$ and the numeric matrix ${\displaystyle H}$. This fact can be seen by rewriting ${\displaystyle {\hat {H}}}$ as

${\displaystyle {\hat {H}}={\begin{pmatrix}\alpha ^{\dagger }&\beta ^{\dagger }\end{pmatrix}}\Gamma _{\pm }U(\Gamma _{\pm }H)U^{-1}{\begin{pmatrix}\alpha \\\beta \end{pmatrix}}}$

and ${\displaystyle \Gamma _{\pm }U(\Gamma _{\pm }H)U^{-1}=D}$ if and only if ${\displaystyle U}$ diagonalizes ${\displaystyle \Gamma _{\pm }H}$, i.e. ${\displaystyle U(\Gamma _{\pm }H)U^{-1}=\Gamma _{\pm }D}$.

Useful properties of Bogoliubov transformations are listed below.

	Boson	Fermion
Transformation matrix	${\displaystyle U={\begin{pmatrix}u&v\\v^{*}&u^{*}\end{pmatrix}}}$	${\displaystyle U={\begin{pmatrix}u&v\\-v^{*}&u^{*}\end{pmatrix}}}$
Inverse transformation matrix	${\displaystyle U^{-1}={\begin{pmatrix}u^{*}&-v\\-v^{*}&u\end{pmatrix}}}$	${\displaystyle U^{-1}={\begin{pmatrix}u^{*}&-v\\v^{*}&u\end{pmatrix}}}$
Gamma	${\displaystyle \Gamma ={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}}$	${\displaystyle \Gamma ={\begin{pmatrix}1&0\\0&1\end{pmatrix}}}$
Diagonalization	${\displaystyle U(\Gamma H)U^{-1}=\Gamma D}$	${\displaystyle UHU^{-1}=D}$

• Holstein–Primakoff transformation
• Jordan–Wigner transformation
• Jordan–Schwinger transformation
• Klein transformation

The whole topic, and a lot of definite applications, are treated in the following textbooks:

• Blaizot, J.-P.; Ripka, G. (1985). Quantum Theory of Finite Systems. MIT Press. ISBN 0-262-02214-1.
• Fetter, A.; Walecka, J. (2003). Quantum Theory of Many-Particle Systems. Dover. ISBN 0-486-42827-3.
• Kittel, Ch. (1987). Quantum theory of solids. Wiley. ISBN 0-471-62412-8.
• Wagner, M. (1986). Unitary Transformations in Solid State Physics. Elsevier Science. ISBN 0-444-86975-1.