In
theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
, the Bogoliubov transformation, also known as the Bogoliubov–Valatin transformation, was independently developed in 1958 by
Nikolay Bogolyubov
Nikolay Nikolayevich Bogolyubov (russian: Никола́й Никола́евич Боголю́бов; 21 August 1909 – 13 February 1992), also transliterated as Bogoliubov and Bogolubov, was a Soviet and Russian mathematician and theoretic ...
and
John George Valatin for finding solutions of
BCS theory
BCS theory or Bardeen–Cooper–Schrieffer theory (named after John Bardeen, Leon Cooper, and John Robert Schrieffer) is the first microscopic theory of superconductivity since Heike Kamerlingh Onnes's 1911 discovery. The theory describes sup ...
in a homogeneous system. The Bogoliubov transformation is an
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
of either the
canonical commutation relation algebra or
canonical anticommutation relation algebra. This induces an autoequivalence on the respective representations. The Bogoliubov transformation is often used to diagonalize
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
s, which yields the stationary solutions of the corresponding
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
. The Bogoliubov transformation is also important for understanding the
Unruh effect
The Unruh effect (also known as the Fulling–Davies–Unruh effect) is a kinematic prediction of quantum field theory that an accelerating observer will observe a thermal bath, like blackbody radiation, whereas an inertial observer would observe ...
,
Hawking radiation
Hawking radiation is theoretical black body radiation that is theorized to be released outside a black hole's event horizon because of relativistic quantum effects. It is named after the physicist Stephen Hawking, who developed a theoretical a ...
, pairing effects in nuclear physics, and many other topics.
The Bogoliubov transformation is often used to diagonalize Hamiltonians, ''with'' a corresponding transformation of the state function. Operator eigenvalues calculated with the diagonalized Hamiltonian on the transformed state function thus are the same as before.
Single bosonic mode example
Consider the canonical
commutation relation
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Group theory
The commutator of two elements, ...
for
bosonic
In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer spi ...
creation and annihilation operators
Creation operators and annihilation operators are mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilation operator (usually d ...
in the
harmonic basis
:
Define a new pair of operators
:
:
for complex numbers ''u'' and ''v'', where the latter is the
Hermitian conjugate
In mathematics, specifically in operator theory, each linear operator A on a Euclidean vector space defines a Hermitian adjoint (or adjoint) operator A^* on that space according to the rule
:\langle Ax,y \rangle = \langle x,A^*y \rangle,
where ...
of the first.
The Bogoliubov transformation is the canonical transformation mapping the operators
and
to
and
. To find the conditions on the constants ''u'' and ''v'' such that the transformation is canonical, the commutator is evaluated, namely,
:
It is then evident that
is the condition for which the transformation is canonical.
Since the form of this condition is suggestive of the
hyperbolic identity
:
the constants and can be readily parametrized as
:
:
This is interpreted as a
linear symplectic transformation of the
phase space. By comparing to the
Bloch–Messiah decomposition, the two angles
and
correspond to the orthogonal symplectic transformations (i.e., rotations) and the
squeezing factor corresponds to the diagonal transformation.
Applications
The most prominent application is by
Nikolai Bogoliubov
Nikolay Nikolayevich Bogolyubov (russian: Никола́й Никола́евич Боголю́бов; 21 August 1909 – 13 February 1992), also transliterated as Bogoliubov and Bogolubov, was a Soviet and Russian mathematician and theoretica ...
himself in the context of
superfluidity
Superfluidity is the characteristic property of a fluid with zero viscosity which therefore flows without any loss of kinetic energy. When stirred, a superfluid forms vortices that continue to rotate indefinitely. Superfluidity occurs in two ...
. Other applications comprise
Hamiltonians and excitations in the theory of
antiferromagnetism
In materials that exhibit antiferromagnetism, the magnetic moments of atoms or molecules, usually related to the spins of electrons, align in a regular pattern with neighboring spins (on different sublattices) pointing in opposite directions. ...
.
[See e.g. the textbook by ]Charles Kittel
Charles Kittel (July 18, 1916 – May 15, 2019) was an American physicist. He was a professor at University of California, Berkeley from 1951 and was professor emeritus from 1978 until his death.
Life and work
Charles Kittel was born in New Yo ...
: ''Quantum theory of solids'', New York, Wiley 1987. When calculating quantum field theory in curved space–times the definition of the vacuum changes, and a Bogoliubov transformation between these different vacua is possible. This is used in the derivation of
Hawking radiation
Hawking radiation is theoretical black body radiation that is theorized to be released outside a black hole's event horizon because of relativistic quantum effects. It is named after the physicist Stephen Hawking, who developed a theoretical a ...
. Bogoliubov transforms are also used extensively in quantum optics, particularly when working with gaussian unitaries (such as beamsplitters, phase shifters, and squeezing operations).
Fermionic mode
For the
anticommutation relations
:
the Bogoliubov transformation is constrained by
. Therefore, the only non-trivial possibility is
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
In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a slight ...
), or (2) for multi-fermionic systems, in which there is more than one type of fermion.
Applications
The most prominent application is again by Nikolai Bogoliubov himself, this time for the
BCS theory
BCS theory or Bardeen–Cooper–Schrieffer theory (named after John Bardeen, Leon Cooper, and John Robert Schrieffer) is the first microscopic theory of superconductivity since Heike Kamerlingh Onnes's 1911 discovery. The theory describes sup ...
of
superconductivity.
The point where the necessity to perform a Bogoliubov transform becomes obvious is that in mean-field approximation the Hamiltonian of the system can be written in both cases as a sum of bilinear terms in the original creation and destruction operators, involving finite
terms, i.e. one must go beyond the usual
Hartree–Fock method
In computational physics and chemistry, the Hartree–Fock (HF) method is a method of approximation for the determination of the wave function and the energy of a quantum many-body system in a stationary state.
The Hartree–Fock method often ...
. In particular, in the mean-field
Bogoliubov–de Gennes Hamiltonian formalism with a superconducting pairing term such as
, the Bogoliubov transformed operators
annihilate and create quasiparticles (each with well-defined energy, momentum and spin but in a quantum superposition of electron and hole state), and have coefficients
and
given by eigenvectors of the Bogoliubov–de Gennes matrix. Also in
nuclear physics
Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions, in addition to the study of other forms of nuclear matter.
Nuclear physics should not be confused with atomic physics, which studies the ...
, this method is applicable, since it may describe the "pairing energy" of nucleons in a heavy element.
Multimode example
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
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
is annihilated by all the annihilation operators:
:
All excited states are obtained as
linear combinations of the ground state excited by some
creation operators
Creation operators and annihilation operators are mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilation operator (usually d ...
:
:
One may redefine the creation and the annihilation operators by a linear redefinition:
:
where the coefficients
must satisfy certain rules to guarantee that the annihilation operators and the creation operators
, defined by the
Hermitian conjugate
In mathematics, specifically in operator theory, each linear operator A on a Euclidean vector space defines a Hermitian adjoint (or adjoint) operator A^* on that space according to the rule
:\langle Ax,y \rangle = \langle x,A^*y \rangle,
where ...
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
is different from the original ground state
, 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 state
In physics, a squeezed coherent state is a quantum state that is usually described by two non-commuting observables having continuous spectra of eigenvalues. Examples are position x and momentum p of a particle, and the (dimension-less) elect ...
s. BCS wave function is an example of squeezed coherent state of fermions.
Unified Matrix description
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
transform as
:
where
is a
matrix. Then naturally
:
For Fermion operators, the requirement of
commutation relations
In quantum mechanics, the canonical commutation relation is the fundamental relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform of another). For example,
hat x,\hat p_ ...
reflects in two requirements for the form of matrix
:
and
:
For Boson operators, the
commutation relations
In quantum mechanics, the canonical commutation relation is the fundamental relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform of another). For example,
hat x,\hat p_ ...
require
:
and
:
These conditions can be written uniformly as
:
where
:
where
applies to Fermions and Bosons, respectively.
Diagonalizing a quadratic Hamiltonian using matrix description
Bogoliubov transformation lets us diagonalize a quadratic Hamiltonian
:
by just diagonalizing the matrix
.
In the notations above, it is important to distinguish the operator
and the numeric matrix
.
This fact can be seen by rewriting
as
:
and
if and only if
diagonalizes
, i.e.
.
Useful properties of Bogoliubov transformations are listed below.
See also
*
Holstein–Primakoff transformation The Holstein–Primakoff transformation in quantum mechanics is a mapping to the spin operators from boson creation and annihilation operators, effectively truncating their infinite-dimensional Fock space to finite-dimensional subspaces.
One impo ...
*
Jordan–Wigner transformation
The Jordan–Wigner transformation is a transformation that maps spin operators onto fermionic creation and annihilation operators. It was proposed by Pascual Jordan and Eugene Wigner for one-dimensional lattice models, but now two-dimensional ana ...
*
Jordan–Schwinger transformation
*
Klein transformation
In quantum field theory, the Klein transformation is a redefinition of the fields to amend the spin-statistics theorem.
Bose–Einstein
Suppose φ and χ are fields such that, if ''x'' and ''y'' are spacelike-separated points and ''i'' and ''j' ...
References
Further reading
The whole topic, and a lot of definite applications, are treated in the following textbooks:
*
*
*
*
{{DEFAULTSORT:Bogoliubov Transformation
Quantum field theory