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Bogoliubov transformation

In theoretical physics, the Bogoliubov transformation, also known as Bogoliubov-Valatin transformation, were independently developed in 1958 by Nikolay Bogolyubov and John George Valatin for finding solutions of BCS theory in a homogeneous system.[1][2] The Bogoliubov transformation is an isomorphism 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 Hamiltonians, which yields the stationary solutions of the corresponding Schrödinger equation. The Bogoliubov transformation is also important for understanding the Unruh effect, Hawking radiation, 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.

Contents

Single bosonic mode exampleEdit

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

 

Define a new pair of operators

 
 

for complex number u and v, where the latter is the Hermitian conjugate 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, viz.

 

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 u and v 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.

ApplicationsEdit

The most prominent application is by Nikolai Bogoliubov himself in the context of superfluidity.[3][4] Other applications comprise Hamiltonians and excitations in the theory of antiferromagnetism.[5] 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.

Fermionic modeEdit

For the anticommutation relation

 ,

the same transformation with u and v becomes

 

To make the transformation canonical, u and v can be parameterized as

 
 

ApplicationsEdit

The most prominent application is again by Nikolai Bogoliubov himself, this time for the BCS theory of superconductivity.[5][6][7][8] 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 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, this method is applicable since it may describe the "pairing energy" of nucleons in a heavy element.[9]

Multimode exampleEdit

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:

 

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

 

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 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 states. BCS wave function is an example of squeezed coherent state of fermions.[10]

ReferencesEdit

  1. ^ Valatin, J. G. (March 1958). "Comments on the theory of superconductivity". Il Nuovo Cimento. 7 (6): 843–857. Bibcode:1958NCim....7..843V. doi:10.1007/bf02745589.
  2. ^ Bogoljubov, N. N. (March 1958). "On a new method in the theory of superconductivity". Il Nuovo Cimento. 7 (6): 794–805. Bibcode:1958NCim....7..794B. doi:10.1007/bf02745585.
  3. ^ N. N. Bogoliubov: On the theory of superfluidity, J. Phys. (USSR), 11, p. 23 (1947), (Izv. Akad. Nauk Ser. Fiz. 11, p. 77 (1947)).
  4. ^ Bogolubov [sic], N. "On the theory of Superfluidity" (PDF). Advances of Physical Sciences. Lebedev Physical Institute. Retrieved 27 April 2017.
  5. ^ a b See e.g. the textbook by Charles Kittel: Quantum theory of solids, New York, Wiley 1987.
  6. ^ Boboliubov, N. N. (1 Jan 1958). "A new method in the theory of superconductivity. I". Soviet Physics (U.S.S.R.) JETP. 7 (1): 41–46.
  7. ^ Bogoliubov, N. N. (July 1958). "A new method in the theory of superconductivity III" (PDF). Soviet Physics (U.S.S.R.) JETP. 34 (7): 51–55.
  8. ^ Bogolyubov, N. N.; Tolmachev, V. V.; Shirkov, D. V. (November 1958). "A new method in the theory of superconductivity". Fortschitte der Physik. 6 (11–12): 605–682. Bibcode:1958ForPh...6..605B. doi:10.1002/prop.19580061102.
  9. ^ Strutinsky, V.M. (April 1967). "Shell effects in nuclear masses and deformation energies". Nuclear Physics A. 95 (2): 420–442. Bibcode:1967NuPhA..95..420S. doi:10.1016/0375-9474(67)90510-6.
  10. ^ Svozil, K. (1990), "Squeezed Fermion states", Phys. Rev. Lett. 65, 3341-3343. doi:10.1103/PhysRevLett.65.3341

Further readingEdit

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

  • J.-P. Blaizot and G. Ripka: Quantum Theory of Finite Systems, MIT Press (1985)
  • A. Fetter and J. Walecka: Quantum Theory of Many-Particle Systems, Dover (2003)
  • Ch. Kittel: Quantum theory of solids, Wiley (1987)
  • M. Wagner, Unitary Transformations in Solid State Physics, Elsevier Science, 1986