Keldysh formalism
   HOME

TheInfoList



OR:

In non-equilibrium physics, the Keldysh formalism is a general framework for describing the
quantum mechanical Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
evolution of a system in a non-equilibrium state or systems subject to time varying external fields (
electrical field An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...
,
magnetic field A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
etc.). Historically, it was foreshadowed by the work of
Julian Schwinger Julian Seymour Schwinger (; February 12, 1918 – July 16, 1994) was a Nobel Prize winning American theoretical physicist. He is best known for his work on quantum electrodynamics (QED), in particular for developing a relativistically invariant ...
and proposed almost simultaneously by
Leonid Keldysh Leonid Veniaminovich Keldysh (; 7 April 1931 – 11 November 2016) was a Soviet and Russian physicist. Keldysh was a professor in the I.E. Tamm Theory division of the Lebedev Physical Institute of the Russian Academy of Sciences in Moscow and a ...
and, separately,
Leo Kadanoff Leo Philip Kadanoff (January 14, 1937 – October 26, 2015) was an American physicist. He was a professor of physics (emeritus from 2004) at the University of Chicago and a former President of the American Physical Society (APS). He contributed t ...
and
Gordon Baym Gordon Alan Baym (born July 1, 1935) is an American theoretical physicist. Biography Born in New York City, he graduated from the Brooklyn Technical High School, and received his undergraduate degree from Cornell University in 1956. He earned hi ...
. It was further developed by later contributors such as O. V. Konstantinov and V. I. Perel. Extensions to driven-dissipative open quantum systems is given not only for bosonic systems, but also for fermionic systems. The Keldysh formalism provides a systematic way to study non-equilibrium systems, usually based on the two-point functions corresponding to excitations in the system. The main mathematical object in the Keldysh formalism is the non-equilibrium
Green's function In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if \operatorname is the linear differential ...
(NEGF), which is a two-point function of particle fields. In this way, it resembles the
Matsubara formalism In thermal quantum field theory, the Matsubara frequency summation (named after Takeo Matsubara) is the summation over discrete imaginary frequencies. It takes the following form :S_\eta = \frac\sum_ g(i\omega_n), where \beta = \hbar / k_ T is t ...
, which is based on equilibrium Green functions in imaginary-time and treats only equilibrium systems.


Time evolution of a quantum system

Consider a general quantum mechanical system. This system has the
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 ...
H_0. Let the initial state of the system be the pure state , n \rangle. If we now add a time-dependent perturbation to this Hamiltonian, say H'(t), the full Hamiltonian is H(t) = H_0+H'(t) and hence the system will evolve in time under the full Hamiltonian. In this section, we will see how time evolution actually works in quantum mechanics. Consider a
Hermitian {{Short description, none Numerous things are named after the French mathematician Charles Hermite (1822–1901): Hermite * Cubic Hermite spline, a type of third-degree spline * Gauss–Hermite quadrature, an extension of Gaussian quadrature meth ...
operator \mathcal. In the
Heisenberg picture In physics, the Heisenberg picture (also called the Heisenberg representation) is a formulation (largely due to Werner Heisenberg in 1925) of quantum mechanics in which the operators (observables and others) incorporate a dependency on time, bu ...
of quantum mechanics, this operator is time-dependent and the state is not. The expectation value of the operator \mathcal(t) is given by :\begin \langle \mathcal(t) \rangle &= \langle n , ^(t,0) \, \mathcal(0) \, U(t,0) , n \rangle\\ \end where, due to time evolution of operators in the Heisenberg picture, \mathcal(t) = U^(t,0) \mathcal(0) U(t, 0). The time-evolution unitary operator U(t_2, t_1) is the
time-ordered In theoretical physics, path-ordering is the procedure (or a meta-operator \mathcal P) that orders a product of operators according to the value of a chosen parameter: :\mathcal P \left\ \equiv O_(\sigma_) O_(\sigma_) \cdots O_(\sigma_). H ...
exponential of an integral, U(t_2,t_1)=T(e^). (Note that if the Hamiltonian at one time commutes with the Hamiltonian at different times, then this can be simplified to U(t_2,t_1)=e^.) For perturbative quantum mechanics and
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, it is often more convenient to use the interaction picture. The interaction picture operator is :\begin \mathcal(t) &= ^(t,0) \, \mathcal(0) \, U_0(t,0), \end where U_0(t_1,t_2) = e^ . Then, defining S(t_1,t_2) = U_0^(t_1,t_2)U(t_1, t_2), we have :\begin \langle \mathcal(t) \rangle &= \langle n , ^(t,0) \mathcal(t) S(t,0) , n \rangle\\ \end Since the time-evolution unitary operators satisfy U(t_3, t_2) U(t_2, t_1) = U(t_3, t_1), the above expression can be rewritten as :\begin \langle \mathcal(t) \rangle &= \langle n , ^(\infty,0) S(\infty, t) \mathcal(t) \, S(t,0) , n \rangle\\ \end, or with \infty replaced by any time value greater than t.


Path ordering on the Keldysh contour

We can write the above expression more succinctly by, purely formally, replacing each operator X(t) with a contour-ordered operator X(c) , such that c parametrizes the contour path on the time axis starting at t=0, proceeding to t=\infty, and then returning to t=0. This path is known as the Keldysh contour. X(c) has the same operator action as X(t) (where t is the time value corresponding to c) but also has the additional information of c (that is, strictly speaking X(c_1) \neq X(c_2) if c_1 \neq c_2 , even if for the corresponding times X(t_1) = X(t_2) ). Then we can introduce notation of path ordering on this contour, by defining \mathcal ( X^(c_1) X^(c_2)\ldots X^(c_n) ) = (\pm 1)^X^(c_) X^(c_)\ldots X^(c_) , where \sigma is a permutation such that c_ < c_ < \ldots c_ , and the plus and minus signs are 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 ...
and
fermionic In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
operators respectively. Note that this is a generalization of
time ordering In theoretical physics, path-ordering is the procedure (or a meta-operator \mathcal P) that orders a product of operators according to the value of a chosen parameter: :\mathcal P \left\ \equiv O_(\sigma_) O_(\sigma_) \cdots O_(\sigma_). H ...
. With this notation, the above time evolution is written as :\begin \langle \mathcal(t) \rangle &= \langle n , \mathcal( \mathcal e^) , n \rangle \end Where c corresponds to the time t on the forward branch of the Keldysh contour, and the integral over c' goes over the entire Keldysh contour. For the rest of this article, as is conventional, we will usually simply use the notation X(t) for X(c) where t is the time corresponding to c, and whether c is on the forward or reverse branch is inferred from context.


Keldysh diagrammatic technique for Green's functions

The non-equilibrium Green's function is defined as \begin iG(x_1, t_1, x_2, t_2)= \langle n , T \psi(x_1,t_1) \psi(x_2,t_2) , n \rangle \end. Or, in the interaction picture, \begin iG(x_1, t_1, x_2, t_2)= \langle n , \mathcal (e^ \psi(x_1,t_1) \psi(x_2,t_2)) , n \rangle \end . We can expand the exponential as a Taylor series to obtain the perturbation series :\sum_^\langle n , \mathcal ((-i\int_t (H'(t', +)+ H'(t',-) )dt')^j \psi(x_1,t_1) \psi(x_2,t_2)) , n \rangle / j! . This is the same procedure as in equilibrium diagrammatic perturbation theory, but with the important difference that both forward and reverse contour branches are included. If, as is often the case, H' is a polynomial or series as a function of the elementary fields \psi, we can organize this perturbation series into monomial terms and apply all possible Wick pairings to the fields in each monomial, obtaining a summation of
Feynman diagrams In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduce ...
. However, the edges of the Feynman diagram correspond to different propagators depending on whether the paired operators come from the forward or reverse branches. Namely, : \langle n , \mathcal \psi (x_1, t_1, +) \psi (x_2, t_2, +), n \rangle \equiv G_0^(x_1,t_1 , x_2, t_2)= \langle n, \mathcal\psi (x_1,t_1) \psi (x_2,t_2), n \rangle : \langle n , \mathcal \psi (x_1, t_1, +) \psi (x_2, t_2, -), n \rangle \equiv G_0^(x_1,t_1 , x_2, t_2)= \langle n, \psi (x_1,t_1) \psi (x_2,t_2), n \rangle : \langle n , \mathcal \psi (x_1, t_1, -) \psi (x_2, t_2, +), n \rangle \equiv G_0^(x_1,t_1 , x_2, t_2)= \pm \langle n, \psi (x_2,t_2)\psi (x_1,t_1), n \rangle : \langle n , \mathcal \psi (x_1, t_1, -) \psi (x_2, t_2, -), n \rangle \equiv G_0^(x_1,t_1 , x_2, t_2)= \langle n, \mathcal\psi (x_1,t_1) \psi (x_2,t_2), n \rangle where the anti-time ordering \mathcal orders operators in the opposite way as time ordering and the \pm sign in G_0^ is for bosonic or fermionic fields. Note that G_0^ is the propagator used in ordinary ground state theory. Thus, Feynman diagrams for correlation functions can be drawn and their values computed the same way as in ground state theory, except with the following modifications to the Feynman rules: Each internal vertex of the diagram is labeled with either + or - , while external vertices are labelled with -. Then each (unrenormalized) edge directed from a vertex a (with position x_a, time t_a and sign s_a) to a vertex b (with position x_b, time t_b and sign s_b) corresponds to the propagator G_0^(x_a,t_a , x_b, t_b). Then the diagram values for each choice of \pm signs (there are 2^ such choices, where v is the number of internal vertices) are all added up to find the total value of the diagram.


See also

*
Spin Hall effect The spin Hall effect (SHE) is a transport phenomenon predicted by Russian physicists Mikhail I. Dyakonov and Vladimir I. Perel in 1971. It consists of the appearance of spin accumulation on the lateral surfaces of an electric current-carrying sa ...
*
Kondo effect In physics, the Kondo effect describes the scattering of conduction electrons in a metal due to magnetic impurities, resulting in a characteristic change i.e. a minimum in electrical resistivity with temperature. The cause of the effect was fir ...


References


Other

# # # # # # #{{Cite journal, last1=Gen, first1=Tatara, last2=Kohno, first2=Hiroshi, last3=Shibata, first3=Junya, date=2008, title=Microscopic approach to current-driven domain wall dynamics, journal=Physics Reports, volume=468, issue=6, pages=213–301, arxiv=0807.2894, doi=10.1016/j.physrep.2008.07.003, bibcode=2008PhR...468..213T, s2cid=119257806 #Gianluca Stefanucci and Robert van Leeuwen (2013). "Nonequilibrium Many-Body Theory of Quantum Systems: A Modern Introduction" (Cambridge University Press, 2013). DOI: https://doi.org/10.1017/CBO9781139023979 #Robert van Leeuwen, Nils Erik Dahlen, Gianluca Stefanucci, Carl-Olof Almbladh and Ulf von Barth, "Introduction to the Keldysh Formalism", Lectures Notes in Physics 706, 33 (2006). arXiv:cond-mat/0506130 Condensed matter physics Electromagnetism