Misplaced Pages

#253746

26-502: Keldysh (Russian: Келдыш ) may refer to: Science [ edit ] Keldysh formalism , for studying non-equilibrium quantum systems Akademik Mstislav Keldysh , a 1980 Russian scientific research vessel renowned for its visits to the wreck of the RMS Titanic Keldysh Institute of Applied Mathematics , a Russian research institute Keldysh (crater) ,

52-688: A crater on the Moon 2186 Keldysh , an asteroid People [ edit ] Leonid Keldysh (1931–2016), Russian physicist, former director of the Lebedev Physical Institute (1988–1994), later a member of the physics faculty at Texas A&M University Mstislav Keldysh (1911–1978), Russian mathematician, president of the Soviet Academy of Sciences (1961–1978) Lyudmila Keldysh (1904–1976), mathematician, sister of Mstislav, wife of Pyotr Novikov Topics referred to by

78-567: A vertex a {\displaystyle a} (with position x a {\displaystyle x_{a}} , time t a {\displaystyle t_{a}} and sign s a {\displaystyle s_{a}} ) to a vertex b {\displaystyle b} (with position x b {\displaystyle x_{b}} , time t b {\displaystyle t_{b}} and sign s b {\displaystyle s_{b}} ) corresponds to

104-440: Is a permutation such that c σ ( 1 ) < c σ ( 2 ) < … c σ ( n ) {\displaystyle c_{\sigma (1)}<c_{\sigma (2)}<\ldots c_{\sigma (n)}} , and the plus and minus signs are for bosonic and fermionic operators respectively. Note that this is a generalization of time ordering . With this notation,

130-493: Is defined as i G ( x 1 , t 1 , x 2 , t 2 ) = ⟨ n | T ψ ( x 1 , t 1 ) ψ ( x 2 , t 2 ) | n ⟩ {\displaystyle {\begin{aligned}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{aligned}}} . Or, in

156-412: Is different from Wikidata All article disambiguation pages All disambiguation pages Keldysh formalism 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

182-522: Is given by where, due to time evolution of operators in the Heisenberg picture, O ( t ) = U † ( t , 0 ) O ( 0 ) U ( t , 0 ) {\displaystyle {\mathcal {O}}(t)=U^{\dagger }(t,0){\mathcal {O}}(0)U(t,0)} . The time-evolution unitary operator U ( t 2 , t 1 ) {\displaystyle U(t_{2},t_{1})}

208-760: Is known as the Keldysh contour. X ( c ) {\displaystyle X(c)} has the same operator action as X ( t ) {\displaystyle X(t)} (where t {\displaystyle t} is the time value corresponding to c {\displaystyle c} ) but also has the additional information of c {\displaystyle c} (that is, strictly speaking X ( c 1 ) ≠ X ( c 2 ) {\displaystyle X(c_{1})\neq X(c_{2})} if c 1 ≠ c 2 {\displaystyle c_{1}\neq c_{2}} , even if for

234-706: Is often more convenient to use the interaction picture . The interaction picture operator is where U 0 ( t 1 , t 2 ) = e − i H 0 ( t 1 − t 2 ) {\displaystyle U_{0}(t_{1},t_{2})=e^{-iH_{0}(t_{1}-t_{2})}} . Then, defining S ( t 1 , t 2 ) = U 0 † ( t 1 , t 2 ) U ( t 1 , t 2 ) , {\displaystyle S(t_{1},t_{2})=U_{0}^{\dagger }(t_{1},t_{2})U(t_{1},t_{2}),} we have Since

260-494: Is the time-ordered exponential of an integral, U ( t 2 , t 1 ) = T ( e − i ∫ t 1 t 2 H ( t ′ ) d t ′ ) . {\displaystyle U(t_{2},t_{1})=T(e^{-i\int _{t_{1}}^{t_{2}}H(t')dt'}).} (Note that if the Hamiltonian at one time commutes with

286-449: The ± {\displaystyle \pm } sign in G 0 − + {\displaystyle G_{0}^{-+}} is for bosonic or fermionic fields. Note that G 0 − − {\displaystyle G_{0}^{--}} is the propagator used in ordinary ground state theory. Thus, Feynman diagrams for correlation functions can be drawn and their values computed

SECTION 10

#1732779658254

312-441: The Hamiltonian at different times, then this can be simplified to U ( t 2 , t 1 ) = e − i ∫ t 1 t 2 H ( t ′ ) d t ′ {\displaystyle U(t_{2},t_{1})=e^{-i\int _{t_{1}}^{t_{2}}H(t')dt'}} .) For perturbative quantum mechanics and quantum field theory , it

338-595: The above expression more succinctly by, purely formally, replacing each operator X ( t ) {\displaystyle X(t)} with a contour-ordered operator X ( c ) {\displaystyle X(c)} , such that c {\displaystyle c} parametrizes the contour path on the time axis starting at t = 0 {\displaystyle t=0} , proceeding to t = ∞ {\displaystyle t=\infty } , and then returning to t = 0 {\displaystyle t=0} . This path

364-408: The above time evolution is written as Where c {\displaystyle c} corresponds to the time t {\displaystyle t} on the forward branch of the Keldysh contour, and the integral over c ′ {\displaystyle c'} goes over the entire Keldysh contour. For the rest of this article, as is conventional, we will usually simply use

390-1114: The corresponding times X ( t 1 ) = X ( t 2 ) {\displaystyle X(t_{1})=X(t_{2})} ). Then we can introduce notation of path ordering on this contour, by defining T c ( X ( 1 ) ( c 1 ) X ( 2 ) ( c 2 ) … X ( n ) ( c n ) ) = ( ± 1 ) σ X ( σ ( 1 ) ) ( c σ ( 1 ) ) X ( σ ( 2 ) ) ( c σ ( 2 ) ) … X ( σ ( n ) ) ( c σ ( n ) ) {\displaystyle {\mathcal {T_{c}}}(X^{(1)}(c_{1})X^{(2)}(c_{2})\ldots X^{(n)}(c_{n}))=(\pm 1)^{\sigma }X^{(\sigma (1))}(c_{\sigma (1)})X^{(\sigma (2))}(c_{\sigma (2)})\ldots X^{(\sigma (n))}(c_{\sigma (n)})} , where σ {\displaystyle \sigma }

416-564: The exponential as a Taylor series to obtain the perturbation series 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 ′ {\displaystyle H'} is a polynomial or series as a function of the elementary fields ψ {\displaystyle \psi } , we can organize this perturbation series into monomial terms and apply all possible Wick pairings to

442-484: The fields in each monomial, obtaining a summation of Feynman diagrams . 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, where the anti-time ordering T ¯ {\displaystyle {\mathcal {\overline {T}}}} orders operators in the opposite way as time ordering and

468-452: The initial state of the system be the pure state | n ⟩ {\displaystyle |n\rangle } . If we now add a time-dependent perturbation to this Hamiltonian, say H ′ ( t ) {\displaystyle H'(t)} , the full Hamiltonian is H ( t ) = H 0 + H ′ ( t ) {\displaystyle H(t)=H_{0}+H'(t)} and hence

494-719: The interaction picture, i G ( x 1 , t 1 , x 2 , t 2 ) = ⟨ n | T c ( e − i ∫ c H ′ ( t ′ ) d t ′ ψ ( x 1 , t 1 ) ψ ( x 2 , t 2 ) ) | n ⟩ {\displaystyle {\begin{aligned}iG(x_{1},t_{1},x_{2},t_{2})=\langle n|{\mathcal {T_{c}}}(e^{-i\int _{c}H'(t')dt'}\psi (x_{1},t_{1})\psi (x_{2},t_{2}))|n\rangle \end{aligned}}} . We can expand

520-431: The notation X ( t ) {\displaystyle X(t)} for X ( c ) {\displaystyle X(c)} where t {\displaystyle t} is the time corresponding to c {\displaystyle c} , and whether c {\displaystyle c} is on the forward or reverse branch is inferred from context. The non-equilibrium Green's function

546-491: The propagator G 0 s a s b ( x a , t a , x b , t b ) {\displaystyle G_{0}^{s_{a}s_{b}}(x_{a},t_{a},x_{b},t_{b})} . Then the diagram values for each choice of ± {\displaystyle \pm } signs (there are 2 v {\displaystyle 2^{v}} such choices, where v {\displaystyle v}

SECTION 20

#1732779658254

572-457: The same term [REDACTED] This disambiguation page lists articles associated with the title Keldysh . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Keldysh&oldid=907302329 " Category : Disambiguation pages Hidden categories: Articles containing Russian-language text Short description

598-407: 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 + {\displaystyle +} or − {\displaystyle -} , while external vertices are labelled with − {\displaystyle -} . Then each (unrenormalized) edge directed from

624-522: 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 operator O {\displaystyle {\mathcal {O}}} . In the Heisenberg picture of quantum mechanics, this operator is time-dependent and the state is not. The expectation value of the operator O ( t ) {\displaystyle {\mathcal {O}}(t)}

650-584: The system. The main mathematical object in the Keldysh formalism is the non-equilibrium Green's function (NEGF), which is a two-point function of particle fields. In this way, it resembles the Matsubara formalism , which is based on equilibrium Green functions in imaginary-time and treats only equilibrium systems. Consider a general quantum mechanical system. This system has the Hamiltonian H 0 {\displaystyle H_{0}} . Let

676-514: The time-evolution unitary operators satisfy U ( t 3 , t 2 ) U ( t 2 , t 1 ) = U ( t 3 , t 1 ) {\displaystyle U(t_{3},t_{2})U(t_{2},t_{1})=U(t_{3},t_{1})} , the above expression can be rewritten as or with ∞ {\displaystyle \infty } replaced by any time value greater than t {\displaystyle t} . We can write

#253746