Schwinger's Quantum Action Principle: From Dirac's formulation through Feynman's path integrals, the Schwinger-Keldysh method, quantum field theory, to source theory
K. A. Milton
备注：
116 pages, submitted to SpringerBriefs. Very substantially updated version of 1402.3762, with new chapters on Quantum Field Theory and Source Theory
Starting from the earlier notions of stationary action principles, we show how Julian Schwinger's Quantum Action Principle descended from Dirac's formulation, which independently led Feynman to his path-integral formulation of quantum mechanics. The connection between the two is brought out, and applications are discussed. The Keldysh-Schwinger time-cycle method of extracting matrix elements in nonequilibrium situations is described. The variational formulation of quantum field theory and the development of source theory constitute the latter part of this work. In this document, derived from Schwinger's lectures over four decades, the continuity of concepts, such as that of Green's functions, becomes apparent.
Starting from the earlier notions of stationary action principles, we show how Schwinger's Quantum Action Principle descended from Dirac's formulation, which independently led Feynman to his path-integral formulation of quantum mechanics. The connection between the two is brought out, and applications are discussed. The Keldysh-Schwinger time-cycle method of extracting matrix elements is described. Part II will discuss the variational formulation of quantum electrodynamics and the development of source theory.
A compact correlation-function expansion is developed for nth order optical susceptibilities in the frequency domain using the Keldysh-Schwinger loop. By not keeping track of the relative time ordering of bra and ket interactions at the two branches of the loop, the resulting expressions contain only n+1 basic terms, compared to the 2n terms required for a fully time-ordered density matrix description. Superoperator Green's function expressions for the nth order suscpeptibility derived using both expansions reflect different types of interferences between pathways .These are demonstrated for c…
We examine, in a quantum mechanical setting, the Hilbert space representation of the BRST symmetry associated with Schwinger-Keldysh path integrals. This structure had been postulated to encode important constraints on influence functionals in coarse-grained systems with dissipation, or in open quantum systems. Operationally, this entails uplifting the standard Schwinger-Keldysh two-copy formalism into superspace by appending BRST ghost degrees of freedom. These statements were previously argued at the level of the correlation functions. We provide herein a complementary perspective by working…
The time evolution of an extended quantum system can be theoretically described in terms of the Schwinger-Keldysh functional integral formalism, whose action conveniently encodes the information about the dynamics. We show here that the action of quantum systems evolving in thermal equilibrium is invariant under a symmetry transformation which distinguishes them from generic open systems. A unitary or dissipative dynamics having this symmetry naturally leads to the emergence of a Gibbs thermal stationary state. Moreover, the fluctuation-dissipation relations characterizing the linear response …
We initiate the study of open quantum field theories using holographic methods. Specifically, we consider a quantum field theory (the system) coupled to a holographic field theory at finite temperature (the environment). We investigate the effects of integrating out the holographic environment with an aim of obtaining an effective dynamics for the resulting open quantum field theory. The influence functionals which enter this open effective action are determined by the real-time (Schwinger-Keldysh) correlation functions of the holographic thermal environment. To evaluate the latter, we exploit…
Motivated by cosmological Hartle-Hawking and microcanonical density matrix prescriptions for the quantum state of the Universe we develop Schwinger-Keldysh in-in formalism for generic nonequilibrium dynamical systems with the initial density matrix. We build the generating functional of in-in Green's functions and expectation values for a generic density matrix of the Gaussian type and show that the requirement of particle interpretation selects a distinguished set of positive/negative frequency basis functions of the wave operator of the theory, which is determined by the density matrix param…
Schwinger-Keldysh Path Integral for Gauge theories
Syracuse University（雪城大学）；Institute for Quantum & Information Sciences, Syracuse University（雪城大学量子与信息科学研究所）；Kavli Institute for the Physics and Mathematics of the Universe (WPI) the University of Tokyo（东京大学宇宙物理数学研究所（WPI）卡弗里研究所）；Perimeter Institute for Theoretical Physics（理论物理 perimeter 研究所）；Abdus Salam Centre for Theoretical Physics, Imperial College（帝国理工学院阿卜杜斯·萨拉姆理论物理中心）；CERCA, Department of Physics, Case Western Reserve University（凯斯西储大学物理系 CERCA）
The Schwinger-Keldysh diagram technique is usually involved in the calculation of real-time in-in correlation functions. In the case of a thermal state, one can analytically continue imaginary-time Matsubara correlation functions to real times. Nevertheless, not all real-time correlation functions usually can be obtained by such procedure. Moreover, numerical analytic continuation is an ill-posed problem. Thus, even in the case of a thermal state one may need for the Schwinger-Keldysh formalism. If the potential of a system admits degenerate minima, instantonic effects enter the game, so one s…
We unveil the universal (model-independent) symmetry satisfied by Schwinger-Keldysh quantum field theories whenever they describe equilibrium dynamics. This is made possible by a generalization of the Schwinger-Keldysh path-integral formalism in which the physical time can be re-parametrized to arbitrary contours in the complex plane. Strong relations between correlation functions, such as the fluctuation-dissipation theorems, are derived as immediate consequences of this symmetry of equilibrium. In this view, quantum non-equilibrium dynamics -- e.g. when driving with a time-dependent potentia…
We studied a QED analog of the Franz-Keldysh effect, and the interplay between the non-perturbative (the Schwinger mechanism) and the perturbative particle production mechanism from the vacuum in the presence of a strong slow field superimposed by a weak field. We found that the Franz-Keldysh effect significantly affects the particle production: (i) the perturbative particle production occurs even below the threshold energy; (ii) the perturbative production becomes the most efficient just above the threshold energy; and (iii) an oscillating behavior appears in the production number above the t…
We present a phase-integral formulation of dynamically assisted Schwinger pair production of scalar charges to find the pair production density under a strong low-frequency field and a weak high-frequency field. The leading WKB action was the Schwinger formula determined by the constant field, whose corrections are determined by the Keldysh parameter for the oscillating field, $mω_{q}/qE_{q}$. We found a systematic expression of the leading WKB action as a power series in the Keldysh parameter, of which coefficients are given as integrals of the product of fields in the complex time domain. Fo…
Schwinger-Keldysh non-perturbative field theory of open quantum systems beyond the Markovian regime: Application to spin-boson and spin-chain-boson models
Department of Physics and Astronomy, University of Delaware（特拉华大学物理与天文学系）；H. H. Wills Physics Laboratory, University of Bristol（布里斯托大学H.H.威尔斯物理实验室）；Department of Mathematical Sciences, University of Delaware（特拉华大学数学科学系）
In order to popularize the so called Schwinger's method we reconsider the Feynman propagator of two non-relativistic systems: a charged particle in a uniform magnetic field and a charged harmonic oscillator in a uniform magnetic field. Instead of solving the Heisenberg equations for the position and the canonical momentum operators, ${\bf R}$ and ${\bf P}$, we apply this method by solving the Heisenberg equations for the gauge invariant operators ${\bf R}$ and $\mathversion{bold}$π$ = {\bf P}-e{\bf A}$, the latter being the mechanical momentum operator. In our procedure we avoid fixing the gau…
We study the dynamics of a quantum Brownian particle weakly coupled to a thermal bath. Working in the Schwinger-Keldysh formalism, we develop an effective action of the particle up to quartic terms. We demonstrate that this quartic effective theory is dual to a stochastic dynamics governed by a non-linear Langevin equation. The Schwinger-Keldysh effective theory, or the equivalent non-linear Langevin dynamics, is insufficient to determine the out of time order correlators (OTOCs) of the particle. To overcome this limitation, we construct an extended effective action in a generalised Schwinger-…
In this paper, we present a formulation of Schwinger's Method for non relativistic propogators in momentum space. While the aforementioned method has been used heavily in some papers to deduce various non relativistic propogators in position space, we show that a variation of method in momentum space works well. We also show how to extract the Propogator for the concerned case for the harmonic oscillator.
The electromagnetic field in a typical geometry of the Casimir effect is described in the Schwinger-Keldysh formalism. The main result is the photon distribution function (Keldysh Green function) in any stationary state of the field. A two-plate geometry with a sliding interface in local equilibrium is studied in detail, and full agreement with the results of Rytov fluctuation electrodynamics is found. As an illustration, plots are shown for a spectrum of the energy density between the plates.
Schwinger pair production in some of space-dependent electromagnetic fields is studied analytically by using worldline instantons formalism for scalar quantum electrodynamics. With the increase of the modified Keldysh parameter $γ_{keb}$, the pair production rate decreases, the region of instanton paths expands regardless of the electromagnetic field form. We also find that all of the paths of the instantons for various electromagnetic fields are exactly in the same plane while with the different elliptical shapes. Moreover, the magnitudes of the normalized instantons action with different ext…
