Classical limit of time-dependent quantum field theory-a Schwinger-Dyson approach (original) (raw)
Related papers
arXiv (Cornell University), 2003
Quantum Field Theory (QFT) makes predictions by combining two sets of assumptions: (1) quantum dynamics, such as a Schrodinger or Liouville equation; (2) quantum measurement, such as stochastic collapse to an eigenfunction of a measurement operator. A previous paper defined a classical density matrix R encoding the statistical moments of an ensemble of states of classical second-order Hamiltonian field theory. It proved Tr(RQ)=E(Q), etc., for the usual field operators as defined by Weinberg, and it proved that those observables of the classical system obey the usual Heisenberg dynamic equation. However, R itself obeys dynamics different from the usual Liouville equation! This paper derives those dynamics, and calculates the discrepancy between CFT and normal form QFT in predicting general observables g(Q,P). There is some preliminary evidence for the conjecture that the discrepancies disappear in equilibrium states (bound states and scattering states) for finite bosonic field theories. Even if not, they appear small enough to warrant reconsideration of CFT as a theory of dynamics.
REMARKS ON THE CLASSICAL LIMIT OF QUANTUM FIELD THEORIES
Recently, there has been an increasing interest in computing quantum mechanical corrections to solutions of classical field equations. In this note, we want to proceed in the opposite way and we summarize theorems about the classical limit of relativistic quantum field models. These results are a byproduct of the so called 'constructive' approach to quantum field theory. After a section on generalities, we discuss in Section 2 the situation where no phase transitions occur in the limit h-+0 and in Section 3 we reformulate one result in the case where such a transition occurs (Glimm et al. [7]). We discuss the validity of the loop expansion. It seems however that the tools to show the rigorous validity of soliton calculations are not yet prepared. l. GENERALITIES
1998
Schwinger's Closed-Time Path (CTP) formalism is an elegant way to insure causal-ity for initial value problems in Quantum Field Theory. Feynman's Path Integral on the other hand is much more amenable than Schwinger's differential approach (related to the Schwinger Dyson equations) for non-perturbative (in coupling constant) expansions such as the large-N expansion. By marrying the CTP formalism with a large-N expansion of Feynman's Path Integral approach, we are for the first time able to study the dynamics of phase transitions in quantum field theory settings. We review the Feynman Path Integral representation for the generating functional for the Green's functions described by an initial density matrix. We then show that the large-N expansion for the path integral forms a natural non-perturbative framework for discussing phase transitions in quantum field theory as well as for giving a space time description of a heavy ion collision. We review results for a tim...
Schwinger-Dyson approach to nonequilibrium classical field theory
Physical Review D, 2001
In this paper we discuss a Schwinger-Dyson [SD] approach for determining the time evolution of the unequal time correlation functions of a non-equilibrium classical field theory, where the classical system is described by an initial density matrix at time t = 0. We focus on λφ 4 field theory in 1+1 space time dimensions where we can perform exact numerical simulations by sampling an ensemble of initial conditions specified by the initial density matrix. We discuss two approaches. The first, the bare vertex approximation [BVA], is based on ignoring vertex corrections to the SD equations in the auxiliary field formalism relevant for 1/N expansions. The second approximation is a related approximation made to the SD equations of the original formulation in terms of φ alone. We compare these SD approximations as well as a Hartree approximation with exact numerical simulations.
Relativistic Quantum Mechanics and Quantum Field Theory
From Nanoscale Systems to Cosmology, 2012
A general formulation of classical relativistic particle mechanics is presented, with an emphasis on the fact that superluminal velocities and nonlocal interactions are compatible with relativity. Then a manifestly relativistic-covariant formulation of relativistic quantum mechanics (QM) of fixed number of particles (with or without spin) is presented, based on many-time wave functions and the spacetime probabilistic interpretation. These results are used to formulate the Bohmian interpretation of relativistic QM in a manifestly relativistic-covariant form. The results are also generalized to quantum field theory (QFT), where quantum states are represented by wave functions depending on an infinite number of spacetime coordinates. The corresponding Bohmian interpretation of QFT describes an infinite number of particle trajectories. Even though the particle trajectories are continuous, the appearance of creation and destruction of a finite number of particles results from quantum theory of measurements describing entanglement with particle detectors.
Quantum field theory from classical statistics
2011
An Ising-type classical statistical model is shown to describe quantum fermions. For a suitable time-evolution law for the probability distribution of the Ising-spins our model describes a quantum field theory for Dirac spinors in external electromagnetic fields, corresponding to a mean field approximation to quantum electrodynamics. All quantum features for the motion of an arbitrary number of electrons and positrons, including the characteristic interference effects for two-fermion states, are described by the classical statistical model. For one-particle states in the non-relativistic approximation we derive the Schrödinger equation for a particle in a potential from the time evolution law for the probability distribution of the Ising-spins. Thus all characteristic quantum features, as interference in a double slit experiment, tunneling or discrete energy levels for stationary states, are derived from a classical statistical ensemble. Concerning the particle-wave-duality of quantum mechanics, the discreteness of particles is traced back to the discreteness of occupation numbers or Ising-spins, while the continuity of the wave function reflects the continuity of the probability distribution for the Ising-spins.
Quantum correlation functions and the classical limit
2000
We study the transition from the full quantum mechanical description of physical systems to an approximate classical stochastic one. Our main tool is the identification of the closed-time-path (CTP) generating functional of Schwinger and Keldysh with the decoherence functional of the consistent histories approach. Given a degree of coarse-graining in which interferences are negligible, we can explicitly write a generating functional for the effective stochastic process in terms of the CTP generating functional. This construction gives particularly simple results for Gaussian processes. The formalism is applied to simple quantum systems, quantum Brownian motion, quantum fields in curved spacetime. Perturbation theory is also explained. We conclude with a discussion on the problem of backreaction of quantum fields in spacetime geometry.
'Asymptotic' Path-Theoretic Representations of Quantum Field Equations
2014
In this essay, I attempt to 1 extend the stochastic representation of (Boos, 2007) to nonrelativistic Hamiltonians HHV t t = +0 with time-varying potentialsVt ; 2 develop ‘stationary phase’ analyses for such potentials which substantiate Feynman’s variational analysis of classical action-functionals as = → 0 ; and generalise such analyses to define 3 ‘asymptotic’ solutions of gravitational and gauge-theoretic field equations associated with action-functionals M = G ∫ G V L and M = GT ∫ GT V L as = → 0 .