A variational principle for bound states in quantum field theory

We develop a Rayleigh-Ritz variational method for estimating relativistic, multi-particle bound state energies in any (weak-coupling) quantum field theory. A comparison is made with bound state energies derived from the Bethe-Salpeter equation in the Wick-Cutkosky model. Possible applications to QCD are discussed.     

A variational approach to second-order multisymplectic field theory

This paper presents a geometric-variational approach to continuous and discrete second-order field theories following the methodology of [Marsden, Patrick, Shkoller, Comm. Math. Phys. 199 (1998) 351–395]. Staying entirely in the Lagrangian framework and letting Y denote the configuration fiber bundle, we show that both the multisymplectic structure on J³Y as well as the Noether theorem arise from the first variation of the action function. We generalize the multisymplectic form formula derived for first-order field theories in [Marsden, Patrick, Shkoller, Comm. Math. Phys. 199 (1998) 351–395], to the case of second-order field theories, and we apply our theory to the Camassa–Holm (CH) equation in both the continuous and discrete settings. Our discretization produces a multisymplectic-momentum integrator, a generalization of the Moser–Veselov rigid body algorithm to the setting of nonlinear PDEs with second-order Lagrangians.     

Semiclassical field theory approach to quantum chaos

We construct a field theory to describe energy averaged quantum statistical properties of systems which are chaotic in their classical limit. An expression for the generating function of general statistical correlators is presented in the form of a functional supermatrix non-linear σ-model where the effective action involves the evolution operator of the classical dynamics. Low-lying degrees of freedom of the field theory are shown to reflect the irreversible classical dynamics describing relaxation of phase space distributions. The validity of this approach is investigated over a wide range of energy scales. As well as recovering the universal long-time behavior characteristic of random matrix ensembles, this approach accounts correctly for the short-time limit yielding results which agree with the diagonal approximation of periodic orbit theory.     

Functional approach to scattering in quantum field theory

A functional approach to scattering theory in quantum field theory is developed by deriving an explicit functional expression fortransition amplitudes. In applications, the formalism avoids dealing with noncommutativity problems of field operators, avoids solving the field equations, avoids dealing with the often quite complicated continual (path) integrals, and avoids combinatoric problems associated with Feynman rules and the old-fashioned Wick's theorem. Finally, it avoids explicitly taking mass shell limits as in the LSZ formalism. The basic idea of the formalism is to use the quantum action principle followed by a systematic analysis of the concept of stimulated emissions as applied to particles of any spin, and is a generalization of an earlier method applied by the author to the much simpler situation of quantum mechanics.     

Zero-time one-particle states in quantum field theory

In the framework of the weakly-coupledP()₂-models we construct perturbation approximations of vectors of a dense set of the state space, especially vectors of the one-particle state subspace, by polynomials of zero-time fields acting on the vacuum state, with rigorous control of the remainders.     

Time-dependent variational method for sine-Gordon quantum field theory

The sine-Gordon model in 1+1 dimensions is studied within the Schrödinger framework for field theory. In particular we evaluate the effective potential and examine the finiteness ofm(t), the soliton mass, for allt.     

A self-consistent approach to quantum field theory for extended particles

A notion of quantum space-time is introduced, physically defined as the totality of all flows of quantum test particles in free fall. In quantum space-time the classical notion of deterministic inertial frames is replaced by that of stochastic frames marked by extended particles. The same particles are used both as markers of quantum space-time points as well as natural clocks, each species of quantum test particle thus providing a standard for space-time measurements. In the considered flat-space case, the fluctuations in coordinate values with respect to stochastic frames are described by coordinate probability amplitudes related to irreducible stochastic phase space representations of the Poincaré group. Lagrangian field theory on quantum space-time is formulated. The ensuing equations of motion for interacting fields contain no singularities in their nonlinear terms, and therefore can be handled by methods borrowed from classical nonlinear analysis.Supported in part by an NSERC grant.     

The entropy bound for local quantum field theory

We investigate the entropy bound for local quantum field theory in this Letter. Both the bosonic and fermionic fields confined to an asymptotically flat space–time are examined. By imposing the non-gravitational collapse condition, we find both of them are limited by the same entropy bound A3/4$A^{3/4}$, where A is the boundary area of the region where the quantum fields are contained in. The gap between this entropy bound and the holographic entropy has been verified.     

A variational approach to distribution function theory

We present a novel formalism for the generation of integral equations for the distribution functions of fluids. It is based on a cumulant expansion for the free energy. Truncation of the expansion at theKth term and minimization of the resulting approximation leads to equations for the distribution functions up toKth order.The formalism is not limited to systems with two-body interactions and does not require the addition of closure relations to yield a complete set of equations. In fact, it automatically generates superposition approximations, such as the Kirkwood three-body superposition approximation or the Fisher-Kopeliovich four-body one.The conceptual approach is adapted from the cluster variation method of lattice theory.     

Holographic entropy bound and local quantum field theory

I show how the holographic entropy bound can be derived from elementary flat-spacetime quantum field theory when the total energy of Fock states is constrained gravitationally. This energy constraint makes the Fock space dimension (whose logarithm is the maximum entropy) finite for both bosons and fermions. Despite the elementary nature of my analysis, it results in an upper limit on entropy in remarkable agreement with the holographic bound, and also provides a microscopic deviation of a more general entropy bound recently introduced by Gour.     

A real-time approach to nonequilibrium quantum field theory of relativistic fields

Summary The basic ingredients of a real-time nonequilibrium quantumfield theoretical approach, based on a generalization of thermofield dynamics, are briefly reviewed. In particular, the possible advantages of its application to relativistic models, as the φ⁴-model for the Higgs field are pointed out. Also the differences between the above approach and others using the Feynman's path integral method are underlined. ?Angelo della Riccia? Fellow.     

Star-product approach to quantum field theory: The free scalar field

The star-quantization of the free scalar field is developed by introducing an integral representation of the normal star-product. A formal connection between the Feynman path integral in the holomorphic representation and the star-exponential is established for the interacting scalar fields.     

Dynamical renormalization group approach to relaxation in quantum field theory

The real time evolution and relaxation of expectation values of quantum fields and of quantum states are computed as initial value problems by implementing the dynamical renormalization group (DRG). Linear response is invoked to set up the renormalized initial value problem to study the dynamics of the expectation value of quantum fields. The perturbative solution of the equations of motion for the field expectation values of quantum fields as well as the evolution of quantum states features secular terms, namely terms that grow in time and invalidate the perturbative expansion for late times. The DRG provides a consistent framework to resum these secular terms and yields a uniform asymptotic expansion at long times. Several relevant cases are studied in detail, including those of threshold infrared divergences which appear in gauge theories at finite temperature and lead to anomalous relaxation. In these cases the DRG is shown to provide a resummation akin to Bloch-Nordsieck but directly in real time and that goes beyond the scope of Bloch-Nordsieck and Dyson resummations. The nature of the resummation program is discussed in several examples. The DRG provides a framework that is consistent, systematic, and easy to implement to study the non-equilibrium relaxational dynamics directly in real time that does not rely on the concept of quasiparticle widths.     

Descending problem in Green's function approach to quantum field theory

The question how to determine lower many-point functions in terms of higher ones, which we call the descending problem, is discussed for the (ø⁴)₁₊₃ model of quantum field theory. Equations to be considered are non-linear non-compact operator equations in complex Banach spaces.Several sufficient sets of conditions for convergence of successive approximation schemes are presented for small values of the renormalised coupling constant. Local uniqueness of solution is proved under certain conditions.     

Bound states and scattering from bound states in the light-front field theory dynamics

Starting from the Weinberg rules we derive a covariant form of the relativistic Schrödinger equation and formulate the bound state problem in the light-front field theory dynamics. We present an explicit rule for embodying the two-body subsystem in the three-body space and demonstrate that the cluster decomposition property is explicitly preserved in the light front field theory dynamics. As an application of these results we write amplitudes forπd→nN ^*, πd→πpn, andπd→πd, in the impulse approximation, in terms of the internal bound state wave functions and two-body reducedt-matrix elements.     

A curiosity concerning the role of coherent states in quantum field theory

In the usual Fock quantisation of fields in Minkowski space-time, one has the result that the expectation value of the quantum Hamiltonian in any coherent state equals the energy of the classical field at which the state is peaked. It is shown that this property can be used tocharacterise the usual Fock representation. It is also pointed out that the entire analysis goes through for a substantially more general class of systems including, in particular, Bose fields in arbitrary stationary space-times.