Global effects in quaternionic quantum field theory

We present some striking global consequences of a model quaternionic quantum field theory which is locally complex. We show how making the quaternionic structure a dynamical quantity naturally leads to the prediction of cosmic strings and nonbaryonic hot dark matter candidates.     

Finite temperature effects in quantum field theory

We consider quantum electrodynamics at finite temperatures. By making use of the real time formalism we compute, on the one-loop level, the finite-temperature correction to the mass of the electron and to the anomalous magnetic moment a_e^th. The gauge-invariant correction to the electron mass is found to be a ten percent effect at a temperature of the order of 2×10¹⁰ K. Some astrophysical implications of this result are briefly discussed. The leading temperature correction to the anomalous magnetic moment of the electron is, at a temperature of 300 K, found to be one order of magnitude smaller than the τ-lepton contribution to a_e^th.     

Duality in quantum field theory

We postulate a convergent version of operator product expansions on the vacuum and explore some of their consequences. They lead to structures much reminiscent of dual resonance models.     

Hyperfunction quantum field theory

The quantum field theory in terms of Fourier hyperfunctions is constructed. The test function space for hyperfunctions does not containC functions with compact support. In spite of this defect the support concept ofH-valued Fourier hyperfunctions allows to formulate the locality axiom for hyperfunction quantum field theory.     

Quaternionic quantum field theory

We show that a quaternionic quantum field theory can be formulated when the numbers of bosonic and fermionic degrees of freedom are equal and the fermions, as well as the bosons, obey a second order wave equation. The theory takes the form of either a functional integral with quaternion-imaginary Lagrangian, or a Schrödinger equation and transformation theory for quaternion-valued wave functions, with a quaternion-imaginary Hamiltonian. The connection between the two formulations is developed in detail, and many related issues, including the breakdown of the correspondence principle and the Hilbert space structure, are discussed.     

Point-form quantum field theory

We examine canonical quantization of relativistic field theories on the forward hyperboloid, a Lorentz-invariant surface of the form x_μx^μ = τ². This choice of quantization surface implies that all components of the 4-momentum operator are affected by interactions (if present), whereas rotation and boost generators remain interaction free—a feature characteristic of Dirac’s “point-form” of relativistic dynamics. Unlike previous attempts to quantize fields on space-time hyperboloids, we keep the usual plane-wave expansion of the field operators and consider evolution of the system generated by the 4-momentum operator. We verify that the Fock-space representations of the Poincaré generators for free scalar and spin-1/2 fields look the same as for equal-time quantization. Scattering is formulated for interacting fields in a covariant interaction picture and it is shown that the familiar perturbative expansion of the S-operator is recovered by our approach. An appendix analyzes special distributions, integrals over the forward hyperboloid, that are used repeatedly in the paper.     

Asymptotic quantum field theory

In this work, we have formulated a quantum field theory with the following features: (1) it is free of divergences and also free of the unphysical procedure of renormalization; (2) the interaction can be specified in our dynamical equation; (3) it gives consistent (renormalized) results for a renormalizable interaction; and (4) it is also applicable to a nonrenormalizable interaction.Our formulation assumes the Bogoliubov causality condition, the strong unitarity condition, the asymptotic condition, the completeness of the in-field (or out-field) and Lorentz covariance. The off-as well as on-mass-shell values play an important role in our formulation and they are treated carefully by employing the operator derivatives with respect to the in-field which satisfies the weak (or ordinary) free-field equation. A dynamical equation which determines all S-matrix elements is developed. The specification of interactions is made in this equation. It is then shown that finite perturbative solutions exist to any order and that the scattering matrix elements can be obtained by imposing suitable boundary conditions. No restrictions on the types of interactions appear in our formulation and, therefore, the same procedure can be carried out for renormalizable as well as for non-renormalizable interactions. For renormalizable interactions, in particular, our dynamical equation reduces to the one previously proposed by Pugh.     

Topological quantum field theory

A twisted version of four dimensional supersymmetric gauge theory is formulated. The model, which refines a nonrelativistic treatment by Atiyah, appears to underlie many recent developments in topology of low dimensional manifolds; the Donaldson polynomial invariants of four manifolds and the Floer groups of three manifolds appear naturally. The model may also be interesting from a physical viewpoint; it is in a sense a generally covariant quantum field theory, albeit one in which general covariance is unbroken, there are no gravitons, and the only excitations are topological.On leave from Department of Physics, Princeton University. Research supported in part by NSF Grants No. 80-19754, 86-16129, 86-20266     

Vacuum Cherenkov effect in logarithmic nonlinear quantum theory

We describe the radiation phenomena which can take place in the physical vacuum such as Cherenkov-type shock waves. Their macroscopical characteristics - cone angle, flash duration, radiation yield and spectral distribution - are computed. It turns out that the radiation yield is proportional to the square of the proper energy scale of the vacuum which serves also as the vacuum instability threshold and the natural ultraviolet cutoff. While the analysis is mainly based on the theory engaging the logarithmic nonlinear quantum wave equation, some of the obtained results must be valid for any Lorentz-invariance violating theory describing the vacuum by (effectively) continuous medium in the long-wavelength approximation.     

Asymptotic estimates in quantum field theory

The Lipatov method for obtaining the asymptotic estimates in perturbation theory is reviewed. The same method is applied to estimate the high order weighted moments of the Green functions for the field theory in a box. It is shown that in this case some negative features of the Lipatov method disappear.Invited talk at the Symposium on Mathematical Methods in the Theory of Elementary Particles, Liblice castle, Czechoslovakia, June 18–23, 1978.     

Arrival time in quantum field theory

Via the proper-time eigenstates (event states) instead of the proper-mass eigenstates (particle states), free-motion time-of-arrival theory for massive spin-1/2 particles is developed at the level of quantum field theory. The approach is based on a position-momentum dual formalism. Within the framework of field quantization, the total time-of-arrival is the sum of the single event-of-arrival contributions, and contains zero-point quantum fluctuations because the clocks under consideration follow the laws of quantum mechanics.     

Scattering theory in quantum field theory in configuration space

The recent analysis of the propagation of relativistic particles inspacetime and their localization problem is used to develop scattering theory in quantum field theory inconfiguration space. An explicit functional expression is derived for the underlying transition amplitudes having a consistent probabilistic interpretation. Some of the basic ingredients in the analysis are the functional approach developed earlier for transition amplitudes and the amplitudes for stimulated emission of particles by external sources in spacetime.