Finite-temperature quantum field theory in Minkowski space

We formulate finite-temperature quantum field theories in Minkowski space (real time) using Feynman path integrals. We show that at non-zero temperature a new field arises which plays the role of a ghost field and is necessary for unambiguous Feynman rules. Consequently, the finite-temperature Lagrangian is different from the zero-temperature one and a new, discrete Z₂ symmetry arises. We discuss the functional formalism and spontaneous symmetry breakdown at finite temperature and also the possibility of spontaneous breakdown of the (thermal) Z₂ symmetry.     

Decoupling theorem of quantum field theory in Minkowski space

The decoupling theorem of quantum field theory with massive particles is proved in Minkowski space when all the masses of the theory are led to go to infinity. The theorem establishes the vanishing property, in the distributional sense, of (absolutely convergent) Feynman amplitudes in a model independent way with subtractions performed at the origin. This extends previous efforts dealing with the proof of the theorem in the Euclidean region.     

A quantum field theory in momentum space of constant curvature

The model of a polarization operator in the rainbow diagram approximation in momentum space of constant curvature is studied. An integral equation is obtained for the absorptive component of the hadron polarization operator in a stereographic parametrization of the de Sitter space. The solution of the integral equation is found for the case of zero mass of the exchange particle. The sections and mean multiplicity of hadron generation in e⁺e^– annihilation are calculated. The infrared divergences which develop in an exact summation of the class of diagrams considered here manifest themselves as a multiplicative renormalization constant.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 20–24, September, 1981.     

Remarks on the quantum field theory in lattice space. II

We consider a strong-coupling approach to ⁴-meson theory as formulated in a lattice space which is of simple cubic type having lattice constanta and total volumeV. Self-adjointness and regularity of the Hamiltonian are established. The strong-coupling perturbation series are examined for the cases with and without mass renormalization. The series for ground state and one-particle state as well as for their energies are shown to converge when the coupling constant is sufficiently large, say > _c (sufficient condition). The bounds _c we have found increase with the total volumeV and/or the cut-off momentuma ^–1. Some other features of the strong-coupling perturbation theory are also discussed.     

Remarks on the quantum field theory in lattice space. I

We calculate the Gelfand functionsE(f,g;a) for quantized field in lattice space,a being the lattice constant. In the limita 0 the functionals take on two different forms depending upon the potentialF[] of the lattice Hamiltonian (coupling between different lattice sites not included). IfF[] is of a short-range type (see text for definition) the limit functional is Gaussian. The corresponding representation of CCR is reducible and its realization apparently non-unique unlessF[] is quadratic. The most natural realization is to represent the field as a linear combination of Fock fields whose masses are given by the excitation energies of the lattice Hamiltonian. IfF[] is of a long-range type, the limit functional takes the more general form once studied byAraki.     

Exactly solvable models of conformal-invariant quantum field theory in D-dimensional space

The method for exact solution of a certain class of models of conformal quantum field theory in D-dimensional Euclidean space is proposed. The method allows one to derive closed differential equations for all the Green functions and also algebraic equations to scale dimensions of all field. A scalar field P of a scale dimension d_p = D − 2 is needed for nontrivial solutions to exist. At D ≠ 2 this field is converted to a constant that coincides with the central charge of two-dimensional theories. A new class of D = 2 models has been obtained, where the infinite-parametric symmetry is not manifest. The two-dimensional Wess-Zumino model is used to illustrate the method of solution.     

Scattering theory and discrete-time quantum walks

We study quantum walks on general graphs from the point of view of scattering theory. For a general finite graph we choose two vertices and attach one half line to each, and consider walks that proceed from one half line, through the graph, to the other. The probability of starting on one line and reaching the other after n steps can be expressed in terms of the transmission amplitude for the graph.     

Intermediate statistics in two space dimensions in a lattice-regularized Hamiltonian quantum field theory

The non-compact Abelian Higgs model with a Chern-Simons term added is investigated within a Hamiltonian formalism, regularized by a spatial lattice. Transmutted matter fields are constructed, which show intermediate statistics interpolating continuously between the Bose and the Fermi case. Their unbounded localization properties are analyzed in detail. Moreover, the braid structure emerging, when exchanges in a product of transmuted fields are performed, is disclosed.     

Zitterbewegung in quantum field theory

Traditionally, the zitterbewegung （ZB） of the Dirac electron has just been studied at the level of quantum mechanics. Seeing the fact that an old interest in ZB has recently been rekindled by the investigations on spintronic, graphene, and superconducting systems, etc., this paper presents a quantum-field-theory investigation on ZB and obtains the conclusion that, the ZB of an electron arises from the influence of virtual electron-positron pairs （or vacuum fluctuations） on the electron.     

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.     

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     

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.