The energy-momentum spectrum in the Yukawa2 quantum field theory

We prove that the Yukawa₂ quantum field theory with periodic boundary conditions satisfies the spectral condition, i.e., the joint spectrum of the energy operatorH and the momentum operatorP is contained in the forward cone. In addition, the -bound is obtained.     

The Yukawa2 quantum field theory: Lorentz invariance

The Yukawa quantum field theory in two-dimensional space-time is considered. It is proved that the CPT invariant states with periodic boundary conditions for the (renormalized) Yukawa₂ model without cutoffs are Lorentz invariant.     

Effective action for the Yukawa2 quantum field theory

Using a rigorous version of the renormalization group we construct the effective action for theY ₂ model. The construction starts with integrating out the bosonic field which eliminates the large fields problem. Studying the soobtained purely fermionic theory proceeds by a series of convergent perturbation expansions. We show that the continuum limit of the effective action exists and its perturbation expansion is Borel summable.     

The FKG inequality for the Yukawa2 quantum field theory

We establish the FKG correlation inequality for the Euclidean scalar Yukawa₂ quantum field model and, when the Fermi mass is zero, for pseudoscalar Yukawa₂. To do so we approximate the quantum field model by a lattice spin system and show that the FKG inequality for this system follows from a positivity condition on the fundamental solution of the Euclidean Dirac equation with external field. We prove this positivity condition by applying the Vekua-Bers theory of generalized analytic functions.Research partially supported by the National Research Council of Canada.Alfred P. Sloan Foundation Fellow.     

Higher order estimates for the Yukawa2 quantum field theory

Uniqueness of KMS states is proved for one-dimensional quantum lattice system. Sakai's theorem on uniqueness of KMS states is generalized to cases of non-commutative generators.     

Volume dependence of Schwinger functions in the Yukawa2 quantum field theory

We prove upper bounds on the partition function and Schwinger functions for the Euclidean Yukawa₂ quantum field theory which depend on the interaction volume Λ only through a term of the form (const)^|Λ|. We also prove a lower bound of the form (const)^|λ| for the partition function. We work throughout in the Matthews-Salam representation with the fermions integrated out.     

On conformal invariance in quantum field theory

We study the transformation law of interacting fields under the universal convering group of the conformal group. It is defined with respect to the representations of the discrete series. These representations are field representations in the sense that they act on finite component fields defined over Minkowski space. The conflict with Einstein causality is avoided as in the case of free fields with canonical dimension. Furthermore, we determine the conformal invariant two-point function of arbitrary spin. Our result coincides with that obtained by Rühl. In particular, we investigate the two-point function of symmetric and traceless tensor fields and give the explicit form of the trace terms.     

Global conformal invariance in quantum field theory

Suppose that there is given a Wightman quantum field theory (QFT) whose Euclidean Green functions are invariant under the Euclidean conformal groupSO _e (5,1). We show that its Hilbert space of physical states carries then a unitary representation of the universal (-sheeted) covering group* of the Minkowskian conformal group SO _e (4, 2)₂. The Wightman functions can be analytically continued to a domain of holomorphy which has as a real boundary an -sheeted covering of Minkowski-spaceM ⁴. It is known that* can act on this space and that admits a globally*-invariant causal ordering; is thus the natural space on which a globally*-invariant local QFT could live. We discuss some of the properties of such a theory, in particular the spectrum of the conformal HamiltonianH=1/2(P ⁰+K ⁰).As a tool we use a generalized Hille-Yosida theorem for Lie semigroups. Such a theorem is stated and proven in Appendix C. It enables us to analytically continue contractive representations of a certain maximal subsemigroup of to unitary representations of*.     

Lorentz covariance of the λ(ϕ4)2 quantum field theory

We prove that the (⁴)₂ quantum field theory model is Lorentz covariant, and that the corresponding theory of bounded observables satisfies all the Haag-Kastler axioms. For each Poincaré transformation {a, } and each bounded regionB of Minkowski space we construct a unitary operatorU which correctly transforms the field bilinear forms:U(x, t)U*=({a, } (x, t)), for (x, t) B. We also consider the von Neumann algebra of local observables, consisting of bounded functions of the field operators (f)= (x, t)f(x, t)dx dt, suppf B. We define a *-isomorphism by setting _{{a, }}(A)=U A U*. The mapping is a representation of the Poincaré group by *-automorphisms of the normed algebra of local observables.Supported in part by the US Air Force Office of Scientific Research, Contract No. 44620-67-C-0029.Alfred P. Sloan Foundation Fellow. Supported in part by the US Air Force Office of Scientific Research, Contract F 44620-70-C-0030.     

Chiral scale and conformal invariance in 2D quantum field theory

It is well known that a local, unitary Poincaré-invariant 2D quantum field theory with a global scaling symmetry and a discrete non-negative spectrum of scaling dimensions necessarily has both a left and a right local conformal symmetry. In this Letter, we consider a chiral situation beginning with only a left global scaling symmetry and do not assume Lorentz invariance. We find that a left conformal symmetry is still implied, while right translations are enhanced either to a right conformal symmetry or a left U(1) Kac-Moody symmetry.     

Scale and conformal invariance in quantum field theory

We study the relation between invariance under rigid and local changes of length scale. In two dimensions, we complete an argument of Zamolodchikov showing that the rigid invariance implies the local under broad conditions. In three or more dimensions we are unable to find either a general proof or a counterexample, but we find some new conformally invariant systems.     

Three-dimensional Gross-Neveu model under the condition of Lorentz invariance violation

A 3-D Gross-Neveu model with Lorentz-violating term b _μ is studied. The effective potential V _eff with a real or imaginary Lorentz-violating term b _μ is calculated. The gap equation is obtained and the influence of the presence of an additional term b _μ on the symmetry of the theory is examined.     

Connection between the eigenvalues of the energy operator and the s matrix in quantum field theory

A formula that connects the eigenvalues of the energy operator of a system of interacting fields with the S matrix generated by the operator of the interaction energy is derived.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 11–15, June, 1980.     

Quantum scalar field in quantum gravity: the propagator and Lorentz invariance in the spherically symmetric case

We recently studied gravity coupled to a scalar field in spherical symmetry using loop quantum gravity techniques. Since there are local degrees of freedom one faces the “problem of dynamics”. We attack it using the “uniform discretization technique”. We find the quantum state that minimizes the value of the master constraint for the case of weak fields and curvatures. The state has the form of a direct product of Gaussians for the gravitational variables times a modified Fock state for the scalar field. In this paper we do three things. First, we verify that the previous state also yields a small value of the master constraint when one polymerizes the scalar field in addition to the gravitational variables. We then study the propagators for the polymerized scalar field in flat space-time using the previously considered ground state in the low energy limit. We discuss the issue of the Lorentz invariance of the whole approach. We note that if one uses real clocks to describe the system, Lorentz invariance violations are small. We discuss the implications of these results in the light of Hořava’s Gravity at the Lifshitz point and of the argument about potential large Lorentz violations in interacting field theories of Collins et al.     

Lorentz invariance of lightcone string field theory: (I). Bosonic closed string

We construct the Lorentz generators for bosonic closed lightcone string field theory with cubic interactions, and prove that the Lorentz algebra closes if the spacetime dimension is 26.