Interest In Quantum Field Theory Research Articles

Multiplication of the distributions (x±i0) z

Abstract. In previous work of the author, a convolution and multiplication product for the set of Associated Homogeneous Distributions (AHDs) with support in ℝ was defined and fully investigated. Here this definition is used to calculate the multiplication product of homogeneous distributions of the form (x±i0) z , for all z ∈ ℂ ${{z\in \mathbb {C}}}$ . Multiplication products of AHDs generally contain an arbitrary constant if the resulting degree of homogeneity is a negative integer, i.e., if it is a critical product. However, critical products of the forms ( x + i 0 ) a . ( x + i 0 ) b $(x+i0) ^{a}\mathbin {\textbf {.}}(x+i0) ^{b}$ and ( x - i 0 ) a . ( x - i 0 ) b $(x-i0) ^{a}\mathbin {\textbf {.}}(x-i0) ^{b}$ , with a + b ∈ ℤ - ${a+b\in \mathbb {Z}_{-}}$ , are exceptionally unique. This fact combined with Sokhotskii–Plemelj expressions then leads to linear dependencies of the arbitrary constants occurring in products like δ ( k ) . δ ( l ) ${{\delta ^{(k)}\mathbin {\textbf {.}}\delta ^{(l)}}}$ , η ( k ) . δ ( l ) ${{\eta ^{(k)}\mathbin {\textbf {.}}\delta ^{(l)}}}$ , δ ( k ) . η ( l ) ${{\delta ^{(k)}\mathbin {\textbf {.}}\eta ^{(l)}}}$ and η ( k ) . η ( l ) ${{\eta ^{(k)}\mathbin {\textbf {.}}\eta ^{(l)}}}$ for all k , l ∈ ℕ ${{k,l\in \mathbb {N}}}$ ( η ≜ 1 π x - 1 ${{\eta \triangleq \frac{1}{\pi }x^{-1}}}$ ). This in turn gives a unique distribution for products like δ ( k ) . η ( l ) + η ( k ) . δ ( l ) ${{\delta ^{(k)}\mathbin {\textbf {.}}\eta ^{(l)}+\eta ^{(k)}\mathbin {\textbf {.}}\delta ^{(l)}}}$ and δ ( k ) . δ ( l ) - η ( k ) . η ( l ) ${{\delta ^{(k)}\mathbin {\textbf {.}}\delta ^{(l)}-\eta ^{(k)}\mathbin {\textbf {.}}\eta ^{(l)}}}$ . The latter two products are of interest in quantum field theory and appear for instance in products of the partial derivatives of the zero-mass two-point Wightman distribution.

On the prefactor in false vacuum decay

This paper gives a survey of recent work in collaboration with G Dunne concerning a new method for computing determinants in quantum field-theoretic applications, using angular momentum cut-off regularization and renormalization. This method is generally applicable to the situation of computing the quantum fluctuations about a classical configuration that has a symmetry allowing the fluctuation operator to be radially separable. There are many such cases of interest in quantum field theory. Here I describe the case of the false vacuum decay rate in a self-interacting scalar field theory modelling the process of nucleation in a four-dimensional spacetime. The rate prefactor involves quantum fluctuations about the classical bounce solution, which is O(4) symmetric. The computational method is based on the Gelfand–Yaglom approach to determinants of ordinary differential operators, suitably extended to higher dimensions using angular momentum cut-off regularization. I also present a simple new formula for the zero-mode contribution to the fluctuation prefactor, expressed entirely in terms of the asymptotic behaviour of the classical bounce solution.

The Hopf algebra structure of multiple harmonic sums

Multiple harmonic sums appear in the perturbative computation of various quantities of interest in quantum field theory. In this article we introduce a class of Hopf algebras that describe the structure of such sums, and develop some of their properties that can be exploited in calculations.

On a class of distributions of interest in quantum field theory

A class of tempered distributions studied by Araki is considered. These distributions are of interest in quantum field theory since they include certain matrix elements of commutators or anticommutators of local quantum field operators. A number of results concerning the null regions in spacetime of such distributions are presented, for the cases when the null regions are causal complements of closed, convex, causally complete subsets of spacetime. Applications to quantum field theory are discussed, and some immediate applications to the theory of the Klein–Gordon equation and the wave equation are noted.

Onp-adic aspects of some perturbation series

Series with factorial terms, which are of potential interest in quantum field theory and string theory, are considered. Divergent series in the real case are usually p-adic convergent. Using simple and number field invariant methods of summation, rational sums are obtained. Sums of the convergent and divergent counterparts of the same series are connected by adelic summability.

The θ = π nonlinear sigma model is massless

We consider here the O(3) nonlinear sigma model in 1 + 1 dimensions with the topological parameter θ = π, which is of considerable interest in quantum field theory as well as the study of chains of half-integral spin. We first define a lattice regulated version of the model and show that it is massless for all values of the coupling g and that it is governed by the k = 1 Wess-zumino-Witten (WZW) fixed point.

Quantum gravity

The nature of the search for a quantum theory of gravity has undergone significant changes over the last few years. This is partly because the success of renormalized Yang-Mills gauge theory has stimulated interest in quantum field theory leading to a number of new ideas (for example instantons, solitons, monopoles, asymptotic freedom) which, focusing as they do on non-perturbative aspects, are potentially of considerable importance in a gravitational context. There has also been the development of supersymmetry and the associated supergravity theories for which the prognosis for quantization is brighter than normal General Relativity. Finally, a major impact was made by Hawking’s (1975) discovery of the thermal radiation produced when a quantum field propagates in a black hole background. This leads to a remarkable synthesis of thermodynamics, quantum theory and general relativity whose significance for physics has still not yet been fully explored. Traditionally, the methods for quantizing the gravitational field have been divided into ‘canonical’ and ‘covariant’ (Isham et al. 1975). A number of years ago the main attack on the canonical front was the quantization of the classical constraints

Functional Differential Calculus of Operators. II

The calculus of operators developed in a previous paper is here carried through in momentum space. The differential quotient, i.e., the derivative of an operator with respect to a free-field operator is expressed algebraically by means of generalized functions and certain commutators. No recourse is made to configuration space but the resultant calculus in p-space is related to the previously developed one in x-space by means of Fourier transforms. The calculus is presented for spins 0, ½, and 1. The difficulties which were encountered before in the calculus for charged vector fields are resolved by working with a field equation which incorporates the supplementary condition needed to eliminate the spin-zero components.

Functional Differential Calculus of Operators

The functional derivative with respect to operators of operator functionals is defined for operators which satisfy certain commutation relations of interest in quantum field theory. From this definition, a functional differential calculus is developed for functionals of tensor as well as spinor fields. It is noted that an implicit definition of the functional derivative can always be given while an explicit one seems to exist only for those operator fields which need not be restricted by supplementary operator conditions, and for which not more than one derivative occurs in the commutation relations.