Zeta-function derivation of Euler-MacLaurin sum rules

The Euler-MacLaurin summation formula and other sum rules of the same type are derived by the -function method for infinite series.     

Boundaries Immersed in a Scalar Quantum Field

We study the interaction between a scalar quantum field $\hat \phi (x)$, and many different boundary configurations constructed from (parallel and orthogonal) thin planar surfaces on which $\hat \phi (x)$ is constrained to vanish, or to satisfy Neumann conditions. For most of these boundaries the Casimir problem has not previously been investigated. We calculate the canonical and improved vacuum stress tensors $ \langle \hat T_{\mu \nu } (x)\rangle\$ and $ \langle \Theta _{\mu \nu (x)} \rangle\$ of $\hat \phi (x)$; for each example. From these we obtain the local Casimir forces on all boundary planes. For massless fields, both vacuum stress tensors yield identical attractive local Casimir forces in all Dirichlet examples considered. This desirable outcome is not a priori obvious, given the quite different features of $ \langle \hat T_{\mu \nu } (x)\rangle\$ and $ \langle \Theta _{\mu \nu (x)} \rangle\$. For Neumann conditions. $ \langle \hat T_{\mu \nu } (x)\rangle\$ and $ \langle \Theta _{\mu \nu (x)} \rangle\$ lead to attractive Casimir stresses which are not always the same. We also consider Dirichlet and Neumann boundaries immersed in a common scalar quantum field, and find that these repel. The extensive catalogue of worked examples presented here belongs to a large class of completely solvable Casimir problems. Casimir forces previously unknown are predicted, among them ones which might be measurable.     

Temperature Dependence of the CPN-1 Model and the Analogy with Quantum Chromodynamics

The two-dimensional CP^N-1 model — a simple field-theoretic analogue of four-dimensional quantum chromodynamics (QCD) — is analysed and reviewed. The major themes are the temperature dependence of the CP^N-1 model, and the analogy between CP^N-1 and QCD. A detailed treatment of the 1/N approximation of the CP^N-1 model is given. The main results emerging from this approximation are discussed at length. These are: asymptotic freedom, dimensional transmutation, confinement and topological charge nonquantization at zero temperature T = O, screening and topological charge quantization at finite temperature T. The analogy with QCD is explained in detail. A new, qualitative, analysis of the CP^N-1 model at finite temperature is introduced. This approach exploits the conformal invariance of the model to “heat” an arbitrary CP^N-1 field from T = O to finite temperature. This is achieved by conformal-transforming the flat Euclidean space-time of the T = O theory to the cylindrical space-time of the finite temperature theory.     

Multimeron solution of the CP n−1 model

An explicit, general multimeron solution of the CP ^n–1 model is given.     

Classical solutions and the energy-momentum tensor

A hyperelliptic two-meron solution of the massless scalar φ^N theory in $\text{n =}\text{2N}\text{(N ? 2)}$ Euclidean dimensions is given. This solution (which interpolates between the two-meron solution and the instanton solution of this theory) is used to illustrate several theory-independent statements which can be made about the energy-momentum tensor for instanton, meron and elliptic meron solutions of all scale invariant classical field theories.     

On the Decay Analysis of Electroproduced Higher-Spin Hadrons

We study exclusive electroproduction of two hadrons where one or both of the hadrons is a higherspin resonance whose decay is analysed. Interesting electroproduction experiments of this type are discussed. A complete formal apparatus is given for the decay analysis of spin-J particles produced in this way with polarized or unpolarized electron beams and targets. The cases J = 1, 3/2, 2 and 5/2 are worked out in detail. A conventional helicity frame analysis in terms of s-channel helicity amplitudes is given. Also we rearrange the formalism for use in the Gottfried-Jackson frame with everything given in terms of t-channel helicity amplitudes. The t-channel formalism makes it possible to completely separate the contributions from longitudinal and transverse virtual photon t-channel helicity states when only the laboratory azimuthal angle between the lepton and hadron planes is variable.