Radiative decays and the SU(6) Lie algebra

We present research on radiative decays of vector [Formula: see text] to pseudoscalar [Formula: see text] particles ([Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] quark system) using broken symmetry techniques in the infinite-momentum frame and equal-time commutation relations and the [Formula: see text] Lie algebra, and conducted without ascribing any specific form to meson quark structure or intra-quark interactions. We utilize the physical electromagnetic current [Formula: see text] including its singlet [Formula: see text] term and focus on the [Formula: see text] 35-plet. We derive new relations involving the electromagnetic current (including its singlet — proportional to the [Formula: see text] singlet). Remarkably, we find that the electromagnetic current singlet plays an intrinsic role in understanding the physics of radiative decays and that the charged and neutral [Formula: see text] meson radiative decays into [Formula: see text] are due entirely to the singlet term in [Formula: see text]. Although there is insufficient radiative decay experimental data available at this time, parametrization of possible predicted values of [Formula: see text] is made. For conciseness and self-containment, we compute all [Formula: see text] Lie algebra simple roots, positive roots, weights and fundamental weights which allow the construction of all [Formula: see text] representations. We also derive all nonzero [Formula: see text] generator commutators and anticommutators — useful for further research on grand unified theories.

VACUUM STABILITY OF THE ${\mathcal{PT}}$-SYMMETRIC (-ϕ4) SCALAR FIELD THEORY

In this paper, we study the vacuum stability of the classical unstable (-ϕ4) scalar field potential. Regarding this, we obtained the effective potential, up to second-order in the coupling, for the theory in 1+1 and 2+1 space–time dimensions. We found that the obtained effective potential is bounded-from-below, which proves the vacuum stability of the theory in space–time dimensions higher than the previously studied 0+1 case. In our calculations, we used the canonical quantization regime in which one deals with operators rather than classical functions used in the path integral formulation. Therefore, the non-Hermiticity of the effective field theory is obvious. Moreover, the method we employ implements the canonical equal-time commutation relations and the Heisenberg picture for the operators. Thus, the metric operator is implemented in the calculations of the transition amplitudes. Accordingly, the method avoids the very complicated calculations needed in other methods for the metric operator. To test the accuracy of our results, we obtained the exponential behavior of the vacuum condensate for small coupling values, which has been obtained in the literature using other methods. We assert that this work is interesting, as all the studies in the literature advocate the stability of the (-ϕ4) theory at the quantum mechanical level while our work extends the argument to the level of field quantization.

PATH INTEGRALS FOR QUASI-HERMITIAN HAMILTONIANS

Inner products in quasi-Hermitian quantum theories, and hence probabilities, are defined through a metric that depends on the details of the Hamiltonians themselves. We shall see that the functional integral for quasi-Hermitian theories, and hence Feynman diagrams, for example, can be calculated without needing to evaluate the metric. The reason turns out be that their derivation is based fundamentally on the Heisenberg equations of motion and the canonical equal-time commutation relations, which retain their standard form. As an application, we show how co-ordinate transformations in the path integral can enable us to recover equivalent Hermitian Hamiltonians.

PARTICLE PRODUCTION AND TRANSPLANCKIAN PROBLEM ON THE NONCOMMUTATIVE PLANE

We consider the coherent state approach to noncommutativity and we derive from it an effective quantum scalar field theory. We show how the noncommutativity can be taken into account by a suitable modification of the Klein–Gordon product, and of the equal-time commutation relations. We prove that, in curved space, the Bogoliubov coefficients are unchanged, hence the number density of the produced particle is the same as for the commutative case. What changes though is the associated energy density, and this offers a simple solution to the transplanckian problem.

THE MASTER WARD IDENTITY

In the framework of perturbative quantum field theory (QFT) we propose a new, universal (re)normalization condition (called 'master Ward identity') which expresses the symmetries of the underlying classical theory. It implies for example the field equations, energy-momentum, charge- and ghost-number conservation, renormalized equal-time commutation relations and BRST-symmetry. It seems that the master Ward identity can nearly always be satisfied, the only exceptions we know are the usual anomalies. We prove the compatibility of the master Ward identity with the other (re)normalization conditions of causal perturbation theory, and for pure massive theories we show that the 'central solution' of Epstein and Glaser fulfills the master Ward identity, if the UV-scaling behavior of its individual terms is not relatively lowered. Application of the master Ward identity to the BRST-current of non-Abelian gauge theories generates an identity (called 'master BRST-identity') which contains the information which is needed for a local construction of the algebra of observables, i.e. the elimination of the unphysical fields and the construction of physical states in the presence of an adiabatically switched off interaction.

GAUGE-INVARIANT FIELDS IN THE TEMPORAL GAUGE, COULOMB-GAUGE FIELDS, AND THE GRIBOV AMBIGUITY

We examine the relation between Coulomb-gauge fields and the gauge-invariant fields constructed in the temporal gauge for two-color QCD by comparing a variety of properties, including their equal-time commutation rules and those of their conjugate chromoelectric fields. We also express the temporal-gauge Hamiltonian in terms of gauge-invariant fields and show that it can be interpreted as a sum of the Coulomb-gauge Hamiltonian and another part that is important for determining the equations of motion of temporal-gauge fields, but that can never affect the time evolution of "physical" state vectors. We also discuss multiplicities of gauge-invariant temporal-gauge fields that belong to different topological sectors and that, in a previous work, were shown to be based on the same underlying gauge-dependent temporal-gauge fields. We argue that these multiplicities of gauge-invariant fields are manifestations of the Gribov ambiguity. We show that the differential equation that bases the multiplicities of gauge-invariant fields on their underlying gauge-dependent temporal-gauge fields has nonlinearities identical to those of the "Gribov" equation, which demonstrates the nonuniqueness of Coulomb-gauge fields. These multiplicities of gauge-invariant fields — and, hence, Gribov copies — appear in the temporal gauge, but only with the imposition of Gauss' law and the implementation of gauge invariance; they do not arise when the theory is represented in terms of gauge-dependent fields and Gauss' law is left unimplemented.

A CALCULATION OF THE MAGNETIC MOMENT OF THE Δ++

A fully relativistic, gauge-invariant, and non-perturbative calculation of the Δ++ magnetic moment, μΔ++, is made using equal-time commutation relations (ETCRs) and the dynamical concepts of asymptotic SU F(2) flavor symmetry and asymptotic level realization. Physical masses of the Δ and nucleon are used in this broken symmetry calculation. It is found that μΔ++=2.04μp, where μp is the proton magnetic moment. This result is very similar to that obtained by using SU(6) ⊗ O(3) symmetry or the static quark model.