Functional Mellin Transforms

Conventional functional/path integrals used in physics often can be defined as infinite-dimensional analogs of Fourier transforms. It turns out that the infinite-dimensional analog of the Mellin transform similarly defines a class of functional integrals. The associated functional integrals, called functional Mellin transforms, are useful tools for probing non-commutative function spaces in general and $C^\ast$-algebras in particular: Functional Mellin transforms can be used to define functional traces, logarithms, and determinants. Several interesting aspects are explored.Comment: This is the second of two papers representing an expanded version of arXiv:1308.106

A Non-standard Standard Model

This paper examines the Standard Model under the strong-electroweak gauge group $SU_S(3)\times U_{EW}(2)$ subject to the condition $u_{EW}(2)\not\cong su_I(2)\oplus u_Y(1)$. Physically, the condition ensures that all electroweak gauge bosons interact with each other prior to symmetry breaking --- as one might expect from $U(2)$ invariance. This represents a crucial shift in the notion of physical gauge bosons: Unlike the Standard Model which posits a change of Lie algebra basis induced by spontaneous symmetry breaking, here the basis is unaltered and $A,\,Z^0,\,W^\pm$ represent (modulo $U_{EW}(2)$ gauge transformations) the physical bosons both \emph{before} and after spontaneous symmetry breaking. Our choice of $u_{EW}(2)$ basis requires some modification of the matter field sector of the Standard Model. Careful attention to the product group structure calls for strong-electroweak degrees of freedom in the $(\mathbf{3},\mathbf{2})$ and the $(\mathbf{3},\overline{\mathbf{2}})$ of $SU_S(3)\times U_{EW}(2)$ that possess integer electric charge just like leptons. These degrees of freedom play the role of quarks, and they lead to a modified Lagrangian that nevertheless reproduces transition rates and cross sections equivalent to the Standard Model. The close resemblance between quark and lepton electroweak doublets in this picture suggests a mechanism for a phase transition between quarks and leptons that stems from the product structure of the gauge group. Our hypothesis is that the strong and electroweak bosons see each other as a source of decoherence. In effect, leptons get identified with the $SU_S(3)$-trace of quark representations. This mechanism allows for possible extensions of the Standard Model that don't require large inclusive multiplets of matter fields. As an example, we propose and investigate a model that turns out to have some promising cosmological implications.Comment: Added to discussion of dark energy. This is an extended version of arXiv:hep-ph/0408266 and arXiv:hep-ph/040830