arXiv subjects
Publications and source records attributed to J. LaChapelle.
Some well-known examples of constrained quantum systems commonly quantized via Feynman path integrals are re-examined using the notion of conditional integrators introduced in [1]. The examples yield some new perspectives on old results. As an interesting new application, the formalism is used to construct a physical model of average prime counting functions modeled as a constrained gamma process.
We define and develop a framework to understand functional integrals as countable families of Banach-valued Haar integrals on locally compact topological groups. The definition forgoes the goal of constructing a genuine measure on an infinite-dimensional space of functions, and instead provides for a topological realization of localization in the infinite-dimensional domain. This yields measurable subspaces that characterize meaningful functional integrals and a scheme that possesses significant potential for representing non-commutative Banach algebras suitable for mathematical physics applications. The framework includes, within a broader structure, other successful approaches that define functional integrals in restricted cases, and it suggests new and potentially useful functional integrals that go beyond the standard Gaussian case. In particular, functional integrals based on skew-Hermitian and Kähler quadratic forms are defined and developed. Also defined are gamma-type and Poisson-type functional integrals based on linear forms suggested by the gamma probability distribution. These non-Gaussian functional integrals are expected to play an important role in generating $C^\ast$-algebras of quantum systems. To illustrate and test the framework, examples and applications are presented in the contexts of quantum field theory and number theory.
Conventional functional/path integrals used in physics are most often defined and understood, either explicitly or implicitly, as the infinite-dimensional analog of Fourier transform. In this paper, the infinite-dimensional analog of Mellin transform is defined and developed. The associated functional integrals are useful tools for probing non-commutative function spaces in general and $C^\ast$-algebras in particular. Functional Mellin transforms are used to define the functional analogs of resolvents, complex powers, traces, logarithms, and determinants. Several aspects of these objects are examined and applied to various constructs in mathematical physics. As substantial applications, we construct Mellin-based QFT generating functionals for bosonic and fermionic degrees of freedom, explore connections between functional complex powers and scattering amplitudes, interpret renormalization from a functional Mellin perspective, define a parameter-dependent entropy that formally justifies the replica trick, and explore $L$-functions associated with functional traces and determinants.
We examine the Standard Model under the electroweak symmetry group $U_{EW}(2)$ subject to the Lie algebra condition $\mathfrak{u}_{EW}(2)\not\cong \mathfrak{su}_{I}(2)\oplus \mathfrak{u}_{Y}(1)$. Physically, the condition ensures that all electroweak gauge bosons interact with each other prior to symmetry breaking. This represents a crucial shift in the identification 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 the physical bosons both before and after spontaneous symmetry breaking. Our choice of $\mathfrak{u}_{EW}(2)$ requires some modification of the matter field representation of the Standard Model. For $U_{EW}(2)$, there are two pertinent representations ${\mathbf{2}}$ and its $U(2)$-conjugate ${\mathbf{2^c}}$ related by a global gauge transformation that squares to minus the identity. The product group structure calls for strong-electroweak degrees of freedom in the $(\mathbf{3},\mathbf{2})$ and the $(\mathbf{3},{\mathbf{2^c}})$ of $SU_C(3)\times U_{EW}(2)$ 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 suggests a mechanism for a speculative phase transition between quarks and leptons that stems from the product structure of the symmetry group. Our hypothesis is that the strong and electroweak bosons see each other as a source of decoherence. In effect, lepton representations get identified with the $SU(3)$-trace-reduced quark representations. This mechanism allows for possible extensions of the Standard Model that don't require large inclusive multiplets of matter fields.
Zeros and poles of $k$-tuple zeta functions, that are defined here implicitly, enable localization onto prime-power $k$-tuples in pair-wise coprime $k$-lattices $\mathfrak{N}_k$. As such, the set of all $\mathfrak{N}_k$ along with their associated zeta functions encode the positive natural numbers $\mathbb{N}_{>1}$. Consequently, counting points of $\mathbb{Z}_{\geq0}$ can be implemented in $\{\mathfrak{N}_k\}$. Exploiting this observation, we derive explicit formulae for counting prime-power $k$-tuples and use them to count lattice points in well-behaved bounded regions in $\mathbb{R}^2$. In particular, we count the lattice points contained in the circle $S^1$. The counting readily extends to well-behaved bounded regions in $\mathbb{R}^n$.
$U(n)$ is a semi-direct product group that is characterized by non-trivial homomorphisms mapping $U(1)$ into the automorphism group of $SU(n)$. For $U(3)$, there are three non-trivial homomorphisms that induce three separate defining representations. In a toy model of $U(3)$ Yang-Mills (endowed with a suitable inner product) coupled to massive fermions, this renders three distinct covariant derivatives acting on a single matter field. By employing a $\mathrm{mod}\,3$ permutation of the vector space carrying the defining representation induced by a ``large'' gauge transformation, the three covariant derivatives and one matter field can alternatively be expressed as a single covariant derivative acting on three distinct species of matter fields possessing the same $U(3)$ quantum numbers. One can interpret this as three species of matter fields in the defining representation.
A key theorem formulated in the context of functional Mellin transforms generalizes the important relationship $\exp\mathrm{tr} M=\det\exp M$. Along with the involution symmetry of the zeta function, the theorem suggests a strategy for tackling the Riemann hypothesis.
We propose $Sp\,(8,\R)$ and its Langlands dual $SO(9,\R)$ as dynamical groups for closed quantum systems. Restricting here to the non-compact group $Sp\,(8,\R)$, the quantum theory is constructed and investigated. The functional Mellin transform plays a prominent role in defining the quantum theory. It provides a bridge between the quantum algebra of observables and the algebra of operators on Hilbert spaces furnishing unitary representations that are induced from a distinguished parabolic subgroup of $Sp\,(8,\C)$. As well, the parabolic subgroup renders a fiber bundle construction that models what can be described as a matrix quantum gauge theory. The formulation is strictly quantum mechanics: no \emph{a priori} space-time is assumed and the only geometrical input comes indirectly from the group manifold. But what appears on the surface to be a fairly simple-minded model turns out to have a capacious structure suggesting some compelling physical interpretations regarding space-time and fundamental interactions.
Restricting attention to kinematics, we develop the $C^\ast$-algebraic quantum mechanics of $Sp(8,\mathbb{C})$. The non-compact group does double duty: it furnishes the quantum Hilbert space through induced representations, and it spawns the quantum $C^\ast$-algebra through a crossed product construction. The crossed product contains operators associated with the lie algebra of $Sp(8,\mathbb{C})$ whose spectra can be interpreted as a $\mathrm{dim}_{\mathbb{C}}=20$ non-commutative phase space with a dynamical, commutative $\mathrm{dim}_{\mathbb{C}}=10$ configuration subspace and an internal $U(4,\mathbb{C})$ symmetry. The construction realizes quantization without first passing through the classical domain, and it exhibits apparent geometry.
We conjecture average counting functions for prime $k$-tuples based on a gamma distribution hypothesis for prime powers. The conjecture is closely related to the Hardy-Littlewood conjecture for $k$-tuples but yields better estimates. Possessing average counting functions along with their corresponding exact counting functions allows to implicitly define pertinent $k$-tuple zeta functions. The $k$-tuple zeta functions in turn allow construction of $k$-tuple analogs of explicit formulae. If the zeros of the (implicitly defined) $k$-tuple zeta can be determined, the explicit formulae should yield a (dis)proof of the $k$-tuple analog of the prime number theorem.
Starting with Zhang's theorem on the infinitude of prime doubles, we give an inductive argument that there exists an infinite number of prime $k$-tuples for at least one admissible set $\mathcal{H}_k=\{h_1,\ldots,h_k\}$ for each $k$.
For gauge theories with direct product internal symmetry groups, the relationship between internal quantum numbers (charges) and coupling strengths is examined. In these types of theories, the Lagrangian density may contain non-trivial factors multiplying the matter field terms, and these factors can modify the interaction coupling strengths i.e., the gauge/matter field vertex factors. Consequently, a matter field can carry a given internal charge yet couple to the associated gauge field with an apparent fractional charge. An example with $SU(3)\otimes U(2)$ symmetry is presented in which the matter fields can have integer U(2) charges but fractional U(2) coupling strengths.
Within the context of the Standard Model, quarks are placed in a $(\mathbf{3},\mathbf{2})\oplus (\mathbf{3},\bar{\mathbf{2}})$ matter field representation of $U_{EW}(2)$. Although the quarks carry unit intrinsic electric charge in this construction, anomaly cancellation constrains the Lagrangian in such a way that the quarks' associated currents couple to the photon with the usual 2/3 and 1/3 fractional electric charge associated with conventional quarks. The resulting model is identical to the Standard Model in the $SU_C(3)$ sector: However, in the $U_{EW}(2)$ sector it is similar but not necessarily equivalent. Off hand, the model appears to be phenomenologically equivalent to the conventional quark model in the electroweak sector for experimental conditions that preclude observation of individual constituent currents. On the other hand, it is conceivable that detailed analyses for electroweak reactions may reveal discrepancies with the Standard Model in high energy and/or large momentum transfer reactions. The possibility of quarks with integer electric charge strongly suggests the notion that leptons and quarks are merely different manifestations of the same underlying field. A speculative model is proposed in which a phase transition is assumed to occur between $SU_C(3)\otimes U_{EM}(1)$ and $U_{EM}(1)$ regimes. This immediately; explains the equality of lepton/quark generations and lepton/hadron electric charge, relates neutrino oscillations to quark flavor mixing, reduces the free parameters of the Standard Model, and renders the issue of quark confinement moot.
Functional integrals are defined in terms of locally compact topological groups and their associated Banach-valued Haar integrals. This approach generalizes the functional integral scheme of Cartier and DeWitt-Morette. The definition allows a construction of functional Mellin transforms. In turn, the functional Mellin transforms can be used to define functional traces, logarithms, and determinants. The associated functional integrals are useful tools for probing function spaces in general and $C^\ast$-algebras in particular. Several interesting aspects are explored.
Analogy with Bayesian inference is used to formulate constraints within a scheme for functional integration proposed by Cartier and DeWitt-Morette. According to the analogy, functional counterparts of conditional and conjugate probability distributions are introduced for integrators. The analysis leads to some new functional integration tools and methods that can be applied to the study of constrained dynamical systems.
If the prime numbers are pseudo-randomly distributed, then analogy with quantum systems suggests that counting primes might be modeled by a non-homogeneous Poisson process. Consequently, postulating underlying gamma statistics, more-or-less standard heuristic arguments borrowed from quantum mechanics in the context of functional integration allows to derive analytic expressions of several average counting functions associated with prime numbers. The expressions are certain sums of incomplete gamma functions that are closely related to logarithmic-type integral functions --- which in turn are well-known to give the asymptotic dependence of the various counting functions up to error terms. The relatively broad success of quantum heuristics applied to functional integrals in general along with the excellent numerical accuracy of the analytic expressions for the average counting functions provide strong evidence of a gamma distribution for prime powers.
Exact summatory functions that count the number of prime $k$-tuples up to some cut-off integer are presented. Related summatory $k$-tuple analogs of the first and second Chebyshev functions are then defined. Using a gamma distribution hypothesis for prime powers, associated average summatory functions are conjectured. With exact and average summatory functions in hand, pertinent $k$-tuple zeta functions can be identified, and Perron's formula allows the formulation of $k$-tuple analogs of explicit formulae. The $k$-tuple zeta functions are then used to make some inferences about $k$-tuple primes.
Exact summatory functions that count the number of prime $k$-tuples up to some cut-off integer are presented. Related $k$-tuple analogs of the first and second Chebyshev functions are then defined.