arXiv subjects

P. D. Jarvis

Publications and source records attributed to P. D. Jarvis.

At least 19 recordsLinked to original sources

arXiv2021

Gauge invariant formulation of the self-interacting Duffin-Kemmer-Petiau equations

We show that the Duffin-Kemmer-Petiau equation, minimally coupled to an Abelian gauge field, can be regarded as a matrix equation for the gauge potential produced internally from the matter fields. This can be solved as a rational expression in terms of currents bilinear in the matter wavefunction, together with a similar expression for the field strength tensor, thus providing a gauge invariant formulation of the self-interacting DKP equations. We give the derivation of this result for the 5 component DKP system, by analogy with the Dirac equation case. To this end, we establish the algebraic structure of the set of bilinear currents, and the properties of the minimal generating set, which consists of two scalars and two four-vectors, together with a single quadratic constraint.

math-ph↗

arXiv2020

Quasicharacters, recoupling calculus and costratifications of lattice quantum gauge theory

We study the algebra of invariant representative functions over the N-fold Cartesian product of copies of a compact Lie group G modulo the action of conjugation by the diagonal subgroup. We construct a basis of invariant representative functions referred to as quasicharacters. The form of the quasicharacters depends on the choice of a reduction scheme. We determine the multiplication law of quasicharacters and express their structure constants in terms of recoupling coefficients. Via this link, the choice of the reduction scheme acquires an interpretation in terms of binary trees. We show explicitly that the structure constants decompose into products over primitive elements of 9j symbol type. For SU(2), everything boils down to the combinatorics of angular momentum theory. In the final part, we show that the above calculus enables us to calculate the matrix elements of bi-invariant operators occuring in quantum lattice gauge theory. In particular, both the quantum Hamiltonian and the orbit type relations may be dealt with in this way, thus, reducing both the construction of the costratification and the study of the spectral problem to numerical problems in linear algebra. We spell out the spectral problem for G=SU(2) and we present sample calculations of matrix elements of orbit type relations for the gauge groups SU(2) and SU(3). The methods developed may be useful in the study of virtually all quantum models with polynomial constraints related to some symmetry.

math-ph↗

arXiv2020

Finite $W$-superalgebras and quadratic spacetime supersymmetries

We consider Lie superalgebras under constraints of Hamiltonian reduction, yielding finite $W$-superalgebras which provide candidates for quadratic spacetime superalgebras. These have an undeformed bosonic symmetry algebra (even generators) graded by a fermionic sector (supersymmetry generators) with anticommutator brackets which are quadratic in the even generators. We analyze the reduction of several Lie superalgebras of type $gl(M|N)$ or $osp(M|2N)$ at the classical (Poisson bracket) level, and also establish their quantum (Lie bracket) equivalents. Purely bosonic extensions are also considered. As a special case we recover a recently identified quadratic superconformal algebra, certain of whose unitary irreducible massless representations (in four dimensions) are "zero-step" multiplets, with no attendant superpartners. Other cases studied include a six dimensional quadratic superconformal algebra with vectorial odd generators, and a variant quadratic superalgebra with undeformed $osp(1|2N)$ singleton supersymmetry, and a triplet of spinorial supercharges.

hep-th↗

arXiv2017

Hidden supersymmetry and quadratic deformations of the space-time conformal superalgebra

We analyze the structure of the family of quadratic superalgebras, introduced in J Phys A 44(23):235205 (2011), for the quadatic deformations of $N = 1$ space-time conformal supersymmetry. We characterize in particular the `zero-step' modules for this case. In such modules, the odd generators vanish identically, and the quadratic superalgebra is realized on a single irreducible representation of the even subalgebra (which is a Lie algebra). In the case under study, the quadratic deformations of $N = 1$ space-time conformal supersymmetry, it is shown that each massless positive energy unitary irreducible representation (in the standard classification of Mack), forms such a zero-step module, for an appropriate parameter choice amongst the quadratic family (with vanishing central charge). For these massless particle multiplets therefore, quadratic supersymmetry is unbroken, in that the supersymmetry generators annihilate \emph{all} physical states (including the vacuum state), while at the same time, superpartners do not exist.

hep-th↗

arXiv2013

Adventures in Invariant Theory

We provide an introduction to enumerating and constructing invariants of group representations via character methods. The problem is contextualised via two case studies arising from our recent work: entanglement measures, for characterising the structure of state spaces for composite quantum systems; and Markov invariants, a robust alternative to parameter-estimation intensive methods of statistical inference in molecular phylogenetics.

q-bio.QM↗

arXiv2012

Algebraic inversion of the Dirac equation for the vector potential in the non-abelian case

We study the Dirac equation for spinor wavefunctions minimally coupled to an external field, from the perspective of an algebraic system of linear equations for the vector potential. By analogy with the method in electromagnetism, which has been well-studied, and leads to classical solutions of the Maxwell-Dirac equations, we set up the formalism for non-abelian gauge symmetry, with the SU(2) group and the case of four-spinor doublets. An extended isospin-charge conjugation operator is defined, enabling the hermiticity constraint on the gauge potential to be imposed in a covariant fashion, and rendering the algebraic system tractable. The outcome is an invertible linear equation for the non-abelian vector potential in terms of bispinor current densities. We show that, via application of suitable extended Fierz identities, the solution of this system for the non-abelian vector potential is a rational expression involving only Pauli scalar and Pauli triplet, Lorentz scalar, vector and axial vector current densities, albeit in the non-closed form of a Neumann series.

hep-th↗

arXiv2011

Projective Representations of the Inhomogeneous Hamilton Group: Noninertial Symmetry in Quantum Mechanics

Symmetries in quantum mechanics are realized by the projective representations of the Lie group as physical states are defined only up to a phase. A cornerstone theorem shows that these representations are equivalent to the unitary representations of the central extension of the group. The formulation of the inertial states of special relativistic quantum mechanics as the projective representations of the inhomogeneous Lorentz group, and its nonrelativistic limit in terms of the Galilei group, are fundamental examples. Interestingly, neither of these symmetries includes the Weyl-Heisenberg group; the hermitian representations of its algebra are the Heisenberg commutation relations that are a foundation of quantum mechanics. The Weyl-Heisenberg group is a one dimensional central extension of the abelian group and its unitary representations are therefore a particular projective representation of the abelian group of translations on phase space. A theorem involving the automorphism group shows that the maximal symmetry that leaves invariant the Heisenberg commutation relations are essentially projective representations of the inhomogeneous symplectic group. In the nonrelativistic domain, we must also have invariance of Newtonian time. This reduces the symmetry group to the inhomogeneous Hamilton group that is a local noninertial symmetry of Hamilton's equations. The projective representations of these groups are calculated using the Mackey theorems for the general case of a nonabelian normal subgroup.

quant-ph↗

arXiv2011

Extended two-level quantum dissipative system from bosonization of the elliptic spin-1/2 Kondo model

We study the elliptic spin-1/2 Kondo model (spin-1/2 fermions in one dimension with fully anisotropic contact interactions with a magnetic impurity) in the light of mappings to bosonic systems using the fermion-boson correspondence and associated unitary transformations. We show that for fixed fermion number, the bosonic system describes a two-level quantum dissipative system with two noninteracting copies of infinitely-degenerate upper and lower levels. In addition to the standard tunnelling transitions, and the transitions driven by the dissipative coupling, there are also bath-mediated transitions between the upper and lower states which simultaneously effect shifts in the horizontal degeneracy label. We speculate that these systems could provide new examples of continuous time quantum random walks, which are exactly solvable.

cond-mat.str-el↗

arXiv2010

The algebra of the general Markov model on phylogenetic trees and networks

It is known that the Kimura 3ST model of sequence evolution on phylogenetic trees can be extended quite naturally to arbitrary split systems. However, this extension relies heavily on mathematical peculiarities of the K3ST model, and providing an analogous augmentation of the general Markov model has thus far been elusive. In this paper we rectify this shortcoming by showing how to extend the general Markov model on trees to to include arbitrary splits; and even further to more general network models. This is achieved by exploring the algebra of the generators of the continuous-time Markov chain together with the "splitting" operator that generates the branching process on phylogenetic trees. For simplicity we proceed by discussing the two state case and note that our results are easily extended to more states with little complication. Intriguingly, upon restriction of the two state general Markov model to the parameter space of the binary symmetric model, our extension is indistinguishable from the previous approach only on trees; as soon as any incompatible splits are introduced the two approaches give rise to differing probability distributions with disparate structure. Through exploration of a simple example, we give a tentative argument that our approach to extending to more general networks has desirable properties that the previous approaches do not share. In particular, our construction allows for the possibility of convergent evolution of previously divergent lineages; a property that is of significant interest for biological applications.

q-bio.PE↗

arXiv2010

A kinematical approach to gravitational lensing using new formulae for refractive index and acceleration

This paper uses the Schwarzschild metric to derive an effective refractive index and acceleration vector that account for relativistic deflection of light rays, in an otherwise classical kinematic framework. The new refractive index and the known path equation are integrated to give accurate results for travel time and deflection angle, respectively. A new formula for coordinate acceleration is derived which describes the path of a massless test particle in the vicinity of a spherically symmetric mass density distribution. A standard ray-shooting technique is used to compare the deflection angle and time delay predicted by this new formula with the previously calculated values, and with standard first order approximations. Finally, the ray shooting method is used in theoretical examples of strong and weak lensing, reproducing known observer-plane caustic patterns for multiple masses.

astro-ph.GA↗

arXiv2010

Markov invariants for phylogenetic rate matrices derived from embedded submodels

We consider novel phylogenetic models with rate matrices that arise via the embedding of a progenitor model on a small number of character states, into a target model on a larger number of character states. Adapting representation-theoretic results from recent investigations of Markov invariants for the general rate matrix model, we give a prescription for identifying and counting Markov invariants for such `symmetric embedded' models, and we provide enumerations of these for low-dimensional cases. The simplest example is a target model on 3 states, constructed from a general 2 state model; the `2->3' embedding. We show that for 2 taxa, there exist two invariants of quadratic degree, that can be used to directly infer pairwise distances from observed sequences under this model. A simple simulation study verifies their theoretical expected values, and suggests that, given the appropriateness of the model class, they have greater statistical power than the standard (log) Det invariant (which is of cubic degree for this case).

q-bio.QM↗

arXiv2010

Exactly solvable three-level quantum dissipative systems via bosonisation of fermion gas-impurity models

We study the relationship between one-dimensional fermion gas-impurity models and quantum dissipative systems, via the method of constructive bosonisation and unitary transformation. Starting from an anisotropic Coqblin-Schrieffer model, a new, exactly solvable, three-level quantum dissipative system is derived as a generalisation of the standard spin-half spin-boson model. The new system has two environmental oscillator baths with ohmic coupling, and admits arbitrary detuning between the three levels. All tunnelling matrix elements are equal, up to one complex phase which is itself a function of the longitudinal and transverse couplings in the integrable limit. Our work underlines the importance of re-examining the detailed structure of fermion-gas impurity models and spin chains, in the light of connections to models for quantum dissipative systems.

quant-ph↗

arXiv2008

Markov invariants, plethysms, and phylogenetics (the long version)

We explore model based techniques of phylogenetic tree inference exercising Markov invariants. Markov invariants are group invariant polynomials and are distinct from what is known in the literature as phylogenetic invariants, although we establish a commonality in some special cases. We show that the simplest Markov invariant forms the foundation of the Log-Det distance measure. We take as our primary tool group representation theory, and show that it provides a general framework for analysing Markov processes on trees. From this algebraic perspective, the inherent symmetries of these processes become apparent, and focusing on plethysms, we are able to define Markov invariants and give existence proofs. We give an explicit technique for constructing the invariants, valid for any number of character states and taxa. For phylogenetic trees with three and four leaves, we demonstrate that the corresponding Markov invariants can be fruitfully exploited in applied phylogenetic studies.

q-bio.PE↗

arXiv2008

Regularised Tripartite Continuous Variable EPR-type States with Wigner Functions and CHSH Violations

We consider tripartite entangled states for continuous variable systems of EPR type, which generalise the famous bipartite CV EPR states (eigenvectors of conjugate choices X1 - X2, P1+ P2, of the systems' relative position and total momentum variables). We give the regularised forms of such tripartite EPR states in second-quantised formulation, and derive their Wigner functions. This is directly compared with the established NOPA-like states from quantum optics. Whereas the multipartite entangled states of NOPA type have singular Wigner functions in the limit of large squeezing, r --> infinity, or tanh r --> 1^- (approaching the EPR states in the bipartite case), our regularised tripartite EPR states show singular behaviour not only in the approach to the EPR-type region (s --> 1 in our notation), but also for an additional, auxiliary regime of the regulator (s --> \sqrt{2}). While the s -->1 limit pertains to tripartite CV states with singular eigenstates of the relative coordinates and remaining squeezed in the total momentum, the s --> \sqrt{2} limit yields singular eigenstates of the total momentum, but squeezed in the relative coordinates. Regarded as expectation values of displaced parity measurements, the tripartite Wigner functions provide the ingredients for generalised CHSH inequalities. Violations of the tripartite CHSH bound (B3 \le 2) are established, with B3 ~ 2.09 in the canonical regime (s -->1^+), as well as B3 ~ 2.32 in the auxiliary regime (s --> \sqrt{2^+}).

quant-ph↗

arXiv2008

Constraint quantisation of a worldline system invariant under reciprocal relativity. II

We consider the world-line quantisation of a system invariant under the symmetries of reciprocal relativity. Imposition of the first class constraint, the generator of local time reparametrisations, on physical states enforces identification of the world-line cosmological constant with a fixed value of the quadratic Casimir of the quaplectic symmetry group Q(3,1) ~ U(3,1) x H(4), the semi-direct product of the pseudo-unitary group with the Weyl-Heisenberg group. In our previous paper, J Phys A 40 (2007) 12095--12111, the `spin' degrees of freedom were handled as covariant oscillators, leading to a unique choice of cosmological constant, required for projecting out negative-norm states from the physical gauge-invariant states. In the present paper the spin degrees of freedom are treated as standard oscillators with positive norm states (wherein Lorentz boosts are not number-conserving in the auxiliary space; reciprocal transformations are of course not spin-conserving in general). As in the covariant approach, the spectrum of the square of the energy-momentum vector is continuous over the entire real line, and thus includes tachyonic (spacelike) and null branches. Adopting standard frames, the Wigner method on each branch is implemented, to decompose the auxiliary space into unitary irreducible representations of the respective little algebras and additional degeneracy algebras. The physical state space is vastly enriched as compared with the covariant approach, and contains towers of integer spin massive states, as well as unconventional massless representations, with continuous Euclidean momentum and arbitrary integer helicity.

hep-th↗

arXiv2008

The molecular asymmetric rigid rotor Hamiltonian as an exactly solvable model

Representations of the rotation group may be formulated in second-quantised language via Schwinger's transcription of angular momentum states onto states of an effective two-dimensional oscillator. In the case of the molecular asymmetric rigid rotor, by projecting onto the state space of rigid body rotations, the standard Ray Hamiltonian $H(1,κ,-1)$ (with asymmetry parameter $1 \ge κ\ge -1$), becomes a quadratic polynomial in the generators of the associated dynamical $su(1,1)$ algebra. We point out that $H(1,κ,-1)$ is in fact quadratic in the Gaudin operators arising from the quasiclassical limit of an associated $su_q(1,1)$ Yang-Baxter algebra. The general asymmetric rigid rotor Hamiltonian is thus an exactly solvable model. This fact has important implications for the structure of the spectrum, as well as for the eigenstates and correlation functions of the model.

math-ph↗

arXiv2005

The Dirichlet Hopf algebra of arithmetics

In this work, we develop systematically the ``Dirichlet Hopf algebra of arithmetics'' by dualizing addition and multiplication maps. We study the additive and multiplicative antipodal convolutions which fail to give rise to Hopf algebra structures, obeying only a weakened (multiplicative) homomorphism axiom. The consequences of the weakened structure, called a Hopf gebra, e.g. on cohomology are explored. This features multiplicativity versus complete multiplicativity of number theoretic arithmetic functions. The deficiency of not being a Hopf algebra is then cured by introducing an `unrenormalized' coproduct and an `unrenormalized' pairing. It is then argued that exactly the failure of the homomorphism property (complete multiplicativity) for non-coprime integers is a blueprint for the problems in quantum field theory (QFT) leading to the need for renormalization. Renormalization turns out to be the morphism from the algebraically sound Hopf algebra to the physical and number theoretically meaningful Hopf gebra. This can be modelled alternatively by employing Rota-Baxter operators. We stress the need for a characteristic-free development where possible, to have a sound starting point for generalizations of the algebraic structures. The last section provides three key applications: symmetric function theory, quantum (matrix) mechanics, and the combinatorics of renormalization in QFT which can be discerned as functorially inherited from the development at the number-theoretic level as outlined here. Hence the occurrence of number theoretic functions in QFT becomes natural.

math-ph↗

arXiv2005

Path integral formulation and Feynman rules for phylogenetic branching models

A dynamical picture of phylogenetic evolution is given in terms of Markov models on a state space, comprising joint probability distributions for character types of taxonomic classes. Phylogenetic branching is a process which augments the number of taxa under consideration, and hence the rank of the underlying joint probability state tensor. We point out the combinatorial necessity for a second-quantised, or Fock space setting, incorporating discrete counting labels for taxa and character types, to allow for a description in the number basis. Rate operators describing both time evolution without branching, and also phylogenetic branching events, are identified. A detailed development of these ideas is given, using standard transcriptions from the microscopic formulation of nonequilibrium reaction-diffusion or birth-death processes. These give the relations between stochastic rate matrices, the matrix elements of the corresponding evolution operators representing them, and the integral kernels needed to implement these as path integrals. The `free' theory (without branching) is solved, and the correct trilinear `interaction' terms (representing branching events) are presented. The full model is developed in perturbation theory via the derivation of explicit Feynman rules which establish that the probabilities (pattern frequencies of leaf colourations) arising as matrix elements of the time evolution operator are identical with those computed via the standard analysis. Simple examples (phylogenetic trees with 2 or 3 leaves), are discussed in detail. Further implications for the work are briefly considered including the role of time reparametrisation covariance.

q-bio.PE↗