Gradings, Braidings, Representations, Paraparticles: some open problems

A long-term research proposal on the algebraic structure, the representations and the possible applications of paraparticle algebras is structured in three modules: The first part stems from an attempt to classify the inequivalent gradings and braided group structures present in the various parastatistical algebraic models. The second part of the proposal aims at refining and utilizing a previously published methodology for the study of the Fock-like representations of the parabosonic algebra, in such a way that it can also be directly applied to the other parastatistics algebras. Finally, in the third part, a couple of Hamiltonians is proposed, and their sutability for modeling the radiation matter interaction via a parastatistical algebraic model is discussed.Comment: 25 pages, some typos correcte

Graded Fock--like representations for a system of algebraically interacting paraparticles

We will present an algebra describing a mixed paraparticle model, known in the bibliography as "The Relative Parabose Set (\textsc{Rpbs})". Focusing in the special case of a single parabosonic and a single parafermionic degree of freedom $P_{BF}^{(1,1)}$, we will study a class of Fock--like representations of this algebra, dependent on a positive integer parameter p (a kind of generalized parastatistics order). Mathematical properties of the Fock--like modules will be investigated for all values of p and constructions such as ladder operators, irreducibility (for the carrier spaces) and Klein group gradings (for both the carrier spaces and the algebra itself) will be established.Comment: 4 pages, 1 ref. updated with respect to the journ. versio

Mass hierarchy, mass gap and corrections to Newton's law on thick branes with Poincare symmetry

We consider a scalar thick brane configuration arising in a 5D theory of gravity coupled to a self-interacting scalar field in a Riemannian manifold. We start from known classical solutions of the corresponding field equations and elaborate on the physics of the transverse traceless modes of linear fluctuations of the classical background, which obey a Schroedinger-like equation. We further consider two special cases in which this equation can be solved analytically for any massive mode with m^2>0, in contrast with numerical approaches, allowing us to study in closed form the massive spectrum of Kaluza-Klein (KK) excitations and to compute the corrections to Newton's law in the thin brane limit. In the first case we consider a solution with a mass gap in the spectrum of KK fluctuations with two bound states - the massless 4D graviton free of tachyonic instabilities and a massive KK excitation - as well as a tower of continuous massive KK modes which obey a Legendre equation. The mass gap is defined by the inverse of the brane thickness, allowing us to get rid of the potentially dangerous multiplicity of arbitrarily light KK modes. It is shown that due to this lucky circumstance, the solution of the mass hierarchy problem is much simpler and transparent than in the (thin) Randall-Sundrum (RS) two-brane configuration. In the second case we present a smooth version of the RS model with a single massless bound state, which accounts for the 4D graviton, and a sector of continuous fluctuation modes with no mass gap, which obey a confluent Heun equation in the Ince limit. (The latter seems to have physical applications for the first time within braneworld models). For this solution the mass hierarchy problem is solved as in the Lykken-Randall model and the model is completely free of naked singularities.Comment: 25 pages in latex, no figures, content changed, corrections to Newton's law included for smooth version of RS model and an author adde

Hopf structures in parafermionic and parabosonic algebras and applications of these algebras in physics

The subject of this thesis is the algebraic study of parafermionic and parabosonic algebras and especially the determination of the various Hopf structures possibly present in them together with indications of possible applications in physics. In chapter 2, an historical introduction is presented together with the first important results regarding the representations of these algebras, in the language they were first stated by Green, Greenberg and Messiah. The rest of the thesis consists of two parts: · Chapter 3, which constitutes a self-contained mathematical introduction into various topics related to the Hopf algebra theory. · Chapters 4,5,6 in which the original results of the thesis are presented: In chapter 4, we lay down the definitions of bosonic and parabosonic algebras in a modern algebraic language and we prove these algebras to be Z2-gr. algebras. The bosonic algebra is also proved to be a quotient algebra of the parabosonic algebra. We show that the notion of Z2 grading is equivalent to a specific action of the Z2 group and we proceed to computing this action. We then proceed to presenting a graded description of the Fock and the Fock-like representations. Finally the parabosonic algebra is proved to be a graded Hopf algebra (or equivalently a braided group). In chapter 5, consequences of the graded Hopf structure are extensively studied: The Green ansatz algebras are shown to be isomorphic to graded tensor products of the bosonic algebra and some progress is made towards a braided construction of the parabosonic Fock-like representations. Some novel generalizations of the Green ansatz are also presented. Finally, the notion of braided group is studied and two methods (one bibliography-based and the second totally new) are presented with the aim of which ordinary Hopf structures for the parabosonic algebra are constructed. A discussion on the comparison of the methods and their results is also supplied. In chapter 6, we contruct realizations of Lie algebras and Lie superalgebras (in both the finite and infinite dim. case) using either parabosonic and parafermionic algebras or mixed algebras which combine parabosonic and parafermionic degrees of freedom. The role played by the various Hopf structures in these realizations together with possible applications in physics and in representation theory are extensively discussed.Το αντικείμενο της διατριβής είναι η αλγεβρική μελέτη των παραφερμιονικών και παραμποζονικών αλγεβρών και πιο συγκεκριμένα η μελέτη των πιθανών Hopf δομών που εμφανίζουν οι άλγεβρες αυτές καθώς και οι πιθανές εφαρμογές στη φυσική. Στο κεφ. 2 παρουσιάζεται μια ιστορική εισαγωγή αλλά και τα πρώτα σημαντικά αποτελέσματα σχετικά με τις αναπαραστάσεις των αλγεβρών αυτών όπως πρωτοδιατυπώθηκαν από τους Green, Greenberg, Messiah. Το υπόλοιπο της διατριβής αποτελείται από δύο τμήματα: · Το κεφ. 3, το οποίο αποτελεί μια αυτόνομη μαθηματική εισαγωγή στις έννοιες των Hopf αλγεβρών. · Τα κεφ. 4,5,6 τα οποία παρουσιάζουν τα πρωτότυπα ερευνητικά αποτελέσματα της διατριβής: Στο κεφ. 4, δίνονται οι ορισμοί της μποζονικής και παραμποζονικής άλγεβρας, αποδεικνύεται οτι οι άλγεβρες αυτές είναι βαθμωτές άλγεβρες και επίσης οτι η μποζονική άλγεβρα αποτελεί άλγεβρα πηλίκο της παραμποζονικής άλγεβρας. Αποδεικνύεται οτι η έννοια της Ζ2 βάθμωσης είναι ισοδύναμη με μια συγκεκριμένη δράση της ομάδας Ζ2 την οποία και υπολογίζουμε. Κατόπιν παρουσιάζεται μια βαθμωτή περιγραφή των Fock και των Fock-like αναπαραστάσεων. Τέλος, αποδεικνύουμε οτι η παραμποζονική άλγεβρα είναι βαθμωτή Hopf άλγεβρα. Στο κεφ. 5, μελετάμε συνέπειες της βαθμωτής Hopf δομής: Αποδεικνύουμε οτι οι άλγεβρες του Green ansatz είναι ισόμορφες με βαθμωτά τανυστικά γινόμενα της μποζονικής άλγεβρας και γίνονται βήματα προς την κατασκευή των παραμποζονικών Fock-like αναπαραστάσεων. Επίσης κατασκευάζονται γενικέυσεις του Green ansatz. Τέλος, μελετούμε την έννοια των πλεκτών ομάδων και παρουσιάζουμε δύο μεθόδους με τις οποίες από μια πλεκτή ομάδα όπως η παραμποζονική άλγεβρα κατασκευάζουμε συνήθεις Hopf άλγεβρες. Συγκρίνουμε τις μεθόδους αυτές. Στο κεφ. 6, κατασκευάζουμε realizations Lie αλγεβρών και super-Lie αλγεβρών χρησιμοποιώντας παραμποζονικές και παραφερμιονκές άλγεβρες αλλά και μικτές άλγεβρες που αναμιγνύουν παραφερμιονικούς και παραμποζονικούς βαθμούς ελευθερίας. Ο ρόλος των διαφόρων Hopf δομών σε αυτές τις realizations καθώς και πιθανές εφαρμογές τους σε προβλήματα φυσικής αλλά και σε προβλήματα θεωρίας αναπαραστάσεων συζητείται εκτενώς