Matej Pavsic

External inversion, internal inversion, and reflection invariance

Having in mind that physical systems have different levels of structure we develop the concept of external, internal and total improper Lorentz transformation (space inversion and time reversal). A particle obtained from the ordinary one by the application of internal space inversion or time reversal is generally a different particle. From this point of view the intrinsic parity of a nuclear particle (‘elementary particle’) is in fact the external intrinsic parity, if we take into account the internal structure of a particle. We show that non-conservation of the external parity does not necessarily imply noninvariance of nature under space inversion. The conventional theory of beta-decay can be corrected by including the internal degrees of freedom to become invariant under total space inversion, though not under the external one.

Spin and electron structure

The recent literature shows a renewed interest, with various independent approaches, in the classical theories for spin. Considering the possible interest of those results, at least for the electron case, we purpose in this paper to explore their physical and mathematical meaning, by the natural and powerful language of Clifford algebras (which, incidentally, will allow us to unify those different approaches). In such theories, the ordinary electron is in general associated to the mean motion of a point–like “constituent” Q, whose trajectory is a cylindrical helix. We find, in particular, that the object Q obeys a new, non-linear Dirac–like equation, such that —when averaging over an internal cycle (which corresponds to linearization)— it transforms into the ordinary Dirac equation (valid for the electron as a whole). PACS nos.: 03.70.+k ; 11.10.Qr ; 14.60.Cd . 0 † Work partially supported by INFN, CNR, MURST; by CAPES, CNPq, FAPESP; and by the Slovenian Ministry of Science and Technology. 0∗ On leave from the J.Stefan Institute; University of Ljubljana; 61111–Ljubljana; Slovenia. 0∗∗ Also: Facolta di Ingegneria, Universita statale di Bergamo, 24044 Dalmine (BG), Italy; and C.C.S., State University at Campinas, 13083-970 Campinas, S.P.; Brazil. 0∗∗∗ On leave from Departamento de Matematica Aplicada — Imecc; UNICAMP; 13084–Campinas, S.P.; Brazil.Abstract The recent literature shows a renewed interest, with various independent approaches, in the classical models for spin. Considering the possible interest of those results, at least for the electron case, we purpose in this paper to explore their physical and mathematical meaning, by the natural and powerful language of Clifford algebras (which, incidentally, will allow us to unify those different approaches). In such models, the ordinary electron is in general associated to the mean motion of a point-like “constituent” Q , whose trajectory is a cylindrical helix. We find, in particular, that the object Q obeys a new, non-linear Dirac-like equation, such that — when averaging over an internal cycle (which corresponds to linearization) — it transforms into the ordinary Dirac equation (valid, of course, for the electron as a whole).

Classical motion of membranes, strings and point particles with extrinsic curvature

Abstract Classical equations of motion for a rigid n -dimensional worldsheet in a curved background spacetime have been derived from the lagrangian which contains the extrinsic curvature. The worldline as a special case for n =1 has been considered and we found out that the corresponding equation of motion is exactly the Papapetrou equation for a spinning particle. Though our particle is point like, it has spin due to the extrinsic curvature term in the action which forces the particle to move along a helical path.

STABLE SELF-INTERACTING PAIS–UHLENBECK OSCILLATOR

It is shown that the interacting Pais–Uhlenbeck (PU) oscillator necessarily leads to a description with a Hamiltonian that contains positive and negative energies associated with two oscillators. Descriptions with a positive definite Hamiltonians, considered by some authors, can hold only for a free PU oscillator. We demonstrate that the solutions of a self-interacting PU oscillator are stable on islands in the parameter space, as already observed in the literature. If we slightly modify the system, by considering a sine interaction term, and/or by taking unequal masses of the two oscillators, then the system is stable on the continents that extend from zero to infinity in the parameter space. Therefore, the PU oscillator is quite acceptable physical system.It is shown that the interacting Pais-Uhlenbeck oscillator necessarily leads to a description with a Hamiltonian that contains positive and negative energies associated with two oscillators. Descriptions with a positive definite Hamiltonians, considered by some authors, can hold only for a free Pais-Uhlenbeck oscillator. We demonstrate that the solutions of a self-interacting Pais-Uhlenbeck oscillator are stable on islands in the parameter space, as already observed in the literature. If we slightly modify the system, by considering a sine interaction term, and/or by taking unequal masses of the two oscillators, then the system is stable on the continents that extend from zero to infinity in the parameter space. Therefore, the Pais-Uhlenbeck oscillator is quite acceptable physical system.

Clifford Algebra of Spacetime and the Conformal Group

We demonstrate the emergence of the conformal group SO(4,2) from the Clifford algebra of spacetime. The latter algebra is a manifold, called Clifford space, which is assumed to be the arena in which physics takes place. A Clifford space does not contain only points (events), but also lines, surfaces, volumes, etc., and thus provides a framework for description of extended objects. A subspace of the Clifford space is the space whose metric is invariant with respect to the conformal group SO(4,2) which can be given either passive or active interpretation. As advocated long ago by one of us, active conformal transformations, including dilatations, imply that sizes of physical objects can change as a result of free motion, without the presence of forces. This theory is conceptually and technically very different from Weyls theory and provides when extended to a curved conformal space a resolution of the long standing problem of realistic masses in Kaluza—Klein theories.

The Quantization of a Point Particle With Extrinsic Curvature Leads to the Dirac Equation

Abstract A rigid point particle, an analogue of the rigid string or membrane, has been considered. Its action containing an extrinsic curvature term has been cast into an equivalent form in which there occurs a one-dimensional vielbein. Classical constraints, hamiltonian, equations of motion and various Poisson bracket relations have been derived. The theory has then been quantized. It has been shown that the equation of state contains the Dirac equation as a special case. The Dirac matrices are just one particular representation of the modified four-velocity operator. Quantum spin has its classical analogue in the helical motion of a rigid point particle. The frequency of the quantum Zitterbewegung turns out to be equal to the frequency of the classical helical motion.

Einstein's gravity from a first order lagrangian in an embedding space

Abstract We formulate a first order action principle in a higher dimensional space M N in which we embed spacetime. The action I is essentially an “area” of a four-dimensional spacetime V 4 weighted with a matter density ω in M N . For a suitably chosen ω we obtain on V 4 a set of worldlines. It is shown that these worldlines are geodesics of V 4 , provided that V 4 is a solution to our variational procedure. Then it follows that our spacetime satisfies the Einstein equations for dust - apart from an additional term with zero covariant divergence. (This extra term was shown in a previous paper to be exactly zero at least in the case of the cosmological dust model.) Thus we establish a remarkable connection of the extrinsic spacetime theory with the intrinsic general relativity. This step appears to be important for quantum gravity.Here we place the Latex typeset of the paper M. Pavsic, Phys. Lett. A116 (1986) 1-5. In the paper we presented the picture that our spacetime is a 3-brane moving in a higher dimensional space. The dynamical equations were derived from the action which is just that for the usual Dirac-Nambu-Goto

Explaining thelarge numbers by a hierarchy of « universes »: A unified theory of strong and gravitational interactions

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Classical theory of a space-time sheet

-brane. We also considered the case where not only one, but many branes of various dimensionalities are present, and showed that their intersections with the 3-brane manifest as matter in 4-dimensional spacetime. We considered a particular case, where the intersections behaved as point particles, and found out that they follow the geodesics on the 3-brane worldsheet (identified with our spacetime). In a series of subsequent papers the original idea has been further improved and developped. This is discussed in a note at the end, where it is also pointed out that such a model resolves the problem of massive matter confinement on the brane, recently discussed by Rubakov et al. and Mueck et al.

Kaluza-Klein theory without extra dimensions: Curved Clifford space

SummaryBy assuming covariance of physical laws under (discrete) dilatations, we succeed in describing strong and gravitational interactions in a unified way. In terms of the (additional, discrete) « dilatational » degree of freedom, our cosmos as well as hadrons can be considered as different states of the same system, or rather assimilar systems. Moreover, a discrete hierarchy can be defined of « universes » which are governed by force fields with strenghts inversely proportional to the « universe » radii. Inside each « universe » an equivalence principle holds, so that its characteristic field can be geometrized there. Wo can thus easilyderive a whole « numerology »,i.e. relations among numbers analogous to the so-called Weyl-Eddington-Dirac « large numbers ». For instance, the « Planck mass » happens to be nothing but the (average) magnitude of thestrong chargeRiassuntoSi riescono a descrivere le interazioni forti e gravitazionali in maniera unificata, postulando la oovarianza delle leggi fisiche rispetto alle dilatazioni spazio-temporali. In funzione del (nuovo,discrete) grado di libertà « dilatazionale », il « cosmo » e gli adroni possono essere considerati come stati differenti dello stesso sistema, o meglio come sistemisimili. Si può inoltre definire una gerarchia discreta di « universi », i quali risultano governati da eampi di forza con «strength » inversamente proporzionale al raggio delF « universo » stesso. Entro ciascun « universo » vale un principio di equivalenza, cosi che il corrispondente campo caratteristico può esservi geometrizzato. Si riesce cosi a dimostrare facilmente un–intera «numerologia », costituita da relazioni tra numeri analoghi ai cosiddetti « grandi numeri » di Weyl-Eddington-Dirac. Per esempio, la « massa di Plank » risulta essere null–altro che il modulo della «carica forte » (media) deiquark degli adroni. La nostra « numerologia », però, lega ilmacrocosmo (gravitazionale) aimicrocosmi forti, piuttosto che a quelli elettromagnetici (come invece nella versione di Dirac). Si suggerisoono per l–interno degli adroni delle equazioniscalate tipo-Einstein (con termine cosmologico), le quali — tra parentesi — producono un « confinamento » (classico) molto naturale per iquark, e sono computabili con la « libertà asintotica ». Infine, nell–ambito di una teoria « a due scale », si propongono delle ulteriori equazioni chea priori forniscono una teoria classica di campo delle interazioni forti (tra adroni). I paragrafl rilevanti sono5.2,7 e8.РезюмеПредполагая ковариантность физических законов относительно расширений, мы предлагаем единое описание сильного и гравитационного взаимодействий. В терминах (дополнительной, дискретной) « продольной » степени свободы наш « космос », а также адроны можно рассматривать как различные состояния одной и той же системы или как подобные системы. Кроме того, может быть определена иерархия « вселенных », которые определяются силовыми полями с интенсивностями обратно пропорциональными радиусу « вселенной ». Внутри каждой « вселенной » действует принцип эквивалентности, так что характеристическое поле может быть определено из геометрических соображений. Таким образом, мы легко можем вывести обычную «нумерацию », т.е. соотношения между числами, аналогичными так называемым большим числам Вейля-Эддингтона-Дир ака. Например, « масса Планка », оказывается, есть (средний) « сильный заряд » адронных кварков. Однако наша « нумерация » связывает (гравитационный) макро-космос с (сильным) микро-космосом, а не с электромагнитным « космосом » (как, например, в варианте Дирака). Предлагаются уравнения эйнштейновского типа (с « космологическим » членом) для сильных взаимодействий, которые дают, случайно, (классическое) удержание кварков и дают, а приори, полевую теорию сильных взаимодействий.