arXiv subjects

Matej Pavsic

Publications and source records attributed to Matej Pavsic.

At least 19 recordsLinked to original sources

arXiv2013

Space Inversion of Spinors Revisited: A Possible Explanation of Chiral Behavior in Weak Interactions

We investigate a model in which spinors are considered as being embedded within the Clifford algebra that operates on them. In Minkowski space $M_{1,3}$, we have four independent 4-component spinors, each living in a different minimal left ideal of $Cl(1,3)$. We show that under space inversion, a spinor of one left ideal transforms into a spinor of another left ideal. This brings novel insight to the role of chirality in weak interactions. We demonstrate the latter role by considering an action for a generalized spinor field $ψ^{αi}$ that has not only a spinor index $α$ but also an extra index $i$ running over four ideals. The covariant derivative of $ψ^{αi}$ contains the generalized spin connection, the extra components of which are interpreted as the SU(2) gauge fields of weak interactions and their generalization. We thus arrive at a system that is left-right symmetric due to the presence of a "parallel sector", postulated a long time ago, that contains mirror particles coupled to mirror SU(2) gauge fields.

hep-th↗

arXiv2011

On the Relativity in Configuration Space: A Renewed Physics In Sight

The idea that possible configurations of a physical system can be represented as points in a multidimensional configuration space ${\cal C}$ is explored. The notion of spacetime, without ${\cal C}$, does not exist in this theory. Spacetime is associated with the degrees of freedom of a chosen single particle within a considered configuration, and is thus a subspace of ${\cal C}$. Finite dimensional configuration spaces of point particles, and infinite dimensional configuration spaces of branes are considered. Multidimensionality of a configuration space has for a consequence the existence of extra interactions, besides the 4D gravity, both at macroscopic and microscopic scales.

gr-qc↗

arXiv2011

Klein-Gordon-Wheeler-DeWitt-Schroedinger Equation

We start from the Einstein-Hilbert action for the gravitational field in the presence of a "point particle" source, and cast the action into the corresponding phase space form. The dynamical variables of such a system satisfy the point particle mass shell constraint, the Hamilton and the momentum constraints of the canonical gravity. In the quantized theory, those constraints become operators that annihilate a state. A state can be represented by a wave functional $Ψ$ that simultaneously satisfies the Klein-Gordon and the Wheeler-DeWitt-Schrödinger equation. The latter equation, besides the term due to gravity, also contains the Schrödinger like term, namely the derivative of $Ψ$ with respect to time, that occurs because of the presence of the point particle. The particle's time coordinate, $X^0$, serves the role of time. Next, we generalize the system to $p$-branes, and find out that for a quantized spacetime filling brane there occurs an effective cosmological constant, proportional to the expectation value of the brane's momentum, a degree of freedom that has two discrete values only, a positive and a negative one. This mechanism could be an explanation for the small cosmological constant that drives the accelerated expansion of the universe.

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arXiv2009

Towards the Unification of Gravity and other Interactions: What has been Missed?

Faced with the persisting problem of the unification of gravity with other fundamental interactions we investigate the possibility of a new paradigm, according to which the basic space of physics is a multidimensional space ${\cal C}$ associated with matter configurations. We consider general relativity in ${\cal C}$. In spacetime, which is a 4-dimensional subspace of ${\cal C}$, we have not only the 4-dimensional gravity, but also other interactions, just as in Kaluza-Klein theories. We then consider a finite dimensional description of extended objects in terms of the center of mass, area, and volume degrees of freedom, which altogether form a 16-dimensional manifold whose tangent space at any point is Clifford algebra Cl(1,3). The latter algebra is very promising for the unification, and it provides description of fermions.

gr-qc↗

arXiv2009

Beyond the Relativistic Point Particle: A Reciprocally Invariant System and its Generalisation

We investigate a reciprocally invariant system proposed by Low and Govaerts et al., whose action contains both the orthogonal and the symplectic forms and is invariant under global $O(2,4)\cap Sp(2,4)$ transformations. We find that the general solution to the classical equations of motion has no linear term in the evolution parameter, $τ$, but only the oscillatory terms, and therefore cannot represent a particle propagating in spacetime. As a remedy, we consider a generalisation of the action by adopting a procedure similar to that of Bars et al., who introduced the concept of a $τ$ derivative that is covariant under local Sp(2) transformations between the phase space variables $x^μ(τ)$ and $p^μ(τ)$. This system, in particular, is similar to a rigid particle whose action contains the extrinsic curvature of the world line, which turns out to be helical in spacetime. Another possible generalisation is the introduction of a symplectic potential proposed by Montesinos. We show how the latter approach is related to Kaluza-Klein theories and to the concept of Clifford space, a manifold whose tangent space at any point is Clifford algebra Cl(8), a promising framework for the unification of particles and forces.

hep-th↗

arXiv2008

A Novel View on the Physical Origin of E8

We consider a straightforward extension of the 4-dimensional spacetime $M_4$ to the space of extended events associated with strings/branes, corresponding to points, lines, areas, 3-volumes, and 4-volumes in $M_4$. All those objects can be elegantly represented by the Clifford numbers $X\equiv x^A γ_A \equiv x^{a_1 ...a_r} γ_{a_1 ...a_r}, r=0,1,2,3,4$. This leads to the concept of the so-called Clifford space ${\cal C}$, a 16-dimensional manifold whose tangent space at every point is the Clifford algebra ${\cal C \ell }(1,3)$. The latter space besides an algebra is also a vector space whose elements can be rotated into each other in two ways: (i) either by the action of the rotation matrices of SO(8,8) on the components $x^A$ or (ii) by the left and right action of the Clifford numbers $R=$exp$ [α^A \gam_A]$ and $S=$exp$ [β^A \gam_A]$ on $X$. In the latter case, one does not recover all possible rotations of the group SO(8,8). This discrepancy between the transformations (i) and (ii) suggests that one should replace the tangent space ${\cal C \ell}(1,3)$ with a vector space $V_{8,8}$ whose basis elements are generators of the Clifford algebra ${\cal C \ell}(8,8)$, which contains the Lie algebra of the exceptional group E$_8$ as a subspace. E$_8$ thus arises from the fact that, just as in the spacetime $M_4$ there are $r$-volumes generated by the tangent vectors of the spacetime, there are $R$-volumes, $R=0,1,2,3,...,16$, in the Clifford space ${\cal C}$, generated by the tangent vectors of ${\cal C}$.

hep-th↗

arXiv2007

Towards a New Paradigm: Relativity in Configuration Space

We consider the possibility that the basic space of physics is not spacetime, but configuration space. We illustrate this on the example with a system of gravitationally interacting point particles. It turns out that such system can be described by the minimal length action in a multidimensional configuration space C with a block diagonal metric. Allowing for more general metrics and curvatures of C, we step beyond the ordinary general relativity in spacetime. The latter theory is then an approximation to the general relativity in C. Other sorts of configuration spaces can also be considered, for instance those associated with extended objects, such as strings and branes. This enables a deeper understanding of the geometric principle behind string theory, and an insight on the occurrence of Yang-Mills and gravitational fields at the `fundamental level'.

gr-qc↗

arXiv2007

On a Unified Theory of Generalized Branes Coupled to Gauge Fields, Including the Gravitational and Kalb-Ramond Fields

We investigate a theory in which fundamental objects are branes described in terms of higher grade coordinates X^{μ_1 ... μ_n} encoding both the motion of a brane as a whole, and its volume evolution. We thus formulate a dynamics which generalizes the dynamics of the usual branes. Geometrically, coordinates X^{μ_1 ... μ_n} and associated coordinate frame fields {γ_{μ_1 ... μ_n}} extend the notion of geometry from spacetime to that of an enlarged space, called Clifford space or C-space. If we start from 4-dimensional spacetime, then the dimension of C-space is 16. The fact that C-space has more than four dimensions suggests that it could serve as a realization of Kaluza-Klein idea. The "extra dimensions" are not just the ordinary extra dimensions, they are related to the volume degrees of freedom, therefore they are physical, and need not be compactified. Gauge fields are due to the metric of Clifford space. It turns out that amongst the latter gauge fields there also exist higher grade, antisymmetric fields of the Kalb-Ramond type, and their non-Abelian generalization. All those fields are naturally coupled to the generalized branes, whose dynamics is given by a generalized Howe-Tucker action in curved C-space.

hep-th↗

arXiv2006

An Extra Structure of Spacetime: A Space of Points, Areas and Volumes

A theory in which points, lines, areas and volumes are on on the same footing is investigated. All those geometric objects form a 16-dimensional manifold, called C-space, which generalizes spacetime. In such higher dimensional space fundamental interactions can be unified \` a la Kaluza-Klein. The ordinary, 4-dimensional, gravity and gauge fields are incorporated in the metric and spin connection, whilst the conserved gauge charges are related to the isometries of curved C-space. It is shown that a conserved generator of an isometry in C-space contains a part with derivatives, which generalizes orbital angular momentum, and a part with the generators of Clifford algebra, which generalizes spin.

gr-qc↗

arXiv2006

The Landscape of Theoretical Physics: A Global View; From Point Particles to the Brane World and Beyond, in Search of a Unifying Principle

This a book is for those who would like to learn something about special and general relativity beyond the usual textbooks, about quantum field theory, the elegant Fock-Schwinger-Stueckelberg proper time formalism, the elegant description of geometry by means of Clifford algebra, about the fascinating possibilities the latter algebra offers in reformulating the existing physical theories, and quantizing them in a natural way. It is shown how Clifford algebra provides much more: it provides room for new physics, with the prospects of resolving certain long standing puzzles. The theory of branes and the idea of how a 3-brane might represent our world is discussed in detail. Much attention is paid to the elegant geometric theory of branes which employs the infinite dimensional space of functions describing branes. Clifford algebra is generalized to the infinite dimensional spaces. In short, this is a book for anybody who would like to explore how the ``theory of everything'' might possibly be formulated. The theory that would describe all the known phenomena, could not be formulated without taking into account ``all'' the theoretical tools which are available. Foundations of those tools and their functional interrelations are described in the book.

gr-qc↗

arXiv2006

Clifford Space as a Generalization of Spacetime: Prospects for Unification in Physics

The geometric calculus based on Clifford algebra is a very useful tool for geometry and physics. It describes a geometric structure which is much richer than the ordinary geometry of spacetime. A Clifford manifold ($C$-space) consists not only of points, but also of 1-loops, 2-loops, etc.. They are associated with multivectors which are the wedge product of the basis vectors, the generators of Clifford algebra. We assume that $C$-space is the true space in which physics takes place and that physical quantities are Clifford algebra valued objects, namely, superpositions of multivectors, called Clifford aggregates or polyvectors. We explore some very promising features of physics in Clifford space, in particular those related to a consistent construction of string theory and quantum field theory.

hep-th↗

arXiv2006

Rigid Particle and its Spin Revisited

The arguments by Pandres that the double valued spherical harmonics provide a basis for the irreducible spinor representation of the three dimensional rotation group are further developed and justified. The usual arguments against the inadmissibility of such functions, concerning hermiticity, orthogonality, behavior under rotations, etc., are all shown to be related to the unsuitable choice of functions representing the states with opposite projections of angular momentum. By a correct choice of functions and definition of inner product those difficulties do not occur. And yet the orbital angular momentum in the ordinary configuration space can have integer eigenvalues only, for the reason which have roots in the nature of quantum mechanics in such space. The situation is different in the velocity space of the rigid particle, whose action contains a term with the extrinsic curvature.

hep-th↗

arXiv2006

Spin Gauge Theory of Gravity in Clifford Space: A Realization of Kaluza-Klein Theory in 4-Dimensional Spacetime

A theory in which 4-dimensional spacetime is generalized to a larger space, namely a 16-dimensional Clifford space (C-space) is investigated. Curved Clifford space can provide a realization of Kaluza-Klein theory. A covariant Dirac equation in curved C-space is explored. The generalized Dirac field is assumed to be a polyvector-valued object (a Clifford number) which can be written as a superposition of four independent spinors, each spanning a different left ideal of Clifford algebra. The general transformations of a polyvector can act from the left and/or from the right, and form a large gauge group which may contain the group U(1)xSU(2)xSU(3) of the standard model. The generalized spin connection in C-space has the properties of Yang-Mills gauge fields. It contains the ordinary spin connection related to gravity (with torsion), and extra parts describing additional interactions, including those described by the antisymmetric Kalb-Ramond fields.

gr-qc↗

arXiv2005

Spin Gauge Theory of Gravity in Clifford Space

A theory in which 16-dimensional curved Clifford space (C-space) provides a realization of Kaluza-Klein theory is investigated. No extra dimensions of spacetime are needed: "extra dimensions" are in C-space. We explore the spin gauge theory in C-space and show that the generalized spin connection contains the usual 4-dimensional gravity and Yang-Mills fields of the U(1)xSU(2)xSU(3) gauge group. The representation space for the latter group is provided by 16-component generalized spinors composed of four usual 4-component spinors, defined geometrically as the members of four independent minimal left ideals of Clifford algebra.

gr-qc↗

arXiv2005

On an Alternative Approach to the Relation between Bosons and Fermions: Employing Clifford Space

We further explore the idea that physics takes place in Clifford space which should be considered as a generalization of spacetime. Following the old observation that spinors can be represented as members of left ideals of Clifford algebra, we point out that the transformations which mix bosons and fermions could be represented by means of operators acting on Clifford algebra-valued (polyvector) fields. A generic polyvector field can be expanded either in terms of bosonic, or in terms of fermionic fields. In particular, a scalar field can transform into a mixture of bosonic and/or fermionic fields.

hep-th↗

arXiv2005

Clifford Space as a Generalization of Spacetime: Prospects for QFT of Point Particles and Strings

The idea that spacetime has to be replaced by Clifford space (C-space) is explored. Quantum field theory (QFT) and string theory are generalized to C-space. It is shown how one can solve the cosmological constant problem and formulate string theory without central terms in the Virasoro algebra by exploiting the peculiar pseudo-Euclidean signature of C-space and the Jackiw definition of the vacuum state. As an introduction into the subject, a toy model of the harmonic oscillator in pseudo-Euclidean space is studied.

hep-th↗

arXiv2003

General Principles of Brane Kinematics and Dynamics

We consider branes as "points" in an infinite dimensional brane space ${\cal M}$ with a prescribed metric. Branes move along the geodesics of ${\cal M}$. For a particular choice of metric the equations of motion are equivalent to the well known equations of the Dirac-Nambu-Goto branes (including strings). Such theory describes "free fall" in ${\cal M}$-space. In the next step the metric of ${\cal M}$-space is given the dynamical role and a corresponding kinetic term is added to the action. So we obtain a background independent brane theory: a space in which branes live is ${\cal M}$-space and it is not given in advance, but comes out as a solution to the equations of motion. The embedding space ("target space") is not separately postulated. It is identified with the brane configuration.

hep-th↗

arXiv2003

How the Geometric Calculus Resolves the Ordering Ambiguity of Quantum Theory in Curved Space

The long standing problem of the ordering ambiguity in the definition of the Hamilton operator for a point particle in curved space is naturally resolved by using the powerful geometric calculus based on Clifford Algebra. The momentum operator is defined to be the vector derivative (the gradient) multiplied by $-i$; it can be expanded in terms of basis vectors $γ_μ$ as $p = -i γ^μ\p_μ$. The product of two such operators is unambiguous, and such is the Hamiltonian which is just the D'Alambert operator in curved space; the curvature scalar term is not present in the Hamiltonian if we confine our consideration to scalar wave functions only. It is also shown that $p$ is Hermitian and self-adjoint operator: the presence of the basis vectors $γ^μ$ compensates the presence of $\sqrt{|g|}$ in the matrix elements and in the scalar product. The expectation value of such operator follows the classical geodetic line.

gr-qc↗