Clifford Algebra, Geometry and 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. Within C-space we can perform the so called polydimensional rotations which reshuffle the multivectors, e.g., a bivector into a vector, etc.. A consequence of such a polydimensional rotation is that the signature can change: it is relative to a chosen set of basis vectors. Another important consequence is that the well known unconstrained Stueckelberg theory is embedded within the constrained theory based on C-space. The essence of the Stueckelberg theory is the existence of an evolution parameter which is invariant under the Lorentz transformations. The latter parameter is interpreted as being the true time - associated with our perception of the passage of time.Comment: 11 pages, Talk presentedat NATO Advanced Research Workshop "The Nature of Time: Geometry, Physics & Perception", May 2002, Tatranska\' a Lomnica, Slovakia; To appear in Proceeding

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'.Comment: 15 pages; Presented at "Time and Matter 2007", 26-31 August 2007, Bled, Sloveni

The Embedding Model of Induced Gravity with Bosonic Sources

We consider a theory in which spacetime is an n-dimensional surface $V_n$ embedded in an $N$-dimensional space $V_N$. In order to enable also the Kaluza-Klein approach we admit $n > 4$. The dynamics is given by the minimal surface action in a curved embedding space. The latter is taken, in our specific model, as being a conformally flat space. In the quantization of the model we start from a generalization of the Howe-Tucker action which depends on the embedding variables ${\eta}^a (x)$ and the (intrinsic) induced metric $g_{\mu \nu}$ on $V_n$. If in the path integral we perform only the functional integration over ${\eta}^a (x)$, we obtain the effective action which functionally depends on $g_{\mu \nu}$ and contains the Ricci scalar $R$ and its higher orders $R^2$ etc. But due to our special choice of the conformal factor in $V_N$ enterig our original action, it turns out that the effective action contains also the source term. The latter is in general that of a $p$-dimensional membrane ($p$-brane); in particular we consider the case of a point particle. Thus, starting from the basic fields ${\eta}^a (x)$, we induce not only the kinetic term for $g_{\mu \nu}$, but also the "matter" source term.Comment: 11 page

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.Comment: 37 page

Pseudo Euclidean-Signature Harmonic Oscillator, Quantum Field Theory and Vanishing Cosmological Constant

The harmonic oscillator in pseudo euclidean space is studied. A straightforward procedure reveals that although such a system may have negative energy, it is stable. In the quantized theory the vacuum state has to be suitably defined and then the zero-point energy corresponding to a positive-signature component is canceled by the one corresponding to a negative-signature component. This principle is then applied to a system of scalar fields. The metric in the space of fields is assumed to have signature (+ + + ... - - -) and it is shown that the vacuum energy, and consequently the cosmological constant, are then exactly zero. The theory also predicts the existence of stable, negative energy field excitations (the so called "exotic matter") which are sources of repulsive gravitational fields, necessary for construction of the time machines and Alcubierre's hyperfast warp drive.Comment: 13 page

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.Comment: 7 page

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.Comment: 27 page

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.Comment: 15 Pages; References added, Introduction and Conclusion complete

Resource Letter on geometrical results for Embeddings and Branes

Due to the recent renewal in the interest for embedded surfaces we provide a list of commented references of interest.Comment: Updated version containing new references. 39 pages, LaTe

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.Comment: 380 pages; Published by Kluwer Academic Publishers, 2001; misprints correcte