We start from a new theory (discussed earlier) in which the arena for physics is not spacetime, but its straightforward extension-the so called Clifford space (C-space), a manifold of points, lines, areas, etc..; physical quantities are Clifford algebra valued objects, called polyvectors. This provides a natural framework for description of supersymmetry, since spinors are just left or right minimal ideals of Clifford algebra. The geometry of curved C-space is investigated. It is shown that the curvature in C-space contains higher orders of the curvature in the underlying ordinary space. A C-space is parametrized not only by 1-vector coordinates x µ but also by the 2-vector coordinates σ µν , 3-vector coordinates σ µνρ , etc., called also holographic coordinates, since they describe the holographic projections of 1-lines, 2-loops, 3-loops, etc., onto the coordinate planes. A remarkable relation between the "area" derivative ∂/∂σ µν and the curvature and torsion is found: if a scalar valued quantity depends on the coordinates σ µν this indicates the presence of torsion, and if a vector valued quantity depends so, this implies non vanishing curvature. We argue that such a deeper understanding of the C-space geometry is a prerequisite for a further development of this new theory which in our opinion will lead us towards a natural and elegant formulation of M-theory.

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.

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. 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)×SU(2)×SU(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.

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

Abstract. Certain difficulties of quantum gravity can be avoided if we embed the spacetime V 4 into a higher dimensional space V N ; then our spacetime is merely a 4-surface in V N . What remains is conceptually not so difficult: just to quantise this 4-surface. Our formal procedure generalises our version of Stueckelberg's proper time method of worldline quantisation. We write the equations of V 4 in the covariant canonical form starting from a model Lagrangian which contains the classical Einstein gravity as a particular case. Then we perform quantisation in the Schrödinger picture by using the concepts of a phase functional and wave functional. As a result we obtain the uncertainty relations which imply that an observer is 'aware' either of a particular spacetime surface and has no information about other spacetime surfaces (which represent alternative histories); or conversely, he loses information about a particular V 4 whilst he obtains some information about other spacetimes (and histories). Equivalently, one cannot measure to an arbitrary precision both the metric on V 4 and matter distribution on various alternative spacetime surfaces. We show how this special case in the 'coordinate' representations can be generalised to an arbitrary vector in an abstract Hilbert space.