arXiv subjects
Publications and source records attributed to Matej Pavšič.
It is shown that the Pais-Uhlenbeck oscillator with damping, considered by Nesterenko, is a special case of a more general oscillator that has not only a first order, but also a third order friction term. If the corresponding damping constants, α and β, are both positive and below certain critical values, then the system is stable. In particular, if α= - β, then we have the unstable Nesterenko's oscillator
We point out that quantum field theories based on the concept of Clifford space and Clifford algebra valued-fields involve both positive and negative energies. This is a consequence of the indefinite signature (p,q) of the Clifford space. When the signature is neutral, p=q, then vacuum energy vanishes and there is no cosmological constant problem. A question of the stability of such theories in the presence of interactions arises. We investigate a toy model of the harmonic oscillator in the space $M_{1,1}$. We have found that in the presence of certain interactions the amplitude of oscillations can remain finite. In general this is not the case and the amplitude grows to infinity, but only when the two frequencies are exactly the same. When they are even slightly different, the amplitude remains finite and the system is stable. We show how such oscillator comes from the Stueckelberg action in curved space, and how it can be generalized to field theories.
We examine the possibility of localized propagating tachyonic fields within a properly extended relativity. A possible extension is to include superluminal transformations and reference frames. This leads to complex 4D spacetime, or real 8D spacetime M_{4,4}. The mass shell constraint in M_{4,4} becomes, after first quantization, the ultrahyperbolic Klein-Gordon equation. The Cauchy problem for such equation is not well posed, because it is not possible to freely specify initial data on a 7D hypersurface of M_{4,4}. We explicitly demonstrate that it is possible to do it on a space-like 4-surface for bradyons, and on a time-like 4-surface for tachyons. But then the evolution of a bradyonic field into the four time-like directions, or the "evolution" of a tachyonic field into the four space-like directions, is not uniquely determined. We argue that this is perhaps no so bad, because in quantum field theory (after second quantization) the classical trajectories of fields are not determined anyway, and so it does not matter, if they are not completely determined already in the first quantized theory. A next possible extension of relativity is to consider 16D Clifford space, C, a space whose elements are oriented r-volumes, r=0,1,2,3,4 of real 4D spacetime. Then the evolution parameter can be associated with an extra light-cone coordinate, e.g., with the sum of the scalar and the pseudoscalar coordinate, and initial data can be given on a light-like hypersurface, in which case the Cauchy problem is well posed. This procedure brings us to the Stueckelberg theory which contains localized propagating tachyons in 4D spacetime.
We consider the ADM splitting of the Einstein-Hilbert action in five dimensions in the presence of matter that can be either a "point particle", or a set of scalar fields. The Hamiltonian, being a linear superposition of constraints, is equal to zero. Upon quantization, we obtain the Schrödinger equation for a wave functional, Ψ, that depends on the matter degrees of freedom, and on the 5D gravity degrees of freedom. After the Kaluza-Klein splitting, the functional Schrödinger equation decomposes so that it contains a part due to 4D gravity, a part due to electrodynamics, and a part due to matter. Depending on choice of the matter term, we obtain two different versions of a modified quantum electrodynamics. In one version, time automatically appears, and there is no problem with infinite vacuum energy density of matter fields, whereas in the other version such problems exist.
We show that the superluminal speeds of the muon neutrinos observed in the OPERA experiment can be explained within a relativity theory with extra time like dimensions. In addition, such theory predicts, the existence of dark matter.
The transition from a classical to quantum theory is investigated within the context of orthogonal and symplectic Clifford algebras, first for particles, and then for fields. It is shown that the generators of Clifford algebras have the role of quantum mechanical operators that satisfy the Heisenberg equations of motion. For quadratic Hamiltonians, the latter equations are obtained from the classical equations of motion, rewritten in terms of the phase space coordinates and the corresponding basis vectors. Then, assuming that such equations hold for arbitrary path, i.e., that coordinates and momenta are undetermined, we arrive at the equations that contains basis vectors and their time derivatives only. According to this approach, quantization of a classical theory, formulated in phase space, is replacement of the phase space variables with the corresponding basis vectors (operators). The basis vectors, transformed into the Witt basis, satisfy the bosonic or fermionic (anti)commutation relations, and can create spinor states of all minimal left ideals of the corresponding Clifford algebra. We consider some specific actions for point particles and fields, formulated in terms of commuting and/or anticommuting phase space variables, together with the corresponding symplectic or orthogonal basis vectors. Finally we discuss why such approach could be useful for grand unification and quantum gravity.