q-deformed Minkowski space based on a q-Lorentz algebra

The Hilbert space representations of a non-commutative -deformed Minkowski space, its momenta and its Lorentz boosts are constructed. The spectrum of the diagonalizable space elements shows a lattice-like structure with accumulation points on the light-cone. Received: 23 January 1997 / Published online: 10 March 1998     

q-deformed phase space and its lattice structure

A q-deformed two-dimensional phase space is studied as a model for a noncommutative phase space. A lattice structure arises that can be interpreted as a spontaneous breaking of a continuous symmetry. The eigenfunctions of a Hamiltonian that lives on such a lattice are derived as wavefunctions in ordinaryx-space.     

q-deformed phase space and its lattice structure

A q-deformed two-dimensional phase space is studied as a model for a noncommutative phase space. A lattice structure arises that can be interpreted as a spontaneous breaking of a continuous symmetry. The eigenfunctions of a Hamiltonian that lives on such a lattice are derived as wavefunctions in ordinaryx-space.     

The geometry of a q-deformed phase space

The geometry of the q-deformed line is studied. A real differential calculus is introduced and the associated algebra of forms represented on a Hilbert space. It is found that there is a natural metric with an associated linear connection which is of zero curvature. The metric, which is formally defined in terms of differential forms, is in this simple case identifiable as an observable. Received: 26 November 1998 / Published online: 27 April 1999     

Free particle in q-deformed configuration space

The SO _q (N)-invariant Schrödinger equation for the free particle is formulated in polar coordinates as a partial differential equation in noncommutative geometry. For each value of the total angular momentum, a Hilbert space of radial functions is constructed as the space of normalizable functions respective to the q-integral. The spectrum of the Hamiltonian is found to be discrete.     

Meron-antimeron solution in Minkowski space

An elementary derivation, using Witten's Ansatz, is given of the elliptic meron-antimeron solution of the (Minkowski) SU(2) gauge theory in the W ₀=0 gauge.     

Algebrodynamics over complex space and phase extension of the Minkowski geometry

First principles should predetermine physical geometry and dynamics both together. In the “algebrodynamics” they follow solely from the properties of biquaternion algebra and the analysis over . We briefly present the algebrodynamics over Minkowski background based on a nonlinear generalization to of the Cauchi-Riemann analyticity conditions. Further, we consider the effective real geometry uniquely resulting from the structure of multiplication and found it to be of the Minkowski type, with an additional phase invariant. Then we pass to study the primordial dynamics that takes place in the complex space and brings into consideration a number of remarkable structures: an ensemble of identical correlated matter pre-elements (“duplicons”), caustic-like signals (interaction carriers), a concept of random complex time resulting in irreversibility of physical time at macrolevel, etc. In partucular, the concept of “dimerous electron” naturally arises in the framework of complex algebrodynamics and, together with the above-mentioned phase invariant, allows for a novel approach to explanation of quantum interference phenomena alternative to recently accepted wave—particle dualism paradigm. The text was submitted by the author in English.     

Quantum complex Minkowski space

The complex Minkowski phase space has the physical interpretation of the phase space of the scalar massive conformal particle. The aim of the paper is the construction and investigation of the quantum complex Minkowski space.     

Conformal gauge fixing in Minkowski space

Conformal gauge fixing on simply connected parts of world sheets in Minkowski space is examined in detail. It is proved that conformal gauge fixing can be imposed, including the usual boundary conditions.     

Solving Bethe-Salpeter equation in Minkowski space

We develop a new method of solving the Bethe-Salpeter (BS) equation in Minkowski space. It is based on projecting the BS equation on the light-front (LF) plane and on the Nakanishi integral representation of the BS amplitude. This method is valid for any kernel given by the irreducible Feynman graphs. For massless ladder exchange, our approach reproduces analytically the Wick-Cutkosky equation. For massive ladder exchange, the numerical results coincide with the ones obtained by Wick rotation.     

Discrete dynamical equations in Minkowski space

Discrete difference equations in Minkowski space are obtained and the discrete Minkowski force is shown to be a four-vector. A transformation from a discrete dynamical equation in Minkowski space to a Lorentz-invariant difference equation in one-dimensional space is given.     

Compact topologies on Minkowski space

Minkowski space can be given topologies which are compact and which have, as their homeomorphism group, the inhomogeneous Lorentz group together with dilatations.     

Cosmological constant and Minkowski space

Using the framework of classical gravitational field theory, it is shown that the equations of Einstein’s General Relativity with a cosmological constant, if requested to be compatible with the Minkowski space, change form and become the equations of Relativistic Theory of Gravitation. These equations, in contrast to General Relativity, lead us to fundamentally different physical conclusions about the Universe’s evolution and Collapse.     

Causal logic of Minkowski space

It is shown that double-orthogonal sets (diamonds) in Minkowski space form an orthomodular complete lattice. Connection with empirical logic of Randall and Foulis is discussed.     

Finite-temperature quantum field theory in Minkowski space

We formulate finite-temperature quantum field theories in Minkowski space (real time) using Feynman path integrals. We show that at non-zero temperature a new field arises which plays the role of a ghost field and is necessary for unambiguous Feynman rules. Consequently, the finite-temperature Lagrangian is different from the zero-temperature one and a new, discrete Z₂ symmetry arises. We discuss the functional formalism and spontaneous symmetry breakdown at finite temperature and also the possibility of spontaneous breakdown of the (thermal) Z₂ symmetry.     

On time-like Willmore surfaces in Minkowski space

We study time-like surfaces in Minkowski space, which are critical points of the Willmore energy. Transforming the fourth order Willmore equation into a quasi-linear, second order hyperbolic system, we prove existence, uniqueness and symmetry properties of such surfaces, subject to geometric initial conditions.     

From unstable Minkowski space to inflation

We present a perturbative approach to the equations describing the behavior of a quantum scalar field in a self-consistently generated Robertson-Walker universe. This approach throws new light on the significance of the Minkowskian instability and on the subtraction procedure which shows that a inflation cosmology is a possible future of the Minkowski space.     

The Zeeman-order topology on Minkowski space

The order topology on Minkowski Space generated by Zeeman's causal order is an elementary example of a topology whose autohomeomorphism group is a Poincaré dilation group.