Quantization of Space and Time in 3 and in 4 Space-time Dimensions

The fact that in Minkowski space, space and time are both quantized does not have to be introduced as a new postulate in physics, but can actually be derived by combining certain features of General Relativity and Quantum Mechanics. This is demonstrated first in a model where particles behave as point defects in 2 space dimensions and 1 time, and then in the real world having 3+1 dimensions. The mechanisms in these two cases are quite different, but the outcomes are similar: space and time form a (non-cummutative) lattice. These notes are short since most of the material discussed in these lectures is based on two earlier papers by the same author (gr-qc/9601014 and gr-qc/9607022), but the exposition given in the end is new.Comment: Lectures held at the NATO Advanced Study Institute on ``Quantum Fields and Quantum Space Time", Carg\`ese, July 22 -- August 3, 1996. 16 pages Plain TeX, 6 Figure

Towards a Simulation of Quantum Computers by Classical Systems

We present a two-dimensional classical stochastic differential equation for a displacement field of a point particle in two dimensions and show that its components define real and imaginary parts of a complex field satisfying the Schroedinger equation of a harmonic oscillator. In this way we derive the discrete oscillator spectrum from classical dynamics. The model is then generalized to an arbitrary potential. This opens up the possibility of efficiently simulating quantum computers with the help of classical systems.Comment: Author Information under http://www.physik.fu-berlin.de/~kleinert/institution.html . Latest update of paper (including all PS fonts) at http://www.physik.fu-berlin.de/~kleinert/kleiner_re324/preprint.htm

The mathematical basis for deterministic quantum mechanics

If there exists a classical, i.e. deterministic theory underlying quantum mechanics, an explanation must be found of the fact that the Hamiltonian, which is defined to be the operator that generates evolution in time, is bounded from below. The mechanism that can produce exactly such a constraint is identified in this paper. It is the fact that not all classical data are registered in the quantum description. Large sets of values of these data are assumed to be indistinguishable, forming equivalence classes. It is argued that this should be attributed to information loss, such as what one might suspect to happen during the formation and annihilation of virtual black holes. The nature of the equivalence classes is further elucidated, as it follows from the positivity of the Hamiltonian. Our world is assumed to consist of a very large number of subsystems that may be regarded as approximately independent, or weakly interacting with one another. As long as two (or more) sectors of our world are treated as being independent, they all must be demanded to be restricted to positive energy states only. What follows from these considerations is a unique definition of energy in the quantum system in terms of the periodicity of the limit cycles of the deterministic model.Comment: 17 pages, 3 figures. Minor corrections, comments and explanations adde

Chiral determinant on the lattice -- Anomalies and Instantons

An expression for the lattice effective action induced by chiral fermions in any even dimensions in terms of an overlap of two states is shown to have promising properties in two and four dimensions: The correct abelian anomaly is reproduced and gauge field configurations with non-zero topological charge are completely suppressed.Comment: 3 pages, ps-fil

Coordinate Independence of of Quantum-Mechanical Path Integrals

We develop simple rules for performing integrals over products of distributions in coordinate space. Such products occur in perturbation expansions of path integrals in curvilinear coordinates, where the interactions contain terms of the form dot q^2 q^n, which give rise to highly singular Feynman integrals. The new rules ensure the invariance of perturbatively defined path integrals under coordinate transformations.Comment: Author Information under http://www.physik.fu-berlin.de/~kleinert/institution.html . Latest update of paper also at http://www.physik.fu-berlin.de/~kleinert/305

Quantum features in statistical observations of "timeless" classical systems

We pursue the view that quantum theory may be an emergent structure related to large space-time scales. In particular, we consider classical Hamiltonian systems in which the intrinsic proper time evolution parameter is related through a probability distribution to the discrete physical time. This is motivated by studies of ``timeless'' reparametrization invariant models, where discrete physical time has recently been constructed based on coarse-graining local observables. Describing such deterministic classical systems with the help of path-integrals, primordial states can naturally be introduced which follow unitary quantum mechanical evolution in suitable limits.Comment: 7 pages. Invited talk at Int. Workshop Trends and Perspectives on Extensive and Non-Extensive Statistical Mechanics, Angra dos Reis (Brazil), Nov. 2003. To appear in Physica

Non-Trivial SU(N) Instantons and Exotic Skyrmions

The classical Yang-Mills equations are solved for arbitrary semi-simple gauge groups in the Schwinger-Fock gauge. A new class of SU(N) instantons is presented which are not embeddings of SU(N-1) instantons but have non-trivial SU(N) color structure and carry winding number $n=N(N^{2}-1)/6$. Explicit configurations are given for SU(3) and SU(4) gauge groups. By means of the Atiyah Manton procedure Skyrmion fields are constructed from the SU(N) instantons. These Skyrmions represent exotic baryon states.Comment: 10 LaTex pages (1 figure available on request), UNITUE-THEP-10-199

Two particle Quantummechanics in 2+1 Gravity using Non Commuting Coordinates

We find that the momentum conjugate to the relative distance between two gravitating particles in their center of mass frame is a hyperbolic angle. This fact strongly suggests that momentum space should be taken to be a hyperboloid. We investigate the effect of quantization on this curved momentum space. The coordinates are represented by non commuting, Hermitian operators on this hyperboloid. We also find that there is a smallest distance between the two particles of one half times the Planck length.Comment: 18 pages Latex, 2 eps figure

2+12+1 Covariant Lattice Theory and t'Hooft's Formulation

We show that 't Hooft's representation of (2+1)-dimensional gravity in terms of flat polygonal tiles is closely related to a gauge-fixed version of the covariant Hamiltonian lattice theory. 't Hooft's gauge is remarkable in that it leads to a Hamiltonian which is a linear sum of vertex Hamiltonians, each of which is defined modulo $2 \pi$. A cyclic Hamiltonian implies that ``time'' is quantized. However, it turns out that this Hamiltonian is {\it constrained}. If one chooses an internal time and solves this constraint for the ``physical Hamiltonian'', the result is not a cyclic function. Even if one quantizes {\it a la Dirac}, the ``internal time'' observable does not acquire a discrete spectrum. We also show that in Euclidean 3-d lattice gravity, ``space'' can be either discrete or continuous depending on the choice of quantization. Finally, we propose a generalization of 't Hooft's gauge for Hamiltonian lattice formulations of topological gravity dimension 4.Comment: 10 pages of text. One figure available from J.A. Zapata upon reques

Scale Anomaly Induced Instanton Interaction

The binary interaction of large size instantons in a SU(2) Yang-Mills theory is obtained from the one-loop effective action for the field strength. The instanton interaction is calculated as a function of the instanton separation and in dependence on radius and relative orientation of the instantons. Two equally oriented instantons with radii large compared with the scale defined by the gluon condensate have purely attractive interaction, whereas the interaction of maximal disoriented instantons is repulsive. We argue that the medium range attractive interaction of the instantons generally holds and is solely due to the instability of the perturbative vacuum.Comment: 11 LaTex pages (3 figures available on request), in press by Physics Letters B, UNITUE-THEP-4-199