Quantum topological geometrodynamics

Journal Article · Mon Sep 01 04:00:00 EDT 1986 · Int. J. Theor. Phys.; (United States)

DOI:https://doi.org/10.1007/BF00668825· OSTI ID:6922091

Pitkaenen, M

The description of 3-space as a spacelike 3-surface X of the space H = M/sup 4/ x CP/sub 2/ (Product of Minkowski space and two-dimensional complex projective space CP/sub 2/) and the idea that particles correspond to 3-surfaces of finite size in H are the basic ingredients of topological geometrodynamics (TGD), an attempt at a geometry-based unification of the fundamental interactions. The observations that the Schrodinger equation can be derived from a variational principle and that existence of a unitary S-matrix follows from the phase symmetry of this action lead to the idea that quantum TGD should be derivable from a quadratic phase-symmetric variational principle for some kind of superfield (describing both fermions and bosons) in the configuration space consisting of the spacelike 3-surfaces of H. This idea as such has not led to a calculable theory. The reason is the wrong realization of the general coordinate invariance. The crucial observation is that the space Map(X,H), the space of maps from an abstract 3-manifold X to H, inherits a coset space structure from H and can be given a Kahler geometry invariant under the local M/sup 4/ x SU(3) an under the group Diff of X diffeomorphisms. The space Map(X,H) is taken as a basic geometric object and general coordinate invariance is realized by requiring that superfields defined in Map(X,H) are diffeo-invariant, so that they can be regarded as fields in Map(X,H)/Diff, the space of surfaces with given manifold topology. Superd'Alembert equations are found to reduce to a simple algebraic condition due to the constant curvature and Kahler properties of Map(X,H). The construction of physical states leads by local M/sup 4/ x SU(3) invariance to a formalism closely resembling the quantization of strings. The pointlike limit of the theory is discussed. Finally, a formal expression for the S-matrix of the theory is derived and general properties of the S-matrix are discussed.

• Research Organization: Framnaesintie, Masala
• OSTI ID: 6922091
• Journal Information: Int. J. Theor. Phys.; (United States), Journal Name: Int. J. Theor. Phys.; (United States) Vol. 25:9; ISSN IJTPB
• Country of Publication: United States
• Language: English

Related Subjects

645400* -- High Energy Physics-- Field Theory
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
ANGULAR MOMENTUM
ANNIHILATION OPERATORS
BOSONS
CHIRAL SYMMETRY
COMMUTATION RELATIONS
FERMIONS
FIELD THEORIES
GAUGE INVARIANCE
GRADED LIE GROUPS
INVARIANCE PRINCIPLES
LIE GROUPS
MAPPING
MATHEMATICAL MANIFOLDS
MATHEMATICAL OPERATORS
MATHEMATICAL SPACE
MATHEMATICS
MATRICES
MINKOWSKI SPACE
PARTICLE PROPERTIES
QUANTIZATION
QUANTUM FIELD THEORY
QUANTUM OPERATORS
S MATRIX
SPACE
SPIN
SU GROUPS
SU-3 GROUPS
SUPERSYMMETRY
SYMMETRY
SYMMETRY GROUPS
TOPOLOGICAL MAPPING
TOPOLOGY
TRANSFORMATIONS
UNIFIED-FIELD THEORIES