*The Set ( Fourier Representation ) *The Group Structure ( Quantum Conditions, Commutation Relations, Graded Lie Algebra ) *The Logic ( Open Sets, Orthomodular Lattice, Information Capacity ) *The Topology ( Denumerability, Measurability ) *The Functional Integral ( Feynman Integral, Fourier Representation, Group Structure, Quantum Holography ).
*Functional Space ( Fourier Representation, Transformations ) *Differential Topology ( Operators ).Quantum Topological Group
*Commutation Relations and Graded Lie Algebra.Graded Lie Manifold
*Fourier representation of the functional space, the operators and transformations require differential topology.Gauge Field
*Unitary Transformations and the Group Structure of the Quantum Space.
Quantum Logic
*Continuous Mappings in the Set Represent Logical Operations.Open Sets and the Orthomodular Lattice
*The Open Sets Represent the Properties of the Orthologic.Structure of the Orthomodular Lattice and Complexity
*Exponential Information Capacity of the Quantum Space is the Source of Complexity.
*Quantum Space ( h Measure ) *Representation Space ( Reisz, < | > ) *Projections ( Subspaces *Fourier Transformations *Translation Operators ) *Functional Integrals ( Functional Space, Convolution Integral ) *Feynman Integral ( Action ) *Extended Symmetry ( T --> T, Extended Poincare ).
The quantum of action h is a measure generates a functional quantum space. The complete Dirac bracket is an abstract representation of the quantum space. Translation operators represent subspaces of the quantum space that are connected via Fourier transformations. Dynamical operators have distributions in functional space via functional integral. This distribution is defined by the classical action. Topological transformations lead to extend Poincare group.Quantum Space
*Quantum of Action is a Measure on the Quantum Set that Generates the Quantum Space.
Quantum space is what mathematicians call functional space, the inner product of p and q is a bigger space than any of p or q. Quantum space is the arena where physical interactions take place. In the space, mappings define two set of dual translations p and q that are subject to the fundamental quantum conditions [p,q]=ih, and these define the structure of the space in terms of function spaces. Subject to Fourier representation there is a set Q(k) that represents a universe of the Fourier components of p and q that structure the space. Mappings on the set can be represented as topological group and logical operations. The algebraic structure factor the open sets that structure the topological group and the orthomodular lattice.Representation Space
*Topology of the Complete Dirac Bracket Represent the Space of Distributions of the Dynamical Operators.
Representation space of the quantum space is an inner product of a Hilbert space H with its dual space H*, < A | A >. According to Reisz representation theorem there is a conjugate-linear isomorphism between H and the dual space of bounded linear functionals on H, but here the term functional refers to the complete bracket.
Topology is defined by the product of an eigenvector with its dual which define a functional space, an antilinear bilinear form < A | A >. Unlike Hilbert space, in this case when an eigenvector grows indefinitely, it still remains within the functional space provided that the product with its dual is a finite measure in the space and this can conveniently be done applying delta functional.Quantum conditions
*Continuous Mappings and Commutation Relations and Projection of the Dynamical Operators.
*Group Structure ( Phase Angle, Compactness ) *Yang-Mills-Einstein Field ( Commutation Relations ).
Applications of quantum topological principles to derive Einstein's gravitational field along with Yang-Mills gauge field from compact group structure.Phase Angles
*Compactness and Measurability are Elements of the Criterion of Convergence.Quantum Topodynamics
*Denumerability.Quantum Cosmology
*The Role that Holographic Representation and Phase Angels Play in Gravitation and Cosmology.