Theory of the Quantum Space

'an outline'

Diaa A Ahmed

e-mail: diahmed@yahoo.com

(Amsterdam 1999)

Theory of the quantum space is founded on the theory of quantum sets. The quantum of action is a measure on the quantum set that generates the quantum space. In quantum space there exist dual spaces p and q defined by translation operators subject to the quantum conditions and their Fourier representation provide the physical elements of the quantum set. The algebraic structure (commutation relations) unfolds structure of the quantum topological group with its graded Lie manifold and gauge field. Representing the continuous mappings as logical operations permit us to represent the algebraic structure as a Hilbert ortholattice with the Fourier components as its rays and the open sets interpreted as properties of the quantum logic. Exponential information capacity of the functional space is the source of complexity. Topology of the complete Dirac bracket represents the space of distributions of the dynamical operators. The criterion for convergence of functional integration performed on a quantum topological group rests on finiteness of the measure, compactness and denumerablity of the space. The role that the holographic representation plays in gravitation in constructing the manifold from interfering rays and representation of functional integrals. Role of translating the group structure into orthomodular structure in quantum computation. Parallel unitary transformations act on the rays and elements of the quantum interference processors.

The Quantum Set

Elements of the Set

*Fourier Representation of the Translation Operators P and Q Subject to Quantum Conditions.

*Cardinality of the Set.

Quantum Mappings

*Continuous Mappings on the Set Represent Group Logical Operation Operation.

Structure of the Open Sets

*Commutation Relations factor the Set into Open Sets that Structure the Quantum Topological Group and the Hilbert Ortholattice.

Quantum Algebraic Structure

*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.

Hilbert Orthomodular Structure

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 Topology

*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.

Functional Integrals

*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.

Quantum Computation

The Group Structure and Coding

Parallel Unitary Transformations and Quantum Interference Processor

Quantum Holography and Functional Integrals

e^-iHt/h |A>

e^-iHt/h? e^ iHt/h; phase angle, vacuum

Seperable Hilbert space, that is, a space that can be spanned by an enumerable infinity of basic forms.

BxA; Is a space much bigger than a Hilbert space.

= d; delta space.

(row) Operator (column); operator acts on a quantum space.