Quantum Topodynamics

Quantum Dynamics of the Space

Diaa A Ahmed

e-mail diahmed@yahoo.com

(Washington, DC October 2005)

Quantum gauge dynamics as the fibre bundle structure of the quantum space. Quantum space as a topological group and a Hilbert ortholattice.

Quantum theory demonstrated that the manifold and the dynamics are connected in the same mathematical manner that vectors and their duals are connected in the theory of functional spaces. "Quantum Topology" is an extension of the space-time manifold into a functional space that incorporates the quantum dynamics; the quantum space. The quantum manifold is expressed as an antilinear-bilinear form. Quantum space is the invariant arena from which physical interactions are projected into a manifold and a field.

Set theory offers a sound mathematical foundation to study topology, gauge group, and quantum logic structures of the quantum space. A coherent theory should find one mathematcal structure on the set that can be represented as topology, gauge group, and quantum logic to account for the dynamics. Fourier analysis gives us an insight into the connectedness of D and Q (F- transformation), and into the projection of rays of D and Q from the quantum space (F-representation). A Quantum Set is defined as the 2-fold infinite set of the dual coordinates of the quantum space D and Q provided by the Fourier representation. Topological structures, both quantum and classical are defined by commutation relations of D and Q. Continuous mathematical transformations on the Quantum Set generates a Topological Group that gives rise to a Compact Group Manifold and a Gauge Field (Fibre Bundle). To take account for the fibre bundle structure of the dynamics in quantum space, the group structure of the quantum set is introduced into the functional integral to formulate "Quantum Topodynamics".

The quantum logic approach to formulate the dynamics is to represent the continuous mathematical operation on the Quantum Set as Logical operation, and to represent the algebraic structure as an orthomodular structure. This approach translates the group structure into the language of quantum logic (quantum numbers; properties) and gives us an insight into quantum computation and, a criterion for the finiteness of the functional integration in the theory on the basis of the global properties of the functional space. An immediate application to this is to use the fibre bundle structure for quantum computation. First, to represent the fundamental logical operation as quantum interference, and then to reflect the group structure in a matrix quantum interference devise, provided we devise an approperiate coding for the quantum numbers. This matrix processor allows NxN gauge potentials to act on the phases of N rays, and the quantum interference of these rays is equivelant to performing functional integration to generate continuous holographic output that represents mathematically the topology of the quantum state being computed.

FUNCTIONAL INTEGRAL OF QUANTUM TOPODYNAMICS

The functional S is expressed in terms of the conjugate dynamical variables; D_l and Q^l ,

the dual 1-forms; D_l and Q^l , are the dual coordinates of the quantum space. D_l and Q^l are represented by their Fourier components,

this Fourier represntation provides us with a connected 2-fold infinite set.

THE QUANTUM SET

We define the 2-fold infinite set of the dual 1-forms; D_l and Q^l, as the fundamental set of the quantum space,

The group operation embedded in the set, generates a group structure for the 1-forms,

these are the basic building blocks of the manifold and the field.

THE QUANTUM TOPOLOGICAL GROUP

The 2-fold infinite set of the dual 1-forms, D_l and Q^l is therefore, a quantum topological group,

the group provides us with a compact graded Lie manifold, the strength of gauge theory lies in the comactness of its group manifold and the subsequent renormalizability of the theory. Gauge transformations were able to eliminate divergent integrals from the theory and renormalize it because of the compactness of the group manifold.

On the other hand, Poincare group does not supply us with a compact group manifold, the translation subgroup is non-compact. This is a major divergence from the basic principles of the standard gauge theory in which integration is performed over a compact group manifold. Therein lies the problem of quantizing gravity; quantum gravity can not be a gauge theory of the Poincare group. My work on "Gauge Theory of Gravitation" is based on this compactness of the group manifold of the theory of gravitation.

QUANTUM TOPOLOGICAL STRUCTURES

The dual 1-forms D _l and Q^l form the open sets of topology and their commutation relations give us the topological structure of the theory.

First Order Quantum Theory: Commutation relations of the dual 1-forms,

give us the structure of the quantum space itself. In the graded manifold,

Q^a and Q_b, are the dual coordianes of the graded manifold.

The gauge principle is a quantum principle that starts from the functional w, to derive the 1-form D_l,

functional integration of these 1-forms should give us the functional, therefore properties of the integrals are determined by topology of the functional. (Functional Analysis, Analysis on Manifolds, Differential Topology).

Second Order Classical Theory: Commutation relations of the 1-forms; D_l and D_n,

give us the 2-form of the classical field F^a_{l n}.

On the other hand, the variational principle starts from the functional S to derive the 2-form F^a_{l n}, therefore it is a classical principle.

HILBERT ORTHOLATTICE

Now, we will consider the quantum logic representation of the set. The fundamental group operation (*) on G (N), represents a logical operation then, the Fourier components of the dual 1-forms D_l and Q^l represent rays of the orthomodular lattice L,

B------------^A------------L

s (g) | | s (g)

B------------^A------------L

the quantum set represents the complete orthomodular lattice L,

{ HOL } = { D_l , Q^l }

the group structure embedded in the dual 1-forms D_l and Q^l, translate into properties of the logic (quantum numbers), and the quantum dynamics represent morphisms in this logical representation.

THE QUANTUM PROCESSOR

Now, we analyse the fibre bundle encoded in the functional integral on the basis of the orthomodular representation,

this will give us an insight into quantum computation.

In the current approach to quantum computation, superposition of quantum states represents the basic logical operation, and the quantum computer is thought of as a kind of super-digital computer; a number (possibly infinite) of parallel digital processors.

Here we follow a different approach based on the orthomodular representation the fibre bundle structure, where quantum interference is the basic logical operation, and the quantum processor is a kind of a matrix quantum interference devise that reflects structure of the fibre bundle; i.e. group structure, provided that we devise an approperiate encoding for the quantum numbers. This matrix processor allows NxN gauge potentials to act on the phases of N rays, and the quantum interference of these rays is equivelant to performing functional integration to generate continuous holographic output that represents mathematically the topology of the quantum state being computed.

To continue ....

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