Quantum Topodynamics

Diaa A Ahmed

e-mail: diahmed@yahoo.com

(Rotterdam March 1997)

The basic set of the quantum space underlies the assumed oneness of the topology, group and logic structure of the quantum space. The elements of the basic set of the quantum space are identified as the Fourier components of dual linear forms D_l , X^l. Dynamics is identified as quantum mappings ( Commutation Relations ). The elements of the quantum set -rays- are the basis of topology and the quantum computer, it encode the logic and perform functional integration, the physical picture based on holography permits for computability and the building up of the quantum computer.

I. THE SET

The infinite set of the Fourier components of dual linear forms D_l , X^l , in which multiplication is defined in terms of topological, group and logical operation.

A. The structure: First Order-Quantum Theory

The fundamental quantum conditions [ D_l , X^l ] = - ( d + i w ) , determines the dynamic structure of the functional space. In group theory, this is realized in graded Lie algebra, where; p_m = - 2 ( g^m c ) - 1 { Q_a , Q_b }, [ Q^a , Q_b ] = - d^a_b. The commutation relations [ T^a , T^b ] = i f^abg T^g , determines the group structure of the 1-form D_l.

B. The Structure: Second Order-Classical Theory

The commutation relations between 1-forms of the quantum field determines the 2-form of the classical field

T^a . F^a_{l n} = i g^-1 [ D_l , D_n ].

Elements of the basic set are the Fourier components of D_l , X^l , they represent states; points or rays as y_ao in H_ao. They are the basic building blocks of logic, topology and group structure.

II. The Logic

A complete orthomodular lattice on the set, where subsets of L , are the open sets of topology and transformations are continuous,

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

s (g) | | s (g)

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

The 1-form D_l , encodes the logical structure a-properties, and dynamics _*-morphisms, of fundamental laws and the fabrics of topology form a quantum computer.

The Quantum Principle: The infinite set of dual linear forms that constitute the functional with there exponential information capacity is the basis of complexity.

III. TOPOLOGY

The set is denumerable ( P < x | ) P | y > = < x | y > , and measurable < A' | A''> = ( A' - A'').

In terms of the Fourier components, the functional space can integrate the dynamics; duality, and provide a global aspect of physics free of singularities and infinities that constitute the limit of using geometry and functions of analysis.

The measure h determines the global property of the functional space, and the forms D_l , ¶ X^l , are the local properties of the functional space. Both the gauge principle and the variational principle start from the functional; S , w , to derive the local 1-form, 2-form. The manifold and the fields are expressible in terms of this 1-form D_l .

IV. THE FUNCTIONAL INTEGRAL

The basic idea here is functional d-space. In mathematics, d is a linear functional on the set of test functions ( d , j ) , which may be interpreted as linear functional on the set of basic vectors forms, | A > , or simply functional d-space, < A | A >. The integral admits representation as Feynman integral,

ò j (q) exp ( i S (q) / h ) d q = j < 0 >.

and the Fourier representation,

½ p ò exp ( i k x ) d k.

and the group structure,

exp { i g T^a . w^a (x)}.

combined we get what we may call a holographic representation,

ò j (q) exp ( ò D_l d q^l ) d q = j < 0 >.

in terms of the Fourier components of D_l.

The continuous infinity of solutions depending on this form -or w^a (x) , the choice of g's, each represent a quantum holograph; an image of a universe existing in functional space. Topodynamics thus means quantum holography. The exponential

{ i g T^a . w^a (x)}, is factored into properties w^a (x) , and bits g's. We may call this a quantum hologram. Although there is an infinite Fourier components of the 1-form D_l , in functional space there arise an infinite component of its dual ¶ X^l , and integration in terms of these twofold infinite components is computable and has topological meaning ( d , w , F , T ). The integral of the 1-form D_l , is a different form; functional. The integral is not affected by the infinite components of D_l , because of its dual X^l , but is determined by the functional; bound by the topology.

Note: the important topological structure is encoded in, exp ( ò D_l d q^l ) , the group and logical structure in D_l = ¶_l - i g W^a_l.

Holography.

Solitons.

DNA.

SQUIDS.

Are candidates for the quantum computer.

[1] Quantum Topology and the references cited therein .

[2] J A Wheeler, It from Bit, Princeton 1991.

[3] L Lopes, Gauge Field Theories, Pergamon 1981.