Differential Topology in Quantum Space

Diaa A Ahmed

e-mail: diahmed@yahoo.com

(Rotterdam July 1997)

To sum up; Continuous Transformations; Poincare-Lorentz, Gauge, Renormalization Group, were needed to supplement functions and their function spaces, in fact it needed a continuous infinity of these and their duals. The concept of functional d-space -as a model- is a central idea to understand the transformations and the physical nature of space. Functions and geometry are no longer representing physical reality.

( The curves ~ , and the waves | , are absent in the diagram, which represents the functional space.) Each point is in fact a ray, a Fourier component of D_l , the metric itself should be represented this way; this is the requirement of classical and quantum field theory, their fundamental Fourier representation and the Fourier representation of the space; an out come of the transformations. The holographic principle ( phases ) recovers the Riemannian Einsteinian structure of the manifold. Charges act as solitons or holograms in the quantum space; project a specific type of a-rays from the functional space through a certain phase angle g w^a(x) , and here again the holographic principle recovers the geometric structure of the fields.

At each different neighboring point of the manifold there exist a locally inertial system of coordinates;

k⁽⁴⁾

k⁽⁵⁾

k⁽⁶⁾

k⁽³⁾

k⁽⁷⁾

k⁽²⁾

k⁽⁸⁾

k⁽¹⁾

h^mn₍₅₎

k⁽⁹⁾

< w^a|

e_b >

d^a_b

h_mn⁽ⁿ⁾

Dual basis, [ Q^a , Q_b ], [ w^a , e_b ], [ g^am , g_mb ], of d^a_b,

The sequence of points, (1), (2),...., there exist a sequence of fields, y⁽¹⁾ , y⁽²⁾ , ..... , a sequence of Minkowskian metrics

h⁽¹⁾, h⁽²⁾,....., and the continuous transformations connects a continuous infinity of these, (1), (2),....., local systems. There is not one system ( say space time ), but one at each point, and translations ( rot ) take place between neighboring systems in functional space. The generators of the transformations, say, [ I + ¶_l ( d x )] , or, [ I + i ¶_l w ( ¶ x^l )] , are the dual basis of functional d-space, < ¶ x^a , ¶ / ¶ x^b > = d^a_b. In the language of quantum mechanics, field theory and Fourier representation, the 1-forms ¶_l, ¶ x^l , are k-waves. There is not a flat space time h^mn, at each neighboring point of the manifold but a k-component of the basis forms ¶_l , ¶ x^l. Quantum mechanics and field theory require basically the language of functional space and differential topology.

[1] Quantum Topology and the references cited therein.

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

[3] MTW, Gravitation, Freeman 1973.