Quantum Topology

Diaa A Ahmed

e-mail: diahmed@yahoo.com

(Cairo June 1989)

"Quantum Topology" deals with the general quantum theory as the theory of quantum space. On the quantum level space-time and energy-momentum forms form a connected manifold; a functional quantum space. Many problems in quantum theory and field theory flow from not perceiving this symmetry and the functional nature of the quantum space.

Both topology, groups and logic are based on the concept of sets. If properties coincide with the open sets of topology, then logic and topology will have the same structure. If transformations are continuous in topology, then we will have topological groups; we can derive fields. Therefore, quantum logic underlies the manifold and the fields and nature is based on the language of quantum logic.

Quantum theory and field theory based on sets and the derived topology, group and logic structures should address the question of computation and the mind; the quantum computer and the quantum mind.

I. INTRODUCTION

Minkowski said: "Henceforth space by itself and time by itself are doomed to fade away into mere shadows and only a union of the two will preserve an independent reality".

We can take a little step and see that the natural extension to that came with the quantum theory; that the space-time form and the conjugate energy-momentum form are not independent realities by themselves, they recede leaving behind a functional space; the quantum space, then topology dominates physics.

The role played by h in quantum theory, analogous to the role of C in relativity theory; it shows us more deeply the connectedness of the world. This view about the quantum is presented in the spirit of Minkowski's radical view of space and time.

A. The Quantum Question

There is a connection between quantum theory and theory of relativity; take the relationship D E D t ³ h , and suppose we have a physical system in the vacuum, its energy will change, it will look as if it went into Lorentzian transformation L E ® E' , its proper time will go into reciprocal transformation L ^-1 T ® T' , such that there product; the action integral itself will remain invariant.

Let the system smoothly vanish ( E₀® 0 ) , now in vacuum, fluctuations of energy will look as it is accompanied by reciprocal fluctuations of time ( as if a gravitational field fluctuate ), such that h will be our invariant.

Energy-momentum fluctuations in vacuum will be accompanied by reciprocal fluctuations of the metric, the whole manifold will go into fluctuations, energy-momentum and space-time lose their independence and become a connected manifold, a functional quantum space. We are left for h alone; the invariant.

Why G? If I may speculate about a ratio between h and cosmological action that makes G a weak coupling on a cosmological scale; just a trace of the direct coupling.

B. Optics Mechanics Analogy

k = ¶ j / ¶ r, p = ¶ S / ¶ r, h k = p,

w = ¶ j / ¶ t, E = ¶ S / ð t, h w = E.

Purely geometric quantities ( j , k , w ), acquired dynamical meaning.

C. Waves, Particles and Probabilities

Waves that do not scatter, particles that do not localize, probabilities that interfere; the quantum sense of these concepts required a generalization beyond the classical meaning of these concepts to some underlying more basic concept.

D. General Relativity

Treating energy momentum form symmetrically with the geometric aspect of nature in the quantum space will lead us from an incomplete geometric view to a complete topological view.

Functional space seems to be the way out of the singularities of space time; functional has no value at separate points.

Quantization.

E. Vacuum State

Quantum fluctuation in the vacuum endowed the vacuum with a complete picture of the dynamics. Fluctuations in energy momentum in vacuum means that the vacuum is not the simple geometric aspect of nature. Simple dimensional analysis shows so; it is h space. It is h that generates the dynamics. We can see that the vacuum is the functional quantum space; the pure quantum aspect of nature.

II. QUANTUM SPACE

T 1. Finiteness of h:

(a) q × p ¹ Æ (means that both q and p are nonembty)

(b) If q × p = Æ then q = Æ or p = Æ

It is the finiteness of hwhich leads us to the functional space. In the classical limit h ® 0 , (b) is valid, p_m , and q^m , split into separate forms; therefore, the classical action can not lead to a functional space.

A 1. h is a finite measure that generates a functional space; the quantum space.

Our starting point is the antilinear bilinear form ( p_m , q^m ) ; a product of forms ò p_m d q^m , that build up the functional h . It is obvious that this form has infinitely many states of representation. We can derive these states from the theory of representation.

III. REPRESENTATION SPACE

T 2. Reisz Representation Theorem:

Each linear functional f on a Hilbert space H can be expressed:

( y_f, y ) is an inner product and yf is an element of H* uniquely determined by,

y_f = ò f (y_f)* y_j d j.

A 2. A product of a Hilbert space H with its dual space H*; in quantum theory < A | A >.

So the states and the Hilbert space of the quantum theory follow directly from the functional representation, with the difference that; only the functional < A | A > is significant.

V. PROJECTIONS

We can build up the space by taking the topological product T₁ x T₂ , of its subspaces.

T 3. Topological Theorem:

The Transformations,

p^{^} : T₁ x T₂ ® T₁,

q^{^} : T₁ x T₂ ® T₂.

defined by,

p^{^} ( x₁ , x₂ ) = X₁,

q^{^} ( x₁ , x₂ ) = X₂.

are continuous; they are projections of the product space into its subspaces.

A 3. In quantum space such mappings can be seen as quantum operators projecting the quantum space into one of its subspaces. We start with Fourier integral relationship,

< p' | X > = h^-½ ò e^-iq'p'/h d q' < q' | X >,

< q' | X > = h^-½ ò e^iq'p'/h d p' < p' | X >.

to derive,

p _m = -i h ¶ / ¶ q ^m ,

q ^m = i h ¶ / ¶ p _m .

and see that; displacements:

- ¶ / ¶ q ^m® p^{^} _m ,

+ ¶ / ¶ p_m ® q^{^ m}.

are the corresponding quantum operators; projections of the quantum space,

i h p^{^}_m ® p_m,

i h q^{^m} ® q^m.

VI. FUNCTIONAL INTEGRALS

We can develop the representation in terms of Dirac d-function: we take the set of basic vector forms and normalize,

< x' | x"> = d ( x' | x"),

the singular point x' = x" , should be seen as a condition that only the product of a basic vector form with its conjugate; x' = x", is permissible and finite ( we may call it conjugate or functional product ), only such product builds up a functional space, the d-function will look like an infinite diagonal matrix, represent such functional.

Now we can see that the requirement of the product being functional normalized the space and removed certain type of infinity.

T 4. The d-Space:

d-function is a functional ( d , | A > ) on the set of basic vector forms. It generates a functional space. Its representative is the antilinear bilinear form < A | A >. We may call it the d-space.

A 4. Any quantum operator should be expanded in d-space. We have the basic equation,

( d , ¦ ) = ò ¦ (x) d (x) d x = f (0).

for such an expansion, instead of the usual < X | F | X > of the quantum theory. It is obvious that such an expansion in functional space will lead to distributions.

In the systems of units h = d = 1 , h-space is not different from d-space. The states themselves transform to quantum operators and this is the physically significant space, the process of normalization carries automatically to the operators themselves. The functional constraint imply that only conjugate interactions are permissible; in the vacuum state, that means that no such thing as interaction between indeterminate number of particles.

VI. FEYNMAN FUNCTIONAL INTEGRAL

We have the basic equation,

( d , j ) = ò j (q) d (q) d q = j (0).

T 5. We can choose a convenient representation of d function in terms of the action integral itself,

< q' | q''> = d ( q' - q'') = e ^is/h.

A 5. This gives us,

ò j (q) e ^is/h d q = j < 0 >.

This allows us to expand any function of a dynamical variable in d space in terms of the action integral. ( Analogous; Fourier Integral Theorem .)

VII. EXTENDED SYMMETRY

Space time is subject to Poincare,

x^m ® x^m' + a^m_n + l xⁿ ,

Extension into the functional space imply other symmetries; the symmetries of the antilinear bilinear form ( p_m , q^m ) ; symmetry between p_m and q^m.

T 6. Topological Theorem:

The graph G , of a continuous transformation,

f : T₁ ® T₂.

is homomorphic with . The graph of the transformation is a subspace of the topological product T₁ x T₂. We have the form

( p_m , q^m ) , p_m's transform as Dirac Majorana spinors p_m = i g^m ¶_m ,

y' (x) = exp ( - i /4 s^mn e_mn ) y (x).

Transformations of the form ( p_m , q^m ) ,

p_m q^m ® p_m' q^m'.

A 6. Will mix four Minkowiski coordinates and four Dirac Majorana spinor translations, the resulting group will be homomorphic with Poincare group. People extensively explored such symmetry under the name "supersymmetry".

C showed us the connectedness of space and time. h showed us the connectedness of p_m and q^m in the quantum space; it shows us the continuum that lies behind discrete nature, reality itself is the quantum space, simple and beautiful.

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