_{1}

The paper utilizes some fundamental results obtained in the context of topological quantum field theory, hyper finite sub-factors, Turaev-Viro machine and four dimensional fusion algebra to shed mathematical, physical and philosophical light on the major problem of cold fusion reactors. In particular, we develop a picture model for the quantum vacuum by using modern transfinite quantum field theory but also guided by philosophical ideas about the picture of the space of logic and reality.

The present work is mainly a continuation of the author’s philosophical thesis that all consistent and correct results obtained in pure mathematics represent correct and consistent facts in the physical world (

This paper is an attempt at a pure mathematical foundation of cold fusion using four dimensional fusion algebra

[

1) Logical pictures can depict the world.

2) A picture is a model of reality.

3) A picture presents a situation in logical space, the existence and non-existence of states of affairs.

The present paper does in fact present two complimentary pictures in

If we follow an admittedly relatively long road from sub-factors to a topological quantum field theory, then our trip will be rewarded by an encounter of the first, second and third type, namely a four dimensional fusion algebra [

The notion of dimensional function was repeatedly encountered in connection with von Neumann-Connes’ work on continuous geometry and more so in noncommutative geometry leading to the bi-dimensions [

where

sion function, which can play a similar important role is that encountered in topological quantum field theory as seen from the view point of the theory of sub-factors. This subject is known as the four dimensional fusion algebra. The dimension function in this case is given by [

It is interesting to see that the union of four quasi “Eigenvalues” of the “Eigen function” leads directly to the five dimensional fractal Kaluza-Klein spacetime or equivalently a fractal de Sitter space which is not directly

obtainable from

The Casimir or ordinary energy density is consequently given by [

exactly as reasoned in previous work using different methods. At this point we should maybe recall that the reason for not being able to measure that dark energy is that it is related to the quantum wave as modelled by the

empty set

is what we perceive as wave collapse. For the sake of completeness we should mention the intimate relation between the adjoint matrix of the Coxeter diagram A_{4} and 4D-fusion algebra [

This section could just have well been entitled “many names for essentially the same energy density”. To see why let us start with the topological Casimir force acting upon the two Casimir uncharged but perfectly conducting

plates. Since the plates are so close that the nano gap may be regarded as almost an empty set

time including spin ½ dimensionality and fractal gaps in it. This means

Now we are used to meeting surprizing results in this field which upon reflection steadily turn out to be not surprizing and this result, which is identical to the density of ordinary energy [

belongs to this category of results which we could call ‘wonder and yet no wonder’. The point is that

i.e.

the energy concentrated at the edge of the universe as per Dvoretzky’s theorem [

The fundamental importance of Hardy’s generic quantum entanglement _{2}, namely [

where n is the number of entangled quantum particles. Setting n = 2 we find the well known Hardy quantum entanglement probability

which gives our E-infinity quantum spacetime core Hausdorff dimension

does not mean P = 0 but gives us an intrinsic topological energy which we recognize as the Casimir topological energy discussed earlier on. It is only when we assume that n is practically infinite that

In view of the Vienna circle [

The present paper is a testament to the author’s deep belief that when it comes to deep questions pertaining to quantum physics, then what is pure mathematics and pure scientific philosophy is basically indistinguishable from pure real physics and its pragmatic implications. The paper argues that the implications of von Neumann- Connes’ dimensional function, as well as the dimensional function of 4 dimensional fusion algebra that makes cold fusion a real pragmatic physical possibility [

In conclusion of our conclusion we wish to salute the late Nobel Laureate J. Schwinger (see

Mohamed S. ElNaschie, (2015) From Fusion Algebra to Cold Fusion or from Pure Reason to Pragmatism. Open Journal of Philosophy,05,319-326. doi: 10.4236/ojpp.2015.56040