It is predicted by this model that separability varies with density and time. It thus follows that there is indistinguishability of some basis vectors of the Lie Hilbert space. It’s
proposed here that separability gives rise to the uncertainty metric.
= 4π
E t
h ≤ Δ · Δ
Ⓗ {Q} Δ Δ
·
E Nh
Where {Q} represents a set of any number of . Since the necessary elements of
V
α
Heisenberg’s equation already are provided for by existing topological descriptions in this paper, it is intuitive to combine the existing ideas, as in the above equation.
Vectors have neighborhoods which are separable according to the overlap of each other’s neighborhoods in time. The separability of Pseudo-Boolean spaces has already
been discussed. However, the Lie Hilbert space’s has not been discussed. It will be proven that multiple separability axioms apply under different conditions of the Lie
Hilbert. In , neighborhood,
N, will be defined as the uncertain region surrounding a point. It is
dimensionally equivalent to
.
Where is a radius in any given
{r } ∏
n
dimensions
α r
α
dimension. That is the geometric description,
N = r dθr
∯
α
Which yields a region topologically equivalent to a sphere external . Because the
X
Boolean
radii extend only into from , there exists a surface on
; a surface which
X
Boolean
X
Boolean
is not a single point, but a curve, indicating that the combination of the surface on and
N is actually topologically equivalent to a torus. The importance of this will
X
Boolean
be discussed later. Quantum uncertainty is expressed as magnitude of indistinguishability of two
neighborhoods. Mathematically,
Ⓗ {Q} Δ Δ
·
E Nh
= ੬ ੬
| |
m l
| |
3. 2. How Particles Move in the Space
What we traditionally consider to be a particle in physics is described mathematically in this new space as “charges” being transferred from one
to another. Particles X
Boolean
move fluently through the space as chains of topological deformations: the speed of a particle is interpreted as the time needed for a deformation of a topology to occur. We
will represent the deformation of a map of one “charge” to another. Thus, the equation for speed of a particle is
l Δ t p
p d v = Δ
=
α
−
β E
Nh
fyb985
Since E will be conserved in a closed system this system is closed on the interval “the whole universe” and is constant, only will change, and will do so according to a
h N
Poisson operator, since anticommutativity is in effect due to the need to conserve a value for the Hamiltonian and since in , products must be Lie like .
3.3. The Lie HIlbert space whose basis is pseudo-Boolean has function-function operators whenever the two functions are contained in separate points which are
Poissonian.
Thus, particle interactions are represented with the Poisson bracket, { } , which is
, p p
α β
equal to the deformation of one to a new form,
X
Boolean
iff Y X : Y → X
: X → Y More generally,
Q} iff Ⓗ{Q}
{Q} {
= ੬
α {Q}
: ... ੬
Q} :
n {Q}
→ { ′ = Ⓗ
′
Which says that the set of all points includes points which are permitted to map with each other to form a permutation of {Q}, called {Q}’.
The “speed of particle” equation given earlier does not necessitate that a particle move only to from one point to an adjacent partner. This is true because: 1 “particles” do not
move in this model, rather pseudo-Boolean spaces are deformed; 2 particles need only move according to the “Speed of Particle” equation based on the Hamiltonian. Thus,
particles may cross large distances based on conservation of global energy, allowing for local violations of the Hamiltonian. More colloquially, “spooky action at a distance” is
clearly permitted by the theory.
This fact at first seems to suggest that {Q}, which is all throughout , is completely
unbound to remain in any particular order. However, this is not the case for three reasons. Firstly, the motion of particles does not constitute the motion of points in a
space in respect to each other, but rather a deformation of the point which, remember, is itself a Pseudo-Boolean space. Secondly, each point’s position is certain to itself and
relative to other points, since its neighborhood is shaped locally like a 4-torus remember from Section 3.1. Thirdly, the definition of dictates that the points maintain the same
“ordering” relative to each other. The points are permitted to experience change overlap
of neighborhood and indistinguishability, but they maintain their order.
3.4. Explanation of Feynman Interactions Under the New System
