Quantum Properties as Emergent from the Structure of Space
0.0. Description
Research Question: Can it be shown that a vector space whose basis is
Pseudo-Boolean could give rise to the four fundamental forces as well as length and time?
Background
String theory is a physical theory which postulates that the fundamental forces can be generated by building matter out of infinitesimal vibrating “strings.” Recently, physicist Erik Verlinde proved that the force of gravity may emerge from the other fundamental properties of matter such that dark matter is no longer theoretically required to accurately describe experimentally observed gravitational effects.
This paper goes one step further to propose that all quantum properties are able to emerge from the structure of space.
This is accomplished by building macroscopic space out of “pseudo-Boolean
subspaces.” This is a one dimensional, n-polar space whose poles only include numbers from 0 to 1, charged respectively to their pole. While a pseudo-Boolean space can have any number of poles, this paper will discuss the case of six pairs of poles. Such a case is a sub-type of the foundational space of M-theory (the leading breach of string theory) and it is shown how the quantum properties can emerge from a single topology.
Some proofs are used here to show the classification of the space proposed here. Most of the proofs are very straightforward and are often trivial. However, sometimes a step is taken based on a somewhat obscure theorem, and the necessary information for these steps are referenced in the citations section.
0.1. Introduction: A Summary
Informally, this paper posits that the macroscopic structure of the space can be
extended over a time interval into vectors by reiterating the random selection of a number 0 ≤ℕ≤ 1. The pseudo-Boolean space is notated as XBoolean and each individual of possibly infinitely many pseudo-Boolean spaces may interact in D4 to D9 with each other. In this way, point values on a Calabi-Yau pseudo-Boolean manifold in a vector space are able to represent point particles externally and strings internally, providing elegant topological and algebraic solutions to problems involving the quantum harmonic oscillator.
The Hilbert space has varied density and separability of its constituent points: this also leads to variable volume of the Hilbert space while maintaining constant energy. It is posited that all the points which make up the Hilbert space (which are themselves the pseudo-Boolean spaces) could occupy the volume in H of a single point and that volume observed in H exists as a result of varied separability of points’ neighborhoods from each other.
1.0. The Structure of Pseudo-Boolean Spaces
1.1. Definitions of Smooth, Countable, and Separable n-Polar Pseudo-Boolean Space
Let
∃
X
Boolean such thatX
Boolean
=
C
(
੬
)
ℝ]01·੬P
1
The Boolean space is defined as a single real number chosen from between 0 and 1 in any of the poles of the space. This definition makes it easy to see how the space
would appear to an observer immediately outside it, but it does not make clear
X
Booleanthe topological mechanisms by which the space chooses values. For the latter, we will use a more comprehensive definition,
(
)
X
Boolean
≡
C
{ ·(1·[
C
(
)]
∧
[
C
(
)] ·[Σ ])}
2
n
੬
P
1
੬
P
1 −1
{
1
ℝ
)
1
From this formula, some special cases occur which will be examined herein. Namely,
, the pseudo-Boolean space behaves like an n/2-dimensional dipolar space
∀
n2∈ ℤ
relative to itself, where n is poles, not dimensions. Where the number of poles in XBoolean
is an even number, the space can be treated internally as having the number of
dipolar, thus any pair of poles forms a dimension and even cases of the number of poles in the Pseudo-Boolean space satisfy npoles/2 dimensions trivially.
, where represents additional iterations of the “choice function,”
∫
X
∂
C
∀
Boolean ∂Cthe pseudo-Boolean space behaves like a 1-dimensional vector relative to a space immediately external itself. This pseudo-Boolean vector extension is used as the vector basis of the Hilbert Lie Space.
, the pseudo-Boolean space behaves like a 3-brane in a space
∭
X
∂
C
∀
Booleanimmediately external itself.
1.2. Demonstration that the n=12 Case of a Pseudo-Boolean Space is a Subset of the Class, D-6 Calabi-Yau Spaces
This special case of a pseudo-Boolean space will now be demonstrated to be
Calabi-Yau, since such a proof makes this theory coherent with existing M-Theory. The proof will be accomplished by showing that the pseudo-Boolean space in question satisfies all requirements to meet the definition of a Calabi-Yau manifold.
In order for a space to be Calabi-Yau, it must be a compactified Kahler manifold whose canonical vector bundle is trivial. For a space to be a Kahler manifold, it must be complex, Riemannian and smooth, and symplectic.
1.3. The Canonical Bundle of X Must be Trivial
The Canonical Vector will be defined,
V
ℂ= Ω
nWhere is the cotangent bundle and n is the number of dimensions. The cotangentΩ bundle may be described in terms of the tangent bundle, provided that
X
Boolean⊂
T
2Where T2 represents the class of Hausdorff-quality separable spaces. When the above equation is true, the tangent and cotangent bundles are mutually isomorphic. It will therefore be beneficial to prove that the pseudo-Boolean space in question is Hausdorff. Prove: ∃neighborhoods of all points. Mathematically, show that for any two elements of the basis of
X
Boolean,
੬m
੬
l=
∅
.
(
)
X
Boolean
≡
C
{ ·(1·[
C
(
)]
∧
[
C
(
)] ·[Σ ])}
2n
੬P1 ੬P1 −1 {1ℝ)
1
Has a differential form which is a Lie Algebra because it is a vector space in which
and . Another demonstration that
V, ] [ℝ , ]
[ ℝ + V = 0 [ ·f V, =ℝ] f· [ V, ℝ ] + · [ V f, ℝ ]
the form is Lie is that in the case of the two products, [ V, ℝ ] and [ℝ , V] , the former creates the vector basis multiplied across the set of real numbers, which is
representative of the curve of XBoolean, whereas the latter represents the set of real numbers multiplied by the vector basis, which represents a ring of vectors not related to the curve of XBoolean. Given that the curve of the Pseudo-Boolean is Lie, it is known that the Pseudo-Boolean is Lie. XBoolean may thus be represented as,
[
V
,
੬
] ≡ Π [ [ ) ,
੬
]
X
Boolean=
P ℝ1 P{
V
∈ ℝ
}
∧
{
੬
m੬
l=
੬m
੬
l}
∧
{
੬
m= },
/
੬
l.
੬
੬m
l=
∅
.
V
⊂
{
T
2}
∧
X
Boolean⊂
{
T
2}
The tangent bundle, , in respect to the basis of the Pseudo-Boolean spacＴ
V
Ｔ
=
∂
ℝ
1
∂
X
Boolean=
In the case of n=1, the cotangent bundle is equal to the canonical vector per the definition of the canonical vector. The cotangent vector is also the dual of
π
.
Ｔ
*:
E
*→
X
Boolean.
Ω
= (
Ｔ
X
Boolean)
: (
E
T
)
→
X
Boolean,
is trivial.E
=
Ｔ
:
X
BooleanΩ
Because of the above three logical demonstrations, we know that the Pseudo-Boolean space is complex, and the product relation with the tangent field satisfies the definition of a smooth Riemannian manifold.
In respect to a Hilbert space, , into which XBoolean is immerse, the tangent bundle will not be trivial. Instead, it will represent the curvature of the knot of the Pseudo-Boolean space’s poles, which are the vector basis of XBooleanfrom within the reference frame of its own space.
is a one-dimensional space built of 12 poles with 6 distinct charges. Thus, poles
X
Booleanoccur in pairs. Since one-dimensionality must be maintained for the theory to work, pairs of poles form “loops,” going from 0 to some charge of 1 to 0 again. This means that
can be thought of as knotted. Charges are allowed to change what pole they
X
Booleanare attached to by going from a 1 to a 0 to a new 1.
2.0. The Structure of a Lie-Hilbert Space with a Basis of
X
Boolean2.1. A Lie Hilbert Space in which a Pseudo-Boolean Space Acts as a Base Vector
Until now, the Pseudo-Boolean space has been discussed almost exclusively. The Pseudo-Boolean space is a way of describing a single-particle space, or Fock space. However, this theory posits that Fock spaces act as points when observed
instantaneously and become vectors when integrated in respect to time. It’s predicted that these vector spaces act as a basis for a macroscopic Lie Hilbert space, .
V
V
..
V
=
a
1+
a
2+ . +
a
nWhich can also be denoted
{
V
})
=
a
· (∑
nα α
And since Vα is the basis vector of , which in this theory has been set as the Pseudo-Boolean Fock space, we can substitute. Given that
X
F ock∈
{
X
Boolean}
,
{
X
})
=
a
· (∑
nα F ockα
Using
Ⓗ
as the Hamiltonian operator and given that
is LieⒽ
= [
(
Ⓗ
: {
X
F ock}), ]
a
This space has highly unusual topological properties of variable density and separability. Separability varies based on neighborhood overlap, but since the neighborhood of points is actually the neighborhood of basis vectors in which varies relative to time,
separability is variable. Density is also variable, and both properties will be discussed below.
2.2. Hamiltonian Operators in the Lie Space
Ⓗ
=
Ⓣ
+
Ⓥ
Where Ⓣ is the kinetic energy operator and Ⓥ is the potential energy operator. The potential energy operator for a system of n-numbered non-interacting particles is traditionally notated as
Ⓥ
(
r
,
t
)
= ∑
n i=1V
iBut it can be notated more conveniently for a Lie space as
[ {
X
} ,
n
]
Ⓥ
= Π
BooleSo that the series Lie Bracket represents the same n non-interacting particles. The Hamiltonian is used to represent the necessary energy to perform a topological deformation of a pseudo-Boolean space within the Lie Hilbert into a topologically equivalent pseudo-Boolean space.
Since deformations can be performed in two directions, this theory provides a solution to the creation of subatomic particles without symmetry breaking.
2.3. The Time Dimension as Being of Variable Density
Assuming dependency of time on shape of space and constancy of speed of light, we can maintain all Relativistic statements about the relation of c, E, and the Planck constant, h, particularly,
. From dimensional analysis,
,
and . The rate of cosmic expansion,c
=
Eh=
t
lc
t
=
lhEwhere is the magnitude of the neighborhood of an N
X
Boole
in , in respect to time is then,
)
dt
dN
=
dN
d
( )
NhE=
h
· (
∂
∂
Ⓣ
N
∂
∂
Ⓥ
N
From this, the rate at which time progresses in a region of space is seen as dependent on the qualities of the neighborhoods within that field, and that the rate of change of those neighborhoods is time-dependent. This formula allows for a universe with time as an emergent phenomena.
3.0. Behaviors of Particles in a Lie Hilbert Space
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
Δ
(
Ⓗ
{
Q
}) Δ
·
NhEWhere {Q} represents a set of any number of
V
α. Since the necessary elements of 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
∏
{
r
}
.
Where is a radius in any givenndimensions
α rα
dimension. That is the geometric description,
N
=
∯
r
αd
(θ/
r
)
Which yields a region topologically equivalent to a sphere external XBoolean. Because the radii extend only into from XBoolean, there exists a surface on XBoolean; a surface which 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
XBoolean
be discussed later.
Quantum uncertainty is expressed as magnitude of indistinguishability of two neighborhoods. Mathematically,
(
Ⓗ
{
Q
}) Δ
Δ
·
NhE=
|
|
੬
m੬
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 XBooleanto another. Particles 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
( ) )
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 ah 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, {pα , pβ} , which is equal to the deformation of one XBooleanto a new form,
iff Y
X :Y →X :X →Y
More generally,
Q
}
iff
Ⓗ
{
Q
}
{
Q
}
{
=
੬
α{Q}: ...
:
੬
n{Q}→ {
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
Particles interact in this theory when one XBoolean is deformed by energy from another. Below are tables of Leptons and Quarks made
Quantum numbers like Baryon and Lepton number which are conserved in Feynman interactions in traditional quantum mechanics are still conserved in this model: they are just conserved through application of the Hamiltonian.
Quantum numbers are represented as Boolean value of an individual pseudo-Boolean space in H.
Charge is described:
c
=
.
( charges ) ∑
( numberofchargedpoles )
This gives all quarks charges of 1e if observed alone, contrary to convention which gives quarks charges of either + 32e or − e. However, quarks are not observed alone, but in
3 1
particles. When quarks combine as shown in the below diagrams (Quark Structure of a Proton and Quark Structure of a Neutron), it is clear that this new model is able to produce identical results to conventional nuclear physics.
Spin is represented by the simple formula,
s
=
.
( charges ) ∑
( numberofchargedpoles )2
There is no inconsistency between spin predictions made by this model and spins observed experimentally or predicted by other contemporary models.
4.0. Conclusion and Evaluation
It has been demonstrated through construction that it is possible to describe all four fundamental forces, as well as length and time, as quantities which emerge from a vector Hilbert space (which is Lie and permits Poisson operators) with Pseudo-Boolean spaces as a vector basis.
This paper is limited by its experimental testability; an experiment to test the structure of space proposed here is as unviable today as is any modern string experiment. A second limit is that the model predicts that outside of the macroscopic universe, the macroscopic universe appears to be a single point. Such a curvature is not predicted by any
strings. Though this theory does not itself mandate high-entropy strings, such entropy is necessary in order to unify all four fields as Verlinde predicted. Otherwise, only the typical three are unified in this paper, though it may be possible for further research to show the possibility of an emergent gravity from this unique spatial structure.
5.0. Citations
Seifart, H and Threlfall, W. A Textbook of Topology and Topology of 3-Dimensional Fibered Spaces. Academic Press, 1980, New York.
Tajtelbaum-Tarski, Alfred. Sur Quelques Théorèms Qui Équivalent à l’Axiome du Choix.
ICM Biblioteka Wirtualna Matematyki, 1931, University of Warsaw.
Barnum, Kevin. The Axiom of Choice and its Implications. University of Chicago, May 2011, Chicago.
New Mexico Tech Mathematics Department. 6.4 Trivial Bundles, from Chapter 6. Fibre Bundles of Topics in Differential Geometry. New Mexico Institute of Mining and
Technology, November 24, 2014, New Mexico.
