Knotted String Foams and Mathematical Physics
Keith Murray
08 September 2017
ABSTRACT
We attempt to show that there is sufficient mathematical evidence to consider knots as “foundational” particles in mathematical physics, by presenting relationships to the mathematics of General Relativity and Yang-Mills.
1. Introduction
Shortly after the onset of the millennium, I was wrestling with a direct quantization of the Einstein and stress-energy tensors which would produce square-integrable wave functions. While I was able to find some results, the interpretation of the wave functions were ambiguous and in limiting cases the equations resembled the Klein-Gordon equations. This caused me to think that I had rediscovered nothing more than an alternative expression for a freely moving photon traveling in curved spacetime. Furthermore, were I to embark on the arduous journey of generalizing these expressions, I may arrive merely at an analog of the Dirac equation on curved spacetime.
Rather than pursuing that path further, I decided it was probably a better idea to investigate what generalizations could be hypothesized for modern objects in mathematical physics. During my survey of objects in mathematical physics, I frequently encountered a generic and recurring theme for which I invoked symbolic logic out of necessity. This was simply a symbolic string:
퐓 = ∅ + 퐇 + 퐀 (1)
Here, T can be a “theory” in the logical or model theoretical sense. The ∅ empty set symbol may literally represent the empty set, the concept of zero, an arbitrarily small epsilon in mathematical analysis, or any other null character which the theory may require. H may be a vector space over a field, module over a ring, or any tensored product thereof; by logical extension, H could be any object we may find in any category. And A is a connection (gauge) or some other object required to satisfy the axioms of the theory with respect to H. Finally, “+” is the familiar plus sign, or Abelian operator; but more generally, it may represent any commutative binary operation, e.g., triples (+, H, A) = (+, A, H).
While this was all fine, in terms of mathematical physics my theoretical boat was slipping further from the safety of the coastline and into the white-capped seas of mathematical logic. To escape this, I then decided to focus on constructing a general framework for elementary particles that would be in alignment with Eqn. 1. The solution my mind produced: a school of fish could be an excellent toy model for elementary particles.
A school of fish is driven by swarm behavior, so immediately we have some nice concepts summarized in a few words: discrete, stochastic, optimization, Markov chains, hitting times. Perhaps spacetime or simply space if it is discrete – and the Loop Quantum Gravity folks will remind us that there is no a priori reason to believe space is discrete, but what the heck – is distinguishable to a test particle because of the way swarms interact. For example, a flock of birds meets a swarm of insects, or
a shoal of fish meets a swarm of bottle-nosed dolphins. These different types of “particles” will interact.
However, then I thought that although swarms (like a bait ball of fish) can have a very recognizable global geometry (a sphere), in the end swarm equations are only behavioral functions. Swarm models of behavior follow very precise codes, and can be simulated using computer programming languages. So while these emulate the sort of behavior one may like to think about for a quantum geometric model of particles, straight away we know that the Lie groups are not required to model swarm behavior. But Lie groups are required to model the interactions of elementary particles.
How to reconcile these thoughts? We do not know what Standard Model point particles are, but the math works in the laboratory. We do not know what strings are, but there is some ancestral relationship with point particle laboratory physics. We do not know what spin foams are, but they can be used to recover General Relativity. We do not know what particles are, really, but we know that particles cannot be modelled by swarm behavior alone. Yet swarms are nice because they have dynamics, they are discrete, and the global geometry is spherical. If only there was a way to connect swarms with Lie algebras, like an embedding! But that yet appears insufficient, because although we can embed swarms in tensored vector spaces or in GL(n)-valued spaces, we still have only sort-of spherical geometry with a swarm, and spherical geometry is only useful for a limited number of physical scenarios – both classical and quantum. What we really need is an idea that includes more general geometric structure, and it would be convenient if there were some pre-established mathematical concepts relating more general geometric structures to physics, in order to avoid reinventing lots of math. So, very close to the swarm idea is the knot!
Knots themselves may be a great way to think about particles, because they have holonomy, inherent relationships to algebra (Jones polynomial, for example), and if they are framed – recall framed knots are more like tubes or rubber bands instead of 1-dimensional strings – then knots could still have the “school of fish” swimming inside the tubes or along frames with a certain periodicity as these “fish” swim around the race track. Also, since knots have holonomy there is a notion of connection, referring back to our simple statement in Eqn. 1, of a logical mathematical physics “theory”. Admittedly, since we do not know much about treating knots themselves as elementary particles, let us rather say we shall treat knots as “foundational” particles, whatever that means, and proceed to lay some groundwork.
2. Knots as Foundational Particles
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3. Categorical Physics
To keep our discussion in order, let us review some basic properties of analysis and establish some of the techniques we will be using.
[For example, the fundamental group is a functor pi1 : Top -> Grp:, A History of n-Categorical Physics, DRAFT VERSION!, John C. Baez]
Example: the fundamental group, with F, G, H other functors.
Categorical String Theory [see Categorical Aspects of Topological QFT, Bruce Bartlett: 3.4.2 TopString, 4.1.3 Triangulated spaces, 4.1.2 The connection as a functor]
Relations to Topos, Sheaves, Sites [see Baez, Gauge Fields, p 333 – 334]
Relationships to Model Theory and Categorical Logic
From Model Theory: Finitary Models can be [First-Order, Categorical Theories]. First order theories cannot be categorical, i.e., they cannot be described up to isomorphism, unless that model is finite. Reworded: given a finite model, a first order theory may be categorical.
Two famous model theoretic theorems deal with the weaker notion of K-categoricity for a cardinal K.
A theory T is called K-categorical if any two models of T that are of cardinality K are isomorphic. For a complete first-order theory T in a finite or countable signature the following conditions are equivalent (Ryll-Nardzewski Theorem):
- T is Aleph0-categorical
- For every natural number n, the Stone Space Sn(T) is finite - For every natural number n, the number of formulas f(x1, x2, x3, … , xn) in n free variables, up to equivalence modulo T, is finite.
ℎ(푥) (5)
4. TOE Unification Model
Let’s start with a note from Wikipedia’s GUT article:
Let’s compare [Quantum Theory from Quantum Gravity, Fotini Markopoulou∗ and Lee Smolin†] the
abstract from Lee Smolin to Mark Hadley.
Winding numbers, Jones Polynomial, Witten, and Standard Model
Spacetime Algebras and Lie Algebras as distinct but related entities. SU(3)SU(2)U(1) can be
found. Spacetime algebra:
[CHERN-SIMONS GAUGE THEORY AS A STRING THEORY, Edward Witten, ⋆School of Natural Sciences, Institute for Advanced Study, Olden Lane, Princeton, NJ 08540]
[Discrete Approaches to Quantum Gravity in Four Dimensions, Renate Loll, Max-Planck-Institut fur Gravitationsphysik, 3 Quantum Regge Calculus, and 4. Dynamical triangulations] [4 Dynamical triangulations ...... 23 4.1 Introduction ...... 23 4.2 Path integral for dynamical triangulations . . 23
4.3 Existence of an exponential bound? ...... 24 4.4 Performing the state sum ...... 24 4.5 The phase structure ...... 25 4.6 Evidence for a second-order transition? . . . . 26 4.7 Influence of the measure ...... 26 4.8 Higher-derivative terms ...... 27 4.9 Coupling to matter fields ...... 27 4.10 Non-spherical lattices ...... 28 4.11 Singular congurations ...... 28 4.12 Renormalization group ...... 28 4.13 Exploring geometric properties ...... 28 4.14 Two-point functions ...... 29 4.15 Summary ...... 30]
[QUASIALGEBRA STRUCTURE OF THE OCTONIONS, Helena Albuquerque, Shahn Majid, December, 1997 – February, 1998; Abstract We show that the octonions are a twisting of the group algebra of Z2 × Z2 × Z2 in the quasitensor category of representations of a quasi-Hopf algebra associated to a group 3-cocycle.]
Octonion X-product and Octonion E8 Lattices, Geoffrey Dixon
Baez, Octonions and relationship to E8.
[A Categorical Equivalence between Generalized Holonomy Maps on a Connected Manifold and Principal Connections on Bundles over that Manifold Sarita Rosenstock1, a) and James Owen Weatherall1, b)]
Spacetime Algebras and Lie Algebras as distinct but related entities. SU(3)SU(2)U(1) can be found. Brief Review: Spacetime algebra:
Clifford Algebra and Discrete Manifolds, and Clifford Algebra Approach to Curvature Extending Finite Element Exterior Calculus to Finite Element Clifford Algebra. The objective of including this in this survey is to think of using discrete forms as manifolds, so the two may be used interchangeably.
From http://math.arizona.edu/~agillette/research/pd11talk.pdf.
Some Clifford/Spacetime calculations arriving at curvature over continuous manifolds are given in:
In: International Journal of Theoretical Physics Vol. 25, No. 6 (1986), 581–588. Curvature Calculations with Spacetime Algebra David Hestenes. http://geocalc.clas.asu.edu/pdf/Curv_cal.pdf.
In particular, where ga are frames defined by ∂ax, and x = x(x0, x1, x2, x3), and the metric tensor is the inner product ga•ga we have the following:
From http://geocalc.clas.asu.edu/pdf/Curv_cal.pdf
Octonionic Clifford Algebras Much like Lie-Algebra valued forms, a la exterior calculus, Clifford Algebras may also be expressed in terms of Octonions. For example, the manuscript “Octonionic representations of Clifford algebras and triality”, by Schray and Manogue, written in 1994, http://arxiv.org/pdf/hep-th/9407179v1.pdf. The abstract follows:
“The theory of representations of Clifford algebras is extended to employ the division algebra of the octonions or Cayley numbers. In particular, questions that arise from the non-associativity and non- commutativity of this division algebra are answered. Octonionic representations for Clifford algebras lead to a notion of octonionic spinors and are used to give octonionic representations of the respective orthogonal groups. Finally, the triality automorphisms are shown to exhibit a manifest Σ3 × SO(8) structure in this framework.”
Discrete Forms versus Charts on an Atlas Trying to map a set of Discrete forms to Charts on an Atlas presents some issues. For one, a single discrete form being a simplicial complex is usually represented with a Euclidean Metric.
(See “Discrete Differential Forms for Computational Modeling”, Mathieu Desbrun Eva Kanso∗ Yiying Tong† Applied Geometry Lab, Caltech)
Another issue is the Dissection Problem, and defining methods of handling curvature on discrete membranes when the metric is not necessarily Euclidean. Perhaps, then we look at Finite Element Clifford Algebras, as mentioned above, and try to deduce some notion of Gaussian Curvature by comparing the curvature expressions in Clifford Algebras on continuous manifolds with the Finite Element Clifford Algebra approach.
Charts on Atlases as Subsets of Topologies Required for Presheaves Since there is a forgetful function F: MAN -> TOP, then the Charts on MAN may suffice as the subsets of a topological space Ua ⊆ X, where X is the space, and the Ua may be used as the subsets for Presheaves.
gent Space on Discrete Forms in Curvilinear coordinates. Consider two discrete (bounded) manifolds, M and N, with cobordism W. If M and N represent 2- dimensional surfaces, multiples of some fundamental area (such as in loop quantum gravity), then consider analyzing tangent spaces at the ((qi)1, (qj)2) curvilinear coordinates on M in a 3-dimensional embedding space. In this case, we consider a small loop around (qi, qj), and using the holonomy of the loop, project onto the 2-dimensional tangent space embedded in three dimensions. The projection onto the tangent space TM appears as an “open” parallelogram with the Lie bracket/commutator describing the failure of commutativity. We can find information regarding the commutator and Clifford algebras in pseudo-Riemannian spaces being related to the curvature tensor in the above reference: International Journal of Theoretical Physics Vol. 25, No. 6 (1986), 581–588. Curvature Calculations with Spacetime Algebra David Hestenes. http://geocalc.clas.asu.edu/pdf/Curv_cal.pdf.
Furthermore, if (qi, qj) ∈ 핆 × 핆, then we need to consult our references on “Octonionic representations of Clifford algebras and triality”, by Schray and Manogue, written in 1994, http://arxiv.org/pdf/hep- th/9407179v1.pdf.
Also, since we are speaking of 2-dimensional surfaces in this example, we may also consider the use of Kahler Manifolds in terms of Quaternionic or Octonionic extensions, if they exist. (To date the author has not seen any evidence of these extensions.) Below is some information on Kahler manifolds from Wikipedia, with reference to their symplectic nature, indicating a relationship to the Weyl Algebra, also known as the Symplectic Clifford Algebra; this is because Weyl Algebras represent the same thing for symplectic bilinear forms that Clifford Algebras represent for non-degenerate symmetric bilinear forms.
Kahler Manifolds (Wikipedia): Symplectic viewpoint
A Kähler manifold is a symplectic manifold equipped with an integrable almost-complex structure which is compatible with the symplectic form.[1] Complex viewpoint
A Kähler manifold is a Hermitian manifold whose associated Hermitian form is closed. The closed Hermitian form is called the Kähler metric. Equivalence of definitions
Every Hermitian manifold is a complex manifold which comes naturally equipped with a Hermitian form and an integrable, almost complex structure . Assuming that is closed, there is a canonical symplectic form defined as which is compatible with , hence satisfying the first definition. On the other hand, any symplectic form compatible with an almost complex structure must be a complex differential form of type , written in a coordinate chart as
for . The added assertions that be real-valued, closed, and non-degenerate guarantee that the define Hermitian forms at each point in .[1] ------
We’ll also note that Kahler Manifolds have a structure such that there may be non-vanishing Ricci Curvature (from the Wikipedia article on Ricci Curvature):
Kähler manifolds
On a Kähler manifold X, the Ricci curvature determines the curvature form of the canonical line bundle (Moroianu 2007, Chapter 12). The canonical line bundle is the top exterior power of the bundle of holomorphic Kähler differentials:
The Levi-Civita connection corresponding to the metric on X gives rise to a connection on κ. The curvature of this connection is the two form defined by
where J is the complex structure map on the tangent bundle determined by the structure of the Kähler manifold. The Ricci form is aclosed two-form. Its cohomology class is, up to a real constant factor, the first Chern class of the canonical bundle, and is therefore a topological invariant of X (for X compact) in the sense that it depends only on the topology of X and the homotopy class of the complex structure. Conversely, the Ricci form determines the Ricci tensor by
In local holomorphic coordinates zα, the Ricci form is given by
where is the Dolbeault operator and
If the Ricci tensor vanishes, then the canonical bundle is flat, so the structure group can be locally reduced to a subgroup of the special linear group SL(n, C). However, Kähler manifolds already possess holonomy in U(n), and so the (restricted) holonomy of a Ricci-flat Kähler manifold is contained in SU(n). Conversely, if the (restricted) holonomy of a 2n-dimensional Riemannian manifold is contained in SU(n), then the manifold is a Ricci-flat Kähler manifold (Kobayashi & Nomizu 1996, IX, §4). ----
Ricci Curvature also appears in the case of 2-dimensional surfaces in a special form; again more information from Wikipedia:
Ricci Curvature wiki article. The Ricci curvature is determined by the sectional curvatures of a Riemannian manifold, but generally contains less information. Indeed, if is a vector of unit length on a Riemannian n-manifold, then Ric(ξ, ξ) is precisely (n − 1) times the average value of the sectional curvature, taken over all the 2- planes containing . There is an (n−2)-dimensional family of such 2-planes, and so only in dimensions 2 and 3 does the Ricci tensor determine the full curvature tensor. A notable exception is when the manifold is given a priori as a hypersurface of Euclidean space. The second fundamental form, which determines the full curvature via the Gauss–Codazzi equation, is itself determined by the Ricci tensor and the principal directions of the hypersurface are also the eigendirections of the Ricci tensor. The tensor was introduced by Ricci for this reason.
Riemann Curvature Tensor wiki article Surfaces For a two-dimensional surface, the Bianchi identities imply that the Riemann tensor can be expressed as
where is the metric tensor and is a function called the Gaussian curvature and a, b, c and d take values either 1 or 2. The Riemann tensor has only one functionally independent component. The Gaussian curvature coincides with the sectional curvature of the surface. It is also exactly half the scalar curvature of the 2-manifold, while the Ricci curvature tensor of the surface is simply given by
Note the similarities between Riemann and Ricci Curvature tensors on surfaces and the Kahler Manifold description of curvature. Note a Laplacian may also be constructed to analyse Gaussian curvature in isothermal coordinates.
Gaussian curvature
In the isothermal coordinates (u, v), the Gaussian curvature takes the simpler form
where .
And the Liouville Equation Article:
In differential geometry, Liouville's equation, named after Joseph Liouville, is the nonlinear partial differential equation satisfied by the conformal factor of a metric on a surface of constant Gaussian curvature K:
where is the flat Laplace operator.
Liouville's equation appears in the study of isothermal coordinates in differential geometry: the independent variables x,y are the coordinates, while f can be described as the conformal factor with respect to the flat metric. Occasionally it is the square that is referred to as the conformal factor, instead of itself.
Note the similarities between the isothermal coordinate equation (conformal) and the solutions to the Schwarzschild line element for Einstein’s vacuum field equations.
In differential geometry, isothermal coordinates on a Riemannian manifold are local coordinates where the metric is conformal to the Euclidean metric. This means that in isothermal coordinates, the Riemannian metric locally has the form.
where is a smooth function.
Schwarzschild line element / Einstein vacuum field solutions with a conformal solution.
“Curvature Form” Wiki article. Definition
Let G be a Lie group with Lie algebra , and P → B be a principal G-bundle. Let ω be an Ehresmann connection on P (which is a -valued one-form on P). Then the curvature form is the -valued 2-form on P defined by
Here stands for exterior derivative, is defined in the article "Lie algebra-valued form" and D denotes the exterior covariant derivative. In other terms,
where X, Y are tangent vectors to P. There is also another expression for Ω:
where hZ means the horizontal component of Z and on the right we identified a vertical vector field and a Lie algebra element generating it (fundamental vector field).[1]
This last observation now leads us in a circle (no pun intended) back to our picture, above, where we were discussing manifolds and cobordisms:
This picture is hinting that we can find a curvature form if we have an Ehresmann connection, and the notion of forms dual to our tangent vectors. Well, we may be able to accomplish all of this, tying together all the ideas from the above, by finding the mathematical equivalences in the following two drawings, of principal curvatures and wedge products, respectively:
Note also from the Gaussian Curvature article the following definition of K:
It is also given by
where is the covariant derivative and g is the metric tensor. At a point p on a regular surface in R3, the Gaussian curvature is also given by
where S is the shape operator.
And compare this to the matrix coefficients expression that follows. Can matrix coefficients be used to recover the commutator of the covariant derivative? If so, then a matrix element in a group G would be equivalent (for some carefully selected group) to the Gaussian curvature * det(g). Perhaps somehow decoupling these quantities would give separable matrices, one the geometric metric tensor and another a matrix algebra large enough to be useful for the Standard Model, such as ℂ⨂ℍ⨂핆:
A matrix coefficient (or matrix element) of a linear representation ρ of a group G on a vector space V is a function fv,η on the group, of the type
where v is a vector in V, η is a continuous linear functional on V, and g is an element of G. This function takes scalar values on G. If V is aHilbert space, then by the Riesz representation theorem, all matrix coefficients have the form
for some vectors v and w in V. For V of finite dimension, and v and w taken from a standard basis, this is actually the function given by the matrix entry in a fixed place. Note, if the Shape Operator expression may be used in some generalized form, and by the property of determinants, and setting v=e1, w=e2, we compare the Gaussian Curvature to the matrix coefficients like this:
det(g) = det(S(p))det(g) = det(S(p)g) = (Sg)11(Sg)22- (Sg)12(Sg)21 = <(D1D2 – D2D1)e1, e2> ?= <(g)e1, e2>
Grassmannian. It can be categorified, so we may be able to find a functorial relationship to Topoi, and also through the Plucker Embedding it is related to both exterior calculus and projective spaces so we can consult Octonionic Clifford Algebras.
The Grassmannian as a Scheme
In the realm of algebraic geometry, the Grassmannian can be constructed as a scheme by expressing it as a representable functor.[2]
Grassmannian as a Representable Functor
Let be a quasi-coherent sheaf on a scheme S. Fix a positive integer r. Then to each S-scheme T, the Grassmannian functor associates the set of quotient modules of
locally free of rank r on T. We denote this set by .
This functor is representable by a separated S-scheme . The latter is projective if is finitely generated. When S is the spectrum of a field k, then the sheaf is given by a vector space V and we recover the usual Grassmannian variety of the dual space of V, namely: Gr(r, V∗). By construction, the Grassmannian scheme is compatible with base changes: for any S-scheme S′, we have a canonical isomorphism
In particular, for any point s of S, the canonical morphism {s} = Spec(k(s)) → S, induces an isomorphism from the fiber to the usual Grassmannian over the residue field k(s).
Universal family
Since the Grassmannian scheme represents a functor, it comes with a universal object, , which is an object of
and therefore a quotient module of , locally free of rank r over . The quotient homomorphism induces a closed immersion from the projective bundle :
For any morphism of S-schemes:
this closed immersion induces a closed immersion
Conversely, any such closed immersion comes from a surjective homomorphism of OT-modules from to a locally free module of rank r.[3]Therefore, the elements of are exactly the projective subbundles of rank r in
Under this identification, when T = S is the spectrum of a field k and is given by a vector space V, the set of rational points correspond to the projective linear subspaces of dimension r − 1 in P(V), and the image of in
is the set
The Plücker embedding
The Plücker embedding is a natural embedding of a Grassmannian into a projective space:
Suppose that W is an r-dimensional subspace of V. To define ψ(W), choose a basis {w1, ..., wr}, of W, and let ψ(W) be the wedge product of these basis elements:
A different basis for W will give a different wedge product, but the two products will differ only by a non- zero scalar (the determinant of the change of basis matrix). Since the right-hand side takes values in a projective space, ψ is well-defined. To see that ψ is an embedding, notice that it is possible to recover W from ψ(W) as the set of all vectors w such that w ∧ ψ(W) = 0. The embedding of the Grassmannian satisfies some very simple quadratic polynomials called the Plücker relations. These show that the Grassmannian embeds as an algebraic subvariety of P(∧rV) and give
another method of constructing the Grassmannian. To state the Plücker relations, choose two r- dimensional subspaces W and Z of V with bases {w1, ..., wr}, and {z1, ..., zr}, respectively. Then, for any integer k ≥ 0, the following equation is true in the homogeneous coordinate ring of P(∧rV):
When dim(V) = 4, and r = 2, the simplest Grassmannian which is not a projective space, the above r reduces to a single equation. Denoting the coordinates of P(∧ V) by X1,2, X1,3, X1,4, X2,3, X2,4, X3,4, we have that Gr(2, V) is defined by the equation
X1,2X3,4 − X1,3X2,4 + X2,3X1,4 = 0. In general, however, many more equations are needed to define the Plücker embedding of a Grassmannian in projective space.
5. Mereotopology and Logic
6. The Schrodinger Equation, Curvature, Feynman Diagrams and Copenhagen
ds^2 = Edx^2 + 2Fdxdy + Gdy^2, and dA = (EG – F^2)^(1/2); this is the first fundamental form of Gaussian curvature. Together with the second fundamental form, and the compatibility conditions known as the Gauss-Codazzi equations, we may recover the Christoffel symbols – or the coefficients of the Affine Connection,
Why Hilbert Spaces and Hilbert Manifolds (Infinite Dimensional) May Be Approximations Consider the escape velocity of an object from Earth’s orbit.
And now let’s perform the same calculations with the gravitational force replaced by the Coulomb force:
F = kQq/r2 , dW = Fdr = - dr(kQq/r2)
2 W = Integral from the ground state radius of hydrogen, R0, to infinity = - kQq/R0 = - ke /R0.
In 2005, I noticed a relationship between the Planck mass and the Fine Structure Constant:
To find the “escape velocity” v for the electron, we can try this relationship:
2 2 2 - ke /R0 = (1/2)mev = Gmp / R0
2 v = (2Gmp / meR0) where mp is the Planck mass.
Covalent Hydrogen Radius 3E-11 electron mass 9.10938E-31 G 6.67408E-11 planck mass 2.17645E-08 fine structure constant 0.007297352 escape velocity 4109007.391 electron escape velocity / speed of light 0.013710402
Since this velocity is about 2 times that of the velocity of an orbiting electron around a stationary proton, per the Bohr model, then it would appear that hydrogen ionizes long before the hydrogen atom could achieve an infinitude of energy levels, per some particle in a box construction. This simple example demonstrates how infinite numbers of energy states and configurations, such as those postulated in a Hilbert Space or Hilbert Manifold, may be exceptionally good approximations to reality. But for a more precise description of physics, we may in fact need to limit ourselves to systems or theories with finitely many states, albeit large.
The reason for this is not simply due to the physical instability of systems that have an infinity of allowable states or energy levels, even if they are limiting as in the case of the Bohr energy levels for the hydrogen atom. The reason is also due to the need to reconcile first-order logics with constructivism, and so reconcile quantum mechanical logics with topological models of particle physics. For example, according to the Lowenheim-Skolem theorem, it is not possible for a set of axioms to characterize Natural numbers, Real numbers, or any other infinite structure. Similarly, the Ryll-Nardzewski theorem says that a complete first-order theory T in a finite or countable signature implies the theory is Aleph- naught categorical, the stone space is finite, the number of formulas in the model in n-free variables is finite up to equivalence modulo T.
Now since quantum mechanical logics are described by lattices, in particular orthomodular lattices, and since complete heyting algebras, frames, and locales describe lattices categorically up to their morphisms, then in order for lattice objects to be categorical theories, some limitations on lattice objects or morphisms must be in place to ensure that theories of lattices remain finitary, and hence categorical.
Solutions to Ordinary Quantum Mechanics (Spherical Harmonics) Yield Gaussian Curvature (Quantum Mechanics →Curvature). Riemannian Manifolds Sufficient to define Orthomodular Lattices for Quantum Logic (Curvature →Quantum Mechanics). The idea here is that we have provided sufficient examples to show that curvature exists on complex, hypercomplex, and complexified manifolds, in addition to the usual pseudo-Riemannian manifolds with which most are familiar. Not only that, but that we can (per Mark Hadley’s research) claim a Riemannian manifold as a sufficient starting place to recover the Orthomodular Lattices needed for Quantum Logic. So, even though the Standard Model, or simply Ordinary Quantum Mechanics, does not directly provide an explanation for gravitation, they do inherently carry the mathematical notions of Gaussian (intrinsic) curvature. Since Gaussian Curvature can be related to both the Riemann curvature tensor and the Ricci curvature tensor, and since both ordinary and relativistic quantum mechanics include the concept of “mass” and Gaussian curvature, then we propose there is sufficient evidence to conclude that the masses in quantum mechanics must interact with the Riemannian manifold structure implied by Gaussian curvature by creating gravitational curvature according to Einstein’s field equations. In summary: quantum mechanical solutions (spherical harmonics) create Gaussian Curvature, which is related mathematically to the Riemann/Ricci curvature tensor; the quantum masses then interact with this curvature tensor due to the Einstein field equations.
This theoretical model has already been supported in part by experiments using “weak measurement” techniques over the past 30 years.
(1987) P. Mittelstaedt; A. Prieur; R. Schieder (1987). "Unsharp particle-wave duality in a photon split-beam experiment". Foundations of Physics 17 (9): 891–03. Bibcode:1987FoPh...17..891M. doi:10.1007/BF00734319.
(1988) D.M. Greenberger and A. Yasin, "Simultaneous wave and particle knowledge in a neutron interferometer", Physics Letters A 128, 391–4 (1988).
(2011) Aephraim Steinberg, Amir Feizpour, Lee Rozema, Dylan Mahler and Alex Hayat; University of Toronto: http://www.physics.utoronto.ca/~aephraim/PWMar13steinberg-final.pdf
Using Markov Chains and Cellular Automata-like partitioning of a Riemannian manifold to produce probability distributions similar to those found in Ordinary Quantum Mechanics.
7. The Riemann Zeta Function, Hypothesis, and Relation to Physics
Groups and Knots related to the Riemann Hypothesis Landau’s function, Sym(G), g(n) being Landau’s function, related to the square root of the inverse (reciprocal?) of the Logarithmic Integral function. (See “Evaluation asymptotique de l’ordre maximum d’un element du group symetrique”, J.P. Massias, J.L. Nicolas, G. Robin.) Since every group G is isomorphic to a subgroup of the symmetric group acting on G by Cayley’s Theorem, then we can relate all the groups used in Superknot theory to the Riemann Hypothesis according to the expression 푙표𝑔(𝑔(푛)) < √퐿𝑖−1(푛).
From a paper by Etienne Ghys, entitled “Lorenz and Modular Flows: A Visual Introduction, A tangled tale linking lattices, knots, templates, and strange attractors.” Note: “It is known that the the family of curves ct tends to fill the space in a “uniform” way but the quantitative estimate of the velocity of this phenomenon is equivalent to the famous Riemann hypothesis”. –See results from Asymptotic behavior of periodic orbits of the horocycle flow and eisenstein series, by Peter Sarnak, Communications on Pure and Applied Mathematics, Volume 34, Issue 6, pages 719–739, November 1981 (Etienne Ghys Unité de Mathématiques Pures et Appliquées de l'E.N.S. de Lyon etienne.ghys at umpa.ens-lyon.fr http://www.umpa.ens-lyon.fr/~ghys/.) Also, see the below reference from SP Lalley (topological entropy of geodesic flow and horocycle flow bears “striking” resemblance to prime number theorem).
8. Conclusion
|휋(푥) − 퐿𝑖(푥)|.
9. References
[1] Edwards, Harold M. (2001), Riemann's Zeta Function. Academic Press (1974 ed.), Dover Publications (2001 ed.)
[2] Schoenfeld, Lowell (1976), "Sharper Bounds for the Chebyshev Functions θ(x) and ψ(x). II", Mathematics of Computation 30 (134): 337–360
[3] Wagon, S. Mathematica in Action. New York: W. H. Freeman, p. 33, 1991.
[4] Shenk, Al, Calculus and Analytic Geometry, Third Edition, 1984, Scott, Foresman, and Company, p. 293.
[5] Derbyshire, J. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. New York: Penguin. P. 114-117 and 373, 2004.
[6] Rosser, Barkley. (1941), Explicit Bounds for Some Functions of Prime Numbers. American Journal of Mathematics, Vol. 63, No. 1, pp. 211-232.
[7] Littlewood, J. E. (1914), "Sur la distribution des nombres premiers", Comptes Rendus 158: 1869– 1872, JFM 45.0305.01
Appendix. Classical Groups From Paul Garrett, The Classical Groups & Domains, April 24, 2005)
Maximal compact Root Name Group Field Form
subgroup system
Special linear SL(n, R) R - SO(n)
Complex special linear SL(n, C) C - SU(n) An−1 Quaternionic special SL(n, H) = H - Sp(n) linear SU∗(2n) (Indefinite) special SO(p, q) R Symmetric S(O(p) × O(q)) orthogonal
Complex special Bm, n=2m+1
SO(n, C) C Symmetric SO(n) orthogonal Dm, n=2m Skew- Symplectic Sp(m, R) R U(m) symmetric Skew-
Complex symplectic Sp(m, C) C Sp(m) Cn symmetric (Indefinite) special SU(p, q) C Hermitian S(U(p) × U(q)) unitary (Indefinite) Sp(p, q) H Hermitian Sp(p) × Sp(q) quaternionic unitary Quaternionic Skew- SO∗(2n) H SO(2n) orthogonal Hermitian