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Archive for June, 2012

CouchDB makes it quite simple to store python code directly in the database that can be used as code-on-demand for the inverse problem of distributed computing.  I did a little test of creating a code database and dumping the solves.py into it but have not tested if the code can be evaluated.  If cvxopt could be used in this way then important techniques like the L1 penalization and LASSO and others could be used by a broader group of people who veer away from the complexity of R or cannot afford MATLAB.  Many Biospect chemists veered away from anything other than linear regression because they felt the learning curve was too steep to justify taking time away from their major focus.  This sort of idea is not very new or different but seems quite feasible even in the context of the structure we are working on for mat2py.  More people would use nontrivial statistical methods if they could just go to some web page and upload their data to a suite of statistical analysis tools without having to learn python or R or MATLAB or cvxopt and obtained interpretable results.

On the problem of cloud service optimization, the more standard approach is something like this: http://clumon.ncsa.illinois.edu/ where we are getting a cleaner system using CouchDB and guarantees on performance and core availability from picloud.  The

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Several central divisions dominate the intellectual landscape, primary ones being science and humanities and science and religion.  Twentieth century science has focused, driven by industry and military, on the subatomic scale, producing, from compliance to an empirical philosophy of constructing reality from interpretation of particular physical phenomena, an incomprehensible reality, while the religions had enjoyed monopoly over authoritative views on metaphysics.  The resulting confusion has produced a situation where metaphysics had to be sidelined by science which did not solve the fundamental problem of understanding the reliable foundations of metaphysics.  A foundational physics cannot avoid the problem of foundation of metaphysics.  Even at the risk of considering the past non-scientific thought on metaphysics as mere prejudice if necessary, it is imperative that science and metaphysics share a coherent foundation jointly.

One of the virtues of an S4 physics is that it could produce a description of physics consistent with the physically observed — obviously no physical theory that cannot explain the landmark experimental discoveries that led to quantum mechanics can lead to an acceptable foundation.  S4 physics provides a very simple explanation of the quantum phenomena geometrically — every geodesic is closed of fixed length in a sphere which leads to quantization — but also provides us the natural mechanism for metaphysical phenomena, which is the higher symmetry of electromagnetism than the U(1) of classical Maxwell theory.

A simple consequence of S4 theory is that the universe has existed eternally, and the possibility that consciousness is itself eternal, and there might exist higher levels of conscious activity than the biological life that has been the focus of sciences.  This reasoning leads to the consideration of human society as natural phenomena in a hierarchy of conscious evolution.  Without a proper metaphysical theory that can at least address our inner experiences as EXTERNAL and objective experiences, we risk misunderstanding the development of consciousness as accidental and isolated, due to the appearance of accidental and exceptional minds such as a Newton, an Einstein or a Dirac.  Although it is a difficult concept to show clearly, the universe contains unimaginable sophistication and intelligence that influence the human society.  Partly the reason for skepticism is that it is easy to slip into a speculative nightmare when one moves away from the concrete measurements of experiments.  But on the other extreme, if it is indeed the case that there are conscious activities that affect us then it is important for us to understand such phenomena.

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I have repeated the arguments for why the universe has four macroscopic spatial dimensions, is compact and stationary and in fact a fixed radius sphere many times earlier since July 2008.  But let us take a quick look at what the Big Bang theory and the expansionary universe models are obstructing.  The major insight into nature that these theories are obstructing is a clear link between subatomic physics and cosmic scale physics.  In the S4 theory I provide the link directly in the radius of the universe being tied to the Planck constant.  It is this identification that allow us to grasp the way in which the geometry of the universe not only dictates quantisation but also provide naturality to the unique force that governs the universe, which is SU(2) electromagnetism.

I am working through the issue of putting a Standard Model in a four dimensional sphere, but here although I am not an expert in particle physics, I can see that there are standard approaches to reducing the dimension from the work on Kaluza-Klein theories as well as by other approaches such as this one.

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In the current physics orthodoxy, there is an insurmountable gap — the cosmological constant problem.  The quantum physics based energy of the vacuum is a hundred order of magnitude larger than the measured value.  I believe the fundamental reason for this is that quantum mechanics is a linear approximation of a correct theory of the universe, which I have claimed is a stationary 4-sphere governed by SU(2) electromagnetism.  So we have a science that is still quite unclear about how the fundamental constitution of the universe and therefore cannot give us the grounds to pursue questions of interest to us, which requires a clearer interpretation of metaphysics from a physical theory.  The answer from something like the Copenhagen interpretation of quantum mechanics is an unsatisfying answer.

My answer has been to take the clear evidence for four macroscopic spatial dimensions seriously and then reconsider the foundational issues for abandoning classical physics.  In fact, it would be interesting to package entire scientific theories and produce methods of testing entire such theories with minimal set of data.

Individual human questions about the nature of existence are often spiritual in nature.  We need a physics and a science that is able to provide foundations for valid answers to metaphysical questions as well.  The current scientific orthodoxy cannot provide real answers to metaphysical questions because it is fundamentally three dimensional in a four dimensional universe.  The top cosmologists of our time are unwilling to accept the evidence for four macroscopic spatial dimensions, so a revolution of the type that is necessary will have a difficult birth.

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11. Quaternion projective spaces in complex matrices

Following Furutani and Tanaka, who provided a Kähler structure on the punctured cotangent bundle on quaternionic projective spaces, we can embed general quaternionic projective spaces {P^n\mathbb{H}} into spaces of complex matrices as follows.

Let {\rho:\mathbb{H}\rightarrow M(2,\mathbb{C})} be the representation given by

\displaystyle \rho(p) = \rho(z + wj) = \begin{array}{cc} z & w\\-\bar{w}&\bar{z} \end{array} \ \ \ \ \ (2)
Introduce a metric on {P^n\mathbb{H}} via the Hopf fibration {\pi:S^{4n+3}\rightarrow P^n\mathbb{H}} where {p\rightarrow[p]} for {p = (p_0,\dots, p_n \in \mathbb{H}^n}. The embedding of {P^n\mathbb{H}} into {M(2n+2,\mathbb{C})} is given by

\displaystyle [p]\rightarrow (P_{ij}) = ( \rho( p_i \bar{p}_j))
The image of this embedding consists of {P\in M(2n+2,\mathbb{C})} satisfying {P^2=P, P^*=P, \trace(P)=2, PJ=J\leftexp{t}{P}}. The canonical one-form can be written as

\displaystyle \theta_{P^n\mathbb{H}}(P,Q) = \frac{1}{2} (d\embedMap}^* \trace(Q dP)
and so the symplectic form can be written

\displaystyle \omega_{P^n\mathbb{H}} = \frac{1}{2} (d\embedMap)^* \trace(dQ\wedge dP).

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\maketitle

1. Introduction

The idea that the universe has more than three dimensions goes back at least to the nineteenth century, although I do not know of a careful scholarly study of the way in which the idea of higher dimensions is explicitly or implicitly present in the thought of Riemann, for example, whose 10 June 1854 presentation, “On the Hypotheses that lie at the foundation of geometry” introduced the notion of hyperspace. The belief that the universe has four macroscopic dimensions was held by Rudolf Steiner, and Nietzsche believed that the apparent reality hides an unseen reality that he makes explicit from his earliest work, {em The Birth of Tragedy}. Modern physical theories typically add microscopic dimensions for various technical reasons. In the early twentieth century, Gunnar Nordstrom, Theodor Kaluza, and Oscar Klein had independently proposed that electromagnetism and gravity can be unified by addition of a spatial dimension. We note as well that William Hamilton spent his entire life from his first discovery of quaternions obsessed with the idea that quaternions will revolutionize physics, and the four dimensional sphere is precisely the quaternionic projective line.

In this note, we take a view that we should have sharp observational evidence for claims for a particular number of dimensions. We present arguments that the universe has precisely four macroscopic spatial dimensions, and that it is a four dimensional sphere. By 1973, a {U(1)\times SU(2) \times SU(3)} gauge theory had succeeded in uniting the electromagnetism and the nuclear forces. Beginning with the description of the universe as a four dimensional sphere of fixed radius, we can note that the natural electromagnetism is {SU(2)} electromagnetism, that properties like spin are most likely effects from a four dimensional electromagnetism, and thus the natural conjecture that the nuclear forces are masking a single force of {SU(2)} electromagnetism.

After giving well-known evidence whose parsimonious interpretation is a macroscopically four space dimensions from observed rotations of regular arrangements that cannot have translation invariance in three dimensions, which have been interpreted as quasicrystals, we sketch why macroscopic four space dimensions would not lead to serious problems with the force law when the total space of the universe is compact. Next we give an explanation of the redshift as an artifact of treating waves on a sphere as waves on a flat space.

Two of the most promising features of a four sphere as a model of the universe are: quantization of inverse wavelengths are automatic on a sphere where every geodesic is closed of the same length, and so quantization of energy is automatic; and second, that all functions and tensors on a sphere wave aspects simply because the possibility of approximation by spherical harmonics. Wave-particle duality thus holds for any space localized object. These features are promising in their ability to infer quantum effects as consequence of the shape of the universe rather than as hypotheses on objects in the small scale.

The currently established model of elementary particles and their interactions is the Standard Model, which can be described as a gauge theory. We recall the mathematical features of gauge theory. We are fortunate to be in a situation where detailed study of a classical field theory of {SU(2)} gauge theory on {S4} had been done by physicists and mathematicians from the 1970s. Instantons for {S4} were constructed by t’Hooft with {5k-3} parameters quite explicitly and Atiyah, Hitchin, and Singer were able to calculate the dimension of the selfdual moduli using the Atiyah-Singer index theorem to obtain {8k-3}, from which they could conclude that t’Hooft solutions did not describe all the instanton solutions with topological charge {k}. Thus once the universe is known to have a four spherical shape, we have some powerful results and analytic tools for {S4} that we may use.

2. Observed symmetries of crystals show that the universe has at least four spatial dimensions

A basic result of crystallography is the crystallographic restriction theorem. This was first proved for arbitrary dimensions by R. Vaidyanathaswamy in 1928 \cite{Vaidyanathaswamy:1928}. According to this theorem, crystals in three dimensions can have rotational symmetries of orders 1, 2, 3, 4, and 6. In four dimensions crystals can have additional symmetries 5, 8, 10 and 12. In six dimensions crystals can have additional symmetries 7, 9, 14, 15, 18, 20, 24, 30.

Since early 1980s when Daniel Shechtman first discovered crystal structures (deemed to be ‘quasicrystals’ but which are much more simply explained as crystals) with rotational symmetries of orders 5 and 10, there has not been a single discovery of crystals with symmetries that would require six dimensions to explain. For example, there has been no crystals found with sevenfold symmetry.

A crystal is modeled via mathematical lattices on {\mathbb{R}^n} which are arrangements of points that, relative to an arbitrarily chosen origin can be written as

\displaystyle r = k_1 a_1 + \cdots + k_n a_n
for a integers {k_1,\dots,k_n} and {a_1,\dots,a_n \in \mathbb{R}^n}. The lattices are translationally invariant. Although ‘quasicrystals’ are not considered crystals,

Their x-ray diffraction patterns has no qualitative difference from diffraction patterns of ‘ordinary’ crystals
The standard models of ‘quasicrystals’ are as slices of higher dimensional crystals. In other words the models we would use if we considered these to be literally higher dimensional crystals are the same as the ones that are used in practice

If we observe a four dimensional crystal with a rotational symmetry that cannot occur for a three dimensional crystal, then we cannot expect to observe three dimensional translation invariance. It then is simply a matter of semantics to call such objects ‘quasicrystals’. A purely parsimonious explanation of the observed crystal symmetries of orders 5, 8, 10, and 12 is that the universe itself has at least four macroscopic spatial dimensions.

Crystal structures are determined by diffraction of electrons, X-rays or neutrons, and studying the interference of phase differences between rays elastically scattered from different atoms in the crystal.

In principle, we could discover crystals with 7 or 9 fold rotational symmetry leading us to conclude by this reasoning that the universe has at least six macroscopic spatial dimensions, but a vigorous search by crystallographers has not produced any such example in 30 years since Shechtman’s groundbreaking discovery.

3. Implications of at least four dimensions

A parsimonious conclusion of at least four spatial dimensions is sufficient reason for us to reconsider the standard interpretations of the dominant physical theories, in particular of quantum mechanics which is numerically one of the most successful scientific theories to date. This is because intuitively at least, the basic reasons for discarding nineteenth century classical physics for quantum mechanics were the explanation of the observed blackbody spectrum which was resolved by Planck simply by introducing quantization of energy and on a four dimensional sphere since every geodesic is closed with the same length, quantization of frequency can be achieved whether we have a classical theory or a special quantum theory, and the second reason that a classical hydrogen atom would be unstable because the electron orbiting a stationary proton would do work and lose electromagnetic energy. This latter problem can find a simple solution in a four dimensional sphere.

As an illustration of the issue of automatic quantization, recall that the standard method of quantization is to replace energy by the operator {i\hbar\partial/\partial t}. Note here that {\hbar} is simply the inverse of the length of a closed geodesic of an {S4(1/h)}-universe. Thus this quantization procedure is producing a linearization after a rescaling. In the four-sphere context, on the other hand, classical physics contains quantized information without linearization. A natural hypothesis is that the physical three dimensional universe is isometrically immersed in {S^4(1/h)} and quantum mechanics arises as the linearization of a classical physical theory on the tangent bundle {TM}.

If these claims are concretely illustrated, then we open up the possibility of a classical physics on a four dimensional physics as a valid description of the actual universe. While such a conclusion might lead to a less sophisticated physical theory than currently held, we recall that the two criteria for ranking scientific theories that do fit the facts are parsimony and predictive power.

4. The universe must be compact

In 1986 Shing-Tung Yau and Peter Li proved that heat kernels on noncompact riemannian manifolds with a lower bound on the Ricci curvature has Gaussian upper bounds. This theorem applies to the observed uniform lower bound observed for the cosmic background radiation at around 2.7 Kelvin. A noncompact universe cannot produce a thermal equilibrium with a uniform lower bound on temperature.

It is well-known that the cosmic background radiation fits a thermal Planck form quite well with temperature around 2.75 K. Although often it is argued that the CBR supports the expansionary cosmological models, the parsimonious assumptions about the shape of the universe is that the fabric of space does not change with time. We shall argue that it is a four dimensional sphere of fixed radius, and in the next section we will show how a redshift can arise in a stationary spherical universe that is mistaken for a flat three dimensional universe.

5. The redshift can be explained as mistaking spherical waves by linear waves

First, recall that the redshift phenomenon is the following. When performing spectral analysis of a light from a distant object, we can detect absorption lines of abundant elements. For example, the absorption lines for hydrogen. The expected wavelength for the H-{\alpha} line in the spectrum is 656.3 nanometers, but the line observed in the spectrum of a distant object could appear at a longer wavelength. In 1929 Hubble discovered a remarkable linear relation between the redshift of spectral lines and the distance of objects. Since then the redshift distance relation has been the central empirical foundation for expansionary universe cosmological models. In this section we provide some observation of how a linear redshift would be expected in a stationary spherical universe where the electromagnetic waves are treated as linear waves.

The standard explanation for the redshift-distance relationship for distant galaxies has been that this is a consequence of a Doppler effect. But there are alternative explanations for this phenomenon. Although our explanation of the redshift is different from the so-called ‘tired light’ hypothesis of Fritz Zwicky, we mention it for historical context. A uniform loss of energy by signals from distant galaxies proportional to the distance travelled would result in a redshift as well. The idea that the photons could lose energy as they travel is not new and is due to Swiss astronomer Fritz Zwicky who suggested the hypothesis in 1929. The mechanism he proposed for loss of energy of photons due to loss of momentum to surrounding masses due to gravitational interactions (\cite{Zwicky:1929}) The “tired light” hypothesis has not been accepted with the major arguments against it being that reducing photon energy would also change its momentum, blurring the light which is not observed, that there is an observed time dilation where a supernova that takes 20 days to decay appears to take 40 days to decay for redshift {z=1}, and that the tired light cannot produce the observed blackbody spectrum correctly, for example by arguments of Edward Wright (\cite{Wright:2008}). The last of these arguments is open to criticism because cosmic background radiation is in thermal equilibrium and the tired light hypothesis pertains to signals from distant galaxies.

5.1. Possible explanation: treating frequency as {n} instead of {\sqrt{n(n+3)}}

In this section we provide a concrete way in which a linear redshift is expected (of the same order as observed) when spherical waves are treated as waves on a flat space. This suggests that the redshift is not a physical phenomenon at all but rather a systematic artifact of treating spherical waves as linear waves. In particular, there need be no energy loss that corresponds to the redshift at all, and no need for an expansionary model of the universe.

To clarify this terminology of spherical versus linear waves, we define linear waves to be the solution of the wave equation on flat Euclidean spaces and spherical waves to be the solutions of the wave equation on spheres. We are particularly interested in the flat space {\mathbf{R}^3} and the four-dimensional sphere {S^4}.

The wave equation

\displaystyle \frac{1}{c^2} \frac{\partial^2 u}{\partial t^2} - \Delta u = 0
can be solved on a four dimensional sphere of radius {a} by separation of variables and an eigenvalue expansion in spherical harmonics. With

\displaystyle \omega_k = \frac{c}{a} \sqrt{k(k+3)}
the solution consists of linear combinations of terms of type

\displaystyle \Phi(x)\exp(-i \omega_k t),
where {\Phi(x)} is a spherical harmonic with eigenvalue {k(k+3)}. The relation between wavelengths and frequencies in this case is not {\nu = c/\lambda} but rather {\nu = \sqrt{n(n+3)}} when {\lambda=c/n}. There is no problem interpreting the term involving {\sqrt{n(n+3)}} as a frequency because in the solution, it appears as a coefficient of time in the circular function {e^{-i\omega t}}.

Let us consider what happens for the case of the H-{\alpha} line at 656 nanometers. Consider the distance of 1 light second, which is obviously many orders of magnitude larger than the wavelength of 656 nanometers. In this case

\displaystyle \nu = 4.573171 \times 10^14
and we can treat {\nu} as an integer with negligible error. We are interested in seeing what happens in a distance of around 106 parsecs, which is the unit of distance used by Hubble. The actual frequency will be {\sqrt{n(n+3)}}, and the error per light second of the wavelength will be

\displaystyle c ( \frac{1}{n} - \frac{1}{\sqrt{n(n+3)}} )
which will be a shift to longer wavelengths. The shift of 656 nm wavelength over distance 106 parsecs would be around 0.023 nanometers, which is small but detectable. Now if we define {z} to be the ratio of this shift to the expected wavelength of 656 nm and multiply {z} by the speed of light {c} we obtain 10,518 km/s which is in the order of magnitude of the velocities that Hubble used in his dataset.

The analysis above indicates that the observed redshift is quite likely explained fully by electromagnetic waves being spherical waves in a four dimensional sphere, and in particular the redshift phenomenon does not necessicitate an expansion of the universe.

We can test this hypothesis that the redshift can be explained as an artifact of mistaking spherical waves as linear waves by checking such a model on a modern dataset containing measurements of distance and their redshifts. We took a dataset of these measurements for 957 cosmic objects from \cite{Amiga}.

6. Avoiding the wrong force law

A basic reason for dismissing four macroscopic spatial dimensions is the idea that in {D} macroscopic dimensions, the force laws would behave as {1/r^{D-1}} which for a universe with four macroscopic spatial dimensions would lead to inconsistency with Newtonian physics. However, the reasoning that is used for Planck scale compactified dimensions, which the Kaluza-Klein theories propose can be extended to a macroscopic compact dimension as well.

Merag Gogberashvili had solved this issue for a noncompact extra dimension in 1998 \cite{Gogberashvili:1998}. L. Randall and R. Sundrum \cite{RandallSundrum:1999} have written of how a model of a 3+1 brane universe where the Standard Model localizes in a higher dimensional space where gravity operates avoids the problem of the wrong force law.

In a {D}-dimensional space with one dimension compactified on a circle of radius {R}, the line element is

\displaystyle dl^2 = g_{ij}dx^i dx^j = dr^2 + r^2 d\Omega_{D-2} + R^2d\alpha^2
with {\sqrt{g} \sim R r^{D-2}}. The force law derived from the potential that solves the Laplace equation is

\displaystyle F \sim \frac{G_D}{2\pi R r^{D-2}} = \tilde{G}/r^{D-2}
where the constant has absorbed the effect of the extra dimension. Our proposal of a universe with four macroscopic spatial dimensions but a sphere of fixed radius {O(1/h)} falls in this category as well, although the additional dimension is not microscopic.

An interesting observation is that in the case of charged particles, the magnitude comparison of Coulomb versus gravitational forces give us the following

\displaystyle \frac{|F_{em}|}{|F_{grav}|} = 4.17 \times 10^42
In the case that the gravitational force law in three macroscopic dimensions is a result of four dimensional force, the gravitational constant {G_3} is a reduction of a four dimensional gravitational constant {G_3 = G_4/(2\pi R)}, which would change the relative strengths of electromagnetic and Coulomb forces to the order {10^8} assuming that the Coulomb force strength remains constant.

7. Special features of four spatial dimensions

The following is material that is well-known to geometers and articulated by Atiyah and Hitchin in a 1978 paper. The basic idea is that electromagnetism is naturally defined on four dimensional manifolds due to the fact that the Hodge *-operator maps 2-forms to 2-forms on a four dimensional manifold and that the curvature of a connection is a 2-form. These features do not exist for higher than four dimensions.

One of the special features of four dimensional compact manifolds is that the rotation group {SO(4)} is locally {Spin(4) = SU\times SU(2)}. In fact, {Spin(4)} is the double-cover of {SO(4)}. The differential 2-forms on a compact four dimensional manifold decompose into self-dual and anti-self dual parts by the eigenspaces of the Hodge *-operator. Electromagnetism can be formulated on a four dimensional manifold by considering a principal {SU(2)}-bundle {P} and considering the electromagnetic potential {A} to be a connection on {P} with curvature {F=dA} satisfying the Yang-Mills equations which say that {F} is anti-self-dual as a 2-form.

8. Some criteria for isometric immersion of a three manifold in a four-dimensional sphere

The general necessary and sufficient conditions for a riemannian {n}-manifold to be isometrically immersed into one of the space forms of dimension {n+1} are the Gauss and Codazzi-Mainardi equations. A different type of necessary and sufficient conditions are given by M-A Lawn and Julien Roth \cite{LawnRoth:2008}. Their condition for isometric immersion into a space form of curvature {\kappa = 4 \eta^2} is the existence of two spinor fields {\phi_1} and {\phi_2} of constant norm that satisfy the equations \begin{align*} D\phi_1 &= (\frac{3}{2}H + 3\eta)\phi_1
D\phi_2 &= -(\frac{3}{2}H + 3\eta)\phi_2 \end{align*} where {H} is a real valued function, which are equivalent to the equations in terms of the spin connection on {M}: \begin{align*} \nabla^{SM}_X\phi_1 &= \frac{1}{2}A(X)\cdot\phi_1 – \eta X\cdot\pi_1
\nabla^{SM}_X\phi_2 &= -\frac{1}{2}A(X)\cdot\phi_1 + \eta X\cdot\pi_2 \end{align*} with {\trace(A) = 2 H}. Either set of conditions then imply the existence of an immersion {F:M \rightarrow S^4(\kappa)} where {\kappa} stands for the curvature rather than the radius.

We have already given direct evidence that the universe is compact and four dimensional. The Lawn-Roth result above is a promising method of showing that an apriori three dimensional universe with two constant norm eigenspinors can be used to isometrically immerse the universe in a four-sphere.

In order to make this idea concrete, consider the proxy for the physical universe to be a 3-dimensional spin manifold for which there exist solutions to the Dirac-Einstein equations for parameters {\epsilon=\pm 1} and {\lambda}: \begin{align*} D\phi &= \lambda\phi
\Ric – (R_g/2)g = \frac{\epsilon}{4}T_\phi \end{align*} These equations are equivalent to the critical point of the functional obtained by integrating

\displaystyle R_g + \epsilon(\lambda(\psi,\psi) - (D_g \psi,\psi))
where {D_g} is the Dirac indexed by the metric. This equivalence was shown by Kim and Friedrich \cite{KimFriedrich:1999}. They show that these equations are equivalent to the generalized Killing spinor equation. The idea is to assume solutions for these equations exist and then use the Lawn-Roth embedding theorem on the spinor energy-momentum tensor to embed {M} in an {S^4} of appropriate radius. Such an exercise would provide theoretical justification for an {S^4}-theory.

9. Restrictions of hypersurfaces of a four-sphere

In the next section we will provide evidence that the gravitational field equations are the Ricci curvature equation of a scaled four sphere. It is useful to have a priori restrictions on submanifolds of a sphere.

A theorem of Hasanis and Vlachos from 2001 tell us that if we assume {M} is isometrically immersed minimal in {S^4} then the Ricci curvature has a supremum at least {1} unless the universal cover of {M} is homeomorphic to {S^3}.

10. Einstein’s gravitational field equations and Gauss equations

Suppose {(S,g_S)} is a four dimensional sphere with metric {g_S}, and {(M,g_M)} is a three dimensional subspace. Then the Gauss equation for the curvature of {M} in terms of the curvature of {S} is

\begin{align*} g_M( \tilde{R}(\tilde{X},\tilde{Y}) \tilde{Z},\tilde{W}) – g_S(R(X,Y)Z,W) & = g_S(B(Y,Z),B(X,W)) – g_S(B(X,Z),B(Y,W)) \end{align*}

and from this we can take contractions in the {Y} and {Z} terms using the formula for the Ricci tensor,

\displaystyle Ric_M(X,Y) = \sum_i g_M(R(X,e_i)e_i, Y)
to obtain

\displaystyle Ric_M(X,Y) - Ric_S(X,Y) = g_S( \sum_{i=1}^3 B(e_i,e_i), B(X,Y)) - \sum_{i=1}^3 g_S(B(e_i,X),B(e_i,Y)) \ \ \ \ \ (1)

This corresponds to the Einstein gravitational field equations for a physical universe {M} where scaling the sphere {S} provides the cosmological constant term as the Ricci curvature of {S} which is 12 times the sectional curvature of {S}. For a sphere of radius {R} the sectional curvature is {1/R^2}. Note that this is a space-only expression for Einstein’s equation.

The quantum field theory estimate of the cosmological constant is in the order of {10^74 GeV^4} while the measured cosmological constant term is in the order {10^{-47} GeV^4}. If we set the radius to be {1/h} with {h} the Planck constant then we obtain the terms {12 h^2} as the Ricci curvature of the ambient sphere. Now Planck’s constant is {4.1356675\times 10^{-24} GeV\cdot s} which produces an order of magnitude match to the measured cosmological constant by {12 h^2}.

The stress-energy tensor due to the currently accepted {U(1)} electromagnetism in free space and flat spacetime is given by

\displaystyle T^{\mu\nu} = \frac{1}{\mu_0}[ F^{\mu\alpha} F^\nu_\alpha - \frac{1}{4}g^{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}
which includes time as a component. In terms of the Poynting vector

\displaystyle S = \frac{1}{\mu_0} E \times B
and the Maxwell stress

\displaystyle \sigma_{ij} = \epsilon_0 E_i E_j + \frac{1}{\mu_0}B_i B_j - \half(\epsilon_0 E^2 + \frac{1}{\mu_0}B^2)\delta_{ij}
the stress energy tensor can be written as a 4×4 matrix. An anlogous expression can be given for {SU(2)} electromagnetism as well. The stress-energy tensor due to an electromagnetic field with its potential given as a connection {A} on a principal {SU(2)}-bundle can be written in terms of the curvature form {F=dA} as

\displaystyle T_{ij} = F_i^kF^{kj} - \frac{1}{4} F^{kl}F_{kl} g_{ij}
The structural resemblance of this form and the second part of the Ricci curvature formula for a submanifold can be assumed not to be accidental. Thus the problem of interest for us is to understand the connection between this expression for the stress-energy tensor and the second fundamental form term of a three dimensional submanifold of a four dimensional sphere.

11. Geometric interpretation of stress-energy tensor

We would like to interpret the gravitational field equations as a literal version of the the equation for the Ricci curvature of a three dimensional submanifold of a four dimensional sphere. A similar exercise was done by \cite{PeiWang:2008} For this we must interpret the field strength tensor as the second fundamental form of an embedding of a three dimensional submanifold in a four-sphere.

Two streams must come together perfectly here. On one hand, O. Hijazi introduced the study of mathematical stress-energy tensor acting on spinors. There is a theorem of Lawn and Roth that tells us how to use these stress-energy tensors to construct embeddings of certain three-manifolds into four dimensional space forms. From this stream we note the useful place of spinors in leading to an embedding of a three-manifold into a four dimensional one. On the other hand there is the stream of Yang-Mills theory which describes equations for critical points of principal {G}-connections by the Yang-Mills functional. These streams merge when the principal {G}-bundle is not arbitrary or auxiliary but the {SU(2)} bundle of the spinor bundle that lifts the {SO(4)} bundle. Concretely, consider a isometrically embedded submanifold {M} with an adapted frame {e_1, e_2, e_3, e_4} with {e_4} normal to {M}. Then consider the three functions

\displaystyle a_i = \langle \nabla_{e_i} e_4, e_i \rangle
We get an {SU(2)} connection when we multiply these by the generators of the Lie algebra. But this {SU(2)} bundle is not auxiliary but tied to the manifold in spinors.

It is worthwhile noting also that when {M} is a submanifold of {S^4} or any other spin four manifold, there is a simple correspondence between spinor bundles of the two spaces: the spinor bundle of {M} is the restriction of the positive spinors on the ambient manifold.

Thus let us assume that on the four-sphere we have an {SU(2)} connection {A} that acts on the orthonormal frame bundle of the manifold. Choose an orthonormal coframe {(\omega_1,\omega_2,\omega_3,\omega_4)} adapted to a submanifold so that {\omega_4} is always normal to {M} and let {A=(A_{ij})} be a matrix of differential 1-forms for which the covariant derivative satisfies

\displaystyle D\omega_i = d\omega_i + \sum_k A^k_i \wedge \omega_k \ \ \ \ \ (2)
The curvature 2-form {F} in this case is

\displaystyle F^j_i = dA^j_i + \sum_k A^j_k \wedge A^k_i
Now we consider what happens when {\omega_4=0}. Taking a second covariant derivative of (2) we find

\displaystyle D^2 \omega_i = \sum_k dA^k_i \wedge \omega_k + \sum_k A^k_i \wedge d\omega_k + + \sum_l A^l_i \wedge d\omega_i + \sum_l \sum_k A^l_k \wedge A^k_i \wedge \omega_k
Now {\omega_4=0} and we can separate the terms containing {d\omega_4}. Now {d\omega_4} can be identified with a second fundamental form term. [This argument needs more work.]

We want to take the analogy of the process of obtaining the induced connection on a hypersurface as the ambient Levi-Civita connection corrected by a potential in the form

\displaystyle \Bar{\nabla}_X = \nabla_X + h(X)
where {h} is the second fundamental form with an {SU(2)} gauge potential which will be given by three coefficients corresponding to the Lie generators of {su(2)}. We would like to identify the {SU(2)} bundle as one of the eigenspaces of the Hodge-* operator acting on 2-forms. We know that abstractly we can define a Dirac operator on spinors and consider a potential term that literally provides a method of embedding the three manifold to a space of constant curvature by the work of geometers, Hijazi et. al. We want to identify the spinors with two forms and at the same time identify the potential term to a second fundamental form of an embedding. Then the Einstein gravitational equations are formally the Ricci curvature equations for a submanifold.

12. The Dirac operator on a sphere

C. Bar has calculated the spectrum of the Dirac operator acting on spinors on spheres and R. Camporesi and A. Higuchi have provided a description of the eigenfunctions of the Dirac operator on spheres and hyperbolic spaces. We follow Camporesi and Higuchi to describe the eigenfunctions of the Dirac operator on spheres.

The Dirac operator acts on the spinor bundle of a spin manifold, which can be described abstractly as the vector bundle associated to the lift of the frame bundle on the manifold, a principal {SO(d)} bundle (here {d} is the dimension of the base manifold) to a {\Spin(d)} bundle. Recall that {\Spin(d)} is the univeral 2-fold covering of {SO(d)}. The spin representation is concretely described using the Gamma matrices.

The Clifford algebra in {d} dimensions is described by matrices {\Gamma^a} satisfying

\displaystyle \Gamma^a\Gamma^b + \Gamma^b\Gamma^a = 2 \delta^{ab} 1,
where the matrices can be chosen to be {2^{[d/2]}} dimensional. Our interest is {d=4} and the corresponding Gamma matrices can be constructed inductively. For {d=2} one takes

\displaystyle \Gamma^2 = \left( \begin{array}{cc} 0 & 1\\ 1 & 0 \end{array} \right), \Gamma^1 = \left( \begin{array}{cc} 0 & i\\ -i & 0 \end{array} \right),
to which one adds for {d=3} the matrix

\displaystyle \Gamma^3 = (-i)\Gamma^1\Gamma^2 = \left( \begin{array}{cc} 1 & 0\\ -1 & 0 \end{array} \right).
For {d=4} let

\displaystyle \Gamma^4 = \left( \begin{array}{cc} 0 & 1\\ 1 & 0 \end{array} \right), \Gamma^j = \left( \begin{array}{cc} 0 & i\tilde{\Gamma}^j\\ -i\tilde{\Gamma}^j & 0 \end{array} \right),
where the {\tilde{\Gamma}} refer to {d=3}.

The matrices

\displaystyle \Sigma^{ab} = \frac{1}{4} [ \Gamma^a, \Gamma^b ]
satisfy the {SO(d)} commutation rules

\displaystyle [ \Sigma^{ab}, \Sigma^{cd} ] = \delta^{bc} \Sigma^{ad} - \delta^{ac} \Sigma^{bd} - \delta^{bd} \Sigma^{ac} + \delta^{ad} \Sigma^{bc},
and gnerate a {2^{d/2}}-dimensional representation of {\Spin(d)}. Since

\displaystyle [\Sigma^{ab},\Gamma^c] = \delta^{bc}\Gamma^a - \delta^{ac}\Gamma^b
Since {\eta = \Gamma^1\cdots\Gamma^d} anticommutes with each of the {\Gamma^a} and commutes with the generators {\Sigma^{ab}}, and

\displaystyle \eta = i^{d/2} \left( \begin{array}{cc} 1 & 0\\ 0 & -1 \end{array} \right).
Since {\eta} is nontrivial, the representation {\tau} with generators {\Sigma^{ab}} is reducible and

\displaystyle \Sigma^{ab} = \left( \begin{array}{cc} \Sigma^{ab}_+ & 0\\ 0 & \Sigma^{ab}_- \end{array} \right).

Staying with general {S^d}, we can write the metric in polar coordinates {(\theta,\omega)} as

\displaystyle ds_d^2 = d\theta^2 + f(\theta)^2 \tilde{g}_{ij} d\omega^i \otimes d\omega^j
Let {\tilde{e}_i} be a frame on {S^{d-1}}, and the Christoffel symbols for this lower dimensional sphere is given as \newcommand{\te}{\tilde{e}} \newcommand{\tC}{\tilde{C}}

\displaystyle \tilde{\omega}_{ijk} = \langle \tilde{\nabla}_{\te_i}\te_j,\te_k\rangle = \frac{1}{2}(\tC_{ijk}-\tC_{ikj}-\tC_{jki}
where

\displaystyle [\te_i,\te_j] = \sum_k \tC_{ijk} \te_k
Now construct a frame on {S^d} by

\displaystyle e_d = \partial/\partial\theta, e_j=(1/f(\theta)) \te_j.
The nonvanishing components of the Christoffel symbols of the Levi Civita connection on {S^d} are

\displaystyle \omega_{ijk} = \frac{1}{f} \tilde{\omega}_{ijk}, \omega_{idk}=\frac{f'}{f}\delta_{ik}=-\omega_{ikd}.
Now a spin connection on {S^d} is induced from the Levi-Civita connection, and the covariant derivative is described by

\displaystyle \nabla_a \psi = e_a \psi -\frac{1}{2}\omega_{abc}\Sigma^{bc}\psi
and the Dirac operator is defined as

\displaystyle D\psi = \Gamma^a\nabla_a \psi.
One can derive the expression: \begin{align*} D\psi &= (\partial_\theta + \frac{d-1}{2}\frac{f’}{f}\Gamma^d\psi + \frac{1}{f}\Gamma^i(\te_i – \frac{1}{2}\tilde{\omega}_{ijk}\Sigma^{jk})\psi
&= (\partial_\theta + \frac{d-1}{2}\frac{f’}{f}\Gamma^d\psi + \frac{1}{f}\left( {cc} 0 & i\tilde{D}
-i\tilde{D} & 0 \right)\psi \end{align*} Now consider the eigenvalue equation

\displaystyle D\psi = i\lambda\psi
by writing {\psi = (\phi_+,\phi_-)^t} so that the eigenvalue equation can be rewritten \begin{align*} (\partial_\theta + \frac{d-1}{2}\phi_- + \frac{1}{f} i\tilde{D}\phi_- = i\lambda\phi_+
(\partial_\theta + \frac{d-1}{2}\phi_+ – \frac{1}{f} i\tilde{D}\phi_+ = i\lambda\phi_- \end{align*} Eliminating {\phi_-} gives the second order equation

\displaystyle \partial_\theta + \frac{d-1}{2}\frac{f'}{f})^2 + \frac{1}{f^2}\tilde{D}^2 \pm \frac{f'}{f^2} i\tilde{D})\phi_\pm = -\lambda^2\phi–\pm.
Camporesi and Higuchi assume solutions {\chi^{\pm}_{lm}} are available for

\displaystyle \tilde{D} \chi^{\pm}_{lm} = \pm i (l+\rho)\chi^{\pm}_{lm}
are available for the lower dimensional Dirac operator and separate variables \begin{align*} \phi_{+nlm} &= \phi_{nl}(\theta)\chi_{lm}^{(-)}(\Omega)
\phi_{+nlm} &= \psi_{nl}(\theta)\chi_{lm}^{(+)}(\Omega) \end{align*} They find that

\displaystyle \phi_{nl}(\theta) = (\cos \frac{\theta}{2})^{l+1}(\sin \frac{\theta}{2})^l P^{(d/2+l-1,d/2+l)}_{d-l}(\cos \theta),
and

\displaystyle \psi_{nl}(\theta) = (\cos \frac{\theta}{2})^{l}(\sin \frac{\theta}{2})^{l+1} P^{(d/2+l,d/2+l-1)}_{d-l}(\cos \theta)

in terms of Jacobi polynomials.

13. Can you hear the shape of the universe?

If the universe is a compact four dimensional riemannian manifold, then we should be able to hear the shape of the universe. We took the hydrogen energy spectrum from the NIST database and fit a series of models which correspond to the eigenvalues of the Laplacian on a sphere of dimension {D}, which are {k(k+D-1)} and obtained fits to the data with R2 exceeding 0.99 and plotted the mean absolute residual divided by mean absolute energy level and obtained the following.

\includegraphics[scale=0.5]{SphericalHarmonicsFitsRelativeResiduals.jpg}

The fits were by linear models:

\displaystyle A_D: E_n = a + b( 1/S_D(n) - 1/S_D(n+1) )
where {S_D(n) = n(n+D-1)} which are the eigenvalues of the Laplacian on a {D}-dimensional sphere. From the plot it is clear that among the spherical harmonics models of different dimensions, the fit by that of dimension 4 is the best.

14. Implications of the shape of the universe being a four dimensional sphere of radius {O(1/h)}

A sphere has the property that every geodesic is a great circle of fixed length. This implies that only possible wavelengths are {2\pi R/N} for integral {N}, and therefore only possible frequencies are integral multiples of {c/2\pi R} where {c} is the speed of light. Therefore one does not require a quantum hypothesis for quantization of energy if {E = h\nu} holds.

Another important feature of a four dimensional sphere is that there are no nonzero spinors satisfying {D\psi = 0} where {D} is the Dirac operator. This fact follows from a fairly standard argument of applying the Weitzenbock formula

\displaystyle D^2 = \nabla^*\nabla + C(K)
where {C(K)} is a universal constant multiple of the scalar curvature which is nonnegative in this case. This implies that there are no massless fermions of a four dimensional sphere.

The natural electromagnetism on the four dimensional sphere is {SU(2)} electromagnetism as the four sphere is identical to the the quaternionic projective line and the projection {\mathbb{H}^2\rightarrow S^4} restricted to pairs of quaternions with unit norm is the Hopf fibration with fiber {SU(2)}. If we accept the Einstein equations as providing a description of gravity, then the unification of gravity and electromagnetism amounts to providing a coherent electromagnetism via connections on this Hopf fibration satisfying the generalized Maxwell’s equations.

In the 1970s Sir Michael Atiyah and others have studied electromagnetism on a four dimensional sphere for mathematical interests, and Atiyah, Drinfeld, Hitchin, and Manin had constructed solutions of instanton equations.

15. Gauge theories for forces

The gravitational field equations for \a three dimensional physical universe we can identify with the Ricci curvature equation for the submanifold as in ((1)) and following Yang and Mills we can recast electromagnetism and other forces via gauge theory. The following explanatory material is taken from 1999 notes by G. Svetlichny and a 1992 paper of D. Gross describing Yang Mills theory. The currently accepted Standard Model contains three dynamical subtheories for quantum electrodynamics, quantum chromodynamics and electroweak theory all based on the gauge principle, that a theory should be invariant under local phase transformations. Note that neither this principle nor gauge theory generally restricts the base space of the universe, although the accepted Standard model is based on a flat three dimensional base space. Therefore the two questions of fundamental interest are: can we recover the results of the Standard Model if the base space is changed to a four-sphere, and whether the geometry of a four-sphere simplifies to a classical theory.

The first gauge theory is classical Maxwell electromagnetism. In appropriate physical units the Maxwell’s equations are: \begin{align*} \nabla\cdot B &= 0
\nabla\times E + \frac{\partial B}{\partial t} &= 0
\nabla\cdot E &= \rho
\nabla\times B – \frac{\partial E}{\partial t} &= J \end{align*} From the homogeneous Maxwell’s equations we conclude that there is function {V} called the scalar potential and a vector field {A} called the vector potential such that {B = \nabla \times A} and {E = -\nabla V - \partial A/\partial t}. Consider the differential 1-form {A = -V dt + A_x dx + A_y dy + A_z dz}. A direct calculation of the field strength {F=dA} gives the coefficients {F_{\mu\nu}}. The homogeneous Maxwell’s equations are {dF=0} and for the inhomogeneous ones, one introduces {j = -\rho dt + J_x dx + J_y dy + J_z dz} and the inhomogeneous Maxwell’s equations are {\delta F = *d* F= j}.

Recall that C. N. Yang and R. Mills introduced nonabelian gauge theories in the 1950s partly as a solution to understanding the surge of discoveries in particle physics at the time. We can reformulate Maxwell’s electromagnetism in terms of the electromagnetic potential {A_\mu} and the electromagnetic field {F_{\mu\nu}}. The standard formulation of Maxwell’s equations produces an abelian {U(1)} gauge theory. Yang and Mills produce the mechanism for a nonabelian gauge theory where the potential {B_\mu} that is a matrix

\displaystyle B_\mu = \frac{1}{2}\sigma_a B^a_\mu
where {\sigma_a} are Pauli matrices. The differences from the abelian case of potential {A_\mu} is that the field strength

\displaystyle F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu
is replaced by

\displaystyle F_{\mu\nu} = \partial_\mu B_\nu - \partial_\nu B_\mu + ig [B_\mu,B_\nu]
Such gauge theories for forces are possible on arbitrary geometric manifolds mathematically by considering a potential to be a connection on a principal {G}-bundle and considering the curvature of the connection to be identical to the field strength.

John Preskill’s Caltech course notes for nonabelian gauge theory provides us with some of the basic issues regarding nonabelian gauge theories. We are interested in gauge theories on a fixed compact manifold, but the basic issues are not different from the standard treatment in the case of a noncompact universe. The minimal coupling prescription is an algorithm to promote a global {U(1)} symmetry to a local {U(1)} symmetry. In that case, one replaces {\partial_\mu \phi} with {D_\mu \phi} where {D_\mu = \partial_\mu - igA_\mu}, and the local transformation are \begin{align*} A_\mu \rightarrow A_\mu + \partial_\mu \omega(x)
\phi(x) \rightarrow \exp(-i e \omega(x)) \phi(x) \end{align*} In the nonabelian case {SU(2)} we may write an infinitesimal transformation as

\displaystyle \Omega^{-1} = 1 - ig\omega^a T^a
where {T^a=\frac{1}{2} \sigma^a} in terms of Pauli matrices {\sigma^a}. Then consider the local transformation

\displaystyle q(x) \rightarrow (1 - i e \omega(x)) q(x)
under which

\displaystyle \partial_\mu q(x) \rightarrow (1-ie\omega(x)) \partial_\mu q - i e \partial_\mu\omega(x) q(x).
In order to construct an invariant Lagrangian we need to cancel the second term by some means. Consider {A_\mu(x) = A_\mu^a(x) T^a} hermitian and traceless and consider {D_\mu = \partial_\mu - i e A_\mu} with the transformation above leads to

\displaystyle D_\mu q \rightarrow (1-ig\omega) D_\mu q + ig( [D_\mu,\omega] + \delta A_\mu)q
and the {D_\mu q} transforms nicely if the second term vanishes or

\displaystyle \delta A_\mu = [ D_\mu, \omega ] = \partial_\mu \omega + ig[A_\mu,\omega]
Under {SU(2)} transformations, we have \begin{align*} q \rightarrow \Omega^{-1} q
ig A_\mu \rightarrow \Omega^{-1} ig A_\mu \Omega – (d \Omega^{-1})\Omega
D_\mu q \rightarrow \Omega^{-1} D_\mu q
F_{\mu\nu} \rightarrow \Omega^{-1} F_{\mu\nu} \Omega \end{align*}

Purely mathematical study of Yang-Mills theory had led to the results for which Simon Donaldson had received his Fields Medals, which are far outside the scope of this note. We can follow T. H. Parker’s study of gauge theories as classical field theories in four dimensions in order to introduce the correspondence between the physical concepts and their mathematical representations. A connection {A} on a principal bundle {P} over a riemannian four manifold {M} is a Yang-Mills connection when it is a critical point of the functional

\displaystyle A \rightarrow \int_M |F_A|^2
Such connections are representatives of forces. The particles are represented by sections {\varphi} of an associated vector bundle with the action

\displaystyle S(A,\varphi) = \int_M |F_A|^2 + |D_A \varphi|^2 - m^2 |\varphi|^2,
where {m} is the mass of the particle. T. H. Parker \cite{Parker:82} extends a seminal result of K. Uhlenbeck who had shown that the Yang-Mills fields cannot have isolated singularities to coupled Yang-Mills equations, both for fermions, based on the Dirac operator acting on bundle-valued spinors and for bosons based on the bundle Laplacian. Thus for compact orientable riemannian manifolds gauge theory has been studied and the mathematical framework exists with reasonable results. The central issue of this note is to point out the evidence that the actual physical universe could be described by this mathematical formalism.

In details taken from T. Parker, let {pi:P\rightarrow M} be a principal bundle with compact structure group {G} and {\rho:G\rightarrow \Aut(V)} be a unitary respresentation with associated vector bundle {E = P \times_\rho V} and let {W} be any bundle associated to the frame bundle of {M}. Let {\cal{M}} be the set of metrics on {M}, let {\cal{A}} be the connections on {P} and let {\cal{E} = \Gamma(E\otimes W)} so that we can write an action

\displaystyle S(g,A,\phi) = \int_M L(g,A,\phi)
for {L} a 4-form constructed from {g}, {A} and {\phi}. An automorphism {f} of {P} is a map for which {f(xg^{-1}) = f(x) g^{-1}}. The subgroup of orientation preserving automorphisms that project to the identity map on {M} can be identified with {P\times_{Ad} G}, and these automorphisms are called gauge transformations {\cal{G}}. The Killing form provides an invariant metric {h} on the adjoint bundle {\cal{g}} and a hermitian metric on {E}.

Consider Lagrangians {L} with the properties of regularity — that in local coordinates {L} should be a universal polynomial in {g}, {h}, {\Gamma} (the Christoffel symbols of {A}), {(\det g)^{-1/2}}, {(\det h)^{-1/2}} and their derivatives; naturality under bundle automorphism {f} and conformal invariance. Then invariant theory can be used to determine the possibilities for {L}. Naturality under orientation preserving diffeomorphisms of {P} implies, by {SO(4)} invariant theory that

\displaystyle L = a_1 |s|^2 + a_2 |B|^2 + a_3 |W^+|^2 + a_4 | W^- |^2 + a_5 \Omega \wedge \Omega + a_6 \Omega \wedge * \Omega,
The Yang-Mills action is

\displaystyle S(g,A) = \int_M \Omega \wedge * \Omega.
The space of connections {\cal{A}} carries a natural differentiable structure on which this is a smooth function, and whose critical points are Yang-Mills fields. In the vector bundle formalism, if {E} is a vector bundle associated to {P} by a locally faithful orthogonal representation of {G}, each connection on {P} corresponds to a differential operator {\nabla:\Gamma(E)\rightarrow \Gamma(\Lambda^1\otimes E)} where

\displaystyle \nabla(f \varphi) = df \otimes \varphi + f \otimes \nabla\varphi
The curvature field

\displaystyle \Omega_{X,Y} = \nabla_X \nabla_Y - \nabla_Y \nabla_X - \nabla_{[X,Y]}
is a two form with values in the skew symmetric endomorphism of {E}. The norm of {\Omega} at a point {x} is given by

\displaystyle \| \Omega \|^2 = \sum_{i<j} \| \Omega_{e_i,e_j} \|^2
where {(e_1,\dots,e_n)} is an orthonormal frame, and the inner product on skew symmetric endomorphisms of {E} is

\displaystyle \langle A, B \rangle = -1/2 \trace(A\circ B)
For any bundle with connection {F}, there is a sequence of differential operators {d^\nabla : \Gamma(\Lambda^p\otimes F) \rightarrow \Gamma(\Lambda^{p+1}\otimes F)} given by

\displaystyle (d^\nabla \varphi)(X_0,\dots,X_p) = \sum_{k=0}^p (-1)^k (\nabla_{X_k}\varphi)(X_0,\dots,\Hat{X_k},\dots,X_p).
We can define adjoints {\delta^\nabla} using the formula

\displaystyle (\delta^\nabla\varphi)(X_1,\dots,X_p) = -\sum_{k=1}^n (\nabla_{e_k}\varphi)(e_k,X_1,\dots,X_p)
The first variation formula for the Yang Mills functional shows that {\nabla} is a critical point if and only if {\delta^\nabla \Omega = 0}, which because of the Bianchi identity {d^\nabla \Omega = 0} is equivalent to {\Delta^\nabla \Omega = 0} where {\Delta^\nabla = d^\nabla \delta^\nabla + \delta^\nabla d^\nabla}.

For four dimensional base space, two-forms break up by the eigenspaces of the {*}-operator. Then {\Omega = \Omega_+ + \Omega_-} is harmonic if and only if both components are harmonic. Bourgaignon, Lawson and Simon show that on a four sphere when {G} is {SU(2)} or {SU(3)} any weakly stable Yang-Mills field is either self-dual or anti-self-dual.

In this setting we have the coupled fermion equations \begin{align*} (d^\nabla)^*\Omega = -1/2\sum \langle \varphi, e^i\cdot\rho(\sigma^\alpha)\varphi\rangle \sigma_\alpha \otimes e_i
D\varphi = m\varphi \end{align*} and the coupled boson equations \begin{align*} (d^\nabla)^*\Omega = J = -\Re \sum \langle \nabla_i \varphi, \rho(\sigma_\alpha)\varphi\rangle \sigma_\alpha \otimes e_i
\nabla^*\nabla \varphi = (s/6)\varphi + a|\varphi|^2\varphi + m^2\varphi. \end{align*}

16. The zero modes and lower bound of the first eigenvalue of a twisted Dirac

On general four dimensional spin manifolds one has the Dirac operator on spinors via the Spin connection. When the scalar curvature is positive, this Dirac operator has zero kernel. On the other hand, one can consider a self dual connection {A} on an auxiliary principal {G}-bundle {P} and using a unitary representation of {G} on a complex vector bundle {E} consider the covariant derivative {\nabla_A}. The twisted Dirac operator is then {D_A:\Gamma(V\otimes E)\rightarrow \Gamma(V\otimes E)} which is {\nabla = \nabla^S\otimes 1 + 1 \otimes \nabla_A} followed by clifford multiplication. This twisted Dirac operator has kernel dimension dictated by the topology of {E} and

\displaystyle \dim \ker D_A = \frac{1}{2} p_1(E)
This is not generally zero even on a four sphere which has constant positive scalar curvature. Atiyah, Hitchin Singer give a Weitzenbock formula and show that when {A} is selfdual, {M} is selfdual and has positive scalar curvature then one can show vanishing of {\psi} with {D_A\psi=0}. Helga Baum has shown that the first eigenvalue of the twisted Dirac operator for positive scalar curvature four manifold satisfies

\displaystyle \lambda_1 \ge sqrt{R_0/3}
and that this On a four sphere universe this provides a gap between the zero and the first eigenvalue for the twisted Dirac operator, which translates to a mass gap that does not translate to {\mathbf{R}^4}. This is an interesting observation as the question of mass gap for nonabelian gauge theories has remained open. This lower bound due to Helga Baum and implicit in Atiyah-Hitchin-Singer is geometric rather than topological. On the other hand, there is a positive dimensional kernel, so for {S4} we have a fairly clear answer for the question of the twisted Dirac spectrum.

The mechanism for nonabelian gauge theories translates without problems to a fixed spherical space. The Standard Model is a {U(1)\times SU(2)\times SU(3)} gauge theory that was successful in uniting the nuclear and electromagnetic forces by 1973. But knowledge that the universe is a four dimensional stationary sphere allows us to re-examine whether electromagnetism itself is an {SU(2)} gauge theory and thus conjecture that there are further simplifications possible, such as realizing that since a classical electron orbiting a classical proton in four dimensions need not be unstable, and therefore it is possible to seek a single force governing nature which are thought to be separate in the nuclear forces.

In the case where the space manifold is the four-sphere, we have a natural principal {SU(2)}-bundle which is the Hopf fibration {S^7\rightarrow S^4}. It is natural to ask what the relation is between electromagnetism in the material universe, which we can identify with a three-dimensional submanifold {M} which from the electromagnetic theory developed from Maxwell, we consider to have {U(1)} symmetry, to {SU(2)} electromagnetism that can be said to arise naturally from the Hopf fibration.

17. The Standard Model in a 4-sphere universe

The Kaluza-Klein theories led to work on adapting the Standard Model to higher dimensions. An example of this approach is the one used by Pomarol and Quiros from 1998 \cite{PomarolQuiros:1998}. In five (including time) dimensions, the vector supermultiplet {(V_M, \lambda^i_L,\Sigma)} of an {SU(N)} gauge theory consists of a vector boson {V_M} a real scalar {\Sigma} and two bispinors {\lambda^i_L} all in the adjoint representation of {SU(N)}. The 5D Lagrangian is given by

\displaystyle \mathcal{L} = \frac{1}{g^2}\trace{ -\frac{1}{2}F^2_{MN} + |D_M\Sigma|^2 + i\bar{\lambda}^i \gamma^M D_M \lambda^i - \lambda^i[\Sigma,\lambda^i]}, \ \ \ \ \ (3)
where {\lambda^i} is the symplectic-Majorana spinor {(\lambda^i_L,\epsilon^{ij}\bar{\lambda}^j_L)T}. The 5D matter supermultiplet {(H_i,\Psi)} consists of two scalar fieds {H_i} and a Dirac spinor {\Psi=(\Psi_L,\Psi_R)^T}. The Lagrangian for the matter supermultiplets interacting with the vector supermultiplet is given by \begin{align*} \mathcal{L} &= |D_M H^a_i|^2 + i \bar{\Psi}_a \gamma^M \Psi^a + hc) – \bar{\Psi}_a \Sigma \Psi^a
&- \bar{H}^i_a \Sigma^2H^a_i – \frac{g^2}{2} \sum_{m,\alpha} [ \bar{H}^i_\alpha(\sigma^m)^j_i T^\alpha H^a_j]^2, \end{align*} where {\sigma} are Pauli matrices. One expects that the same type of analysis as Pomarol and Quiros can be applied to produce a Standard Model on the 4-sphere.

18. Quaternion projective spaces in complex matrices

Following Furutani and Tanaka, who provided a Kähler structure on the punctured cotangent bundle on quaternionic projective spaces, we can embed general quaternionic projective spaces {P^n\mathbb{H}} into spaces of complex matrices as follows.

Let {\rho:\mathbb{H}\rightarrow M(2,\mathbb{C})} be the representation given by

\displaystyle \rho(p) = \rho(z + wj) = \begin{array}{cc} z & w\\-\bar{w}&\bar{z} \end{array} \ \ \ \ \ (4)
Introduce a metric on {P^n\mathbb{H}} via the Hopf fibration {\pi:S^{4n+3}\rightarrow P^n\mathbb{H}} where {p\rightarrow[p]} for {p = (p_0,\dots, p_n) \in \mathbb{H}^n}. The embedding of {P^n\mathbb{H}} into {M(2n+2,\mathbb{C})} is given by

\displaystyle [p]\rightarrow (P_{ij}) = ( \rho( p_i \bar{p}_j))
The image of this embedding consists of {P\in M(2n+2,\mathbb{C})} satisfying {P^2=P, P^*=P, \trace(P)=2, PJ=J\leftexp{t}{P}}. The canonical one-form can be written as

\displaystyle \theta_{P^n\mathbb{H}}(P,Q) = \frac{1}{2} (d\embedMap)^* \trace(Q dP)
and so the symplectic form can be written

\displaystyle \omega_{P^n\mathbb{H}} = \frac{1}{2} (d\embedMap)^* \trace(dQ\wedge dP).
The symplectic form allows us to specify classical mechanics on quaternionic projective spaces and we may focus attention to the case of {P^1\mathbb{H} = S^4}.

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