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## GRIGORYAN’S METHOD FOR GAUSSIAN UPPER BOUNDS TO TIME-FRACTIONAL DIFFUSIONS

Alexander Grigoryan’s method (GrigoryanGaussianBounds)for Gaussian upper bounds on the heat kernel translates on diagonal bounds $p(t,x,x) \le C/f(t)$ into Gaussian upper bounds using riemannian distance function $d(x,y)$ which was originally observed by Varadhan in 1966, that heat kernels should have $\lim_{t\rightarrow 0} 2t \log(p(t,x,y)) = -d^2(x,y)$.  What happens when we replace the parabolic equation of the heat equation with a time-fractional equation?  If we use $D^\beta_t$ the Caputo derivative

$D^\beta_t f = \int_0^t (t-s)^{-\beta} f'(s) ds$

then there is a well-known relationship between solution of the Cauchy problem for the non-fractional standard Cauchy problem for the parabolic equation and the fractional one: if $u(t,x)$ solves the nonfractional problem $\partial_t u = L u$ and $u(0,x) = \phi(x)$ with $L$ a second order differential equation such as the case of interest on riemannian manifolds the Laplace-Beltrami operator, and $v(t,x)$ is the solution of the corresponding fractional equation $D^\beta_t v = L v$ then

$v(t,x) = \int_0^{\infty} u( (t/s)^\beta, x) g_{\beta}(s) ds$

First, let’s get some more information regarding the density function $g_{\beta}(s)$.  It is the inverse stable subordinator density whose Laplace transform is $exp(-cs^\beta)$.  This has been precisely been analyzed by Mainardi-Luchko-Pagnini (fundamentalSolutionSTfrac):

Suppose then that $p(t,x,x)$ is the parabolic kernel and $q(t,x,x)$ is the fractional parabolic kernel.  Assume $f( (t/s)^\beta ) \le A s^{-\beta} f(t^\beta)$ as a convenient regularity condition and $p(t,x,x) \le C/f(t)$.  Then

$q(t,x,x) = \int_0^{\infty} p(t,x,x) g_{\beta}(s) ds$

$\le C/f(t^\beta) \int_0^\infty A s^{-\beta-1} M_{\beta}(cs^{-\beta}) ds$

$\le C (1/\Gamma(1-\beta)) 1/f(t^\beta)$

The last line follows from changing variables $u=cs^{-\beta}$ and integrating the Wright function using the (4.28) formula from the Mainardi-Luchko-Pagnini paper.

## ELECTROMAGNETISM IS PROBABLY NOT A U(1) GAUGE THEORY

The Bohm-Aharanov effect can be simply understood to mean that electromagnetic potential has physical meaning.  It is well-known that Maxwell’s equations are invariant under phase translations.  Now in S4(1/h) universe, take a small neighborhood of the physical universe (a hypersurface).  You have approximately a trivial circle bundle if the neighborhood is sufficiently small and close enough to flat; the fibers are circles of radius $1/h$.  One possible explanation of Bohm-Aharanov is that electromagnetism is not in fact described by Maxwell’s equations which are gauge invariant but described by some variation of it which are gauge invariant only when the hypersurface is sufficiently nice.  This is an interesting technical question.  In any case, there is no reason to exclude the possibility of a macroscopic fourth space dimension that is ‘purely electromagnetic’ in the same sense that electromagnetic potentials are real in Bohm-Aharanov effects.  It is claimed that Kaluza-Klein type theories must have very tiny extra dimension by experiments in particle accerators but this is not quite right.  A purely electromagnetic extra dimension can describe the phases of electromagnetism while not being reduced to the type of extra physical dimension that was tested.

## ELECTROMAGNETISM IS PROBABLY NOT A U(1) GAUGE THEORY

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### zulfikar.ahmed@gmail.com<zulfikar.ahmed@gmail.com>

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Gauge theories originate with Hermann Weyl’s 1918 attempt to produce a unified field theory combining gravity and electromagnetism followed by careful work with Pauli who originally had scoffed at his attempt to transition from mathematics to physics.  A standard account is attached.  The main idea is that if M is a three dimensional manifold, the electromagnetic potential will be a connection A (1-form) on a principal circle-bundle, the electromagnetic field will be F=dA which is then gauge-invariant since translation of A by any exact 1-form leaves F invariant.

Now consider a different picture of the universe:  M is a hypersurface of S4(1/h) and electromagnetic potential is not a connection on a circle bundle but represents a normal variation of the physical universe: consider the vector potential A as taking values in the normal bundle of M which is topologically trivial, which is equivalent to a metric deformation of M by exp(A).  Now the electromagnetic field F is essentially the second fundamental form of the deformed metric that is not ‘gauge invariant’ at all.  Note that the Bohm-Aharanov effect shows that A is ‘physical’ and therefore can be thought as evidence against gauge invariance of electromagnetism, so in this picture we have an electromagnetism that is not gauge-invariant but similar to U(1) gauge theory picture in that there is a pseudo circle bundle — the normal circle of radius 1/h at each point in M from the embedding M in S4(1/h).  It is not a real circle bundle obviously because there can be intersections of ‘fibers’ over different point.

Now the spinor bundle  P of S4(1/h) restricted to any hypersurface is canonically identified in a standard manner (see O. Hijazi or T. Friedrich): the restriction of P to M is naturally isomorphic to either the intrinsic spinor bundle or its double.  The Lagrangian for a fermion in terms of the Dirac operator D on the hypersurface is

L = <psi, D psi> – m<psi, psi>

with the inner product in the appropropriate metric.  It is possible that the anomalous magnetic moment of the electron could be produced in this sort of setting without appeal to complicated computations — an example attempt is attached.
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## ROBERT REILLY VARIATION FORMULA TO DETERMINE S4 GRAVITATIONAL FIELD EQUATIONS

The standard Einstein-Hilbert variation involves all possible deformations of a metric on a three-manifold; it was David Hilbert who had derived the Euler-Lagrange equations for the action that is the integral of scalar curvature.  For S4 physics, we want to consider the variation of the integral of the scalar curvature only by deformations that are still hypersurfaces.  In general this is a fairly messy computation (see for example Yano: ).  However, it happens that the the scalar curvature for a hypersurface of a four-sphere can be written as

$S = \bar{S} + cRic(N,N) + c(9H^2 - ||A||^2)$

where $\bar{S}$ is the constant scalar curvature of the ambient four-sphere, $H$ is the mean curvature and $\| A \|^2$ is the length of the second fundamental form.  For this form, we have a functional that can be expressed in terms of the elementary symmetric polynomials of the principal curvatures, i.e. the eigenvalues of the second fundamental form.  Write $S_0 = 1, S_1=H, S_2 = k_1 k_2 + k_2 k_3 + k_1 k_3, S_3 = k_1 k_2 k_3$.  For functions of these elementary symmetric functions, the Euler-Lagrange equation for the action $f(S_1,S_2,S_3)$ have been determined by Robert Reilly (see Reilly)

We have $f(S_1,S_2) = aS_1^2 + bS_2$ and the Euler-Lagrange equation will be

$(2aS_1)_{,ij}T^{ij}_0-S_1 f + (S_1^2 - 2S_2)2 a S_1 + (S_2 S_1 - 3 S_3) b + 2 a S_1 3c + 2bc S_1 + 6 ac S_1 + 6 c a S_1 + 2bc S_1 = 0$

Simplifying, let $x=S_1$ and $y=S_2$ then

$(2ax)_{,ij} T^{ij}_0 = (4a+b)y - c(4b+6a) - ax^2$

## RESTRICTION OF REAL KILLING SPINORS TO HYPERSURFACES

A general fact about spin structures on hypersurfaces $M\i N^{n+1}$ is that the intrinsic spin structure is related to the restriction of the spin structure on the ambient manifold as follows: $\Sigma N|_M = \Sigma M$ when the dimension of $M$ is even; otherwise one takes the splitting $\Sigma N = \Sigma^+ N \oplus \Sigma^- N$ based on the $+/-$-eigenvalues of the action of the complex volume form $i^{[n+1/2]}\omega$ and finds $\Sigma M = \Sigma^+ N$.  One can then check that spinor fields $\psi$ on $N$ restrict to $M$ via $\psi\rightarrow \psi^*=\nu \cdot \psi$.  The spinorial Gauss formula is

$(\nabla^N \psi)^* = \nabla \psi + \frac{1}{2} h(X)\cdot \psi$

where $h(X)$ is the second fundamental form.  Real spinor fields with parameter $\lambda$ on the ambient manifold satisfy $\nabla^N_X \psi = \lambda X\cdot \psi$.  These exist with $\lambda = \pm h$ (here we abuse notation and mean a Planck constant for $h$) for Slatex S^4(1/h)\$. Let the energy-momentum tensor for a spinor field be defined as

$T^{\psi}(X,Y) = Re\langle X\cdot\nabla_Y \psi + Y\cdot \nabla_X \psi, \psi / |\psi|^2 \rangle$

We want to show that when

$2 T^{\psi} = h + 6 \lambda$

For a tangential orthonormal frame $e_1, e_2, e_3$ of the three-manifold, we apply the spinorial Gauss formula with $X=e_i$, Clifford multiply with $e_j$ and take inner product with $\psi$.  We obtain

$Re\langle e_j \nabla^N_{e_i}\psi+ e_i \nabla^N_{e_j}\psi, \psi \rangle = Re\langle e_j \nabla_{e_i} \psi + e_i \nabla_{e_j} \psi, \psi\rangle + \frac{1}{2} ( e_j \cdot h(e_i) + e_i \cdot h(e_j))\cdot \psi, \psi\rangle$

On the first part use the real Killing spinor condition and on the right use the symmetry of $h(\cdot)$ and on both sides use the Clifford relation $e_i e_j + e_j e_i = \delta_{ij}$ and sum over $i,j=1,2,3$.  The result is

$-3\lambda |\psi|^2 = T^{\psi} |\psi|^2 - \frac{1}{2} \sum_{i,j} h_{ij} |\psi|^2$

The importance of this computation is that it tells us that the energy-momentum tensor defined in terms of the spinor field can be expressed as a second fundamental form with the lambda term.  The lambda term will directly correspond to a cosmological constant in Einstein’s gravitational field equations for example in the approach of Friedrich and Kim on solving Dirac-Einstein equations (MorelEnergyMomentumSecondFundamentalForm2003EinsteinDiracRiemannianSpin)

## GEOMETRIC QUANTIZATION FOR S4 PHYSICS

I happen to have a firm conviction that quantum mechanics is incorrect as a description of the actual universe regardless of its empirical accuracy.  On the other hand, it is probably a theory that is a correct approximate linear model.  Now geometric quantization is an approach to link classical to quantum mechanics.  In the axiomatic geometric quantization, functions $f\in C^{\infty}(M)$ are mapped to operators $\hat{f}$ satisfying some axioms: (a) linearity, (b) constant functions get mapped to constant times identity, (c) Poisson brackets map as $\{ f_1, f_2 \} = f_3$ go to $[\hat{f_1},\hat{f_2}] = -i\hbar \hat{f_3}$.  Now in geometric quantization on a line bundle $B\rightarrow \Simga$ for a symplectic $\Sigma$ with $\hat{f} s = -i\hbar\nabla_{X_f} s + f s$.  Let’s consider the physical space $M\i S^4(1/h)$ and $\Sigma = T^*M$ which is symplectic.  There’s a natural line bunlde for $M$ which is the normal bundle of the embedding, so we can get the geometric quantization ‘prequantum bundle’ here canonically.

See Ritter for the geometric quantization fundamentals:  RitterGeometricQuantization.

The key issue in geometric quantization is the need for polarizations.  Now uncertainty principles exist for compact riemannian manifolds as you can find in the monumental thesis of Erb (UncertaintyPrinciplesRiemannianManifolds).  So the basic idea is that geometric quantization specialized to a hypersurface of the 4-sphere might allow us to dispense with quantum mechanical interpretation altogether and recognize that all quantum mechanical features arise from the geometry of the three dimensional physical universe embedded in an S4 universe.

## A PRIORI ESTIMATES FOR FRACTIONAL HEAT EQUATION

For the classical heat and parabolic equations, a priori estimates are extremely standard and can be found in Chapter 7 of L. C. Evans’ book and many others (evans-pde)  For time-fractional heat equation this I believe is new but not very different.  The essential difference is that one uses a weighted $L^2$ norm in the time direction.  Let $0<\alpha<1$ and let $I^{\alpha}_t$ be the fractional integral

$I_t^{\alpha} f(t) = \int_0^t (t-s)^{\alpha-1} f(t)$

which is the inverse of the fractional derivative $D^{\alpha}_t$.  Consider

$D^{\alpha}_t u - \Delta u= f$ with initial data $u(\cdot,0)=g$.  Rigorous a priori estimates can use a finite dimensional approximation using eigenfunctions of Laplacian as a basis on a bounded domain in $\mathbf{R}^n$ as in Evans’ book but the key computation is the following where we assume that there is a smooth solution with sufficiently fast decay as $|x|\rightarrow \infty$.

First, let $X=\mathbf{R}^n$ and

$\int_X f^2 dx = \int_X (D^{\alpha}_t u)^2 - 2\Delta u D^{\alpha}_t u + (\Delta u)^2 dx$

Now $D^{\alpha}_t (|Du|^2) = 2 Du DD^{\alpha}_t u$ since we are using the Caputo derivative.  So multiply $D^{\alpha}_t u - \Delta u = f$ with a test function and integrate by parts and set the test function to $u$:

$\frac{1}{2} D^{\alpha}_t \int_X u^2 dx + \int_X |Du|^2 = \int_X fu dx$ (1)

Now apply $I^{\alpha}_t$ first to the term on the right and use Cauchy-Schwartz:

$I^{\alpha}_t \int_X fu dx \le (I^{\alpha}_t \int_X f^2)^{1/2}(I^{\alpha}_t \int_X u^2)^{1/2}$

$\cdot \le \frac{1}{4 \epsilon} I^{\alpha}_t \int_X f^2 + \epsilon C(T) \int_X u^2 dx$

where $C(T) = I^{\alpha}_T(1) = \int_0^T (T-s)^{\alpha-1} ds$.  Choose $\epsilon C(T) = 1/4$ and plug in the estimate into (1).  Obtain

$\frac{1}{4}\sup_{[0,T]} \int_X u^2 + I^{\alpha}_t \int_X |Du|^2 \le C(I^{\alpha}_t\int_X f^2 dx + \frac{1}{2} \int_X g^2 dx)$

So what is happening here that is different from the standard case is that instead of the usual Lebesgue measure for time we have a weighted time measure; otherwise the argument is the same.  Now the problem is that arguments involving the Gronwall lemma are a bit more complicated but there exists a Gronwall lemma proved for time-fractional functions in 2006 which could apply: fractionalGronwallInequality2006.

For $0 < \alpha < 1$ and bounded domains in $\mathbf{R}^N$ the hard analysis for divergence form time-fractional diffusion equations
${}^{RL}D^\alpha_t u - div( A(t,x)Du) = 0$
along with a weak Harnack inequality and applications to a maximum principle and a Liouville-type theorem is proven by Rico Zacher (ZacherFracHarnack2010).  He follows the Moser iteration technique and uses Yosida approximation scheme for the Riemann-Liouville fractional derivative.  His methods yield the same results for the Caputo derivative which differs from the RL derivative by a fixed function of time only:  ${}^{RL}D^{\alpha}_t u = {}^CD^{\alpha}_t u - g(t) u(0)$.  For $1<\alpha<2$ existence and uniqueness for the fractional wave equation are addressed using entirely different techniques by Kian and Yamamoto (Kian-Yamamoto-SemilinearFracWave2015)  The latter authors work in the Hilbert space spanned the eigenfunctions of the second order operator and prove a Strichartz estimate directly using estimates of the Mittag-Leffler function.