On a Final Theory of Mathematics and Physics by Michael Rios

To introduce more physical ideas into this subject, in particular magic SUGRA I have been musing over this for a while. I think it is important to keep in contact with physical ideas. I do at the end here consider the prospect that this connects with J3 and E6. This has to do with the general uncertainty principle and the tension on the brane of our universe. If the tension is zero then the fluctuations in the spacetime are tiny. However, if we are observing spacetime under a high magnification of a Heisenberg microscope then we are imposing high transverse momenta and the tension is huge. Under those conditions the fluctuations are quite large.

We might think of the situation like this. Suppose the uncertainty principle is

О"pО"x = Д§ + F(x)О"xО"t.

This is a version of the uncertainty principle modified for string theory

О"pО"x = Д§ + ОјО"x^2,

where ОјО"x = F(x)О"x/c = F(x)О"t. The constant Ој has been replaced with a force that depends upon distance or a region in space/spacetime. I have though a term F(x) that is written to indicate force. This leads to the spread in the width of the uncertainty

О"x = О"pc/2F(x) +/- c sqrt{О"p^2 - 4Д§F(x)/c}/2F(x)

From О"pО"x = Д§ + F(x)О"xО"t we can divide by О"t with О"x/О"t ~ c to get

F(x)О"x = О"pc - Д§/О"x

where the LHS is the work equivalent due to this force. We may write this according to О"W = О"pc - Д§/О"x, so that the work is seen to increase with О"p and decrease with О"x. This may also be expressed to connect the momentum-position uncertainty with a "quantum work-time uncertainty" so that

О"WО"t = О"pО"x - Д§.

For F = 0 then О"pО"x = Д§ and the LHS is zero. This illustrates how О"WО"t is associated with an increase in uncertainty.

If you think this looks similar to the entropy force of gravity results you are not too far off! The entropy force of gravity concerns the transverse displacement of a holographic screen. Here we are concerned with the stretching or longitudinal motion along the screen. The entropy force of gravity F = T∇S determines work as

О"W = FО"x = T∇SО"x = TО"S.

We employ E = (1/2)NkT which equals Mc^2, We then see that temperature is T = Mc^2/Nk. This is inserted into the equation for the entropy force

FО"x = (2Mc^2/Nk)О"S

The motion of the holographic screen by a distance О"x = Д§/mc results in

F = (Mmc^3/Nk Д§)О"S

and we use N = A/4L_p^2 = πR^2/(Għ/c^3). We then have that

F = 2GMm/(kR^2)О"S

Where we then assign the unit of entropy О"S = k/2 to get the result.

The NS5-brane has spacetime of four dimensions on a D-brane sandwiched between two other branes embedded in 5 dimensions. This sandwich is contained in an anti-de Sitter spacetime (AdS_5) of five dimensions. In four dimensions of spacetime a static gravity field in 3-dimensional space has the property that a two-dimensional surface bounding a mass will have the same number of line of flux through it independent of the radial size of the surface. Thus the force F multiplied by the surface area 4ПЂR^2 is a constant and equal to the gravity constant in 4-dimensions multiplied by the masses

GMm = 4ПЂFR^2,

Which leads to Newton's law of gravity

F = GMm/R^2

The 4ПЂ has been "absorbed into the definition of G. In five dimensions the surface bounding the mass is not two dimensional but three dimensional, which gives us the law

G_5Mm = 4ПЂFR^3

And so the force law in five dimensions is

F = G_5Mm/R^3.

The G_5 is the gravitational constant in five dimensions, and it reflects how the branes are "warped" in the embedding space by interacting with each other. G_5 in these units is Nm^3/kg^2 (N = Newton, m = meter, kg = kilogram). The relationship with the standard gravitational constant G with units Nm^2/kg^2 is that G_5 = GL, where L is a unit of length.

The brane tension, or equivalently the tension of strings attaching them together, is T = kL, for k = constant. The string length is L = 1/sqrt{R}, for R the curvature of the brane which we write as L = L_0^2/s, for L_0 the zero tension string with F(x) = 0 and s the "warp length," which is something that Randall worked out.

G_4 = (4ПЂ/3)О»G_5^2 = G_5/s

It is kind of neat. The big question I have is whether or not our four dimensional spacetime is not in fact on the conformal boundary of the AdS_5. This does go somewhat into the idea about changing gravity, which appeals to the physics of our brane in or on the AdS_5 and how gravity is related to the brane tension.

We also see that the tension depends upon the length L and for the tension at a minimum the Newton gravitational constant G_4 is small. For T or О» large the gravitational constant is huge. We then have a renormalization group flow for the gravitational constant. Also in addition G_4 = (Planck area)c^3/Д§, and for the tension large this is larger and the area of a Planck unit is larger. This is due to an impulse F(x)О"x = О"pc - Д§/О"x that "stretches" the horizon or Planck unit of area. This force is what adjusts the warp length.

Quantum gravity involves noncommutative geometry. In particular there are coordinates of spacetime that are not commensurate or which have an uncertainty relationship. This can be seen with the argument Bohr gave at the Solovay conference in 1930. Einstein argued that a box containing a photon could have a door opened and the mass of the box plus photon measured. This would get around the uncertainty principle. However, the uncertainty in momentum is О"p = TgО"m, for T an interval of time, g the acceleration of gravity and О"m the uncertainty in mass. The spread in distance is О"x = cО"t, and the product in О"pО"x = TgcО"mО"t = (Tg/c)О"mc^2О"t. With О"pО"x = Д§ then О"mc^2О"t = Д§ and the uncertainty in time is О"t = (Tg/c^2)О"x.

We may now consider this with the uncertainty in time and radial distance with the metric coefficients. We have then that t' = t sqrt{1 - 2GM/r'c^2}, and for О"t = t" - t' measured between two small radial distances in a Schwarzschild metric small enough then

О"t = t(sqrt{1 - 2GM/r"c^2} - sqrt{1 - 2GM/r'c^2}) ~ t(GM/c^2)(1/r" - 1/r').

For r" - r' = О"r then

О"t ~ -tGMО"r/(rc)^2,

We now consider the spread in the potential V = -GM/r, О"V = GMО"r/r^2 - GО"M/r. This leads to

О"t ~ (t/c^2)(GО"M/r + О"V)

We now let cО"t = О"x, for x a distance related to the radius. The time t is equated to GM/rc, as above. In addition the fluctuation in the radius and mass are for the Schwarzshild radius equal, so the uncertainty in the potential GО"M/r =~ О"V. This gives us

О"x ~ 2GО"M(t/rc) = 2GО"EV(t/rc^3).

A multiplication by О"t is

О"xО"t ~ 2GО"EО"t(t/rc^3) = 2GД§(t/rc^3).

The t and r are equated and this leads to

О"rО"t = 2GД§/c^4 = 8.5x10^{-75}ms.

This is a Planck unit of area divided by c. From О"pО"x = Д§ + F(x)О"xО"t it is easy to see that

О"pО"x = Д§ + (GД§/c^4)F(x)

= Д§(1 + GF(x)/c^4)

We then see that the uncertainty in the spacetime variables is due to this force. This force is what is responsible for the brane tension. If this tension is zero or minimal then the fluctuations on our brane are minimal. Measurements across huge distances are using a telescope to measure across vast distances. An extreme high energy experiment imparts huge transverse momenta/energy and increases the brane tension.

The entropy force of gravity involves motion of a holographic screen along a direction perpendicular to the screen. Here the concern is with the dynamics transverse to the motion derived by the entropy force of gravity. This means that we have a general problem with fluid dynamics. This seems to suggest that maybe the general form of fluid dynamics is given by the 3x3 matrix, maybe diagonalized by the E6.

LC

The paper http://arxiv.org/pdf/1407.5977v2.pdf by Bonezzi, Corradini and Waldron is pretty interesting. Much of the start seems to hinge around

1/sqrt{det(H)} = exp(-ВЅ Tr log(H)).

We have then that

I* = ½ Tr(H) = ½ ∫_0^∞ ds/s exp(sH).

This is an elliptic integral that diverges at s = 0. We then set the lower integration s = 1, which corresponds to the area of the string world sheet with s = t/x, for t the time and x the spatial coordinate on the sheet. The s = t/x is a form of the S-duality of the string, or a form of t and s channels in Mandelstam variables are equivalent.

The same seems plausible with the Grassmannian coordinates c. The equation 13

∫Dχexp(-Tr∫d^3cχF_A) = sum_A'δ(A - A')det[δF_A/δA]

is integrated over the space of Grassmannian variables. This could be represented with a string target map. The Grassmannian coordinates c^a = ОґОё^a are anticommuting variables and the target map is given by a determinant of the c^a's, similar to a Slater determinant of fermion states, that defines the string world sheet.

LC

I am looking into Motives. I have found the following http://www.ams.org/notices/200410/what-is.pdf to be of some help. The Wikipedia article assumes too much understanding of the preliminaries.

I do see that this has some connection to groupoids and magma constructions. I also think this has something to do with nature of the Leech lattice in J3(O). The 24-cell, indeed the very number 24, keeps cropping up. The F4 group has as its roots space the 24-cell, and I have been working on something involving F4 and its B4 reduction with respect to projective varieties of rays. These as putative hidden variables fail to define a Ω [φ, ψ] function as unit for a ψ-epistemic or 0 for a ψ-ontic theory. The F4 also diagonalizes J3(O).

There seems to be some sort of relationship of the sort

Leech lattice ---------------------------- modular forms

Golay Code ---------------------------- Elliptic curve (exotic varieties, Goppa codes)

Sporadic groups ------------------------ Special functions ( Jacobi θ-functions, Ramanujan mock Θ-function)

The space of quaternions H = {pm 1. pm i, pm j, pm k} ---> SL(2, 3) is 24 dimensional. In the sequence

F4: B4 --- > F_{52/16} ---> OP^2

the F4 when modded by the quaternion space, is then the space of J3(O) on the RHS minus the pairs of roots which is an elliptic curve.

My main interest is to connect the M_{24} in this system, but this is just one rung on the whole sporadic group ladder leading up to the FG or monster group.

LC

This mathematics of motives is really a set of varieties that connect cohomological systems. I had a course in algebraic topology, I have a masters in math, and have used it some, though again I am more of a physicist.

What is interesting is that this approach with algebraic varieties is something I worked with looking at nonlocal hidden variables in QM. These have in four dimensions 6 collineations of 24 vectors, 48 in total with their duals, and this is the root space of F4. In three dimensions this is reduced to 3 collineations that share a normalizer, and there is an automorphism that connects these. As a result there is no "frame independent" manner in which the correlation function can be evaluated. There is then no manner by which we can establish the epistemic or ontic nature of the quantum. This puts a lot of, and I hope all, quantum interpretations into the junk heap. These automorphisms are on projective varieties. This system of 3 collineations is a form of Mermin's trick with 9 observables.

The full spacetime form involves the F4, and this is the automorphism of J3. So this seems to be meshing together in some physical manner. The F4 is the real eigenvalued diagonalizer of the "3x3." E6, and its variants, are involved with "real plus imaginary" eigenvalues, which seems to connect to quantum field theory in curved spacetime. The imaginary evs appear to be Bogoliubov coefficients for quantum fields in curved spacetime. E7 and E8 then connects with quantum gravity where

e8 --- > 1 ⊕ 56 ⊕ (133 ⊕ 1) ⊕ 56 ⊕ 1

for the 133 ---- > e7, and the 1 a graviton and the other 1 the Taub-NUT mag-graviton dual, and the E7 corresponds to states that connect with gravity.

There is the general 3x3 of O's or the vector E8 plus the two spinor E8s in a SUSY setting. The systems for evaluating this matrix then looks like the hierarchy F4, E6, E7 and E8, which takes us from standard quantum field theory, to QFT in curved spacetime and up to I think quantum gravity.

There seems to be a lot here I yet do not quite understand. I am trying to rely on physics to fill in the gaps. The modular functions, such as the Jacobi θ-functions obey diff-eqs that are a form of the heat equation, or Schrodinger equation in a complex form, which can evaluate a heat kernel. I am not sure but I have thought these are some sort of underlying unital structure. The modular functions (forms) the conserve quantum information, which is another way of looking at SLOCC algebras.

Cheers LC

The paper by Yang-Hui He & John McKay centers initially upon the linear fractional group. This is of course an aspect of T-duality in string theory. It is also a part of what I have worked with respect to quantum phases of the event horizon. A highly accelerated observer that hovers closer to the horizon must do so under increasing acceleration. The distance to the horizon is d = c^2/a, where near the Planck acceleration a =~ 10^{53}m/s^2 the observer is on the holographic screen of the stretched horizon. In this limit there is a quantum critical phase transition. The phase transition changes the horizon from a conducting phase to a symmetry protected topological phase similar to a Mott insulator.

The SPT is an energy gapped phase at zero temperature, which in our case is near the Hagedorn temperature, where the heat capacity of event horizons is negative. The SPT states with a symmetry given by group G have topological orders given by the cohomology H^2(G, U(1)). For the case d = 2 the 2-space plus ime model with the Euclideanized G = SO(3) gives the 2 + 1 spin Hall effect and he time reverse symmetry group Z. The projective representation of this group is found by the quotient with the normalizer of that group PG = G/N with the map π : G = PG that defines a bundle of lift elements. These elements λ obey for g \in G the rule λ (g; g') = σ(g; g') λ (g) λ (g'), where σ(g; g') is a Schur multiplier. The projective representation of the group \mods out rq' the action of this normalizer. This is a cocycle in the cohomology H^2(G; N) with the normalizer N = U(1).

The Schur multiplier is a way in which the projective realization of a group can be represented according to the second cohomology that is an abelian group that defines a covering or line bundle. The projective representation is a homomorphism into the projective linear group PGL(n, K),specifically for K = R or C, and for n = 2. A field F is mapped into K, the group C and B have the sequence 1 --- > F --- > C --- > G --- > 1 for C the centeral of G and this sequence is the central extension of the group G. The following diagram ensues

F ---- > C ------------- > G

| | |

K --- > GL(n, K) --- > PGL(n, K)

The group SO(2, 1) ~ SU(1; 1) is homomorphic to SL(2, R) x SL(2, R)/Z_2. The linear fractional group SL(2, Z) the gives the braid group as

B_3 ------------ > PSL(2, Z)

| |

SL(2, R) ------- > PSL(2, R),

where vertical lines are down arrows. The braid group is then the central extension of the linear fractional or modular group PSL(2, Z). The braid group B_3 then contains a center which under this map gives PSL(2, Z) ~ B_3/C. This center is under the correspondence between C and G = B_3 with the normalizer N = U(1) projective realization PSL(2, R) = SL(2, R)/U(1) a set of elements that definnes the modular group as a projective realization of the braid group.

I like trying to make contact with physics, and I have this idea that physics found in things like condensed matter or quantum optics can be general principles that occur in other forms. I think in addition that this physics with accelerated frames means that physics in inertial and accelerated frames are ultimately equivalent.

LC