You still have a funny issue with computing something as variable in this way as the generator of a partition function. The usual way of computing things is to derive things from the partition function and from there determine such variations. In effect what you are doing puts this upside down a bit. Your analysis is guaranteed to be criticised by reviewers for this.
Cheers LC
Ah! Lawrence,
Just when I thought we had this issue resolved you come back with more Delphic pronouncements. What am I "...computing something as variable ..."? What "partition function" I am using when I am not using any? Can there be any other way than the "usual way of computing things..."? And what is wrong with "... what you are doing puts this upside down a bit", if there are no mathematical flaws in the arguments and the 'physics of the physical makes sense'? Isn't this in fact how we progress?
I can understand the need to "see the epicycles" in the "ellipses that I am showing". It may be a necessary and comforting exercise for those that know how to use epicycles to calculate orbits. I confess I don't. Perhaps you can show us where they lie! Were you to do that, nothing in what I have done will get any clearer. But what has been done may get clearer.
Constantinos
On page 3 of your FQXi paper you have the definition of the Planck formula according to what amounts to being a Laplace transform. Whether you admit it or not this is a sort of partition function calculation. You have the distribution e^{νu} where u is "time," which by implication is associated with a temperature. You have by implication a relationship u < -- > 1/T. In fact if you changed the sign of the exponential distribution to e^{-νu} and changed the limits on the integration things would be more in line with standard formulation.
This is somewhat non-standard to say the least, For instance, the temperature in the Boltzmann factor is an "unknown," and its properties are computed after the fact by calculations on the e^{-E/kT}. What you are doing is somewhat off the standard procedures, which are certain to raise the eyebrows of reviewers.
Cheers LC
Dear Lawrence,
From what you write, it is clear that you have found a connection between the derivation of Planck's Law in my essay and statistical thermodynamics. That is wonderful! I do hope this does raise some eyebrows with insightful delight. I just find it amazing that Planck's Law can be derived more simply using continuous methods. And isn't this what we badly need in Physics? Perhaps this connection that you found may provide us with greater insight that leads to a 'physical view that makes sense'.
Thanks for that!
Constantinos
Hi Lawrence,
I agree with you in general, although I use different language and a different method of getting to the same conclusions. I always have to read your stuff over and over to make sure the connection is there, not because it's opaque but because you pack a lot of meaning into fairly spare symbols.
To this point of whether string theory solutions are real, or mathematical artifacts (quantum amplitudes), I think it just doesn't matter. That is, to make an analogy, in E = mc^2, c^2 is also a mathematical artifact; i.e., it doesn't affect the meaning of the statement E = m. Similarly, when we speak of the fundamental domain of string theory as a hyberbolic 2-space, the range of solutions can be infinite without violating the meaning of string theory as a unifying principle that retrodicts physical results. (By the same comparison, the theory of common ancestry known as evolution is a retrodictive theory, because we cannot observe random mutation and natural selection in real time, though we can model it and compare the model to past events.)
So a 10 dimension boundary condition should be sufficient to limit the domain solutions if the real solutions can be shown finite given the boundary -- which is precisely within the spirit of Einstein's "finite but unbounded" relativistic universe. A finite set of solutions obtained by construction rather than heuristics should give us the means to at least perform thought experiments and create accurate computer models.
So ... going back to my research, which I am in the process of expanding and trying to compact (wish I had your talent for that!) into more comprehensible and standard terms:
My energies have been directed toward a tight proof of the 10 dimension limit. A key idea is the construction of the modulus 12 subgroup of Sophie Germain primes (detailed in my time barrier paper) such that the 2 dimension submanifold self similar to the mod 12 subgroup possesses properties of linear independence over Q and infinite orientability on C*. This latter property allows n-dimension continuation over non-compact manifolds.
The role of the mod 12 structure is then clear, as an eternally recurring zero of an ordered set of kissing spheres admitting non-lattice numbers. That is, every sphere, S > 1, contains at least one 12-vertex lattice, S^2 --> S^oo.
To elaborate, supposes -- instead of analyzing by differentiation and integration -- we allow discrete elements of a contnuous function to self organize in the n dimension Euclidean space. The least 3 dimension sphere, S^2, with 2^n degrees of freedom -- 3(2^n) -- is the zeroth member of S^n, for being the least of the elliptic (zero) manifolds. Lesser Euclidean dimensions have hyperbolic (negative) characteristics and parabolic (positive) manifolds are hyperspatial.
Expressing as a table:
Geometry Order Dimension Topology
point - 3 0 none
line - 2 1 S^0
plane - 1 2 S^1
sphere 0 3 S^2
hypersphere 1 4 S^3
(I hope the table doesn't break up. I can't preview on this computer.)
Best,
Tom
Of course the standard Planck formula derivation involves the Boltzmann factor and summations over discrete energy steps. What you have is similar, but with a continuous approach (which can just be an approximation) with a Laplace transform.
Cheers LC
Tom,
The use of sphere packing is a sort of physical argument with respect to qubits. The Planck scale is a limit to the scale where one can identify a qubit. It is not possible to identify quantum information on a scale smaller than the Planck unit. Sphere packing is a configuration where each sphere with a qubit defines a quantum error correction code. The polytope is then a root configuration for the quantum error correction code. The fun group is the F_4 group, which has the root space of the 24-cell. The 24-cell has several representations, where the D_4 or SO(8)xS^3 is an interesting realization. The B_4 decomposition has a bearing on the BFSS M-matrix in M-theory.
If one works up the E_8, the most important thing which comes from this is the Freudenthal cubic system. Take a look at Phillip Gibbs' paper where he works out the cubic elliptic structure for the hyperdeterminant of the 3-qubit system. The 3-qubit system turns out to have a correspondence with BPS black hole types. I discuss that some below. The Freudenthal system is a general Jordan matrix algebraic result for the determinant and trace system of the heterotic group E_8 in a 3x3 matrix system. The hyperdeterminant for the 3-qubit system in the SLOCC group (see below on SLOCC group) defines a cubic form G_abcdψ^aψ^bψ^c = 0, which corresponds to a quadratic matrix M^{ab} = G^{abcd} ψ^bψ^c. This defines an elliptic curve from the hyperdeterminant of the cubic 2x2x2 form G_abcdψ^d
y^2 = det(M) = Det[G_abcdψ^d] Det means hyperdeterminant
This will be modular of course due to the A. Wiles proof of the Tanayama-Shimira conjecture. An explicit realization of this modularity comes from the equivalency with the AdS_n, and in particular with the near horizon condition AdS_{n+2} - -> AdS_2xS^n, which is conformal QM SL(2,R). This is the modular group, or its discrete subgroup SL(2,Z) defines the braid group.
The 4-qubit system describes extremal black holes, or there is a mathematical correspondence. The SLOCC group is a 2x2x2x2 matrix structure, which is entirely different again. This is given by a 4-qubit SLOCC group (see named section below). The M^{ab} matrix above is an SO(4) and the product is contained in SO(8) ~ SO(4,C). The moduli group is then SO(8)/SO(4)xSO(4) as a coset structure. The 4-qubit system is then given by a Cayley hyperdeterminant polynomai of degree 4.
So what does this have to do with my paper? The AdS_n space is the quotient
AdS_n = SO(n-1,2)/SO(n-1,1)
and the AdS_4 near a black hole horizon, assuming a black hole is contained in the spacetime, is AdS_2xS^2. There is the part about Taub-NUT spacetime in my paper, which has a correspondence with the AdS. The spacetime AdS_4xS^7 then has isometries SO(2,3)xSO(8). The S^7 is fixed by the G_2 group on SO(8). The G_2 group is the stabilizer group of F_4 (above) in the E_8. The spin(7) subgroup in SO(8) is then fixes an S^7, spin(7)/G_2 ~ S^7. Now on the 11-dimensional spacetime here we consider the lightcone or infinite momentum gauge to reduce the S^7 to S^6. This is then the Calabi-Yau manifold of compactification.
This CY manifold has an action on the AdS_4, now reduced to the AdS_2xS^2. The CY manifold is the space of Dp-brane wrapping. Duff et al (referenced below) work this for a 0,2,4,6 torus and even number branes. However, this clearly can be generalized. The result is that the electric and magnetic charges define a generator of a partition function with the appropriate BPS charges. The AdS_2 part corresponds to a Taub-NUT charge. This is equivalent to CFT_1, which turns out by Sen to be equivalent to the Hartle-Hawking states of the Wheeler DeWitt equation HΨ[g] = 0 The resulting partition function is an Eisenstein series realization. The BPS charges associated with the BPS - 3-qubit correspondence then construct a partition function. Further, the 3-form version of this is a partition function for the integers.
The F_4 or 24-cell turns out to have under it constant action with G_2 a form of CY manifold structure for h_{1,1}. This then connects back to the system of sphere packing in four dimensions.
This is then an eigenspectrum for the states of the universe. Due to the AdS_2xS^2 structure near a black hole horizon this is also related to the spectrum or counting of microstates on the event horizon of a black hole. A black hole has a set of quantal units of area on its event horizon. These are identified with quantum states, and are a partition of integers for counting the number of states on a black hole horizon. The area of a black hole is composed of little quanta of areas given by a sum of integers n_i >= 0,
A = 4π a(n_1 + n_2 + ... n_m)
where this total number N = n_1 + n_2 + ... n_m can be written according to the integer partition. Another way of thinking about this is that the string modes can exist in a distribution which is an integer parition. This is the holographic principle in action, where the event horizon or stretched horizon is composed of a "gas" of strings.
The density of states for a string is tr(w^N) , which for N = sum_nα_{-n}α_n the string number operator. Given there are 24 string operator the computation of this generating function is tr(w^N) = f(w)^{-24} for
F(w) = Π_{n=1}^∞(1 - w^n)
This is a form of the Dedekind η-function and the remaining calculation leads to a form of the Hardy-Ramanujan approximation for the integer partitions. Recent results by Ono, Brunier, Folsom, and Kent in the role of modular forms in number theory has result in an exact theory for integer partitions. What I am working towards is a string theoretic version of this theorem.
Cheers LC
The 3-qubit SLOCC group:
Based on a paper "Four-qubit entanglement from string theory" L. Borsten, D. Dahanayake, M. J. Duff, A. Marrani, W. Rubens.
In general we have Stochastic Local Operations and Classical Communication (SLOCC) in entanglement and the teleportation of states. Two states are SLOCC related by a teleporation if they can be inter-converted to each other in a reversible manner with some probability of success. This uses group theory, where the group G_{SLOCC} for this process is an N-partite system of q-bits with some group GL(2,C). The states further transform as a (2,2,...,2).
G_{SLOCC} = SL(2,C)_1(x)SL(2,C)_2(x) ... (x)SL(2,C)_N
where the composite state
|ψ_{12...N}> = SL(2,C)_1(x)SL(2,C)_2(x) ... (x)SL(2,C)_N|φ_{12...N}>
So this is an N-partite quantum information system where the entanglements are determined by the group element G_{SLOCC} and polynomials of this group. This is the moduli space for black holes composed of Q-bits and the U-duality group.
For a 2 Q-bit system this construction is apparent. You have a stat of the form sum_{ij}a_{ij}|i,j> for i and j running form 0 to 1. The elements a_{ij} transform as (2,2) of the G_{SLOCC}. The invariant element is the determinant of these matrices so det(a_{ij}) transformed under the G_{SLOCC} into
det(a_{ij}) - -> det(a'_{ij}) = det(U_{i'i}a_{ij}U'*_{j'j}) = det(a_{ij})
with the obvious result on the determinant of a product that the transformation elements have unit determinant. The entanglement entropy is given by this measure so S_{ij} = 4|det(a_{ij})|^2. For multipartite systems the same rule generally applies, but the matrix interpretation is different. For an N-partite system the entanglement entropy is given by a 2x2x...x2 (N times) set of elements. This then leads to the entangled states |00> + |11> and |01> + |10> (without normalization) for singlet and triplet entangled states.
For a 3 q-bit system things are more difficult. The amplitudes are elements a_{ijk} --- a 2x2x2 elements that is not a matrix. We have no diagonalization procedure here. We then have to focus on invariants. The determinant is replaced by a hyperdeterminant that transforms as a (2,2,2), and there are elements σ_i σ_j and σ_k, co-invariants, that transform as (3,1,1) (1,3,1) and (1,1,3) of the G){SLOCC}. So these four then construct entanglement measures S_{ijk}, S_{ij}, S_{ik} and S_{jk}.
The 4-qubit SLOCC
The four Q-bit system is G_{SLOCC} = SL(2,C)^4(four products of SL(2,C)) with 4x3 = 12 complex parameters and the state variable is ψ_{ABCD} and has 2^4 = 16 complex parameters. The entanglement of 4 Q-bits is then 16 - 12 = 4 coset space given by
(C^2)^4/(SL(2,C)^4
Now the entanglement space is not a discrete set as seen with the 3 Q-bit system, but is now given by a continuous set of parameters in this coset group. There are now an infinite number of possibilities. The 4 Q-bit has G_{SLOCC} flows or orbits of the state ψ_{ABCD} SO(4,R) ~ SL(2,C), and we can convert the SL(2,C)^4 into SO(4)^4 and use SO(4,C) = SO(4)^2. The orbits of the G_{SLOCC} are the an SO(8) conjugacy class. The set of nilpotent orbits is a classification of SO(8,C). With extremal black holes the condition is given by these nilpotent orbits on the moduli space. A nilpotent orbit is where there is a group G with algebra g, then for a \in G and b \in g then the adjoint action of a and b is
b - -> b' = aba^{-1}
A nilpotent orbit is given by b^n = bb ...b (n times) and this is stationary as it is clear than b'^n = b^n, and are a fixed point in the moduli space. This Lie group homomorphism SL(2,C)^2 ~ SO(4,C) converts ψ_{ABCD} to a (4,4) of SO(4,C)^2, so then under SL(2,C)^2 ψ_{ABCD} transforms as
ψ_{ABCD} - -> ψ'_{ABCD} = U_A(x)U_B(x)U_C(x)U_D ψ_{ABCD}
= [U_A(x)U_B(x)] ψ_{ABCD}[U_C(x)U_D ψ_{ABCD} ]^T
= M_{AB}ψ_{(AB)(CD)}N^T_{CD}
and is a convenient transformation Mψ N^T = ψ' for U in SL(2,C) and M and N in SO(4,C).
Dear Lawrence,
You seem to see in my derivation of Planck's Formula Laplace transform. That may lead you to some deeper insights, but from my perspective I don't see the point. I just lose the physical meaning of that math.
All the results in my essay have a clear and simple physical meaning and are mathematically argued. What is the physical meaning of the Laplace transform you are arguing I inadvertently used in my derivation of Planck's formula? And does this also show what I am showing? Namely, that Planck's formula is a mathematical tautology that describes the interaction of measurement. This is why the experimental blackbody spectrum is indistinguishable from the one obtained from Planck's formula.
I do not argue with the mathematics used in physics. Rather, I argue with the physics in the mathematics used. What motivates me in this intellectual venture is my desire to understand physics physically. Modern Physics lacks physical meaning that makes sense. The 'man in the street' knows more about 'time travel', for example, than the theorist who with mathematical certainty asserts it.
We need 'physical realism'. I show in my essay this is possible. My derivation of Planck's Formula avoids energy quanta and discrete statistics. In this view, we gain a clearer understanding of what the Formula actually means.
Constantinos
Lawrence,
You hit the problem of string theory experimental falsification right on the head. String theory does not live in the domain where Planck's constant resides.
Where quantum information is not an irreducible given but rather the result of hyperbolic geometric projection -- which is my strategy -- we can speak of normalization on S^3, which is the first member of my kissing order calculation. It's long been known that low dimension topology has some rather unique properties, which is part of the reason it took so long to prove the Poincare Conjecture, and why we have Calabi-Yau manifolds.
By setting the threshold of observation at 3 dimensions (instead of 2) we get "3 for 2" in a manner of speaking, and a projection of length 1 to 4 dimensions. IOW, 3 1 dimensions in a kinetic theory recovers Minkowski space by a strict algebraic method. This method has the added advantage of projective determinacy to the Lebesgue measure domain (where we can deal with PLanck's constant), because by construction, all even dimension models (up to our limit of 9 1) reduce to a 0 1 spacetime. But look (table in previous post) -- the first even dimension domain is - 3 1, leaving the - 2 hyperbolic fundamental domain of string theory. The mathematics is sound, because when we allow the point a - 3 geometry, and knowing that complex analysis interprets the point as a line, we get the string domain naturally coupled with a time metric, without having to impose boundary conditions. Then with analytical continuation over n-dimension manifolds, we get irreversible dynamics while nevertheless preserving by Poincare Recurrence the classical time symmetry of 3 1, as a result of recursion native to the real number line. (PR is a better tool than straight commutative symmetry, because no experiment could in principle reverse the time metric along the same path, since the experimenter and apparatus are never at rest relative to the metric.)
The next major task, as alluded to previously, is to fix a finite range of vacuum solutions in the Lebesgue measure domain and put together a program to test them (even if only in thought experiment). It would be serendipitous if the range turned out to to be [0, 1] :-)
I'll look up the Hardy-Ramanujan reference to see if I can connect to it.
Best,
Tom
This sounds a bit like the paper of yours I read over a year ago. It seems that this PR idea is meant to conserve information. Is the kissing number meant as a way of error correction code for this?
Cheers LC
Lawrence,
That is correct. Remember, though, that in my construction, information and time are identical. The redundancy of information recurrence in a dynamical system provides sufficient randomization in a nearly closed ensemble, to contribute useful feedback for error correction.
So time symmetry/information conservation in the 3 1 domain characterizes classical physics -- while gravity is the result of n-dimension dissipation over manifolds n > 4. My specifically physical definition of time: "n-dimension infinitely orientable metric on self avoiding random walk" implies that dissipation over Riemannian manifolds in Euclidean space (all Riemannian manifolds are orientiable) increases entropy toward the center of mass which in sphere kissing terms is the center sphere. All the information we can recover, though, is on the manifold of S^3, which leads to holography.
Do you remember back in the article introducing Verlinde's results to FQXi, that Lubos Motl objected to the entropic gravity model on the grounds that gravity is symmetric? I allowed that he is right -- and right he is -- it's a problem I recognized years ago, and for which I supplied an algebraic model (see my "Time Counts" FQXi essay and my "time barrier" preprint) that shows a very slight asymmetry using the escape velocity scalar. Briefly, every point of spacetime has a unique and changing value of escape velocity; every instantaneous gravitational relation between any 2 bodies, however, is projectively asymmetric by a tiny but nonzero amount. I mean, that instead of taking c^2 as a spacetime area, I take the value c as a metric and calculate the spatial area projected on each point -- represented by the difference between antipodal escape velocities. This method has the virtue not only of explaining Einstein's approach to "the relativistic theory of the asymmetric field," it explains noncommutative geometry in physical terms. That is, backward-forward projection is not arithmetically commutative. As a consequence, we need the closed algebra of C* to fully capture time and information conservation in both classical and quantum mechanics.
As noted previously, I've identified a 10 dimension (9 1) limit to the 4 dimension (3 1) horizon outlined above. This limit is an extradimensional analogue to 3 1 Minkowski space.
Yes, I get your Hardy-Ramanujan connection now. Note in my ICCS 2006 paper (5.5.2), the linear transformation map A --> B[(0)mod 2] to the even part of B and the odd part of A, and A --> B[(1)mod2] to the odd part of B and the even part of A. As a result, the unit diameter of S^3 has no zero dimension center point. That would leave the embedded hyperbolic space where your string theoretic model can live, projecting holographic information to the S^3 boundary.
All best,
Tom
Dear Lawrence,
thanks for the reply but I'm a particles in empty space man. The Archimedes screw idea does away with a spacetime continuum. I have a mechanism to explain the orbit of Mercury quandry as well, it's the 'inclination hypothesis' i.e. that gravity is stronger towards the plane of rotation of a celestial body.
Best wishes,
Alan
I got to this a bit late, so I will only comment on one thing. The Verlinde result involves the dynamics induced by an entropy of the holographic screen. Motl misinterpreted the whole thing by saying that a dynamical orbit would be irreversible. Not everything Motl says is right, and frankly when it comes to the political and environmental stuff he says things which are mendacious.
More later & cheers
LC
Lawrence,
Mendacious in his politics, perhaps. Not naive in physics, though. As I remember it, Lubos's argument was not directed at classical time reverse symmetry, but at the equivalence principle as applied to quantum mechanics, where T = 1.
In this domain, he is certainly correct. The symmetric spacetime field implies instantaneous reversibility in the dynamic quantum configuration space. If one wants entropy (irreversible dynamics) where information loss implies time dependence, there is no instant (time interval) short enough to violate information (energy) conservation. Quantum mechanical unitarity still applies.
As Bell's theorem explicitly shows that quantum configuration space cannot map to physical space without a nonlocal model, if information is lost, it must be lost in the classical domain. That's where I get my model of instantaeous gravitational energy exchange, with precisely calculated information loss between bodies. With the assumption that this slight loss among bodies of ponderable mass sums to 1 over the entire quantum universe, it becomes an interesting question of whether the information is recovered in the quantum domain. I think personally that the holographic dynamic does preserve all 4-dimension information (on the principle that entropy decrease in d =< 4 is at the expense of entropy increase in d > 4, given dissipation over n-dimension manifolds); however, I don't have to go that far to make the point: Einstein's failed research in the relativistic theory of the non-symmetric field was not wrong-headed. The quantum field cannot maintain symmetry with time dependence, and cannot lose symmetry without it. Poincare recurrence provides dynamic symmetry in the classical limit without losing quantum mechanical unitarity where T = 1.
Tom
Tom, In reference to your last post:
The Verlinde result involves the screen, or in effect the event horizon. The Birckhoff theorem tells us that if you place a black screen around any spherically symmetric gravity field, that the actual configuration of the source of that field does not matter, and all are equivalent to a black hole. So classically we have that a black hole is the fundamental source, which has a gravity field at any radius outside of its event horizon which is equivalent to a spherically symmetric mass at that radius. So everything seems perfect, there is no problem, for our entire GR works fine within this cloak or screen.
However, what happens with extreme curvatures? Consider any disk region of space with a radius r. Curvature is a field associated with a 2-form that is dual to this area, a bit like an electric or magnetic vector field passing through this area. The curvature has the effect of cutting a small wedge, with an angle θ, out of the disk and gluing the two sides together. The disk is now a sort of little cone, which we may smooth out to eliminate the cusp. The curvature scalar is R = θ/2πr^2 in this little area of radius r. Now let us suppose we have a standing wave on this little disk, say there is an "atom" with a wave function there. This wedge which is cut out adjusts the wave slightly if θ 2π and the curvature scalar is r ~ 1/L_p^2. Quantum fields are horrendously scattered around in this region.
A black hole has this event horizon, and for a modest black hole the horizon may have sufficient curvature that a little bit of this wave scattering by curvature is going on. Consider the case where the wave which is scattered is the Dirac field. The Dirac field is the "square root" of E^2 - p^2 = m^2 in quantized form. This means the spinor fields have positive and negative energy solutions. The energy-momentum of the field upon scatter changes its root value. This is the Dirac sea, and the curvature of spacetime is perturbing the Dirac sea of negative energy states and the particle states of positive energy. This means that Dirac particles of negative energy can be scattered out of the Dirac sea by the spacetime curvature into a real particle state. This is Hawking radiation, and now the screen has a certain particle production which occurs with some chemical potential μ and a thermodynamic content e^{μN} so that μN = E/kT. The energy is then E = μN/β, and this is the energy of entropy. This is the entropy of the "screen" or the event horizon.
So if you have some system which is interacting with the black hole, and couples with these quantum particles produced, then you have an entropy content associated with the force of gravity. This is physically where the Verlinde result is most relevant. It is not something which changes the entropy of a system falling here on Earth, or which is going to result in some irreversible mechanics with the interferometer of a quantum particle falling in a gravity field here on Earth. If that quantum is on a geodesic which takes it close of a quantum black hole the story is different!
The emission of a particle by a black hole adjusts the horizon. The negative mass particle is associated with a Boulware vacuum state with a negative stress-energy, and during this scattering period the vacuum state (Dirac sea occupation zone) is replaced with a real particle state. Also the event horizon adjusts into an apparent horizon configuration for a small period of time Δt ~ ħ/mc^2, which becomes a "real event horizon" for a time t >> Δt. This is the motion of the screen part of the dynamics, which is associated with the E = ∫F*dr, and how this "force" is associated with an entropy.
There is then this apparent problem with time reversibility. However, the structure of quantum gravity is not unitary in the standard sense. Things are modular, and the states of quantum gravity are determined by modular forms. Modular forms, in particular the Jacobi θ-functions and Mock Ramanujan θ-functions have the symmetries of the Heisenberg group. Further, the Heisenberg group is parabolic and it is embedded in a coset which is equivalent to the light cone structure of spacetime. So the Heisenberg group pertains to conformal quantum mechanics on AdS_2. So quantum physics does persist, and there is a conservation of quantum information. The time reversibility is a sort of Wheeler Delayed Choice thing, for a particle the interacts with a black hole is both stuck on the horizon as seen by an external observer and on the singularity as seen by an infalling observer. The two apparent events are really the identical event. The Hawking radiation demolishes the states of the system as witnessed by the exterior observer, and this is the same event witnessed by the infalling observer who witnesses the system demolished by the singularity. This is even though these two copies of the events are separated by vast periods of time, they are the same event and there is a time symmetry here.
This does get into the issue of the configuration space of a quantum state. However, I am beginning to lose energy here, and so I am going to bring this to a close. The quantum state of the system has then different spacetimes for the same configuration.
As for Lubos, the guy is very quick with knowledge and analytical abilities. The thing which disturbs me about him is his great willingness to promote falsehoods, largely with respect to this climate change stuff. This does not seem to impact string theory and the rest, but it does make me wonder to what degree he is willing to distort things to prevent him from admitting error. He said sometime back, "I am always right. If you are ever in doubt look at the previous sentence." The greatest fallacy anyone can make is a claim of infallibility.
Cheers LC
Cheers LC
Hi ,
You mix too much without a real universal sorting.The aim is not to enumerate a serie of ideas, furtermore perhaps 1 on 2 is right,for a kind of credibility for maths. You confound a little the computing and the physics.Your convergences are rare dear thinkers.It is probably due to a bad utilization of maths methods.And a bad understanding of generality of course.That doesn't mean you aren't skillings, no just you lack a little of whole point of vue.It is the reason why you invent decoherences and not real deterministioc convergences.You can invent extradimensions of your pc , that won't change the universal 3D dynamic and its motions of rotations by spheres.
Study a little the biology dear thinkers!!! Really you shall see the real finite proportions towards the Planck scale.The mass is the mass and it is real, the computing is totally different.as is physics /maths.
Regards
Steve
Philip Gibbs posted 24-Cell and the Kochen Specker theorem, which is a continuation of correspondence between n-partite qubit entanglements and BPS black holes. This concerns an interesting paper which illustrates how the Kochen-Specker theorem, similar to the Bell theorem, is a consequence of the symmetries of the 24-Cell.
There is in my opinion a "stringy" interpretation of this. F_4 is the isometry group of the projective plane over the octonions. There are extensions to this where the bi-ocotonions CxO have the isometry group E_6, HxO has E_7 and OxO has E_8. This forms the basis of the "magic square." F_4 plays a prominent role in the bi-octonions, which is J^3(O) or the Jordan algebra as the automorphism which preserves the determinant of the Jordan matrix
The exceptional group G_2 is the automorphism on O, or equivalently that F_4xG_2 defines a centralizer on E_8. The fibration G_2 --> S^7 is completed with SO(8), where the three O's satisfy the triality condition in SO(8). The G_2 fixes a vector basis in S^7 according to the triality condition on vectors V \in J^3(O) and spinors θ in O, t:Vxθ_1xθ_2 --> R. The triality group is spin(8) and a subgroup spin(7) will fix a vector in V and a spinor in θ_1. To fix a vector in spin(7) the transitive action of spin(7) on the 7-sphere with spin(7)/G_2 = S^7 with dimensions
dim(G_2) = dim(spin(7)) - dim(S^7) = 21 - 7 = 14.
The G_2 group in a sense fixes a frame on the octonions, and has features similar to a gauge group. The double covering so(O) ~= so(8) and the inclusion g_2 \subset spin(8) determines the homomorphism g_2 hook--> spin(8) --> so(O). The 1-1 inclusion of g_2 in so(O) maps a 14 dimensional group into a 28 dimensional group. This construction is remarkably similar to the moduli space construction of Duff et al. .
The so(9) is not the most general symmetry of J^3(0). There exists a permutation on the three scalars z_0, z_1,z_2 and {\cal O}^3. This means there is an additional automorphism so(3). The more general automorphism is then F_4. The quotient between the 52 dimensional F_4 and the 36 dimensional so(9) ~ B_4 defines the short exact sequence
F_4/B_4:1 --> spin(9) --> F_{52\16} --> {\cal O}P^2 --> 1,
where F_{52\16} means F_4 restricted to 36 dimensions, which are the kernel of the map to the 16 dimensional Moufang or Cayley plane OP^2. Geometrically the F_4 define the symmetry of the 24cell, called the icositetrachoron or polyoctahedron, according to 24 octahedral cells. The B_4 also defines a more restricted symmetry on the 24 cell according to 16 tetrahedral cells and 8 octahedral cells. The 8 octahedral cells define the {\bf 8}_0, or so(8) in the J^3(O), while the 16 tetrahedral cells are mapped to the OP^2. This means on the algebraic level f_4 ~= so(8)()V()θ_1()θ_2 [here () = \oplus], which explicitly describes the triality condition the three octonions with the so(8). More generally according to octonions f_4 ~= so(O)()O^3, and f_4 diagonalizes the Jordan cubic matrix.
The 24-cell has the largest group representation F_4 in 52 dimensions, of which the SO(9) in 36 dimension defines a short exact sequence between spin(9) and the Moufang plane OP^2. B_4 ~ SO(9) defines the symmetry of the 24-cell by 16 tetrahedral and 8 octahedral cells. The elements of the exceptional Jordan matrix is composed of elements V_{ab} which are accompanies by 16 superpartners θ_{ab}, where the indices a and b indicate internal elements which transform these elements to N\times N matrices in SU(N). The SU(N) may be contained in the HxO, with E_7 structure, for N = 8, where SU(8) exists in the 64 dimensional quaternion-octonionic space. This obtains for a single D-brane, in particular here a D0-brane, where for N > 1 this gauge group is SU(8)^N , or the embedding group SU(8N). The Lagrangian assumes the form
L = (1/2)(tr(∂_μV_i)^2 - (1/2g)tr[V_i, V_j]^2 - 2{θ-bar}_iγ_j[θ^i, V^j]),
where integer the indices i, j denote the matrix indices.. Here the superpartners to the vectors V transform as spinors under the SO(9) transverse rotations, and the matrices V_{ab}, θ_{ab} (vectors and spinors in J^3(O)) are components in a 10 dimensional super Yang-Mills space. This lagrangian is applied as the SO(9) theory in the BFSS.
The 36 sets of 4 mutually orthogonal rays is contained in F_{52\16} above. The short exact sequence defines the f_4 ~= so(9)()S^9 in the short exact sequence above. This means the K-S theorem is a consequence of qubit structure which has the Eguchi-Kostant isomorphism with black holes.
Cheers LC
Dear Lawrence
To my mind 24 Cell theory-redundant theory.
I can imagine 12 Cell theory because in Standard Model well known 2 facts:
1.As is well known all matter is made of 12 particles: 6 quarks and 6 leptons.
2.Standard Model describes their electromagnetic, weak and strong interactions, 12 leptons (including antiparticles), 12 quarks (including antiparticles) and 12 bosons (8 gluons Three gauge boson and a photon).
Yuri
Tha naonymous above is me, the author. I sometimes find these web pages do not always automatically reference the author and put anon instead.
The number of gluons in the SU(3) QCD gauge theory is due to the dimension of the group dim(S((n)) = n^2 - 1, which is 8. This emerges from the 8()1 irreducible representation of the group, here () means oplus. The other representation is 3(x)3, where (x) means otimes, which describes doublets for quarks and leptons.
F_4 is a much larger group, and has dimension dim(F_4) = 52. 48 of this 52 are the roots of the F_4 as a complex or self dual system. This is considerably larger than the SU(3). The F_4 lattice is a four dimensional body-centered cubic lattice. This lattice defines a ring called the Hurwitz quaternion ring. The 24 Hurwitz quaternions of norm 1 form the 24-cell. The Weyl group of F_4 is the symmetry group of the 24-cell of F_4 which is generated by reflections through the hyperplanes orthogonal to the F_4 roots. This is a solvable group of order 1152. This can be seen in that each of the 48 roots has 24 reflections. These are the Hurwitz integral quaterions.
Cheers LC
Dear LC
If Feynman, and earlier Stueckelberg, proposed an interpretation of the positron as an electron moving backward in time, all anti-particle will be redundant and N=12 would be right solution.
Yuri
