Archive for April, 2007

GR Revisited

Sly old Carl Brannen has been busy discussing Schwarzschild black holes in flat space at PF. It seems that particle physics just wasn’t interesting enough for him.

Regulars at PF often use cool signatures on their posts. For example, turbo-1 quotes Einstein (1924):

“The ether of general relativity therefore differs from that of classical mechanics or the special theory of relativity respectively, in so far as it is not ‘absolute’, but is determined in its locally variable properties by ponderable matter.”

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Sparring Sparling IV

Since comments on blogs float like corks on the sea, I will bookmark here the interesting remarks of Matti Pitkanen and Carl Brannen on the Abell cluster A586, which is a particularly spherical galaxy cluster. The authors of this recent paper find that, for A586, the ratio of kinetic energy density to potential energy density is $-0.76 \pm 0.05$. This is quite different to the value of $-0.5$ expected from the usual virial theorem.

Ignoring interpretations involving the Dark Force, Carl pointed out that the actual value of $-0.75$ was precisely three times the expected value. This may be explained by a higher value for the speed of light, which fits nicely into the preon particle physics, but the actual value of $c$ is not really relevant.

Kinetic energy goes as $v^2 = v_{1}^{2} + v_{2}^{2} + v_{3}^{2}$ for a 3-vector $v$. Thus for the three speeds of light, forming a 3-vector $(c_1, c_2, c_3)$, the kinetic energies will add. But if we ignore the three speeds, and substitute instead a value which is the length of $(c_1, c_1, c_1)$, then we would have $c = \sqrt{3} c_1$. In the first instance the kinetic energy goes as $\frac{v^2}{c_{1}^{2}}$, while in the second it goes as $\frac{1}{3} \frac{v^2}{c_{1}^{2}}$. This is why kinetic energy at small cosmological scales is incorrectly defined by a factor of 3. Of course, at small scales this doesn’t matter.

This links the mass generators of M theory, which obey the tripled Pauli statistics, to the three time coordinates.

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Mass Gap Revisited

There appears to be a lot of work involved in completely solving the mass gap Millennium Problem. The official problem statement, by Jaffe and Witten, asks:

“Prove that for any compact simple gauge group $G$, a non-trivial quantum Yang-Mills theory exists on $\mathbb{R}^4$ and has a mass gap $\Delta > 0$. Existence includes establishing axiomatic properties at least as strong as those cited in [Osterwalder + Schrader, Streater + Wightman].”

Unlike the purely mathematical problems, not all terms in this statement are well defined. What is a Yang-Mills theory? The article suggests that a rigorous formulation of QCD would be adequate. The remaining difficulty with the statement is the one thing that people seem so comfortable with: the use of $\mathbb{R}^4$.

In M theory, as in Algebraic Geometry, we don’t like fixing number fields unnecessarily. In fact, since the numbers depend on the class of experimental question, we had better go a long way with the rationals before we even contemplate any kind of continuum limit. Fortunately, Grothendieck understood this a long time ago. Now when we discuss twistor geometry and the division algebras, we understand that these things are simply convenient models for the underlying operadic axioms.

It is still necessary, however, to display in detail how this geometry arises from the axioms. This is why we need Motivic Cohomology, because without it we will never get close to anything looking like a path integral.

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Sparring Sparling III

Nigel Cook made an interesting comment about the three times. In particular, he pointed out that “physically, every body which has gained gravitational potential energy, has undergone contraction and time dilation, just as an accelerating body does. This is the equivalence principle of general relativity.”

Note that by considering stationary objects in the Sparling metric, in the approximation of one $c$ value, we obtain

$s^2 = c^2 (t^2 + u^2 + v^2)$

which is basically Louise’s equation, provided we set a time parameter $T$ to be the length of the time 3-vector. Unfortunately, Lunsford does not appear to have a blog, but here is a link to Nigel’s discussion on this matter.

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Sparring Sparling II

G. Sparling may like to discuss three time dimensions, but as Louise Riofrio and Alain Connes often like to point out, we should understand how time is emergent, how the arrow of time arises from the expansion of space.

In lessons on special relativity we always use 1-dimensional spaces, because the relative motion of two frames happens along a line, conveniently called an $x$ axis. The Lorentz transformation relates $x$ and $t$ whilst leaving $y$ and $z$ unchanged. The metric considered by Sparling is simply a threefold version of the pair $(x,t)$, namely

$s^2 = x^2 + y^2 + z^2 – t^2 – u^2 – v^2$

where $t$, $u$ and $v$ are all time coordinates. Using imaginary time, which is only proper, the 1-dimensional case looks like the equation for a circle in the complex plane. Similarly, one might interpret the full metric as a 5-sphere in $\mathbb{C}^3$.

But what happened to the speed of light? The metric should really look like

$s^2 = x^2 + y^2 + z^2 – c_{1}^2 t^2 – c_{2}^2 u^2 – c_{3}^2 v^2$

where the $c_i$ form a 3-vector, analogous to the 3-vector for mass in M theory. Deviations from $c_i = 1$ will introduce some ellipticity into the geometry of the hypersurface. Now the homework exercise is to write out $E = m c^2$ three times.

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Sparring Sparling

Thanks to a commenter at Not Even Wrong we have this link to an article about the work of G. Sparling of twistor fame. The article is about 3 space and 3 time dimensions, and it contains this diagram which refers to three copies of twistor space. Here is the arxiv link to the paper. The article also mentions triality, Jordan algebras, category theory and condensed matter physics. It feels like Christmas!

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The Maypole

The strange people who have been hanging around this blog for a while will recall a paper by Mulase and Waldron on matrix models and quaternionic graphs.

In particular, T duality appears between the symplectic and orthogonal integrals. This involves a doubling in the size of the matrices being considered. For this reason, it might be interesting to investigate the doubling of matrix sizes in the honeycomb geometry.

Recall that in the 3×3 case, a single central hexagon appears. For 4×4 matrices, there are three central hexagons. In general, the number of hexagons is the sum of $1,2,3, \cdots , N-2$ for $NxN$ matrices, which is equal to $\frac{1}{2} N(N – 1)$. Observe that as $N \rightarrow \infty$ the increase in the number of hexagons obtained by doubling the matrix size is fourfold, since for the $\frac{N}{2}$ case the total is $\frac{1}{8} (N^2 – 2N)$. For any $N$, the number of additional hexagons is given by $\frac{1}{8} (3 N^2 – 2N)$.

By the way, the maypole is a dance (that my childhood ballet class used to perform each year) in which ribbons are knotted.

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M Theory Lesson 45

Tao mentions a paper by Speyer on a proof of the honeycomb theorem that uses no representation theory at all. It contains a theorem by Klyachko [1] which states that the additive problem is solvable for spectra $(\lambda, \mu, \nu)$ iff the multiplicative problem is solvable for $(e^{\lambda}, e^{\mu}, e^{\nu})$.

Kholodenko attacks the Gromov-Witten invariants via the multiplicative problem. The 3×3 relation

$\lambda_1 + \lambda_2 + \lambda_3 + \mu_1 + \mu_2 + \mu_3 = \nu_1 + \nu_2 + \nu_3$

is replaced by the generalised expression

$\lambda_1 + \lambda_2 + \lambda_3 + \mu_1 + \mu_2 + \mu_3 = \nu_1 + \nu_2 + \nu_3 + N (d_1 + d_2 + d_3)$

where the $d_i$ are associated to punctures on a sphere. Let $d$ be the sum of the $d_i$. Fusion rules then belong to quantum cohomology

$\sigma_{a} * \sigma_{b} = \sum_{d,c} q^d C_{ab}^{c} (d) \sigma_{c}$

with a new kind of product for classes. These coefficients give the Gromov-Witten invariants in the genus zero, three point case. In terms of monodromy matrices, Kholodenko writes

$\prod_{i = 1}^{n} \textrm{exp} (2 \pi i \frac{A_i}{d_i}) = \textrm{exp} (2 \pi i I)$

where the $A_i$ are diagonalisable matrices that produce an eigenvalue set.

[1] A. Klyachko, Lin. Alg. Appl. 319 (2000) 37-59

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Quantum Symmetry

Recall that Coquereaux likes looking at generalised Dynkin diagrams. In a paper on SU(3) graphs there is the example of the $\varepsilon_{21}$ graph, which generalises the diagram for $E_8$. This graph has 24 vertices in three sets of eight. Edge orientation is omitted from the diagram. Triality arises from considering the vertex labels mod 3. Complex conjugation corresponds to the symmetry of the central horizontal axis. There is a norm associated to this graph, which takes the value $0.5(1 + \sqrt{2} + \sqrt{6})$. The adjacency matrix for such a graph encodes an associative algebra structure which is compatible with the fusion algebra in a suitable way.

Apparently Ocneanu worked out many details of these quantum symmetries, but his work is largely unpublished. Meanwhile at The Cafe there is an amusing discussion in a comment thread initiated by the probably not-so-innocent David Corfield.

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Internalisation

An arrow f in any category C can be turned into an arrow in some category of categories by taking the functor from the category 2 into C which picks out the arrow f. Similarly, in the category of categories, one can replace a square by a quadruple of squares. The arrow 0 in the diagram represents the source of an arrow, whereas the arrow 1 is the target. Similarly, a triangle is really a hexagon, and so on.

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