The Art of Math

Anyone who wants to see some more technical details can go here.

Anyone who wants to see some more technical details can go here:

http://golem.ph.utexas.edu/category/2010/10/the_art_of_math.html

Mathematics is not just an art but to some extent also a useful tool. Did technology benefit from what belongs to the category theory on higher level as for instance theory of distributions, aleph_2 or hyperreal numbers? Perhaps it did not yet and will never do. I which John Baez good luck. In general, shouldn't those who are highly gifted and well educated in mathematics so try and live up to the obligation checking foundational questions of appropriateness for physics?

I am somewhat familiar with Grothendieck topos theory. I have also read Baez' papers that connect the Cayley number system of reals to octonions and Clifford algebraic extensions with superymmetry. I have a number of technical questions with those, but will defer them until later. To me the central question is whether category theory indicates whether Heisenberg groups have some categorical equivalence with light cone structure. In other word I suspect that quantum mechanics and general relativity are equivalent in a categorical system.

Lawrence B. Crowell

Lawrence,

Do you mean equivalent or dual? It's impossible for quantum mechanics and general relativity to be equivalent in any theory. I can only see them dual under isomorphism in a scale invariant background space. I don't see how category theory can solve the problem of background independence.

Tom

It does sound crazy to speculate that general relativity and quantum mechanics are equivalent in some ways. To say they are dual is another way of maybe saying equivalent. In an S-dual setting with a magnetic monopole the electric and magnetic fields are completely equivalent in form.

Cheers LC

I think, as you suspect, that if you design these two theories into a categorical system they could be made equivalent. That system may be able to "unify" these two apparently different descriptions of matter, however, it obviously wouldn't be able to unify matter itself. Thus, that equivalence would be akin to simply saying that matter is matter. The real trick, in my opinion, is to find some kind of relationship between quantum mechanics and general relativity that can be rendered as something that is measurable--rather than trying to find a way to equate these two models.

Lawrence,

I don't think isomorphism allows the kind of equivalence that one finds between orthogonally propagating fields. Because category theory is a theory of isomorphic relations, it assumes that relation is more fundamental than equivalence. I mean, specifically: an equation assumes properties of equality, reflexivity and transitivity. An isomorphic relation does not assume transitivity--consider Lee Smolin's view of a pure relational model*:

"R1: There is no background...

R2: The fundamental properties of the elementary entities consist entirely in relationships between these elementary entities...

R3: The relationships are not fixed, but evolve according to law. Time is nothing but changes in the relationships, and consists of nothing but their ordering..."

So things in relation are not necessarily equivalent.

Isomorphisms are static representations, but one would need demonstrate an evolutionary principle that obviates a background and drives sytem change in a self-organizing manner. This is the view I personally favor, and which led me to the conclusion that time (the least action principle) is the only fundamental property driving change in the universe. I agree in principle with Fotini Markopoulou that "Space does not exist, so time can." That's why my model is purely algebraic in real (Lebesgue measure) terms, crossing over from analysis by deriving from the complex plane a numerically precise epsilon term (eq. 4, ICCS 2006), that inserted into an otherwise perfectly ordered set of n-dimension Euclidean kissing spheres ("time barrier" preprint), leads to a dissipative n-dimensional system. This comports with the Jacobson-Verlinde model of entropic gravity, where gravity and (physical) information are identical. In my model, time, gravity and information are _all_ identical.

There is no background, because point-line duality and the Bekenstein-Mayo result that black holes form a 1-dimension information channel** support self organization independent of background space, requiring only a single quantum fluctuation in imaginary time, and allowing that the time metric is n-dimensional continuous, on a random, self avoiding walk.

Tom

*[2005 preprint] "The case for background independence." arXiv:hep-th/0507235v1

** "Black Holes are One-Dimensional." General Relativity

and Gravitation 33;12, December (2001)

Time exists, time is real,

Time is numerical order of change.

Time is exclusively a mathematical quantity.

Mathematic is real......Attachment #1: 1_Time_measured_with_Clocks_12.10..pdf

The categorical equivalency I think that might exist is between a discrete system of quantum mechanics and light cones. The discrete system of QM is the addition of paths, similar to a Feynman path integral, and concatenations of field configurations on paths. These are an algebraic field (,*). The AdS spacetime under a quotient with a discrete Klienian group has a similar structure. The categorical equivalency might exist on a certain logic level. Clearly there is no equivalency between QM and GR with regards to their differential equations. The Fotini paper has some relevance to this.

I think the Smolin background independence argument has been superseded by the AdS/CFT developments. In other words the "background" is given by the structure of elementary particles, which exist on the boundary of the AdS. The conformal completion of the AdS is then an Einstein spacetime or a de Sitter spacetime of one dimension lower. In other words the background is determined by the physics of elementary particles.

Cheers LC

Has Lee Smolin conceded this case?

I get confused over the common interchangeable use of "correspondence" and "duality" as if they are the same thing. I can't wrap my mind around that -- duality, e.g. point-line duality in geometry, is a closed relation. If correspondence were such a closed relation, it would be duality. I don't see the identity.

The key point, I think, is Smolin's emphasis on "... the real meaning of background independence, which is that fixed classical fields or global symmetries play absolutely no role in the formulation of the dynamics or observables of the theory" in his response to Joe Polchinski's response to Smolin's The Trouble with Physics here

That criticism is answered in my own theory by the introduction of non-local measure criteria. True nonperturbative correspondence would be the duality of dynamically interacting fields. We've had this discussion before, about the gulf between geometric symmetry and physical dynamics. Einstein overcame it -- isn't it time for an encore?

Tom

Of course nobody has conceded anything. Yet the natural background does appear to be based on the AdS spacetime. Why this is so probably requires understanding string theory at the Hagedorn temperature. At that temperature T ~ 1/L_{string} strings merge into a "gemish," which mimics the physics of quark-gluon plasmas. So the occurrence of the AdS is due to a phase transition, probably a quantum critical point. This would mean in analogue with a QCD meson, two quark connected by a flux tube of gluons, that D0-branes, partons or particles underlie string theory.

Cheers LC

The natural background is still the background. Background independence would require dynamically interacting spacetime, to preserve relativity. I sincerely would like to see string theory succeed; however, unless there is a higher order of mechanism than phase transition or symmetry breaking, I can't see that we have more than a static representation of physics with the action artificially inserted, as in that classic cartoon: "Here, a miracle happens."I have the same problem with Garrett Lisi's program (or any lattice symmetry) -- geometric models are just not self actualizing.

Dynamic self organization of space and time based on least action eliminates such assumptions. The range of the action is self limiting within the domain of the observables; i.e., what we see is the least of all possible particle trajectories, at any arbitrarily chosen moment. Let string theorists design a thought experiment that incorporates least action on the boundary of the hyperbolic string domain and the disjoint parabolic surface, and I think unique solutions will be easier to come by. (My own n-dimensional Euclidean kissing spheres model is a step in that direction.)

Tom

Tom,

The underlying question is whether gravity can at all be quantized in a complete form. To be honest I don't think so. The AdS spacetime, with its Einstein spacetime boundary and equivalent QCD or CFT physics, I think emerges as a phase transition. The AdS spacetime in two dimensions is an SL(2, R) group, the conformal group of quantum mechanics, and which defines classical flows that obey the sine-Gordon equation. Zamolodchikov proved how this system is S-dual to Fermi-Dirac Lagrangian with a quartic potential. The quartic potential defines Bogoliubov coefficients, which define the thermodynamics of Hawking radiation. If this is the case gravitation is formally only a semi-classical field, computable to a small finite loop order. String theory tends to reflect this to a degree, where the string defines a quantum correction to a background. The loop variable approach runs into some difficulty as well, where the Imirzi-Barbero parameter ambiguity with gamma = 3/8 instead of 1/4, and the break up of Lorentz symmetry at small distances (ruled out observationally) suggests that something funny is going on as well.

Underneath the string are D0-branes or partons. For an open strung the endpoints are analogous to quarks in a meson and the string is composed of D0-branes analogous to gluons. The closed string is analogous to a glue-ball observed in di-jet hadron scattering. String theory emerged as a way of working hadron physics, where the Hagedorn temperature was the phase transition to the quark-gluon plasma phase. Similarly the Hagedorn temperature T = 1/L_{string} is a phase transition temperature from the underlying structure of partons, partons as Fermi-Dirac particles plus bosons --- a sort of SU(N) QCD theory, to the extended structures as strings and Dp-branes for p > 0.

As for Lisis' theory, the biggest problem is that he frames the graviton with other fields in a way the Coleman-Mandula theorem rules out. Lisi's theory is clever in a formal sense with representation theory, but it suffers from this and other difficulties. The only way to frame the graviton with other fields is with supersymmetry, which is a compelling reason for thinking SUSY is real physics.

Cheers LC

I never said supersymmetry isn't real physics. In fact, I think supersymmetry is demanded by real physics -- not as a primary phenomenon, but a result.

Whatever quantum field theory has to recommend it as a background independent theory, I think at this point we have to recognize the field source as hyperbolic and extradimensional in principle while remaining true to relativity -- i.e., by not allowing the field an independent physical reality.

Tom

At the risk of making some mistake I am going to extraploate things a bit. The background turns out to be somewhat fixed, or determined by other physics. The AdS/CFT correspondence holds that the AdS_{n+1} has a boundary defined by a horizon-like condition on one of the time variables. This construction is an Einstein spacetime E_n with a conformal quantum theory that has the same content as the AdS. The AdS_4 and AdS_5 are particularly interesting, but let me look at AdS_4. This has a boundary that is a 2 space plus time boundary. The BTZ black hole in 2+1 spacetime has zero entropy with maximal BPS charge. In the interior of the AdS_4 spacetime the BTZ black hole has a correlated extremal black hole with AdS_2xS^2 structure. This is the extremal condition on an ordinary black hole in 3+1 spacetime. The AdS_2 ~ SL(2,R) which is the elementary s = ½ conformal group of quantum mechanics. Further, the AdS_2 spacetime, the Poincare disk defines geodesics which preserve volume in a manner that obeys the sine-Gordon equation. The dual to this is a Thirring model of a Fermi-Dirac field. This is very interesting, for it means there is naturally embedded in the AdS spacetime a supersymmetric structure that is inherited at the boundary. In fact Maldecena's theory is naturally N = 4 supersymmetry.

The boundary of the AdS_{n+1} is a spacetime of zero or positive constant curvature. If the boundary has zero curvature the AdS spacetime is a cylinder, where the boundary is the spacetime and the interior the holographic content. The soliton and Fermi-Dirac field contents are projected onto the boundary as supersymmetric partners. These fields have enormous quantum fluctuations near the Planck scale, and this leads to the 120 order of magnitude problem with the cosmological constant. However, these enormous quantum fluctuations may only exist in this interior region. A particle may access this enormous quantum vacuum by transforming to its supersymmetric partner, but it transitions through supersymmetry space or equivalently through this interior. This only happens at 10^{-17}cm or the TeV scale in energy. At lower energy physics on the boundary does not interact with these quantum fluctuations because supersymmetry is broken and decoupled from this vacuum gemish. Supersymmetry generators Q are such that Q^2 = 0, for they are fermionic. Q^2 = 0 defines a cohomology, so there is some topology here. The breaking of supersymmetry is equivalent to a surgical change in topology, and the violation of an Euler index. The Euler index is a quantum number, and if we are to presume quantum numbers (information) is conserved it must be countered elsewhere. It is assumed on the boundary in the transition of the spherical space to a flat space. So the can is peeled open and the outer boundary stretched out into an infinite spacetime. The AdS Poincare disk is rolled out onto the Poincare half-plane.

Cheers LC

Lawrence,

I think we are very close, though our mathematical methods differ.

If you mean by Euler index, the Euler characteristic (Xi) or Euler number, then sidebar 2 (pp 32-35) in "time barrier" explains the internal quantum fluctuation and topology with surgery that you describe:

With the 6 points of a 3-ball embedded on the 10 dimensional manifold, just a single quantum fluctuation at the sub Planck scale on the interior point of the 3-axis intersection (which we may interpret as a quantum field potential at the string scale) is destabilizing, resulting in an interior split of the Xi = -3 object (hyperbolic)into a normalized Xi = 2 2 (parabolic) by the Banach-Tarski construction. We find then that the interior plane, by fixed-point hyperbolic projection is open (flat, infinite) and the external manifold (Xi = 2) is closed (finite, bounded at the Planck energy), implying particle properties. So we have a fully relativistic model of spacetime-particle dynamic interaction, with quantum field string energy in the 10-dimension limit.

The deeper implication is that spacetime structure itself originates at the sub Planck scale. IOW, we don't need the assumption of an infinitely dense matter state at the origin (big bang). All we need is a self organized state of space and time; i.e., spacetime is a structure of measure zero in an n-dimensional scale invariant quantum field. This gives the big bang cosmology a probability 1.0, independent of any specific spacetime point. Which means that time alone seeds particle action at the string scale -- the 2 2 topology of disjoint 3-balls is self-sustaining in a scale invariant evolving complex system, finite and unbounded, consistent with GR. I show a very slight information loss by n-dimensional dissipation, because time (and gravity) is identical to physical information and the time metric is n-dimension continuous by analytic continuation.

I am pretty confident that my model can also deal with the mass hierarchy problem. Randall and Sundrum have already shown that the problem can be met by an extradimensional theory in a fully relativistic theory. And that's where I think we're all in agreement -- that any comprehensive unity of QM and GR has to be fully relativistic. My own aesthetic sense is closer to Smolin, though -- I want full background independence as well.

Tom

I had this idea at one time about using Banach Tarski theorem to duplicate Planck volumes and the like. I still speculate that exactly at the Planck scale physics in a sense ends and maybe there is that sort of complete chaos.

Cheer LC

I think complete chaos is actually the beginning of physics. Perfect randomization, after all, guarantees at least one ordered sequence. We know by Brouwer's fixed point theorem that however scrambled the topology, we can count on a point set.

Tom