A Question
Today at lunch I was asked one of the questions that nonsense theorists are often asked: so what does this have to do with the real world? Of course, one could always launch into a (now fashionable) tirade about protocols for quantum information, or two dimensional systems and topological quantum field theories. However, since the conversation was set more in the context of quantum gravity, and the asker was mostly looking for a very simple, one line answer (after having already suffered a five minute introduction to category theory), I was at a loss to find the right words.
So here is the challenge: can you summarize categorical quantum gravity in 20 catchy words or less? We assume that our readers will not be captivated by statements along the lines of Everything is made of Strings or, more pertinently, the speed of light varies (although that is, of course, true). Rather, the phrase should capture the potential of quantum gravity to describe aspects of the world completely outside the domain of established physical theory.
So here is the challenge: can you summarize categorical quantum gravity in 20 catchy words or less? We assume that our readers will not be captivated by statements along the lines of Everything is made of Strings or, more pertinently, the speed of light varies (although that is, of course, true). Rather, the phrase should capture the potential of quantum gravity to describe aspects of the world completely outside the domain of established physical theory.
posted by Kea | 2:15 AM
16 Comments:
It's a bit like asking J. J. Thomson in 1897 what electron research would be useful for in the real world. We cannot yet comprehend nor foresee the vast implications of categorical quantum gravity.
That's sweet, kneemo, and that is the sentiment that I tried to express, but of course it probably comes across as a cop out.
symmetrical monoidal category.
An = Dn only for A3 = D3= Conformal Group.
Gauging Conformal Group = Gravity.
In the above, I sort of cheated to get within the 20-word limit by considering finite-dim as one word and = as not a word,
and by leaving details to the following references:
"Physics, Topology, Logic and Computation: A Rosetta Stone" by John Baez and Mike Stay (see page 33)
"Group Theor" by Predrag Cvitanovic (see section 15.4) - this has nice category-like diagrams that he calls bird-tracks.
"Unification and Supersymmetry" by Rabindra Mohapatra (Springer) (section 14.6) - using a MacDowell-Mansouri type mechanism to get gravity from the conformal group.
Tony Smith
PS - If you are asked "why the An and Dn series of Lie groups?",
an answer is that the An include the gauge groups of the Standard Model and the Dn give you spinors (for fermion-type things), so only for An and Dn do you get both, and that is only for A3 = D3,
and
you could go on to say that you get the Standard Model U(1) and SU(2) and SU(3) by considering the categorical bird-track diagrams for the A0 and A1 and A2 subgroups of A4.
Sorry for a typo:
I said " the A0 and A1 and A2 subgroups of A4"
when I should have said
" the A0 and A1 and A2 subgroups of A3".
Of course,
A3 = SU(2,2) = Spin(2,4) = D3 in non-compact version
and
A3 = SU(4) = Spin(6) = D3 in compact version,
which
is where I got to thinking about the 4 of SU(4).
Tony Smith
Sorry for an even worse typo:
My first comment here should have been about
this 20-word statement:
"Lie group finite-dim representations = compact symmetrical monoidal category.
An = Dn only for A3 = D3= Conformal Group.
Gauging Conformal Group = Gravity"
Tony Smith
Kea said it best, the speed of light varies. Thsi could lead to applications that today's science couldn't imagine. Just for one, energy from Black Holes could make nucllear fusion look crude.
Categorical quantum gravity should categorize and then sum graviton-exchange Feynman diagrams for all gravity charges (mass/energy like photons, electrons, etc.).
Category theory boils down the essence of the information content of quantum mechanics to the least mathematical structure possible.
Uh, since I don't understand category theory much I don't know how true this is, but as long as the audience doesn't understand it either, it might fly.
My perspective is something like this: we do not know the ontological "substance" of things very well. We arguably have some direct knowledge of what's occurring in the individual consciousness from moment to moment, but reality as a whole is conceived in ways which are inferential constructions from the concepts and percepts available to the individual consciousness. So I agree with the positivistic analysis which says that physics is in part a method for predicting sense experiences, while disagreeing that this is an appropriate point at which to stop thinking.
N-category theory, as a recursively constructed theory of the composition of (n-1)-morphisms, is a sort of general theory of the formal possibilities of structure and relation which is ontologically neutral. So it's very interesting that you naturally get n-simplexes coming out of it, and can build up various "quantum geometries" from them. I'd say, therefore, that the primary interest to me of categorical quantum gravity is as a step towards a theory of the formal ontology of nature, in that the empirical phenomenon of gravity may emerge very naturally from the gemeric combinatorics of structure.
Thanks everybody, but I suspect that none of these responses would have been acceptable, with the exception of Louise's. As kneemo points out, people want to know they can have futuristic technology (beam me up, Scotty) but we are not in a good position to imagine exactly what this might be. Louise's idea is very concrete, easily understandable, and can be stated in few words.
I will summarize: there is no categorized quantum gravity, because the rules of QM are not even well defined yet.
I mean, defined in the categorical way.
There are three approaches to quantum mechanics, Heisenberg's matrix mechanics, Schroedinger's equation and its solutions, and Feynman's path integrals.
The only one which is physically deep and leads to any understanding is Feynman's path integrals; the others are just statistical approximations. With path integrals you can see what you are doing because you are summing interaction graphs with varying geometry (for simple tree interactions, such as Moller scattering of electrons or the diffraction of light by water which Feynman illustrates brilliantly in his 1985 book QED) or both varying geometry and also varying the interaction by including spacetime loops for pair production and annihilation (which don't occur in the doubel slit experiment, refraction of light by water, or low energy electron scattering because pair production requires strong electric fields to occur).
So Categorical theory should be applied to replace the path integral. Path integrals are inherently flawed in the way the integrate continuously variable differential equations (lagrangian equations) for field interactions. This is both physically and mathematically wrong, because nature is discrete and thus discontinuous. Instead of integrating, there should be a discrete summation. Categorical theory should offer a means to do this. I'm assuming that Categorical theory is to physics today, what group theory was to physics in the 30s when Weyl and Wigner were developing and trying to popularize it (long before it was used with Noether's theorem to derive the Yang-Mills equation for charge-carrying field quanta, let alone the full Standard Model).
Somehow, the path integral should be replaced with a deeper understanding, which will throw more light on the mechanisms for cutoffs and renormaliation. Clearly these problems are due to formulation of the path integral which ignores the discrete nature of interactions. Although a perturbative expansion for any given path integral (containing successively more complex spacetime loops) results in a sequence of terms eqch discretely corresponding to a separate Feynman diagram interaction, it's physically wrong to use integrate a differential equation for which is only approximating discrete events. A true mathematical model of quantum fields should be discrete in mathematical nature, like the phenomena it models, not continuously variable!
You can go on barking for ever, but it does not mean an apple will ever fall on your head.
Qubit,
Should I presume that your insult is directed to my comment?
I'm sorry for being born if that helps. But is no reason why gravity isn't simple in nature, being mediated by radiation (gravitons) exchanged by gravitational charges (mass and energy, like photons and electrons).
The falling apple is forced to accelerate due to graviton exchange. Feynman's path integral sums a lot of graviton interactions by weighting them according to their influence. Many cancel out due to geometric reasons. E.g., if equal amounts of graviton exchange with distant masses occurs to the left and right of the apple, it is not accelerated right to left. The asymmetry is vertical.
String theorists begin with the Fietz-Pauli argument that quantum gravity is due to only the apple and the earth, thus ignoring the surrounding mass of 9 × 10^21 stars, totalling 3 × 10^52 kg.
By ignoring the 3 × 10^52 kg observable mass around us and assuming that the apple only exchanges gravitons with the earth, Fierz and Pauli found that gravitons would need to be spin-2 (180 degrees rotational symmetry, so outgoing and incoming gravitons look identical):
‘In the particular case of spin 2, rest-mass zero, the equations agree in the force-free case with Einstein’s equations for gravitational waves in general relativity in first approximation ...’
– Conclusion of the paper by M. Fierz and W. Pauli, ‘On relativistic wave equations for particles of arbitrary spin in an electromagnetic field’, Proc. Roy. Soc. London, volume A173, pp. 211-232 (1939).
This is where string theory starts, building on error. What's needed is a correct summation of graviton exchanges. I can do it geometrically using various mathematical tricks, but don't have the time to build up an elaborate mathematical obfuscation that looks professionally impressive to mainstream physicists. It would be great if Categorical theorists could sort out quantum gravity!
No!!! Definity not! It not insult to you at all, I've never even read your post! Its my 20 word contribution.
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