Categories si! Computational logic, I don't think so. CDT is interesting but not basic. I still like Thiemann and the Phoenix program, in spite of the non-covariance of canonical quantization. But if the Phoenix crashes there are still spin foam path integrals. Therefore loops.
String Theory and M-theory get absorbed into Matrix theory in the nonperturbative regime. But Matrix theory is just an example of noncommutative geometry (or more properly nonassociative geometry). Loops, in holonomy flux *-algebra form, has much in common with Matrix theory. This is why I explored a Jordan GNS construction for a Jordan holonomy flux *-algebra. A Yang-Mills/diffeomorphism invariant state seems to exist in that case.
Categories nicely surface in noncommutative geometry as well, via K-homology. This is because an element of the K-homology group can be obtained by an object of the derived category of coherent sheaves (hep-th/9902116). K-homology, in Matrix theory, describes D-brane worldvolume configurations.
It is a challenge to generate a CDT from noncommutative geometry, though I'm not certain a strict triangulation would emerge. I'm even less sure about the role of time in a physical theory based on noncommutative geometry.
I'm voting something else entirely for now, in hope that all the above will become facets of a deeper, PF theory.
computation defines the basic units of existence [bits] and causality [logic]- nothing can be more basic/fundamental than bits that are either on or off- and no causality or dynamics more fundamental than Boolean logic which defines any possible action/relationship- all these other approaches- and indeed any possible theory of quantum gravity is merely a sub-set of possible computations- there are no conceivable systems that cannot be perfectly described and rendered through computation- therefore any formalism 'more fundamental' than computation could not posess any causal structure or quantifiable properties- it could have no form or relationships because all possible forms and relationships are types of computations- so computation is axiomatically fundamental- that is the most fundamental definable/quantifiable structure/system possible
edit:I think it is important to realize that ALL possible theories of QG-strings/loops/CDT/quantum geometry/black-hole thermodynamics- are in fact subsets or types of Computation/categories/causal sets- they are different kinds of algorithms built up from observation an mathematics-
the sooner you see this the sooner you will realize that the right path to Quantum Gravity is not endless piddling with different kinds of algorithms- instead look at the wiring of the 'computer' itself and see what algorithms naturally emerge from it
I don't see why the majority of people on this sub-forum are so anti-stringy. M-Theory has many more researchers than any other approach to quantum gravity, and it is probably for a reason. No doubt it is going through a hard moment now with the string landscape, but that doesn't mean it is wrong, maybe just that it has been heading temporarily down the wrong path. Probably when a nonperturbative formulation of M-theory is found it will provide the most unified and fundamental framework of theoretical physics.
String theory/ M-theory have branes and strings as submanifolds imbedded in a background spacetime which is not explained. What obviously needs to be done is to explain where spacetime came from to begin with and how particles arise from that. CDT seems to be the most direct approach. I think that strings and LQG will turn out to be some holomorphic properties of spacetime distortions.
Whichever one provides actual experimental data - perferably lots of it. The choice list seems light on topics that include that though...
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