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Homemathematical programs in quantum gravity

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# mathematical programs in quantum gravity

There are several distinct research programs aimed at developing the mathematical foundations of quantum gravity theories. These include, but are not limited to, the following.

# 0.1 Mathematical programs developments in quantum gravity

1. The twistors program applied to an open curved space-time (see refs. [1, 2]), (which is presumably a globally hyperbolic, relativistic space-time). This may also include the idea of developing a

*‘sheaf cohomology’*for twistors (see ref. [2]) but still needs to justify the assumption in this approach of a charged, fundamental fermion of spin-3/2 of undefined mass and unitary ‘homogeneity’ (which has not been observed so far);2. The

*supergravity*theory program, which is consistent with supersymmetry and superalgebra, and utilizes*graded Lie algebras*and*matter-coupled superfields*in the presence of*weak*gravitational fields;3. The no boundary (closed),

*continuous*space-time programme (ref. [1]) in quantum cosmology, concerned with singularities, such as black and ‘white’ holes; S. W. Hawking combines, joins, or glues an initially flat Euclidean metric with convex Lorentzian metrics in the expanding, and then contracting, space-times with a very small value of Einstein’s cosmological ‘constant’. Such Hawking, double-pear shaped, space-times also have an initial Weyl tensor value close to zero and, ultimately, a largely fluctuating Weyl tensor during the ‘final crunch’ of our universe, presumed to determine the irreversible arrow of time; furthermore, an observer will always be able to access through measurements only*a limited part*of the global space-times in our universe;4. The TQFT/HQFT approach that aims at finding the topological invariants of a manifold embedded in an abstract vector space related to the statistical mechanics problem of defining extensions of the partition function for many-particle quantum systems;

5. The string and superstring theories/M-theory that ‘live’ in higher dimensional spaces (e.g., $n\geq 6$, preferred $n-dim=11$), and can be considered to be topological representations of physical entities that vibrate, are quantized, interact, and that might also be able to predict fundamental masses relevant to quantum particles;

6. The ‘categorification’ and groupoidification programs ([3, 4]) that aims to deal with quantum field and QG problems at the abstract level of categories and functors in what seems to be mostly a global approach;

7. The ‘monoidal category’ and valuation approach initiated by Isham to the quantum measurement problem and its possible solution through local-to-global, finite constructions in small categories.

# References

- 1
S.Hawkings. 2004.
*The beginning of time*. - 2 R. Penrose. 2000. Shadows of the mind., Cambridge University Press: Cambridge, UK.
- 3
Baez, J. and Dolan, J., 1998b,
*“Categorification”, Higher Category Theory, Contemporary Mathematics*, 230, Providence:*AMS*, 1-36. - 4
Baez, J. and Dolan, J., 2001, From Finite Sets to Feynman Diagrams, in
*Mathematics Unlimited – 2001 and Beyond*, Berlin: Springer, pp. 29–50.

## Mathematics Subject Classification

18D25*no label found*18-00

*no label found*55U99

*no label found*81-00

*no label found*81P05

*no label found*81Q05

*no label found*

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