Exotic differentiable structures and general relativity

Brans, Carl H.; Randall, Duane

doi:10.1007/bf00758828

Cited by 26 publications

(23 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In that case, Mostow rigidity [49] implies the topological invariance of the volume (see Proposition 6.1)! Thus, there is no scaling of the volume so that the contribution to the expectation value of the volume (8) cannot be neglected (see the examples (14,16)). Furthermore, if we consider the the scaling parameter ℓ defined by (12) and argue via a consistent quantum field theory based on the Chern-Simons action then Witten [47] showed that this parameter must be quantized.…”

Section: Resultsmentioning

confidence: 99%

“…How does the observable or its expectation value depend on the smoothness structure? In the case of the exotic R 4 the first question was discussed by Brans and Randall [16] and later by Brans [17,18] alone to guess, that exotic smoothness can be a source of non-standard solutions of Einsteins equation. The author published an article [19] to show the influence of the differential structure to GRT for compact manifolds of simple type.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Exotic smoothness and quantum gravity

Asselmeyer-Maluga¹

2010

Class. Quantum Grav.

View full text Add to dashboard Cite

Abstract. Since the first work on exotic smoothness in physics, it was folklore to assume a direct influence of exotic smoothness to quantum gravity. Thus, the negative result of Duston [1] was a surprise. A closer look into the semi-classical approach uncovered the implicit assumption of a close connection between geometry and smoothness structure. But both structures, geometry and smoothness, are independent of each other.In this paper we calculate the "smoothness structure" part of the path integral in quantum gravity assuming that the "sum over geometries" is already given. For that purpose we use the knot surgery of Fintushel and Stern applied to the class E(n) of elliptic surfaces. We mainly focus our attention to the K3 surfaces E(2). Then we assume that every exotic smoothness structure of the K3 surface can be generated by knot or link surgery a la Fintushel and Stern. The results are applied to the calculation of expectation values. Here we discuss the two observables, volume and Wilson loop, for the construction of an exotic 4-manifold using the knot 5 2 and the Whitehead link W h. By using Mostow rigidity, we obtain a topological contribution to the expectation value of the volume. Furthermore we obtain a justification of area quantization.

show abstract

Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Exotic smoothness and quantum gravity

Asselmeyer-Maluga¹

2010

Class. Quantum Grav.

View full text Add to dashboard Cite

show abstract

“…A lot of work was done in the last decades to fulfill this goal. It starts with the work of Brans and Randall [32] and of Brans alone [29,30,31] where the special situation in exotic 4-manifolds (in particular the exotic R 4 ) was explained. One main result of this time was the Brans conjecture: exotic smoothness can serve as an additional source of gravity.…”

Section: Introductionmentioning

confidence: 99%

Smooth Quantum Gravity: Exotic Smoothness and Quantum Gravity

Asselmeyer-Maluga

2016

Fundamental Theories of Physics

View full text Add to dashboard Cite

Over the last two decades, many unexpected relations between exotic smoothness, e.g. exotic R 4 , and quantum field theory were found. Some of these relations are rooted in a relation to superstring theory and quantum gravity. Therefore one would expect that exotic smoothness is directly related to the quantization of general relativity. In this article we will support this conjecture and develop a new approach to quantum gravity called smooth quantum gravity by using smooth 4-manifolds with an exotic smoothness structure. In particular we discuss the appearance of a wildly embedded 3-manifold which we identify with a quantum state. Furthermore, we analyze this quantum state by using foliation theory and relate it to an element in an operator algebra. Then we describe a set of geometric, non-commutative operators, the skein algebra, which can be used to determine the geometry of a 3-manifold. This operator algebra can be understood as a deformation quantization of the classical Poisson algebra of observables given by holonomies. The structure of this operator algebra induces an action by using the quantized calculus of Connes. The scaling behavior of this action is analyzed to obtain the classical theory of General Relativity (GRT) for large scales. This approach has some obvious properties: there are non-linear gravitons, a connection to lattice gauge field theory and a dimensional reduction from 4D to 2D. Some cosmological consequences like the appearance of an inflationary phase are also discussed. At the end we will get the simple picture that the change from the standard R 4 to the exotic R 4 is a quantization of geometry.

show abstract

“…(3) The role of exotic smooth structures in physics has been discussed in several papers. (1)(2)(3)11,26,27,33,37) Taking spacetime regions, modelled by Sh(B) Takeuti's topos, one can explain the appearance of QM effects in short distances for a classical observer. Besides, some difficulties of QFT and QG, like divergencies, (non)renormalization, definition of functional integration measure, are explained as connected with the extrapolation of classical logic, assigned to an observer, over intuitionistic regions of spacetime modeled by different topoi.…”

Section: Discussion and Perspectivesmentioning

confidence: 99%

A Model for Spacetime: The Role of Interpretation in Some Grothendieck Topoi

Król

2006

Found Phys

View full text Add to dashboard Cite

We analyse the proposition that the spacetime structure is modified at short distances or at high energies due to weakening of classical logic. The logic assigned to the regions of spacetime is intuitionistic logic of some topoi. Several cases of special topoi are considered. The quantum mechanical effects can be generated by such semi-classical spacetimes. The issues of: background independence and general relativity covariance, field theoretic renormalization of divergent expressions, the existence and definition of path integral measures, are briefly discussed in the proposal. The connection with some problems in foundations of mathematics and differential topology are also discussed.

show abstract

Exotic differentiable structures and general relativity

Cited by 26 publications

References 17 publications

Exotic smoothness and quantum gravity

Exotic smoothness and quantum gravity

Smooth Quantum Gravity: Exotic Smoothness and Quantum Gravity

A Model for Spacetime: The Role of Interpretation in Some Grothendieck Topoi

Contact Info

Product

Resources

About