Abstract
Topology and geometry have played an important role in our theoretical understanding of quantum field theories. One of the most interesting applications of topology has been the quantization of certain coupling constants. In this paper, we present a general framework for understanding the quantization itself in the light of group cohomology. This analysis of the cohomological aspects of physics leads to reconsider the very foundations of mechanics in a new light.
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Notes
- 1.
This of course is another matter and not our topic here. However, it is important to recognise that the significance of non-abelian gauge theories was revived with the proof given by the renormalisation of non-abeliennes gauge theories by Hooft (1971, 1994) and Hooft and Veltman (1972) (1971–72). This lead to the resurrection of previous papers on the subject and of the standard theory with the gauge group SU(2)xU(1) electromagnetic and weak interactions and also of quantum chronodynamics (QCD) with the gauge group SU(3), the preferred model in confining quarks that are considered responsible for strong interactions.
- 2.
Listing was the first to use the term Topology (actually he used Topologie for he wrote in German) in a letter to a friend in 1836 see Pont (1974), and Listing (1847, 1861). Remark that the term Topology was not commonly present in literature prior to the 1920s (instead we find the Latin terms geometria situs and analysis situs). The first important use of a topological argument could be argued to go back as far as Euler’s solution (Euler 1736) to Konigsberg’s bridge problem in 1736. During the elaboration of his answer, Euler had the idea to associate a graph to the problem giving birth to what we now call graph theory and so, with this example, he presented one of the first combinatorial topological problems. Today graph theory is a subject in its own right.
- 3.
The homology appeared as a redoubling of abstraction; The homological forms have doubled the algebra of the geometric forms which they enveloped. We can distinguish quite clearly two movements: a birth of geometry or algebra followed by homological stabilization. From a logical point of view, a geometric object and an homological object have the same nature.
- 4.
We note sometimes this sequence as (\(M_*, \partial _*)\) (or \(M_*,d_*\)).
- 5.
Usually the non vanishing of a cohomology class in algebra, geometry, and topology, express some “failure”. Indeed, Often in math you wish something were true, but in general it is not. But, the quantification of how badly it fails, help us towards finding out a more precise statement that holds generally. The size (or dimension) of the corresponding cohomology group is a measurement of how many ways things can go wrong. If it is nice or if you can understand it completely, then you may be able to analyze all the possible failure modes exhaustively, and use that to prove something interesting. This idea can be applied in an amazingly broad set of contexts. This explain in some way the use of Cohomology to describe quantization.
- 6.
In deciding to extend the concepts of homology and cohomology outside the ideal world of mathematics, We are led to accept the use of some analogies. In physics as in mathematics, a universal concept of homological object does not exist.
- 7.
W. S. Massey (1920) listed five definitions of fibre space (Massey 1999): (a) fibre bundles in the American sense (Steenrod 1951); (b) fibre spaces in the sense of Ehresmann and Feldbau (Ehresmann 1934; Feldbau 1939; c) fibre spaces as defined by the French school (Cartan 1956; d) fibre spaces in the sense of Hurewicz and Steenrod (Steenrod 1951), and (e) fibre spaces in the sense of Serre (1951). Each of these competing definitions developed out of a mix of examples and problems of interest to the research community in topology, often marked by a national character. We will consider the origins of each of these strands and the relations among them.
- 8.
Also, they should be square integrable in an appropriate sense. This requires discussing how we equip the space of sections with a Hilbert space structure and what measure we integrate against.
- 9.
The cycles carried by a surface \(\Sigma \) are formal combinations of manifolds of dimension 3 bordered by \(\Sigma \); The partition function Z defines a form of intersection on cycles and homology occurs when we quotient by the kernel of this form.
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Acknowledgements
For the invitation to contribute to this volume and for discussion I thank Shyam Wuppuluri.
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Kouneiher, J. (2018). Topological Foundations of Physics. In: Wuppuluri, S., Doria, F. (eds) The Map and the Territory. The Frontiers Collection. Springer, Cham. https://doi.org/10.1007/978-3-319-72478-2_13
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