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Adding an answer (I hope) to the Quantic side of the question.
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Eric Arnéo Vespira Kengne
Eric Arnéo Vespira Kengne
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The answer given by @user1504 to the question [1] illustrates in a clear way, (being a profane in quantum theory, I could be mistaken), the link between the problem of Mass-Gap and the mathematics, including the holomony, the theory of hilbertian measured spaces...

However, continuing with the example of the geometry defined by a metric, the quantisation of such theory corresponds to the quantisation of gravitation (at least when the metric is lorentzian). In this context, a link between the Mass-Gap problem and mathematics is the proof by an effective method (quantification method), of the positive mass theorem. The positivity of the mass is intimately linked to the problem of the classification of differential varieties. This classification involves the most sofisticated devices of geometric analysis such as: the Atiyah-Singer index theorem, the Dirac operators, Seiberg-Witten operators, the variation calculus, the Ricci Flow, ... Cf [2; 3; 4].

Note. The positive mass theorem of J. Lohkamp asserts that: under the hypothesis of the dominant energy condition, the total mass for any casenon-vacuous isolated relativistic gravitational systems should be positive. Cf introduction of [3] for a bright presentation of all of concepts (and very more). It's at this point, I think (and if I well understand the answer of @user1504 given to the question [1]), that a constructive proof (quantised proof) of a positive mass theorem (perhaps without any hypothesis on the energy regime) is linked (is equivalent?) to the Mass-gap problem for the quantised gravity (if any...).

I would like to thank very much @user1504 for his answer to question [1] and for his remark on the previous version of this answer.

The quantum theory seems so fascinating, unfortunately, it falls out of my field of competence (for now).

As I said in the previous version of this answer, these ideas are borrowed from the works of so many geometers that I can not precisely say of whom these ideas are (if I have grasped something about these works).
That

However, I will name a few, without any intention to underestimate someone (by omission): A. Banyaga, R. Bryant, Y. Choquet-Bruhat, D. Christodoulou, P. T. Chrusciel, S. Donaldson, M. Gromov (The Father-Christmas in problems and results of geometry), N. Hitchin, S. Klainerman, C. LeBrun, P. Michor, R. Penrose, S. T. Yau.

That they all, all my recognition and my admiration.

[1 ]Statement of Millenium Problem: Yang-Mills Theory and Mass Gap

[2 ]https://arxiv.org/pdf/1704.05490

[ 3]https://arxiv.org/pdf/1612.07505

[4 ]https://arxiv.org/pdf/1908.10612

In any case, these ideas are borrowed from the works of so many geometers that I can not precisely say of whom these ideas are (if I have grasped something about these works).
That they all, all my recognition and my admiration.

The answer given by @user1504 to the question [1] illustrates in a clear way, (being a profane in quantum theory, I could be mistaken), the link between the problem of Mass-Gap and the mathematics, including the holomony, the theory of hilbertian measured spaces...

However, continuing with the example of the geometry defined by a metric, the quantisation of such theory corresponds to the quantisation of gravitation (at least when the metric is lorentzian). In this context, a link between the Mass-Gap problem and mathematics is the proof by an effective method (quantification method), of the positive mass theorem. The positivity of the mass is intimately linked to the problem of the classification of differential varieties. This classification involves the most sofisticated devices of geometric analysis such as: the Atiyah-Singer index theorem, the Dirac operators, Seiberg-Witten operators, the variation calculus, the Ricci Flow, ... Cf [2; 3; 4].

Note. The positive mass theorem of J. Lohkamp asserts that: under the hypothesis of the dominant energy condition, the total mass for any non-vacuous isolated relativistic gravitational systems should be positive. Cf introduction of [3] for a bright presentation of all of concepts (and very more). It's at this point, I think (and if I well understand the answer of @user1504 given to the question [1]), that a constructive proof (quantised proof) of a positive mass theorem (perhaps without any hypothesis on the energy regime) is linked (is equivalent?) to the Mass-gap problem for the quantised gravity (if any...).

I would like to thank very much @user1504 for his answer to question [1] and for his remark on the previous version of this answer.

The quantum theory seems so fascinating, unfortunately, it falls out of my field of competence (for now).

As I said in the previous version of this answer, these ideas are borrowed from the works of so many geometers that I can not precisely say of whom these ideas are (if I have grasped something about these works).

However, I will name a few, without any intention to underestimate someone (by omission): A. Banyaga, R. Bryant, Y. Choquet-Bruhat, D. Christodoulou, P. T. Chrusciel, S. Donaldson, M. Gromov (The Father-Christmas in problems and results of geometry), N. Hitchin, S. Klainerman, C. LeBrun, P. Michor, R. Penrose, S. T. Yau.

That they all, all my recognition and my admiration.

[1 ]Statement of Millenium Problem: Yang-Mills Theory and Mass Gap

[2 ]https://arxiv.org/pdf/1704.05490

[ 3]https://arxiv.org/pdf/1612.07505

[4 ]https://arxiv.org/pdf/1908.10612

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Eric Arnéo Vespira Kengne
Eric Arnéo Vespira Kengne
  • 81
  • 7

A geometry (differential say) is essentially about the study of a class of connections on a differentiable fiber area (the E. Cartan's viewpoint). This class is defined in general by imposing that the associated "covariant derivation" preserves a structure; the structure itself is very often represented by a tensor (said structure tensor); the best example being the metric tensor defining a pseudo-riemannian structure on a smooth manifold (when the topological conditions are matched).
The problem may then arise to seek the canonical form of the associated quadratic form. This problem is essentially equivalent to the search for a suitable frame; i.e, the search for a gauge potential for the group of the rotations of the metric.

The Yang-Mills equations system is of Euler-Lagrange type. The action density being a "norm" of the curvature of a class of connections. As a result, the Yang-Mills potential field (in fact, the Yang-Mills connection 1-form), solution of the Yang-Mills equations is an "extremalization" of a connection in the class of connections, producing by the associated covariant derivation, the Yang-Mills Strength Field.
This is an account about the fact that Yang-Mills is linked to the search for the best connection (from my viewpoint).

An extremalized connection 1-form necessarily makes it easier, to approach a "canonical form". With well-designed constraints, it is possible to obtain in principle (I think), the canonical form itself.
This is an account about the fact that Yang-Mills is linked to the search for a canonical form of a given (or even looking for) geometry, would only it be in the case where the geometry is defined by a pseudo-riemannian metric structure. This understood in the sense that we give in paragraph 1.

In any case, these ideas are borrowed from the works of so many geometers that I can not precisely say of whom these ideas are (if I have grasped something about these works).
That they all, all my recognition and my admiration.

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