If the solution to Yang Mills equations in a local neighbourhood of space -time yields two different gauge fileds, are they both necessarily related by a gauge transformation in that neighbourhood.

If that is true, I require a complete proof and it would be helpful if somebody can suggest a good reference for this.

*Posted By :*Venkateshan

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Venky, I am not very sure this is what you want. The short-answer to your question is yes, locally you can formulate a kind of uniqueness theorem -but you have to qualify it with appropriate conditions on the domain and the kind of initial data you are solving for.

There is a paper by Choquet-Bruhat, Yvonne; Christodoulou, Demetrios-

Existence of global solutions of the Yang-Mills, Higgs and spinor field equations in $3+1$ dimensions. Annales Scientifiques de l’École Normale Supérieure Sér. 4, 14 no. 4 (1981), p. 481-506 which deals with this question. Even if it by itself is not useful, you might find the references there useful.

Then, there are papers by Douglas M. Eardley and Vincent Moncrief

The global existence of Yang-Mills-Higgs fields in $4$-dimensional Minkowski space. I. Local existence and smoothness properties

Source: Comm. Math. Phys. 83, no. 2 (1982), 171–191

and The global existence of Yang-Mills-Higgs fields in $4$-dimensional Minkowski space. II. Completion of proof

Source: Comm. Math. Phys. 83, no. 2 (1982), 193–212

And further, the results of the previous paper has been extended to generic spacetimes in the paperGlobal existence of solutions of the Yang-Mills equations on globally hyperbolic four dimensional Lorentzian manifolds by Piotr T. Chru’sciel and Jalal Shatah

And yeah, all these are results about the

classicalYang-Mills theory. As far as I understand, any result about its quantum counterpart are more difficult to establish – in fact, as you might already know Quantum Yang-Mills theory is literally one of the million-dollar problems listed with the likes of P vs NP, Poincare Conjecture and Riemann Hyposthesis.I hope this helped. And by the way, I don’ t understand very well the mathematical structures which enter into the papers above. So, don’t ask me anything about the fine details of the proofs – for one, I’m yet to sit and read functional analysis properly to be comfortable with Sobolev spaces 😉