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    $\begingroup$ "t's massively overcounting them and you have no way to control the manner in which it does that." but what if there is a method that would make us able to control this overcounting? how can you be so sure that there is no way to do this? this is the exact point of my question. $\endgroup$
    – Yossarian
    Commented Jan 19, 2017 at 15:10
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    $\begingroup$ @AnarchistBirdsWorshipFungus There is a method to control it. It's called gauge fixing, counting only one configuration per gauge orbit. Everything you do to control the overcounting will be functionally equivalent to it. $\endgroup$
    – ACuriousMind ♦
    Commented Jan 19, 2017 at 15:10
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    $\begingroup$ sure, but maybe you can control it without fixing the gauge, somehow. Can you prove me that this is imposible? $\endgroup$
    – Yossarian
    Commented Jan 19, 2017 at 15:11
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    $\begingroup$ @AnarchistBirdsWorshipFungus No, because you haven't proposed any manner different from gauge fixing for doing so. I cannot disprove things when I have no idea what they even are. $\endgroup$
    – ACuriousMind ♦
    Commented Jan 19, 2017 at 15:13
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    $\begingroup$ In lattice gauge theory, it is possible to compute Euclidean path integrals without gauge fixing. (See answer below.) As you say, this is equivalent to computing on the quotient space or on a gauge slice, because the physics doesn't see the gauge transformations. But the answer to OP's question is "yes". $\endgroup$
    – user1504
    Commented Jan 21, 2017 at 3:23