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  • $\begingroup$ Thank you very much for your answer. The book of Henneaux and Teitelboim has been on my radar for a while, it's probably time to read it. What I think to have learned from your answer, as well as the link provided by Qmechanic in his second comment to the question, is that a gauge theory in standard terminology can mean different things and some theory being a gauge theory is not equivalent to it having gauge transformations. As I understand further, according to your definition every field theory has gauge symmetries. Im not sure I understand the criterion though. $\endgroup$
    – Adomas Baliuka
    Commented May 26, 2017 at 8:41
  • $\begingroup$ If one has a function from the space of fields to itself, $\phi\mapsto\phi'$, how does one decide if it depends non-trivially on spacetime? Clearly the requirement "$\phi'(x) $ should only depend on $\phi(x ) $" (equivalently, given the transformation and field value at a single point $x $ one can compute the transformed field value at $x $) is not what one wants to define trivial dependence. Poincare transformation do not satisfy above requirement. Should one say that "there exists $y $ such that $\phi'(x) $ depends only on $\phi (x) $ where $y $ can be deduced from the transformation map? $\endgroup$
    – Adomas Baliuka
    Commented May 26, 2017 at 8:49
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    $\begingroup$ @AdomasBaliuka A generic (infinitesimal) field transformation dependent on a parameter function $\epsilon(x)$ looks like $\delta\phi = \epsilon (\Delta \phi) + (\partial^\mu \epsilon) (\Delta \phi)_\mu + (\partial^\mu \partial^\nu \epsilon) (\Delta \phi)_{\mu\nu}+\dots$, where the $\Delta\phi_{\mu_1\dots\mu_n}$ are some expressions (polynomials) in the fields. A gauge transformation is one where $\epsilon$ is not constant. $\endgroup$
    – ACuriousMind ♦
    Commented May 26, 2017 at 12:21
  • $\begingroup$ Is that the most general transformation to first order? I'm at a loss at how you would write non-local (which I don't know how to define, just examples) transformations in that form. Say for example $\phi\mapsto$ Fourier transform of $\phi$, or some other integral transform, perhaprs non-linear. One does not usually consider such transformations, but I think that's besides the point, or alternatively I would want a definition of what kinds of transformations are usefull to consider and WHY. $\endgroup$
    – Adomas Baliuka
    Commented May 27, 2017 at 23:06
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    $\begingroup$ @AdomasBaliuka Note that I wrote infinitesimal. Something like a Fourier transform has no infinitesimal version, it's a discrete, not a continuous transformation. We are often only considering continuous transformations/symmetries in physics because Noether's first and second theorem require such continuous symmetries. $\endgroup$
    – ACuriousMind ♦
    Commented May 28, 2017 at 14:11