Timeline for What is a gauge theory?
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| May 29, 2017 at 16:42 | comment | added | ACuriousMind♦ | @AdomasBaliuka You seem to be using "local"/"non-local" in a different manner from physicists: A local transformation in the physics sense is one where $\epsilon$ is not constant. (No one calls the others "non-local", they're "global" instead.) Your "non-local transformations" like convolutions with a Gaußian are simply not what physicists consider when they think about "symmetries of the action". If you want to know why not, then that's an entirely different question and you should ask it separately. | |
| May 28, 2017 at 17:26 | comment | added | Adomas Baliuka | Then how about convolution with a gaussian of width $\epsilon$? This operation acts as identity for $\epsilon \to 0$ and should thus have an infinitesimal version?! Basically I'm puzzled why $\delta\phi(x)$ only depends on the values and derrivatives of $\epsilon$ and the values of $\phi$ AT THE SAME POINT $x$. Can you point towards sources that cover such transformations in some generality? I've only ever seen examples. | |
| May 28, 2017 at 14:11 | comment | added | ACuriousMind♦ | @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. | |
| May 27, 2017 at 23:06 | comment | added | Adomas Baliuka | 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. | |
| May 27, 2017 at 8:08 | vote | accept | Adomas Baliuka | ||
| May 26, 2017 at 12:21 | comment | added | ACuriousMind♦ | @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. | |
| May 26, 2017 at 8:49 | comment | added | Adomas Baliuka | 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? | |
| May 26, 2017 at 8:41 | comment | added | Adomas Baliuka | 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. | |
| May 25, 2017 at 21:48 | history | answered | ACuriousMind♦ | CC BY-SA 3.0 |