Timeline for Promoting of global symmetry to local one: what's the motivation?
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
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| Jul 9, 2023 at 10:34 | vote | accept | user267839 | ||
| Jul 8, 2023 at 11:22 | comment | added | user267839 | a typo: it was meant $\theta(x) \neq 0 $ in last comment | |
| Jul 7, 2023 at 8:56 | comment | added | user267839 | space dependent $\theta(x)$. Can then the "locality" principle rephrased as that since the interaction is caused by observer standing at point $x_0$, only $\theta(x) \neq 1$ with $x$ in neighborhood of $x_0$? Or does the "locality of interaction" does not mean this? (as before, the question is not if from "actual" research viewpoint this make sense, but from viewpoint of Yang Mills philosophy as a "thought experiment" assuming they were right) | |
| Jul 7, 2023 at 8:50 | comment | added | user267839 | If I understood your explanations so far correctly, could you check if following "consequence" is correct in sense of this Yang & Mills philosophy: say we have a state described by wave function $\phi$ and an observer stands at point $x_0$ and stays there. Then the observer interacts somehow physically with this state described by the wave function above. What can we say follwing the Yang Mills philosophy how this interaction (which should then "local") is described as transformation of $\phi$. Assume for simplicity the "form" of this trafo is $\psi(x) \rightarrow e^{-i\theta(x)}\phi(x)$ with | |
| Jul 5, 2023 at 15:07 | comment | added | user267839 | And then the original Yang & Mills philosophy tells that those of second type, the "physical" ones, can only be local, right? For example if a symmetry transformation caused by an observer ( or any physically caused process) would be constant, eg $\phi(x) \rightarrow \phi(x)+c$, then it would cause in spirit of this Yang Mills mantra an contradiction simply because it transforms at any point of spacetime as $\phi(x)+c$, so the effect is not local in sense above, simply since it acts nontrivially at every point in spacetime. That' s the logic behind it or do I missing the point? | |
| Jul 5, 2023 at 15:00 | comment | added | user267839 | let me try to rephrase how I understood you: you use the same notion $\phi(x) \rightarrow T(\phi(x))$ for the term "symmetry transformation", but the message is that one should keep in mind to carefully distinguish between two meanings of it: as an abstract transformation in pure mathematical sense, not coming from a physical process like interaction with an observer (these in case of beeing global provide exactly conserved objects; so are in some sense "intrinsical" symmetries), and those which "happen" as physical processes, eg as an interaction with observer. | |
| Jul 5, 2023 at 1:31 | comment | added | Andrew | My point is that you shouldn't think of a global symmetry transformation like $\phi(x) \rightarrow \phi(x) + c$ as an actual physical process, but a mathematical transformation. If that transformation leaves the underlying Lagrangian invariant, it implies the existence of some useful tools that can be used like a conserved Noether current. | |
| Jul 5, 2023 at 1:30 | comment | added | Andrew | @user267839 A global symmetry requires transforming a field or set of fields $\phi_a(x)$ by some constant amount at every point in spacetime, for example in a shift symmetry we would shift $\phi(x) \rightarrow \phi(x) + c$ for a spacetime independent constant $c$. That requires transforming $\phi(x)$ at every spacetime point $x$. If you imagine that this transformation corresponds to an actual physical process, you need a huge army to shift the field at every point. Any individual observer can only interact with the field in a neighborhood around where they are located (that's locality). | |
| Jul 4, 2023 at 23:14 | comment | added | user267839 | alright, I understand that the original motivation of Yang & Mills is no longer valid today. But what I still would like to understand is the precise meaning of following two phrases you used in your answer dealing with the historical part. Firstly, what precisely Yang and Mills understood by the term "locality" in this context? And secondly, what does the phrase that the "observer is only capable of transforming the fields locally" mean in more down to earth terms? | |
| Jul 4, 2023 at 20:42 | comment | added | Andrew | @user267839 The context of that quote is that historically, Yang and Mills believed symmetries should not be global. That's not how we think of things today. If a theory is invariant under a continuous global symmetry, then there will be various consequences covered in a QFT course: you'll be able to derive a Noether current associated with that symmetry, the correlation functions will transform linearly under the action of the global symmetry group, etc. | |
| Jul 3, 2023 at 14:38 | comment | added | user267839 | So far I understand it, Yang and Mills originally seeked for a theory beeing local and Lorentz invariant. And they conjectured that such theory cannot have global symmetry if it should be compatible with phyical observations, that's what you mean in your answer? My concern is just that I not understand your phrase that " an observer is only capable of transforming the fields locally" as statement. Could you formalize it? Does the statement as such says in physical terms that " any observable couldn't be invariant with respect to glocal symmetries, only the local ones"? | |
| Jul 3, 2023 at 14:18 | comment | added | user267839 | yes, but I would like from didactical point of view firstly take a glace at the original motivation, before proceeding to the fact that it finally turned out to be wrong. Do you know the original source where the motivation is discussed? My concern is really just about the correct phrasing of the original motivation (which as you said turned out to be wrong). Could you check if I phrase the orginal motivation correctly? | |
| Jul 3, 2023 at 13:48 | comment | added | Andrew | @user267839 It's perfectly possible to have a local and Lorentz invariant theory with a global symmetry and well-defined observables. As I said, the historical motivation of Yang and Mills to consider local symmetries is no longer considered to be correct, even though the resulting Yang-Mills theory is correct. There's a lot of examples of wrong motivations leading to correct results in the history of QFT, like the Dirac's sea of negative energy fermions. | |
| Jul 2, 2023 at 23:24 | comment | added | user267839 | Does it mean that if we require locality & Lorentz invariance, then for a (matter) field with dynamics described by a Lagrangian having a global symmetry the concept of observables would be not well defined? Or do I misunderstand this point? | |
| Jul 2, 2023 at 23:07 | comment | added | user267839 | in the second paragraph: what do you mean by the statement that an observer is only capable of transforming the field locally? Why is this a direct consequence of (of course resonable) assumptions that locality principle & Lorentz invariance should be satisfied by the theory? | |
| Jul 2, 2023 at 22:44 | history | answered | Andrew | CC BY-SA 4.0 |