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We say that something is symmetric if there is some transformation we can perform on that object that leaves some property unchanged. The set of symmetry transformations of an object form a group, and the name of this group is used as the name of the symmetry of the object.

It's dilatation symmetry $\leftrightarrow$ scale invariant. …
answered Oct 4 '18 by jak
I'm trying to build some intuition for a very particular definition of the notions global and local gauge symmetries. The definition goes as follows and appears, for example, in "Quantum Field Theory …
asked Oct 2 '18 by jak
The punchline of Goldstone's theorem is well known. When a continuous symmetry breaks necessarily, new massless (or light, if the symmetry is not exact) scalar particles appear in the spectrum of … possible excitations. There is one scalar particle—called a Nambu–Goldstone boson—for each generator of the symmetry that is broken, i.e., that does not preserve the ground state. The Nambu–Goldstone …
asked Nov 16 '17 by jak
It's not true that we only care about groups. Instead, for most applications Lie algebras are actually more important. To quote Ed Witten in his review on "Physics and Geometry" Experiment tells …
I'm referring here to invariance of the Lagrangian under Lorentz transformations. There are two possibilities: Physics does not depend on the way we describe it (passive symmetry). We can choose … transformations. Physics is the same everywhere, at any time (active symmetry). Another perspective would be that translation invariance in time and space means that physics is the same in the whole universe …
asked Jan 21 '15 by jak
Of course, strictly speaking, a symmetry is always a transformation that leaves a given object unchanged. But I'm curious why observable symmetries of physical systems are exactly those … transformations that map solutions of the equation of motion into other solutions. Concretely, given some differential equation $$\mathcal D \Lambda (x) =0$$ a symmetry of the system described by this …
In analogy with the more common notion "isospin space", I would say the corresponding space for $U(1)$ gauge symmetry is "charge space". For example, one book which uses this notion is Particle Astrophysics by Donald H. Perkins. …
answered Oct 8 '18 by jak
An important aspect in the Hamiltonian formulation of Classical Mechanics are canonical transformations which provide maps between different sets of canonical coordinates. These canonical coordinates …
The standard name seems to be point transformations. And there do not seem to be any restrictions. To quote from page 120 in Lagrangian and Hamiltonian Mechanics by Melvin G. Calkin: "We saw that …
The symmetry connected to Baryon/Lepton Number conservation is, as far as I understand, global U(1) symmetry (which is called here global gauge invariance). Does anyone know of an explicit computation of this, using Noether etc.? Any idea, link or book recommendation would be much appreciated …
asked Sep 25 '14 by jak
$(=diagonal generator) gets a vev. There are four such generators in$24$. Which fermions do get a mass after symmetry breaking and is there any difference if the Higgs field corresponding to$H_1 … Cartan generators get a vev. Is this correct? My problem with this observation is that the subgroup we are breaking to depends heavily on the choice of the Higgs field. The subgroup after symmetry
asked Jul 22 '15 by jak
For a given gauge symmetry $G$, we get via Noether's theorem conservation laws $$\partial_\mu j^\mu = 0 .$$ Do these conservation laws still hold, when $G$ gets broken spontaneously through a non-zero vacuum expecation value of some scalar field? …
Since our fundamental laws are invariant under rotations. Hence, we say that spacetime isotropic. Now Quantum Mechanics is invariant under (global) complex rotations ($U(1)$ transformations. Hence, w …
The conjugate momentum density, following as a conserved quantity with Noethers Theorem, from invariance under displacement of the field itself, i.e. $\Phi \rightarrow \Phi'=\Phi + \epsilon$, is given …
Symmetries correspond to specific properties of the space in question. translation symmetry $\leftrightarrow$ homogeneity, rotational symmetry $\leftrightarrow$ isotropy What property is related to invariance under dilatations? …