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The time reversal and chiral symmetry are special because they are antiunitary symmetries, in contrast to the other unitary symmetries like translation and rotation symmetries. Antiunitary symmetry operation involves complex conjugation of the wave function of the system, which is a non-trivial operation beyond unitary transforms. Unitary symmetries are ...


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Your issue is that this interview was not transcribed by a physicist! What he said was "Gauge Symmetry" not "Gate Symmetry". Your googling should work better now, and here is one place to start: https://en.wikipedia.org/wiki/Gauge_theory


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I think the first helpful fact to clarify is that there are two different kinds of topological phases: there are so-called Symmetry Protected Topological (SPT) Phases (displaying 'symmetry protected topological order') and there are (intrinsic) Topological Phases (displaying '(intrinsic) topological order'). As some quick examples: topological insulators and ...


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Going the way stated in the question's title is easy: The Euler-Lagrange condition is, inherently, a condition on the action -- the statement is that the classical path is the path for which the action takes a minimum value for the path. Since this is a statement about the value of the action, and the action is Lorentz-invariant, then this minimum value is ...


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We do observe spontaneous symmetry restorations in nature. This is called an emergent symmetry. See e.g. this post. A system posses an emergent symmetry if it appears symmetric at large (coarse-grained) scales although the apparent symmetry is explicitly broken by the microscopic description (typically the Hamiltonian or Lagrangian). I can give two examples ...


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The precise statement of "self-adjoint operators generate continuous unitary symmetries" is Stone's theorem. It guarantees that there is a bijection between self-adjoint operators $O$ on a Hilbert space and unitary strongly continuous one-parameter groups $U(t)$ that is given by $O\mapsto \mathrm{e}^{\mathrm{i}tO}$. The definition of the exponential for an ...


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When we say that a lattice has a particular symmetry we mean that the lattice is mapped onto itself by the symmetry. So if I have a (2d) material which has inversion symmetry in the bulk and which has an atom at a point $(x,y)$ then inversion symmetry tells me that there is another, identical atom at $(-x, -y)$. At the surface, however, this is no longer ...


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You can only distinguish the sublattices in this case because you've tagged them A,B. The process of inversion only exchanges identical carbon with carbon, leaving the crystal physically unchanged. If you gave me a crystal with one orientation and I then returned it to you without telling you whether or not it's been inverted, you'd have no way of knowing. ...


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If $\rho$ is a generic physical quantity, e.g. mass density in this case, then spherical symmetry is represented in the form of $\rho = \rho(\lvert \vec r\rvert)$ and not $\rho = \rho(\vec r)$ with $\vec r$ being the position vector of the point at which the quantity $\rho$ is being measured.It is assumed that the center of mass of the distribution coincides ...


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Of course an anomalous global symmetry destroys the associated Ward identity, but...we don't care so much about that. The Ward identity of global symmetries is not needed for consistency of the theory. However, a broken local Ward identity completely destroys the associated gauge theory, in particular since the decoupling of the unphysical degrees of freedom ...


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For a charge distribution $\rho({\bf r'})$, the electric field at ${\bf r}$ is $${\bf E}({\bf r})=k\iiint\frac{\rho({\bf r'})}{|{\bf r}-{\bf r'}|^2}\hat{\bf R}d^3{\bf r'}$$ where ${\bf R}={\bf r}-{\bf r'}$. I think the OP's claim is at positions where $\rho \ne 0$, the above integral is infinite because the integrand blows up at ${\bf R}={\bf 0}$. I think ...


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Your thinking is correct. At least, if the sphere is made up of small point charges, then the field will be infinite as you approach them. There is a point you are missing when you say that this contradicts Gauss' law: Gauss' law only gives you the flux of the field. To get the field of the sphere from it in the textbook-way, you have to use symmetry. You ...



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