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 Jul 21 comment How to prove that a spacetime is maximally symmetric? Ok. Thanks for the clarification. Jul 21 comment How to prove that a spacetime is maximally symmetric? Sorry if its a trivial question but I don't understand why you didn't consider the possibilities $x^2+y^2+z^2+w^2-t^2=-r_0^2$ and $x^2+y^2+z^2-t_1^2-t_2^2=r_0^2$? Nov 15 comment Path integral derivation of the state-operator correspondence in a CFT The space of boundary conditions at the origin is anyway much smaller than on a circle of nonzero radius and hence can't be the whole configuration space. Nov 15 comment Path integral derivation of the state-operator correspondence in a CFT @Prahar ya you are right. I read that in a hurry. Perhaps the argument is that to each primary state we can assign a local field (through some algorithm that i don't remember) and then fields corresponding to other states can be generated by applying differential operators (Ln's) to the primary fields. But, I have never encountered any rigorous proof of these statements. Nov 15 comment Path integral derivation of the state-operator correspondence in a CFT I mean the map may not be surjective Nov 15 comment Path integral derivation of the state-operator correspondence in a CFT Also, the correspondence is 1-1 in one direction i.e. to each local functional we can assign a state on the boundary. However i am not sure if to each state specified on the boundary we can construct a local functional or not. Nov 15 comment Path integral derivation of the state-operator correspondence in a CFT @Prahar the wavefunctions are of the form $\psi(\phi_i(\sigma))$ where $\phi_i(\sigma)$ is field specified on a circle of nonzero radius. On a circle of zero radius (i.e. a point) there are no (nontrivial) boundary conditions to be specified and we can at most associate a local functional depending upon the value of the field and its derivatives at that point. So by taking the r->0 limit of a wave function assigned to the inner boundary of an annulus we may only get a local functional at the origin and not a wave function. Nov 15 comment Path integral derivation of the state-operator correspondence in a CFT @Prahar At the origin $\psi(\phi_i)$ is not a wave function. Its rather a local functional of the field. Wave functions are assigned to proper boundaries. However, I am not sure if the above map $T_D$ is invertible. Aug 20 comment The states of the adjoint representation correspond to the generators The matrices $T_a$ span a Lie algebra $g$ which, in particular, is also a vector space(or state space). The adjoint representation is a representation of $g$ on itself. That is, the states are again the matrices $T_b$ (or in bra notation $|T_b>$), and a matrix $T_a$ acts on state $|T_b>$ as $T_a|T_b>=|[T_a,T_b]>=if_{abc}|T_c>$. Jul 16 comment Is there a physical system whose phase space is the torus? +1: Just a minor point: mathematically one may consider non-relativistic massless particles. But no such particles are found in nature. Jul 16 comment Is there a physical system whose phase space is the torus? Oh sorry, I was confusing 2-torus with double-torus i.e. genus 2-surface. Thanks. Jul 16 comment Is there a physical system whose phase space is the torus? Yeah, but usually the space of flat connections is defined as $(\pi_{1}(\Sigma)\to G)/G$ i.e. space of group homomorphisms from the first fundamental group to G, quotiented by conjugation. Since U(1) is abelian so conjugation is trivial. On the other hand maps from the fundamental group of 2-torus to U(1) is 3-dimensional. Jul 15 comment Is there a physical system whose phase space is the torus? @Ryan How do we see that the moduli space of U(1) flat connections on a 2-torus is itself a 2-torus ? Since there are 4 homology cycles with one relation between them, it seems that the dimension of the space should be three rather than two. Can you please point out what I am missing. Jul 15 comment Is there a physical system whose phase space is the torus? Jun 20 comment Are there massless bosons at scales above electroweak scale? TwoBs I think, due to the locality property of quantum field theories, what I said can't be correct. Do you agree with what @anna v is saying? Jun 20 comment Are there massless bosons at scales above electroweak scale? I am bit confused about your comment about the difference between early universe and LHC. Do you mean following - "We are stuck in a vacuum where the higgs field has a specific vacuum expectation value. We can't come out of this vacuum by feeding energy into electroweak fields only in a local region of spacetime. However if we do so globally at each point of space, then above the electroweak scale the gauge fields will be exactly massless". Jun 17 comment Intersecting Wilson loops in 2D Yang-Mills @ACuriousMind Use formula 3.30 for an intersecting loop in the notes by Moore-Cordes-Ramgoolam, and try to do the group integrations using equation (7) in the above answer. I think it should give 6j symbol at the vertex. I too will try to do this calculations if I get time. Jun 16 comment Why didn't electroweak symmetry breaking happen earlier than it did? "The electroweak transition is a phase transition" - But not in the usual sense. Elecroweak theory is a quantum field theory in Minkowski space and its statistical interpretation makes sense in 4D Euclidean space rather than in 3D space as for ordinary transitions. We can say that initially electroweak fields carried a very high energy and hence SU(2) gauge symmetry was manifest, but after a while the system lost its energy to other fields and fell into a particular choice of vacuum out of infinitely many choices. This is what is called electroweak symmetry breaking. Jun 12 comment What does it mean that there is no mathematical proof for confinement? Yes, but I don't understand how the existence of mass gap can imply non-existence of such states. Jun 12 comment What does it mean that there is no mathematical proof for confinement? I mean a bound state which is not in singlet representation of global SU(3). Can't such states exist ?