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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?
Related : what are some mechanics examples with a globally non generic symplecic structure
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 ?
Jun
12
comment What does it mean that there is no mathematical proof for confinement?
Why a non-singlet bound state imply free gluons?
Jun
12
comment What does it mean that there is no mathematical proof for confinement?
If we assume that finite energy states are global-SU(3) singlets then it would imply that there is a mass gap. However, I don't understand how the converse is true. There may be a mass gap but, in principle, there may still exist finite energy states which are not global-SU(3) singlets. No?