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May
21
comment If a symmetry operator S in a QFT annihilates the vacuum, why does S preserve the space of 1-particle states?
@Void 2) Assuming Lorentz symmetry, the renormalization factors Z in ψ R =Z(ψ)ψ are Lorentz scalars, thus nothing essential changes in the analysis when the true renormalized field is used: Its transformation properties remain the same.
May
21
comment If a symmetry operator S in a QFT annihilates the vacuum, why does S preserve the space of 1-particle states?
@Void 1) Please think for a moment of $Q$ as the electric charge operator, you can write it uzing the Gauss' law as a surface integral of the electric field over a very large sphere at infinity. Due to the large distance these fields do not produce singularities when multiplied by other fields, thus the only singualrities coming from the commutator are those due to the local field $\psi$, thus the commutator itself is also a local field at $x$.
May
5
comment The $U(1)$ charge of a representation
@JakobH The generator of the U(1) charge $Y_{\gamma}$ needs by definition to commute with all root generators $E_{\gamma}$ of the unremoved nodes. The Cartan-Weyl generator $H_i$ corresponding to the removed node does not possess this property, but its dual (called a coweight) does. The duality transformation can be accomplished on weight space by means of the metric tensor.
Mar
5
comment The Aharonov-Bohm effect is purely classical, right?
@levitopher you can obtain the semiclassical aproximation (including the A-B effect) from the path integral, but Tuyman's theory requires even less structure than needed for the path integral. He does not require the symplectic form to be integral, thus does not impose the Dirac's quantization condition.
Nov
26
comment What are orbifolds and why are they useful and interesting for physics?
@Siva This observation is based on the solution of Schrödinger equation on a two dimensional cone where the energy eigenfunctions become more concentrated around the tip as the cone's half angle becomes smaller.
Oct
30
comment Aharonov-Bohm Effect and Flux Quantization in superconductors
They showed that the Shrödinger wave function of a single particle acquires a phase under a translation and a boost and this is alright, but, under a sequence of transformations: translation, boost, reverse translation, reverse boost, the overall phase does not vanish even though we returned to the initial frame of reference. This is a multivalued function related to the nontrivial central extension of the Galilean group.
Oct
30
comment Aharonov-Bohm Effect and Flux Quantization in superconductors
@jinawee Wigner worked on the representations of the Galilean group together with Inönü in their article "Representations of the Galilei Group". Please see the article on page 359 of Wigner's collected work: books.google.co.il/….
Oct
8
comment Spacetime Torsion, the Spin tensor, and intrinsic spin in Einstein-Cartan theory
Sorry, the first sentence in my comments should be: The scalar and Dirac field examples were not given for the purpose of showing how to couple torsion to classical fields, they were given as examples of matter.
Oct
8
comment Spacetime Torsion, the Spin tensor, and intrinsic spin in Einstein-Cartan theory
The answer to your second comment is that nothing was added by hand, the coupling term is obtained according to the rules of the minimal coupling. The only manipulation done is to separate the symmetric part of the affine connection and include it in the spin connection, and writing the antisymmetrical components separately giving the interaction term. In summary, the coupling of the matter fields to Einstein-Cartan theory can be viewed as a stage in the determination of its spin tensor.
Oct
8
comment Spacetime Torsion, the Spin tensor, and intrinsic spin in Einstein-Cartan theory
This is given in equation (2.20) in Shapiro. The spin tensor is also analogous to the energy momentum tensor being a Noether current of the Lorentz symmetry as the energy momentum tensor is the Noether current of the translation symmetry. (Together they generate the Poincare symmetry).
Oct
8
comment Spacetime Torsion, the Spin tensor, and intrinsic spin in Einstein-Cartan theory
This is exactly analogous to the energy momentum tensor: In order to obtain the energy momentum tensor, first one minimally couples the matter theory to gravity, and then varies the Lagrangian with respect to the metric. The analogy for the spin tensor: first one couples the matter theory to gravity with torsion (i.e., with a nonsymmetrical affine connection), then varies the Lagrangian with respect to the contorsion.
Oct
8
comment Spacetime Torsion, the Spin tensor, and intrinsic spin in Einstein-Cartan theory
The scalar and Dirac field examples were given for the purpose of showing how to couple torsion to classical fields, they were given as examples of matter, please see that these examples are given in Shapiro's review in section 2.3 titled as "Interaction of torsion with matter". The interaction appears as a term of the form: Spin tensor $\times$ contorsion tensor as can be seen in the Dirac example.
Aug
4
comment What does it mean to renormalize an effective field theory?
@ New_new_newbie It may happen that some process (like photon-photon scattering in QED) does not exist at the tree level approximation, then you can say that radiative corrections predict new physics. But apart from that, the control of the quantum corrections is mainly required to ensure that these corrections will not spoil the leading order.
Jul
3
comment Expressing an adjoint representation Wilson line in terms of the fundamental representation
Sorry, $E^{ij}$ is the matrix with all elements zero except a unit at the place $i,j$
Jun
5
comment The $U(1)$ charge of a representation
@user41746, cont.Now, Slansky insists to identify the highest weight of the $SU(5) $ representation with the highest weight of the quark representation after the symmetry breaking. For that he needs to act on the weights with a “projection matrix” given in equation 6.7. After the action of the projection matrix the fourth component of the weights is lost, but nevertheless, the dual U(1) vector can be obtained by insisting that it has a constant scalar product with each subrepresentation weights. As you can see all the exercises lead to the same result for the hypercharges.
Jun
5
comment The $U(1)$ charge of a representation
@user41746, Sorry I noticed that I used $a_1, a_2$ for the $SU(3)$ labels and $a_4$ for the $SU(2)$ and in your example you used $a_3, a_4$ for $SU(3)$ labels and $a_1$ for SU(2). Thus you should use the second row in $G$ for your example. You will get the same result with the normalization factor $\frac{1}{3}$.
Jun
5
comment The $U(1)$ charge of a representation
@user41746 $SU(5)$ is $A_4$, so the metric tensor should be $4\times 4$ matrix, please see the first matrix in table 7 page 82. Its third row for n=4 is just the given vector. I can’t see this exercise on page 84 (This page belongs to the appendix and it contains tables 10 and 11a. However, on page 16 equation (3.3), Slansky obtained the same hypercharges as in the question.
May
22
comment The derivation of the Belinfante-Rosenfeld tensor
@ramanujan_dirac: The energy momentum tensor constitutes of the source term in Einstein equations: $R_{\mu\nu}-g_{\mu\nu}R = \frac{8\pi G}{c^4}T_{\mu\nu}$, and both the Ricci curvature and the metric tensor are symmetric.
May
1
comment The curvature of the space of commuting hermitian matrices
@vesofilev: Sorry for not understanding you. The following reference arxiv.org/abs/0801.2913 contains a parameterization of the matrix $U$ for the flag manifold $\frac{SU(3)}{U(1)\times U(1)}$ on page 9, the third paragraph. (The authors call it the dressing matrix).
Apr
30
comment The curvature of the space of commuting hermitian matrices
@vesofilev: The $SU(2)$ case is a special case. Take for example the space of one commuting matrix in $SU(3)$, the spaces obtained from the adjoint action on $diag(0,0,1)$ and $diag(0,1,1)$ are disconnected.