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Dec
15
comment How do higher-order optical chiralities look like?
Please, see the following recent review by T.G. Philibin where the conserved quantities generalizing the zilch are given in equations (21-22). These quantities correspond to the Noether symmetries given in equation (19). arxiv.org/abs/1303.0687
Nov
26
comment Energy in dynamical variational principle
This method is called instanton calculus. It is really not trivial to implement and requires special expertise, but it led to tremendously important discoveries in physics. The article given at the end of this note is my favorite reference doing instanton calculus on geometrical manifolds of the type obtained in the variational approach. However, it treats a much simpler problem (a single spin) than the Bose-Hubbard model. I'll try to write for you a more comprehensive answer (with more references) in the next few weeks if I can. arxiv.org/abs/cond-mat/0111139v1
Nov
26
comment Energy in dynamical variational principle
The main reason for the existence of this correction is that, while classically, all the states on the classical solution path in the trial wave function manifold are degenerate in energy, quantum mechanically there should be a unique ground state, which is obtained from quantum splitting of this degeneracy. The computation of the correction to the ground energy, indeed involves imaginary time replacement. …
Nov
26
comment Energy in dynamical variational principle
If the trial wave function manifold is close enough to the ground state, then the conserved energy value on the classical solution is a good approximation of the ground state energy. (Quantum) corrections to this first approximation can be obtained from quantizing the effective theory defined by the Lagrangian $\mathcal{L}$. Since the manifold of trial wave functions is in general non-Euclidean, one needs more general quantization techniques than canonical quantization to perform this task (e.g. geometric quantization). …
Nov
24
comment Energy in dynamical variational principle
Certainly, $E(f_n, f_n^*)$ is the Hamiltonian of the mechanical system defined by the (time independent) Lagrangian $\mathcal{L}(f_n, f_n^*, \dot{f}_n, \dot{f}_n^*)$. Thus it conserved. Please see my answer physics.stackexchange.com/questions/197297/…, where you can think of the parameter vector $R$ as the set of coefficients $f_n, f_n*$. Technically, the conservation stems from the antisymmetry of the Berry curvature.
Nov
11
comment What does it mean for an action to be defined “on-shell”?
... However there are difficulties in attaining the space time supersymmetry at the quantum level. One major recent advancement in this direction is the introduction of pure spinor Green-Schwarz superstrings.
Nov
11
comment What does it mean for an action to be defined “on-shell”?
When the supersymmetry algebra does not close of-shell, then supersymmetry will not be preserved in the quantum theory. Thus even if the theory can be quantized, it would not be supersymmetric at the quantum level. The situation in supersymmetric gauge theories is even more complicated, because there are cases where the closure of the BRST algebra requires the closure of supersymmetry algebra. Strings viewed as 2 dimensional sigma models can be quantized while preserving the world sheet supersymmetry (It is not technically easy in the Neveu-Schwarz case). ...
Sep
17
comment Operator norm directly from phase space representation of photonic quantum operator
@Cosmas Zachos Thank you for your remarks Prof. Zachos. It is a great pleasure to have you on physics S.E.
Aug
9
comment How to include Berry connection in Hamiltonian?
@Chinmayee The answer of your first question is positive $\mathcal{H}(R) = E(R)$, which is the Hamiltonian function on the parameter space. For your second question I added an update to the answer.
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.