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Jul
8
comment Spontaneous symmetry breaking in classical mechanics, quantum mechanics and quantum field theory
@BebopButUnsteady In the buckling problem, only two modes participate (for example the straight and the one period modes). This is the reason that one can simulate the buckling mechanism using rigid elements together with (a finite number of ) springs.
Jul
7
comment Spontaneous symmetry breaking in classical mechanics, quantum mechanics and quantum field theory
@drake I have changed "invariant solutions", to "time invariant solutions". This means that the classical distribution on the phase space does not change in time. This definition avoids the mention of vacuum. Actually, the Klein-Gordon example is appropriate here, because the invariant solution $\phi = 0$ indicates that the Lorentz invariance is not broken, however it is not a "vacuum", since the Hamiltonian is not bounded from below.
Jun
30
comment Spontaneous symmetry breaking in classical mechanics, quantum mechanics and quantum field theory
Of course, I changed the wording to hopefully avoid confusion.
Jun
9
comment Classical and quantum anomalies
@drake: There is a great new article (arxiv.org/abs/1305.1955v1) by Michael Stone and Vatsal Dwivedi in which they classically derive the Abelian and non-Abelian chiral anomlies using a single fermion classical model without second quantization, Grasmann variables, the path integral measure etc.
May
28
comment Calculation of the spherical harmonic sum in the propagator of the particle on a sphere
@ramanujan_dirac : It means that it is a fractional derivative. One way to understand it is to think about the Fourier transform. The derivative becomes a multiplication by the dual variable $p$ in the Fourier representation. A half-derivative is the multiplication by $p^{\frac{1}{2}}$. Now, a multiplication in the Fourier (momentum) space is a convolution in the position space. This explains the definition 8.14 of the fractional derivative in Comporesi.
May
21
comment Coherent $U(N)$ intertwiners in Loop Quantum Gravity (LQG) and a measure on the Grassmannian
@levitopher cont. In fact, the procedure adopted by Qmechanic is a part of the "Faddeev-Popov" procedure for reducing gauge symmetries, and in principle it can be completed to get a BRST invariant integrand.
May
21
comment Coherent $U(N)$ intertwiners in Loop Quantum Gravity (LQG) and a measure on the Grassmannian
@levitopher Yes, the $U(2)$ volume in this case is a component of the integration over $GL(2)$. In fact $Z$, or ( $u_1$ and $u_2$) are redundant for the Grassmannian, even with the constraints because they sum up to $2N-4$ coordinates, while the dimension of the Grassmannian is $2N-8$. Another way to look at it is that the constraints are gauge fixing conditions and $U(2)$ represents the gauge symmetry. This is exactly the meaning of equation 82 in the article.
Apr
25
comment What does the sum of two qubits tell about their correlations?
Cont. What is important is that on each manifold type there is an explicit parameterization of the 4-dimensional state vector (in terms of 3, 4, and 5 parameters respectively). (The generic case is not written explicitly in the article but can be easily worked out). Given the state vector, the $S_z$ probability distribution can be explicitly computed for each type of orbit and compared to the experimental distribution
Apr
25
comment What does the sum of two qubits tell about their correlations?
The idea is to use the classification given in arxiv.org/abs/quant-ph/0006068 (Geometry of entangled states) by Marek Kuś Karol Žyczkowski. The manifolds of equal entanglement of the pure two-qubit system fall into three strata parameterized by a single parameter $\theta$. The nongeneric orbits are $\mathbb{R}P^3$ and $S^2 \times S^2 $ correspond to $\theta = \frac{\pi}{2}, 0 $ respectively. The generic orbits correspond to $0 < \theta <\frac{\pi}{2}$ are five dimensional twisted $S^2$ bundles over $\mathbb{R}P^3$.
Apr
25
comment What does the sum of two qubits tell about their correlations?
Will the special case of a pure (but otherwise arbitrarily entangled) composite system be of interest to you.
Apr
22
comment Aharonov-Bohm Effect and Integer Quantum Hall Effect
@ramanujan_dirac, Yes your explanation is correct and conforms with Laughlin's original argument.
Apr
22
comment Aharonov-Bohm Effect and Integer Quantum Hall Effect
@ramanujan_dirac: (cont.) The gauge transformation $\phi$ must be a true function on the circle, i.e. $\phi(0) = \phi(2\pi)$. Thus we can choose a (large i.e., nonhomotopic to the identity) gauge transformation $\phi = n \theta$ to remove the integer part of the flux, such that only its fractional part will enter the Schroedinger equation.
Apr
22
comment Aharonov-Bohm Effect and Integer Quantum Hall Effect
@ramanujan_dirac: The Schroedinger equation a particle moving on a circular ring is invariant under the gauge transformation $\psi\rightarrow e^{i\phi} \psi$, $ A_{\theta }= A_{\theta } + \frac{c \hbar}{eR} \partial_{\theta}\phi$.
Apr
17
comment When can a global symmetry be gauged?
@Tomáš Brauner: Yes, the obstruction to gauging can be understood intuitively by means of the Dirac's constraint theory. If the theory can be gauged, the currents couple to the gauge field. Since the time component of the gauge field is non-dynamical, the charge densities of the symmetry currents will become constraints, i.e., must become zero on the gauge surface which can be thought as a reformulation of the theory with gauge invariant fields. But, then how can the bracket of two vanishing quantities give a nonzero constant.
Apr
17
comment When can a global symmetry be gauged?
Tomáš Brauner: Regarding the remark by drake, Please see section 7.5 of the lecture notes by: Riccardo Rattazzi: itp.epfl.ch/files/content/sites/itp/files/groups/ITP-unit/…. @drake: The scalar Stueckelberg field is real and so is its shifting transformation. The extension to complex values, together with the fact that the Schrodinger Lagrangian is linear in the time derivatives make the shifting transformation anomalous.
Apr
8
comment How does the quantum path integral relate to the quantization of energy?
@drake I have placed a list in a file exchange server: fileconvoy.com/…
Mar
31
comment Aharonov-Casher effect for charged particles
@rubenvb:I have added an update answering your question.
Mar
20
comment Validity of naively computing the de Broglie wavelength of a macroscopic object
@John Rennie You are correct, but I don't know if it is due to the present limitations in knowledge or technology or there is a fundamental theoretical limitation of decoherence. In the first case it might be possible in the future to increase the experimental bounds where interference is observable.
Mar
20
comment Validity of naively computing the de Broglie wavelength of a macroscopic object
Anna cont. It is true that in the composite case the decoherence conditions are harder to achieve, because not only can this system loose coherence by being "kicked" by interacting particles but also it can loose coherence by random excitations of its internal degrees of freedom for example when the thermal excitations exceed its vibrational energies.
Mar
20
comment Validity of naively computing the de Broglie wavelength of a macroscopic object
@Anna cont. When conditions are provided such that the system stays in a single (ground state) internal state, it is possible to approximate it by its center of mass quantum dynamics governed by its de Broglie wavelength, for example in the case of the buckyball and other large molecule interferometry.