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Nov
28
comment Proof that eigenvalues of Fermionic creation/annihilation operators are Grassman numbers
@Echows I recommend to start from the following review arxiv.org/abs/math-ph/0202026v1 by Cartier, DeWitt-Morette, Ihl and Sämann
Nov
12
comment Boson calculus and Maximum Weight State
Yes, we use commutators to shift $b_2$. In fact all the commutators are trivial and the operators commute because the raising operators do not contain $b_1$ which is the only operator not commuting with $b_1^{\dagger}$
Oct
31
comment Is the third spin vector of a photon always suppressed?
@WetSavannaAnimal aka Rod Vance Electromagnetic waves in waveguides propagate in TE and TM modes where either the magnetic the electric field have components along the direction of propagation.
Oct
25
comment What are orbifolds and why are they useful and interesting for physics?
@Nick Kidman $\mathbb{C}^2/~$: $\theta$ ~ $\theta + \alpha$
Oct
18
comment Spin tensor and Lorentz group operator in bispinor case
If the problem is a sign problem, please notice that the $\Psi$s are Grassmann variables and they acquire minus signs when they are commuted.
Sep
13
comment Non-associative operators in Physics
This phenomenon was discovered by Roman Jackiw: inspirehep.net/record/204787?ln=en. It happens for a system of a particle moving in field of a magnetic monopole.
Aug
21
comment What is the algebraic property that corresponds to a topological term?
2) When you say that you want to go from quantum to classical, I assume that you mean that the quantum theory is defined by means of a path integral, but in general path integrals are ambiguous, and also it is hard to see from them the quantum linear structure that you want to dequantize, this is why I think that methods of geometric quantization are superior.
Aug
21
comment What is the algebraic property that corresponds to a topological term?
@BebopButUnsteady Yes, that what I am trying to tell. The quantization point of view is quite unifying of various physical ideas. But, let me first remark: 1) The WZW terms already change the symplectic structure in the classical theory, not only that but they can spoil the closure of the Jacobi identities resulting nonassociativity and lack of Hilbert space representation of the quantum theory.
Aug
13
comment Why helicity is proportional to the spin of particle and has two values?
@PhysiXxx The helicity eigenvalue in contrast to that of the spin projection indicates the identity of the particle. For example a photon with a positive helicity is a different particle from a photon with a negative helicity. This is because different helicities correspond to different representations of the Poincare' group. This can also be seen in the photon's Fock space where different helicity photons have different creation and annihilation operators.
Aug
13
comment Why helicity is proportional to the spin of particle and has two values?
@PhysiXxx The way I see it is that the computation of the helicity operator is classical, and since the energy of a massless particle is never zero, one may divide by it. After obtaining the classical expression, we quantize according the rules of canonical quantization
Aug
12
comment Why helicity is proportional to the spin of particle and has two values?
@PhysiXxx I have added an update showing the computation of the helicity in 3-vector notation. The generalization of the case of the electromagnetic field (that I know of) requires the use of spinor notation of the massless field equations. It requires more than a few lines. I'll try to find a good reference.
Aug
7
comment Large and small gauge transformations?
@Hunter The identity component $\mathcal{G}_0$ that you gauge away is almost the whole gauge group, it is an infinite dimensional group. The group of large gauge transformation is only $\mathbb{Z}$, it is discrete. Thus I think that you may consider this as a fine tuning of the gauge principle.
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