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Dec
11
comment Beyond WKB approximation for energies
@Emilio You may find these lecture note by B.C. Hall useful: math.cinvestav.mx/~NorteSur/Taller10/notas/Hall.pdf
Nov
20
comment Finding the energy levels of an electron in a plane perpendicular to a uniform magnetic field
If I may add, if you write $a = x+ip$ and $a^{\dagger} = x-ip$, you get the harmonic oscillator Hamiltonian in the usual representation, but the harmonic osillator coordinates $x$ and $p$ are not the original coordinates that you started with.
Nov
20
comment Finding the energy levels of an electron in a plane perpendicular to a uniform magnetic field
If you substitute the expressions of $a$ and $a^{\dagger}$ given in the answer in the Hamiltonian $H=\hbar\omega(a a^{\dagger}+\frac{1}{2})$ you get the Hamiltonian you started with expressed in terms of $X$ and $P$.
Nov
20
comment Finding the energy levels of an electron in a plane perpendicular to a uniform magnetic field
The Hamiltonian expressed in terms of the creation and annihilation operators has exactly the form of the harmonic oscillator Hamiltonian. Thus the energy levels are exactly equal to those of the harmonic oscillator, however with infinite degeneracy per level (The harmonic oscillator energy levels are nondegenerate). The advantage of using this method is that it allows an algebraic solution of the energy levels (i.e. without solving differential equations), please see the quantum harmonic oscillator Wikipedia page: en.wikipedia.org/wiki/Quantum_harmonic_oscillator
Nov
19
comment What are the solution spaces of Nonlinear Schrödinger equations?
@Arnold The following work by Forger and Romero establishes the equivalence betwee Crnkovic-Witten-Zuckerman and the Peierls brackets and relates them to the multisymplectic approach arxiv.org/pdf/math-ph/0408008.pdf
Nov
19
comment Interacting representation of the Poincaré group
cont., In 4D scalar theory, in the free case, the (free) Poincaré generators can be expressed in terms the field's creation and anihilation operators and their action on the corresponding (free) Fock space is completely known. In addition approximate representations can be constructed order by order in perturbation theory in which the interacting Poincaré generators will have higher polynomial dependence on the creation and anihilation operators (acting on the free Fock space), but an exact representation is not known.
Nov
19
comment Interacting representation of the Poincaré group
@BGal One can easily find an interaction term, and construct the expressions of the interacting algebra in terms of the field variables. For example in a scalar field theory \mathcal{H} = \lambda \phi^4 is a possible interaction term, and it is easy to exactly construct interacting Poincaré generators in terms of the field and its derivatives. Finding a representation on the other hand would mean to find a Hilbert space on which the action of the interacting Poincaré generators is completely known.
Nov
18
comment What are the solution spaces of Nonlinear Schrödinger equations?
@Arnold, please see the Crncovic-Witten and Zuckerman's articles given in Urs Schreiber's answer physics.stackexchange.com/questions/26883/…. The Crncovic-Witten's link is not working, but you can find their article in the book: books.google.co.il/…
Nov
18
comment What are the solution spaces of Nonlinear Schrödinger equations?
@Arnold, Please see for example fiz.uni.opole.pl/pgar/documents/IJMPA87.pdf by Piotr Garbaczewski
Nov
6
comment Symmetries of spacetime and objects over it
@Qmechanic Thank you
Oct
16
comment Mathematically challenging areas in Quantum information theory and quantum cryptography
Coherent states which are widely used in quantum cryptography, have a geometric origin (Bargmann space). Please see for example Ma's thesis: web.williams.edu/go/math/sjmiller/public_html/crypto/handouts/…
Oct
4
comment How to construct the charge conjugation matrix for any given dimension?
I can show you how to construct in this particular case, but can you please check your calculations because in your expressions $(\Gamma^1)^2=(\Gamma^2)^2=1$ while $(\Gamma^3)^2=(\Gamma^4)^2=-1$, i.e., you are working in a signature $(1, 1, -1, -1)$. Is this really the case you need.
Sep
23
comment Killing vectors for SO(3) (rotational) symmetry
cont. For the computation of the Killing vectors according to the given Wikipedia page, one needs in advance to construct the invariant metric. Furthermore, given the invariant metric, the Killing vector components satisfy differential equations which are harder to solve than the algebraic equations in the method described in the answer.
Sep
23
comment Killing vectors for SO(3) (rotational) symmetry
@ramanujan_dirac: Actually, for the construction described above, one does not need to know any property of the Maurer-Cartan form except its evaluation by the insertion of the Euler parameter formula of $g$ into its definition. This form has many interseting properties and applications, please see for further reading Shlomo Sternberg lectures: math.harvard.edu/~shlomo/docs/lie_algebras.pdf.
Sep
6
comment Aharonov-Bohm Effect and Flux Quantization in superconductors
Yes, this is exactly the definition of multivaluedness. Take for example the "funtion" $e^{\frac{i\theta}{2}}$ on the circle, it is multiple valued since it takes two different values at $\theta = 2\pi$ and $\theta = 4\pi$ which are the same physical point. Its modulus is a true function on the circle. Of course, the modulous operation cancels only a single global phase and if the wave function is a superposition, the relative phases will still exist. This is the reason why the wave function "feels" the topology in the Aharonov-Bohm effect.
Sep
5
comment classical dynamics on group manifold SU(2)
I added an update answering the question about the hydrogen atom
Aug
16
comment Why do some anomalies (only) lead to inconsistent quantum field theories
Sorry for the late response, I added an update to the answer addressing your question and relating to Drake's remark. I have edited the account about the Mickelsson’s project which did not reflect my opinion on the importance of this approach which I think that is very valuable.
Aug
13
comment First class and second class constraints
When all the 5 constraints are first class, then one has to choose 5 functions on the phase space (gauge fixing conditions)$\chi_{\alpha}$, such that the matrix $M_{\alpha\beta} = \{\chi_{\alpha}, \phi_{\beta}\}|_{\Sigma^{\prime}}$ is every where nonsingular on the constraint plus gauge fixing surface $\Sigma^{\prime}$
Aug
2
comment Classical and quantum anomalies
Actually, I have a little more information on this subject. I'll try to add an update when I can
Aug
2
comment Classical and quantum anomalies
(cont.) webcache.googleusercontent.com/…