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Oct 16 |
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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/… |
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Oct 4 |
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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. |
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Sep 23 |
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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. |
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Sep 23 |
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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. |
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Sep 6 |
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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. |
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Sep 5 |
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classical dynamics on group manifold SU(2) I added an update answering the question about the hydrogen atom |
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Aug 16 |
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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. |
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Aug 13 |
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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 $\Sigma^{\prime}$ |
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Aug 2 |
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Classical and quantum anomalies Actually, I have a little more information on this subject. I'll try to add an update when I can |
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Aug 2 |
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Classical and quantum anomalies (cont.) webcache.googleusercontent.com/… |
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Aug 2 |
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Classical and quantum anomalies (cont.) But, if one uses the Sugawara construction for the Virasoro generators from the current generators, the Virasoro central charge (in the Poisson brackets) will be zero. However, in the Liouville theory one already obtains a nonvanishing Virasoro central charge in the Poisson brackets, please see the following article by Toppan: |
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Aug 2 |
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Classical and quantum anomalies Here are some examples referring to both your questions. In the WZNW model in two dimensions, one gets the Kac-Moody algebra at the classical level (Poisson brackets) with the correct central charge. In fact this computation was performed by Witten in his original article: projecteuclid.org/…. For a more clear derivation please see: citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.148.8249 |
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Aug 1 |
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Classical and quantum anomalies I have added an update with the elaboration of the spin case |
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Jul 31 |
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Classical and quantum anomalies When the representation on the wave functions is a ray representation,it is an indication of an anomaly, because the symmetry group is not represented faithfully. Of course group extensions can be studied in general separately from quantum mechanics. I'll update my answer tomorrow and add a few more references. |
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Jul 31 |
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Gauge fixing choice for the gauge field $A_0$ (cont.), please see the following review by: ANTTI SALMELA ethesis.helsinki.fi/julkaisut/mat/fysik/vk/salmela/gausssla.pdf |
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Jul 31 |
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Gauge fixing choice for the gauge field $A_0$ But if one subtitutes $A_0 = 0$ in the Lagrangian before the computation of the equations of motion, then equation of motion of $A_0$, namely, the Gauss law will be lost, because one will not have an $A_0$ in the Lagrangian to take a variation with respect to it. In this case one has to impose the Gauss law "by hand". |
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Jul 24 |
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What does symplecticity imply? (cont.) There are also advantages in Poisson geometry, since the symplectic leaves of a Poisson manifold may not be manifolds, and also it is more natural to perform stability analysis within Poisson geometry. |
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Jul 24 |
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What does symplecticity imply? @Arnold (in addition to Jonathan's remark). The same happens for the ideal fluid example. The motion is restricted to a single coadjoint orbit of the volume preserving diffeomorphism group. Actually, the integral curves of any Hamiltonian on any Poisson manifold are restricted to a single symplectic leaf. The importance of Poisson geometry lies in that it includes the solutions of all possible initial conditions in the classical case and all inequivalent quantizations after quantization. |
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Jul 24 |
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What does symplecticity imply? @Arnold Of course, but symplectic geometry remains important even within Poisson geometry (I think that this is the reason that they don't have an Arxiv subject classification of Poisson geometry). The rigid body example you gave can be formulated on the symplectic manifold $T^*SO(3)$ (Euler angles + angular momenta), then Poisson reduced to $\mathbf{so(3)^*}$ a Poisson non symplectic manifold, but the dynamics actually takes place on a single coadjoint orbit: $S^2$, agian a symplectic manifold. |
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Jul 23 |
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when is the stationary phase approximation exact? @Arnold. The generalization of the Duistermaat-Heckman theorem to field theories is only to the physics level of rigor. Actually, the field theory examples for which this formalism was applied constitute of finite dimensional reduced phase spaces. For example the Chern-Simons theory whose reduced phase space is the moduli space of flat connections, or the coherent state path integral which is reduced to the lowest landau level by supersymmetry. I think that the first one who in roduced this generalization is Antti Niemi, please see for example Arxiv: hep-th/9301059 (He has also earlier works). |
