/sci/ - Science & Math

>>11504172
Introduction to Lie Algebras and Representation Theory by Humphreys

>>11314238
Try Guillemin & Sternbergs' Symplectic Techniques in Physics.
>>11314984
What makes you think leptons have internal degrees of freedom? Actually asking.

>>11156151
Sure it does. Sturm-Liouville is one of the most accessible and useful ways to find ONBs and [math]N[/math]-representations for infinite-dimensional Hilbert spaces. This is worlds away from linear independence in the trivial, tiny world of polynomials.

>>11037854
I think the problem is more with how physics is much more disjointed than math. Idiots can always be ignored but a general can't really be kept if no one really cares about what you're doing. The general level of /pg/ visitors is also a problem, which drives away anyone who can keep a decent discussion.
Take a /pg/ thread that showed up a couple months prior for instance; it was pretty frustrating for me to find that a supposedly "serious" poster didn't even understand how the Higgs mechanism works and the rationale behind spontaneous symmetry breaking, but I suppose my expectations were just too high.

>>11027237
I meant eigensectors of [math]\gamma_5[/math], of course.
>>11027251
Enough with the f-slur. It's personally offensive.

>>10737896
>tried teaching a close friend and colleague of mine some results in a recent article I've read about gauging finite symmetries in spin-TQFT and classification SET phases
>end up spending the entire night teaching him [math]K[/math]-theory instead

>>10223294
My gripe with ST isn't that it's "useless".
>>10223296
Kobayashi is a good start.

>>10112457
A vector field. Let [math]M[/math] be a smooth [math]n[/math]-dimensional manifold, then there exists charts [math](U,\alpha)[/math] about each point [math]p \in M[/math] such that [math]T_pM = C^\infty(M) \otimes \{\partial_i\}_i[/math] as a free [math]C^\infty(M)[/math] module, where [math]\partial_i = \frac{\partial}{\partial x_i}[/math] whence [math]\alpha(p) = x[/math] for all [math]p \in U[/math].
>>10113145
It depends on what topology you endow on [math]V^*[/math]. In general you can give [math]V^*[/math] the weak limit topology [math]f_n \rightharpoonup f \in V^* \iff |f_n(x) - f(x)|_V \xrightarrow[n\rightarrow \infty]{} 0[/math], and this gets you shit like the Dirac delta distribution, which is obviously not in the Schwarz space.
For [math]V \cong V^*[/math], [math]V[/math] needs to be separable. [math]V[/math] with uncountable basis need not have this isomorphism (see Schwarz functions on non-compact spaces).

>>10061781
Yes. For any orientable manifold [math]M[/math] the normal space [math]N_xM[/math] at [math]x\in M[/math] can be defined as the spaces of normal vectors perpendicular to every vector in the tangent space [math]T_xM[/math], where the notion of "perpendicular" comes from the orientation on [math]M[/math]. In fact, orientability is equivalent to the existence of a globally defined normal bundle [math]NM[/math]. The vector normal to a trajectory [math]t\mapsto x(t)[/math] is just [math]n \in N_{x(t)}M[/math] such that [math]n \perp X_f[/math], where [math]X_f[/math] is the Hamiltonian vector field generating the trajectory.
>>10063202
Suppose [math]B = \{e_1,\dots,e_n\}[/math] spans the vector space [math]V[/math] over a feld [math]k[/math], then there is an isomorphism [math]\operatorname{End}(V) \cong V^* \otimes V[/math] between linear self-maps of [math]V[/math] with the matrix representation, where [math]L \in V^*\otimes V[/math]. Endow [math]V[/math] with an inner product [math](\cdot,\cdot):V\times V\rightarrow k[/math], then [math]L\in \operatorname{End}(V)[/math] is a coordinate transform iff [math](Lx,Ly) = (x,y) = (x,L^*Ly)[/math] for every [math]x,y\in V[/math], hence [math]L \in O(n)[/math] necessarily is an element of the orthogonal group. Since [math]O(n)\hookrightarrow \operatorname{End}(V)[/math] embeds into the group of self-maps, the above isomorphism descends to a representation [math]\rho:O(n)\rightarrow SL_k(n) \subset V^*\otimes V[/math]. It is always possible to express coordinate transforms by a matrix of unit determinant. You probably did something wrong.
>>10064773
>Then their composition h = g o f is biholomorphic
Only on the Riemann sphere [math]\mathbb{C}\cup \{\infty\}[/math]. Remember, Mobius transforms generates the group of conformal transformations in [math]\mathbb{C}[/math], the element [math]\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}[/math] of which has no inverse unless you append the point at infinity.

>>9985068
Quantum mechanics is nothing but classical mechanics with a Hermitian L^2 line bundle (called the prequantum bundle), and classical mechanics is nothing but symplectic geometry. Hence do understand quantum mechanics you just need to understand the geometry and topology of bundles on symplectic spaces, which is as simple as it gets in my opinion.

>>9842062
>I didn’t know they pay you
Let me guess, you're in Europe? It's customary to be subsidised in the NA and if they don't offer you any funding then that's basically a rejection letter.
>Demanding it seems pretty cruel to me desu
That's because the faculties here don't fuck around and actually pay us for our work.
Now I'm not saying you can't change your interests further down the line. I myself did my (paid) masters in condensed matter but now I'm doing a PhD in TQFT. But this doesn't change the fact that I knew I wanted to study topological superconductors going into my masters.

>>9557192
Start one. There had been like 2 or 3 successful ones before they were ruined by autists from here.

>>9461012
(Pre-)Quantization just ultimately a replacement of the Poisson bracket on [math]C^\infty[/math] functions with the Lie bracket on the Lie algebra of operators on some Hilbert space. This promotes your classical observables to Hermitian operators the eigenvalues of which are the actual measurable quantities.
This amounts to finding conditions on your symplectic manifold [math](M,\omega)[/math] on which you can centrally extend the Lie algebra of [math]C^\infty[/math] functions to that of automorphisms of the Hermitian line bundle [math]B\rightarrow M[/math] on [math]M[/math]. Conventionally this is done with the integrality condition [math]\omega \in H^2(M,\mathbb{Z})[/math] where the curvature of the connection on [math]B[/math] is [math]\frac{1}{\hbar}\omega[/math] but Kostant's construction gives a more general form of this procedure.
A sense in which a quantization scheme is "nice" may be the fact that for free fields you can decompose field operators into creation/annihilation operators (generators of the Heisenberg algebra) and everything you've learned from basic QM falls through, and you can construct S-matrix elements as usual (modulo some more axioms you have to assume such as asymptotic completeness of your Hilbert space). Of course this can't always be done and this depends on whether if Kostant's construction gives you a representation of Heisenberg algebra or not.

>>9451589
>How should I start?
Read Geometric Quantization by Woodhouse. Essential quantization amounts to finding a Hermitian line bundle on your symplectic manifold with a curvature lying in the integral lattice of the second Cech cohomology class. This guarantees that any Lie group of symplectomorphisms on your symplectic manifold can be centrally extended to quantomorphisms on the Hermitian line bundle that preserves the connection via Kostant's construction, so your space of sections quotiented out by the polarization defined via the orbits of the quantomorphisms defines a proper Hilbert space for wavefunctions.
Now to find the orbitals you define a [math]U(1)[/math] Hermitian circle bundle on the symplectic manifold [math]S^2[/math] with the usual Liouville symplectic form and construct a representation of the Hamiltonian on the vector space of wavefunctions. The eigenfunctions form a complete (since the space is Hilbert) discrete (since [math]S^2[/math] is compact) set of orthonormal (since the eigenproblem is of Sturm-Liouville type) basis functions that the anon here >>9451603 typed out. The [math]l[/math] and [math]m[/math] indices give you the orbials.
I hope this clears up any confusion.

>>9395131
No. Sounds like what people would gather from skimming Griffith. Go read an actual QM book like Townsend, Sakurai or Landau-Lifshitz.

>>9392561
>Do you have any good overarching intuitions
Not really, I just know that integrating out momentum shells to get renorm flows isn't a proper group operation since the Hubbard-Stratonovich transform doesn't have unique inverses, so this whole thing isn't very well understood mathematically (by myself at least) and I try not to have any impressions about what renormalization actually is before I do.

>>9355014
>using incorrect intuition to understand something yields "spooky" results
Wow who would've thunk??