/sci/ - Science & Math

>>11439758
No, it means the spinor part is trivial, i.e. transforms under the trivial representation [math]{\bf 1}[/math] of your gauge group [math]G[/math]. Generally, a spinor transforms as [math]\psi \mapsto \rho(g) \cdot \psi[/math] with Clifford multiplication, which requires a "matrix" structure on [math]\psi[/math]. The trivial representation on the other hand literally just gives you [math]\psi \mapsto \psi [/math] so no matrix structure is required.
Given this you can build up your effective action [math]S[\psi] = \int_M \mathcal{L}(J^\infty\psi)[/math] in grand Ginzburg-Landau style with a [math]G[/math]-invariant Lagrangian density [math]\mathcal{L}(J^\infty\psi^g) = \mathcal{L}(J^\infty \psi)[/math].
>>11444654
Remember that if the Gaussian surface [math]S[/math] is given by the zero locus of a harmonic map [math]\phi[/math] then [math]\partial \phi[/math] induces an orthonormal frame on [math]TS[/math] with respect to the Riemannian metric as [math]g[/math] on [math]S[/math]. As [math]S[/math] is 2D, the two-form [math]d\phi \wedge d\phi[/math] is a top-degree exterior form and hence defines a normal form [math]n(S)[/math] in the normal bundle [math]NS[/math]. Its local derivative [math]-\nabla_s n(S)[/math] is the shape operator. As you have perhaps noticed, the matrix [math]d\phi_s \odot d\phi_s[/math] is the Hessian that tells you the curvature at [math]s\in S[/math].
Now as you move in [math]S[/math], [math]n_s(S)[/math] gives a vector normal to the surface at [math]s \in S[/math], which scales with the curvature; eigenvalues and eigenvectors of the shape operator then also scales with those of the Hessian, which tells you how much and towards where the curvature of [math]S[/math] is changing.
>>11455059
Generally you must have [math]\delta(\sum_\text{incoming}p - \sum_\text{outgoing}p)[/math] to preserve 4-momentum. Also I don't think a vertex like [math](A\wedge \ast A )dA[/math] is gauge invariant (modulo boundary terms), is it?

>>11403033
Quantum mechanically they are scattered off the atom. Some weak interaction (like [math]n\rightarrow p+ e^-[/math]) must occur in order for the neutron to be absorbed, but this typically doesn't happen unless at quite high energies.
>>11403156
Given a measure space [math](\Omega,\mu)[/math], historically the space of generalized functions is the [math]L^2[/math]-closure of [math]C^\infty_c(\Omega)[/math], or a sequence [math](\varphi_n)_n\subset C^\infty_c(\Omega)[/math] such that [math] \int_\Omega \varphi_n^* f\rightarrow \int_\Omega \varphi^* f < \infty[/math] for all [math]f\in C^\infty_c(\Omega)[/math]. Depending on the measure [math]\mu[/math] and properties of the measure space [math]\Omega[/math], this is subtly different from the modern understanding of distributions as the topological dual [math]\mathcal{S}'[/math] of [math]\mathcal{S} =C^\infty_c[/math] (which is what you are thinking of when you say "it's like a function[al] whose input is another function"), however, as not all distributions are generalized functions, i.e. not all distributions admit a representation as an integration. You typically need [math]\Omega[/math] to be at least paracompact and finitely measurable.
In addition, how "nice" (in terms of its regularity) [math]\mathcal{S}'[/math] is depends again on the measure space as well as the topology (weak-, strong- or Mackey) you endow on it. For certain spaces they can very different. It'd be a slight misunderstanding to say that "distributions are [...] 'well-behaved' functionals" (quotes mine) since they are by definition the entire linear dual; their regularity is controlled by the topology you endow, not by how you artificially pick them out.
>>11403437
The algebraic (multiplicative) properties of [math]\mathbb{C}[/math] are endowed from those of [math]\mathbb{R}[/math] by taking the algebraic completion. The polar representation then falls out like a stillborn.
>>11403504
No.

Thread's been kind of simple isn't it?
>>11255167
Discretize the EoM for Earth's orbit and put the endpoints at the solutions.

>>9057993
I assume you've done Feynman diagrams?
Check out the books by Strocchi and Fujiwara for info on non-perturbative methods in QFT.
>>9058237
>Superconductivity falls easily into gauge theories
No, I wouldn't say "easily"; there is no general procedure that maps second-quantized Hamiltonians into quantum field theories. The only one that's been rigorously formalized is the Haldane mapping which maps the universality class of Ising-type models into those of [math]\phi^4[/math] theories.
Superconductivity however can be described and characterized by category theory (https://arxiv.org/abs/1506.05805)), which [math]is[/math] related to holonomies seen in gauge theories. The braiding matrices arising from the holonomies are also used to define Wilson loop variables a la string theory (or vice versa, see https://arxiv.org/abs/0707.1889).).
>>9058265
If you're interested in QFT from the constructive/algebraic point of view Baez has a good book on it. Strocchi's book on symmetry breaking also delves into von Neumann algebras and how they're used to generalize Goldstone's and Neother's theorems via spontaneous symmetry breaking.

>>9022031
Let [math]A_n[/math] be the algebra of formal power series over the determinants [math]X^{ij},~ i\neqj[/math] over [math]\mathbb{C}[/math] modulo the ideal generated by [math]X^{ij} - X^{ji},~[X^{ij}+X^{jk},X^{ik}][/math] and [math][X^{ij},X^{kl}][/math] with all indices distinct. Given [math]\kappa \neq 0[/math] consider [math]\omega = \frac{1}{\kappa}X^{ij}\omega_{ij}[/math] where [math]\omega[/math] is a monodromy connection.
Put [eqn]\theta(\gamma) = 1 + \sum_{m=0}^{\infty}\int_{\gamma}\omega^m[/eqn] where the product is taken as an iterated integral, then [math]\theta:P_n \rightarrow A_n[/math] is a representation of the pure braid group.
Now if [math]\omega[/math] is flat (like the KZ connection), then [math]\theta[/math] is a single-valued representation that depends only on the homotopy of the loop [math]\gamma[/math], and [eqn]\exp\left(-i\int_\gamma \theta(\gamma)\right)[/eqn] defines a 1-dimensional unitary representation of the holonomy of the principal bundle defined by the connection [math]\omega[/math].
The idea I have is to extend this expression to non-flat [math]\omega[/math], which I believe would likely get me terms that depend on the base points of [math]\gamma[/math] in the unitary representation. Using matrices to represent this kind of branching, we could in theory construct a non-Abelian unitary representation of the pure braid group.

After some thought, the points [math]\{p_i\}[/math] could just be attached to the the arcs [math]L = \bigcup_i L_i[/math] inside the decorated 3-manifold [math]M\in \mathscr{B}[/math] that end on [math]\partial M[/math] after chopping [math]M[/math] up with a cut system of discs [math]C_i[/math] with [math]C_i \cap L_j \neq \emptyset \iff i \neq j[/math]. This means that I could go straight to enriching the space structure with CFTs on [math]\{p_i\}[/math].
I think the best and most straightforward way to do this is to equip each [math]\Sigma \in \mathscr{A}[/math] with an Laurent series affine Lie algebra-valued KZ 1-form [math]\omega \in \hat{\mathfrak{g}}_{\mathbb{C}}(t_1,\dots,t_{2g}) \otimes \Lambda^1(\Sigma)[/math] such that the embedding [math]\operatorname{Hom}(V_{\mu_1}\otimes V_{\mu_1^*}\otimes,\dots,\otimes V_{\mu_g}\otimes V_{\mu_g^*},\mathbb{C})
\hookrightarrow V_{\mu_1 \mu_1^*\dots \mu_g\mu_g^*}[/math], whence [math]\Phi \in V_{\mu_1 \mu_1^*\dots \mu_g\mu_g^*} \iff d\Phi = \omega \Phi[/math], exists, and that [math]\omega[/math] induces a flat connection [math]\nabla = d - \omega[/math] on a fibration of the conformal blocks [math]\mathcal{E}(p_1,\dots,p_{2g}) \rightarrow V_{\mu_1 \mu_1^*\dots \mu_g\mu_g^*}[/math]. This will ensure that [math]\Phi[/math] is a conformal block invariant under the diagonal action of the affine Lie algebra [math]\hat{\mathfrak{g}}_{\mathbb{C}}(t_1,\dots,t_{2g})[/math], which will in turn generate the conformal algebra. The TQFT functor will then map cobordisms to homomorphisms of the space of conformal blocks.
This characterization of the CFT [math]\mathscr{T}(M,\omega)[/math] by the conformal algebra is enough to guarantee naturality and uniqueness, and the only major thing to do afterwards is to make sure that Witten's invariant can be obtained from the operator invariant of this TQFT.