/sci/ - Science & Math

>>11541378
In QED, scattering amplitudes are computed via the path-integral [math]W[\phi,\Sigma] = \int_{\Phi|_\Sigma =\phi} D\Phi e^{-S[\Phi]}[/math], which receives radiative corrections through Feynman diagrams. Tree-levels amplitudes typically involves a 4-integration over [math]\tilde{G}_0[/math], the Fourier transform of the free propagator/inverse Laplacian. [math]G_0[/math] is in general only a distribution, hence we encounter branch cuts in the complex [math]p_0[/math] plane, such as in the segment between [math]\pm m[/math] for time-ordered [math]G_0[/math]. We can send this segment into [math](0,i\infty)[/math] by a conformal transform, and picking up the residue at the branch point gives rise to a multi-valued amplitude [math]\mathcal{M}({\bf k}-{\bf k}')=\mathcal{M}(|{\bf k}|\cos\phi)[/math] across [math]\phi = \pi/2[/math]. This does not mean [math]\mathcal{M}(\phi)[/math] is undefined for all [math]\phi > \pi/2[/math], but only that it comes from another branch of [math]\mathcal{M}[/math] that the physical scattering process does not detect.
>>11542233
Note that the Tietze extension [math]\tilde{f}[/math] of [math]f[/math] is bounded at infinity since [math]f[/math] is bounded, hence we may extend [math]\tilde{F}:\mathbb{R}^n\cup \{\infty\} \rightarrow \mathbb{R}[/math] to a function on the one-point compactification [math]\mathbb{R}^n\cup \{\infty\} \simeq S^n[/math]. [math]\tilde{F}\in \Gamma(S^n,\mathbb{R} \times S^n)[/math] defines a global section [math]F[/math] on the tautological line bundle [math]S^n\times \mathbb{R}\rightarrow S^n[/math], whose image [math]F(S^n)[/math] is precisely the graph [math]\Gamma_{\tilde{F}}[/math]. Now since [math]F[/math] is a global section and [math]S^n \hookrightarrow \mathbb{R}^{n+1}[/math] is a codim-1 embedding, this induces a codim-1 embedding of the image [math]F(S^n) \hookrightarrow \mathbb{R}^{n+1}[/math]. Now just poke a hole at [math]\infty[/math] and restrict to [math]F[/math].

>>11492400
There is no internal symmetry that leads to the conservation of baryon number via Noether; in fact anomalies such as chiral-ABJ in SM can violate it.

>>8989236
Not funny at all. Number theory is pure wank and it still haven't found itself an application to physics.
Also some fields of mathematics are starting to be inspired by physics (i.e. TQFT and quantum algebra); mathematicians will be doing physics before they know it.