/sci/ - Science & Math

>>11495338
>lattice
>QFT
>FT
Anon... time to stop pretending you know what you're talking about. You've tried it before in /mg/ and it's not gonna work again here.

>>11484992
No, my post is fine. Don't worry about it little boy, it's not for you.
>>11485009
Yeah that seems ok.

>>11483287
[math]\int\frac{dx}{\sqrt{2\pi}} e^{ikx}\delta(x)= 1[/math]. The plane wave is the Fourier transform of the delta function. It's not square integrable in the conventional sense but only as a distribution.

>>11457286
>That's not what he asked at all,
That's exactly what he asked, hun. To quote, "... how does that translate into analysis on [math]\mathbb{C}[/math] being different than analysis on [math]\mathbb{R}^2[/math]?"
>big abstract picture
Now THIS is not what he ask for at all.
>>11457316
There's no payoff.

>>11437535
Suppose [math]u: M\rightarrow N[/math] is a smooth map and [math]p:x\mapsto x(t)[/math] a path on [math]M[/math], then we push [math]p[/math] into [math]N[/math] by [math]u^*p(t) = u(x(t))[/math]. Now just apply the chain rule [math]\dot{x} = \dot{u} \cdot dx(u)[/math], which taking away [math]x[/math] and in coordinates reads [math]\frac{d}{dt} = \partial_t + \dot{u}_i \cdot \nabla[/math].
Now in general [math]\nabla u[/math] is tangent to the [math]level~sets[/math] of [math]u[/math], on which [math]u[/math] is constant, so as [math]x[/math] varies [math](\nabla u)(x) \cdot u(x) = 0[/math]. This gives us [math]\nabla(\rho u^2) = u\cdot \nabla(\rho u)[/math] by Leibniz.

>>11417104
[math]\mathcal{B}[/math] are the bounded linear operators and [math]\mathcal{U}[/math] are the unitaries. In fact if [math]\mathcal{H}[/math] is equipped with a sesquilinear form then they form [math]C^*[/math]-algebras, which is the backbone of QM. See Von Neumann's book on the mathematical foundations.
>>11417104
>Does it make sense to think like this when I'm thinking of a Pauli matrix acting on a spin state?
Yes, [math]\operatorname{Lie}G[/math] is by definition the tangent space of [math]G[/math] at the identity. If you think about [math]G[/math] as actually moving your state on the Bloch sphere then you [math]have[/math] to think about the generators as tangent fields.
>I still don't know what "gauging" means
Gauging makes global symmetries local. This is achieved by forcing parallel transport to have a horizonal lift, which can be done by minimal coupling to a connection. The entire point is that gauging makes local both the Lie group and the Lie algebra.
Don't worry about this for the moment. Focus on understanding elementary QM first.

>>11406517
>fate
More Remilia's field desu. I just do boundaries.

>>11325686
It may probably have some influence on how SRE phases are defined on lattice systems, but I doubt it could touch anything in the continuum limit. The reason I say this is because SRE phases are defined (according to Wen) as equivalence classes under local unitary transforms, which can in turn be represented as finite-depth quantum circuits. See https://arxiv.org/abs/1004.3835

Try out gay dating first, you may just be a gay man. You can never really fully detransition once you've started on that train.
But if you have already talked to therapists and fully sure about your decisions then go for it. Can't be worse than dealing with dysphoria I imagine.

>>11059858
Here's hoping this thread is better than the last one.

>>11007153
https://en.wikipedia.org/wiki/Parallelizable_manifold
[math]K[/math]-theory provides an obstruction to this.

>>10964038
I'm not the "I/we're not a 'he'" poster, but I appreciate your gratitude.

>>10939039
I don't know who you are but please don't interrupt when adults are talking.

>>10351621
Yes I've dated an experimentalist before

>>10180830
Gluing lemma.
>is it fair
Not sufficient

>>9231719
These 6j symbols are what's more generally referred to as fusion matrices for a conformal field theory. Let [math](\mathscr{V},\{V_i\}_i)[/math] be a simple modular category with simple objects [math]V_i[/math] and some fixed set of fusion rules, then there exists a covariant functor [math]F:\text{Rib}_\mathscr{V} \rightarrow \mathscr{V}[/math] from the ribbon category based on [math]\mathscr{V}[/math] to [math]\mathscr{V}[/math] called the operator invariant. This functor allows you to cast fusion rules in the form of ribbon graphs as well as express some results regarding the dimensions and traces of objects in [math]\mathscr{V}[/math] as knot invariants. This allows you to prove fusion relations such as the pentagon/hexagon relations, Verlinde-Seiberg formula and Vafa's theorem with braids and knots. In addition if [math]\mathscr{V}[/math] is unitary metaplectic then these fusion matrices form a [math]\mathbb{C}[/math]-vector space called the space of conformal blocks. In this case the 6j-symbols are basically instructions for fusing the simple objects of [math]\mathscr{V}[/math] as Hilbert spaces and you can consider it as relations between knots or ribbon graphs WLOG. For instance, the fusion matrix [math] F_{pq}\left[\begin{bmatrix} i & j \\ k & l\end{bmatrix}\right] [/math] can be considered as a consistency condition between fusing objects [math]i,j,k[/math] by fusing [math]i,j[/math] first into [math]p[/math] and then fusing [math]p,k[/math] into [math]l[/math] or first fusing [math]j,k[/math] into [math]q[/math] then fusing [math]i,q[/math] into [math]l[/math].

Algebraic topology and homotopy/homology can be used to describe defects in disordered media: https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.51.591..
Given the stablizing subgroup [math]H< G[/math] of the symmetry group that fixes the space of low energy states [math]R \subset \mathcal{H}[/math], the topological orders of the theory is characterized by [math]\pi_n(R/H)[/math], where [math]n=0[/math] characterizes a planar defect (e.g. domain walls), [math]n=1[/math] a line defect (e.g. cosmological strings) and [math]n=2[/math] a point defect (e.g. Dirac monopoles). This is also related to the Goldstone-Nambu theorem (where non-trivial topological orders can cause a massless zero mode to form) and the Mermin-Wagner theorem (where long range topological ordering cannot exist in a conformally invariant theory for dimensions larger than 2).