/sci/ - Science & Math

>>12565714
>Is that related
Of course, we can gauge higher-categorical symmetries [math]G\in {\bf Grpd}_n[/math] in a TQFT [math]Z:{\bf Bord}_{d+1} \rightarrow {\bf Vect}_\mathbb{C}[/math] just the same via the delooping [math]B:{\bf Grpd}_n \rightarrow {\bf CW}_\ast[/math]. Holography then comes from the correspondence between Levin-Wen models on [math]{\bf Rep}_n(G\rtimes \mathbb{Z}/2^F)[/math] and DW-like TQFT [math]\widehat{\Omega}^{\text{Spin}}(B(G\rtimes\mathbb{Z}/2^F))[/math], which is a form of higher Fourier-Mukai transform.

>>12510323
Yes of course, I just told you.

>>11491319
>You dont think a triangle inequality can be set up?
I'm not sure if the triangle inequality is useful here because right at the boundary the distance to the original neighborhood becomes infinitely small.
Hausdorff means that you can separate distinct points with their neighborhoods, but you really need to know how sequences work here. In essence, the argument is that you are able to bound every term in the sequence away from [math]every[/math] single point in [math]N_r(x)[/math] by some [math]\epsilon >0[/math]; in particular, they're bounded away from [math]x[/math] by [math]\epsilon + r[/math]. This should tell you that the limit of [math]y_n\rightarrow y[/math] will never "leak" into [math]N_r(x)[/math].

>>11487047
By a standard result the regularity of [math]f[/math] determines the (polynomial) boundedness of [math]\mathcal{F}f[/math], but yes, it is generally not compactly supported; this is why Schwarz picked to work with the tempered test functions instead.
>I'm interested in the transform of [math]\cos(kξ)^{−1}[/math], not [math]\cos(kξ)[/math]
I'm aware. I merely wanted to illustrate that, with how ill-behaved [math]\sec[/math] is on [math]\mathbb{R}[/math], it can't be more well-behaved than [math]\cos[/math] and even just Fourier transforming [math]\cos[/math] takes you into the distributions.
>How can I do it?
By residue theorem [math]\int_\mathbb{C} dz f(z)\csc \pi z = \sum_{n\in\mathbb{Z}}f(z) [/math] for holomorphic [math]f[/math] that vanish at (complex) infinity. Complexify [math]x\rightarrow z[/math], shift [math]z \rightarrow z+\frac{\pi}{2}[/math] then evaluate the Fourier transform with a Poisson-like sum [math]\sum_{n\in\mathbb{Z}} f(n)e^{in\xi}[/math]. This is just a sketch btw, you can work out the details.
>how do I evaluate the sum
Depends on what [math]f[/math] is, but in general dunno.

>>11454232
That sounds great sweetie! Come and apply to one of the schools near IQC. Most QC people here do TQC or error correction.
Also read up on Nayak:
https://arxiv.org/abs/0707.1889

>>11423646
It is not just a mathematical curiosity. Plenty of physical systems are described by non-linear PDEs, such as KdV and the coupled oscillator. They also exhibit behaviours, such as shocks and caustics, that [math]cannot[/math] be described by their linearizations. Knowing how to solve them is what sets apart a plumber and an aerospace engineer.

>>11416857
>it serves no purpose
It serves EXACTLY every purpose. The Hamiltonian is fundamental in the sense that it determines everything in the system: we take its image as the Hilbert space [math]\mathcal{H}[/math] of states, and we require the observables, as a self-adjoint (SA) subalgebra in [math]\mathcal{B}(\mathcal{H})[/math], to contain it. To study properties of symmetries [math]G[/math] we must find a unitary irrep of it in [math]\mathcal{U}(\mathcal{H})[/math], which is a different space than the observables. This is why we use Stone's to consider their SA generators "[math]\operatorname{Lie}G[/math]" in order to have the commutation relations [math][H,\operatorname{Lie}G]=0[/math] well-defined. This wouldn't have made any sense if the domains of [math]H[/math] and [math]g\in G[/math] don't intersect, i.e. when the unitary rep of [math]g[/math] lies outside of the observable algebra.
This method also lets us consider discrete symmetries like PCT.
>Can I still picture this like I can picture a rotation?
Look up the Bloch sphere.
>I think it's relevant here
It is EXACTLY the opposite. If a symmetry is local then so are Lie algebras and their Lie groups. Gauging is fundamentally a distinct operation from taking [math]\exp[/math]. The reason why QFT people like the Lie algebra more is because the space of (associated) principal [math]G[/math]-connections is (locally) isomorphic to the affine space [math]\Omega^1_\text{aff}(M,\operatorname{Lie}G)[/math] of 1-forms, which lets them do minimal coupling very easily. In fact I have personally never encountered a case where the Lie group acts globally while the Lie algebra acts locally, or vice versa.

I think the problem here is your QM class focuses too much on computations and not nearly enough on the underlying theory. It'd be good for you to read Townsend or Cohen-Tannoudji.

>>11324331
Good point, but Stone's gives us necessary and sufficient condition for the existence of an ESA generator [math]A[/math] for the one-parameter unitary, while the anon who asked the question [math]started[/math] from the generator [math]\partial[/math].

>>11318083
If [math]G[/math] has no self-loops, let [math]n_\pm(v)[/math] be the number of incoming (outgoing) edges at the vertex [math]v\in V[/math], put [math]b_0(G) = \frac{1}{2}\sum_{v\in V}\left(n_+(v) - n_-(v)\right)[/math].
Consider [math]G[/math] as an oriented 1-skeleton of a 2D CW complex [math]X[/math]. The invariant [math]b_0(G)[/math] is the same as the invariant[math]\operatorname{rank}H_0(X,\mathbb{Z})[/math] of the cellular homology of [math]X[/math].

>>11287850
It's a fair connection to make, especially since most get their first contact with QM through classical mech considerations.

>>11268190
No problem hun.
>trying to mix stuff is fucky
I understand perfectly how you feel; mathphys is such a discipline after all.

>>11184236
You're quite welcome hun.

>>11154126
>Is this idea good enough, then?
Not quite, you still need the data of [math]A[/math] to make the statement that "every morphism [math]B\rightarrow P[/math] has an inverse morphism". More precisely, [math]given[/math] the morphism [math]B\xrightarrow{\phi} P[/math] with [math]\operatorname{ker}\phi = A[/math], we have a section [math]\psi: P\rightarrow B[/math] such that [math]\phi\psi = \operatorname{id}_P[/math]. This implies that the section implicitly depends on what [math]A[/math] (and hence what [math]\phi[/math]) is, and the statement about projectivity of [math]P[/math] is for [math]every[/math] such possible [math]A[/math] if you hold [math]B[/math] fixed.

>>11101749
Superconductivity.
>>11102458
State sum constructions [math]Z(M)[/math] for the TQFT partition function [math]Z[/math] on a closed compact [math]n[/math]-bordism [math]M[/math] are typically expressed as sums over topological invariants. Examples include graph invariants in 2D, knot invariants in 3D and Seiberg-Witten and Witten-Dijkgraaf for spin/+Abelian gauge-TQFTs, respectively. What I meant was that if TMFs do indeed capture deformation classes of conformal net bundles, then the topological data of sugras as a TQFT [math]Z(M)[/math] can be expressed in terms of invariants constructed from TMFs as a cohomology theory over some complicated scheme.

>>11065571
Legendre transform, as well as many operations that physicists do that seem "questionable" to the untrained eye, is made rigorous by contact geometry. As long as [math]T^*M[/math] forms a symplectic manifold, the contact manifold can be constructed from pulling back a line bundle [math]L \times T^*M [/math] by some smooth map [math]f:M\rightarrow M[/math] and everything follows.

>>11060251
>>11055211
Because the picture shows physical tools becoming important in mathematics over the past few decades and vice versa. Examples include Seiberg-Witten monopoles, the KZB equations and string theory. In addition, many mathematical conditions can be translated to "existence" statements in physics (e.g. existence of vortex solutions to HYMH is equivalent to the Deligne stability of the line bundle).
>>11055136
Not quite. The reason that string theory, as inherently a CFT, admits such nice descriptions is that the theory itself has a very structured and well-controlled algebra of quantum operators (viz. Virasoros and affine Lies). This doesn't solve any open questions regarding convergence issues in general QFTs, where the quantum operators form general von Neumann nets over spacetime geometries. In other words, string theory and CFTs in general are very special cases of QFTs where we may leverage the massive conformal symmetry group to explicitly evaluate physical quantities.
From a TQFT perspective, the reason why we can evaluate [math]Z[/math] explicitly for strings and SUSYstrings is because TMFs has the conformal nets of free fermions as geometric realizations (similar to how [math]ku[/math]-theory has Clifford-module bundles as geometric realization). This allows the TQFT with target into the 3-category of vN alg. of boundary conditions to be expressed in terms of topological/geometric invariants such as TMF Euler classes. This, as well as an explicit non-Abelian localization formula for the partition function, is still not proven, but we have seen numerous examples in which this is true. Hence why we suspect that the string theory partition function does indeed allow localization (the so-called "finite dimensional functional integral" in your words).

>>11028700
Yep, I was banned for posting yaoi of Yachie on /jp/.
>>11028918
Think of it like this: if [math]S^1 \times S^2 \simeq S^3[/math], then they are also homotopic over [math]S^2[/math] and the fibre bundle [math]0\rightarrow S^1 \rightarrow S^3 \rightarrow S^2 \rightarrow 0[/math] is weakly homotopically equivalent to the trivial bundle [math]0\rightarrow S^1\rightarrow S^2 \times S^1 \rightarrow S^2 \rightarrow 0[/math]. Of course this would imply that all sections [math]S^2 \rightarrow S^3[/math] are trivializable and so the long exact sequence in homotopy gives [math]\pi_3(S^2) = 0[/math], which contradicts the fact that [math]\pi_3(S^2)=\mathbb{Z}[/math] with the generator given by the Hopf map.
>>11030837
https://en.wikipedia.org/wiki/Hadamard_product_(matrices)
>>11031472
Anon, I can't stop you from embarrassing yourself in the future but please:
1. understand what Born-von Karmen boundary conditions are, and
2. study spectral theorems for bounded elliptic compact operators on the n-torii
before making a post like this next time.
>>11033002
From an effective perspective, [math]\hbar[/math] serves as an energy scale the separates the physics of the classical from the quantum, similar to how the Planck mass [math]M_P[/math] separates the physics of the Newtonian from the Einsteinian GR. The main tenet of effective field theory is that the physics at different scales decouple, and that every theory we devise is an effective theory of another.
>>11035216
Denote by [math]P[/math] the projection onto an [math]n[/math]-dimensional subspace of [math]C^1([a,b])[/math]. Your goal is to show that [math]A[/math] is invertible iff it's an automorphism on [math]\operatorname{im}P[/math].

>>11022741
Maybe the fact that [math]BG \cong K(G,1)[/math] would help?

>>11006340
Because [math]dJ[/math] is not [math]\operatorname{div}J[/math], [math]d\ast J[/math] is. Remember, [math]J[/math] is a 1-form and [math]d[/math] is a graded differential of degree 1, so [math]dJ[/math] is a 2-form while [math]d\ast J[/math] is a 4-form. On orientable 4D spacetime, [math]\Omega^0 \cong \Omega^4[/math] where the isomorphism is given by multiplicaiton by the volume form [math]d\operatorname{vol}\in\Omega^4[/math] so [math]d\ast J (d\operatorname{vol})^{-1}= \operatorname{div}J[/math] is an actual number.

>>10961067
>sushi
That's no way to refer to a lady's private parts, is it?

>>10958660
Smooth maps are homotopic to (piece-wise) continuous maps for closed [math]M[/math]. Without additional structure on [math]M[/math] (namely being a locally convex Banach) we cannot use resolution of unity and Hahn-Banach to glue extensions together.

>>10938947
>I'm not asking for a statistical approach to answering the question.
So you're not looking for an answer at all. Or do you have some sort of quantum vortex supercomputer that can store every single qubit in the universe?
>everyone can agree there is no symmetry in the observable universe.
Who's this "everyone"? What happened to Lorentz? Gauge symmetries? Even at the cosmic level, the Einstein action is invariant under reparameterizations of the metric. Where do you think these symmetries went?
>why is the universe not fractal-like?
This is asking for a very specific symmetry, namely scaling symmetry. Scaling is extremely special and only shows up in e.g. systems near criticality, for which conformal symmetry emerges. In fact, it can be taken as the definition: scaling symmetry emerges iff the system reaches criticality, as scaling directly implies the proliferation of some macroscopic state throughout the system. In this sense, it would be quite far-fetched to expect the observable universe now, even as a statistical system, to have scaling symmetry.

There have been people, especially string theorists, who argue that scaling emerges as an internal symmetry of a "hidden" fundamental degree of freedom (i.e. strings), for which CFT was used extensively to make computations. The problem with this is that the theory [math]presumes[/math] criticality of the universe's fundamental degrees of freedom; this is where the (SUSY)-string transition occurs in which strings condense into actual particles and their worldsheets become heavily compactified. A priori, we have nothing to compare the coherence length of this phase transition to, and hence no understanding of the RG "time scale" to determine if our universe is actually near criticality, let along a SUSY string criticality. There hasn't even been proof that such a RG flow into string criticality is necessary.
Besides, you wouldn't be able to see this with deep space pictures anyway.

>>10913241
By Sobolev embedding [math]W^{n,1} \hookrightarrow C^0[/math], solutions to weak EL [math]\int_\Omega dx \delta F = 0[/math] achieved in [math]W^{n,1}[/math] is automatically achieved in [math]C^0[/math]. In addition, the solution is unique (uniqueness of weak limit [math]u_n \harpoon u[/math] in [math]W^{n,1|[/math]). This means that once you've found a solution, you can stop looking.

For the problem of geodesics, the kernel [math]F \propto \sqrt{-g}[/math], where [math]g[/math] is the metric tensor on [math]\Omega[/math]. For sufficiently "nice" spaces [math]\Omega[/math] or smooth manifolds ([math]n \neq 3,4[/math]), [math]F[/math] is Caratheodory and hence the weak EL is [math]W^{n,1}[/math]-solvable.

>>10887018
My fiance will be finishing up his clinical psych PhD a year after I finish mine so if I can't get a postdoc after that I can just be his trophy husband.