/sci/ - Science & Math

>>11006433
>>11006463
>>11006475
Good work organizing these. It makes /sqt/ a better place.
>>10991538
https://en.wikipedia.org/wiki/Topological_data_analysis
https://en.wikipedia.org/wiki/Persistent_homology
These concepts can see application in economics, or anything related to Big Data, really.
>>10986936
Hahn-Banach.
https://en.wikipedia.org/wiki/Hahn%E2%80%93Banach_theorem
It's not exactly taking a limit, but a trick of partition of unity by taking advantage of uniform boundedness.
There is also Tietze extension
https://en.wikipedia.org/wiki/Tietze_extension_theorem,
which is an application of Hahn-Banach by using the Minkowski gauge [math]\rho_A[/math] to bound [math]f[/math].
>>10984785
This might not be as stupid as you first thought. Floer cohomology is computing something similar to this, though not with [math]any[/math] ODE, but one generated by a Morse function [math]f:X\rightarrow \mathbb{R}[/math]. To be specific, the stable, unstable and irreducible critical points of [math]f[/math] form Floer cycles [math]\hat{C}(X),\breve{C}(X), \overline{C}(X)[/math] as graded (by the index) Abelian [math]\mathbb{Z}[/math]-modules, respectively. With the boundary [math]\partial a = \sum_b |M(a,b)| b[/math], where [math]M(a,b)[/math] is the space of trajectories generated by [math]f[/math] connecting the critical points [math]a[/math] and [math]b[/math], we can define the Floer cohomology groups [math]H = C/\operatorname{im}\partial[/math]. For "nice" [math]X[/math] (e.g. finite CW), these Floer cohomology groups are actually isomorphic to singular [math]H(X), H(X,\partial X)[/math] and [math]H(\partial X)[/math] by universal coefficient theorem.

>>11001137
Collect everything in a single Module.
>>11003519
You are correct. The "for all finite collections of [math]X[/math]" type of statements are how one extends finite-dim notions to infinite-dim cases formally without any additional assumptions.
Without a norm (hence a Banach space) you cannot make precise the notion of convergence. Even with the slightly weaker notion of seminorms, we still directly work with finite subsets of the family for convergence/uniformity statements.
>>11005347
The Fourier transform is a morphism [math]\mathcal{F}:X \mapsto \operatorname{Hom}_\mathbb{C}(X,U(1))[/math] into the space of irreducible characters of a compact Abelian group [math]X[/math]. Pontrjagyn duality then states that [math]\mathcal{F}[/math] is idempotent.
>>11005863
Apply [math]d\ast [/math] to Maxwell's equation [math]d\ast F = J[/math] to obtain [math]d\ast d \ast F = d\ast J = d \delta F[/math] as [math]\ast d \ast = \delta[/math] is the codifferential. However since [math]F[/math] is exact with [math]dA=F[/math], [math]dF = 0[/math] hence [math]d\delta F = (d\delta + \delta d)F = \Delta F[/math], where [math]\Delta[/math] is the Laplacian. The Lorentz gauge then implies that [math]F = F_\text{harm}[/math] is harmonic hence [math]\Delta F = d\ast J = 0[/math].

>>10958134
https://arxiv.org/abs/math/0111118

>>10957120
https://en.wikipedia.org/wiki/Tietze_extension_theorem

>>10949872
Gauging global finite [math]G[/math] symmetries in fermionic TQFT.
https://arxiv.org/abs/1812.11959
3D TQFTs with boundary/defect data.
https://arxiv.org/abs/1710.10214
Complete classification of extended invertible TQFTs.
https://arxiv.org/abs/1712.08029
Complete symmetry classification of topological orders.
https://arxiv.org/abs/1604.06527
Classification of (3+1)D liquid topological orders.
https://arxiv.org/abs/1704.04221
https://arxiv.org/abs/1801.08530
Twisted TQFT from crystallographic T-duality in CFT.
https://arxiv.org/abs/1806.11385

>>10935107
It's called the Neumann series
https://en.wikipedia.org/wiki/Neumann_series..
It's used and abused in QFT, in particular to derive the Dyson equation [math]G^{-1} = G_0^{-1} + \Sigma[/math]

>>10915042
>applications to 2d percolation in statistical physics
Please tell me more.

https://www.youtube.com/watch?v=KDJ6Wbzgy3E
https://www.youtube.com/watch?v=bGv_dGJhVQA
https://www.youtube.com/watch?v=3u-unvYedx8
>>10904791
Good post.

>>10898843
[math]\hbar\rightarrow 0[/math]

>>10889010
https://www.youtube.com/watch?v=Ty2wQU3PEoo
>>10890104
In general renormalization is the procedure of integrating out high-energy/hard modes, or coarse-graining out short-range modes in real space. Let [math]\psi\in \Gamma(M,E)[/math] be a section of an appropriate vector bundle [math]E\rightarrow M[/math] and [math]\mathcal{L}\in\Omega^d(J^\infty E)[/math] the Lagrangian density. By integrating along the fibres of [math]J^\infty E[/math] we get the action [math]S[/math] as a functional on jets of the space of sections of [math]E[/math].
Decompose [math]\psi = \psi_< + \psi_>[/math] such that [math]\psi_>\in L_\text{loc}[/math] has compact support on the momentum shell [math]b\Lambda < k < \Lambda[/math] for some RG parameter [math]b[/math]. Suppose a functional Liouville measure [math]\int_\mathcal{D} d\mu[/math] on the space [math]\mathcal{D}[/math] of sections of [math]E[/math], we may write [math]d\mu = d\mu_> d\mu_<[/math] such that the partition function reads [math]Z = \int d\mu_< e^{-S_0[\psi_<]}\int d\mu_> \exp\left( -S[\psi_<,\psi_>]\right) = \int d\mu_< e^{- S_0[\psi_<] + W_\text{eff}[\psi_<]}[/math], where [math]W_\text{eff}[/math] is the effective quantum action that takes into account the "average" interaction of hard modes with soft modes [math]\psi_<[/math].
By computing [math]W_\text{eff}[/math] perturbatively, you may bring [math]S_0 + W[/math] into a form [math]S_\text{eff}[/math], which belongs to the same polynomial class of [math]S[/math] as a functional on jets. You can now rescale the jets of [math]\psi[/math] and the parameters [math]\alpha[/math] in [math]Z[/math] such that [math]Z = \int d\mu' e^{-S_0'}[/math], where the prime indicates dependence on rescaled parameters and fields.
This rescaling then generates a flow [math]b \mapsto \alpha(b)[/math], for which [math]\frac{d\alpha}{db} = \beta(\alpha)[/math]. This is the definition of the [math]\beta[/math] function, as the generator of RG flow.

>>10858165
Brezis.

>>10835530
Nakahara. Proofs there are coordinate-free.

>>10830266
Waves are points in a Banach sigma algebra space [math](\mathcal{A},\Gamma,v)[/math], topologized symplectically through the underlying symplectic manifold of [math]\mathcal{A}[/math] and equipped with an action by the representation [math]\Gamma[/math] of the Weyl algebra [math]W[/math] and a distinguished cyclic vacuum vector [math]v[/math]. Particles are elements in the [math]\mathbb{C}[/math]-linear tuple [math](\mathcal{F},\pi,\nu)[/math] composed of a Fock space [math]\mathcal{F} = \bigotimes_n \mathcal{H}_n[/math] cyclicly generated under a representation [math]\pi[/math] of the Heisenber algebra [math]H[/math] by a distinguished vacuum vector [math]\nu[/math]. The particle-wave duality is a compact orthogonal (for bosons) or unitary (for fermions) transformation [math]T:(\mathcal{A},\Gamma,v)\rightarrow (\mathcal{F},\pi,\nu)[/math] such that [math]T\Gamma(W) \subset \pi(H)[/math] and [math]T(v) = \nu[/math].

>>10812778
A monopole is a [math]\pi_2[/math]-topological defect in a [math]U(1)[/math] gauge theory. Given a vector bundle [math]E\rightarrow M[/math] with compact Lie structure group [math]G[/math], monopoles are critical points of the Yang-Mills-[math]G[/math] functional [math]S[\phi,A] = |F\wedge \star F|^2_{L^2} + |D\phi|^2_{L^2} + \Delta[/math] such that the solutions have homotopy type [math]\mathbb{Z}[/math] in [math]\pi_2[/math], where [math]D[/math] [math]F\in\Omega^2(M,\operatorname{End}E)[/math] is the [math]G[/math]-invariant curvature 2-form and [math]D[/math] is the covariant derivative. Here [math]\Delta[/math] is a generic renormalization term including the Reidemeister /Ray-Singer torsion of [math]E[/math].
https://ncatlab.org/nlab/show/Seiberg-Witten+theory
https://ncatlab.org/nlab/show/Kaluza-Klein+monopole
Incidentally, solutions that have non-trivial homotopy type in [math]\pi_1[/math] are called vortices.
>>10812830
https://www.nature.com/articles/326367a0
https://arxiv.org/abs/1705.06657
https://arxiv.org/abs/1705.05162

>>10799774
[math]U_q[/math] deforms relations of [math]\mathfrak{g}[/math] like how first quantization deforms the Poisson algebra of [math]C^\infty[/math] functions on a symplectic manifold; it eseentially gives you the operator algebra on quantum states. One place I have seen [math]U_q[/math] show up explicitly is when its generators are being used as a state sum for the correlation functions of a CFT in the case [math]\mathfrak{g} = \mathfrak{sl}_2[/math]. Personally I have noticed that the relations in certain quantum aglebras look like Skein relations of knots, and there's a very concrete way of understanding correlations in CFT as knot invariants (see e.g. https://scinapse.io/papers/201303897).).

What's your favorite mechanism of spontaneous chiral symmetry breaking that solves the strong CP problem?

>>10708901
>symplectification of higher-dimensional Chern-Simon/WZW theories
>classification of 4D TQFTs
>generalize Levin-Wen to 3D G-TQFTs for arbitrary finite G
>construct [math]a[/math] consistent state sum for 4D TQFT
>generalize the TQFT/CFT correspondence to Lie/pro-fintie G's
>construct spin-TQFTs from generalized multi-fusion categories
>prove generalized tangle hypothesis
>prove cobordism hypothesis for spin-bordism groups
>prove that topological phases in geometric (not necessarily topological) field theories are determined by the free part of [math]\operatorname{Hom}(\Omega^G,U(1))[/math]
>prove modularity of 6D superconformal
>prove AdS/CFT
Take your pick.

>>10361104
Yes. Topology is beautiful and no amount of slander will change that.

>>10264677
>is ultimately based on topological qft
Incorrect. There are topological codes that do NOT admit a field theoretic description, such as Haah's cube code or the [math]\mathbb{Z}_2[/math] lattice gauge theory. These models have fracton phases with a sub-extensive ground state degeneracy that blows up when you take the thermodynamic limit, which means that there is no coherent theory to describe them in the continuum.
These fracton codes can use the mobile intersections of submanifolds as qubits. It's one of the most interesting things in TQC at the moment.
>it's still a bona fide qft
Incorrect. Something that does not allow a continuum description in the first place cannot be a field theory, classical or otherwise.
>Quantum chemistry simulation
Which is a completely different field than TQC.
>2nd quantized forms
Again, that's only applicable when you have a coherent classical field theory in the first place, which quantum chem have but fracton phases do not. This is borderline irrelevant to what I was talking about.

>>10220787
Let's work with indistinguishable particles for clarity. In general there are two approaches to understand aynons; the approach I described and the braid group approach; here I'll describe the latter since I have already described the former.
Let [math]M[/math] be a smooth manifold and put [math]Q_N(M) = (M^N\setminus\Delta)/S_N[/math] be the configuration space on [math]M[/math], where [math]\Delta[/math] is the set of diagonals in [math]M^N[/math]. We define the braid group [math]B_N=\pi_1(Q_N(M))[/math] as the fundamental groups of the configuration space.
Now a wavefunction is defined as a section of a (prequantum) Hermitian line bundle [math]H\rightarrow Q_N(M)[/math], on which [math]B_N[/math] acts. To characterize the [math]U(1)[/math]-representations of this action, one can WLOG put particles [math](z_1,\dots,z_N) \in Q_N(M)[/math] on a straight line and draw braid diagrams as they time evolve; the braiding operation gives you the generators [math]\sigma_i[/math] of [math]B_N[/math]. For [math]M = \mathbb{C}[/math] we have the Artin group [math]B_N = \langle \sigma_i\mid \begin{cases}\sigma_i\sigma_j\sigma_i = \sigma_j &; |i-j|<2\\ \sigma_i\sigma_j = \sigma_j\sigma_i &; |i-j|\geq 2\end{cases}\rangle[/math], whose Abelianization [math][B_N]_\text{Ab}\cong U(1)[/math] characterizes the [math]U(1)[/math]-reps. However for [math]M=S^2[/math], we have the condition that [math]\prod_{i=1}^N\sigma_i\prod_{j=0}^{N-1}\sigma_{N-j} = e[/math], hence the [math]U(1)[/math]-reps are labeled by, and hence particles on the sphere can acquire phases of, [math]N[/math]-th roots of unity.
Now notice that we can unbraid any links in [math]M = \mathbb{R}^3[/math], so we have another relation [math]\sigma_i^2 = e[/math] in the braid group. The [math]U(1)[/math]-reps are then labeled by [math]e^{i0} = 1[/math] and [math]e^{i\pi} = -1[/math]. This is why we only get bosons and fermions in 3D.

>>10217385
>since the other particles cannot occupy the same space
This is incorrect. Bosons can occupy the same spacce while Fermions can do the same as long as they have distinct labels (e.g. spin, momentum, etc.).
It'd be good for you to learn what "observation" actually means in the context of decoherence.
>>10217675
Because the angular momentum is conserved. In a Keplerian system, the coadjoint orbits of the group of rotational symmetry [math]SO(3)[/math] determines the trajectories and an your symplectic manifold foiliates through these orbits. Since the angular momenta [math]L_{1,2,3}[/math] generate [math]\mathfrak{so}(3)\cong\mathfrak{su}(2)[/math], the coadjoint orbits are labeled by these angular momenta and, as physical trajectories can't leave the leafs of these orbits, they have to have the same value of [math]L_i[/math] throughout. This makes planetary orbits unable to change its direction perpendicular to its orbital plane, since doing so would change the direction of the angular momentum.

>>10207395
"Moments" actually refer to angular momentum, not torque.
Any motion a rigid body undergoes can be generated by the generators [math]\{p_i,L_i\}_{i=1}^n[/math] of the Lie algebra of the Euclidean group [math]\mathbb{E}(n)[/math]. This means that any external influence, be it forces or toques, must give rise to a change in these generators. These are the Euler equations.
In addition, the fact that the moments [math]L_i[/math] and momentum [math]p_i[/math] commute in the Lie algebra means that any external change in the inertia of the rigid body decomposes into a linear part and an angular part. These are the forces and torques on the body.
>>10207720
Looks alright. These ladder operators [math]L_\pm[/math] furnish a representation of the Lie algebra [math]\mathfrak{su}(2)[/math] on the sphere. By a fundamental result in conformal field theory, a singular vector (the cyclic vacuum) exists in the highest weight Verma module [math]V_\mu = \oplus_{\lambda \leq \mu}V_{\lambda^\dagger} \otimes V_\lambda [/math] (the Fock space of momentum eigenstates) only if the conformal blocks [math]V_\lambda[/math] (the span of the momentum eigenstates) satisfying the quantum Clebsch-Gordan conditions, i.e. the [math]\lambda[/math]'s satisfy a certain algebraic condition before the [math]V_\lambda[/math]'s can be combined into a Verma module. From this you can obtain the coefficients in front of your eigenstates.

>>10195622
The Pointyn vector is the wedge [math]F \wedge \ast F[/math], where [math]F[/math] is the [math]U(1)[/math] curvature 2-form [math]F = dA[/math] (whose components are the field strength tensor) and [math]A[/math] is the [math]U(1)[/math] connection 1-form. Now from Maxwell's equations we know that [math]dF = 0[/math] while [math]d\ast F = J[/math], and for [math]V[/math] a 3-cycle in Euclidean space, the pullback [math]\iota^*d[/math] by the inclusion [math]\iota:V \hookrightarrow \mathbb{M}[/math] into 4D Minkowski spacetime. Hence by using Stokes's theorem [eqn]\int_S F\wedge \ast F = \int_V \iota^*d(F\wedge \ast F) = \int_V \left(\iota^*d F \wedge \ast F + F\wedge \iota^* d\ast F\right).[/eqn]
(Strictly speaking I should also include [math]S[/math] into the Minkowski space but I'm too lazy)
Now [math]\iota^*d[/math] is proportional to the time derivative, and for time-independent fields we can
1. add a time derivative and use Leibniz's law for [math]d = d_t + \iota^*d[/math] to pull out [math]d_t[/math] in the first term, and
2. add the time-derivative of [math]J[/math] in the second to get
[eqn]\int_S F\wedge dF = d_t\int_V F\wedge \ast F + \int_V F \wedge J.[/eqn]
Hope this clears everything up.

>>10148157
The Kaluza-Klein monopole in string theory is made from the Hopf fibration. String theory is not my expertise though so I'll give you another example.
Suppose you have Helium-3 in its chiral superfluid-A1 phase, then its BdG Hamiltonian is parameterized by the Bloch vectors [math]{\bf k}\in\mathbb{T}^3[/math] such that [math]H = \begin{pmatrix} \epsilon_k + k_z & k_x - ik_y \\ k_x - ik_y & \epsilon_k - k_z\end{pmatrix}[/math], where [math]\epsilon_k^2 = |{\bf k}|^2 - \mu[/math] is the dispersion. By tuning the chemical potential [math]\mu[/math], you can tune the magnitude of the 3-dimensional [math]{\bf d}[/math]-vector without interfering with its topological order, hence He3-A1 is characterized by homotopy classes of maps [math]{\bf d}[/math] into the [math]2[/math]-sphere [math]S^2[/math]. Now suppose we add a vortex (i.e. string defect) through the superconducting order parameter along [math]\hat{z}[/math] - via a magnetic field threading the superfluid He3, for instance - then the [math]SO(3)[/math] symmetry is broken down to [math]SO(2)[/math] for which [math]\hat{\bf d}[/math] is parameterized by
1. the orbital coordinate [math]\phi \in S^1[/math], and
2. the "unbroken" Bloch vectors [math]{\bf k}^\perp = k_x,k_y[/math], i.e. these crystal momenta stay good quantum numbers even in the presence of the vortex, compactified such that [math]{\bf k}^\perp \in S^2[/math] and [math]{\bf d}[/math] is defined at [math]\infty[/math],
then we see that [math]\hat{\bf d} \in [S^2\times S^1,S^2][/math]. Hence the Pontrjagyn invariant [math]\mu[/math], or equivalently the Pontrjagyn index [math]\mathbb{Z}_2[/math], characterizes the topological orders of the superfluid He3-A1 in the presence of a vortex.