/sci/ - Science & Math

>>11552182
A spin structure is an isomorphism [math]TM \xrightarrow{\sim} \operatorname{Cliff}(V)[/math] to a Clifford bundle; this means that every fibre in the tangent bundle [math]T_xM \cong V_x[/math] is (up to isomorphism) a representation space [math]V_x[/math] for a Clifford group, which captures the algebraic properties of spin. This endows [math]M[/math] with a spinor bundle [math]S \rightarrow M[/math].
The vierbeins [math]e[/math] form an orthogonal frame of [math]TM[/math], and the form of the metric tensor [math]g[/math] depends on this choice. The choice natural to the canonical frame [math]dx \in TM[/math] was picked out to be the "spacetime frame", while the above isomorphism [math]TM \cong \operatorname{Cliff}(V)[/math] rotates it into the "Lorentz frame"; the vierbeins [math]e[/math] implement this rotation: [math]g_{\mu\nu} = e_\mu^a e_\nu^b g_{ab}[/math]. This equation can be seen as a necessary consistency condition for the existence of a spin structure.
To see this, suppose [math]V[/math] is taken to be flat, then [math]e[/math] can be seen as a "square root" of the metric tensor, which defines a global section of a "square-root spin bundle" [math]\mathcal{S}[/math] for which [math]\mathcal{S}^{\otimes 2} \cong S[/math]. This implies that the obstruction class for spin structures, i.e. the characteristic second Stiefel-Whitney class [math]f^*w_2 \in H^2(M,\mathbb{Z}_2)[/math], where [math]f: M\rightarrow BSO(1,n)[/math] is the classifying map, vanishes.

>>11503067
Consider minimization of the (occupation of) Coulomb potential [math]\exp\left[-\sum\limits_{i<j}^N q_{ij}\ln|z_i-z_j| \right][/math] in the 2D plane [math]\mathbb{C}[/math]; for uniform charges [math]q_{ij} = 1[/math], the minimal configuration gives us an arrangement of points such that the inter-particle distance [math]|z_i-z_j|[/math] is maximized. Hence let us consider [math]Z_N[\{z_i\}_{i},g] = e^{-\sum\limits_{i<j}^N\ln |z_i-z_j|_g}[/math], where [math]|z|_g = \sqrt{g_{z\overline{z}}|z|^2}[/math] is the norm on the Riemann sphere [math]\overline{\mathbb{C}}[/math] with respect to a metric [math]g_{z\overline{z}}[/math] of positive scalar curvature. The goal is then to minimize the average [math]\overline{Z}_N[g] = \sum_{\{z_i\}_i\in Q_N(\overline{\mathbb{C}})}Z_N[\{z_i\}_i,g][/math] as a functional on the moduli space [math]\mathscr{M}_0[/math] of conformal classes of metrics, where [math]Q_N(\overline{\mathbb{C}}) = (\overline{\mathbb{C}}^N\setminus\Delta)/S_N[/math] is the configuration space, and find the asymptotics [math]\overline{Z}^*[/math] of [math]\inf \overline{Z}_N[/math]. Suppose a minimizer [math]g^*\in\mathscr{M}_0[/math] exists such that [math]\inf \overline{Z}_N = \overline{Z}_N[g^*][/math] is achieved with some [math]g^*[/math] such that the inter-particle distance [math]|z_i-z_j|_{g^*}\sim r_*[/math] is approximately uniform (it'd be interesting to prove this), then each summand reads [math]Z_N[\{z_i\}_i,g^*] = \frac{1}{N!}\prod_{i<j}^N\frac{1}{|z_i^*-z_j^*|_{g^*}} \sim \frac{r_{*}^{-N}}{N!}[/math], hence [math]\overline{Z}_N[g^*] \sim \sum_{k\leq N} \frac{r_*^{-k}}{k!} \rightarrow e^{-r_*}[/math], and the asymptotic dependence is suppressed exponentially.

>>11426147
In essence, it's the minimum number of spatial dimensions for the compatibility of spatial Lorentz symmetry and internal conformal symmetry. If you construct a [math]classical[/math] string theory with the Nambu-Goto action, canonical quantization takes coefficients of the canonical fields (as symplectic coordinates on the worldsheet) to conformal generators, which has Virasoro relation [math][L_n,L_m] = 2nL_0 + \frac{d-2}{12}n(n^2-1)[/math]. Indeed, if you interpret, as in a conventional CFT, the central charge as counting the number of chiral zero modes, then it stands to reason that string theory, as a background spacetime theory, have chiral zero modes associated with the number of available spacetime dimensions you can move in.
However, canonical quantization does [math]not[/math] preserve the emergent Lorentz invarance generated by the canonical fields. A computation of [math][M_i,M_j][/math] yields a term [math]\frac{d-2}{24}[/math] aside from the desired classical result, [math]precisely[/math] due to the existence of the central charge. This is the reason why we need [math]d = 26[/math] for bosonic strings.
When you add anticommutative fermions, however, you can "shrink" the conformal algebra by grading it against the fermion counting operator [math](-1)^F[/math], which increases your central charge to [math]\frac{d-2}{8}[/math].

>>11351025
You can just consider crossed products [math]V\otimes_\mathbb{C} V^*\subset \bigoplus_n V^{\otimes n}[/math] where [math]\mathbb{C}[/math] acts by skew-conjugation. This gets you a sesquiinear product which you can just evaluate into [math]\mathbb{C}[/math] and then quotient out the entire thing [math]\bigoplus_n V^{\otimes n}[/math] by the ideal generated by its preimage (i.e. elements in [math]V\otimes_\mathbb{C} V^*[/math] with well-defined inner products). As long as this product structure inherits Cauchy-Schwarz and polarization identity then it will automatically induce a norm and orthogonal projectors.
>Does this have a name
Don't know but just know that you're working [math]a~priori[/math] with [math]V[/math] this algebra coincides with its own *-irrep in [math]V[/math] (which induces a Hilbert structure), and no representation of it will be faithful on any proper subspace.
>algebra homomorphism
I was referring to the quptienting projection.

I'm back.
>>11306824
Kind of whipped this up in 5 mins so there are a few typos.
1. it should be [math]c^2 = 2 + \{C,C^*\}[/math], not sure why the braces didn't show up.
2. The spectral bound holds only for ESA operators, or symmetric operators with a self-adjoint extension.
3. We actually don't need [math]\operatorname{dim}\mathcal{H}[/math] in our bound, [math]K[/math] suffices.
4. It should be [math]|C|^2[/math] under the square root, so the final bound should read
[eqn]\langle \psi, A\psi\rangle + \langle \psi,B\psi\rangle \leq \frac{\sqrt{2}}{2}\left(K_a\sqrt{1+|A|^2} + K_b\sqrt{1+|B|^2}\right).[/eqn]

>>11184399
Very good question anon.
In general QM is the study of *-representations of Von Neumann algebras, and the representation space is the relevant Hilbert space. In this context, a natural choice for this representation space is given by the GNS construction, which gives a *-representation for any [math]C^*[/math]-algebra [math]\mathcal{A}[/math] on a Hilbert space [math]\mathcal{H}[/math] with a cyclic vacuum vector [math]v\in\mathcal{H}[/math]. What this means is that [math]\mathcal{A}v[/math] is dense in [math]\mathcal{H}[/math], and if [math]\mathcal{A}[/math] is finitely or polynomially generated, then [math]\mathcal{A}v[/math] forms a discrete ONB for which [math]\mathcal{H}[/math] becomes separable. Examples of this type include the many-body space of occupied states in condensed matter or QFT on compact manifolds.
Non-separable Hilbert spaces then becomes relevant for QFTs on non-paracompact-manifolds; the continuous degrees of freedom given by the fields form operator-valued distributions, and there are no good [math]N[/math]-representations for our ONB on non-paracompact manifolds, as elliptic operators such as the Laplacian [math]\Delta[/math] achieve a non-discrete spectrum. The standard practice of working with the Fourier basis [math]e^{ipx}[/math] is especially troublesome, as [math]N[/math]-representations give the Fourier kernels only on the compact [math]n[/math]-torus.
This is why so many divergence problems arise in the continuum limit [math]a\rightarrow 0[/math] or the thermodynamic limit [math]V\rightarrow \infty[/math], where [math]a,V[/math] are the lattice spacing and the system volume, respectively. This is both a curse and a blessing, for we must regularize or renormalize quantities in these limits but the divergences actually tell us about critical points. If the Hilbert space changes character, it is natural to assume that the system also acquires new ground states upon which phase transition occurs.

>>11105842
When you write down the Pauli matrices you're selecting a specific irrep of [math]\mathfrak{su}(2)[/math] labeled by a half-integer (for spins) or an integer (for orbitals) [math]j[/math]. Think about what this [math]j[/math] is when you say "each [math]J[/math]-matrices are [math]2\times 2[/math]".
Note that each irrep gives a homomorphism [math]SU(2) \xrightarrow{\rho_j} GL_{2j-1}(V)[/math] into a [math]2j+1[/math]-dimensional vector space because we have [math]2j+1[/math] roots distinguished by the Weyl group of [math]\mathfrak{su}(2)[/math]. This leads to, given a unique singular highest-weight vector [math]|0\rangle[/math] with [math]J_-|0 \rangle = 0[/math], [math]2j+1[/math] number of linearly-independent eigenvectors with weights [math]|m| \leq j[/math]. How many of these eigenvectors do you think you have when [math]j[/math] is such that "the [math]J[/math]-matrices are [math]2\times 2[/math]"?

>>11056809
Low-dim AdS/CFT need not have physical relevance through only cosmology.
https://arxiv.org/abs/0901.0924
Holography maps charged anti-de Sitter blackholes onto Quantum Hall states. This opens doors to many applications of SUSYstrings and superconformals to strongly-correlated condensed matter systems.
In fact, a twisted variant of homological mirror symmetry has been formulated through the study of topological phases enriched by non-symmorphic space groups (https://arxiv.org/abs/1806.11385).). The third string revolution is going to be spearheaded by condensed matter theorists and you can quote me on this.

>>11045348
Yeah a physicist probably wouldn't have posted a thumbnail

Get better at LA, do some basic applied complex analysis and diff eq, learn lagrangian mechanics, maybe read a chapter or two of dummit and foote or something

>>10918096
>but not between distributions themselves in general
Well, I guess that depends on what you mean by "in general". It works as long as one of the distributions belongs to [math]\mathcal{S}'_c[/math], i.e. has compact support.
https://ncatlab.org/nlab/show/convolution+product+of+distributions
There is also a way to loosen the regularity conditions so that [math]S[/math] takes distributions in [math]L_\text{loc}[/math], namely those that have evaluates to [math]\int_\omega dx \phi f [/math] on test functions [math]f[/math] and all [math]\omega \subset\subset \Omega[/math] compactly contained in [math]\Omega[/math]. Fixing [math]\omega[/math] lets you work with [math]\phi \in \mathcal{S}'_c[/math]; of course the problem then becomes finding [math]\phi[/math] that optimizes [math]S[/math] for all [math]\omega\subset\subset\Omega[/math].

Again, because of Sobolev embedding (and the fact that [math]\mathcal{S}'[/math] is fuck-huge compared to the usual "nice" Sobolev spaces like [math]H^1[/math]), if you can find optimizers without distribution theory then you won't find any new ones with it. You'd probably need to look at something non-linear like KdV in order to see the full strength of distribution theory.

>>10913876
To understand that, there are a few facts you have to accept first:
1. EM wave travels at [math]c = \frac{1]{\sqrt{\mu_0\epsilon_0}}[/math] in vacuum.
2. All physical processes should be Lorentz invariant (does not care about what reference frame you use).
3. Light is the free propagation of EM waves.
Using the free form of Maxewll's equations in the Lorentz gauge [math]d\ast A = 0[/math], we have [math]\square A = 0[/math] where [math]\square = d\delta + \delta d = \frac{1}{c^2}\partial_t^2 - \nabla^2[/math] is the Laplace operator on Minkowski space (d'Alembertian). This proves the first and third statement.
Second statement is a fundamental principle in physics, you'll just have to take it on faith.

Now think about what happens when a charged particle is moving (on a train, for instance) in the presence of a magnetic field [math]B_0[/math]. In the frame outside of the particle, the moving charge generates a magnetic field[math]B[/math], while in the frame of the particle no magnetic field is generated, but the moving background field[math]B_0[/math] will generate an electric field [math]E[/math]. The equivalence between these scenarios is the celebrated unification of electricity and magnetism.
Now if the speeds [math]c,c'[/math] for the propagation of these fields are different in different reference frames, then you'd end up with two distinct Maxwell's equations, which leads to two different Lorentz forces acting on the particle. This contradicts statement 2, as different forces give different trajectories for the particle, which cannot be achieved by merely a change in the reference frame. Hence [math]c=c'[/math], proving the Lorentz invariance of Maxwell's equations.

>>10887337
The Abelian Yang-Mills [math]U(1)[/math] gauge theory, describing the quantum dynamics of EM, was the first instance in which Atiyah-Bott localization was applied to moduli spaces of connections. Looking back it's quite trivial but it did lay the groundwork for e.g. non-Abelian localization and Hermitian Yang-Mills theory.

>>10760982
Well if you have the intuition for Riesz on Hilbert spaces like [math]L^2[/math] then some of that intuition might not carry over to the more general weak version. For instance, consider the Lagrandian density [math]\mathcal{L} \in \Omega^n(J^\infty E)[/math] on the jet bundle [math]J^\infty E[/math] of a vector bundle [math]E\rightarrow M[/math] whose sections [math]\Gamma(M)[/math] forms a Banach space (like [math]L^p[/math] for some [math]0 \leq p \leq \infty[/math]). Formally taking a functional derivative along the direction [math]v \in T_u(\Gamma(M))[/math] at [math]u \in \Gamma(M)[/math] yields [math]\delta_v \mathcal{L}[u][/math], whence optimization gives the strong Euler-Lagrange equation [math]\delta_v \mathcal{L}[u] = 0[/math] for all [math]v \in T_u(\Gamma(E))[/math].
Now by fibre integration along the jet bundle we can form the action [math]S = \int_M \mathcal{L}[/math] and obtain the weak EL equations [math]d_vS[u][/math]. In this case, if we wish to apply Lax-Milgram to [math]dS[/math], we need to be able to identify [math]T^*(\Gamma(M))[/math] with [math]\Gamma^*(M)[/math] (which can be done locally by regular Riesz in case [math]\Gamma(M)[/math] is Hilbert) so that we can define the bilinear form [math]a(u,v) = d_vS[u][/math] on [math]\Gamma(M) \otimes \Gamma^*(M)[/math], or at least in the neighborhood of a solution [math]u[/math] of the weak EL equation: [math]d_vS[u] = 0[/math] for all [math]v \in \Gamma(M)[/math]. This means that we need to identify [math]T_u^*(\Gamma(M))[/math] with [math]\operatorname{ker}\mathcal{L}[/math] as a functional operator, which in general is not true if [math]\mathcal{L}[/math] is not sufficiently regular, and Lax-Milgrim cannot be applied to yield a solution.

>>10710348
Thanks anon. Been kind of busy lately what with the quals and paper writing, but I have more time now for the next 2 months.

>>10379533
だが断る

>>10337556
The Cartan model for [math]G[/math]-equivariant cohomology was fucking based.
http://www.math.toronto.edu/mein/research/enc.pdf
Can't do Wess-Zumino-Witten without it imo.
>>10340584
What I know is that in certain percolation/MBL models the critical exponent [math]\gamma \leq \frac{2}{d}[/math]. Since presumably for strongly-enough-correlated phenomena the conformal scaling relations don't necessarily hold, the paper may imply a similar inequality for the magnetic critical exponent [math]\Delta[/math].

>>10337176
Noether's theorem (or generalized Noether's theorem) states that for any generator [math]\tau^a[/math] of the compact Lie group [math]G[/math] for which the Lagrangian [math]L\in \Omega^p(\mathcal{J},M)[/math] (as a differential p-form on the jet bundle of sufficient regularity) is invariant under, there exists a conserved current [math]J^a[/math] satisfying [math]\operatorname{div}J^a = 0[/math], and whose charge [math]Q^a = \int_V J^a[/math] is a constant of the motion.
(Topological) K theory is the study of isomorphism classes of vector bundles. Since for each vector bundle [math]E\rightarrow M[/math] there exists [math]n \in \mathbb{Z}[/math] such that [math]E \oplus (n) \rightarrow M[/math] is trivial, there exists a notion of "addition" [math]+[/math] (distinct from the Whitney sum [math]\oplus[/math], but no unique "subtraction") on the Abelian monoid [math]A(M)[/math] of equivalence classes of vector bundles on [math]M[/math]. By quotienting out the relation [math][E\oplus F] - [E] + [F][/math] for vector bundles [math]E,F[/math] on [math]M[/math] we get an Abelian group [math]K(M)[/math], called the K group of [math]M[/math].

>>10264508
>QFT + GR
Dead field. You have to understand that theoretical physics relies on experimental data to fit and create models. In fields where experimental data is literally impossible to obtain you'd have to compete with mathematicians for funding.
>more experimental side
Prepare to do more number crunching than a programmer.
>just QFT for QC and QI.
QC does not utilize QFT unless you're doing tensor networks, and that has the same convergence issue as QFT ON TOP of the decoherence issue of QC. Conventional QC typically takes more CS theory than physics to do research in, the amount of quantum it takes is just elementary QM.
Topological QC on the other hand is a relatively new field that might be worthwhile to go into. Again, it doesn't utilize QFT but it does apply some of the more mathematical techniques used and concepts in the theory of strongly-correlated matter to build quantum algorithms.
t. Mathematical physics PhD