/sci/ - Science & Math

Sphericity condition in fusion 1-categories
>the snake
In fusion 2-categories
>the swallowtail
How lovely! What'd be the name for this in 3-categories? Perhaps
>the cinnamon roll

>>12573975
A path integral, subject to boundary conditions [math]\Sigma[/math], is a Liouville probability expectation [math]\mathbb{E}_\Sigma[O]\equiv e^{i\Psi_\Sigma[O]} = \int d\mu_\Sigma(\psi,A) O(\psi,A)[/math] over the space [math]\mathcal{D}[/math] of the degrees of freedom, be it connections [math]A[/math] (gauge fields) or particles [math]\psi[/math] (spinors/tensorial fields) or any other dynamical things you may put into your theory. In Euclidean field theory, you're right in that the integration is weighted by the Gibbs measure [math]d\mu_{\psi,A}=d\psi dA \exp -iS_\Sigma[\psi,A][/math], where the Boltzmann factor [math]\exp -S[/math] tells you the probability of a configuration [math]\psi,A[/math] (satisfying boundary conditions [math]\Sigma[/math]) being occupied.
Now the problem is the "Liouville" part: we need the WKB wavefunction [math]\Psi[/math] to describe dynamics, which means that we must put a symplectic or at least a Poisson structure on the moduli space [math]\mathcal{D}/\mathcal{G}[/math], which is usually infinite dimensional and not even a manifold. For general interacting QFTs (i.e. not a TQFT/CFT/susyST) this is still an open problem, because finding the appropriate [math]d\mu[/math] is equivalent to quantizing interacting fields. For free theories [math]S_\Sigma \sim |\mathcal{A}\psi|^2 + |F|^2[/math], however, you are able to leverage the zeta-determinant/heat kernel technique to evaluate the path integral. This involves an expansion in [math]\Sigma[/math]-eigenmodes of [math]\mathcal{A}[/math] in order to decompose [math]d\mu \propto \prod_n d\psi_n da_n[/math]. You can then endow each eigenmode an individual Poisson structure when quantizing.

>>12561686
Any quantum effect makes your phase space
non-commutative; this is true in QFT/QG and QM/CM. There's no "analogy" here, they're real.

>>11889425
You need continuity or PL-smoothness for Fourier inversion.
>makes no use of the provided hint
Are you sure?
>>11901629
Microstates vs macrostates, hun.
>>11903986
Decoherence is not at all an open question sweetie.

>>11523395
That forbids homotopies, not diffeomorphisms. Your argument relies on Brouwer's: maps on an open set [math]V[/math] minus one point can have no fixed points, which implies the same for [math]U[/math] if a homotopy exists, but that's a contradiction since you're sending a boundary point of [math]U[/math] to the fixed point in [math]V[/math]. This is in fact not sufficient to prove that there is no diffeomorphism between [math]U[/math] and [math]V[/math] however, since homotopies need not be diffeomorphisms and taking out a point changes more than just the smooth structure. I wanted to emphasize why corners (and only corners) matter in [math]{\bf Diff}[/math] but not in [math]{\bf Top}[/math] or even [math]{\bf hTop}[/math] precisely due to this fact.
>>11529267
This follows from standard results in homotopy theory [math]\mathbb{R}^n\setminus \ast \simeq S^{n-1}[/math] and [math]\pi_n(S^n) \neq 0[/math]. In particular by just taking a cellular decomposition [math]S^n = \ast \cup_f D^n[/math] it follows that [math]H^\ast(S^n,\mathbb{Z}) \equiv \mathbb{Z}[e_0,e_n][/math] with free generators [math]e_0,e_n[/math] at degrees 0 and [math]n[/math]. This is more refined than using de Rham, since Poincare duality (in the way you're using) requires [math]\mathbb{R}[/math]/char-0 field-valued-coefficients, which kills all torsion.

>>11493307
Indeed, that's why you shrink the dimension of the local Hilbert space along with it, at least formally. Here's what I mean: instead of taking point-wise local finite-dimensional Hilbert spaces and observables in our definition, we take them associated to each space-like open set [math]U\subset\mathbb{M}[/math] in the continuum spacetime [math]\mathbb{M}[/math]. We must do this, in fact, otherwise our QM definition above would lead to non-local interactions (things that couple faster than light) at some point in the continuum limit.
Since Minkowski spacetime is locally compact, we can find countably many such local space-like covers such that [math]\mathcal{H} = \bigotimes_{U\in\mathcal{U}}\mathcal{H}_U[/math] is at most seperable. This is the Haag-Kaster construction of QFT, via local operator nets.

>>11464170
Hint: take [math]t[/math] small enough so that [math]d(tx,0) < d(h,0)[/math]. Remember that metrics are homogeneous so [math]d(\lambda x ,\lambda y) = |\lambda| d(x,y)[/math] for any scalar [math]\lambda\in\mathbb{R}[/math].
>>11464210
I'm confused. [math]t[/math] scales both [math]r[/math] and [math]\theta[/math] so you still approach [math](0,0)[/math]. The function you defined is by definition multivalued, so you cannot use the identification [math]0 \sim 2\pi[/math] on [math]\theta[/math] in the image. This means that [math]t\theta \xrightarrow[t\rightarrow0]{}0[/math] unambiguously, not to [math]2\pi[/math].
Think of it as [math]\ln[/math] being a conformal transform from the Riemann sphere [math]\overline{\mathbb{C}}[/math] to the semi-infinite strip [math]\mathbb{R}\times [0,2\pi][/math] on the principal branch. You only get back the entirety of [math]\mathbb{C}[/math] if you glue every single branch of [math]\ln[/math] together.

>>11456158
Some European/Asian universities have no application fee for their PhD programs. Try them again next year hun.
>>11456568
Having a complex structure gets you a Hodge decomposition [math]\Omega^n_\text{dR} = \bigoplus_{n = p+q}\Omega^{p,q}_\text{Dol}[/math] of the forms. You can't do this in [math]\mathbb{R}^2[/math] because you can have harmonic maps that don't satisfy Cauchy-Riemann.

>>11425549
Do you "immediately" see why the average of a function [math]f\in L^p(\Omega,\mu)[/math] is defined by [math]\mu(f) = \int_\Omega d\mu(x) f(x)[/math]? Perhaps you should think for longer, sweetie.
>how to compute it for those given poses
I don't know what a pose is. Given a representation [math]\rho: G\rightarrow GL_\mathbb{R}(V)[/math] into some matrix group, you'll be able to parameterize [math]G[/math] with [math]\mathbb{R}^n[/math] for [math]n = \operatorname{rank}\rho[/math]. This transforms the Haar measure and its integration [math]\int_G d\mu(g) \rightarrow\int_{\rho(G)} d^n x[/math], or a regular nD integral over the embedding [math]\rho(G)\subset \mathbb{R}^n[/math].
>algorithm
What's that? I don't know what that is.

>>11423617
Variational calculus is how mathematicians actually solve optimization problems. You probably have seen that certain possibly non-linear PDEs can be expressed as the (strong) Euler-Lagrange (sEL) equations [math]L(\{J^k u\}_{0\leq k \leq m}) = 0[/math] for a functional [math]S[/math], as a point-wise version of the zero variation [math]\delta S = 0[/math] condition. Here [math]J^ku[/math] denotes the [math]k[/math]-jets of [math]u[/math], so [math]L[/math] requires [math]m[/math]-differentiability of [math]u[/math] to be a well-defined map.
However, functions [math]u[/math] for which [math]S[u][/math] is well-defined in general has much less regularity conditions on it than the strong EL. For instance, we only require [math]u\in H^1[/math] for [math]S = -\int dx |\nabla u|^2 < \infty[/math] while we need [math]u \in C^2[/math] for [math]\nabla^2 u[/math] to make sense, and [math]H^1[/math] is much [math]much[/math] larger than [math]C^2[/math]. This means that for generic [math]S[/math], optimization [math]\delta S=0[/math] may be achieved by functions for which you cannot even write down a strong-EL for. You cannot use techniques in PDEs, let along those you learn in an undergrad class, to solve these optimization problems. This is the basis and motivation for the theory of distributions.

>>11403520
This anon >>11403524 does raise a good geometric point:
[math]\mathbb{C} = \mathbb{R}\otimes \mathbb{C}[/math] is the complexification, so their isometry groups [math]\operatorname{Isom}\mathbb{R}_\mathbb{C} \cong \operatorname{Isom}\mathbb{C}[/math] are also complexifications of each other. If we just strip away the translations we have [math]O(1)_\mathbb{C} = (\mathbb{Z}_2)_\mathbb{C} \cong SO(2) = U(1)[/math] so inversions in [math]\mathbb{R}[/math] becomes the circle group under complexification.

>>11378740
Not if you actually know that whoever edged anon out this year is a one-in-a-million genius.
>>11378751
It might just be that you didn't do as much/well as you think. Have you done any research? Publications? Clear/reasonable academic goals?
What specifically do you want to study anon? Write a proposal to me as a practice.

>>11378008
By "3x2" you mean spin -1 with spin-1/2? In that case the fact that [math]\mathfrak{su}(2)[/math] has the decomposition [math]{\bf j}_1\otimes{\bf j}_2 = \bigoplus_{k\geq|j_1-j_2|}^{j_1+j_2}{\bf k}[/math], means [math]{\bf 1}\otimes {\bf \frac{1}{2}} = {\bf \frac{1}{2}} \oplus {\bf \frac{3}{2}}[/math] whence you can use the usual Pauli matrices for these irreps to compute the Clebsch-Gordan's. Specifically if [math]V_j = \bigoplus_{|m_j|\leq j}V_{j,m_j}[/math] is the Verma module for the [math]j[/math]-th irrep of [math]SU(2)[/math] spanned by the basis [math]|j,m_j\rangle[/math], then the Clebsch-Gordan's consists of the coefficients that rotate [math]|{\bf 1},m_1\rangle \otimes |{\bf\frac{1}{2}},m_{\frac{1}{2}}\rangle[/math] into [math]|{\bf \frac{1}{2}},m_{\frac{1}{2}}'\rangle \oplus |{\bf \frac{3}{2}},m_{\frac{3}{2}}\rangle[/math], which is accomplished by certain products of elements of the spin-[math]j=\frac{1}{2},\frac{3}{2},1[/math] Pauli matrices.
I'm pretty sure there are tables out there for these values, but this is how you compute them yourself.
>symmetric or antisymmetric
In dimensions at least 3, we can use spin-statistics theorem to find a one-to-one correspondence between half-spins and fermions (antisymmetric), and integral-spins and bosons (symmetric). In 2D ayons/semions exist, depending on the topology of the configuration space, so there is no straightforward correspondence between irreps of [math]\mathfrak{su}(2)[/math] and the statistics.

>>11331885
Use Koopman's lemma. The map [math]T[/math] pulls back to a unitary [math]T^*\in \mathcal{U}(L^2(\mathbb{C},dz))[/math] such that [math](T^*f)(z) = f(Tz)[/math] such that it is ergodic iff 1 is a simple eigenvalue. It then suffices to examine the equation [math]T^*f(z) = f(z) = f(e^{i2\pi\theta}z)[/math] which implies [math]f\in L^2(\mathbb{C},dz)[/math] defines a branched cover of [math]\mathbb{C}[/math] with each branch being "twisted" by [math]\theta[/math]; more specifically, the preimage of [math]z=1[/math], say, under [math]f[/math] contains all points [math]e^{in\theta}[/math] for [math]n\in\mathbb{Z}[/math] labeling the branches, if you will. Since [math]\{e^{in\theta}\mid n\in\mathbb{Z}\}[/math] is dense in [math]S^1\subset\mathbb{C}[/math] for irrational [math]\theta[/math] you are done.
>>11332154
That's not a good idea; contradiction forces you to begin with the non-ergodicity of [math]T[/math] for [math]\theta\in\mathbb{Q}[/math] which is in general not really a tractable condition. Points [math]z[/math] returning infinitely many times under [math]T[/math] is [math]not[/math] equivalent to non-ergodicity.

>>11316326
Their half-lives are extremely short due to their large mass, but they nevertheless constitute alternative scattering channels in high-energy processes. In other words, microscopic scattering events consists of these three generational channels for which the first generation would be "less frequent" than if the other generations' particles did not exist. This may have a very small effect in each scattering event, but on a larger scale (such as star formation) it can have significant consequences.
These heavy excitations also contribute to asymptotic freedom but that's more of a theoretical point.

>>11307385
Honestly? I'd go live with my fiance.

>>11238921
>>11246118
"Energy" is represented by [math]H\in C^\infty(M)[/math] on a symplectic manifold [math](M,\omega)[/math], called a Hamiltonian function. The Hamiltonian vector field [math]X_H[/math] satisfies [math]dH + \iota_{X_H}\omega = 0[/math], and it always exists and is unique. In quantum mechanics, the energy is represented by the a local Hermitian operator [math]H[/math] called the Hamiltonian, which generates the ESA operator of time evolution [math]U(t) = \exp\left(-i Ht\right)[/math] with a dense domain of definition.
In either case, the kinetic energy [math]K[/math] is defined as the "maximal" term in [math]H[/math] that is bilinear; namely [math]K = Q_q(q,q) + Q_p(p,p)[/math] for some ([math]c[/math]-number or operator) quadratic forms [math]Q_{q,p}[/math], invariant under symmetries of [math]H[/math], and such that the potential energy [math]H - K[/math] cannot be written in terms of a single dependence on [math]Q_{q,p}[/math]. By this definition, the potential energy always exists.

>>11173409
First, the functor [math]M \otimes_R \cdot [/math] is right exact. However, for [math]{\bf Tor}^n_R(M,N) = 0[/math] for every [math]n[/math] it means that the derived sequence [math]\rightarrow M\otimes_R P\rightarrow M\otimes_R N \rightarrow 0[/math] is itself exact. If the chain maps are identified as [math]\partial_n \otimes \operatorname{id}_N[/math] then this implies the exactness of the projective resolution [math]\rightarrow P\rightarrow N \rightarrow 0[/math] itself, which I don't think it's the case. The converse might be true if the statement is [math]for ~all ~R[/math]-modules [math]N[/math].
>>11177262
No. If [math]\operatorname{dim}P = 3[/math] and [math]\operatorname{dim}M = 2[/math], a projection [math]P \rightarrow M[/math] does not in general split globally. Both the torus and the Klein bottle project over the base circle [math]S^1[/math] and have circle fibres [math]S^1[/math], but they are topologically distinct; the former splits globally but the latter does not.
>>11179481
Information is defined as the von Neumann entropy [math]S=-\operatorname{tr}\rho \rho \ln \rho[/math], where [math]\rho[/math] is the state operator for the quantum system [math]B[/math]. What "information is stored on the surface" means is that [math]S_A = -\operatorname{tr}_A \rho \ln \rho \sim \operatorname{vol}\partial A[/math], where [math]\operatorname{tr}_A[/math] is the partial trace over [math]A[/math]. In other words the "partial entropy" of the subsystem [math]A\subset B[/math] scales with the volume of the surface of [math]A[/math]. This is in general true for short-range entangled states [math]\rho[/math] with discrete spectral gap.
>>11182368
Given a parameterization [math]t\mapsto x(t) \in C[/math] of the [math]1[/math]-cycle [math]C[/math], we can write [math]\int_C F = \int_C dx F(x) = \int_0^1 dt \iota_{x'(t)}(F(x(t)))[/math], where [math]\iota_V: T^*M\rightarrow \mathbb{R}[/math] is the inner product by the vector field [math]V[/math] on forms.

>>11147127
Antimatter are [math]CT[/math]-partners of matter, where [math]C[/math] and [math]T[/math] are charge conjugation and time reversal operators, respectively. The reason that [math]u + u^\dagger \rightarrow 2\gamma[/math] vertoces are allowed is because you have the the term [math]\psi^\dagger \not A \psi[/math] in the QED Lagrangian.
Remember, bosons have integer spin, and the Abelian [math]U(1)[/math] gauge boson (photon) has spin 0. This means that they transform under trivial representations of [math]\mathfrak{so}(1,3)[/math] for which [math]C[/math] and [math]T[/math] are both the identity.
>>11147731
In terms of plain relativistic QM, the Heisenberg algebra and its representation Fock space gives the proper setting for us to talk about "particles". For a particle to exist at different times, it must be an excitation of a non-local second-quantized field operator [math]\psi[/math] for which [math]\langle \psi^\dagger(t)\psi(t')\rangle \sim \text{const}[/math]. If this particle is not part of the vacuum (namely if [math]\psi \not\in P_0[/math] where [math]P_0[/math] is the orthogonal projection onto [math]\operatorname{ker}H[/math] with [math]H[/math] some Hamiltonian involving [math]\psi[/math]), then it necessarily violates the Lieb-Robinson bound [math]\langle[\tau^H_t(\psi),\psi]\rangle \sim \text{const}[/math], where [math]\tau_t^H(A) = e^{-iHt}Ae^{iHt}[/math] is the Heisenberg dynamics generated by [math]H[/math]. Unless the time direction is compactified to [math]S^1[/math] and [math]t = t' \mod 2\pi[/math] like in a time crystal, you best have a good reason for violating the Lieb-Robinson bound.
>>11149157
The adjoint representation is the natural action [math]h\mapsto ghg^{-1}[/math] of [math]G[/math] on [math]\mathfrak{g}[/math]. Think of this as the conjugation action on [math]G[/math] near the identity; naively it "rotates" the tangent space.

>>11106286
I spoke a little too soon back there. On a more formal level, the unitary algebra [math]\mathfrak{u}(N)[/math] as well as its oriented counterpart [math]\mathfrak{su}(N)[/math] are thought of as an algebra on [math]N^2[/math],[math]N^2-1[/math] generators satisfying certain commutation (Clifford) relations (of course, this was not the historic approach). The algebra is completely characterized by these relations (the universal property), regardless of the rank of the representations you have in mind.
Now when you're thinking explicitly of matrices to describe [math]\mathfrak{su}(N)[/math], you're already thinking of its "minimal" irrep which has rank [math]N[/math] (you can prove this for general Clifford groups in fact, that you can represent [math]\operatorname{Cliff}_{0,N}[/math] with a rank-[math]N[/math] matrix algebra). We can certainly find matrix algebras of rank larger than [math]N[/math] that both satisfy the Clifford relations as well as fixing no subspace, hence they nevertheless form irreps.
>Thanks for your patient responses so far
You're very welcome dear.

>>11101679
>Makes perfect sense.
It sure does :))).
https://arxiv.org/abs/1103.4187
The algebra of quantum operators for sugras form bundles of conformal nets, and was claimed in the above to be a geometric realization for TMFs. This means that sugras can be classified, at least topologically, by TQFTs with state sum given by TMF-Euler classes. If we know the classification of sugras from some [math]C[/math]-theorem for superconformals, we can extract their topological data with TMFs.
My post wasn't meant for you to feel insecure, honey, but please understand that the math I'm describing here does require specialized technical terms in order to convey. Please don't take it personally.

>>11099551
Yep, already on the mailing list.
>>11100826
>nonlinear regularization
Tell me more.

>>11064805
>>11065131
Hello, the TQFT Yukariposter here. Penrose has little to do with much of recent development in TQFT; CFT through string theory perhaps, but not nearly to the extent where he would have a reference.
In general, LSM/HOLSM-type (Hastings-Oshikawa) theorems give necessary conditions for the existence of gapped symmetric ground states (GSs) in generic fermionic systems (liquid or otherwise), and is in fact a very powerful tool for understanding how topological phases arise. The short-range entangled and tensor network states responsible for topological phases in (1+1)D are examples of such symmetric gapped GSs.
The reason why HOLSMs are much more powerful than Chern-Weil (when applied to Chern-Simons-like, [math]F[/math] and [math]F^2[/math], theories) is because it is non-perturbative, in the sense that the topological properties of the GS can be extracted directly from [math]ab initio[/math] interacting models instead through the (dubious) construction of an effective topological field theory Lagrangian first. Besides, non-liquid topological phases and those with sub-extensive GS degeneracy in (2+1)D have no well-defined continuum limit, and hence no notion of a "TQFT". HOLSMs can tackle these cases.
Of course, this comes with the price of not knowing exactly what the topological invariant characterizing such phases are, but works toward anomalous textures have proposed twisted equivariant/group cohomology theories as candidates. This, as well as understanding the bulk-edge Gysin map on them, is my current project.
>>11065227
For foundation, I have found Lehmann's "Mathematical Methods of Many-Body Quantum Field Theory" and Sénéchal's "Theoretical Methods for Strongly Correlated Electrons" (first few chapters) to be good reads, especially for those with a mathematical slant.

>>11012573