/sci/ - Science & Math

>>11477490
>Give me the final consensus /sci/ is Qunatum computing a meme?
Yes, but it's still useful, feasible, and interesting.

Quantum Computing seems to work, but it's going to take a long time to develop machines reliable enough to factor large coprimes. Other use cases will also take time. Unless a super fast quantum mechanics simulator is discovered for classical computers, quantum computing will provide a unique speedup over classical computers.

>>11453591
Haag theorem states that interacting QFT is not unitarily equivalent to (products of) single-particle QFT. This means that most of scattering calculations done by physicists is wrong and there's no such thing as "the interacting picture".
Given sufficient regularity of the interacting potential, however, Goldstone's theorem in fact proves the asymptotic free-ness of interacting states in the [math]|t|\rightarrow\infty[/math] limit. So the best we can do is to add the axiom of asymptotic completeness to the Sterater-Wightman axioms of QFT and compute [math]S[/math]-matrix elements with far future/past single-particle scattering states. We can't probe intermediate scattering events unless someone develops an interacting/many-body QFT ab initio.

>>11286654
As long as you don't care about the classical limit, there is no need for "quantization" in the strict mathematical sense. Von Neumann's QM text directly constructs states from the *-irreps of the Heisenberg/Weyl algebra. The reason why deformation quantization is such a rich and complex field is because you're studying how to make the CCR/ACR algebra jive well with the classical Poisson structure. There is no need for that in algebraic QM; the density distributions are exactly what you get from the *-irreps and nothing more, what's "classical" is merely a matter of opinion.
>>11287611
Try Zinn-Justin.

>>10527558
Your intuition is correct, since the t-channel for [math]ee \rightarrow \mu\mu[/math] will involve electron-muon bilinears in the amplitude which vanish due to them being in different lepton families.
>>10531310
[math]x^2 + y^2 = 1 - z^2 = (1-z)(1+z)[/math]. You get the latter in the denominator when you compose the two charts.
>>10532008
By Parseval [math]\mathcal{F} \in U(L^2)[/math] preserves the Hermitian [math]L^2[/math] inner product. What this means is that, treating [math]f: X \rightarrow \mathbb{C}[/math] as a [math]L^2[/math] section of a Hilbert line bundle [math]\mathbb{C}\rightarrow X[/math], [math]\mathcal{F}[/math] descends to an automorphism on the space of unit-norm sections [math]f \in S^\infty[/math].
>>10533346
It's minimal coupling. The canonical momentum [math]p \mapsto p - qA[/math] under minimal coupling and when you do Legender transform back to [math]\dot{q}[/math] you get those terms in the Lagrangian. The reason why the canonical momentum acquires a term [math]-qA[/math] is because the [math]U(1)[/math]-principal gauge bundle defines a connection [math]\nabla -qA[/math] on the symplectic manifold [math](M,\omega)[/math] and the symplectic form [math]\omega[/math] needs to be covariantly closed or trajectories in the base space [math]X[/math] of the [math]U(1)[/math]-principal bundle won't lift to a Hamiltonian trajectory on [math]M[/math]. This condition leads to [math]\omega = dq \wedge d\tilde{p}[/math] where [math]\tilde{p} = p - qA[/math].
>>10535506
Any [math]L^2[/math] section of a prequantum bundle prior to doing holomorphic polarization.
>>10536648
Use Kirchoff's formula: [math]E(x) = \frac{1}{4\pi \epsilon_0}(P_x\ast \rho)[/math] where [math]P_x[/math] is the Poisson kernel and [math]\ast[/math] is the convolution operator.

>>9392325
It is.
https://en.wikipedia.org/wiki/Wigner%27s_theorem
https://en.wikipedia.org/wiki/Gelfand%E2%80%93Naimark%E2%80%93Segal_construction

>>9338839
The axiom of microcausality states that given [math]A(x),B(y)
\in \mathcal{A}(U)[/math] observables in the local algebra [math]\mathcal{A}(U)[/math] of [math]U \subset \mathbb{M}[/math] if [math]|x - y|_{\mathbb{M}} < 0[/math] then the vacuum expectation value satisfies [math]\langle A(x)B(y)\rangle = \langle A(x) \rangle\langle B(y)\rangle[/math]. This (along with other axioms in Wightman QFT) implies the locality of the observables in [math]\mathcal{A}(U)[/math] by the clustering theorem [math]\lim_{t \rightarrow \pm\infty}\langle A(x,t)B(y,t)\rangle = \langle A_\pm(x)\rangle\langle B_\pm(y)\rangle[/math] of asymptotic states. This means that the states on which the observables act cannot be entangled if the observables space-like separated. The experiments of Bell et al. shows that Bohmiam mechanics give states that directly contradict this statement. This is also why you can't send information faster than the speed of light with entanglement.
If you don't accept this explanation then you need to take your problem up with Wightman, though I doubt his ghost would suffer a moment with you.

>>9265707
Something like [math]\overleftrightarrow{\partial}[/math]? This means
[eqn]
\hat\psi \overleftrightarrow \psi = \hat\psi \partial \psi + (\partial \hat\psi)\psi
[/eqn]

>>9058158
>Peskin and Schroeder
It's a great intro book; other good ones are Bjorken and Drell, and Srednicki.
Don't read Zee btw it's shit.

>>9027966
>though from what I've heard it's difficult as all hell to get in
That's correct, and it's extremely competitive. Your CV and recs have to be stellar in order to get into Perimeter, since they'd only take in students that they're willing to give substantial fundings for. I've tried when I was looking for masters but to no avail.

>>9000156
You mean [math]SL(2,\mathbb{Z})[/math]? It's the subgroup of the conformal group that fixes fractional quantum Hall states and dual quantum Hall states are related by exactly [math]ad - bc = 1 \mod (2\pi)[/math].