/sci/ - Science & Math

>>11452481
Alright. Thanks for looking.

>>11322775
That's not the point; you [math]definitely[/math] want your fields to form reps of [math]\operatorname{Spin}(1,3)[/math], regardless if Streater-Wightman axioms are able to reproduce the full power of QFT. No one said that those are complete, but they (as well as the additional completeness assumption on the Hilbert space) for sure form a lower bound.
>extremely basic scenarios
If you consider a full theory of free fields, which forms the basis of literally every scatter computation, to be "extremely basic", then I really don't know what to tell you.
>Weren't those uncapable of describing anything
No axiomatic approach to general QFT (i.e. not TQFT/CFT) has yet been completely able to capture what QFTists are doing. The closest we have is Haag-Kastler's local operator nets and Strocchi-Swieca/Osterwalder-Schrader's analytic constructions of Wightman/Schwinger functions, and these in some scenarios the regularity requirements are too stringent to produce any useful [math]S[/math]-matrix.
Aside from obvious obstructions like Haag's theorem, we can't find the Poisson structures needed for Kostant-Souriau-Sardanashvily geometric quantization of jets, and Zhang-Baez's AQFT struggles to get even basic classical limits like WKB. One needs to strike the perfect balance between formalism and applicability in order to produce useful axiomatization of QFT.
The gap is much less significant than you think, however, as CFTs (IR fixed points of QFTs) can in fact be fully captured by the mathematical theory of Friedan-Shenkar's gauge system. Not to mention Atiyah-Lurie's axiomatization of TQFTs has produced much progress in both physics and mathematics.

>>11290322
No, the physics general died because of schizos and cranks like that dude who denies QM.

>>11245983
Read Di Francesco

>>9532631
>In the first place, Berezin integration has nothing to do with sums.
Why should they?
>Should we still think of a grassmann valued field as taking some stochastic sequence of configurations, each of which has some contribution to the observable we calculate?
Yes? Berezin integration is a linear functional from the algebra of Grassmann numbers to [math]\mathbb{C}[/math] so I don't see why not?
>It seems not so,
Why?
>but then the entire notion of "path integration" seems only to apply to bosons
No? It applies to any particle with any spin statistics given sufficient regularity conditions (i.e. Swieca-Strozzi type or when there's some non-trivial topology/geometry to exploit, like in TQFTs or CFTs).
>and most particles are not even bosons.
In dimension larger than two there are only bosons and fermions.
>In the second place, it doesn't even seem like the concept of a "grassmann valued field" means anything in terms of quantum states.
Except it does? They're representations of the generators of an infinite dimensional Clifford algebra.
>In the case of bosons, a field s(x) is normal (so [s(x), s*(x)] = 0; where s*(x) is the Hermitian conjugate of s(x)) and at a fixed time t = 0, [s(x),s(y)] = 0.
That's not what a boson is at all. Bosons satisfy the Heisenberg algebra [math][\psi^\dagger(x),\psi(y)]_{x^0 = y^0} = \delta({\bf x}-\bf{y}) \neq 0[/math]. In general bosons are operator-valued distributions, and so are fermions.
>Because of these two commutation rules, it is possible to simultaneously diagonalize the s(x)'s at each spacetime point
That's not true at all. There'd be no off-diagonal S-matrix elements otherwise.

>>9461105
>is it really that well understood in the physics community?
The short answer is no. The long answer is because we haven't completely understood what constitutes an appropriate framework for rigorous QFT yet, which is why we get things like Haag's theorem and regularization problems showing up when dealing with ill-defined S-matrices.
Geometric quantization and AQFT are ways to construct rigorous frameworks for QFT. Though AQFT is more like a "safe than sorry" approach to QFT; it's nice and all but it can't deal with anything other than free fields. If I were to put money on which approach would lead to a better understanding of the formalism of QFT I'd probably go with geometric quantization, though that is not to say that AQFT isn't interesting in its own right.

>>9431879
>What is the "most correct" theory we have now?
Nothing is and will be "correct". Everything we will ever develop are just effective field theories for other effective field theories at yet higher/lower energy hierarchies.

>>9408607
Because space-like separated states cannot in general give rise to entangled expectations. This is one of the consequences of the axioms of Wightman QFT. And since Wightman axioms (+ asymptotic completeness) underlies the mathematical foundations of some of the most successful physical theories so far in the history of mankind it's pretty unlikely that the Wightman axioms are wrong just because it forbids FTL entanglement.

>>9391202
At some point the distinction vanishes. If you're looking for physics background I suggest you read Landau-Lifshitz.
>obfuscated by layers upon layers of notation, convention and assumptions
Example? I've never seen this happen.
>Is it just a barrier that i'll get through with more energy, or do i lack some fundamental component to be useful in physics?
Hard to say without an example.

>>9348432
It is sensible to interpret it that way, though I'd say that they're entirely distinct things, just named similarly. The mathematical statement of particle-wave duality is the existence of a unitary morphism taking boson fields [math](\Gamma,A,\omega,v_0)[/math] to an [math]N[/math]-particle Fock space [math]F[/math] that maps the vacuum field [math]v_0\in\Gamma[/math] to the cyclic vacuum state [math]|0\rangle[/math] in the Fock space. Since this statement doesn't directly concern energies I'd be hard-pressed to say that the continuous/discrete dichotomy is due to particle-wave duality.

>>9338546
>Non-locality in general, reguardless of what quantum interpretation you ascribe to, is still a fairly strange and unexplained phenomenon.
It isn't. Microcausality can be formalized rigorously with von Neumann algebras, which implies locality of observables. In fact this is one of the Wightman axioms of QFT; you wouldn't have had any problems with if you knew the mathematics behind it.