/sci/ - Science & Math

>>9770355
It depends on both what the 1-parameter (semi)group of time evolution is and what you consider "physical" trajectories. If the 1-parameter group is not Lie or if you have caustics along the physical trajectory then [math]\hat{H}[/math] in general need not be complete or bounded.
In the GNS construction of the Fock space [math]\mathcal{F}[/math], there is a theorem that states the number operator [math]\hat{N}[/math] is bounded iff a unique vacuum [math]|0\rangle \in \mathcal{F}[/math] exists. So once you can find a distinguished vacuum vector, then you can build up the usual SHO Hamiltonian without worrying about boundedness.
>>9770415
That's right, and also it'd be a problem if solutions exists in a nice space, e.g. [math]H^1[/math], only for some range of values of a parameter in the PDE. I've had a colleague asking me about why he sees finite-proper time blow-ups in the numerical solutions to his Einstein field equations at certain values of the cosmological constant, even if he fixed a nice initial condition.
>>9770581
Guillemin & Sternberg, Woodhouse, von Neumann

>>9526006
>but I don't think he had any effect on anything tangible.
So you don't think developing Feynman diagrams in conformal field theories as knot invariants as "tangible"? What kind of experimentalist (i.e. retarded) world view do you have?
>Index theorems and whatnot are a good way to see and organize the math around the tools for toy models in current field theory work
Atiyah-Singer allows you to construct topologically non-trivial sigma models that can describe physical phenomena from the quantum Hall effect to cosmic strings. Atiyah-Bott/Deligne-Vergne allows you to use the stationary phase approximation as exact solutions for the stationary points of Feynman path integrals. How are these not tangible?
>but it's rather unrelated to e.g. like the 1930 quantum electrodynamics have on laser physics, or like the stat. mech models have on conductor development.
No shit. The latter things you mentioned are physical constructs, developed by physicists. >Meanwhile, without Minkowski it would have taken a while till spacetime would have been conceptualized like it is now
What a hilariously simple criterion. You might as well nominate Poincare since his name is on the group of isometries for the Minkowski space.
>taken a while
I highly doubt that. Some simple diff geo wasn't beyond even someone as stupid as Einstein.
>and that's physics theories of which we can argue that they really apply.
What are you even trying to say? That most physical theories are required to be relativistic? Do you even know what the Osterwalder-Schrader axioms and the reconstruction theorems are?
Why do undergrads keep thinking they have anything substantial to say regarding these things? They'd only embarrass themselves.

>>9239395
Is it really any wonder why it's always people who have no idea what they're talking about bashing bra-ket notation?

>>9101575
>Ya got prelims next semester or just going straight into advanced topics?
Prelims (i.e. core courses) were already taken care of by my MSc. It's all research from this point on, and maybe an interesting course here and there.
>Hopefully after the summer is over most of the of the really annoying shitposters will leave or at the very least be less active.
And we both know that the cuckold mods wouldn't do jack shit about that.
I'll try to keep discussions afloat in the physics general (if I ever get the time to make one). Hopefully the retards steer clear of it.

>>9059842
To me there's no sense in worrying about having a physical interpretation when it all just comes down to whether you accept the metaphysical ontology offered by QM (i.e. that all objects are described by their quantum states and not their position/momenta or amplitude/frequency) or not .
The one thing to understand is that duality has a very rigorous mathematical formalization given in constructive QFT a la Baez. Duality is defined to be the theorem that there exists an inner product-preserving orthogonal/unitary map from free boson/fermion fields defined in terms of operator-valued probability distributions to tensor products of Hilbert spaces satisfying bosonic/fermionic statistics, respectively. The former is a formalization of "wave" and the latter that of "particles". There's really no other more precise meaning to "duality".