Need more glimpses of QFT in my life. Is this a type of formalism that extends across multiple field theories? The article details the case of the stationary, bound electron, interacting with EM fields (the stationary particle allowing for just the "E" requirement of "EM", presumably?), but can a Hamiltonian treatment arrive at the same sort of re-formulation of Heisenberg's results for fields like early universe quark gluon plasma, or weak force-dominated nuclear meltdown? Obviously, gravity is out, but I would guess the standard model covers the electroweak-to-strong forces. I'm like 99.5% sure. edit: experimentalist shaming should be every theorist's cathartic, collective pastime, seriously. :) My understanding is that Lagrangian mechanics better accommodates for Newtonian/classical-scale dynamics because the difference in potential vs. kinetic energy is much more defined than at the quantum scale, where total energy is a more useful framing. Thus, the Hamiltonian. I'm leaning towards thinking that this zero-point field approach (fluctuations in energy above some equilibrium "zero") also reinforces the same idea. But my understanding could be wrong. Always. If the vacuum field fluctuations are CMB energy density level-ish (3 K or whatever thermal wavelengths for almost any particle resonance not too relevant except for maybe... neutrinos[?] ), I dunno, for most environments hospitable to our lifeform, seems inconsequential. Like everything else, gets weird near black holes, with blueshift. You know. Was great to see conjugate/Hilbert space treatment of Fourier equivalence from a new approach. Still working on the maths. Can't say I'm done with it until things 100% click. Could be never.
Yes, it underlines them, so to speak. You can even derive the standard quantum mechanics representation from it. It's also often clearer to operate on symmetries and prove conjectures for entire classes of objects. SQMR: the classical algebra of functions of position, covariant for the group of shifts in space. As a rule, you'd use Lagrangian over Hamiltonian whenever you want to keep the momentum of a system you're describing fixed to a value; convient from particle collision physics to path integral formulation. Hamiltonians are convenient because they themselves are explicitly a conserved value (total energy), and with their solutions usually being first-order, it's also easier to find their time evolution. Going between L and H formalism is as simple as applying Legendre transform, and they describe the same physics. Sometimes, one is more convenient to work with, that's all. Sometimes one is trivial to solve but problematic to interpret, and vice versa. I can, and will, go on if provoked. More to come, I need to check some stuff to avoid long-form redaction later.Is this a type of formalism that extends across multiple field theories?
My understanding is that Lagrangian mechanics better accommodates for Newtonian/classical-scale dynamics because the difference in potential vs. kinetic energy is much more defined than at the quantum scale, where total energy is a more useful framing.
I should have been more clear, sorry. It sounded like I was asking if QFT is a formulation for all the fields (save gravity), but I meant; Can we recover the same relationships between x and p if we give a treatment of, for example, how a quark interacts with a gluon? Obviously, the answer is yes. Doubly obvious, since we've unified the fields (...save gravity). Duh and sorry. It's funny because also in my field, we minimize or maximize observables (a lot of times I minimize "Faraday residue", the residual in Faraday's Law, from E-field & B-field measurements) across a dataset to yield the most suitable transformation matrix; finding the relationship between local geometric orientation of a boundary layer and the global coordinate system. The process is reducible to an eigenvector/eigenvalue formulation. But SQMR (a special case of QMR?) sounds like a generalized version of this(?). provoking intensifiesYes, [the formalism] underlines them, so to speak. You can even derive the standard quantum mechanics representation from it.
SQMR
I can, and will, go on if provoked.
Eh, I'm not entirely sure here? QCD is an odd beast, one where at the same time you have gauge invariant gluon spin distribution, but no 3-direction projection of gluon spin can be gauge invariant by itself. Unfortunately, I only know enough to have semi-educated doubts about general statement. Discipline divided by common maths and insular lingo. SQMR is a shorthand for the "Consider the Hilbert space of quadratically integrable functions of positions phi(q), Integral[phi(q)^2, {q, R^n}] < Infinity..." you see in most textbooks. Still reading up on stuff though.Can we recover the same relationships between x and p if we give a treatment of, for example, how a quark interacts with a gluon? Obviously, the answer is yes. Doubly obvious, since we've unified the fields (...save gravity). Duh and sorry.
But SQMR (a special case of QMR?) sounds like a generalized version of this(?).