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  1. first, if i understand the Lagrangian formalism, then any potential, of any form or origin, would wind up incorporated into the time component, of the (generalized) derivative... e.g. a hypothetical exponential-well, for a mathematical model, of a strong-force potential... second, i perceive a potential problem, in properly interpreting the symbols, since [math]|| d^4x \sqrt{det(g)} ||[/math] = length4 so, seemingly, for a Schwarzschild metric, [math]det(g) = r^4 sin^2(\theta)[/math], so that [math]d^4x \sqrt{det(g)} = c \; dt \; dr \; d\theta \; d\phi \; r^2 \; sin(\theta)[/math] so, si
  2. http://en.wikipedia.org/wiki/Hamilton–Jacobi_equation The "Classical" HJ equation [math]H + \frac{\partial S}{\partial t} = 0[/math] where the action [math]S = \int L dt[/math] gives rise to [math]\Psi = \Psi_0 e^{i \frac{S}{\hbar}}[/math] so the phase of wave-functions, at some particular point, is equal to the time integral, of the action, at that point, from [math]t=-\infty[/math] to the current time ?
  3. [math] \left( E - q \Phi \right) \Psi_+ - c \vec{\sigma}\circ\left(\hat{p}-q \vec{A}\right) \Psi_- = m c^2 \Psi_+[/math] [math] - \left( E - q \Phi \right) \Psi_- + c \vec{\sigma}\circ\left(\hat{p}-q \vec{A}\right) \Psi_+ = m c^2 \Psi_-[/math] If, for each component, of the matter spinor, [math]\Psi_+ \rightarrow \Psi_+' e^{- \imath \frac{m c^2}{\hbar}t}[/math], then [math]\hat{E} \Psi_+ = e^{-\imath \frac{m c^2}{\hbar}t} \hat{E} \Psi_+' + m c^2 \Psi_+[/math]. So, seemingly, the Dirac equation is (nearly) reducible to that of a ("pseudo-")massless particle. Next, re-arranging terms,
  4. http://en.wikipedia.org/wiki/Resonance-enhanced_multiphoton_ionization intuitively, for one electron, to simultaneously absorb two (or more) photons, would require the simultaneous overlap, of all of the involved wave-functions (in a quantum equivalent, of a multi-body collision). If so, then that would inform you of the necessary photon density (assuming photon wave-functions are of order their wavelengths across). If so, then the multiple simultaneous overlap of several wave-functions seems somewhat similar, to the "Rydberg blockade" effect, crucial to the formation of "photonic mol
  5. http://en.wikipedia.org/wiki/Dirac_equation#Comparison_with_the_Pauli_theory [math] \left( E - q \Phi \right) \Psi_+ - c \vec{\sigma}\circ\left(\hat{p}-q \vec{A}\right) \Psi_- = m c^2 \Psi_+[/math] [math] - \left( E - q \Phi \right) \Psi_- + c \vec{\sigma}\circ\left(\hat{p}-q \vec{A}\right) \Psi_+ = m c^2 \Psi_-[/math] If no anti-matter exists, i.e. [math]\Psi_- = 0[/math], then matter must exist in a "super-conducting state", to wit [math]\left( \vec{p} - q \vec{A} \right) = 0[/math]. So, oppositely, if matter does not propagate in such a "super-conducting BCS-similar state", the
  6. Table of Fundamental Fermions charge (col) vs. Hyper-charge (row) [math]\bordermatrix{ ~ & -1 & -\frac{2}{3} & -\frac{1}{3} & 0 & +\frac{1}{3} & +\frac{2}{3} & +1 \cr +1 & ~ & \bar{u} & ~ & \bar{\nu} & \bar{d} & ~ & \bar{e} \cr -1 & e & ~ & d & \nu & ~ & u & ~ \cr}[/math] for Fermions spinning in [math]\langle +\hat{z} \rangle[/math], Fermions' charge-4-vectors are: [math]\bordermatrix{ ~ & \hat{x} & \hat{y} & \langle \hat{z} \rangle & \hat{w} \cr e & -1 & -1 & \langl
  7. http://en.wikipedia.org/wiki/Quantum_electrodynamics#Equations_of_motion When you take the variational derivative, of the QED Lagrangian, with respect to the wave function [math]\Psi[/math]... why doesn't the derivative include terms, due to the conjugate transpose of the wave function [math]\bar{\Psi}[/math] ? In Classical analogy, for a Lagrangian with the KE term [math]\left( \frac{1}{2m} \vec{p}^T \circ \vec{p} \right)[/math], derivatives with respect to momentum would include (one) terms, from the transpose of momentum, which is essentially the same mathematical object
  8. is the following derivation correct (ignoring for simplicity's sake a few factors of c): for a single charged particle, instantaneously located at [math]\vec{r}[/math], in the Classical limit, the fields from said particle, at point [math]\vec{r}'[/math]: [math]\Phi(r') \propto \frac{q}{r}[/math] [math]r = \sqrt{ \left( \vec{r}' - \vec{r} \right) \circ \left( \vec{r}' - \vec{r} \right) }[/math] [math]\vec{A}(r') = \vec{v} \Phi(r')[/math] the potential generate the force fields: [math]\vec{E} = -\nabla \Phi - \frac{\partial \vec{A}}{\partial t} = -\nabla \Phi - \vec{a} \
  9. working backwards, from the Klein-Gordon equation... is the generalized mass-energy relation, for curved spacetime, full of EM fields: [math]\left(P - e A\right)^{\mu} g_{\mu \nu} \left( P - e A \right)^{\nu} = m^2[/math]
  10. minimal coupling would seem appropriate, for single particle wave-function solutions, given the "low energy" limit ? if the KG generalizes, in the presence of EM fields, to [math]D^{\mu} \rightarrow D^{\mu} - e A^{\mu}[/math]... then would any energy potential, be treated, similarly, inserted into the time component of the generalized differential operator ?
  11. Q1: trying to denote spin, w/ subscripts, what prevents the "trifurcation" of photons: [math]\gamma_{+\hbar} \longrightarrow \gamma_{+\hbar} + \gamma_{-\hbar} + \gamma_{+\hbar}[/math] The above hypothetical photon decay could conserve all quantum numbers, as well as spin, and energy + momentum. So, what prevents (presumably?) such processes ? Q2: If photons are spin=1... then why don't electrons, interacting w/ protons in atoms, constantly spin flip, each time they emit or absorb virtual photons ? Q2': If virtual bosons can exist "off mass shell", w/ non-Einstein-equation-c
  12. skipping subscripts, then, the hyper-charge of normal anti-neutrinos is +1 (the opposite of normal neutrinos)... and normal pions are spin-less un-net-hyper-charged particles (+0). So, if backwards electrons still carried the same hyper-charge, as normal electrons... then the following decay could be quickly accounted for: [math]e_R^- + \bar{\nu} \longrightarrow \pi^-[/math] If the Standard Model is completely correct, and hyper-charge is an invariant quantity under Parity, then the above decay poses no problems, and the appropriate page on Wikipedia would be better if duly updated.
  13. what about the generalized Klein-Gordon approach [math]D^{\mu}D_{\mu} \Psi = m^2 \Psi[/math] [math]\longrightarrow[/math] [math]D^{\mu} \eta_{\mu \nu} D^{\nu} \Psi = m^2 \Psi[/math] (inserting Minkowski metric matrix) [math]\longrightarrow[/math] [math]D^{\mu} g_{\mu \nu} D^{\nu} \Psi = m^2 \Psi[/math] (swapping out Minkowski, inserting GR metric matrix) ? The statements of AJB seem completely correct, only somewhat understated... at an age of ~1 TPL, energy densities w/in the universe were ~1 EPL, so since 1 planck time, the whole history of the complete cosmos has been
  14. you would want to write, for right-handed anti-neutrinos [math]\bar{\nu}_L[/math] ?
  15. perhaps important, is the fact that the eigenstates of the Weak interaction, are not the eigenstates of mass (or other Force interactions?). So, quarks emitted from gluons, would not be the same quark-states, emitted from W bosons (which would be "mixed" states, according to the appropriate mixing matrices, e.g. CKM)
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