Lock_charge_wave_2026-11-12, trial 1: the longitudinal (charge-carrying) branch of the quon lattice

  path G-X [100]: k from 0 to 1.0000 pi/d, 41 points
       t = 0.00   w_L^2/w_p^2 = 1.000000   hbar w_L =  361.331 keV   v_g/c = +0.00120   |e.k_hat| = 1.000000
       t = 0.25   w_L^2/w_p^2 = 1.015354   hbar w_L =  364.094 keV   v_g/c = +0.02142   |e.k_hat| = 1.000000
       t = 0.50   w_L^2/w_p^2 = 1.052294   hbar w_L =  370.658 keV   v_g/c = +0.02962   |e.k_hat| = 1.000000
       t = 0.75   w_L^2/w_p^2 = 1.089059   hbar w_L =  377.078 keV   v_g/c = +0.02049   |e.k_hat| = 1.000000
       t = 1.00   w_L^2/w_p^2 = 1.104236   hbar w_L =  379.696 keV   v_g/c = +0.00113   |e.k_hat| = 1.000000
  1    G-X [100]: band of hbar w_L (keV)                                              361.331 to 379.696
  2    G-X [100]: largest |v_g|/c                                                     0.02962
  3    G-X [100]: least |e.k_hat| on the branch (1 = purely longitudinal)             1.000000

  path G-M [110]: k from 0 to 1.4142 pi/d, 41 points
       t = 0.00   w_L^2/w_p^2 = 1.000000   hbar w_L =  361.331 keV   v_g/c = -0.00287   |e.k_hat| = 1.000000
       t = 0.25   w_L^2/w_p^2 = 0.946330   hbar w_L =  351.501 keV   v_g/c = -0.05693   |e.k_hat| = 1.000000
       t = 0.50   w_L^2/w_p^2 = 0.800018   hbar w_L =  323.188 keV   v_g/c = -0.10365   |e.k_hat| = 1.000000
       t = 0.75   w_L^2/w_p^2 = 0.626949   hbar w_L =  286.102 keV   v_g/c = -0.09776   |e.k_hat| = 1.000000
       t = 1.00   w_L^2/w_p^2 = 0.546345   hbar w_L =  267.078 keV   v_g/c = -0.00624   |e.k_hat| = 0.707107
  1    G-M [110]: band of hbar w_L (keV)                                              267.078 to 361.331
  2    G-M [110]: largest |v_g|/c                                                     0.11124
  3    G-M [110]: least |e.k_hat| on the branch (1 = purely longitudinal)             0.707107

  path G-R [111]: k from 0 to 1.7321 pi/d, 41 points
       t = 0.00   w_L^2/w_p^2 = 1.000000   hbar w_L =  361.331 keV   v_g/c = -0.00537   |e.k_hat| = 1.000000
       t = 0.25   w_L^2/w_p^2 = 0.882900   hbar w_L =  339.517 keV   v_g/c = -0.09955   |e.k_hat| = 1.000000
       t = 0.50   w_L^2/w_p^2 = 0.621018   hbar w_L =  284.746 keV   v_g/c = -0.14657   |e.k_hat| = 1.000000
       t = 0.75   w_L^2/w_p^2 = 0.404530   hbar w_L =  229.816 keV   v_g/c = -0.09874   |e.k_hat| = 1.000000
       t = 1.00   w_L^2/w_p^2 = 0.333333   hbar w_L =  208.615 keV   v_g/c = -0.00507   |e.k_hat| = 0.618665
  1    G-R [111]: band of hbar w_L (keV)                                              208.615 to 361.331
  2    G-R [111]: largest |v_g|/c                                                     0.14657
  3    G-R [111]: least |e.k_hat| on the branch (1 = purely longitudinal)             0.618665

  4    overall band of the charge-carrying branch (keV)                               208.615 to 379.696
  5    overall largest group velocity |v_g|/c                                         0.14657
  6    magnetic field of a longitudinal wave: curl E = 0, so dB/dt = 0                none
  7    P1 (travels at c at all energies)                                              FAIL
  8    P2 (B perpendicular to the Poynting vector, |B| = |E|/c)                       FAIL (B = 0)
  9    P3 (two transverse polarizations)                                              FAIL (one longitudinal branch)
