Eikonal Blog

2010.09.15

Feynman derivation of Maxwell equations

  • “On Feynman’s Approach to the Foundations of Gauge Theory” by M. C. Land, N. M. Shnerb and L. P. Horwitz; 2 Aug 1993. – 36 p. – http://cdsweb.cern.ch/record/568394/:
      Abstract: In 1948, Feynman showed Dyson how the Lorentz force and Maxwell equations could be derived from commutation relations coordinates and velocities. Several authors noted that the derived equations are not Lorentz covariant and so are not the standard Maxwell theory. In particular, Hojman and Shepley proved that the existence of commutation relations is a strong assumption, sufficient to determine the corresponding action, which for Feynman’s derivation is of Newtonian form. Tanimura generalized Feynman’s derivation to a Lorentz covariant form, however, this derivation does not lead to the standard Maxwell theory either. Tanimura’s force equation depends on a fifth ({\it scalar}) electromagnetic potential, and the invariant evolution parameter cannot be consistently identified with the proper time of the particle motion. Moreover, the derivation cannot be made reparameterization invariant; the scalar potential causes violations of the mass-shell constraint which this invariance should guarantee. In this paper, we examine Tanimura’s derivation in the framework of the proper time method in relativistic mechanics, and use the technique of Hojman and Shepley to study the unconstrained commutation relations. We show that Tanimura’s result then corresponds to the five-dimensional electromagnetic theory previously derived from a Stueckelberg-type quantum theory in which one gauges the invariant parameter in the proper time method. This theory provides the final step in Feynman’s program of deriving the Maxwell theory from commutation relations; the Maxwell theory emerges as the “correlation limit” of a more general gauge theory, in which it is properly contained.

2010.02.05

Equations of electro+magneto-dynamics in 3+1 dimensions

Filed under: physics — Tags: , — sandokan65 @ 22:49

Maxwell equations:

  • \nabla \vec{D} = \rho_e,
  • \nabla \vec{B} = \rho_m,
  • \partial_t  \vec{B} + \nabla \times \vec{E} = - \vec{j}_m,
  • \partial_t  \vec{D} - \nabla \times \vec{H} = - \vec{j}_e.

Integral form of Maxwell equations:

  • \oint_S \vec{D}\cdot d\vec{S} = Q_e :\equiv \int_V \rho_e dV,
  • \oint_S \vec{B}\cdot d\vec{S} = Q_m :\equiv \int_V \rho_m dV,
  • \oint_l  \vec{E} \cdot d\vec{l} = - \int_S ( \vec{j}_m + \partial_t  \vec{B})
  • \oint_l  \vec{H} \cdot d\vec{l} = + \int_S ( \vec{j}_e + \partial_t  \vec{D})

Moments of the field (aka Energy-Momentum(-Pressure) complex of the electromagnetic field):

  • Flow of energy (Poynting vector): \vec{\Gamma}:\equiv\vec{E}\times\vec{H},
  • Linear momentum of field: \vec{G}:\equiv \vec{B} \times\vec{D},
  • Stress-pressure tensor: {\cal T}:\equiv \frac12 \vec{E}\otimes\vec{D} + \frac12 \vec{H}\otimes\vec{B} - \frac14 {\bf 1} \frac12 (\vec{E}\cdot\vec{D} + \vec{H}\cdot\vec{B}).

Material equations: in the case of the weak fields
the polarizable and magnetizable environment yields the linear responses:

  • \vec{D} = \epsilon \vec{E} + \vec{P} = \hat{\epsilon} \vec{E},
  • \vec{B} = \mu \vec{H} + \mu \vec{M} = \hat{\mu} \vec{H},

where the linear coefficients are the permittivity (\epsilon) and the the permeability (\mu), and the polarization and the magnetization of the substance are measured by vector fields \vec{P} and \vec{M}.

When \rho_m\ne 0 or \vec{\jmath}_m\ne 0 one can not use the signgle complex of potentials (\Phi, \vec{A}) (as used in ordinary electrodynamics), but should add one more complex of potentials (\Psi, \vec{F}). Then:

  • \vec{E} = - \partial_t \vec{A} -\nabla \Phi -c^2 \nabla \times \vec{F},
  • \vec{D} = - \partial_t \vec{L} -\nabla \Psi - \nabla \times \vec{U},
  • \vec{H} = - \partial_t \vec{U} -\nabla \phi + c^2 \nabla \times \vec{L},
  • \vec{B} = - \partial_t \vec{F} -\nabla \psi + \nabla \times \vec{A}.

The source constrainst:

  • \partial_t \rho_e + \nabla \cdot \vec{\jmath}_e = 0,
  • \partial_t \rho_m + \nabla \cdot \vec{\jmath}_m = 0.

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