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6X AX action principle angular momentum anticommutation relations antisymmetric matrix antisymmetric tensor arbitrary bilinear concomitant boson commutation relations condition consider const contribute corresponding degrees of freedom denotes dependence derivatives Dirac displacement dt dt dt|t dynamical variables electric and magnetic energy density commutator equations of constraint equations of motion expression fermion fields field equations finite number form the commutation function gauge invariant gauge transformation given Gq(t H dt Hence Hermitian matrix Hermitian operator imaginary independent infinite line infinitesimal transformations kinematic Lagrangian density last lecture loop Lorentz group Lorentz invariance Lorentz transformation magnetic charge non-Hermitian nonsingular notation number of degrees path of integration pk(t positive definite Quantum Mechanics real symmetric relativistic requires rotations satisfied simplest choice skew Hermitian space spin spinor fields symmetric and real Tok(x transverse unitary transformations V X H variations 6X verify write