Abstract
For an arbitrary electromagnetic field, we define a prepotential S, which is a complex-valued function of spacetime. The prepotential is a modification of the two scalar potential functions introduced by E. T. Whittaker. The prepotential is Lorentz covariant under a spin half representation. For a moving charge and any observer, we obtain a complex dimensionless scalar. The prepotential is a function of this dimensionless scalar. The prepotential S of an arbitrary electromagnetic field is described as an integral over the charges generating the field. The Faraday vector at each point may be derived from S by a convolution of the differential operator with the alpha matrices of Dirac. Some explicit examples will be calculated. We also present the Maxwell equations for the prepotential.
Highlights
Introduction and motivationIn general, the electromagnetic field tensor F, expressed by a 4 × 4 antisymmetric matrix, is used to describe the electromagnetic field intensity
We show that for each such vector, there is a complex dimensionless scalar which is invariant under a certain representation of the Lorentz group
We show that the matrices occurring in the description of the connection of the prepotential to the field are a representation of the Dirac α-matrices
Summary
The electromagnetic field tensor F , expressed by a 4 × 4 antisymmetric matrix, is used to describe the electromagnetic field intensity. Whittaker introduced [24] two scalar potential functions He was able to reduce the degrees of freedom of the electromagnetic field description to 2. In classical mechanics we have rotating forces, which are described by two-forms Such forces cannot be expressed as derivatives of a scalar potential. The prepotential must be complex-valued, and we will need to define a Lorentz invariant conjugation of the gradient of the prepotential in order to obtain the 4-potential of the field. Another important property of a prepotential of an electromagnetic field is its locality. We show that the matrices occurring in the description of the connection of the prepotential to the field are a representation of the Dirac α-matrices
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