with the conservation of probability current and density following from the Schrödinger equation:
The fact that the density is positive definite and convected according to this continuity equation, implies that we may integrate the density over a certain domain and set the total to 1, and this condition will be maintained by the conservation law. A proper relativistic theory with a probability density current must also share this feature. Now, if we wish to maintain the notion of a convected density, then we must generalize the Schrödinger expression of the density and current so that the space and time derivatives again enter symmetrically in relation to the scalar wave function. We are allowed to keep the Schrödinger expression for the current, but must replace by probability density by the symmetrically formed expression
which now becomes the 4th component of a space-time vector, and the entire 4-current density has the relativistically covariant expression
1. What exactly are and ?
2. Are they tensors?
3. If they are, how are they defined?