Superenergy tensors for a massive scalar field

Abstract

We define a general class of superenergy tensors of even rank 2(n+1) for a real massive scalar field propagating in Minkowski spacetime. In the case where n=1, we establish that this class is a two-parameter family, which reduces to a unique tensor W(up to a constant factor) when the complete symmetry on the four indices is required. We show that the superenergy density $W^{\al\ba\ga\da}u_{\al}u_{\ba}u_{\ga}u_{\da}$ relative to any timelike unit vector $u$ is positive definite and that the supermomentum density $W^{\al\ba\ga\da}u_{\ba}u_{\ga}u_{\da}$ is a timelike or a null vector ($W^{\al\ba\ga\da}$ stands for W). Next, we find an infinite set of conserved tensors $U_{(p,q)}$ of rank 2+p+q, that we call weak superenergy tensors of order n when p=q=n. We show that $U_{(1,1)}$ and W yield the same total superenergy and the same total supermomentum. Then, using the canonical quantization scheme, we construct explicitly the superhamiltonian and the supermomentum operators corresponding to W and to each weak superenergy tensor $U_{(n,n)}$. Finally, we exhibit a two-parameter family of superenergy tensors for an electromagnetic field and for a gravitational field.