Addendum to vector calculus: fields as co-chains of differential operators

Abstract

Let D be the set of differential operators on functions defined on R3, Let alpha (p), p=0,1,2 and 3 be scalar, vector, pseudovector and density fields on R3, respectively. Then alpha (p) can be regarded as a multilinear map Dp to D. Let r0,...,rp in R3 and let (r0,...,rp) be the Euclidean p-simplex having these vertices. Then the integral of alpha (p) over (r0,...,rp) is a function F( alpha (p)) (r0,...,rp) which satisfies Fd alpha = delta ASFalpha . Here d alpha (p) means grad alpha (0), curl alpha (1) or div alpha (2), and delta AS is a natural cohomological operator (the usual Alexander-Spanier co-boundary operator). For any function F(r0,...,rp) there is a natural map Phi F:Dp to D which satisfies Phi ( delta ASF)= delta H Phi F Where delta H is the Hochschild co-boundary for co-chains Phi F on D. Thus, when alpha (p) is regarded as being the cochain Phi (F alpha (p)) on D, grad, curl and div all become Hochschild co-boundary operators: grad alpha (0)(H) is the commutator of the operator H with the function alpha (0) and curl alpha (1)(H1,H2) measures the amount by which alpha (1) fails to be a derivation on D. If div alpha (2)=0 then alpha (2) provides a deformation of the composition product on D. This new viewpoint of fields as operator-valued maps of p-tuples of operators has implications in several areas of physics and mathematics. One consequence is that the Hamiltonian in quantum mechanics may be regarded as its own probability current density operator. Another is that Maxwell's equations describe the algebraic character of the electric and magnetic fields E and B regarded as co-chains on D. We give some explicit formulae for alpha (p)(H1,...,Hp).