Manuscript XI-B@1 · XI-B.tex · section · 063b989e
Off-shell and causal presentations
The polynomial field algebra and the free differential › Off-shell and causal presentations · lines 785–824
XI-B@1#eq:causal-even, XI-B@1#eq:causal-aux, XI-B@1#eq:causal-ghost
Let be the complex graded commutative polynomial algebra generated by compact interior smearings of the five independent fields. Products are finite. They have smooth finite-rank multivariable kernels, so no coincident-field composite is needed to define them.
This is an off-shell algebra: no field equation is used in Lemma lem:nilpotence. A second, free-shell algebra is obtained by imposing the linear equations and causal commutation relations. Making that second presentation does not retroactively impose free equations on classical relational functionals.
Put
The full causal matrix is the retarded-minus-advanced inverse of the action Hessian. Its nonzero blocks are
The appropriate vector Kronecker symbols accompany the ghost blocks. These are causal commutator kernels, not Hadamard two-point functions.
Displayed formulas in this passage
XI-B@1#eq:causal-even · line 806
XI-B@1#eq:causal-aux · line 806
XI-B@1#eq:causal-ghost · line 806
unlabeled · line 800