The Story of Superalgebras of Field Operators

The objects subsequently called superalgebras were introduced in the works [1,2]. The starting point was an attempt to answer the question of what the local interactions of quantum fields are in the Heisenberg picture the (anti)commutators of fields shifted in time are not c-numbers in the presence of interaction. In [1] the conjecture was made that in a suitable formalism of the fields, after restriction by 3-linear vertices, some averaging of local operators and in first order of expansion in Δt of the local algebra of the fields A_i, one can give the closed form

where are certain matrix operators including the matrix of interaction parameters. In doing so the commutator operation will connect Bose (d) and Fermi (f) elements of the algebra by the rules

which leads directly to the superalgebra structure. Actually, for fields (as distinct from currents) it must be the algebra of causal connection of two different timelike points and the commutator operation must be constructed in terms...