Borchers algebra

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Borchers algebra

A Borchers algebra (also called the Borchers–Uhlmann algebra or BU-algebra) is the tensor algebra built from a vector space, usually a space of smooth test functions. It provides an abstract setting for studying products of field operators.

History and idea: The concept was developed by H. J. Borchers in 1962. He showed that the Wightman distributions of a quantum field—the vacuum expectations of products of fields—can be interpreted as a state on the Borchers algebra. This state is called a Wightman functional. With a BU-algebra and a state, one can often construct an O*-algebra (a type of operator algebra).

Locality: The BU-algebra associated with a quantum field theory has a locality (or commutator) ideal generated by elements of the form ab − ba when a and b have spacelike-separated support. The Wightman functional vanishes on this ideal, which Expresses the locality axiom of quantum field theory.

References: H. J. Borchers, 1962, “On structure of the algebra of field operators.”; Jakob Yngvason, 2009, “The Borchers–Uhlmann Algebra and its Descendants.”