Borchers algebra
Encyclopedia
In mathematics, a Borchers algebra or Borchers–Uhlmann algebra or BU-algebra is the tensor algebra
of a vector space
, often a space of smooth test functions. They were studied by , who showed that the Wightman distributions of a quantum field could be interpreted as a state
, called a Wightman functional, on a Borchers algebra. A Borchers algebra with a state can often be used to construct an O*-algebra.
The Borchers algebra of a quantum field theory has an ideal called the locality ideal, generated by elements of the form ab−ba for a and b having spacelike-separated support. The Wightman functional of a quantum field theory vanishes on the locality ideal, which is equivalent to the locality axiom for quantum field theory.
Tensor algebra
In mathematics, the tensor algebra of a vector space V, denoted T or T•, is the algebra of tensors on V with multiplication being the tensor product...
of a vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
, often a space of smooth test functions. They were studied by , who showed that the Wightman distributions of a quantum field could be interpreted as a state
State (functional analysis)
In functional analysis, a state on a C*-algebra is a positive linear functional of norm 1. The set of states of a C*-algebra A, sometimes denoted by S, is always a convex set. The extremal points of S are called pure states...
, called a Wightman functional, on a Borchers algebra. A Borchers algebra with a state can often be used to construct an O*-algebra.
The Borchers algebra of a quantum field theory has an ideal called the locality ideal, generated by elements of the form ab−ba for a and b having spacelike-separated support. The Wightman functional of a quantum field theory vanishes on the locality ideal, which is equivalent to the locality axiom for quantum field theory.