In mathematics, a Borchers algebra, Borchers–Uhlmann algebra, or BU-algebra is the
tensor algebra
In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra over a field, algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', ...
of a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
, often a space of smooth
test function
In mathematical analysis, a bump function (also called a test function) is a function f : \Reals^n \to \Reals on a Euclidean space \Reals^n which is both smooth (in the sense of having continuous derivatives of all orders) and compactly suppor ...
s. They were studied by , who showed that the
Wightman distributions of a
quantum field
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct physical models of subatom ...
could be interpreted as a
state
State most commonly refers to:
* State (polity), a centralized political organization that regulates law and society within a territory
**Sovereign state, a sovereign polity in international law, commonly referred to as a country
**Nation state, a ...
, 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.
References
*
External links
*
Operator algebras
Axiomatic quantum field theory
{{mathanalysis-stub