A new functional calculus for non commuting operators

A fascinating topic!

The functional calculus for non-commuting operators is a relatively new area of research, and there are several approaches to develop such a calculus. Here, I'll provide an overview of some of the recent developments:

Motivation

In quantum mechanics, operators do not necessarily commute with each other. For example, the position and momentum operators do not commute. This non-commutativity leads to difficulties in defining a functional calculus for non-commuting operators. The traditional functional calculus, developed by von Neumann and Stone, relies heavily on the commutativity of operators.

Recent developments

Several approaches have been proposed to develop a functional calculus for non-commuting operators:

  1. Non-commutative functional calculus: This approach, developed by A. Connes and M. Rieffel, uses the theory of non-commutative geometry to define a functional calculus for non-commuting operators. The key idea is to replace the usual algebra of functions on a manifold by a non-commutative algebra of operators.
  2. Operator-valued distributions: This approach, developed by J. B. Conway and M. A. Rieffel, uses operator-valued distributions to define a functional calculus for non-commuting operators. The idea is to extend the traditional functional calculus to operators that do not commute with each other.
  3. Deformation quantization: This approach, developed by M. Kontsevich and D. Kazhdan, uses deformation quantization to define a functional calculus for non-commuting operators. The idea is to deform the traditional functional calculus by introducing a non-commutative parameter.
  4. Non-commutative spectral theory: This approach, developed by M. Rieffel and others, uses non-commutative spectral theory to define a functional calculus for non-commuting operators. The idea is to extend the traditional spectral theory to operators that do not commute with each other.

Key results

Some of the key results in this area include:

Open problems

Despite these advances, there are still many open problems in this area, including:

Applications

The development of a functional calculus for non-commuting operators has many potential applications in physics, mathematics, and engineering, including:

In conclusion, the development of a functional calculus for non-commuting operators is an active area of research, with many potential applications in physics, mathematics, and engineering. While there have been many advances in this area, there are still many open problems to be solved.