anticommutate

In mathematics, particularly in linear algebra and quantum mechanics, two operators A and B are said to anticommute if their commutator is zero. The commutator of two operators is defined as [A, B] = AB - BA. Therefore, anticommutation means that AB - BA = 0, which simplifies to AB = BA.

This property of anticommutation is fundamental in certain areas of physics, most notably in quantum field theory and quantum mechanics. For instance, the anticommutation relations are central to the formulation of fermionic quantum fields. Fermions are particles that obey Fermi-Dirac statistics, and their creation and annihilation operators satisfy specific anticommutation relations. These relations ensure that the Pauli exclusion principle is inherently satisfied, meaning that no two identical fermions can occupy the same quantum state.

The anticommutation relation for fermionic creation operators, denoted by c†, and annihilation operators, denoted by c, is typically written as {c†i, c†j} = 0, {ci, cj} = 0, and {ci, c†j} = δij, where δij is the Kronecker delta. The curly braces denote the anticommutator, defined as {A, B} = AB + BA. Thus, the condition for anticommutation is {A, B} = 0.

If two operators anticommute, it means that the order in which they are applied does not matter for their product, similar to commuting operators, but the definition involves a sign change. Specifically, if AB = -BA, then the operators anticommute. This is the more common definition used in contexts like quantum mechanics when referring to anticommutation relations.

Understanding anticommutation is crucial for comprehending the behavior of quantum systems, particularly those involving identical particles. It is a key concept that distinguishes the quantum mechanical description of fermions from bosons.