PhyslibAlpha.QuantumMechanics.OperatorAlgebra.Observables.Lie

Lie structure on observables

The observables of a complex C⋆-algebra carry the Lie bracket

⁅a, b⁆ = -(i / 2) (ab - ba),

equivalently the imaginary part of their product.

Together with the Jordan product, this gives the antisymmetric and symmetric parts of observable multiplication. The Lie bracket measures noncommutativity and governs infinitesimal unitary dynamics.

Lie bracket

Lie ring

Real Lie algebra

Elementary identities

2 declarations

theorem

The product of observables $a$ and $b$ is an observable $\iff ab = ba$

#isSelfAdjoint_mul_iff_commute

Let $A$ be a complex $C^*$ -algebra. For any two observables $a, b \in A$ (which are self-adjoint elements, i.e., $a^* = a$ and $b^* = b$ ), their product $ab$ is an observable if and only if $a$ and $b$ commute, such that $ab = ba$ .

theorem

Decomposition of the product of observables into Jordan and Lie parts: $ab = a \odot b + i \unicode{x2045}a, b\unicode{x2046}$

#mul_decomposition

Let $A$ be a complex $C^*$ -algebra. For any two observables $a, b \in A$ , their product in the algebra can be decomposed into its Jordan and Lie parts as \[ab = (a \odot b) + i \unicode{x2045}a, b\unicode{x2046}\] where $a \odot b$ is the Jordan product (the symmetric part) and $\unicode{x2045}a, b\unicode{x2046}$ is the Lie bracket (the imaginary part).