In [quantum Mechanics](Quantum%20Mechanics%20(index).md) commutators between [observables](Observable.md) determine whether one measurement will influence the result of another measurement. A pair of observables whose measurements influence each other are referred to as [_incompatible_](conjugate%20observables%20in%20quantum%20mechanics.md) while those whose measurements don't influence each other are [_compatible_.](compatible%20observables.md) Since [commutators](Commutators%20in%20quantum%20mechanics.md) are an algebraic property of [observable](Observable.md) pairs, they also appear as algebraic properties in [second quantization](second%20quantization), namely for [bosonic creation and annihilation operators](creation%20and%20annihilation%20operators.md#Bosonic%20creation%20and%20annihilation%20operators), which themselves also give rise to observables. # Commuting and non-commuting pairs of observables ## Quantum uncertainty relations ([... see more](Quantum%20uncertainty%20relations.md)) # Lie brackets in quantum mechanics Algebraically, commutators in [quantum mechanics](Quantum%20Mechanics%20(index).md) are also [Lie brackets,](Lie%20bracket.md) meaning that they associated with [Lie algebras.](Lie%20algebras.md) %%There needs to be a separate entry for quantum symmetries and Lie Groups in quantum mechanics.%% # Commutation relations with both observables and time evolution operators %%this is specifically when we deal with commutators between time evolution operators see this question as well https://physics.stackexchange.com/questions/83791/does-the-time-evolution-operator-commute-with-any-other-operators%% %% # Commutation relations in 2nd quantization [second quantization](second%20quantization) # Commutation relations with density matrices %%Density matrices are not observables - or at least not necessarily so. so there has to be some meaning to commutation relations that include density matrices that needs to be discussed here.%% #QuantumMechanics/MathematicalFoundations #QuantumMechanics/QuantumMeasurement/QuantumObservables