The observables [$\hat{q}$](Position%20Operator.md) and [$\hat{p}$](Quantum%20Mechanics/Quantum%20Measurement/Momentum%20Operator.md) are an example of [conjugate observables](conjugate%20observables%20in%20quantum%20mechanics.md) in quantum mechanics and thus they form a _position-momentum [commutation relation](Commutators%20in%20quantum%20mechanics.md#Commuting%20and%20non-commuting%20pairs%20of%20observables)._ Due to its importance, this relation is known as the _canonical commutation relation_$[\hat{q},\hat{p}]=i\hbar\hat{\mathbb{1}}$ which may be written as $[\hat{q},\hat{p}]=i\hbar$ since applying the identity operator is equivalent to multiplying by $1$. This relation is also written in terms of different components of position or momentum on different axes such that $[\hat{q}_i,\hat{p}_j]=i\hbar\delta_{ij}$ where $\delta_{ij}$ is the [Kronecker delta.](Kronecker%20delta.md) This means that $[\hat{q}_i,\hat{p}_j]=0$ for $i=j.$ ^a96259 If we then consider the [position-Momentum commutation relation](Position-Momentum%20Commutators.md) relation along the same spatial axis, the relation reduces to $[\hat{x},\hat{p}]=i\hbar.$ %%many textbooks give [x,x] and [p,p] as commutators as well.%% # Position-Momentum commutators as Lie Brakets The [canonical commutation relation](Position-Momentum%20Commutators.md) is the [Lie bracket](Commutators%20in%20quantum%20mechanics.md#Lie%20brackets%20in%20quantum%20mechanics) that is associated with the [Heisenberg algebra,](Heisenberg%20group.md#Heisenberg%20algebra) # position-momentum commutator for a quantized field --- # Proofs and examples ## Proof that the momentum and position operators are infinite dimensional Since the proof that both the [position operator](Position%20Operator.md) and [momentum operator](Momentum%20Operator.md) are [infinite dimensional](Hilbert%20space%20dimension%20in%20quantum%20mechanics.md#Infinite%20dimensional%20Hilbert%20spaces%20in%20quantum%20mechanics) is derived from the properties of the [position-momentum commutators,](Position-Momentum%20Commutators.md) we will provide it here:   #QuantumMechanics/QuantumMeasurement/QuantumObservables