§ DICTIONARY · CONCEPT
Poisson bracket
The antisymmetric bilinear {f, g} = Σ (∂f/∂q·∂g/∂p − ∂f/∂p·∂g/∂q). Every observable evolves as df/dt = {f, H}.
§ 01
Definition
The Poisson bracket of two phase-space functions f and g is the antisymmetric bilinear operation {f, g} = Σᵢ (∂f/∂qᵢ · ∂g/∂pᵢ − ∂f/∂pᵢ · ∂g/∂qᵢ). Hamilton's equations themselves can be written q̇ = {q, H}, ṗ = {p, H}, and every observable f that does not explicitly depend on time evolves as df/dt = {f, H}.
Dirac's canonical quantisation maps the Poisson bracket to the quantum commutator: {f, g}_classical ↔ (1/iℏ)[f̂, ĝ]_quantum. This is not a derivation but a dictionary — it is why position and momentum, whose classical Poisson bracket is {q, p} = 1, have quantum commutator [q̂, p̂] = iℏ.
§ 02