Weyl quantization and symbol composition

For a compactly supported smooth symbol a, define its Weyl operator by the displayed integral. General symbol classes use the same oscillatory formula with the regularization discussed previously. The factors match the negative-exponent Fourier transform from the first lesson. A real symbol gives a formally symmetric operator; concluding self-adjointness for an unbounded operator still requires a domain argument. On Schwartz functions, position Q=x and momentum P=hD_x are well-defined and permit the polynomial calculations below.

Weyl quantization of xξ is the average of QP and PQ. The two ordered products differ by an h-sized constant, so merely saying replace ξ by hD does not determine the answer. The midpoint convention is especially useful because adjunction corresponds to complex conjugation of the symbol, and its product expansion treats the two factors symmetrically. Polynomial identities can be exact on the Schwartz domain even when the polynomials are not globally bounded order-zero symbols.

Use {a,b}=∂_ξa·∂_xb−∂_xa·∂_ξb. Thus {ξ,x}=1 and {x,ξ}=−1. With this convention the Hamiltonian vector field acts by H_p a={p,a}, giving xdot=∂_ξp and ξdot=−∂_xp. Some physics conventions reverse the order in the definition of the bracket. Either convention is workable, but changing only the bracket while retaining the old commutator coefficient or flow equation gives inconsistent dynamics.

For h-independent smooth symbols with fixed compact phase-space support, the Weyl product begins with ab+(h/(2i)){a,b}. The next terms involve higher derivatives and have h-uniform remainder bounds in the symbol calculus. Antisymmetrizing cancels even powers, so the commutator is (h/i) times the quantized bracket plus an O(h³) operator remainder in this compact-symbol setting. A leading O(h) symbol statement and an operator-norm remainder are different claims; the stated support and derivative assumptions justify passing between them here.

For linear symbols the derivative expansion terminates, and canonical commutators are exact. The result [P,Q]=−ih matches {ξ,x}=1 and h/i=−ih. This check fixes the later Heisenberg sign: the derivative of U*AU for U=exp(−itP_h/h) is (i/h)[P_h,U*AU], whose principal symbol evolves by {p,a}. Symmetric ordering is therefore part of a coherent package connecting algebra, Fourier transforms, and classical flow.

A linear transformation of phase space need not preserve this package. In one conjugate coordinate pair, a real linear map is symplectic exactly when its determinant is one. A shear preserves the canonical bracket, while stretching position without the reciprocal momentum adjustment does not. The deterministic matrix lab checks this elementary distinction; it does not implement metaplectic operators or prove the general covariance theorem. Nonlinear canonical transformations need Fourier integral operators and cannot be replaced by arbitrary substitutions into operator products.