Canonical commutation relation
This relation is attributed to Werner Heisenberg, Max Born and Pascual Jordan (1925),[1][2] who called it a "quantum condition" serving as a postulate of the theory; it was noted by E. Kennard (1927)[3] to imply the Heisenberg uncertainty principle.The Stone–von Neumann theorem gives a uniqueness result for operators satisfying (an exponentiated form of) the canonical commutation relation.However, an analogous relation exists, which is obtained by replacing the commutator with the Poisson bracket multiplied byThis observation led Dirac to propose that the quantum counterpartsIn 1946, Hip Groenewold demonstrated that a general systematic correspondence between quantum commutators and Poisson brackets could not hold consistently.[4][5] However, he further appreciated that such a systematic correspondence does, in fact, exist between the quantum commutator and a deformation of the Poisson bracket, today called the Moyal bracket, and, in general, quantum operators and classical observables and distributions in phase space.He thus finally elucidated the consistent correspondence mechanism, the Wigner–Weyl transform, that underlies an alternate equivalent mathematical representation of quantum mechanics known as deformation quantization.The time derivative of a quantum state is represented by the operatorIn order for that to reconcile in the classical limit with Hamilton's equations of motion,generated by exponentiation of the 3-dimensional Lie algebra determined by the commutation relationHowever, n can be arbitrarily large, so at least one operator cannot be bounded, and the dimension of the underlying Hilbert space cannot be finite.If the operators satisfy the Weyl relations (an exponentiated version of the canonical commutation relations, described below) then as a consequence of the Stone–von Neumann theorem, both operators must be unbounded.Still, these canonical commutation relations can be rendered somewhat "tamer" by writing them in terms of the (bounded) unitary operatorsOne can easily reformulate the Weyl relations in terms of the representations of the Heisenberg group.The uniqueness of the canonical commutation relations—in the form of the Weyl relations—is then guaranteed by the Stone–von Neumann theorem.[8] Since, as we have noted, any operators satisfying the canonical commutation relations must be unbounded, the Baker–Campbell–Hausdorff formula does not apply without additional domain assumptions.[9] (These same operators give a counterexample to the naive form of the uncertainty principle.)These technical issues are the reason that the Stone–von Neumann theorem is formulated in terms of the Weyl relations.A discrete version of the Weyl relations, in which the parameters s and t range over, can be realized on a finite-dimensional Hilbert space by means of the clock and shift matrices.valid for the quantization of the simplest classical system, can be generalized to the case of an arbitrary LagrangianThis definition of the canonical momentum ensures that one of the Euler–Lagrange equations has the formHowever, in the presence of an electromagnetic field, the canonical momentum p is not gauge invariant.The non-relativistic Hamiltonian for a quantized charged particle of mass m in a classical electromagnetic field is (in cgs units)All such nontrivial commutation relations for pairs of operators lead to corresponding uncertainty relations,[12] involving positive semi-definite expectation contributions by their respective commutators and anticommutators.Substituting for A and B (and taking care with the analysis) yield Heisenberg's familiar uncertainty relation for x and p, as usual.For the angular momentum operators Lx = y pz − z py, etc., one has thatis the Levi-Civita symbol and simply reverses the sign of the answer under pairwise interchange of the indices.Here, for Lx and Ly ,[12] in angular momentum multiplets ψ = |ℓ,m⟩, one has, for the transverse components of the Casimir invariant Lx2 + Ly2+ Lz2, the z-symmetric relations as well as ⟨Lx⟩ = ⟨Ly⟩ = 0 .Consequently, the above inequality applied to this commutation relation specifies