In quantum mechanics, the canonical commutation relation is the fundamental relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform of another). For example,
between the position operator x and momentum operator px in the x direction of a point particle in one dimension, where [x , px] = xpx − pxx is the commutator of x and px, i is the imaginary unit, and ℏ is the reduced Planck constanth/2π, and is the unit operator. In general, position and momentum are vectors of operators and their commutation relation between different components of position and momentum can be expressed as
By contrast, in classical physics, all observables commute and the commutator would be zero. However, an analogous relation exists, which is obtained by replacing the commutator with the Poisson bracket multiplied by iℏ,
This observation led Dirac to propose that the quantum counterparts , ĝ of classical observables f, g satisfy
In 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.[4][6]
Derivation from Hamiltonian mechanics
According to the correspondence principle, in certain limits the quantum equations of states must approach Hamilton's equations of motion. The latter state the following relation between the generalized coordinate q (e.g. position) and the generalized momentum p:
In quantum mechanics the Hamiltonian , (generalized) coordinate and (generalized) momentum are all linear operators.
The time derivative of a quantum state is - (by Schrödinger equation). Equivalently, since the operators are not explicitly time-dependent, they can be seen to be evolving in time (see Heisenberg picture) according to their commutation relation with the Hamiltonian:
In order for that to reconcile in the classical limit with Hamilton's equations of motion, must depend entirely on the appearance of in the Hamiltonian and must depend entirely on the appearance of in the Hamiltonian. Further, since the Hamiltonian operator depends on the (generalized) coordinate and momentum operators, it can be viewed as a functional, and we may write (using functional derivatives):
In order to obtain the classical limit we must then have
Weyl relations
The group generated by exponentiation of the 3-dimensional Lie algebra determined by the commutation relation is called the Heisenberg group. This group can be realized as the group of upper triangular matrices with ones on the diagonal.[7]
Alternately, if and were bounded operators, note that , hence the operator norms would satisfy
so that, for any n,
However, 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 operators and . The resulting braiding relations for these operators are the so-called Weyl relations
These relations may be thought of as an exponentiated version of the canonical commutation relations; they reflect that translations in position and translations in momentum do not commute. One 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.
It is important to note that for technical reasons, the Weyl relations are not strictly equivalent to the canonical commutation relation . If and were bounded operators, then a special case of the Baker–Campbell–Hausdorff formula would allow one to "exponentiate" the canonical commutation relations to the Weyl relations.[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. Indeed, counterexamples exist satisfying the canonical commutation relations but not the Weyl relations.[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.
Generalizations
The simple formula
valid for the quantization of the simplest classical system, can be generalized to the case of an arbitrary Lagrangian.[10] We identify canonical coordinates (such as x in the example above, or a field Φ(x) in the case of quantum field theory) and canonical momentaπx (in the example above it is p, or more generally, some functions involving the derivatives of the canonical coordinates with respect to time):
This definition of the canonical momentum ensures that one of the Euler–Lagrange equations has the form
The canonical commutation relations then amount to
Canonical quantization is applied, by definition, on canonical coordinates. However, in the presence of an electromagnetic field, the canonical momentum p is not gauge invariant. The correct gauge-invariant momentum (or "kinetic momentum") is
where q is the particle's electric charge, A is the vector potential, and c is the speed of light. Although the quantity pkin is the "physical momentum", in that it is the quantity to be identified with momentum in laboratory experiments, it does not satisfy the canonical commutation relations; only the canonical momentum does that. This can be seen as follows.
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. In general, for two Hermitian operatorsA and B, consider expectation values in a system in the state ψ, the variances around the corresponding expectation values being (ΔA)2 ≡ ⟨(A − ⟨A⟩)2⟩, etc.
Then
where [A, B] ≡ A B − B A is the commutator of A and B, and {A, B} ≡ A B + B A is the anticommutator.
Substituting for A and B (and taking care with the analysis) yield Heisenberg's familiar uncertainty relation for x and p, as usual.
Uncertainty relation for angular momentum operators
For the angular momentum operators Lx = y pz − z py, etc., one has that
where is the Levi-Civita symbol and simply reverses the sign of the answer under pairwise interchange of the indices. An analogous relation holds for the spin operators.
Here, for Lx and Ly,[12] in angular momentum multiplets ψ = |ℓ,m⟩, one has, for the transverse components of the Casimir invariantLx2 + Ly2+ Lz2, the z-symmetric relations
⟨Lx2⟩ = ⟨Ly2⟩ = (ℓ (ℓ + 1) − m2) ℏ2/2 ,
as well as ⟨Lx⟩ = ⟨Ly⟩ = 0 .
Consequently, the above inequality applied to this commutation relation specifies
hence
and therefore
so, then, it yields useful constraints such as a lower bound on the Casimir invariant: ℓ (ℓ + 1) ≥ |m| (|m| + 1), and hence ℓ ≥ |m|, among others.
Hall, Brian C. (2013), Quantum Theory for Mathematicians, Graduate Texts in Mathematics, vol. 267, Springer.
Hall, Brian C. (2015), Lie Groups, Lie Algebras and Representations, An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer.