Canonical commutation relation

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] = x pxpx x is the commutator of x and px, i is the imaginary unit, and is the reduced Planck constant h/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

where is the Kronecker delta.

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.

Relation to classical mechanics edit

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 i,


This observation led Dirac to propose that the quantum counterparts  , 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 edit

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 edit

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]

According to the standard mathematical formulation of quantum mechanics, quantum observables such as   and   should be represented as self-adjoint operators on some Hilbert space. It is relatively easy to see that two operators satisfying the above canonical commutation relations cannot both be bounded. Certainly, if   and   were trace class operators, the relation   gives a nonzero number on the right and zero on the left.

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 edit

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 momenta πx (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

where δij is the Kronecker delta.

Further, it can be shown that


Using  , it can be shown that by mathematical induction

generally known as McCoy's formula.[11]

Gauge invariance edit

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

  (SI units)        (cgs units),

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)

where A is the three-vector potential and φ is the scalar potential. This form of the Hamiltonian, as well as the Schrödinger equation = iħ∂ψ/∂t, the Maxwell equations and the Lorentz force law are invariant under the gauge transformation
and Λ = Λ(x,t) is the gauge function.

The angular momentum operator is

and obeys the canonical quantization relations
defining the Lie algebra for so(3), where   is the Levi-Civita symbol. Under gauge transformations, the angular momentum transforms as

The gauge-invariant angular momentum (or "kinetic angular momentum") is given by

which has the commutation relations
is the magnetic field. The inequivalence of these two formulations shows up in the Zeeman effect and the Aharonov–Bohm effect.

Uncertainty relation and commutators edit

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 operators A and B, consider expectation values in a system in the state ψ, the variances around the corresponding expectation values being A)2 ≡ ⟨(A − ⟨A⟩)2, etc.


where [A, B] ≡ A BB A is the commutator of A and B, and {A, B} ≡ A B + B A is the anticommutator.

This follows through use of the Cauchy–Schwarz inequality, since |⟨A2⟩| |⟨B2⟩| ≥ |⟨A B⟩|2, and A B = ([A, B] + {A, B})/2 ; and similarly for the shifted operators A − ⟨A and B − ⟨B. (Cf. uncertainty principle derivations.)

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 edit

For the angular momentum operators Lx = y pzz 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 = |,m, one has, for the transverse components of the Casimir invariant Lx2 + Ly2+ Lz2, the z-symmetric relations

Lx2⟩ = ⟨Ly2⟩ = ( ( + 1) − m2) ℏ2/2 ,

as well as Lx⟩ = ⟨Ly⟩ = 0 .

Consequently, the above inequality applied to this commutation relation specifies

and therefore
so, then, it yields useful constraints such as a lower bound on the Casimir invariant:  ( + 1) ≥ |m| (|m| + 1), and hence ≥ |m|, among others.

See also edit

References edit

  1. ^ "The Development of Quantum Mechanics".
  2. ^ Born, M.; Jordan, P. (1925). "Zur Quantenmechanik". Zeitschrift für Physik. 34 (1): 858–888. Bibcode:1925ZPhy...34..858B. doi:10.1007/BF01328531. S2CID 186114542.
  3. ^ Kennard, E. H. (1927). "Zur Quantenmechanik einfacher Bewegungstypen". Zeitschrift für Physik. 44 (4–5): 326–352. Bibcode:1927ZPhy...44..326K. doi:10.1007/BF01391200. S2CID 121626384.
  4. ^ a b Groenewold, H. J. (1946). "On the principles of elementary quantum mechanics". Physica. 12 (7): 405–460. Bibcode:1946Phy....12..405G. doi:10.1016/S0031-8914(46)80059-4.
  5. ^ Hall 2013 Theorem 13.13
  6. ^ Curtright, T. L.; Zachos, C. K. (2012). "Quantum Mechanics in Phase Space". Asia Pacific Physics Newsletter. 01: 37–46. arXiv:1104.5269. doi:10.1142/S2251158X12000069. S2CID 119230734.
  7. ^ Hall 2015 Section 1.2.6 and Proposition 3.26
  8. ^ See Section 5.2 of Hall 2015 for an elementary derivation
  9. ^ Hall 2013 Example 14.5
  10. ^ Townsend, J. S. (2000). A Modern Approach to Quantum Mechanics. Sausalito, CA: University Science Books. ISBN 1-891389-13-0.
  11. ^ McCoy, N. H. (1929), "On commutation formulas in the algebra of quantum mechanics", Transactions of the American Mathematical Society 31 (4), 793-806 online
  12. ^ a b Robertson, H. P. (1929). "The Uncertainty Principle". Physical Review. 34 (1): 163–164. Bibcode:1929PhRv...34..163R. doi:10.1103/PhysRev.34.163.
  • 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.