Canonical commutation relation

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 $i\backslash hbar$,

$$\backslash \{x,p\backslash \}\; =\; 1\; \backslash ,\; .$$

This observation led Dirac to propose that the quantum counterparts $\backslash hat\{f\}$, $\backslash hat\{g\}$ of classical observables {{mvar|f}}, {{mvar|g}} satisfy

$$[\backslash hat\; f,\backslash hat\; g]=\; i\backslash hbar\backslash widehat\{\backslash \{f,g\backslash \}\}\; \backslash ,\; .$$

In 1946, Hip Groenewold demonstrated that a general systematic correspondence between quantum commutators and Poisson brackets could not hold consistently.{{Cite journal | last1 = Groenewold | first1 = H. J. | title = On the principles of elementary quantum mechanics | doi = 10.1016/S0031-8914(46)80059-4 | journal = Physica | volume = 12 | issue = 7 | pages = 405–460 | year = 1946 |bibcode = 1946Phy....12..405G }}{{harvnb|Hall|2013}} Theorem 13.13

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.{{Cite journal | last1 = Curtright | first1 = T. L. | last2 = Zachos | first2 = C. K. | doi = 10.1142/S2251158X12000069 | title = Quantum Mechanics in Phase Space | journal = Asia Pacific Physics Newsletter | volume = 01 | pages = 37–46 | year = 2012 | arxiv = 1104.5269 | s2cid = 119230734 }}

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:

$$\backslash begin\{cases\}$$

\dot{q} = \frac{\partial H}{\partial p} = \{q, H\}; \\

\dot{p} = -\frac{\partial H}{\partial q} = \{p, H\}.

\end{cases}

In quantum mechanics the Hamiltonian $\backslash hat\{H\}$, (generalized) coordinate $\backslash hat\{Q\}$ and (generalized) momentum $\backslash hat\{P\}$ are all linear operators.

The time derivative of a quantum state is represented by the operator $-i\backslash hat\{H\}/\backslash hbar$ (by the Schrödinger equation). Equivalently, since in the Schrödinger picture the operators are not explicitly time-dependent, the operators can be seen to be evolving in time (for a contrary perspective where the operators are time dependent, see Heisenberg picture) according to their commutation relation with the Hamiltonian:

$$\backslash frac\; \{d\backslash hat\{Q\}\}\{dt\}\; =\; \backslash frac\; \{i\}\{\backslash hbar\}\; [\backslash hat\{H\},\backslash hat\{Q\}]$$

$$\backslash frac\; \{d\backslash hat\{P\}\}\{dt\}\; =\; \backslash frac\; \{i\}\{\backslash hbar\}\; [\backslash hat\{H\},\backslash hat\{P\}]\; \backslash ,\backslash ,\; .$$

In order for that to reconcile in the classical limit with Hamilton's equations of motion, $[\backslash hat\{H\},\backslash hat\{Q\}]$ must depend entirely on the appearance of $\backslash hat\{P\}$ in the Hamiltonian and $[\backslash hat\{H\},\backslash hat\{P\}]$ must depend entirely on the appearance of $\backslash hat\{Q\}$ 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):

$$[\backslash hat\{H\},\backslash hat\{Q\}]\; =\; \backslash frac\; \{\backslash delta\; \backslash hat\{H\}\}\{\backslash delta\; \backslash hat\{P\}\}\; \backslash cdot\; [\backslash hat\{P\},\backslash hat\{Q\}]$$

$$[\backslash hat\{H\},\backslash hat\{P\}]\; =\; \backslash frac\; \{\backslash delta\; \backslash hat\{H\}\}\{\backslash delta\; \backslash hat\{Q\}\}\; \backslash cdot\; [\backslash hat\{Q\},\backslash hat\{P\}]\; \backslash ,\; .$$

In order to obtain the classical limit we must then have

$$[\backslash hat\{Q\},\backslash hat\{P\}]\; =\; i\; \backslash hbar\; ~\; I.$$

The group $H\_3(\backslash mathbb\{R\})$ generated by exponentiation of the 3-dimensional Lie algebra determined by the commutation relation $[\backslash hat\{x\},\backslash hat\{p\}]=i\backslash hbar$ is called the Heisenberg group. This group can be realized as the group of $3\backslash times\; 3$ upper triangular matrices with ones on the diagonal.{{harvnb|Hall|2015}} Section 1.2.6 and Proposition 3.26

According to the standard mathematical formulation of quantum mechanics, quantum observables such as $\backslash hat\{x\}$ and $\backslash hat\{p\}$ 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 $\backslash hat\{x\}$ and $\backslash hat\{p\}$ were trace class operators, the relation $\backslash operatorname\{Tr\}(AB)=\backslash operatorname\{Tr\}(BA)$ gives a nonzero number on the right and zero on the left.

Alternately, if $\backslash hat\{x\}$ and $\backslash hat\{p\}$ were bounded operators, note that $[\backslash hat\{x\}^n,\backslash hat\{p\}]=i\backslash hbar\; n\; \backslash hat\{x\}^\{n-1\}$, hence the operator norms would satisfy

$$2\; \backslash left\backslash |\backslash hat\{p\}\backslash right\backslash |\; \backslash left\backslash |\backslash hat\{x\}^\{n-1\}\backslash right\backslash |\; \backslash left\backslash |\backslash hat\{x\}\backslash right\backslash |\; \backslash geq\; n\; \backslash hbar\; \backslash left\backslash |\backslash hat\{x\}^\{n-1\}\backslash right\backslash |,$$ so that, for any n,

$$2\; \backslash left\backslash |\backslash hat\{p\}\backslash right\backslash |\; \backslash left\backslash |\backslash hat\{x\}\backslash right\backslash |\; \backslash geq\; n\; \backslash hbar$$

However, {{mvar|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 $\backslash exp(it\backslash hat\{x\})$ and $\backslash exp(is\backslash hat\{p\})$. The resulting braiding relations for these operators are the so-called Weyl relations

$$\backslash exp(it\backslash hat\{x\})\backslash exp(is\backslash hat\{p\})=\backslash exp(-ist\backslash hbar)\backslash exp(is\backslash hat\{p\})\backslash exp(it\backslash hat\{x\}).$$

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.

For technical reasons, the Weyl relations are not strictly equivalent to the canonical commutation relation $[\backslash hat\{x\},\backslash hat\{p\}]=i\backslash hbar$. If $\backslash hat\{x\}$ and $\backslash hat\{p\}$ 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.See Section 5.2 of {{harvnb|Hall|2015}} for an elementary derivation 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.{{harvnb|Hall|2013}} Example 14.5 (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 $\backslash mathbb\{Z\}/n$, can be realized on a finite-dimensional Hilbert space by means of the clock and shift matrices.

It can be shown that

$$[F(\backslash vec\{x\}),p\_i]\; =\; i\backslash hbar\backslash frac\{\backslash partial\; F(\backslash vec\{x\})\}\{\backslash partial\; x\_i\};\; \backslash qquad\; [x\_i,\; F(\backslash vec\{p\})]\; =\; i\backslash hbar\backslash frac\{\backslash partial\; F(\backslash vec\{p\})\}\{\backslash partial\; p\_i\}.$$

Using $C\_\{n+1\}^\{k\}\; =\; C\_\{n\}^\{k\}\; +\; C\_\{n\}^\{k-1\}$, it can be shown that by mathematical induction

$$\backslash left[\backslash hat\{x\}^n,\backslash hat\{p\}^m\backslash right]\; =\; \backslash sum\_\{k=1\}^\{\backslash min\backslash left(m,n\backslash right)\}\{\; \backslash frac\{-\backslash left(-i\; \backslash hbar\backslash right)^k\; n!m!\}\{k!\backslash left(n-k\backslash right)!\backslash left(m-k\backslash right)!\}\; \backslash hat\{x\}^\{n-k\}\; \backslash hat\{p\}^\{m-k\}\}\; =\; \backslash sum\_\{k=1\}^\{\backslash min\backslash left(m,n\backslash right)\}\{\; \backslash frac\{\backslash left(i\; \backslash hbar\backslash right)^k\; n!m!\}\{k!\backslash left(n-k\backslash right)!\backslash left(m-k\backslash right)!\}\; \backslash hat\{p\}^\{m-k\}\backslash hat\{x\}^\{n-k\}\}\; ,$$

generally known as McCoy's formula.McCoy, N. H. (1929), "On commutation formulas in the algebra of quantum mechanics", Transactions of the American Mathematical Society 31 (4), 793-806 [https://pdfs.semanticscholar.org/1bc1/688c10bbb6d6630e647f675695a822f2a380.pdf online]

In addition, the simple formula

$$[x,p]\; =\; i\backslash hbar\; \backslash ,\; \backslash mathbb\{I\}\; ~,$$

valid for the quantization of the simplest classical system, can be generalized to the case of an arbitrary Lagrangian $\{\backslash mathcal\; L\}$.{{cite book |first=J. S. |last=Townsend |title=A Modern Approach to Quantum Mechanics |url=https://archive.org/details/modernapproachto0000town |url-access=registration |publisher=University Science Books |location=Sausalito, CA |year=2000 |isbn=1-891389-13-0 }} We identify canonical coordinates (such as {{mvar|x}} in the example above, or a field {{math|Φ(x)}} in the case of quantum field theory) and canonical momenta {{math|π_x}} (in the example above it is {{mvar|p}}, or more generally, some functions involving the derivatives of the canonical coordinates with respect to time):

$$\backslash pi\_i\; \backslash \; \backslash stackrel\{\backslash mathrm\{def\}\}\{=\}\backslash \; \backslash frac\{\backslash partial\; \{\backslash mathcal\; L\}\}\{\backslash partial(\backslash partial\; x\_i\; /\; \backslash partial\; t)\}.$$

This definition of the canonical momentum ensures that one of the Euler–Lagrange equations has the form

$$\backslash frac\{\backslash partial\}\{\backslash partial\; t\}\; \backslash pi\_i\; =\; \backslash frac\{\backslash partial\; \{\backslash mathcal\; L\}\}\{\backslash partial\; x\_i\}.$$

The canonical commutation relations then amount to

$$[x\_i,\backslash pi\_j]\; =\; i\backslash hbar\backslash delta\_\{ij\}\; \backslash ,$$

where {{math|δ_ij}} is the Kronecker delta.

Canonical quantization is applied, by definition, on canonical coordinates. However, in the presence of an electromagnetic field, the canonical momentum {{mvar|p}} is not gauge invariant. The correct gauge-invariant momentum (or "kinetic momentum") is

: $p\_\backslash text\{kin\}\; =\; p\; -\; qA\; \backslash ,\backslash !$ (SI units) {{spaces|4}} $p\_\backslash text\{kin\}\; =\; p\; -\; \backslash frac\{qA\}\{c\}\; \backslash ,\backslash !$ (cgs units),

where {{mvar|q}} is the particle's electric charge, {{mvar|A}} is the vector potential, and {{math|c}} is the speed of light. Although the quantity {{math|p_kin}} 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 {{mvar|m}} in a classical electromagnetic field is (in cgs units)

$$H=\backslash frac\{1\}\{2m\}\; \backslash left(p-\backslash frac\{qA\}\{c\}\backslash right)^2\; +q\backslash phi$$

where {{mvar|A}} is the three-vector potential and {{mvar|φ}} is the scalar potential. This form of the Hamiltonian, as well as the Schrödinger equation {{math|1=Hψ = iħ∂ψ/∂t}}, the Maxwell equations and the Lorentz force law are invariant under the gauge transformation

$$A\backslash to\; A\text{'}\; =\; A+\backslash nabla\; \backslash Lambda$$

$$\backslash phi\backslash to\; \backslash phi\text{'}\; =\; \backslash phi-\backslash frac\{1\}\{c\}\; \backslash frac\{\backslash partial\; \backslash Lambda\}\{\backslash partial\; t\}$$

$$\backslash psi\; \backslash to\; \backslash psi\text{'}\; =\; U\backslash psi$$

$$H\backslash to\; H\text{'}\; =\; U\; H\; U^\backslash dagger,$$

where $$U=\backslash exp\; \backslash left(\; \backslash frac\{iq\backslash Lambda\}\{\backslash hbar\; c\}\backslash right)$$ and {{math|1=Λ = Λ(x,t)}} is the gauge function.

The angular momentum operator is

$$L=r\; \backslash times\; p\; \backslash ,\backslash !$$

and obeys the canonical quantization relations

$$[L\_i,\; L\_j]=\; i\backslash hbar\; \{\backslash epsilon\_\{ijk\}\}\; L\_k$$

defining the Lie algebra for so(3), where $\backslash epsilon\_\{ijk\}$ is the Levi-Civita symbol. Under gauge transformations, the angular momentum transforms as

$$\backslash langle\; \backslash psi\; \backslash vert\; L\; \backslash vert\; \backslash psi\; \backslash rangle\; \backslash to$$

\langle \psi^\prime \vert L^\prime \vert \psi^\prime \rangle =

\langle \psi \vert L \vert \psi \rangle +

\frac {q}{\hbar c} \langle \psi \vert r \times \nabla \Lambda \vert \psi \rangle \, .

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

$$K=r\; \backslash times\; \backslash left(p-\backslash frac\{qA\}\{c\}\backslash right),$$

which has the commutation relations

$$[K\_i,K\_j]=i\backslash hbar\; \{\backslash epsilon\_\{ij\}\}^\{\backslash ,k\}$$

\left(K_k+\frac{q\hbar}{c} x_k

\left(x \cdot B\right)\right)

where $$B=\backslash nabla\; \backslash times\; A$$ is the magnetic field. The inequivalence of these two formulations shows up in the Zeeman effect and the Aharonov–Bohm effect.

All such nontrivial commutation relations for pairs of operators lead to corresponding uncertainty relations,{{cite journal |first=H. P. |last=Robertson |title=The Uncertainty Principle |journal=Physical Review |volume=34 |issue=1 |year=1929 |pages=163–164 |doi=10.1103/PhysRev.34.163 |bibcode = 1929PhRv...34..163R }} involving positive semi-definite expectation contributions by their respective commutators and anticommutators. In general, for two Hermitian operators {{mvar|A}} and {{mvar|B}}, consider expectation values in a system in the state {{mvar|ψ}}, the variances around the corresponding expectation values being {{math|1=(ΔA)² ≡ {{langle}}(A − {{langle}}A{{rangle}})²{{rangle}}}}, etc.

Then

$$\backslash Delta\; A\; \backslash ,\; \backslash Delta\; B\; \backslash geq\; \backslash frac\{1\}\{2\}\; \backslash sqrt\{\; \backslash left|\backslash left\backslash langle\backslash left[\{A\},\{B\}\backslash right]\backslash right\backslash rangle\; \backslash right|^2\; +\; \backslash left|\backslash left\backslash langle\backslash left\backslash \{\; A-\backslash langle\; A\backslash rangle\; ,B-\backslash langle\; B\backslash rangle\; \backslash right\backslash \}\; \backslash right\backslash rangle\; \backslash right|^2\}\; ,$$

where {{math|1=[A, B] ≡ A B − B A}} is the commutator of {{mvar|A}} and {{mvar|B}}, and {{math|1={A, B} ≡ A B + B A}} is the anticommutator.

This follows through use of the Cauchy–Schwarz inequality, since

{{math|{{!}}{{langle}}A²{{rangle}}{{!}} {{!}}{{langle}}B²{{rangle}}{{!}} ≥ {{!}}{{langle}}A B{{rangle}}{{!}}²}}, and {{math|1=A B = ([A, B] + {A, B})/2 }}; and similarly for the shifted operators {{math|A − {{langle}}A{{rangle}}}} and {{math|B − {{langle}}B{{rangle}}}}. (Cf. uncertainty principle derivations.)

Substituting for {{mvar|A}} and {{mvar|B}} (and taking care with the analysis) yield Heisenberg's familiar uncertainty relation for {{mvar|x}} and {{mvar|p}}, as usual.

For the angular momentum operators {{math|1=L_x = y p_z − z p_y}}, etc., one has that

$$[\{L\_x\},\; \{L\_y\}]\; =\; i\; \backslash hbar\; \backslash epsilon\_\{xyz\}\; \{L\_z\},$$

where $\backslash epsilon\_\{xyz\}$ 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 {{mvar|L_x}} and {{mvar|L_y }}, in angular momentum multiplets {{math|1=ψ = {{!}}{{ell}},m{{rangle}}}}, one has, for the transverse components of the Casimir invariant {{math|L_x² + L_y²+ L_z²}}, the {{mvar|z}}-symmetric relations

:{{math|1={{langle}}L_x²{{rangle}} = {{langle}}L_y²{{rangle}} = ({{ell}} ({{ell}} + 1) − m²) ℏ²/2 }},

as well as {{math|1={{langle}}L_x{{rangle}} = {{langle}}L_y{{rangle}} = 0 }}.

Consequently, the above inequality applied to this commutation relation specifies

$$\backslash Delta\; L\_x\; \backslash ,\; \backslash Delta\; L\_y\; \backslash geq\; \backslash frac\{1\}\{2\}\; \backslash sqrt\{\backslash hbar^2|\backslash langle\; L\_z\; \backslash rangle|^2\}~,$$

hence

$$\backslash sqrt$$

and therefore

$$\backslash ell(\backslash ell+1)-m^2\backslash geq\; |m|\; ~,$$

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

• Canonical quantization
• CCR and CAR algebras
• Conformastatic spacetimes
• Lie derivative
• Moyal bracket
• Stone–von Neumann theorem

{{reflist}}

• {{citation|first=Brian C.|last=Hall|title=Quantum Theory for Mathematicians|series=Graduate Texts in Mathematics|volume=267 |publisher=Springer|year=2013}}.
• {{citation|first=Brian C.|last=Hall|title=Lie Groups, Lie Algebras and Representations, An Elementary Introduction|series=Graduate Texts in Mathematics|edition=2nd|volume=222 |publisher=Springer|year=2015}}.

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