Editing Canonical commutation relation

{{Short description|Relation satisfied by conjugate variables in quantum mechanics}}

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,
<math display="block">[\hat x,\hat p_x] = i\hbar \mathbb{I}</math>

between the position operator {{mvar|x}} and momentum operator {{mvar|p<sub>x</sub>}} in the {{mvar|x}} direction of a point particle in one dimension, where {{math|1= [''x'' , ''p''<sub>''x''</sub>] = ''x'' ''p''<sub>''x''</sub> − ''p''<sub>''x''</sub> ''x''}} is the [[Commutator#Ring theory|commutator]] of {{mvar|x}} and {{mvar|p<sub>x</sub> }}, {{mvar|i}} is the [[imaginary unit]], and {{math|ℏ}} is the [[reduced Planck constant]] {{math|''h''/2&pi;}}, and <math> \mathbb{I}</math> 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
<math display="block">[\hat x_i,\hat p_j] = i\hbar \delta_{ij},</math>
where <math>\delta_{ij}</math> is the [[Kronecker delta]].

This relation is attributed to [[Werner Heisenberg]], [[Max Born]] and [[Pascual Jordan]] (1925),<ref>{{cite web |title=The Development of Quantum Mechanics|url=https://www.heisenberg-gesellschaft.de/3-the-development-of-quantum-mechanics-1925-ndash-1927.html}}</ref><ref>{{Cite journal | last1 = Born | first1 = M. | last2 = Jordan | first2 = P. | doi = 10.1007/BF01328531 | title = Zur Quantenmechanik | journal = Zeitschrift für Physik | volume = 34 | pages = 858–888 | year = 1925 | issue = 1 |bibcode = 1925ZPhy...34..858B | s2cid = 186114542 }}</ref> who called it a "quantum condition" serving as a postulate of the theory; it was noted by [[Earle Hesse Kennard|E. Kennard]] (1927)<ref>{{Cite journal | last1 = Kennard | first1 = E. H. | title = Zur Quantenmechanik einfacher Bewegungstypen | doi = 10.1007/BF01391200 | journal = Zeitschrift für Physik | volume = 44 | issue = 4–5 | pages = 326–352 | year = 1927 |bibcode = 1927ZPhy...44..326K | s2cid = 121626384 }}</ref> to imply the [[Werner Heisenberg|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 ==
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 <math>i\hbar</math>,
<math display="block">\{x,p\} = 1 \, .</math>

This observation led [[Paul Dirac|Dirac]] to propose that the quantum counterparts <math>\hat{f}</math>, <math>\hat{g}</math> of classical observables {{mvar|f}}, {{mvar|g}} satisfy
<math display="block">[\hat f,\hat g]= i\hbar\widehat{\{f,g\}} \, .</math>

In 1946, [[Hilbrand J. Groenewold|Hip Groenewold]] demonstrated that a ''general systematic correspondence'' between quantum commutators and Poisson brackets could not hold consistently.<ref name="groenewold">{{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 }}</ref><ref>{{harvnb|Hall|2013}} Theorem 13.13</ref>

However, he further appreciated that such a systematic correspondence does, in fact, exist between the quantum commutator and a ''[[Deformation theory|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 [[Phase-space formulation|deformation quantization]].<ref name="groenewold"/><ref>{{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 }}</ref>

=== Derivation from Hamiltonian mechanics ===
According to the [[correspondence principle]], in certain limits the quantum equations of states must approach [[Poisson bracket#Hamilton's equations of motion|Hamilton's equations of motion]]. The latter state the following relation between the generalized coordinate ''q'' (e.g. position) and the generalized momentum ''p'':
<math display="block">\begin{cases}
  \dot{q} =  \frac{\partial H}{\partial p} = \{q, H\}; \\
  \dot{p} = -\frac{\partial H}{\partial q} = \{p, H\}.
\end{cases}</math>

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

The time derivative of a quantum state is represented by the operator <math>-i\hat{H}/\hbar</math> (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:
<math display="block">\frac {d\hat{Q}}{dt} = \frac {i}{\hbar} [\hat{H},\hat{Q}]</math>
<math display="block">\frac {d\hat{P}}{dt} = \frac {i}{\hbar} [\hat{H},\hat{P}] \,\, .</math>

In order for that to reconcile in the classical limit with Hamilton's equations of motion, <math> [\hat{H},\hat{Q}]</math> must depend entirely on the appearance of <math>\hat{P}</math> in the Hamiltonian and <math>[\hat{H},\hat{P}]</math> must depend entirely on the appearance of <math>\hat{Q}</math> 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 derivative]]s):
<math display="block">[\hat{H},\hat{Q}] = \frac {\delta \hat{H}}{\delta \hat{P}} \cdot [\hat{P},\hat{Q}]</math>
<math display="block">[\hat{H},\hat{P}] = \frac {\delta \hat{H}}{\delta \hat{Q}} \cdot [\hat{Q},\hat{P}] \, . </math>

In order to obtain the classical limit we must then have
<math display="block"> [\hat{Q},\hat{P}] = i \hbar ~ I.</math>

== Weyl relations ==
The [[Lie group|group]] <math>H_3(\mathbb{R})</math> generated by [[exponential map (Lie theory)|exponentiation]] of the 3-dimensional [[Lie algebra]] determined by the commutation relation <math>[\hat{x},\hat{p}]=i\hbar</math> is called the [[Heisenberg group]]. This group can be realized as the group of <math>3\times 3</math> upper triangular matrices with ones on the diagonal.<ref>{{harvnb|Hall|2015}} Section 1.2.6 and Proposition 3.26</ref>

According to the standard [[mathematical formulation of quantum mechanics]], quantum observables such as <math>\hat{x}</math> and <math>\hat{p}</math> should be represented as [[self-adjoint operator]]s on some [[Hilbert space]].  It is relatively easy to see that two [[operator (mathematics)|operator]]s satisfying the above canonical commutation relations cannot both be [[bounded operator|bounded]]. Certainly, if <math>\hat{x}</math> and <math>\hat{p}</math> were [[trace class]] operators, the relation <math>\operatorname{Tr}(AB)=\operatorname{Tr}(BA)</math> gives a nonzero number on the right and zero on the left.

Alternately, if <math>\hat{x}</math> and <math>\hat{p}</math> were bounded operators, note that <math>[\hat{x}^n,\hat{p}]=i\hbar n \hat{x}^{n-1}</math>, hence the operator norms would satisfy
<math display="block">2 \left\|\hat{p}\right\| \left\|\hat{x}^{n-1}\right\| \left\|\hat{x}\right\|  \geq n \hbar \left\|\hat{x}^{n-1}\right\|,</math> so that, for any ''n'',
<math display="block">2 \left\|\hat{p}\right\| \left\|\hat{x}\right\| \geq n \hbar</math>
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 operator]]s <math>\exp(it\hat{x})</math> and <math>\exp(is\hat{p})</math>. The resulting braiding relations for these operators are the so-called [[Stone–von Neumann theorem|Weyl relations]]
<math display="block">\exp(it\hat{x})\exp(is\hat{p})=\exp(-ist\hbar)\exp(is\hat{p})\exp(it\hat{x}).</math>
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 [[Stone–von Neumann theorem#The Heisenberg group|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 <math>[\hat{x},\hat{p}]=i\hbar</math>. If <math>\hat{x}</math> and <math>\hat{p}</math> 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.<ref>See Section 5.2 of {{harvnb|Hall|2015}} for an elementary derivation</ref> 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.<ref>{{harvnb|Hall|2013}} Example 14.5</ref> (These same operators give a [[Uncertainty principle#A counterexample|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 <math>\mathbb{Z}/n</math>, can be realized on a finite-dimensional Hilbert space by means of the [[Generalizations of Pauli matrices#Construction: The clock and shift matrices|clock and shift matrices]].

== Generalizations ==
It can be shown that
<math display="block">[F(\vec{x}),p_i] = i\hbar\frac{\partial F(\vec{x})}{\partial x_i}; \qquad [x_i, F(\vec{p})] = i\hbar\frac{\partial F(\vec{p})}{\partial p_i}.</math>

Using <math>C_{n+1}^{k} = C_{n}^{k} + C_{n}^{k-1}</math>, it can be shown that by [[mathematical induction]]
<math display="block">\left[\hat{x}^n,\hat{p}^m\right] = \sum_{k=1}^{\min\left(m,n\right)}{ \frac{-\left(-i \hbar\right)^k n!m!}{k!\left(n-k\right)!\left(m-k\right)!} \hat{x}^{n-k} \hat{p}^{m-k}} = \sum_{k=1}^{\min\left(m,n\right)}{ \frac{\left(i \hbar\right)^k n!m!}{k!\left(n-k\right)!\left(m-k\right)!} \hat{p}^{m-k}\hat{x}^{n-k}} ,</math>
generally known as McCoy's formula.<ref>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]</ref>

In addition, the simple formula
<math display="block">[x,p] = i\hbar \, \mathbb{I} ~,</math>
valid for the [[Canonical quantization|quantization]] of the simplest classical system, can be generalized to the case of an arbitrary [[Lagrangian (field theory)|Lagrangian]] <math>{\mathcal L}</math>.<ref name="town">{{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 }}</ref> 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|&pi;<sub>''x''</sub>}} (in the example above it is {{mvar|p}}, or more generally, some functions involving the [[derivative]]s of the canonical coordinates with respect to time):
<math display="block">\pi_i \ \stackrel{\mathrm{def}}{=}\ \frac{\partial {\mathcal L}}{\partial(\partial x_i / \partial t)}.</math>

This definition of the canonical momentum ensures that one of the [[Euler–Lagrange equation]]s has the form
<math display="block">\frac{\partial}{\partial t} \pi_i = \frac{\partial {\mathcal L}}{\partial x_i}.</math>

The canonical commutation relations then amount to
<math display="block">[x_i,\pi_j] = i\hbar\delta_{ij} \, </math>
where {{math|''δ''<sub>''ij''</sub>}} is the [[Kronecker delta]].

== Gauge invariance ==
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
: <math>p_\text{kin} = p - qA \,\!</math>  ([[SI units]]) {{spaces|4}} <math>p_\text{kin} = p - \frac{qA}{c} \,\!</math>  ([[Gaussian units|cgs units]]),
where {{mvar|q}} is the particle's [[electric charge]], {{mvar|A}} is the [[Magnetic vector potential|vector potential]], and {{math|''c''}} is the [[speed of light]]. Although the quantity {{math|''p''<sub>kin</sub>}} 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 (quantum mechanics)|Hamiltonian]] for a quantized charged particle of mass {{mvar|m}} in a classical electromagnetic field is (in cgs units)
<math display="block">H=\frac{1}{2m} \left(p-\frac{qA}{c}\right)^2 +q\phi</math>
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 equation]]s and the [[Lorentz force law]] are invariant under the gauge transformation
<math display="block">A\to A' = A+\nabla \Lambda</math>
<math display="block">\phi\to \phi' = \phi-\frac{1}{c} \frac{\partial \Lambda}{\partial t}</math>
<math display="block">\psi \to \psi' = U\psi</math>
<math display="block">H\to H' = U H U^\dagger,</math>
where <math display="block">U=\exp \left( \frac{iq\Lambda}{\hbar c}\right)</math> and {{math|1=Λ = Λ(''x'',''t'')}} is the gauge function.

The [[angular momentum operator]] is
<math display="block">L=r \times p \,\!</math>
and obeys the canonical quantization relations
<math display="block">[L_i, L_j]= i\hbar {\epsilon_{ijk}} L_k</math>
defining the [[Lie algebra]] for [[so(3)]], where <math>\epsilon_{ijk}</math> is the [[Levi-Civita symbol]].  Under gauge transformations, the angular momentum transforms as
<math display="block"> \langle \psi \vert L \vert \psi \rangle \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 \, .
</math>

The gauge-invariant angular momentum (or "kinetic angular momentum") is given by
<math display="block">K=r \times \left(p-\frac{qA}{c}\right),</math>
which has the commutation relations
<math display="block">[K_i,K_j]=i\hbar {\epsilon_{ij}}^{\,k}
\left(K_k+\frac{q\hbar}{c} x_k
\left(x \cdot B\right)\right)</math>
where <math display="block">B=\nabla \times A</math> 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 ==
All such nontrivial commutation relations for pairs of operators lead to corresponding [[uncertainty principle|uncertainty relations]],<ref name="robertson">{{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 }}</ref> involving positive semi-definite expectation contributions by their respective commutators and anticommutators.  In general, for two [[Self-adjoint operator|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=(Δ''A'')<sup>2</sup> &equiv; {{langle}}(''A'' − {{langle}}''A''{{rangle}})<sup>2</sup>{{rangle}}}}, etc.

Then
<math display="block"> \Delta A \, \Delta B \geq \frac{1}{2} \sqrt{ \left|\left\langle\left[{A},{B}\right]\right\rangle \right|^2 + \left|\left\langle\left\{ A-\langle A\rangle ,B-\langle B\rangle \right\} \right\rangle \right|^2} ,</math>
where {{math|1=[''A'', ''B''] &equiv; ''A B'' &minus; ''B A''}} is the [[Commutator#Ring theory|commutator]] of {{mvar|A}} and {{mvar|B}}, and {{math|1={''A'', ''B''} &equiv; ''A B'' + ''B A''}} is the [[anticommutator]].

This follows through use of the [[Cauchy–Schwarz inequality]], since
{{math|{{!}}{{langle}}''A''<sup>2</sup>{{rangle}}{{!}} {{!}}{{langle}}''B''<sup>2</sup>{{rangle}}{{!}} &ge; {{!}}{{langle}}''A B''{{rangle}}{{!}}<sup>2</sup>}}, 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.

== Uncertainty relation for angular momentum operators ==
For the angular momentum operators {{math|1=''L''<sub>''x''</sub> = ''y p<sub>z</sub>'' − ''z p<sub>y</sub>''}}, etc., one has that
<math display="block"> [{L_x}, {L_y}] = i \hbar \epsilon_{xyz} {L_z}, </math>
where <math>\epsilon_{xyz}</math> 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 (physics)|spin]] operators.

Here, for {{mvar|L<sub>x</sub>}} and {{mvar|L<sub>y</sub> }},<ref name="robertson" /> in angular momentum multiplets {{math|1=''ψ'' = {{!}}''{{ell}}'',''m''{{rangle}}}}, one has, for the transverse components of the [[Casimir invariant]] {{math|''L<sub>x</sub>''<sup>2</sup> + ''L<sub>y</sub>''<sup>2</sup>+ ''L<sub>z</sub>''<sup>2</sup>}}, the {{mvar|z}}-symmetric relations
:{{math|1={{langle}}''L<sub>x</sub>''<sup>2</sup>{{rangle}} = {{langle}}''L<sub>y</sub>''<sup>2</sup>{{rangle}} = (''{{ell}}'' (''{{ell}}'' + 1) − ''m''<sup>2</sup>) ℏ<sup>2</sup>/2 }},
as well as {{math|1={{langle}}''L<sub>x</sub>''{{rangle}} = {{langle}}''L<sub>y</sub>''{{rangle}} = 0 }}.

Consequently, the above inequality applied to this commutation relation specifies
<math display="block">\Delta L_x \, \Delta L_y \geq \frac{1}{2} \sqrt{\hbar^2|\langle L_z \rangle|^2}~, </math>
hence
<math display="block">\sqrt {|\langle L_x^2\rangle \langle L_y^2\rangle |} \geq \frac{\hbar^2}{2} \vert m\vert</math>
and therefore
<math display="block">\ell(\ell+1)-m^2\geq |m| ~,</math>
so, then, it yields useful constraints such as a lower bound on the [[Casimir invariant]]: {{math|''{{ell}}'' (''{{ell}}'' + 1) &ge; {{pipe}}''m''{{pipe}} ({{pipe}}''m''{{pipe}} + 1)}}, and hence {{math|''{{ell}}'' &ge; {{pipe}}''m''{{pipe}}}}, among others.

== See also ==
* [[Canonical quantization]]
* [[CCR and CAR algebras]]
* [[Conformastatic spacetimes]]
* [[Lie derivative]]
* [[Moyal bracket]]
* [[Stone–von Neumann theorem]]

== References ==
{{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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