Canonical Commutation Relations

Formula

For one canonical coordinate $X$ and conjugate momentum $P$,

$$[X,P] = i\hbar I.$$

For $n$ Cartesian degrees of freedom,

$$[X_i,P_j] = i\hbar\delta_{ij}I,$$ $$[X_i,X_j]=0, \qquad [P_i,P_j]=0.$$

For distinguishable particles labeled by $\alpha$ and $\beta$,

$$[X_{\alpha i},P_{\beta j}] = i\hbar \delta_{\alpha\beta} \delta_{ij}I.$$

Operators belonging to different particle labels or distinct Cartesian canonical pairs commute in the standard unconstrained theory.

At a Glance

Form	Relation
One canonical pair	$[X,P]=i\hbar I$
Cartesian components	$[X_i,P_j]=i\hbar\delta_{ij}I$
Coordinate components	$[X_i,X_j]=0$
Canonical momentum components	$[P_i,P_j]=0$
Many particles	$[X_{\alpha i},P_{\beta j}]=i\hbar\delta_{\alpha\beta}\delta_{ij}I$
Symplectic form	$[Z_a,Z_b]=i\hbar\Omega_{ab}I$
Weyl form	$T(a)M(b)=e^{-iab/\hbar}M(b)T(a)$
Uncertainty consequence	$\Delta X\,\Delta P\geq\hbar/2$

The identity operator is often omitted from the notation. Restoring it is useful when comparing operator types or testing finite-dimensional approximations.

Meaning

The canonical commutation relations encode the structure of a conjugate position-momentum pair. They imply that:

• $X$ and $P$ are incompatible sharp observables;
• $P$ generates translations of position;
• $X$ generates translations of momentum;
• position and momentum representations are related by a Fourier transform;
• position and momentum standard deviations obey the Heisenberg bound;
• commutators with functions of $X$ and $P$ act like canonical derivatives under suitable conditions.

The relation is an operator statement. It does not say that numerical measurement outcomes $x$ and $p$ fail to commute.

Schrödinger Representation

On

$$\mathcal H = L^2(\mathbb R,dx),$$

the standard position-space realization is

$$(X\psi)(x) = x\psi(x),$$ $$(P\psi)(x) = -i\hbar \frac{d\psi}{dx}(x).$$

On a sufficiently regular common domain,

$$([X,P]\psi)(x) = i\hbar\psi(x).$$

In momentum representation, using

$$\langle x\rvert p\rangle = \frac{1}{\sqrt{2\pi\hbar}} \exp\left( \frac{ipx}{\hbar} \right),$$

the same operators act as

$$(P\widetilde\psi)(p) = p\widetilde\psi(p),$$ $$(X\widetilde\psi)(p) = i\hbar \frac{d\widetilde\psi}{dp}(p).$$

The derivative signs track the Fourier-kernel convention. A source using the opposite exponential signs must change the corresponding representation formulas consistently.

Translation Form

Define the active position-translation operator

$$T(a) = \exp\left( -\frac{iaP}{\hbar} \right).$$

It acts as

$$(T(a)\psi)(x) = \psi(x-a)$$

and satisfies

$$T^\dagger(a)XT(a) = X+aI.$$

Define the momentum-translation or modulation operator

$$M(b) = \exp\left( \frac{ibX}{\hbar} \right).$$

Then

$$M^\dagger(b)PM(b) = P+bI.$$

The two shifts obey the Weyl relation

$$T(a)M(b) = e^{-iab/\hbar} M(b)T(a).$$

These signs are tied to the displayed definitions. Active-versus-passive conventions or different generator signs produce equivalent formulas with corresponding sign changes.

Why the Weyl Form Matters

The operators $X$ and $P$ are unbounded and are not defined on every Hilbert-space vector. Their exponentials $T(a)$ and $M(b)$ are bounded unitary operators defined everywhere. For rigorous representation theory, the strongly continuous Weyl relations are often cleaner than treating the raw commutator as an unrestricted identity.

The Weyl phase is dimensionless because

$$[ab]=[\text{action}]=[\hbar].$$

The phase records the noncommutativity of translations along conjugate phase-space directions.

Symplectic Form

Collect the canonical operators into

$$Z = \left( X_1,\ldots,X_n, P_1,\ldots,P_n \right)^{\mathsf T}.$$

Then

$$[Z_a,Z_b] = i\hbar \Omega_{ab}I,$$

with

$$\Omega = \begin{pmatrix} 0&I_n\\ -I_n&0 \end{pmatrix}.$$

A real linear transformation

$$Z' = SZ$$

preserves the canonical relations when

$$S\Omega S^{\mathsf T} = \Omega.$$

This compact form is standard for coupled oscillators, Gaussian states, continuous-variable quantum information, and phase-space methods.

Useful Consequences

Using the product rule for commutators,

$$[A,BC] = [A,B]C +B[A,C],$$

the canonical relation gives

$$[X,P^n] = i\hbar nP^{n-1},$$ $$[P,X^n] = -i\hbar nX^{n-1}.$$

For suitable functions and on an appropriate common domain,

$$[X,f(P)] = i\hbar f'(P),$$ $$[P,g(X)] = -i\hbar g'(X).$$

These identities are exact for polynomials on a common invariant domain. For more general functions, the functional calculus, differentiability, and domain conditions must be checked.

For

$$H = \frac{P^2}{2m} +V(X),$$

one obtains

$$[X,H] = i\hbar\frac{P}{m},$$ $$[P,H] = -i\hbar V'(X),$$

which yield the corresponding Heisenberg equations.

Uncertainty Consequence

The Robertson relation is

$$\Delta A\,\Delta B \geq \frac12 \lvert \langle[A,B]\rangle \rvert.$$

For a canonical pair,

$$\Delta X\,\Delta P \geq \frac{\hbar}{2}.$$

This is a state-dependent preparation-uncertainty statement. It is not, by itself, a statement about instrument resolution, measurement disturbance, or simultaneous numerical readout errors.

The bound requires finite variances and the domain conditions needed for the commutator argument. Formal insertion of $[X,P]=i\hbar I$ does not rescue a state for which $\Delta X$ or $\Delta P$ is undefined.

Canonical and Kinetic Momentum

The displayed relations use canonical momentum. For a particle of charge $q$ in a vector potential,

$$\boldsymbol{\Pi} = \mathbf P -q\mathbf A(\mathbf X,t)$$

is kinetic momentum. Its components satisfy

$$[\Pi_i,\Pi_j] = iq\hbar \epsilon_{ijk} B_k(\mathbf X,t).$$

Position still obeys

$$[X_i,\Pi_j] = i\hbar\delta_{ij}I$$

when $\mathbf A$ is a function of position. Thus the vanishing momentum-component commutator in the canonical table must not be transferred blindly to kinetic momentum in a magnetic field.

Magnetic translations can also replace ordinary translations as the symmetry generators. The relevant choice depends on the Hamiltonian and gauge structure.

Classical Correspondence

Classical canonical variables satisfy

$$\{q_i,p_j\}_{\mathrm{PB}} = \delta_{ij}.$$

Canonical quantization motivates the schematic correspondence

$$\{F,G\}_{\mathrm{PB}} \longrightarrow \frac{1}{i\hbar} [\widehat F,\widehat G].$$

This is structurally useful for basic variables and suitable observables. It is not a universal quantization algorithm: nonlinear functions introduce operator-ordering ambiguities, and no prescription preserves all Poisson brackets exactly.

Dimensionless Conventions

If

$$Q = \frac{X}{x_0}, \qquad K = \frac{x_0P}{\hbar},$$

then

$$[Q,K]=iI.$$

Natural-unit conventions may also set $\hbar=1$ and write

$$[X,P]=iI.$$

Before restoring dimensions, determine whether the variables themselves were rescaled or only $\hbar$ was suppressed. The product of the physical coordinate and physical conjugate momentum must have action units.

Domains and Boundary Conditions

The formal commutator is meaningful on vectors for which both products $XP$ and $PX$ exist:

$$\psi \in \mathcal D(XP) \cap \mathcal D(PX).$$

A practical calculation often uses a dense invariant core such as the Schwartz space on $\mathbb R$, where multiplication and differentiation preserve regularity.

On a finite interval, the derivative expression

$$-i\hbar\frac{d}{dx}$$

can have different self-adjoint domains depending on boundary conditions. The bounded coordinate and a chosen self-adjoint momentum do not necessarily satisfy the naive commutator on a domain preserved by both. On a circle, a globally defined angle observable has additional periodicity and branch subtleties; exponentiated relations can be the safer language.

The equation should therefore be read as the canonical algebra of the specified representation, not as permission to ignore the configuration space.

No Exact Finite-Dimensional Pair

Finite matrices cannot satisfy

$$[X,P]=i\hbar I$$

exactly. For $d\times d$ matrices,

$$\operatorname{Tr}[X,P]=0$$

by cyclicity, whereas

$$\operatorname{Tr}(i\hbar I_d) = i\hbar d \neq0.$$

A truncated oscillator or grid representation can reproduce low-energy matrix elements approximately, but it must violate the canonical relation somewhere. The defect often becomes largest near the truncation boundary.

Symbols

Symbol	Mathematical type	Meaning
$X_i$	Self-adjoint operator	Canonical coordinate
$P_i$	Self-adjoint operator	Canonical conjugate momentum
$I$	Identity operator	Identity on the relevant Hilbert space
$\delta_{ij}$	Kronecker delta	Selects conjugate Cartesian components
$\delta_{\alpha\beta}$	Kronecker delta	Selects the same particle label
$\hbar$	Positive constant with action units	Reduced Planck constant
$T(a)$	Unitary operator	Active position translation
$M(b)$	Unitary operator	Momentum translation or modulation
$\Omega$	Antisymmetric matrix	Standard symplectic form
$\boldsymbol{\Pi}$	Operator vector	Kinetic momentum

Hats are often used as $\hat x$ and $\hat p$ when the distinction from classical variables or eigenvalues matters. This card uses capitals $X$ and $P$ to emphasize operator type.

Units and Dimensions

The commutator has action units:

$$[[X,P]] = [X][P] = [\hbar].$$

For position and linear momentum,

$$[X]=L, \qquad [P]=MLT^{-1},$$

so

$$[X][P] = ML^2T^{-1}.$$

The indices and Kronecker deltas are dimensionless. In the Weyl relation, $a$ has position units and $b$ has momentum units.

Assumptions

• $X_i$ and $P_i$ form canonical Cartesian pairs.
• The raw commutators are evaluated on a suitable common invariant domain.
• The representation uses the standard quantum of action $\hbar$.
• The displayed vanishing momentum-component relation refers to canonical momentum.
• Constraints, compact configuration spaces, gauge fields, and reduced phase spaces have been treated before declaring variables canonical.
• Weyl operators are strongly continuous when self-adjoint generators are inferred.

Validity and Limitations

The canonical algebra is exact for ordinary unconstrained nonrelativistic canonical degrees of freedom in a valid representation. It does not automatically apply unchanged to:

• finite-dimensional Hilbert spaces;
• angle variables and compact configuration spaces;
• constrained systems before reduction or Dirac-bracket quantization;
• kinetic momentum components in magnetic fields;
• noncommutative-coordinate models;
• projected band subspaces, where effective coordinates can acquire nontrivial commutators;
• infinitely many degrees of freedom, where representation uniqueness can fail.

The Stone–von Neumann theorem gives the finite-degree-of-freedom uniqueness statement for regular irreducible Weyl representations and states its hypotheses.

Calculation Checks

• Antisymmetry requires

$$[P,X]=-i\hbar I.$$

• For $i\neq j$, the Cartesian component commutator should vanish.
• Acting on a smooth test function should reproduce $i\hbar\psi$.
• Both sides must have action units.
• Translation signs should agree among the exponential, wavefunction action, and conjugated coordinate.
• A proposed finite-matrix realization must fail the trace test somewhere.
• Replacing canonical momentum by kinetic momentum should trigger a magnetic-field commutator check.
• Setting $\hbar=1$ should be accompanied by a clear dimensional or rescaling convention.

Minimal Representation Check

For a smooth test function $\psi$,

$$(XP\psi)(x) = -i\hbar x\psi'(x),$$

whereas

$$(PX\psi)(x) = -i\hbar \left[ \psi(x)+x\psi'(x) \right].$$

Subtracting gives

$$([X,P]\psi)(x) = i\hbar\psi(x).$$

This verifies the differential expression on the selected test-function domain. It does not establish self-adjointness or global domain invariance by itself.

Derivation and Canonical Home

Canonical Commutation Relations owns the Schrödinger derivation, Weyl form, translation meaning, domain qualifications, symplectic notation, and finite-dimensional obstruction.

Translations and Momentum develops the symmetry interpretation. Heisenberg Group owns the group structure, and Position–Momentum Representations owns the Fourier-related realizations.

Common Mistakes

• Omitting the identity operator and then treating the right-hand side as an ordinary scalar in a type-sensitive argument.
• Treating the relation as an exact finite-dimensional matrix identity.
• Applying the derivative representation without specifying a domain and boundary conditions.
• Reversing $[X,P]$ without reversing the sign.
• Mixing active and passive translation signs.
• Confusing canonical momentum with kinetic momentum in a gauge field.
• Assuming all functions obey derivative commutator identities without regularity or domain conditions.
• Inferring simultaneous numerical measurement values from an operator relation.
• Treating canonical quantization as an ordering-independent map for every classical observable.
• Applying Cartesian relations directly to constrained, angular, or projected coordinates.

Related Formulas

• Commutator Identities
• Uncertainty Relations
• Baker–Campbell–Hausdorff
• Position Operator Symbol
• Momentum Operator Symbol
• Commutator Table
• Minimal Coupling

References

• H. Weyl, The Theory of Groups and Quantum Mechanics, Dover, 1950, ch. IV.
• M. H. Stone, “Linear Transformations in Hilbert Space. III. Operational Methods and Group Theory,” Proceedings of the National Academy of Sciences 16, 172–175 (1930).
• J. von Neumann, “Die Eindeutigkeit der Schrödingerschen Operatoren,” Mathematische Annalen 104, 570–578 (1931).
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994, chs. 7 and 12.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, rev. ed., Academic Press, 1980, sec. VIII.5.