Draft:Covariant four momentum operator commutator

The covariant four-momentum operator commutator appears in quantum field theory when extending the usual momentum operator to include interactions and gauge fields in a manifestly covariant form. In the presence of gauge fields, partial derivatives are replaced with covariant derivatives, and this replacement modifies the commutation relations of momentum operators, encoding the gauge field strength or curvature of the underlying connection.

Overview

In standard (non-relativistic) quantum mechanics, the canonical momentum operator is given by

{\hat {\mathbf {p} }}\;=\;-\,i\hbar \,\nabla .

In relativistic quantum field theory (QFT), one upgrades to the four-momentum operator ${\hat {P}}^{\mu }$ (with index $\mu$ running over spacetime components). When fields interact with gauge potentials such as the electromagnetic 4-potential $A^{\mu }$ , the canonical derivative $\partial _{\mu }$ is replaced by the covariant derivative $D_{\mu }$ :

D_{\mu }\;=\;\partial _{\mu }\;+\;i\,g\,A_{\mu },

where $g$ is the coupling constant (e.g., electric charge $q$ or $e$ in QED). The corresponding covariant four-momentum operator then becomes

{\hat {P}}_{\mu }\;=\;-\,i\hbar \,D_{\mu }.

Commutator Definition

The covariant four-momentum operator commutator is:

{\bigl [}{\hat {P}}_{\mu },\;{\hat {P}}_{\nu }{\bigr ]}\;=\;{\bigl [}-\,i\hbar \,D_{\mu },\;-\,i\hbar \,D_{\nu }{\bigr ]}.

Expanding $D_{\mu }$ in terms of $\partial _{\mu }$ and $A_{\mu }$ , one obtains

{\bigl [}D_{\mu },\;D_{\nu }{\bigr ]}\;=\;i\,g\,F_{\mu \nu },

where $F_{\mu \nu }$ is the field strength tensor (for example, the electromagnetic field tensor in QED or the non-Abelian field strength in more general gauge theories). Consequently,

{\bigl [}{\hat {P}}_{\mu },\;{\hat {P}}_{\nu }{\bigr ]}\;=\;-\,\hbar ^{2}\,{\bigl [}D_{\mu },\;D_{\nu }{\bigr ]}\;=\;-\,\hbar ^{2}\,{\bigl (}i\,g\,F_{\mu \nu }{\bigr )}\;=\;-\,i\,\hbar ^{2}\,g\,F_{\mu \nu }.

This result shows that the nontrivial commutator of covariant momentum operators directly encodes the field strength tensor $F_{\mu \nu }$ , thereby linking the geometry of the gauge field (or curvature) to the algebra of momentum operators in a quantum theory.

Physical Interpretation

In an Abelian gauge theory such as Quantum Electrodynamics (QED), $F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }$ represents the usual electromagnetic field tensor. The commutator $[{\hat {P}}_{\mu },\,{\hat {P}}_{\nu }]$ is proportional to $F_{\mu \nu }$ , illustrating how charged particles “feel” the electromagnetic field as modifications of their momentum space.

In a non-Abelian gauge theory (e.g., Quantum Chromodynamics), the covariant derivative becomes $D_{\mu }=\partial _{\mu }+i\,g\,A_{\mu }^{a}\,T^{a}$ , where $T^{a}$ are the generators of the gauge group (such as SU(3) for QCD). The field strength tensor $F_{\mu \nu }^{a}$ includes commutator terms among these generators, reflecting the self-interactions of non-Abelian gauge bosons.

Mathematical Formulation

When written explicitly, the covariant derivative in non-Abelian gauge theory is:

D_{\mu }\;=\;\partial _{\mu }\,\mathbf {1} \;+\;i\,g\,A_{\mu }^{a}\,T^{a},

with $\mathbf {1}$ the identity in the representation space of the fields. The commutator is:

{\bigl [}D_{\mu },\;D_{\nu }{\bigr ]}\;=\;i\,g\,{\bigl (}\partial _{\mu }A_{\nu }^{a}-\partial _{\nu }A_{\mu }^{a}\;+\;g\,f^{abc}\,A_{\mu }^{b}\,A_{\nu }^{c}{\bigr )}\,T^{a},

where $f^{abc}$ are the structure constants of the gauge group. One then identifies the field strength tensor $F_{\mu \nu }^{a}$ by comparing to:

F_{\mu \nu }^{a}\;=\;\partial _{\mu }A_{\nu }^{a}-\partial _{\nu }A_{\mu }^{a}\;+\;g\,f^{abc}\,A_{\mu }^{b}\,A_{\nu }^{c}.

Hence, the covariant four-momentum operator commutator in a non-Abelian theory is:

{\bigl [}{\hat {P}}_{\mu },\;{\hat {P}}_{\nu }{\bigr ]}\;=\;-\,i\,\hbar ^{2}\,g\,F_{\mu \nu }^{a}\,T^{a},

explicitly displaying how gauge curvature emerges in the momentum-space algebra.

Applications

Quantum Electrodynamics (QED): Describes how electrons, positrons, and photons interact via an Abelian gauge field. The commutator $[{\hat {P}}_{\mu },{\hat {P}}_{\nu }]$ captures the electromagnetic field tensor, playing a central role in analyzing phenomena such as the minimal coupling to $A_{\mu }$ .
Quantum Chromodynamics (QCD): Non-Abelian SU(3) gauge theory of quarks and gluons. The commutator structure is more complex, reflecting color charge interactions and gluon self-interactions.
General Gauge Theories: Any gauge-invariant quantum field theory relies on covariant derivatives and the associated commutators to define how fields transform and how interactions arise.

Draft:Covariant four momentum operator commutator

Overview

Commutator Definition

Physical Interpretation

Mathematical Formulation

Applications

See Also

References

Further Reading