Draft:Covariant four momentum operator commutator

• Comment: It both needs sources and to be rewritten so it is not textbook material, see WP:NOTTEXTBOOK. Ldm1954 (talk) 13:40, 24 March 2025 (UTC)

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A covariant four-momentum operator commutator is the non-zero commutator of a momentum operator in a curved spacetime, arising from the presence of spacetime curvature. It generalizes the usual flat-spacetime momentum operator algebra to include geometric effects described by the Riemann curvature tensor.

Overview

In flat Minkowski space, the canonical four-momentum operator in quantum field theory is given by: ${\hat {p}}_{\mu }=-\,i\hbar \,\partial _{\mu },$ whose components all commute: $[\,{\hat {p}}_{\mu },\,{\hat {p}}_{\nu }\,]=0.$

When extended to a curved spacetime background, one replaces the partial derivatives with covariant derivatives $\nabla _{\mu }$. The resulting covariant four-momentum operator is ${\hat {P}}_{\mu }=-\,i\hbar \,\nabla _{\mu }.$ Unlike the flat-spacetime case, these operators in general do not commute.

Definition

The commutator of the covariant four-momentum operators ${\hat {P}}_{\mu }$ is defined by: $[\,{\hat {P}}_{\mu },\,{\hat {P}}_{\nu }\,]\;=\;{\hat {P}}_{\mu }\,{\hat {P}}_{\nu }\;-\;{\hat {P}}_{\nu }\,{\hat {P}}_{\mu }.$ Because $\nabla _{\mu }$ includes the affine connection (or Christoffel symbols), this commutator encodes the curvature of the underlying spacetime.

Expression in terms of the Riemann curvature

A key result is: $[\,{\hat {P}}_{\mu },\,{\hat {P}}_{\nu }\,]\,{\hat {T}}^{\alpha }\;=\;-\,\hbar ^{2}\,R_{\ \beta \mu \nu }^{\alpha }\,{\hat {T}}^{\beta },$ for any vector operator field ${\hat {T}}^{\alpha }$. Here, $R_{\ \beta \mu \nu }^{\alpha }$ is the Riemann curvature tensor, which measures the failure of second covariant derivatives to commute: $[\,\nabla _{\mu },\,\nabla _{\nu }\,]\,T^{\alpha }\;=\;R_{\ \beta \mu \nu }^{\alpha }\,T^{\beta }.$ In flat spacetime, $R_{\ \beta \mu \nu }^{\alpha }=0$, and the commutator vanishes.

Connection to tensor potentials

Because the Riemann curvature tensor can be written in terms of the Christoffel symbols $\Gamma _{\mu \nu }^{\lambda }$, the commutator may also be expressed as: $[\,{\hat {P}}_{\mu },\,{\hat {P}}_{\nu }\,]\;=\;-\,\hbar ^{2}\;{\Bigl (}\partial _{\mu }\Gamma _{\nu }\;-\;\partial _{\nu }\Gamma _{\mu }\;+\;[\Gamma _{\mu },\,\Gamma _{\nu }]{\Bigr )},$ where each $\Gamma _{\mu }$ is a matrix-valued connection component. This formulation closely parallels the structure of non-Abelian gauge theory, where curvature (or field strength) arises from the commutator of covariant derivatives.

Physical significance

• The non-commuting nature of ${\hat {P}}_{\mu }$ operators reflects the intrinsic curvature of the spacetime manifold.
• In quantum field theory in curved spacetime, it implies that local definitions of particle momentum depend on how one parallel-transports state vectors along the manifold.
• The result has direct analogies with gauge theory: the curvature plays a role analogous to non-Abelian field strength, and the commutator measures “holonomies” in momentum space.
• In quantum gravity approaches, promoting $\Gamma _{\mu \nu }^{\lambda }$ and $g_{\mu \nu }$ to quantum operators implies that the momentum algebra itself can fluctuate dynamically.

Related equations

• The Covariant Heisenberg equation generalizes the usual Heisenberg equation of motion to curved spacetime, relying on these covariant momentum commutators.
• The commutator also appears in second-order forms such as

$[\,\nabla _{\mu },\,\nabla _{\nu }\,]\,{\hat {O}}\;=\;R({\hat {O}}),$ highlighting how quantum fields experience curvature.

See also

• Covariant derivative
• General covariance
• Heisenberg picture
• Non-Abelian gauge theory
• Riemann curvature tensor

References

• Birrell, N. D.; Davies, P. C. W. (1982). Quantum Fields in Curved Space. Cambridge University Press. ISBN 978-0-521-23385-7.
• Parker, L.; Toms, D. (2009). Quantum Field Theory in Curved Spacetime: Quantized Fields and Gravity. Cambridge University Press. ISBN 978-0-521-87787-7.
• Wald, R. M. (1994). Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics. University of Chicago Press. ISBN 978-0-226-87012-7.

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• Quantum field theory
• Quantum gravity
• General relativity
• Mathematical physics