Draft:Covariant Heisenberg equation

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The covariant Heisenberg equation (sometimes referred to as the relativistic Heisenberg equation or covariant equation of motion) is a formulation of the Heisenberg equation of motion that is made compatible with the principles of special and/or general relativity. Unlike the standard Heisenberg equation, which is often expressed using time as the sole evolution parameter, the covariant form treats spacetime coordinates on equal footing and incorporates covariant derivatives, ensuring consistency with relativistic transformations of fields and operators.

Overview

In non-relativistic quantum mechanics, the Heisenberg equation of motion for an operator ${\hat {A}}$ is: ${\frac {d{\hat {A}}}{dt}}\;=\;{\frac {i}{\hbar }}{\bigl [}{\hat {H}},\,{\hat {A}}{\bigr ]}\;+\;\left({\frac {\partial {\hat {A}}}{\partial t}}\right)_{\mathrm {explicit} },$ where ${\hat {H}}$ is the Hamiltonian operator, $\hbar$ is the reduced Planck constant, and the second term on the right-hand side represents explicit time dependence of ${\hat {A}}$.

To make this formalism compatible with special relativity, time $t$ is replaced with a suitable relativistic parameter or the proper time $\tau$, and partial derivatives are replaced with covariant derivatives in the context of quantum field theory. This ensures that the resulting equations transform appropriately under Lorentz transformations or, in curved spacetime, under general coordinate transformations.

Mathematical formulation

In the covariant Heisenberg picture, one often treats field operators ${\hat {\phi }}(x)$ defined at spacetime points $x^{\mu }$. A general symbolic form of the covariant Heisenberg equation can be written as: $\nabla _{\mu }\,{\hat {\phi }}(x)\;=\;{\frac {i}{\hbar }}\,{\bigl [}{\hat {\mathcal {H}}}_{\mu }(x),\,{\hat {\phi }}(x){\bigr ]}\;+\;\left({\frac {\partial {\hat {\phi }}(x)}{\partial x^{\mu }}}\right)_{\mathrm {explicit} },$ where:

• $\nabla _{\mu }$ is the covariant derivative with respect to the spacetime coordinate $x^{\mu }$.
• ${\hat {\mathcal {H}}}_{\mu }(x)$ is related to the energy-momentum tensor or Hamiltonian density in a relativistic quantum field theory.
• The commutator reflects the canonical (anti)commutation relations of the fields or operators involved.
• The term $\left({\frac {\partial {\hat {\phi }}(x)}{\partial x^{\mu }}}\right)_{\mathrm {explicit} }$ accounts for any additional explicit dependence on the coordinates.

When the theory is formulated in curved spacetime, the covariant derivative also includes the effects of the metric connection, ensuring that the formulation respects local Lorentz invariance and general covariance.

Applications

• Quantum Field Theory (QFT): The covariant form is used extensively in canonical quantization of fields, where field operators at different spacetime points must transform covariantly under Lorentz or general coordinate transformations.
• Curved Spacetime and Quantum Gravity Research: In semiclassical treatments of gravity or other curved-background field theories, the covariant Heisenberg equation guides how quantum fields evolve in a background metric, forming the basis of studies in cosmology and black hole physics.
• High-Energy Physics and Particle Physics: Covariant formulations allow consistent treatment of scattering processes in relativistic collider experiments, since operators can be tracked throughout spacetime rather than in a single global time variable.

Historical context

The original Heisenberg equation was introduced by Werner Heisenberg in 1925 as part of the matrix formulation of quantum mechanics. The need for a covariant generalization became apparent with the development of relativistic quantum mechanics and, later, quantum field theory in the late 1920s and 1930s by Paul Dirac, Wolfgang Pauli, Eugene Wigner, and others. As physicists moved toward a field-theoretic viewpoint, ensuring that equations remained valid under Lorentz transformations was essential. Thus, the covariant Heisenberg equation emerged naturally from canonical quantization procedures, where fields are promoted to operators and must obey fully relativistic commutation relations.

See also

• Heisenberg picture
• Canonical quantization
• Quantum field theory
• Relativistic quantum mechanics
• Dirac equation

Further reading

• Weinberg, Steven (1995). The Quantum Theory of Fields, Vol. 1. Cambridge University Press.
• Bjorken, James D. and Drell, Sidney D. (1964). Relativistic Quantum Mechanics. McGraw-Hill.
• Itzykson, Claude and Zuber, Jean-Bernard (1980). Quantum Field Theory. McGraw-Hill.

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Categories:

• Quantum mechanics
• Quantum field theory
• Theoretical physics