[Paper Review] Pictures and equations of motion in Lagrangian quantum field theory

Bozhidar Z. Iliev|ArXiv.org|Feb 1, 2003

Quantum Mechanics and Applications30 references3 citations

TL;DR

This paper introduces the momentum picture—a novel formulation in Lagrangian quantum field theory where state vectors depend on spacetime while field operators remain constant, leading to algebraic equations of motion. It generalizes standard pictures (Schrödinger, Heisenberg, interaction) into a covariant framework, deriving equations of motion for arbitrary polynomial or convergent power-series Lagrangians, with the momentum picture yielding particularly simple, algebraic dynamics for field operators.

ABSTRACT

The Heisenberg, interaction, and Schrödinger pictures of motion are considered in Lagrangian (canonical) quantum field theory. The equations of motion (for state vectors and field operators) are derived for arbitrary Lagrangians which are polynomial or convergent power series in field operators and their first derivatives. The general links between different time-dependent pictures of motion are derived. It is pointed that all of them admit covariant formulation, similar to the one of interaction picture. A new picture, called the momentum picture, is proposed. It is a 4-dimensional analogue of the Schrödinger picture of quantum mechanics as in it the state vectors are spacetime-dependent, while the field operators are constant relative to the spacetime. The equations of motion in momentum picture are derived and partially discussed. In particular, the ones for the field operators turn to be of algebraic type. The general idea of covariant pictures of motion is presented. The equations of motion in these pictures are derived.

Motivation & Objective

• To systematically develop and unify the formalism of different pictures of motion in Lagrangian quantum field theory.
• To derive equations of motion (Euler-Lagrange and Heisenberg-type) in all standard pictures—Heisenberg, interaction, Schrödinger—for general Lagrangians.
• To propose and formalize a new picture, the momentum picture, where state vectors are spacetime-dependent while field operators are constant.
• To establish a covariant formulation of time-dependent pictures, extending the interaction picture to a relativistically invariant framework.
• To demonstrate that the momentum picture yields algebraic equations of motion for field operators, simplifying their dynamics.

Proposed method

• Uses unitary transformations to relate different pictures of motion, preserving matrix elements of operators.
• Derives equations of motion via the Euler-Lagrange equations applied to general Lagrangians that are polynomial or convergent power series in fields and their first derivatives.
• Introduces the momentum picture by defining a spacetime-dependent unitary transformation that shifts time dependence from field operators to state vectors.
• Derives the Heisenberg and momentum picture equations of motion using canonical commutation relations and the Lagrangian formalism.
• Establishes a general framework for covariant pictures using spacetime-dependent unitary operators, such as those based on momentum, angular momentum, or charge generators.
• Applies Noether's theorem to define conserved quantities (e.g., momentum, angular momentum, charge) in each picture, ensuring consistency with relativistic invariance.

Experimental results

Research questions

• RQ1 How can the standard pictures of motion (Schrödinger, Heisenberg, interaction) be systematically generalized to a covariant framework in quantum field theory?
• RQ2 What are the equations of motion for field operators and state vectors in the proposed momentum picture, and how do they differ from those in other pictures?
• RQ3 Can the momentum picture be formulated in a manifestly Lorentz-covariant way, and what are its dynamical implications?
• RQ4 What is the role of unitary transformations in relating different pictures, and how do they preserve physical observables?
• RQ5 How do the equations of motion simplify in the momentum picture, particularly in terms of algebraic versus differential structures?

Key findings

• The momentum picture is introduced as a 4-dimensional analogue of the Schrödinger picture, where state vectors are spacetime-dependent and field operators are constant.
• In the momentum picture, the equations of motion for field operators reduce to algebraic equations, significantly simplifying their dynamics compared to differential equations in other pictures.
• The Heisenberg, interaction, and Schrödinger pictures admit covariant formulations similar to the interaction picture, preserving Lorentz invariance.
• The general framework of covariant pictures is established, with unitary operators parameterized by spacetime-dependent functions of momentum, angular momentum, or charge generators.
• The momentum picture is shown to be consistent with the canonical formalism and Noether's theorem, ensuring conservation laws are preserved.
• The paper demonstrates that the transition between pictures is achieved via unitary transformations that leave all physical matrix elements invariant, preserving the theory's physical content.