[Paper Review] On momentum operator in quantum field theory

Bozhidar Z. Iliev|ArXiv.org|Jun 2, 2002

Quantum Mechanics and Applications8 references3 citations

TL;DR

This paper investigates the distinction between two definitions of the momentum operator in quantum field theory: the canonical (physical) momentum operator derived from Noether's theorem via the energy-momentum tensor, and the translation (mathematical) momentum operator defined as the generator of spacetime translations on the operator space. The key result is that these two operators are generally inequivalent, with the mathematical momentum operator vanishing on physical states unless the 3-momentum is zero, revealing an incompatibility between quantum field theory and standard quantum mechanics postulates.

ABSTRACT

The interrelations between the two definitions of momentum operator, via the canonical energy-momentum tensorial operator and as translation operator (on the operator space), are studied in quantum field theory. These definitions give rise to similar but, generally, different momentum operators, each of them having its own place in the theory. Some speculations on the relations between quantum field theory and quantum mechanics are presented.

Motivation & Objective

• To clarify the conceptual and mathematical distinction between two definitions of the momentum operator in quantum field theory.
• To analyze the conditions under which the canonical (physical) and translation (mathematical) momentum operators coincide or differ.
• To investigate the foundational incompatibility between quantum field theory and standard quantum mechanics, particularly regarding momentum operators.
• To explore the implications of this distinction for the relationship between non-relativistic and relativistic quantum mechanics and quantum field theory.

Proposed method

• Derives the canonical momentum operator from the Noether current associated with spacetime translation invariance of the Lagrangian.
• Defines the translation momentum operator as the generator of the Poincaré group action on the space of field operators in the Heisenberg picture.
• Uses the commutation relations between field operators and the translation generator to derive the relation between the two momentum operators.
• Applies the Heisenberg picture formalism to analyze the time evolution and transformation properties of field operators under translations.
• Compares the resulting momentum operators by examining their action on physical states and their consistency with the Schrödinger equation.
• Analyzes the inconsistency arising when imposing the Schrödinger equation on a QFT framework, leading to the conclusion that only zero 3-momentum states are compatible.

Experimental results

Research questions

• RQ1 How do the canonical momentum operator (from Noether's theorem) and the translation momentum operator (from group representation theory) relate in quantum field theory?
• RQ2 Under what conditions are the canonical and translation momentum operators equivalent?
• RQ3 Why does the standard Schrödinger equation in quantum mechanics conflict with the structure of quantum field theory when applied to momentum operators?
• RQ4 What is the physical significance of the 4-momentum operator being defined up to a constant vector in the translation definition?
• RQ5 How does the incompatibility between the momentum operators affect the derivation of quantum mechanics from quantum field theory?

Key findings

• The canonical momentum operator, derived from the energy-momentum tensor, is conserved and corresponds to physical momentum in the system.
• The translation momentum operator, defined as the generator of spacetime translations on the operator space, is generally inequivalent to the canonical momentum operator.
• The two operators coincide only if the 3-momentum is zero, which leads to a contradiction when applied to non-zero momentum states in quantum mechanics.
• The Schrödinger equation in quantum mechanics is incompatible with the full structure of quantum field theory unless the 3-momentum is zero.
• The mathematical momentum operator vanishes on physical states unless the momentum is zero, indicating a fundamental incompatibility between the postulates of quantum mechanics and quantum field theory.
• The paper concludes that quantum mechanics cannot be consistently derived from quantum field theory by simply restricting to the Schrödinger equation, due to this momentum operator mismatch.