# Lagrangian quantum field theory in momentum picture. IV. Commutation relations for free fields

## TL;DR

This paper investigates algebraic commutation relations in free quantum fields, showing that standard bilinear relations are uniquely consistent with all physical and mathematical constraints, including vacuum state considerations.

## Contribution

It demonstrates that standard bilinear commutation relations are the only consistent choice under the imposed physical and mathematical conditions for free fields.

## Key findings

- Standard bilinear relations satisfy all conditions.
- Paracommutation relations are the most general compatible with Heisenberg equations.
- Redefinition of dynamical variables aligns with normal ordering.

## Abstract

Possible (algebraic) commutation relations in the Lagrangian quantum theory of free (scalar, spinor and vector) fields are considered from mathematical view-point. As sources of these relations are employed the Heisenberg equations/relations for the dynamical variables and a specific condition for uniqueness of the operators of the dynamical variables (with respect to some class of Lagrangians). The paracommutation relations or some their generalizations are pointed as the most general ones that entail the validity of all Heisenberg equations. The simultaneous fulfillment of the Heisenberg equations and the uniqueness requirement turn to be impossible. This problem is solved via a redefinition of the dynamical variables, similar to the normal ordering procedure and containing it as a special case. That implies corresponding changes in the admissible commutation relations. The introduction of the concept of the vacuum makes narrow the class of the possible commutation relations; in particular, the mentioned redefinition of the dynamical variables is reduced to normal ordering. As a last restriction on that class is imposed the requirement for existing of an effective procedure for calculating vacuum mean values. The standard bilinear commutation relations are pointed as the only known ones that satisfy all of the mentioned conditions and do not contradict to the existing data.