[Paper Review] What is "Relativistic Canonical Quantization"?

D. A. Arbatsky|ArXiv.org|Mar 11, 2005

Cosmology and Gravitation Theories1 references3 citations

TL;DR

This paper introduces Relativistic Canonical Quantization (RCQ) as a rigorously invariant framework for quantizing relativistic fields by formulating Hamiltonian mechanics on an invariant phase space equipped with a Poincaré-invariant symplectic structure. The method constructs quantized fields from classical solutions using group-theoretic methods, yielding explicitly relativistic-invariant commutation relations and naturally accommodating indefinite metric spaces—providing a mathematically sound foundation that generalizes Noether’s theorem and resolves long-standing issues in electromagnetic field quantization.

ABSTRACT

The purpose of this review is to give the most popular description of the scheme of quantization of relativistic fields that was named relativistic canonical quantization (RCQ). I do not give here the full exact account of this scheme. But with the help of this review any physicist, even not a specialist in the relativistic quantum theory, will be able to get a general view of the content of RCQ, of its connection with other known approaches, of its novelty and of its fruitfulness.

Motivation & Objective

• To provide a clear, accessible overview of Relativistic Canonical Quantization (RCQ) for non-specialists in relativistic quantum theory.
• To establish a relativistic-invariant formulation of Hamiltonian mechanics by generalizing phase space, symplectic structure, and time evolution to solution space.
• To demonstrate how RCQ naturally incorporates Poincaré symmetry and leads to a mathematically rigorous quantization procedure without ad hoc postulates.
• To resolve the long-standing problem of indefinite metric spaces in quantum electrodynamics by showing they are not an obstacle but a natural feature of the construction.
• To unify and generalize existing quantization methods by embedding them within a single, coherent, and invariant formalism.

Proposed method

• Define the invariant phase space as the set of all solutions to the classical field equations, which serves as the configuration space for the Hamiltonian formalism.
• Equip the invariant phase space with a symplectic structure derived from the Lagrangian, ensuring Poincaré invariance.
• Introduce the canonical action of the Poincaré group on the invariant phase space, preserving both the symplectic structure and the linear structure of the solution space.
• Treat linear relativistic fields as defining symplectic representations of the Poincaré group on the invariant phase space.
• Construct the quantum field via standard algebraic methods (e.g., universal enveloping algebras), using the classical field values and symplectic structure as input.
• Ensure all quantum properties—such as commutation relations and conserved charges—follow constructively from the geometric and group-theoretic structure, not postulated.

Experimental results

Research questions

• RQ1 How can Hamiltonian formalism be made manifestly relativistic-invariant in field theory?
• RQ2 What is the role of the solution space (invariant phase space) in defining a relativistic canonical structure?
• RQ3 How does the Poincaré group act on the space of classical field solutions in a way that preserves symplectic structure?
• RQ4 Can quantization be performed without assuming a positive-definite inner product, particularly for fields like the electromagnetic potential?
• RQ5 How does RCQ provide a rigorous, constructive analog of Noether’s theorem in quantum field theory?

Key findings

• RCQ provides a fully relativistic-invariant quantization scheme by formulating Hamiltonian mechanics on the space of classical field solutions.
• The invariant phase space naturally carries a symplectic structure derived from the Lagrangian, which is preserved under Poincaré transformations.
• The Poincaré group acts canonically on the invariant phase space as a group of symplectic transformations, enabling a group-theoretic approach to quantization.
• Quantum commutation relations and conserved charges emerge constructively from the geometric structure, not by postulate, thus generalizing Noether’s theorem rigorously.
• The method naturally accommodates indefinite metric spaces, resolving the long-standing issue of quantizing the electromagnetic field without requiring a Hilbert space structure.
• For linear fields, RCQ yields equivalent results to standard quantization when a positive-definite metric is assumed, but provides a more fundamental and systematic framework without relying on classical analogies or ad hoc assumptions.