[Paper Review] Quantization of the Myers-Pospelov model: a progress report

Carlos M. Reyes, Luis F. Urrutia+1 more|ArXiv.org|Dec 20, 2007

Noncommutative and Quantum Gravity Theories13 references3 citations

TL;DR

This paper presents a perturbative quantization of the Myers-Pospelov model's modified electrodynamics, treating Lorentz invariance violation (LIV) as a higher-order time derivative (HOTD) perturbation over standard QED. By applying a modified perturbation scheme that ensures a positive-definite Hamiltonian and redefining fields to eliminate unphysical degrees of freedom, the authors derive modified normal mode expansions, free propagators, and interaction terms—laying the foundation for future radiative corrections while ensuring consistency with standard QED in the limit of vanishing LIV parameters.

ABSTRACT

The Myers-Pospelov (MP) model is an effective field theory, including dimension five operators, which describes the phenomenology of active Lorentz invariance violation produced by a preferred reference frame. We concentrate here in the case of the modified electrodynamics. The point of view taken in this work is that the Lorentz violating part of the action in the MP model, which includes higher order time derivative (HOTD) operators, is to be considered as a perturbation over the dynamics described by standard Electrodynamics, particularly in the quantum case. In order to cope with the challenges posed by HOTD theories it will be necessary to deal with a modified perturbation scheme which is well described in the literature. We apply such methods to this specific model providing a quantization of the free sector of the theory. The calculation of interacting processes, together with radiative corrections, is beyond the scope of the present article.

Motivation & Objective

• To develop a consistent quantum field theory framework for the Myers-Pospelov model’s modified electrodynamics, treating Lorentz invariance violation (LIV) as a perturbation over standard QED.
• To resolve the quantization challenges of higher-order time derivative (HOTD) theories, particularly the unbounded Hamiltonian, by applying a known modified perturbation procedure.
• To ensure the resulting quantum theory reduces to standard QED when LIV parameters vanish, preserving experimental consistency.
• To derive the free sector of the quantum theory, including modified dispersion relations, propagators, and normal mode expansions, as a prerequisite for future interacting processes.

Proposed method

• Adopt a classical formulation of the Myers-Pospelov model with dimension-five operators introducing LIV via a preferred frame, using a fixed vector $ n^\mu = (1, \mathbf{0}) $.
• Apply the field redefinition and effective theory reduction method from Ref. [8] to transform the HOTD classical action into a form with standard time derivative structure and modified interactions.
• Use the resulting modified classical theory as the starting point for canonical quantization, preserving the physical content while ensuring a positive-definite Hamiltonian.
• Identify modified normal mode expansions for the fields, incorporating LIV corrections to lowest order in the parameters $ \xi, \eta_1, \eta_2 $.
• Derive free-field propagators that reflect the modified dispersion relations due to LIV, essential for perturbative calculations.
• Implement a Pauli-Villars-type regulator with scale $ \bar{M} \ll M_{QG} $ to suppress high-momentum modes and ensure the correct $ \Xi \to 0 $ limit to standard QED.

Experimental results

Research questions

• RQ1 How can a higher-order time derivative (HOTD) theory like the Myers-Pospelov model be consistently quantized when its Hamiltonian is not bounded from below?
• RQ2 What is the correct perturbative procedure to treat Lorentz invariance violation as a small correction to standard QED without introducing spurious degrees of freedom?
• RQ3 How can the quantum theory be constructed such that it reduces to standard QED in the limit where LIV parameters vanish?
• RQ4 What are the modified free propagators and normal mode expansions in the presence of dimension-five LIV operators in electrodynamics?
• RQ5 How can a regulator be introduced that ensures the correct continuum limit to QED while respecting the effective field theory structure?

Key findings

• The authors successfully derive a modified normal mode expansion for the photon field in the Myers-Pospelov model, incorporating LIV corrections to lowest order in the parameters $ \xi, \eta_1, \eta_2 $.
• The free-field propagators are obtained with modified dispersion relations due to the dimension-five LIV operators, reflecting the altered propagation of excitations.
• The procedure ensures that the zeroth-order Hamiltonian is positive-definite, resolving a key obstacle in quantizing HOTD theories.
• A Pauli-Villars-type regulator with mass scale $ \bar{M} $ is introduced to suppress high-momentum modes and ensure the correct $ \Xi \to 0 $ limit to standard QED.
• The resulting quantum theory is consistent with standard QED in the limit of vanishing LIV parameters, satisfying the requirement of experimental compatibility.
• The framework sets the stage for future calculations of radiative corrections, including self-energies, which may reveal fine-tuning issues in LIV models.