[Paper Review] Quantization of kappa-deformed free fields and kappa-deformed oscillators

Marcin Daszkiewicz, J. Lukierski+1 more|ArXiv.org|Dec 3, 2007

Noncommutative and Quantum Gravity Theories5 references3 citations

TL;DR

This paper proposes a deformed quantization procedure for $κ$-deformed free scalar fields using a $κ$-deformed star product and oscillator algebra, ensuring consistent $κ$-causality and preserving standard bosonic statistics. The key result is a $κ$-deformed Pauli-Jordan function $Δ_{\kappa}(x;m^2)$ that replaces the standard commutator, derived from a $κ$-deformed mass Casimir and maintaining classical four-momentum addition for multi-particle states.

ABSTRACT

We describe the deformed E.T. quantization rules for kappa-deformed free quantum fields, and relate these rules with the kappa-deformed algebra of field oscillators.

Motivation & Objective

• To develop a consistent quantization framework for $κ$-deformed free quantum fields that respects the $κ$-Poincaré symmetry and noncommutative spacetime structure.
• To resolve the issue of non-locality and causality in $κ$-deformed field theories by introducing a $κ$-deformed commutator function $Δ_{\kappa}(x;m^2)$.
• To ensure that multi-particle states constructed from $κ$-deformed oscillators exhibit standard bosonic symmetry and Abelian four-momentum addition, despite the nontrivial coproduct in the $κ$-Poincaré algebra.
• To establish a correspondence between $κ$-deformed field commutators and a deformed oscillator algebra using a $κ$-deformed product $∘_{\kappa}$, ensuring consistency with the field equations and on-shell conditions.

Proposed method

• Introduce a $κ$-deformed star product $∗_{\kappa}$ for field operators, defined via a non-local exponential factor involving the $κ$-deformed momentum space structure.
• Define a $κ$-deformed oscillator algebra using a $∘_{\kappa}$-product that modifies the standard commutation relations between creation and annihilation operators, with phase factors dependent on momenta and the $κ$-scale.
• Construct the field operator as a Fourier integral over $κ$-deformed oscillators, with the on-shell condition replaced by the $κ$-deformed mass Casimir $C_2^{\kappa}(p)$.
• Derive the $κ$-deformed commutator function $Δ_{\kappa}(x;m^2)$ as the Wightman function, replacing the standard Pauli-Jordan function and encoding $κ$-causality.
• Ensure that the $κ$-deformed oscillator algebra is associative and compatible with the $κ$-Poincaré Hopf algebra structure, particularly the coproduct for four-momenta.
• Demonstrate that multi-particle states built from $κ$-deformed oscillators satisfy standard bosonic symmetry and classical Abelian momentum addition, despite the nontrivial coproduct.

Experimental results

Research questions

• RQ1 How can a consistent field quantization procedure be formulated for $κ$-deformed free fields that respects the $κ$-Poincaré symmetry and noncommutative spacetime?
• RQ2 What is the appropriate $κ$-deformed commutator function that generalizes the standard Pauli-Jordan function and ensures $κ$-causality?
• RQ3 How can $κ$-deformed creation and annihilation operators be defined such that multi-particle states maintain standard bosonic symmetry and classical momentum addition?
• RQ4 What is the role of the $κ$-deformed product $∘_{\kappa}$ in relating the field algebra to the oscillator algebra in the deformed framework?
• RQ5 Can the $κ$-deformed field theory be constructed so that the field equations and on-shell conditions are preserved while incorporating the $κ$-deformation of the mass Casimir?

Key findings

• The paper derives a $κ$-deformed Pauli-Jordan function $Δ_{\kappa}(x;m^2) = \frac{i}{(2\pi)^3}\int d^4p~\epsilon(p_0)~\delta(C_2^{\kappa}(p)-m^2)e^{ipx}$, which replaces the standard commutator and encodes $κ$-causality.
• The $κ$-deformed oscillator algebra is defined via a $∘_{\kappa}$-product that modifies the commutation relations with phase factors $e^{\pm \frac{i}{2}p\theta q}$, ensuring consistency with the field star product.
• Multi-particle states constructed from $κ$-deformed oscillators satisfy standard bosonic symmetry $|\vec{p},\vec{q}\rangle = |\vec{q},\vec{p}\rangle$ and have four-momenta that add via the Abelian law $p_i + q_i$, despite the non-Abelian coproduct in the $κ$-Poincaré algebra.
• The field operator is expressed as a Fourier integral over $κ$-deformed oscillators, with the on-shell condition replaced by the $κ$-deformed mass Casimir $C_2^{\kappa}(p)$, ensuring consistency with the deformed field equations.
• The $κ$-deformed star product $∗_{\kappa}$ and oscillator product $∘_{\kappa}$ are shown to be consistent with each other, leading to a closed algebraic structure for the deformed field theory.
• The formalism preserves the standard Fock space structure for multi-particle states, with the only modification being the momentum dependence of the oscillator operators through the $κ$-deformed momentum mapping $\vec{p} \to \vec{\cal P} = e^{-q_0/(2\kappa)}\vec{p}$.