[Paper Review] Non-Unitary and Unitary Transitions in Generalized Quantum Mechanics and Information Problem Solving

Alexander Shalyt-Margolin|ArXiv.org|Sep 25, 2003

Noncommutative and Quantum Gravity Theories9 references3 citations

TL;DR

This paper proposes that unitarity in quantum gravity is preserved through nonunitary transitions between generalized quantum mechanics (QMFL) and standard quantum mechanics (QM), driven by the presence of black holes. By deforming the density matrix and Heisenberg algebra with a dimensionless parameter $\alpha = l_{\text{min}}^2/x^2$, the framework shows that information is conserved across the Big Bang (QMFL → QM) and black hole formation (QM → QMFL), resolving Hawking's information paradox via a closed, symmetric cycle of transitions.

ABSTRACT

The present work is a study of the unitarity problem for Quantum Mechanics at Planck Scale considered as Quantum Mechanics with Fundamental Length (QMFL).In the process QMFL is described as deformation of a well-known Quantum Mechanics (QM). Similar to previous works of the author, the basic approach is based on deformation of the density matrix (density pro-matrix) with concurrent development of the wave function deformation in the respective Schrodinger picture. It is demonstrated that the existence of black holes in the suggested approach in the end twice results in nonunitary transitions (first after the Big Bang of QMFL to QM, and then when on trapping of the matter into the black hole the situation is just the opposite - from QM to QMFL)and hence in recovery of the unitarity. In parallel this problem is considered in the deformation terms of Heisenberg algebra, showing the identity of the basic results. From this an explicit solution for Hawking's informaion paradox has been derived

Motivation & Objective

• To resolve the information paradox in black hole physics by reinterpreting unitarity in the context of quantum mechanics with a fundamental length (QMFL).
• To establish a consistent framework for quantum gravity at Planck scale using deformation of the density matrix and Heisenberg algebra.
• To demonstrate that black hole formation and the Big Bang are nonunitary transitions that collectively restore unitarity and information conservation.
• To unify two approaches—density matrix deformation and Heisenberg algebra deformation—into a coherent picture of quantum gravity.
• To provide a first-principles derivation of the Bekenstein-Hawking entropy formula and extend the concept of entropy density per unit minimum area.

Proposed method

• Introduces a dimensionless deformation parameter $\alpha = l_{\text{min}}^2/x^2$ with $0 < \alpha \leq 1/4$ to describe quantum mechanics with a fundamental length (QMFL), replacing dimensional parameters.
• Defines a deformed density pro-matrix $\rho(\alpha) = \sum_i \omega_i(\alpha) |i\rangle\langle i|$ satisfying $\text{Sp}[\rho(\alpha)] - \text{Sp}^2[\rho(\alpha)] \approx \alpha$, ensuring finite trace and continuity to standard QM as $\alpha \to 0$.
• Derives the trace of the density pro-matrix as $\text{Sp}[\rho(\alpha)] \approx \frac{1}{2} + \sqrt{\frac{1}{4} - \alpha}$, which reduces to 1 in the $\alpha \to 0$ limit, recovering standard QM.
• Applies the same deformation to the Heisenberg algebra, introducing $\kappa$-deformed commutators: $[x_i, x_j] = -\frac{\hbar^2}{\kappa^2} i\epsilon_{ijk} J_k$ and $[x_i, p_j] = i\hbar \delta_{ij} (1 + \frac{E^2}{\kappa^2})^{1/2}$, with $\kappa \to \infty$ recovering standard commutation relations.
• Models the Big Bang as a nonunitary transition from QMFL ($\kappa \sim M_p$) to QM ($\kappa = \infty$), and black hole formation as the reverse process, forming a closed cycle.
• Demonstrates that entropy density at initial and final singularities ($\alpha \approx 1/4$) is identical: $S^{\text{in}} = S^{\text{out}} = S^{\alpha=1/4}_{1/4}$, implying information preservation.

Experimental results

Research questions

• RQ1 Can unitarity be preserved in quantum gravity despite nonunitary transitions at the Planck scale?
• RQ2 How does the existence of black holes affect the conservation of quantum information in a generalized quantum mechanics framework?
• RQ3 What is the role of the deformation parameter $\alpha = l_{\text{min}}^2/x^2$ in ensuring consistency between QMFL and standard QM?
• RQ4 Can the Bekenstein-Hawking entropy formula be derived from first principles within this deformed quantum framework?
• RQ5 Is there a duality or equivalence between the density matrix deformation and Heisenberg algebra deformation approaches in resolving the information paradox?

Key findings

• The transition from QMFL to standard QM after the Big Bang is nonunitary, but the reverse transition during black hole formation restores unitarity, preserving total information.
• The entropy density at the initial singularity ($\alpha \approx 1/4$) equals that at the final state ($\alpha \approx 1/4$), with $S^{\text{in}} = S^{\text{out}} = S^{\alpha=1/4}_{1/4}$, indicating information conservation.
• The deformed density matrix formalism yields $\text{Sp}[\rho(\alpha)] \approx \frac{1}{2} + \sqrt{\frac{1}{4} - \alpha}$, which smoothly reduces to 1 as $\alpha \to 0$, recovering standard QM.
• The $\kappa$-deformed Heisenberg algebra reproduces the same physical results as the density matrix deformation, confirming consistency between the two approaches.
• The information paradox is resolved: black holes do not destroy information but instead trigger a reverse nonunitary transition that restores unitarity and information content.
• The framework provides a first-principles derivation of the Bekenstein-Hawking entropy formula and introduces a refined notion of entropy density per unit minimum area.