[Paper Review] Fractal thermodynamics and ninionic statistics of coherent rotational states: realization via imaginary angular rotation in imaginary time formalism

M. N. Chernodub|arXiv (Cornell University)|Oct 11, 2022

Quantum, superfluid, helium dynamics31 references4 citations

TL;DR

This paper proposes 'ninions'—a novel type of particle statistics in 3+1 dimensions that interpolates between bosonic, fermionic, and ghost-like distributions via imaginary angular rotation in the Euclidean imaginary-time formalism. The rotwisted boundary conditions induce a continuous statistical parameter χ, leading to fractal thermodynamics and a no-go theorem for analytical continuation from imaginary to real rotation, with potential implications for dark energy and quantum simulations.

ABSTRACT

We suggest the existence of systems in which the statistics of a particle changes with the quantum level it occupies. The occupation numbers in thermal equilibrium depend on a continuous statistical parameter that interpolates between bosonic or fermionic and ghost-like statistical distributions. We call such particle states ``ninions'': they are different from anyons and can exist in 3+1 dimensions. We suggest that ninions can be associated with coherent angular momentum states. In the Euclidean imaginary-time formalism, the ninionic statistics can be implemented via the rotwisted boundary conditions, which are associated with the rigid global rotation of the system with an imaginary angular frequency. The imaginary rotation is characterized by a PT-symmetric non-Hermitian Hamiltonian and possesses a well-defined thermodynamic limit. The physics of ninions in thermal equilibrium is accessible for numerical simulations on Euclidean lattices. We provide a no-go theorem on the absence of analytical continuation between real and imaginary rotations in the thermodynamic limit. The ground state of ninions shares similarity with the $θ$-vacuum in QCD. The ninions can produce negative pressure and energy, similar to the Casimir effect and the cosmological dark energy. In the thermodynamic limit, the dependence of thermal energy of free ninions on the statistical parameter is a fractal.

Motivation & Objective

• To explore the existence of particle statistics that vary with quantum level, beyond standard bosonic and fermionic statistics.
• To establish a framework for non-integer, continuous statistics in 3+1 dimensions using imaginary angular momentum and rotwisted boundary conditions.
• To demonstrate that such systems exhibit fractal thermodynamic behavior and are describable via non-Hermitian PT-symmetric Hamiltonians.
• To connect the statistical parameter χ to the θ-vacuum structure in QCD and explore its role as an independent coupling in thermodynamics.
• To propose a realization of ninions in coherent angular momentum states and assess their relevance to cosmological dark energy and numerical simulations.

Proposed method

• Formulate the system using the Euclidean imaginary-time formalism with rotwisted boundary conditions, where the field acquires a phase shift under a full compactified time evolution.
• Introduce a statistical parameter χ via the rotwisted boundary condition, which generalizes periodicity and allows interpolation between bosonic, fermionic, and ghost-like statistics.
• Implement the formalism in the path integral using a non-Hermitian PT-symmetric Hamiltonian that ensures unitary time evolution and well-defined thermodynamic limit.
• Derive the occupation number distribution for ninions, showing its dependence on χ and the quantum state, with a functional form related to the Thomae function.
• Analyze the thermodynamic limit, demonstrating that the energy dependence on χ is fractal and discontinuous, implying non-analyticity.
• Identify the ground state of free ninions as analogous to the QCD θ-vacuum, with χ playing the role of the topological angle.

Experimental results

Research questions

• RQ1 Can particle statistics in 3+1 dimensions be continuously interpolated beyond bosonic and fermionic statistics?
• RQ2 How does the introduction of imaginary angular rotation via rotwisted boundary conditions affect the statistical distribution of particles?
• RQ3 What is the nature of the thermodynamic limit for systems with such generalized statistics?
• RQ4 Can the statistical parameter χ be interpreted as an independent coupling in thermodynamics, akin to the θ parameter in QCD?
• RQ5 Is there a no-go theorem for analytical continuation between real and imaginary rotations when defined via boundary conditions?

Key findings

• The occupation number of ninions depends on the quantum state and the continuous statistical parameter χ, leading to a level-dependent statistics distinct from anyons.
• The statistical parameter χ is implemented via rotwisted boundary conditions in imaginary time, which correspond to a PT-symmetric non-Hermitian Hamiltonian with unitary evolution.
• The thermodynamic energy of free ninions exhibits a fractal dependence on χ, described by the Thomae function, which is discontinuous at every rational point.
• A no-go theorem is established: analytical continuation from imaginary to real rotation is impossible if the rotation is defined via boundary conditions in the thermodynamic limit.
• The ground state of free ninions resembles the QCD θ-vacuum, with χ acting as a topological parameter analogous to the θ angle.
• Certain ninionic states exhibit negative pressure and energy, suggesting a possible connection to cosmological dark energy.