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Complex Quantum Network Manifolds in Dimension d > 2 are Scale-Free.

Ginestra Bianconi1, Christoph Rahmede2

  • 1School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, United Kingdom.

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|September 11, 2015
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This study introduces Complex Quantum Network Manifolds (CQNM) for describing discrete geometries in quantum gravity. CQNM exhibit scale-free properties for dimensions greater than two, leading to emergent quantum statistics like Fermi-Dirac and Bose-Einstein distributions.

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Area of Science:

  • Quantum Gravity
  • Theoretical Physics
  • Complex Networks

Background:

  • Existing quantum gravity approaches often model discrete spaces as homogeneous network manifolds.
  • There is a need for new theoretical frameworks to describe the quantum nature of discrete geometries.

Purpose of the Study:

  • To define Complex Quantum Network Manifolds (CQNM) for describing quantum network state evolution.
  • To investigate the structural properties and emergent statistics of CQNM.

Main Methods:

  • Construction of CQNM from growing simplicial complexes of dimension d.
  • Analysis of network homogeneity and scale-free properties based on dimension.
  • Definition of generalized degrees for δ-faces within d-dimensional CQNM.

Main Results:

  • CQNM are homogeneous for d=2 and scale-free for d>2, exhibiting inhomogeneities typical of complex networks.
  • Spontaneous emergence of quantum statistics from the self-organized evolution of CQNM.
  • Generalized degrees follow Fermi-Dirac, Boltzmann, or Bose-Einstein distributions depending on the dimension of the δ-faces.

Conclusions:

  • CQNM provide a novel framework for quantum gravity, offering insights into discrete geometry.
  • The scale-free nature of CQNM for d>2 has significant implications for understanding complex network structures in physics.
  • The emergence of different quantum statistics from CQNM highlights their potential to unify diverse physical phenomena.