Velocity polytopes of periodic graphs and a no-go theorem for digital physics

Abstract A periodic graph in dimension d is a directed graph with a free action of Z d with only finitely many orbits. It can conveniently be represented in terms of an associated finite graph with weights in Z d , corresponding to a Z d -bundle with connection. Here we use the weight sums along cycles in this associated graph to construct a certain polytope in R d , which we regard as a geometrical invariant associated to the periodic graph. It is the unit ball of a norm on R d describing the large-scale geometry of the graph. It has a physical interpretation as the set of attainable velocities of a particle on the graph which can hop along one edge per timestep. Since a polytope necessarily has distinguished directions, there is no periodic graph for which this velocity set is isotropic. In the context of classical physics, this can be viewed as a no-go theorem for the emergence of an isotropic space from a discrete structure.

[1]  Luis J. Garay.  Quantum Gravity and Minimum Length , 1995 .

[2]  Giacomo Mauro D'Ariano.  The Quantum Field as a Quantum Computer , 2010 .

[3]  J. Eon.  Topological density of nets: a direct calculation. , 2004, Acta crystallographica. Section A, Foundations of crystallography.

[4]  Chris D. Godsil,et al.  Periodic Graphs , 2008, Electron. J. Comb..

[5]  W. Klee.  Crystallographic nets and their quotient graphs , 2004 .

[6]  Michael O'Keeffe,et al.  Crystal nets as graphs: Terminology and definitions , 2005 .

[7]  J. Eon.  Euclidian embeddings of periodic nets: definition of a topologically induced complete set of geometric descriptors for crystal structures. , 2011, Acta crystallographica. Section A, Foundations of crystallography.

[8]  V. Blatov,et al.  Periodic‐Graph Approaches in Crystal Structure Prediction , 2010 .

[9]  Egon Wanke,et al.  Minimum Cost Paths in Periodic Graphs , 1995, SIAM J. Comput..

[10]  Simone Severini,et al.  Quantum bose-hubbard model with an evolving graph as a toy model for emergent spacetime , 2009, 0911.5075.

[11]  Jonathan L. Gross,et al.  Topological Graph Theory , 1987, Handbook of Graph Theory.

[12]  E. Fredkin.  Digital mechanics: an informational process based on reversible universal cellular automata , 1990 .

[13]  Shinya Fujita,et al.  Note on Highly Connected Monochromatic Subgraphs in 2-Colored Complete Graphs , 2011, Electron. J. Comb..

[14]  M. Fekete.  Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten , 1918 .

[15]  R. Ho.  Algebraic Topology , 2022 .

[16]  Crystal topologies--the achievable and inevitable symmetries. , 2009, Acta crystallographica. Section A, Foundations of crystallography.

[17]  Simone Severini,et al.  Quantum graphity: A model of emergent locality , 2008, 0801.0861.

[18]  Th. Hahn,et al.  Nomenclature and generation of three-periodic nets: the vector method , 1984 .

[19]  L. Collatz.  Spektren periodischer Graphen , 1978 .

[20]  T. Ando,et al.  Energy bands of carbon nanotubes in magnetic fields , 1996 .

[21]  Jonathan L. Gross,et al.  Generating all graph coverings by permutation voltage assignments , 1977, Discret. Math..

[22]  Giacomo Mauro DʼAriano.  The quantum field as a quantum computer , 2012 .

[23]  H. Weyl,et al.  Philosophy of Mathematics and Natural Science , 1950 .

[24]  Norman Biggs.  Algebraic Graph Theory: Index , 1974 .

[25]  Jonathan L. Gross,et al.  Voltage graphs , 1974, Discret. Math..

[26]  Giacomo Mauro D'Ariano,et al.  Emergence of space–time from topologically homogeneous causal networks , 2013 .

[27]  Edith Cohen,et al.  Recognizing Properties of Periodic Graphs , 1990, Applied Geometry And Discrete Mathematics.

[28]  Gerard 't Hooft,et al.  The fundamental nature of space and time , 2009 .