Topological Decompositions of the Pauli Group and their Influence on Dynamical Systems

In the present paper we show that it is possible to obtain the well known Pauli group P = 〈 X , Y , Z | X 2 = Y 2 = Z 2 = 1,( Y Z ) 4 = ( Z X ) 4 = ( X Y ) 4 = 1〉 of order 16 as an appropriate quotient group of two distinct spaces of orbits of the three dimensional sphere S 3 . The first of these spaces of orbits is realized via an action of the quaternion group Q 8 on S 3 ; the second one via an action of the cyclic group of order four ℤ ( 4 ) $\mathbb {Z}(4)$ on S 3 . We deduce a result of decomposition of P of topological nature and then we find, in connection with the theory of pseudo-fermions, a possible physical interpretation of this decomposition.

[1]  F. Bagarello Damping and pseudo-fermions , 2012, 1212.3663.

[2]  On topological actions of finite groups on S^3 , 2016, 1606.07626.

[3]  B. Zimmermann On hyperbolic knots with homeomorphic cyclic branched coverings , 1998 .

[4]  F. Russo,et al.  Realization of Lie algebras of high dimension via pseudo-bosonic operators , 2020, 2002.09727.

[5]  Quantum mechanics on Riemannian manifold in Schwinger's quantization approach I , 2001, hep-th/0102139.

[6]  On finite groups acting on spheres and finite subgroups of orthogonal groups , 2011, 1108.2602.

[7]  Karl H. Hofmann,et al.  The Structure of Compact Groups , 2020 .

[8]  B. Zimmermann On topological actions of finite, non-standard groups on spheres , 2016, 1602.04599.

[9]  Daniel Gottesman,et al.  Stabilizer Codes and Quantum Error Correction , 1997, quant-ph/9705052.

[10]  F. Russo,et al.  On the presence of families of pseudo-bosons in nilpotent Lie algebras of arbitrary corank , 2018, Journal of Geometry and Physics.

[11]  M. Kibler Variations on a theme of Heisenberg, Pauli and Weyl , 2008, 0807.2837.

[12]  Fabio Bagarello Linear pseudo-fermions , 2012 .

[13]  B. Zimmermann,et al.  On hyperelliptic involutions of hyperbolic 3-manifolds , 2001 .

[14]  E. Knill Non-binary unitary error bases and quantum codes , 1996, quant-ph/9608048.

[15]  J. Linnett,et al.  Quantum mechanics , 1975, Nature.

[16]  Greenberg Particles with small violations of Fermi or Bose statistics. , 1991, Physical review. D, Particles and fields.

[17]  F. Bagarello,et al.  Model pseudofermionic systems: Connections with exceptional points , 2014, 1402.6201.

[18]  Joseph E. Borzellino Riemannian Geometry of Orbifolds , 1992 .

[19]  F. Russo,et al.  A description of pseudo-bosons in terms of nilpotent Lie algebras , 2017, 1712.04631.

[20]  Andrea Rocchetto,et al.  Decomposition of Pauli groups via weak central products. , 2019, 1911.10158.

[21]  Sidney A. Morris,et al.  The Structure of Compact Groups: A Primer for Students - A Handbook for the Expert , 2006 .

[22]  J. Provost,et al.  Riemannian structure on manifolds of quantum states , 1980 .

[23]  Robert Kohl,et al.  An Introduction To Algebraic Topology , 2016 .

[24]  Daniel Gottesman Fault-Tolerant Quantum Computation with Higher-Dimensional Systems , 1998, QCQC.

[25]  Daniel W. Hook,et al.  PT Symmetry , 2018 .

[26]  J. Ward,et al.  On Finite Soluble Groups , 1969, Journal of the Australian Mathematical Society.

[27]  D. Burago,et al.  A Course in Metric Geometry , 2001 .

[28]  R. Ho Algebraic Topology , 2022 .

[29]  R. Mohapatra,et al.  Infinite statistics and a possible small violation of the Pauli principle , 1990 .

[30]  Yusef Maleki,et al.  Para-Grassmannian Coherent and Squeezed States for Pseudo-Hermitian q-Oscillator and their Entanglement , 2011, 1108.5005.

[31]  Wolfgang Pauli Zur Quantenmechanik des magnetischen Elektrons , .

[32]  Fabio Bagarello,et al.  Deformed Canonical (anti‐)commutation relations and non‐self‐adjoint hamiltonians , 2015 .

[33]  D. A. Trifonov,et al.  Fermionic coherent states for pseudo-Hermitian two-level systems , 2006, quant-ph/0608177.

[34]  Erwin Schrödinger International,et al.  Supported by the Austrian Federal Ministry of Education, Science and Culture , 1689 .

[35]  Charles F. Miller,et al.  Combinatorial Group Theory , 2002 .