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 | X2 = Y2 = Z2 = 1,(Y Z)4 = (ZX)4 = (XY )4 = 1〉 of order 16 as an appropriate quotient group of two distinct spaces of orbits of the three dimensional sphere S3. The first of these spaces of orbits is realized via an action of the quaternion group Q8 on S3; the second one via an action of the cyclic group of order four \(\mathbb {Z}(4)\) on S3. 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.


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F.B. acknowledges support from Palermo University and from the Gruppo Nazionale di Fisica Matematica of the I.N.d.A.M. The other two authors (Y.B. and F.G.R.) thank Shuttleworth Postgraduate Scholarship Programme 2019 and NRF for grants no. 118517 and 113144.


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The present appendix is meant to make the paper self-contained, by giving some essential definitions and results on pseudo-fermions. We consider two operators a and b, acting on the Hilbert space \({\mathscr{H}}=\Bbb C^{2}\), which satisfy the following rules, [6]:


where {x,y} = xy + yx is the anti-commutator between x and y. We first observe that a non zero vector φ0 exists in \({\mathscr{H}}\) such that aφ0 = 0. Similarly, a non zero vector Ψ0 exists in \({\mathscr{H}}\) such that \(b^{\dagger } {\Psi }_{0}=0\). This is because the kernels of a and b are non-trivial.

It is now possible to deduce the following results. We first introduce the following non zero vectors

$$ {\varphi}_1:=b {\varphi}_0,\quad {\Psi}_{1}=a^{\dagger} {\Psi}_{0}, $$

as well as the non self-adjoint operators

$$ N=ba,\quad N^{\dagger}=a^{\dagger} b^{\dagger}. $$

Of course, it makes no sense to consider \(b^{n} {\varphi }_{0}\) or \({\left (a^{\dag }\right )}^n {\Psi }_{0}\) for n ≥ 2, since all these vectors are automatically zero. This is analogous to what happens for ordinary fermions. Let now introduce the self-adjoint operators Sφ and SΨ via their action on a generic \(f\in {\mathscr{H}}\):

$$ S_{\varphi} f=\sum\limits_{n=0}^1\left<{\varphi}_{n},f\right> {\varphi}_{n}, \quad S_{\Psi} f=\sum\limits_{n=0}^{1}\left<{\Psi}_{n},f\right> {\Psi}_{n}. $$

The following results can be easily proved:

  • $$ a\varphi_{1}=\varphi_{0},\quad b^{\dagger}{\Psi}_{1}={\Psi}_0. $$
  • $$ N{\varphi}_{n}=n{\varphi}_{n},\quad N^{\dagger} {\Psi}_{n}=n{\Psi}_{n}, \ \text{for} \ n=0,1. $$
  • If the normalizations of φ0 and Ψ0 are chosen in such a way that \(\left <\varphi _{0},{\Psi }_{0}\right >=1\), then

    $$ \left<\varphi_k,{\Psi}_{n}\right>=\delta_{k,n}, \ \text{for} \ k,n=0,1. $$
  • Sφ and SΨ are bounded, strictly positive, self-adjoint, and invertible. They satisfy

    $$ \|S_{\varphi}\|\leqslant\|\varphi_0\|^2+\|\varphi_1\|^2, \quad \|S_{\Psi}\|\leqslant\|{\Psi}_0\|^2+\|{\Psi}_{1}\|^{2}, $$
    $$ S_{\varphi} {\Psi}_{n}={\varphi}_{n},\qquad S_{\Psi} {\varphi}_{n}={\Psi}_{n}, $$

    for n = 0, 1, as well as \(S_{\varphi }=S_{\Psi }^{-1}\) and the following intertwining relations

    $$ S_{\Psi} N=N^{\dagger} S_{\Psi},\qquad S_{\varphi} N^{\dagger} =N S_{\varphi}. $$

Notice that, being biorthogonal, the vectors of both \(\mathcal {F}_{\varphi }\) and \(\mathcal {F}_{\Psi }\) are linearly independent. Hence φ0 and φ1 are two linearly independent vectors in a two-dimensional Hilbert space, so that \(\mathcal {F}_{\varphi }\) is a basis for \({\mathscr{H}}\). The same argument obviously can be used for \(\mathcal {F}_{\Psi }\). More than this: both these sets are also Riesz bases. We refer to [5] for more details.

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Bagarello, F., Bavuma, Y. & Russo, F.G. Topological Decompositions of the Pauli Group and their Influence on Dynamical Systems. Math Phys Anal Geom 24, 16 (2021).

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  • Pauli groups
  • Actions of groups
  • Hamiltonians
  • Pseudo-fermions
  • Central products

Mathematics Subject Classification (2010)

  • Primary: 57M07
  • 57M60; Secondary: 81R05
  • 22E70