Topological decompositions of the Pauli group and their influence on dynamical systems. (English) Zbl 1472.57027
In this article, we will see that it is possible to realize a manifold which has the Pauli group \(P = \langle X, Y, Z \mid X^2 = Y^2 = Z^2 = 1, (YZ)^4 = (ZX)^4 = (XY)^4 = 1 \rangle\) as its fundamental group. The Pauli group \(P\) is a group of order \(16\) introduced by [W. Pauli jun., Z. Phys. 43, 601–623 (1927; JFM 53.0858.02)]. Specific expressions of the three generators are:
\[
X = \begin{bmatrix} 0&1 \\ 1&0 \end{bmatrix}, \ Y = \begin{bmatrix} 0&-i\\ i&0 \end{bmatrix}\text{ and } Z = \begin{bmatrix} 1&0\\ 0&-1 \end{bmatrix}.
\]
It is easy to check that \(X^2 = Y^2 = Z^2 = I\), where \(I\) is the identity matrix.
The authors’ main result is the theorem below. Note that \(S^3\) indicates the three-sphere. Further, the quaternion group and a cyclic group of order four, denoted by \(\mathbb Q_8\) and \(\mathbb Z_4\) respectively, operate on \(S^3\). Both finite actions are isometries on \(S^3\), hence the quotient spaces \(U = S^3/Q_8\) and \(V = S^3/\mathbb Z_4\) are both elliptic three-manifolds. More specifically, \(U\) is a quaternion manifold which is a family of the prism manifolds and \(V\) is a lens space \(L(4, 1)\).
Theorem. There exist two compact path connected quotient spaces \(U = S^3/Q_8\) and \(V = S^3/\mathbb Z_4\) such that the following conditions are true:
In the first statement, the familiar assumptions of the Seifert and van Kampen Theorem are satisfied. For example, [C. Kosniowski, A first course in algebraic topology. (1980; Zbl 0441.55001), Chapter 23] is a handy reference source regarding the theorem we are using here. In fact, its proof is a repeated use of the Seifert and van Kampen Theorem involving algebraic manipulations of elements in \(\pi_1(U) \cong Q_8\), \(\pi_1(V) \cong \mathbb Z_4\) and \(\pi_1(U \cap V) = \mathbb Z_2\) in order to obtain the appropriate relations.
Moreover, the article discusses connections with physics or quantum mechanics, with an emphasis on pseudo-bosons and pseudo-fermions.
It is easy to check that \(X^2 = Y^2 = Z^2 = I\), where \(I\) is the identity matrix.
The authors’ main result is the theorem below. Note that \(S^3\) indicates the three-sphere. Further, the quaternion group and a cyclic group of order four, denoted by \(\mathbb Q_8\) and \(\mathbb Z_4\) respectively, operate on \(S^3\). Both finite actions are isometries on \(S^3\), hence the quotient spaces \(U = S^3/Q_8\) and \(V = S^3/\mathbb Z_4\) are both elliptic three-manifolds. More specifically, \(U\) is a quaternion manifold which is a family of the prism manifolds and \(V\) is a lens space \(L(4, 1)\).
Theorem. There exist two compact path connected quotient spaces \(U = S^3/Q_8\) and \(V = S^3/\mathbb Z_4\) such that the following conditions are true:
- 1.
- \(U \cup V\) is a compact path connected space with \(U \cap V \neq \emptyset\), \(\pi_1(U \cap V) \cong \mathbb Z_2\) and \(P \cong \pi_1(U \cup V) / N\) for some normal subgroup \(N\) of \(\pi_1(U \cup V)\).
- 2.
- \(U \# V\) is a Riemannian manifold of \(dim(U \# V) = 3\) and \(P \cong \pi_1(U \# V) / L\) for some normal subgroup \(L\) of \(\pi_1(U \# V)\), where \(\#\) denotes the usual connected sum of manifolds.
In the first statement, the familiar assumptions of the Seifert and van Kampen Theorem are satisfied. For example, [C. Kosniowski, A first course in algebraic topology. (1980; Zbl 0441.55001), Chapter 23] is a handy reference source regarding the theorem we are using here. In fact, its proof is a repeated use of the Seifert and van Kampen Theorem involving algebraic manipulations of elements in \(\pi_1(U) \cong Q_8\), \(\pi_1(V) \cong \mathbb Z_4\) and \(\pi_1(U \cap V) = \mathbb Z_2\) in order to obtain the appropriate relations.
Moreover, the article discusses connections with physics or quantum mechanics, with an emphasis on pseudo-bosons and pseudo-fermions.
Reviewer: Ryo Ohashi (Wilkes-Barre)
MSC:
| 57M07 | Topological methods in group theory |
| 57M60 | Group actions on manifolds and cell complexes in low dimensions |
| 81R05 | Finite-dimensional groups and algebras motivated by physics and their representations |
| 22E70 | Applications of Lie groups to the sciences; explicit representations |
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