In quantum physics, the choice of basis is crucial for formulation. The generalization of the Pauli matrices via the Kronecker product, known as Pauli strings, is typically restricted to 2 ⁿ ‐dimensional systems. This paper explores extending this generalization to n ‐dimensional systems, where n is an integer, n > 2, to construct n × n ‐Kronecker–Pauli matrices. We establish a method for constructing a set of Kronecker–Pauli matrices for any dimension by examining the method for constructing the specific cases of 3 × 3 and 5 × 5 Kronecker–Pauli matrices. Another possible method for constructing a set of Kronecker–Pauli matrices, by using the Weyl operators, is discussed.

In quantum physics, the choice of basis is crucial for formulation. The generalization of the Pauli matrices via the Kronecker product, known as Pauli strings, is typically restricted to \(2^{n}\) dimensional systems. This paper explores extending this generalization to \(N\)-dimensional systems, where \(N\) is a prime integer, to construct \(N \times N\)-Kronecker-Pauli matrices. We begin by examining the specific cases of \(3 \times 3\) and \(5 \times 5\) Kronecker-Pauli matrices, with the goal of the purpose constructing a set of \(N \times N\)-Kronecker-Pauli matrices for any prime integer \(N\). Another possible method for constructing a set of Kronecker-Pauli matrices is discussed.

A sign operator of energy, analogous to the helicity operator, but in the direction of what we call energy vector has been introduced. It is possible that there may be physical phenomena where energy vector should be considered. However, to write a wave function this energy vector needs a time vector. But, unlike the energy vector the time vector has no physical meaning yet. To make physical senses of the components of the time vector, the time dilation in special relativity has been studied and also the components of the time vector have been related to the tunneling times when an electron crosses a potential barrier. Physical results for quantum tunneling time will not be limited to this study.

This work represents the expression of the eigenvalues, eigenstates of an hamiltonian operator obtained from the Dirac equation and the application of the eigenstates of this hamiltonian for formulating the transition probability of two flavor neutrino oscillations.

Within the framework of quantum theory, we review the Chrestenson operator, the Weyl operator basis, and the Kronecker-Pauli operator basis in $d$-dimensional Hilbert spaces using Dirac notation, where $d$ is a prime integer strictly greater than 2. We establish a new algebraic relation connecting these operators and present the cases $d=3$ and $d=5$ as illustrative examples.

In quantum mechanics, classical matrix bases such as the Pauli matrices are often generalized to higher dimensions. So, it is useful to express their corresponding operators using the Dirac Bra and Ket. In this paper, to express the corresponding operators we review the Kronecker-Pauli matrices and how to construct them for an N-dimensional system, with N a prime integer, N>2. Then, we give the expression of the Kronecker-Pauli operators and show that their matrices with respect to the standard basis fulfill the conditions to form a set of Kronecker-Pauli matrices. Relationship between the Kronecker-Pauli operators and the Weyl operators has been studied.