The well-known Schrödinger and Dirac time evolution equations in quantum mechanics (QM) both require the Pauli spin matrices (σx, σy, σz) to produce accurate results for probabilities of real-world events. These Pauli spin matrices reveal the eight vertices of a cube. In experiments, QM equations with the Pauli spinors may represent physical dynamics in a three-dimensional Euclidian space. Thus, the Pauli spinors in fact define a cubic lattice. In sum, QM equations assume a cubic lattice is a fundamental feature of physical systems. Consequences of this QM assumption have been explored in binary mechanics, documenting a transition from partial, incomplete QM to full, complete QM.
Using the full quantizaion postulates of binary mechanics (BM), energy was quantized at a new microcopic level and spatial structures were defined [1] (Fig.2).
Three spot units may be assembled along the axes of an absolute reference frame to form a spot. Finally, eight spots form the spot cube spatial object based on the eight possible permutations itemized in Fig. 3 (from Table 1 in [3]). The bit function B (eq. 2 in [1]) replaces the QM wave function %psi;.
Pauli Spin Matrices
As presented previously [3], three pairs of equations establish that the Pauli matrices define a cubic structure in space.
First, σx inverts Ji parity in spatial dimensions i,
Bc(I1K)i = σxBc(I0K)i (2)
Second, σy inverts Ki parity in spatial dimensions i,
Bc(IJ1)i = σyBc(IJ0)i (4)
Third, σz inverts Ii parity along any spot unit axis i in S,
Bc(1JK)i = σzBc(0JK)i (6)
Discussion
Since each Pauli matrix is associated with one of three spatial dimensions x, y and z, Eqs. 1 - 6 establish that the Pauli matrices define the eight vertices of a cubic spatial object.
Each Pauli matrix defines a [0,1] position swap. The position swaps in Eqs. 1- 6 occur when the complex representation Bc of a spot unit is premultiplied by a Pauli matrix. In effect, these equations change focus from one spot to another (columns in Fig. 3). There may be substantial scientific significance of this fact since the eight spots map to eight elementary particles (Fig. 2). Thus, while the electron may be the key player in many experiments, there may also be participation of other elementary particles in the observed effects. Is this the most significant role of the Pauli matrices in the Schrödinger and Dirac time evolution equations? The present analysis suggests that other particles may participate in production of experimental results.
Pauli spin matrices (σx, σy, σz) are three complex, Hermitian, and unitary matrices. The Pauli matrices are often used in QM literature to represent the spin-1/2 operators. In addition in this article, the Pauli matrices were found to specify a cubic spatial structure.
As discussed previously [4], the Pauli spin matrix set in the Schrödinger and Dirac equations have produced excellent results, say, for electron behavior. Physics literature presents a variety of somewhat cryptic explanations of why use of these matrices works to produce results as predicted.
In outdated QM literature, the time development equations might wrongly portray complex interactions among the eight BM elementary particles represented in the spot cube as the behavior of a single particle type, such as the electron.
If this is true, the bottom line is that QM is simply unable to represent correctly events at the BM level of microscopic fineness. This possible QM fail is a major limitation for continued usage of QM formalism in scientific and commercial applications. In contrast, BM formalism (eqs. 1-17 in [1]) provides an unprecedented means to achieve substantially greater accuracy and resolution in physics experiments and in commercial design and engineering tasks.
References
[1] Keene, J. J. "Binary mechanics postulates" JBinMech November, 2020.
[2] Keene, J. J. "How to derive the primary and secondary physical constants" JBinMech March, 2025.
[3] Keene, J. J. "Binary mechanics" JBinMech July, 2010.
[4] Keene, J. J. "Quantum technology advance" JBinMech September, 2025.
© 2026 James J Keene
