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The Dirac equation is a relativistic wave equation that describes all spin ½ particles, such as electrons and quarks and is consistent with both the principles of quantum mechanics and the theory of special relativity. In order to combine these two theories, Dirac developed a set of 4 X 4 matrices, known as the Dirac matrices, that satisfied certain mathematical conditions allowing a solution to the Dirac equation. In the paper, the author extends these 4 X 4 matrices into an n-dimensional representation and then uses these to develop a new n-dimensional operator whose properties may have possible applications in quantum physics.
“Higher-dimensional matrices are already used in relativistically invariant wave equations in arbitrary space-time dimensions, notably in superstring theory. Our physical space is observed to have three spatial dimensions and, along with time, is a four-dimensional continuum known as space time. However, nothing prevents a theory from including more than 4 dimensions, indeed, string theory generally requires space time to have 10 dimensions.”, said Darbyshire.
Since the groups generated by these higher-dimensional matrices are all the same, it is possible to look for similarity transformations that connects them. This transformation is generated by a respective charge conjugation matrix. C-symmetry refers to the symmetry of physical laws under a charge conjugation transformation in which electromagnetism, gravity and the strong interaction all obey, but weak interactions violate. Interestingly, this relates back to the mathematical analysis used in the paper for the development of the new n-dimensional operator.
“Further work needs to be developed on the operator to uncover more mathematical properties that relate to its symmetry, but our initial results show some exciting potential applications; some beyond quantum physics”, said Darbyshire.
Dr P. M. Darbyshire is Technical Director, Computational Biophysics Group, Algenet Cancer Research, Nottingham. UK.
Extending the 4 X 4 Darbyshire Operator Using n-Dimensional Dirac Matrices