One of the Most Ignored Equations in Physics

We have all heard of the Schrodinger, Dirac and Klein Gordon equations and these equations are extensively used in modern physics. In addition, almost every book on non-relativistic and relativistic quantum mechanics discusses these equations. There is however one equation that is rarely discussed in contemporary modern physics literature and that is the Levy-Leblond equation [1]. The only well known book that, in my knowledge, introduces this equation is Greiner's advanced quantum mechanics book. In this article, I intend to highlight how important this equation can be in contemporary physics.

I claim this because I did not know about this equation as well until I discovered it myself [2]. During my PhD, I went on a quest to discover new ideas and this equation was one of the ideas I discovered and got the paper published as well. It was after around two years of publishing this idea and a follow up paper when I came to know that this equation was proposed around 50 years ago by Prof. Levy-Leblond. My approach in the paper, however, was quite different and, in particular, I solved a well known problem in quantum mechanics namely the step potential problem which was not done by Prof. Levy-Leblond in his original paper. More interestingly, the editor of the journal I published in was a well known Nobel laureate in physics. One of the referees was so excited about the equation that he/she called it something new after a 100 years. I contacted several well known physicists and they all suggested new insights that I can get from these equations but none of them knew about the Levy-Leblond equation.

After finding out about Levy-Leblond's equation I contacted Prof. Levy-Leblond and he acknowledged in his reply that independent discoveries are always made in science. He was also amazed that the editor and the referees did not know about this equation. This experience taught me the political dynamics of the physics community and how the community can have its own biases. Its amazing that equations such as the Schrodinger, Dirac and Klein Gordon equations are so popular and frequently appear in lists of beautiful equations in physics, whereas the Levy-Leblond equation is rarely mentioned. In my view, I think one of the reasons physicists and in particular particle physicist ignored it was that it is not Lorentz invariant.

The Levy-Leblond equation introduces spin in the non-relativistic limit and can lead to several new insights as I have shown in my papers. It is the analog of Dirac equation for non-relativistic quantum mechanics. I believe that this equation may be useful in contemporary fields like spintronics which focus on manipulating the spin of non-relativistic electrons. I have continued to work on this equation [3,4,5] and keep coming across referees who do not know about this equation and think that the calculations have already been explored without giving any strong references. I hope that this article brings attention to this beautiful equation as well and we as physicists learn to avoid such mistakes in the future! 

[1] https://projecteuclid.org/euclid.cmp/1103840281

[2] https://link.springer.com/article/10.1007%2Fs10701-015-9944-z

[3] http://iopscience.iop.org/article/10.1088/0256-307X/34/5/050301/meta

[4] http://www.ijqf.org/archives/3574

[5] https://arxiv.org/abs/1705.10409