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How Dirac's Seminal Contributions Pave The How Dirac’s Seminal Contributions Pave the Way for Comprehending Nature’s Deeper Designs Mani L. Bhaumik Department of Physics, University of California, Los Angeles. E-mail: [email protected] Editors: Zvi Bern & Danko Georgiev Article history: Submitted on October 27, 2019; Accepted on December 6, 2019; Published on December 17, 2019. redible reasons are presented to reveal that was the totally unexpected existence of anti-particles, many of the lingering century old enigmas, sur- culminating in the eventual development of the quan- Crounding the behavior of at least an individual tum field theory of the Standard Model that reveals quantum particle, can be comprehended in terms of the deepest secrets of the universe known to date. an objectively real specific wave function. This wave Quanta 2019; 8: 88–100. function is gleaned from the single particle energy- momentum eigenstate offered by the theory of space filling universal quantum fields that is an inevitable 1 Introduction outcome of Dirac’s pioneering masterpiece. Exam- ples of these well-known enigmas are wave particle Just after getting his Ph.D. in 1926, the same year in duality, the de Broglie hypothesis, the uncertainty which physics started to advance at breakneck speed fol- principle, wave function collapse, and predictions of lowing the successful formulation of quantum physics measurement outcomes in terms of probability in- to resolve the perplexities of the atomic domain, Paul stead of certainty. Paul Dirac successfully incorpo- Dirac entered the field of research as a freshly minted rated special theory of relativity into quantum me- scientific prodigy. Within a mere couple of years, he chanics for the first time. This was accomplished fashioned an elegant equation, ever to be known as the through his ingenious use of matrices that allowed iconic Dirac equation of the electron. In contrast to exten- the equations of motion to maintain the necessary sive earlier efforts, his ingenious use of a special type of first order time derivative feature necessary for posi- non-commutative 4 × 4 matrices allowed the relativistic tive probability density. The ensuing Dirac equation quantum mechanical equations of motion to maintain the for the electron led to the recognition of the mystify- sought after hallmark first order time derivative charac- ing quantized spin and magnetic moment as intrinsic teristic of the Schrodinger¨ nonrelativistic wave equation. properties in contrast to earlier ad hoc assumptions. The solution of his relativistic equation for the hydro- gen atom produced results in perfect agreement with This is an open access article distributed under the terms experimental data available at the time. The most far of the Creative Commons Attribution License CC-BY-3.0, which permits unrestricted use, distribution, and reproduction in any medium, reaching prediction of the celebrated Dirac equation provided the original author and source are credited. Quanta j DOI: 10.12743/quanta.v8i1.96 December 2019 j Volume 8 j Issue 1 j Page 88 This was a brilliant scheme for successfully combining mentum operators, the special theory of relativity with quantum mechan- @ @ ics to explain the behavior of electrons more accurately, E = {~ ; p = −{~ (2) avoiding the various problems encountered in earlier in- @t @x vestigations. ensuing in First we present an extended overview of the emer- @2 @2 gence of Dirac’s seminal equation, which had a profound − ~2 = m2c4 − ~2c2 : (3) 2 0 2 impact for a large variety of topics including, most im- @t @x portantly, that seeded the eventual development of the For a wave function , dividing equation (3) by ~2c2 quantum field theory of the Standard Model of particle and rearranging the terms, the Klein–Gordon equation physics. We will then focus on how the principles of becomes the quantum field theory can be utilized for successfully 1 @2 m2c2 uncovering the century old enigmas surrounding quantum − r2 + 0 = 0: (4) physics. c2 @t2 ~2 In order to fully appreciate Dirac’s momentous contri- The equation (4) can be written in a graceful form bution, it would be helpful to briefly portray the particu- larly confounding circumstances of the period following + µ2 = 0 (5) the epochal unveiling of the Schrodinger’s¨ quantum me- m0c chanical wave equation. As early as in 1887, while trying where µ = ~ and is the d’Alembert operator, to find a precise standard of length for use in their in- 1 @2 tended detection of ether drift, Albert A. Michelson and = − r2: (6) 2 2 Edward W. Morley [1] observed that the red hydrogen c @t spectral line is actually a closely spaced double line. The Soon the equation became the subject of numerous pa- accuracy of measurement was so impressive that Michel- pers and was considered by many to be the correct and son received the 1907 physics Nobel Prize for precision natural generalization of the Schrodinger¨ equation. How- metrology and spectroscopy—and not, as might be com- ever, although it had a mathematical and aesthetic appeal, monly assumed, for the celebrated null result for ether before long the limitation of its range of applicability be- drift that eventually provided one of the underpinnings came clear. To begin with, just like Schrodinger’s¨ earlier for Einstein’s formulation of special relativity. unpublished work, the equation failed to reproduce Som- The remarkable effectiveness of incorporating the spe- merfeld’s formula for the hydrogen spectrum that was in cial theory of relativity by Arnold Sommerfeld into the exact agreement with experiments. Possibly one of the old quantum theory, before Schrodinger,¨ was already evi- biggest disappointments with the Klein–Gordon equation dent in providing a more accurate agreement with exper- was its failure to account for the spin of the electron, as imental results of the hydrogen spectrum. Hence it was was also true of the nonrelativistic Schrodinger¨ equation. natural to seek a relativistic quantum mechanical wave However, it has been revived for use in the quantum field equation. In fact, in his own attempt to formulate the theory of spin 0 particles such as the Higgs boson. quantum mechanical wave equation, Schrodinger¨ first Even before the emergence of the brave new world constructed a relativistic equation. He did not, however, of contemporary quantum physics, some experimental publish this formulation since it did not agree with exper- observations were very baffling and required the intro- imental results. duction of electron spin to make sense of them. These are exemplified by the Stern–Gerlach experiment, the anomalous Zeeman effect, and Wolfgang Pauli’s need for 2 The Klein–Gordon Equation two additional quantum numbers to complete his Pauli exclusion principle, which provides a natural explanation In 1926, a number of authors independently formulated a for the periodic table of atoms. relativistic wave equation, although the priority in publi- In 1922, Otto Stern and Walther Gerlach conducted cation belongs to Oscar Klein [2]. It is known today as a rather remarkable experiment [3] to test the Bohr– the Klein–Gordon equation. Sommerfeld model of the atom of the old quantum theory. The Klein–Gordon equation is derived using the rela- A beam of silver atoms having a single electron in their tivistic mass-energy equation outer shell was passed through a spatially varying mag- 2 2 4 2 2 netic field, which deflected them before they struck a E = m0c + p c (1) detector screen. Stern and Gerlach found that the elec- and drawing on the quantum mechanical energy and mo- trons were deflected discretely into only two clusters, a Quanta j DOI: 10.12743/quanta.v8i1.96 December 2019 j Volume 8 j Issue 1 j Page 89 spectacular evidence of space quantization in an atom, Pauli also presented a modified Schrodinger¨ equation 1 supporting the evolving quantum paradigm at the time. for spin 2 particles taking into account the interaction Niels Bohr was so startled that he personally wrote to of the particle’s spin magnetic moment with an external Gerlach magnetic field caused by the orbital motion of the elec- tron [7]. This was a somewhat ad hoc modification to the I would be grateful if you or Stern could let Schrodinger¨ equation to explain the existence of a doublet me know, in a few lines, whether you interpret of additional energy levels attributable to the presence your experimental results in this way that the of spin. The natural occurrence of the electron spin and atoms are oriented only parallel or opposed, but its magnetic moment still remained shrouded in a great not normal to the field, as one could provide mystery and the scientific community continued to devote theoretical reasons for the latter assertion. [4] significant efforts to find a cogent explanation of the co- nundrum. The relativistic Klein–Gordon equation offered Furthermore, in some experiments with atoms in a mag- no help either. netic field, spectral lines were observed to split into four, Perhaps the major concern with the Klein–Gordon six, or even more lines and some triplets showed wider equation was that the probability density given by the spacing than expected. These deviations were labeled equation was not positive-definite. Apparently Dirac was “anomalous Zeeman effect” and were completely incom- the first to realize the problem with the probability inter- prehensible to early investigators. pretation for equations with second-order time derivatives. Meanwhile Pauli devised the Pauli Exclusion Principle Also, because special relativity requires treating time and in 1925. The principle states that no two electrons can space on equal footing, Dirac reasoned that the equation share the same quantum state at the same time [5]. This has to be first order not only in time derivatives, but also in means that no two electrons in a single atom can have spatial derivatives.
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