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  • The Dirac equation arose out of the need for a quantum mechanical wave equation that treats position and time on an equal basis, just as relativity expects them to be. Now the Schrodinger equation is second order in space and first order in time rendering it non-relativistic.

  • One possibility for converting the non-relativistic Schrödinger equation for a free particle to a relativistic equation is to replace the classical energy $H=\frac{p^2}{2m}$by the relativistic energy $H=\sqrt{\mathbf p^2c^2+m^2c^4}$ and replacing $\mathbf p \rightarrow -i \hbar\nabla.$

  • Dirac’s solution was essentially to turn the problem on its head. Rather than trying to find the square root of an existing quantity, he postulated that the quantity inside the square root is the perfect square of an expression that is linear in the momentum, so that both space and time are involved as first order. $$H = -i \vec \alpha.\mathbf p + \beta m$$

  • My question is could we have also chosen a scenario where we had both space and time derivatives of second order instead of linear equations? Apart from possibly making the life complex, would there be any physical considerations not allowing this formulation?

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The scenario in which both time and space derivatives where second order was firstly studied by Klein-Gordon. The Klein-Gordon equation was in fact a first try at a relativistic quantum theory, but it had some problems. One of those problems, which really bothered Dirac, was the fact that the probability given by the wavefunction wasn't positive definite. The current density associated to the Klein-Gordon field is in fact given by $$\phi^\dagger \overset{\leftrightarrow}{\partial_\mu}\phi = \phi^\dagger\partial_\mu\phi-\phi\partial_\mu\phi^\dagger$$ which is easy to see how it could become negative.

Beaside this, there was the problem of negative energy solution, which was not really solved by Dirac, but it didn't bother him very much, his solution to that was the famous Dirac sea, which now is not a widely accepted idea and we look more at the Feynman solution to that.

Anyway, Dirac thought that to solve the problem in the not positive definite probability, was to treat both time and space on the same footing as first order quantities. This in fact solved the trick.

You can find this and many more interesting historical facts in the first chapter of Weinberg, volume one.

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  • $\begingroup$ Thanks, I have read the history and why did we need an alternative formulation from KG equation. My question is why only linear order in space and time? Why not second order? It could be difficult and so on, but is there a physical reasoning to avoid higher orders? $\endgroup$ – Abhay Hegde Apr 26 '20 at 7:53
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    $\begingroup$ The second order solution is KG's one. Why not third, fourth, and so on order? Surely because it does not make any physical sense since the energy-momentum relation is somewhat linear in both energy and momentum $\endgroup$ – Davide Morgante Apr 26 '20 at 8:03
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The Klein-Gordon equation does not have a meaningful probability interpretation. The problem goes further than the generation of negative probabilities, and Dirac was convinced that a first order equation was necessary. Although there is no record that Dirac had proved this, he had almost certainly played around with the formulae to convince himself that it was true. It can now be proven as a simple corollary of Stone's theorem, proven in 1932, after establishing that the probability interpretation means that time evolution is unitary and satisfies the conditions for Stone's theorem.

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