# Dirac Relativistic Wave Equation

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1. Aug 10, 2015

### John Rampton

I would like people's opinions on why the negative energy solutions of Dirac's Relativistic Wave equation were simply ignored in 1934 to make things fit. Another related question is with the energy conservation laws as they stand. Why in pair production from a photon at 1.022MeV forming a positron and electron each of 0.511MeV does the angular momentum component of each created particle (which also has an energy contribution) simply get ignored to make things fit energy conservation

2. Aug 10, 2015

### stevendaryl

Staff Emeritus
I wouldn't say that the negative energy solutions of the Dirac equation were ignored. Dirac took those solutions pretty seriously, and used them to predict the existence of a positron. Originally, Dirac's interpretation involved the assumption that what we call "vacuum" is actually filled with negative-energy electrons (that is, he assumed that all negative-energy states were filled). In his interpretation, a high-energy photon of energy $2 m_e c^2$ ($m_e$ is the mass of an electron) can cause an negative-energy electron with energy $-m_e c^2$ to become a positive-energy electron with energy $+m_e c^2$. This would produce two things: a "hole" in the negative energy states, and a positive-energy electron. Dirac argued that a "hole" in the negative-energy states would look like a positively charged particle, the positron. So raising the energy of the negative-energy electron would appear to produce an electron/positron pair.

This framework was enormously successful, although clunky, with its unobservable "sea" of negative-energy electrons. But it led to a more elegant field-theoretic view that eventually became QED (quantum electrodynamics).

Dirac's idea of "holes" in an otherwise filled set of energy states appearing like positively charged particles is still used in solid-state physics, where the filled states form the "Fermi sea", rather than the vacuum.

3. Aug 10, 2015

### John Rampton

I don't understand why the spin angular momentum component and contribution to energy equation, in pairs of particles created in pair production, seem to be conveniently ignored

4. Aug 10, 2015

### stevendaryl

Staff Emeritus
It's obviously a conspiracy by the military/industrial/physics complex.

What do you mean? The Dirac equation is a theory (or part of Dirac's theory of electrons). A theory is a guess about the way things work. It's an educated guess, and a guess that is empirically testable. But it's a guess. It could be proved wrong. Dirac hypothesized that the total energy $E$ of a free electron with wave function $\psi$ is given by:

$E \psi = (-i \hbar \nabla \cdot \alpha + \beta m c^2) \psi$

There is no specific term corresponding to the energy due to spin angular momentum, but there is no experimental suggestion that such a term is needed. Because every electron has the same magnitude for spin angular momentum, such a term would make a constant difference, and would be unobservable.

5. Aug 10, 2015

### Teek

I only just started reading about QFT and partially the Dirac Equation, but I thought that in a wave solution of the Klein Gordon (which is also a solution of Dirac Eq.), in the complex exponential of the Lorentz invariant quantity pμxμ, the negative energy was accounted for by Feynman by reversing the direction of time such that (-E)(-t) = Et and it corresponded to an antiparticle. When time is reversed the spin and all other momenta change sign, right?

6. Aug 10, 2015

### John Rampton

Thanks for your replies but I don't agree or else there are other things to be answered here.

If you say -E(-t) = E(t) then how can this be. Energy is a scalar quantity so should have no direction which would seem to be implicit with the equation quoted, Energy reverses with negative time

Also this may contravene the 2nd law of thermodynamics

Replies appreciated I know this is a difficult concept