# Dirac's equation and anti-matter

1. Aug 24, 2013

### Salamon

Can someone explain how Dirac was able to deduce that anti-matter exists? How did this follow naturally from Dirac's equation? Did Dirac have to derive his equation or was it just an empirical law of nature like Newton's gravity or Einstein's Field equations?

2. Aug 25, 2013

### tom.stoer

Dirac's intention was to find a relativistic generalization of the Schrödinger equation which is first order in time derivative. In order to do that he had to introduce new 4-component objects called spinors. At his time this was pioneering guess-work. Today I would say that the Dirac equation follows (almost) uniquely from symmetry considerations, i.e. from the requirement of a Lorentz-covariant wave equation for spin 1/2 fields. Einstein's field equations (GR) a not required, SR is sufficient.

Antimatter followed from the equation

$E^2 = (mc^2)^2 + (pc)^2$

which has two roots, i.e. allowes for both positive and negative energy solutions. In addition some handwaving arguments like the Dirac sea, absence of an electron with negative energy equals presence of a positron with positive energy etc. is required. Today the framework of QFT is much more satisfactory to deal with the Dirac equation and antimatter, however one cannot fully avoid the Dirac sea which appears in normal ordering (regularization).

3. Aug 25, 2013

### mpv_plate

Doesn't antimatter follow from the 4-component spinor? 2 components are for matter, the other 2 for antimatter.

4. Aug 25, 2013

### tom.stoer

That's a specific representation for Dirac-spinors. But antimatter exists for scalar fields as well, therefore its existence does not require antimatter.