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Monday, November 26, 2007

Dirac's equation

Why does Dirac’s equation hold? Despite an all too prevalent belief it is not some strange property of nature. It is a trivial property of geometry. Considering only space transformations, ignoring interactions and internal symmetry, objects (thus free) belong to states of the Poincaré group. This has two invariants (like the rotation group has one, the total angular momentum). For a massive object these are the mass and spin in the rest frame. Knowing these the object is completely determined. Thus two equations, not one, are needed to determine an object. For spin- 1/2 , only, these two can be replaced by one, Dirac’s equation. Why is this? The momentum, p_µ is a four-vector. There is another four-vector, g_ µ. Thus µpµ is an invariant. It is a property of the object, and we give that property the name mass. Thus g_µp_µ = m, (1) which is Dirac’s equation. It gives the mass of the object, and the spin, 1/2 . This is only possible because of the g_µ’s. These form a Clifford algebra and there is (up to inversions) only one for each dimension. This is then the reason for Dirac’s equation, and only for a single spin.1/2

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