Locality of Dirac Field Operators

Proof:  Using the identity (true in a particular basis, so true in any basis),
and (from the solution of the Dirac Equation)
we find that

Similarly,

We have
where ^T denotes that α and β are transposed. Using the resolution of unity and the solution of the Dirac equation,
Likewise for the antiparticle,
Substituting p → −p at x⁰ = y⁰
Adding at x⁰ = y⁰ gives the equal time anticommutator,
As required.


Proof:  From the above formulae,
and
The anticommutator is found by adding
using the generalised scaling property of the delta function applied to the mass shell condition. The integral is Lorentz invariant and is zero when x⁰ − y⁰ = 0. We conclude that it is zero whenever x − y is spacelike.

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Conserved Current

Proof of Lemma:
Take the Hermitian conjugate and postmultiply by γ⁰,
So, by commuting the terms,

Proof of theorem:  We have, from Ehrenfest’s theorem, that the expectation of the current density obeys
The first term is zero by the lemma. From the solution of the Dirac equation, we have that
Taking the Hermitian conjugate, post multiplying by γ⁰, and using the conjugacy relation, γ⁰γ^a†γ⁰ = γ^a gives
Similarly
and
Current density is given by
Normal ordering transposes the creation and annihilation operators, but not the spin indices. Expanding the creation and annihilation operators in terms of momentum and differentiating the particle term gives,
The other terms are found to be zero in the same way.

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Maxwell’s Equation’s

Proof:  We have,
Hence,

The first term gives the electrostatic equation
The last three terms give
These equations are summarised in the Ampère-Maxwell law

From the definition of the Faraday tensor,
we have the identity,
a, b, c = 0, 2, 3 gives
a, b, c = 0, 3, 1 gives
a, b, c = 0, 1, 2 gives
These three equations are summarised in Faraday’s law,
a, b, c = 1, 2, 3 gives the magnetostatic law,
Thus, in a treatment based on the magnetic laws are mathematical identities, not physical laws.

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