QuantumPhysics.dvi

The lowest dimensional representations play an ubiquitousrole in physics, (0,0) dim = 1 real scalar ( 1 2 ,0) dim = 2 complex le ...

22 The Dirac Field and the Dirac Equation Among the relativistic wave equations, we have not yet obtained one suitable for spin ...

whereIis the unit matrix in the Dirac representation space. Given the Dirac matrices, which we shall realize shortly in explicit ...

22.2 Explicit representation of the Dirac algebra From the definition ofγ^5 , we have, tr(γ^5 ) =−itr(γ^0 γ^1 γ^2 γ^3 ). By usin ...

The generators of the Lorentz algebra in this basis are givenby S 0 i = 1 2 γ 0 γi=− 1 2 γ^0 γi= 1 2 ( −σi 0 0 σi ) Sij = Sij= 1 ...

The relation, as it was derived above, holds for Λ close to theidentity. Given the overall covariance of the tensors of the last ...

We begin by checking that the free Dirac equation is indeed Lorentz invariant. We shall show that ( γμ∂′μ−m ) ψ′(x′) =D(Λ) (γμ∂μ ...

the free Dirac equation reads, ( −m σμ∂μ ̄σμ∂μ −m )( ψL ψR ) = 0 (22.41) Separating the two components, σμ∂μψR−mψL = 0 σ ̄μ∂μψL− ...

Eliminating one or the other components, gives the standardenergy-momentum-mass relation, (k^20 −~k^2 −m^2 )uL,R(~k) = 0. To sol ...

To calculate the combination−γ^0 D(Λ)†γ^0 , we take Λ infinitesimally close to the identity, so that D(Λ) = I+ 1 2 ωμνSμν+O(ω^2 ...

22.7 The free Dirac action and Hamiltonian The field equations (γμ∂μ−m)ψ= 0 may be derived from an action principle. The action ...

The key property is that thegauge covariant derivativeDμ=∂μ+iqAμofψtransforms exactly asψdoes under gauge transformations. Indee ...

23 Quantization of the Dirac Field We proceed to quantizing first the free Dirac field, and shall then motivate the quantization ...

of time”, instead of reversing the sign of the energy. Let us see how this would work in practice. Creating a particle with posi ...

The correct interpretation is that these “negative energy solutions” actually describe the anti- particle of the electron, i.e. ...

It will be convenient to parametrize the corresponding solutions in a basis where the chirality matrix is diagonal. In this basi ...

The calculation of the first line, for example, proceeds as follows, ̄ur(k)us(k) =u†(k)γ^0 vs(k) = ( ξr†( √ k·σ) −iξr†( √ k·σ ̄) ...

Evaluating the time derivative, we find, i∂ 0 ψ= ∫ d^3 k (2π)^3 1 √ 2 k^0 ∑ s=1, 2 { k^0 us(k)bs(k)e−ik·x−k^0 vs(k)d†s(k)e+ik·x ...

Using the anti-commutation relations andbanddoscillators, we obtain contributions of either 0 orδs,s′(2π)^3 δ(~k−~k′). The expre ...

equation for potential scattering, and was solved there by the introduction of a small imaginary part. The same works here, and ...