Quantum Mechanics for Mathematicians

46.5 Space-time symmetries The choice of Coulomb gauge nicely isolates the two physical degrees of freedom that describe photons ...

Taking Ĥ=^1 2 ∫ R^3 :(|Ê|^2 +|B̂|^2 ):d^3 x= 1 2 ∫ R^3 : ( | ∂ ∂t |^2 +|∇×Â|^2 ) :d^3 x = ∫ R^3 ωp(a† 1 (p)a 1 (p) +a† 2 (p ...

46.5.3 Rotations We will not go through the exercise of constructing the angular momentum operator̂Jfor the electromagnetic fiel ...

Maxwell’s equations become just the standard massless wave equation. Since ∇×B− ∂E ∂t =∇×∇×A+ ∂^2 A ∂t^2 − ∂ ∂t ∇A 0 =∇(∇·A)−∇^2 ...

The Lorenz gauge condition is needed to get Maxwell’s equations, but it cannot be imposed as an operator condition − ∂ 0 ∂t ...

One can take elements ofH 1 to beC^4 -valued functionsα= (α 0 (p),α(p)) onR^3 with Lorentz invariant indefinite inner product ...

This sort of covariant quantization method is often referred to as the “Gupta- Bleuler” method, and is described in more detail ...

gauge. For a general discussion of constrained Hamiltonian systems and their quantization, with details for the cases of the ele ...

Chapter 47 The Dirac Equation and Spin 1 2 Fields The space of solutions to the Klein-Gordon equation gives an irreducible repre ...

Cliff(3,1) is isomorphic to the algebraM(4,R) of 4 by 4 real matrices. Several conventional identifications of the generatorsγjw ...

The Dirac equation (γ 0 ∂ ∂x 0 +γ·∇−m)ψ(x) = 0 can be written in the form of a Schr ̈odinger equation as i ∂ ∂t ψ(t,x) =HDψ(t,x) ...

The four dimensional Fourier transform of a solution is of the form ψ ̃(p) =^1 (2π)^2 ∫ R^4 e−i(−p^0 x^0 +p·x)ψ(x)d^4 x =θ(p 0 ) ...

as follows γ 0 M= ( 0 −iσ 2 −iσ 2 0 ) , γ 1 M= ( σ 3 0 0 σ 3 ) γM 2 = ( 0 iσ 2 −iσ 2 0 ) , γM 3 = ( −σ 1 0 0 −σ 1 ) Quadratic co ...

theγ 0 γare symmetric, and the derivative is antisymmetric. Applying the finite dimensional theorem 30.1 in this infinite dimens ...

47.2.1 Majorana spinor fields in momentum space Recall that in the case of the real relativistic scalar field studied in chapter ...

u−( 0 )e−imt+u−( 0 )eimt=     0 cos(mt) −sin(mt) 0     correspond physically to a relativistic spin^12 particle of massm ...

space does not completely resolve the problem. Using theα+(p),α−(p) allows for an explicitly positive-definite inner product, wh ...

47.3 Weyl spinors For the casem= 0 of the Dirac equation, it turns out that there is an interesting operator acting on the space ...

( ∂ ∂t −σ·∇ ) ψL=imψR Whenm= 0 the equations decouple and one can consistently restrict atten- tion to just right-handed or left ...

where Λ is an element ofSpin(3,1) andS(Λ)∈SL(2,C) is the (^12 ,0) repre- sentation (see chapter 41, whereS(Λ) = Ω). Λ·xis theSpi ...