0.1 Klein-Gordon Equation 0.2 Dirac Equation

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0.7 Dirac Equation as a Field Equation Although the Dirac equation is somewhat sucessful as a one particle equation, the idea of a filled sea of negative energy states is bizarre, and is also not applicable to particles of integral spin that do not satisfy the exclusion principle. The integral spin particles also do not have a consistent quantum treatment with positive definite probability density. There were other difficulties such a the Klein paradox. Therefore the Dirac equation came to be regarded as a field equation to be subjected to a form of canonical quantization just like the scalar and electromagnetic fields. To realize this, we must have a Lagrangian for the Dirac equation. A Lagrangian density that serves is L = iψγ µ ∂ µ ψ − mψψ (17) It is easily verified that the equations of motion arising from this are the Dirac equation and its conjugate: i∂ µ ψγ µ + mψ = 0 When we come to construct the Hamiltonian, however, there are pathological elements in the calculation. For example, the momentum conjugate to ψ is identically zero, and the momentum conjugater to ψ is Π = ∂L ∂∂ 0 ψ = iψγ 0 Since the momentum conjugate to ψ is iψ † , a coordinate, we are dealing with a constrained system. Though there are sophisticated methods to deal with such constraints, we will follow a simple practical method and consider only ψ to be a coordinate in the Lagrangian. When we construct the Hamiltonian as H = Π∂ 0 ψ − L, We find that the resulting Hamiltonian does not involve time derivatives, and that these could not be eliminated anyway as in the previous examples because the canonical momentum does not contain time derivatives. Despite these peculiarities, the Hamiltonian density derived above leads to the Hamiltonian ∫ H = d 3 xψ ( † iγ 0 γ i ∂ i + γ 0 m ) ψ, 12
expressed in terms of the fields. Although this looks like the expectation value of the Dirac Hamiltonian in the single particle theory, it now signifies the field Hamiltonian with ψ and ψ interpreted as fields to be quantized later. 0.7.1 Dirac Field in Fourier Space To formulate the theory in Fourier space, we begin by expressing ψ as a sum of free particle solutions of the free Dirac equation. Denote the two positive energy solutions by ψ = u s ( ⃗ k)e i⃗ k⃗x e −iωt and the two negative ones by ψ = ṽ s ( ⃗ k)e i⃗ k⃗x e iωt with the usual definition of ω and s = 1, 2. The tilde is used because we will later wish to redefine the negative energy states to make them more easily identifiable as antiparticle states. Because u s and ṽ s are eigenstates of the Hermitian H belonging to different eigenvalues, they satisfy u † s (⃗ k)ṽ s ′( ⃗ k) = 0. They can also be chosen orthogonal among themselves by appropriate choices of the spinor χ in Eq.(??). This can be done simply by choosing the χ s ’s to be orthogonal. However, unless the χ’s are eigenstates of the helicity h = σ · ˆp which is conserved by the free H, the resulting u’s or v’s will not be eigenstates of any component of the spin even though χ is always such an eigenstate. In fact, the components of the spin are not conserved by the free Dirac Hamiltonian unless they are in the direction of the momentum. We normalize the states to satisfy the orthonormality relations u † s( ⃗ k)u s ′( ⃗ k) = 2ωδ ss ′, ṽ † s (⃗ k)ṽ s ′( ⃗ k) = 2ωδ ss ′, u † s (⃗ k)ṽ s ′( ⃗ k) = 0. Beginning with the superposition ∫ ( ψ(x) = a s ( ⃗ k)u s ( ⃗ k)e i⃗k·⃗x + ˜b s ( ⃗ k)ṽ s ( ⃗ ) k)e i⃗ k·⃗x ˜dk ∑ s where the quantities with tildes will be modified in anticipation of their role as antiparticle amplitudes. Since an antiparticle associated with a negative 13
Page 1 and 2: [revised Oct 15, 2010] 0.1 Klein-Go
Page 3 and 4: where ψ 0 is a constant 4-componen
Page 5 and 6: and for boosts Σ 0i = − 1 2 ( 0
Page 7 and 8: In higher order, the equivalence of
Page 9 and 10: is conserved. The operator P define
Page 11: and a similar series for G with coe
Page 15: This Lagrangian leads to solutions

0.1 Klein-Gordon Equation 0.2 Dirac Equation

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