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1.4 the Dirac Equation 14 Relativistic Quantum Mechanics to the vacuum polarization by creation of virtual particle-antiparticle pairs which didn’t influence probability currents because there was not enough energy in the system to promote them to become real particles. Looking for an analogy, one can probably think about the Lamb shift in atomic physics where the vacuum polarizations affects energy levels. The problem of RQM is now clear: the formalism describes one parti- cle (or a fixed number of particles) but physics needs many particles; the number of which can’t be fixed; particles can be created and particles can be annihilated. One needs RQF to describe such physics. RQM can be used, as long as the number of particles is fixed. Considering the limit of the uncertainty relation p s ¯h, we see that the pair cre- ation which starts at p mc sets the∼ limit on the s ¯h/(mc). So as long as we are studying ∼ physics at a scale bigger than h/ ¯∼(mc), known as the Compton length, RQM can be applied. Atomic physics is an ex- ample when this condition is fulfilled. But RQM can also be applied for many processes of high energy particle physics. A representative exam- ple could be the electron-positron annihilation producing hadrons; many + hadrons. A fundamental process in that case is e + e− q + ¯q. The number of particles is 2 and is fixed and the change from→ 2 leptons to 2 quarks can be handled by RQM. A fragmentation of quarks to hadrons is taking place on a different, much slower time scale and therefore can + be separated from the fundamental process of the e + e− annihilation. Is there any limit on p? How well can one measure momentum? In the non-relativistic quantum mechanics, momentum can be measured with any precision but due to the limit speed < c, ¯h p t , ∼ c see the Introduction chapter in (5). Infinite precision p 0 requires infinite measurement time t . → → ∞ 1.4 The Dirac equation This is the main section of the book on RQM. First, we will consider different representations of the Dirac eqn, a probability current and bi- linear covariants. Then, we will find wave functions describing free spin 1/2 particles like electrons, discuss chirality and helicity operators before applying them to describe the Standard Model (SM) interactions at the limit when masses can be neglected. Electromagnetic interactions will be introduced via, so called, minimal coupling and the non-relativistic limit will be obtained. ξ Combining ξα and η into one Dirac spinor Ψ, Ψ = , the Dirac β˙ η eqn 1.12 can be written as 0 p + pσ 0 Ψ = mΨ. (1.30) p pσ 0 0 − Instead of Ψ, we could use Ψ = UΨ, where U is a unitary operator. In the new basis, eqn 1.30 would look different. So, in general, the Dirac 1.4 The Dirac equation 15 eqn can be written as (γp m)Ψ = 0 (1.31) − where ∂ γp γµp = p γ0 pγ = iγ0 + iγ . (1.32) ≡ µ 0 − ∂t ∇ Comparing eqns 1.31 and 1.30 one can see that in the basis or repre- sentation which was used to get eqn 1.30, 0 1 0 σ γ0 = , γ = . (1.33) 1 0 σ −0 This representation in known as the Weyl or symmetric or chiral 26 26 α representation . Multiplying eqn 1.31 by γp from the left, one gets In some books, ξ and ηβ˙ swap representation independent constraint on γ matrices: places leading to different space-like γ matrices; multiplied by -1. γµγν + γν γµ = 2gµν . (1.34) The matrix γ0 is Hermitian and the matrices γ are anti-Hermitian27 : (γ0)2 = 1 γ0 = γ0, γ† = γ (1.35) (γ1)2 = (γ2)2 = (γ3)2 = 1 † − − 1 γ = UγU † = UγU − Applying the Hermitian conjugation to eqn 1.31, using properties of the γ matrices given by eqn 1.35 and after some algebra28 one gets the 27in any representation. adjoint Dirac equation 28Eqn 1.31 is Ψ(¯ γp + m) = 0 (1.36) ∂Ψ ∂Ψ iγ0 + iγk mΨ = 0. where the adjoint spinor ∂t ∂xk − 0 Ψ¯ Ψ†γ (1.37) Applying one gets ≡ † ∂Ψ ∂Ψ and p acts on the left. i † γ0 + i † ( γk) + mΨ = 0 k † Multiplying eqn 1.31 by Ψ¯ from the left and eqn 1.36 by Ψ from the ∂t ∂x − 0 right and adding resulting equations, one gets and multiplying by γ from the right, using eqn 1.34, gives ¯ µ ¯ µ ¯ µ ∂Ψ¯ ∂Ψ¯ Ψγ ∂µΨ + (∂µΨ)γ Ψ = ∂µ(Ψγ Ψ) = 0, i γ0 + i γk + mΨ¯ = 0. ∂t ∂xk µ which is a continuity equation, ∂µj = 0, for the probability current 4-vector jµ = Ψ¯ γµΨ. (1.38) The probability density 4 ρ j0 = Ψ¯ γ0Ψ = Ψ 2 (1.39) ≡ | i| i=1 is a time-like component of the probability current, it is positive definite α and it has similar form to the non-relativistic expression. (ξ )∗ transforms like a dotted spinor One should note that Ψ¯ is a natural partner to Ψ to form objects of and (ηβ˙ )∗ transforms like an undotted well defined transformation properties like a scalar ΨΨ¯ or the probability spinor current 4-vector or so called bilinear covariants. The matrix γ0 which is sitting inside Ψ¯ is swapping spinors inside the bi-spinor such a way that the dotted spinor is meeting the dotted one and the undotted is meeting 16 Relativistic Quantum Mechanics undotted one; as it should be, the dotted index is contracted with the dotted one and the undotted index is contracted with the undotted one. The Hamiltonian H of the Dirac eqn one gets multiplying eqn 1.31 by γ0 from the left and separating the time derivative: ∂Ψ HΨ = i , (1.40) ∂t where H = α p + βm (1.41) · 0 and α = γ0γ, β = γ . (1.42) αiαj + αj αi = 2δij βα + αβ = 0 Matrices α and β are Hermitian and in the Weyl representation are β2 = 1 σ 0 0 1 given by α = , β = . (1.43) 0 σ 1 0 − The Weyl representation is very well suited at the ultra-relativistic limit, when the mass can be neglected because the Dirac bispinor effec- tively is reduced to the Weyl spinor; one spinorial component vanishes. At the non-relativistic limit, however, both spinor components of the Dirac bispinor contribute equally in the Weyl representation and there- fore in the non-relativistic limit another representation, called the stan- dard or the Dirac representation, is more suitable (this representation is the most common in textbooks). A transformation from the Weyl representation to the Dirac represen- tation is a Unitary transformation 1 1 1 U = , √2 1 1 − which gives ϕ ξ 1 ξ + η Ψ(Dirac) = = UΨ(W eyl) = U = . χ η √2 ξ η − 1 The transformation of γ matrices; γ(Dirac) = Uγ(W eyl)U − gives 1 0 0 σ 0 σ γ0 = β = , γ = , α = . The 0 1 σ 0 σ 0 − − Dirac eqn in the Dirac (standard) representation is then given by Eϕ p σχ = mϕ − · Eχ + p σϕ = mχ. (1.44) − · In the non-relativistic limit, χ 0 and the Dirac spinor becomes effec- tively a two component Pauli spinor.→ Foldy and Wouthuysen transformetin and the representation should be briefly described here, showing two decoupled eqns for two compo- nent spinors; the non-relativistic limit and problems with the Lorentz transformation and introduction of interactions. 1.5 Gauge symmetry 17 From here there is a smooth transition to (7) which should become the main textbook. For further reading, RQM part of (8) volume I is recommended. 1.4.1 Free particle solutions 1.4.2 Chiriality = helicity 1.4.3 Helicity conservation and interactions via currents 1.4.4 C, P, T 1.4.5 Electromagnetic interactions and the Referencesnon-relativistic limit 1.5 Gauge symmetry Steane A.M., Relativity made relatively easy, to be published. Schutz B.F., A First Course in General Relativity, Cambridge Univer- sity Press (1985). Misner C.W., Thorne K.S. and Wheeler J.A., Gravitation, W.H. Free- man and Company (1995). , neutron interference experiment. Landau course, Bierestecki W.B., Lifshitz E.M. and Pitajewski L.P., Relativistic Quantum Theory part I. Maggiore M., A Modern Introduction to Quantum Field Theory, Ox- ford University Press (2005). Halzen F. and Martin A.D., Quarks & Leptons: An Introductory 7 Course in Modern Particle Physics, Wiley (1984). Aitchison I.J.R. and Hey A.J.G., Gauge Theories in Particle Physics, 8 3rd edition, Institute of Physics Publishing (2003). Perkins D.H., Introduction to High Energy Physics, 3rd edition, Addison–Wesley (1987). Perkins D.H., Introduction to High Energy Physics, 4th edition, Cam- bridge University Press (2000)..
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