Introduction to Quantum Electrodynamics Peter …

1 Introduction to Quantum ElectrodynamicsPeter Pre snajderThese are lecture notes devoted to introductory chapters of QuantumElectrodynamics (QED). The notes consist of two chapters:1. The Dirac field and the relativistic invariance- The Lorentz transformations and relativistic fields- The Dirac equation and its solutions, polarization sums- Dirac field quantization, field energy and momentum, charge- Fermions, the Dirac field propagator2. Quantum Electrodynamics and Feynman rules- QED equations of motion, Gauss law, Coulomb gauge- Free transversal electromagnetic field and its quantization- The interaction picture and the perturbation theory- Self-interacting scalar field, Feynman rules- QED in Coulomb gauge, gauge invariance- The relativistic formalism and Feynman rulesThis two chapters should be followed by a part devoted to simple appli-cations of Feynman perturbation technique:3. Elementary processes in QED- Scattering amplitudes and the differential cross-section- Kinematics of binary processes, decay rate of an unstable particle- The scatteringe e+ +, the square of the scattering amplitude- Unpolarized scattering and its differential cross-section- The scatteringe e , the square of the scattering amplitude- Crossing symmetry, Mandelstam variables, crossed channels- Compton scatteringe e 1- The polarization sum for photons - Ward identity- Klein-Nishina formula for the cross-section of unpolarizede scattering- The annihilatione e , crossing symmetry and cross-section1 The Dirac field and its relativistic Lorentz transformationsWe shall label the points of the Minkowski space-time as follows:x= (x ) = (x0, x1, x2, x3) = (t, ~x),( )wheret=x0denotes time and~x= (x1, x2, x3) labels the space scalar product of two 4-vectorsx= (x ) ay= (y ) in Minkowskispace-time is given y =x y ,( )wherey = y , y = y.

2 We lower or rise the indices with the help of the relativistic metric tensor( ) = diag(1, 1, 1, 1) or its inverse ( ) = diag(1, 1, 1, 1)): = = ,where is the Kronecker symbol defined by: = 1 for = and = 0for 6= . We adopt theEinstein summation convention: we sum over the2same repeated upper and lower indices, y =x0y0+x1y1+x2y2+ us consider the linear transformation which preserves the relativisticscalar y of any two 4-vectorsxandx:x 7 x , y 7 y .( )Let us rewrite the scalar product in matrix y, whereydenotes the column with 4 componentsy ,xTis a row with 4 componentsx and is 4 4 matrix with elements . Similarly the transformation law inmatrix notation can be written as follows:x7 x = xay7 y = ywhere is the 4 4 matrix with elements . The invariance of the scalarproducts induces a constraint on admissible matrices :xT y=x T y =xT T y T = .( )Here, Tis the transposed matrix of the matrix . Such matrices form a Liegroup, calledLorentz group.

3 The elements of the Lorentz group which canbe expressed in exponential form = exp ( i J) = (exp ( i J)) ( )The symbol in the exponent is a real number andJis 4 4 matrix satisfyingconditionJT + J= 0 JT= J .( )This condition is a direct consequence of ( ) and ( ). There are 6 inde-pendent 4 4 matricesJ =J satisfying ( ). Their matrix elementsare given as:(J ) =i( ).( )In this form we can freely rise and lower all indices simultaneously on any proper Lorentz transformation = exp ( iJ) the exponentJisgiven as a linear combinationJ=12 J = 0,1,2,3,( )specified by 6 real parameters = . The matrixJ generate Lorentztransformations in ( )-plane in Minkowski space:3 Three generatorsJij,i, j= 1,2,3, generate rotations in 3-space (forits specification we need 3 parameters - 3 Euler angles , , ); Three generatorsJ0j,j= 1,2,3, generate boosts (the transformationto a system moving with a speed~vwith respect to the original referenceframe - this requires again 3 parameters).

4 For infinitesimal Lorentz transformation, specified by infinitesimal pa-rameters we obtainx 7 (exp ( i2 J )) x =( i2(J ) +..)x =x i2(J ) x +..x 7 x + x , = .In the last step we have used the explicit formula ( ).It can be easily shown that the matricesJ satisfy the commutationrelations for Lorentz group generators, defining relations of Lie algebraso(3,1):[J , J ] = i( J J + J J ).( )Finally, we point out that it holdsJ x =i( x x ).( )This formula is equivalent to the relationx 7 x , = exp ( i2 J )( )which tell us that 4 real numbersx= (x ), = 0,1,2,3,transform asrelativistic Relativistic scalar fields. The relativistic scalar field (x) isdescribed by a (real or complex) function defined in all points of Minkowskispace-time:x7 (x). By definition, under Lorentz transformationx7 xthe field (x) is transforming in the following way: (x)7 T( ) (x) = ( 1x).( )4In ( ) 1denotes the inverse matrix of the matrix . The symbolT( ) represents the linear operator defined by the last equation.

5 The as-signment 7 T( ) defines the Lorentz group representation because it copies the group product:T( 1)T( 2) =T( 1 2), T(1) = symbol1denotes the 4 4 unit matrix (corresponding to the unity ingroup) and the symbolIis the unit operator corresponding to the identitymap: (x)7 (x).Under infinitesimal Lorentz transformationx 7 x + x the fieldtransforms as follows (x)7 ( 1x) = (x x )= (x) i2 (J )(x).Comparing the last expression with the Taylor expansion of the field on afirst line, we obtain the formula for the generator of Lorentz transformationsJ which acts on fields as a 1-st order differential operator:J = i(x x ), = ,( )where = x . It can be easily shown that the differential operatorsJ = J again satisfy the commutation relations ( ) for Lorentz relativistic fields. Let us considern-component field (x) = 1(x).. n(x) with components a(x),a= 1, .. , n. We shall assume that under Lorentztransformationx xthe field components transform as follows: a(x)7 Sab( ) b( 1x) (T( ) (x))a.

6 ( )The mapping (x)7 T( ) (x) will generate the Lorentz group represen-tation:T( 1)T( 2) =T( 1 2), T(1) =I5exactly, when 7 Sba( ) will be the (n n)-matrix representation of theLorentz groupSab( 1)Sbc( 2) =Sac( 1 2), Sab(1) = : As an important example of multi-component field can servetherelativistic vector fieldV (x) which under Lorentz transformations mapsas follows:V (x)7 V ( 1x).( )We leave as an exercise to derive, in this case, the form of the differentialoperatorsJ generating Lorentz Particles and relativistic fields. In the framework of quantumtheory, the relativistic particle with massmand 4-momentump= (E~p, ~p), , with the 3-momentum~pand energyE~p, is described by the de Brogliewave function1(2 )3/2e (2 )3/2e iEpt+i~p.~x, Ep E~p= ~p2+m2.( )Below we shall frequently use the notationEpinstead ofE~p(a similar sim-plification we shall frequently use for some other quantities too).The set of such particles with different momenta, which do not possessother internal degrees of freedom, is described by relativistic scalar field (x) =1(2 )3 d3~p 2Ep(ape +b pe+ ), p= (Ep, ~p),( ) the scalar field is represented by a complex linear combination of deBroglie wave functions and complex conjugated wave functions (the numeri-cal factor in front of the integral and the factor 2 Epin the measure repre-sent just a convenient normalization): The first integrand describes a system of free particles with 3-momenta~p, the complex coefficientap a~pis proportional to the probability ampli-tude that thea-particle with 3-momentum~pis contained in the ensemble ofparticles.

7 6 The second integrand is a linear combination of complex conjugatedwave functions ofb-particles with 3-momenta~p, the coefficientbp b~pis pro-portional to the probability amplitude that theb-particle with 3-momentum~pis contained in the ensemble of : Forcomplexscalar fields the coefficientsapabpare independent(there is no relation among them). Simply, we have two sorts of particles:a-particles andb-particles which areantiparticlestoa-particles. Particles andantiparticles have the same mass but they possess opposite electric charge(and all other charges they possess are opposite too).The reality condition (x) = (x) forrealscalar field implies constraintap=b p. The system contains one kind of particles: the particleais identicalto its antiparticle, the particles possess zero free scalar field is a solution ofKlein-Gordon equation( +m2) (x) = 0.( )The relativistic invariance of Klein-Gordon equation. Under Lorentztransformationx7 xthe scalar field (x) (x)7 (x) = ( 1x),kde ( 1x) =1(2 )3 d3~p 2Ep(ape ip.)

8 1x+b pe+ip. 1x).Taking into account, that under substitutionp = pd3~p2Ep=d3~p 2Ep ,( )and thatp. 1x=pT 1x=pT T x= ( p)T x, we can write (x) =1(2 )3 d3~p 2Ep (a p e ip .x+b p e+ip .x),( )wherea p = Ep Epap, b p = Ep E pbp, p = p .( )7We see that the transformed field (x) is again a solution of the Klein-Gordon equation ( ):( +m2) (x) = 0,however, with expansion coefficients transformed according to ( ).We have constructed a representation of the Lorentz group realized inspace of field configurations. In fact, we have two independent unitary rep-resentations: the first one in the space of particle configurations and theother one in the space of antiparticle configurations (the coefficientsapabpare independent, and the unitarity is due to the positivity of the integralmeasure). The Dirac equationThe free particle relativistic equation, the Klein-Gordon equation, hadbeen first suggested by E. Schr odinger. He used the quantization rule: theenergyEand 3-momentum~pshould be replaced in the Hamiltonian (formulafor the energy) by differential operatorsE7 i t, pj7 i j,( )where j= xj,j= 1,2,3.

9 However, he found problems with relativis-tic formulation of the hydrogen atom problem. Therefore, he applied therule ( ) within the non-relativistic formula for the electron energy mov-ing in the Coulomb field of proton. He solved and published his well-knownSchr odinger relativistic version of the Quantum equation of motion was publishedafter by Klein and Gordon (the equation was known to V. A. Fock too). Thefact, that due to relativistic invariance the Klein-Gordon equation containedsecond ordertime derivative (beside second order position derivatives) leadsto various complications with Quantum -mechanical interpretation of the fundamental problems motivated P. A. M. Dirac to search for arelativistic equation describing the free particle with massmcontaining justthefirst orderspace-time derivatives. Dirac postulated the equation in theform(i m) (x) = 0, = x ,( )8with the coefficients being unknown constant determined s from the requirement that ( ) should containparticle-like solutions, , the solutions of the first order Dirac equation( ) should simultaneously satisfy the second order Klein-Gordon ( ) with the operator (i +m) we obtain second orderequation(i +m)(i m) (x) = 0,( +m2) (x) = 0.

10 ( )Since the differential operator commutes with any other we can write =12{ , } .Equation ( ) will be consistent with Klein-Gordon equation ( ) pro-vided the coefficients satisfyanticommutationrelations:{ , } + = 2 .( )This follows directly from =12{ , } = = .Equations ( ) represent the well-known defining relations for the (real)Clifford algebra generators: pre 6= the generators anti-commute: = , and they are normalized by: ( 0)2= +1 and ( i)2= 1,i= 1,2, algebra of -matrices ( ) may be realized in terms of 4 4 complexmatrices, so called Dirac matrices. In the Weyl (chiral) basis they are givenin 2 2 block form by the formulas: 0=(0 11 0), j=(0 j j0), j= 1,2,3.( )All entering elements are 2 2 matrices:0is the zero matrix,1is the unitmatrix and j,j= 1,2,3, are the Pauli matrices: 1=(0 11 0), 2=(0 ii0), 3=(1 00 1).( )9 Note: Dirac used a different realization of -matrices: 0=(1 00 1), j=(0 j j0), j= 1,2,3.( )This Dirac (or standard) realization of -matrices is equivalent to the Weylrealization.