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Chapter 5 Elements of Quantum Electrodynamics

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Chapter 5 Elements of Quantum Electrodynamics Chapter 5 Elements of quantum electrodynamics 5.1 Quantum mechanical equations of motion In quantum mechanics I & II, the correspondence principle played a central role. It is in a sense the recipe to quantize a system whose Hamiltonian is known. It consists in the following two substitution rules : #» p i (momentum), (5.!) �−→ − ∇ E i∂ (energy). (5.") �−→ t #or nonrelati$istic quantum mechanics we get the celebrated Schr¨odinger equation, ! #» i∂ ψ%Hψ, withH% &V( x), ( .3) t − "m� #» whose free solution (V( x) () is, ≡ #» #» #» 2 #» i(Et p x) p ψ( x , t)%Ce − · withE% . "m The relati$istic $ersion of the energy-momentum relationship is however, #» E2 % p 2 &m 2, (5.+) from which we get, using again the correspondence principle Eq. ( .1), ∂2ψ%( &m 2)ψ. (5. ) − t −� - CHAPTER 5. ELEMENTS OF QUANTUM ELECTRODYNAMICS .t this point we de/ne some important symbols which will follow us troughout the rest of this lecture, ∂ :% (∂ , ), (5.-) µ t ∇ ∂µ :%g µν∂ %(∂ , ), (5.0) ν t −∇ :%∂ ∂µ %∂ 2 . (5.1) � µ t − � With this notation we can then reformulate Eq. ( .5) to get the Klein-Gordon equation, 2 �&m ψ%( , (5.3) with solutions, � � #» #» #» i(Et p x) #»2 2 ψ( x , t)%Ce − · withE% p &m . ± � 2e see that in this case it is possible to ha$e negati$e energy eigen$alues, a fact not arising with the nonrelati$istic case. As in the case of the 4chr5odinger equation ( .3) we can formulate a continuity equation. )o do so we multiply the 6lein*Gordon equation (5.3) by the left withψ ∗ and its conjugate, 2 (�&m )ψ ∗ % 0, byψ and then subtract both equations to get, µ µ (%ψ ∗∂ ∂µψ ψ∂ ∂µψ∗ µ − %∂ (ψ∗∂ ψ ψ∂ ψ∗) µ − µ ∂ (ψ∗∂ ψ ψ∂ ψ∗)& (ψ ∗ ψ ψ ψ ∗)%(. (5.!() ⇒ t t − t ∇· ∇ − ∇ 2e would like to interpretψ ∗∂tψ ψ∂ tψ∗ in Eq. ( .!() as a probability density, or more exactly, − ρ%i(ψ ∗∂ ψ ψ∂ ψ∗), t − t which is not a positi$e de/nite quantity (as we can con$ince oursel$es by computingρ for the plane wa$e solution), and hence cannot be interpreted as a probability density as it was the case in :;. When computing the continuity equation for the 4chr5odinger equation, where such a problem does not arise, we see that the problem lies essentially in the presence of a second order time deri$ati$e in the Klein-Gordon equation. 2e now make a big step, by imposing that our equation of motion only contains a first order time deri$ati$e. 4ince we want a <orentz in$ariant equation of motion, we conclude that only a linear dependence on is allowed. #ollowing = irac>s intuition, we make the ansatz, ∇ #» (iγµ∂ m)ψ%(iγ 0∂ &i γ m)ψ%(. (5.!!) µ − t ·∇− 5.1. QUANTUM MECHANICAL EQUATIONS OF MOTION 0 )urning back to the correspondence principle we remark that, #» #» (γ0E γ p m)ψ%( (5.!") − · − #» #» (γ 0E γ p m) 2ψ%(, ⇒ − · − #» which must stay compatible with the mass-shell relation,E 2 % p 2 &m 2. This implies that theγ µ>s cannot be numbers since it would then be impossible to satisfy, 3 2 #» #» ! #» ( γ p) 2 % γip p 2, · i ∝ � i=1 � � so we let them ben n matrices, for ann which is still to be dete rmined. × 2e now deri$e relations that theγ µ>s must full/ll, so that the mass-shell relation remains true. #rom Eq. (5.!2), and again with the correspondence principle, we must ha$e, 0 1 #» 0 1 i∂t =(γ )− γ ( i ) +(γ )− m · − ∇#» E p 3 ����2 ! � ��0 �1 i 0 1 j 0 1 j 0 1 i ∂ % (γ )− γ (γ )− γ &(γ )− γ (γ )− γ ∂ ∂ (5.!') ⇒ − t − " i j i,j=1 E2 � � � 3 ���� 0 1 i 0 1 0 1 0 1 i i m (γ )− γ (γ )− &(γ )− (γ )− γ ∂ (5.!+) − · i i=1 2 �0 1� 0 1 � &m (γ )− (γ )− (5.! ) ! 2 =(∂i∂i &m ) (5.!-) =( &m2). −�#» p 2 ���� Comparing Eqs. (5.!6) and (5.!5) we conclude that, 0 1 0 1 ! 0 1 0 (γ )− (γ )− % (γ )− %γ . ( .!7) ⇒ =efining a, b :% ab& ba and comparing ,q. (5.! -) and Eq. ( .!+) we get { } 0 i 0 1 ! i 0 γ γ (γ )− %( γ , γ %(.( .!1) ⇒{ } Finally, comparing Eq. (5.!-) and ,q. ( .!3), we ha$e, ! 0 1 i 0 1 j 0 1 j 0 1 i ! i j (γ )− γ (γ )− γ &(γ )− γ (γ )− γ %δ γ , γ % "δ . (5.!3) −" ij ⇒ { } − ij � � 1 CHAPTER 5. ELEMENTS OF QUANTUM ELECTRODYNAMICS 2e can summarize Eqs. (5.!0), (5.!8) and (5.!9) in the Clifford algebra of theγ- matrices, γµ, γν %"g µν , (5."() { } whereg 00 % 1,g ii % ! and all the other elem ents $anish. − Important facts : )he eigen$alues ofγ 0 can only be ! and those ofγ i i and the γ*matrices ha$e $anishing trace : ± ± )rγ i % )r (γ0γ0γi)% )r (γ 0γiγ0)% )rγ i )rγ i %(, − − ⇒ 0 0 i i 1 i 0 i 1 0 0 )rγ % )r (γ γ (γ )− )% )r (γ γ (γ )− )% )r (γ ) )rγ %(. − − ⇒ The eigen$alue property ofγ 0 implies with the last equation that the dimensionn of the γ*matrices must be even. #orn % " there are no matric es satisfying Eq. ( ."(), as can be checked by direct compu* tation. #orn % + there are many p ossibilities. The most common choice in textbooks is the =irac*@auli representation : ( (σ i γ0 % σ 3 % , γi %σ i (iσ 2)% , (5."1) ⊗ ( ⊗ σi ( � − � � − � with the @auli matrices, !( (! ( i !( σ0 %% , σ1 % , σ2 % , σ3 % , (! !( i(− ( ! � � � � � � � − � and the Kronecker product of " "-matrices, × b A b A A B% 11 12 . ⊗ b21A b22A � � <ooking at the !irac equation (iγµ∂ m)ψ%( (5."") µ − we see thatψ is no longer a function but a $ector, called "#-)spinor, ψ1 ψ ψ% 2 . ψ3 ψ 4 #or +*spinors, there are two types of ad8oints, namely, 5.2. SOLUTIONS OF THE DIRAC EQUATION 3 the hermitian adjointψ † %(ψ∗ ψ∗ ψ∗, ψ∗), and • 1 2 3 4 0 the =irac adjoint ψA :%ψ †γ %(ψ∗ ψ∗ ψ ∗, ψ ∗). • 1 2 − 3 − 4 Note that ψA satisfies a dirac equation of its own, µ i∂µψγA &m ψA%(. (5."') 2e now focus our attention on the continuity equation for the =irac /eld. #rom Eqs. (5."") and (5."3), 0 i 0 iψ†(∂ ψ)% iψ†γ γ ∂ &ψ †γ m ψ, t − i � � and its hermitian con8ugate, 0 i 0 i(∂ ψ†)ψ% i(∂ ψ†)γ γ &ψ †γ m ψ, − t i � � we get the diCerence, 0 i 0 i ∂ (ψ†ψ)% (∂ ψ†)γ γ ψ&ψ †γ γ i(∂ ψ) , t − i i ∂ (ψγA 0ψ)% ∂ (ψγA iψ). (5."+) t − �i � 2e identify the components as, #» #» ρ% ψγA 0ψ, j% ψAγ ψ, or interpreting them as components of a +*$ector as in classical electrodynamics, jµ % ψγA µψ, (5." ) we see that Eq. (5."4) can be reexpressed in the manifestly co$ariant form, µ ∂µj %( . (5."-) 5.2 Solutions of the Dirac equation Before we look at the solutions of the free =irac equation, we introduce the slash notation µ for contraction with theγ*matrices : a/:%γ aµ. The Dirac equation then reads (i∂/ m)ψ% 0. − -( CHAPTER 5. ELEMENTS OF QUANTUM ELECTRODYNAMICS 5.2.1 &ree particle at rest In the rest frame of a particle, the =irac equation reduces to, 0 iγ ∂tψ% mψ, for which we /nd four linearly indepedent solutions, namely, ! ( imt ( imt ! ψ % e− , ψ % e− ,E%m (particles) 1 ( 2 ( ( ( ( ( +imt ( +imt ( ψ3 % e , ψ4 % e ,E% m (antiparticles). ! ( − ( ! 5.2.2 &ree particle In order to preser$e the <orentz in$ariance of a solution, it must only depend on <orentz µ scalars E quantities which are in$ariant under <orentz transformations E likep x%p µx . 2e make the ansatz, · ip x 0 ψ1,2 % e− · u (p), p >( ± +ip x 0 ψ3,4 % e · v ( p), p <(. ∓ − Pluging those ansatz in the =irac equation, we get, (p/ m)u (p)%Au(p/ m)%(,( ."0) − ± ± − (p/&m)v (p)%Av(p/&m)%(,( ."1) ± ∓ where we replaced p p in the second equation, ha$ing thusp 0 > ( in both cases now. − �→ 5.2.3 Explicit form ofu andv As checked in the exercices, the explicit form for theu andv functions are, 0 #» χ#» u (p)% p &m σ p± , ( ."9) ± p0+· m χ � #» #» ± � � σ p 0· χ v (p)% p0 &m p +m ∓ , ( .'0) ± χ � � ∓ � 5.2. SOLUTIONS OF THE DIRAC EQUATION -! ! ( whereχ + % corresponds to a Fspin upG state andχ % to a Fspin downG ( − ! state. � � � � 2e note on the way that the application, #» #» #» i p3 p1 ip2 p σ p%σ pi % − , �−→ · p1 & ip2 p3 � − � defines an isomorphism between the $ector spaces of '*$ectors and hermitian " "* matrices. × 5.2.4 )perators on spinor spaces *amiltonian The Hamiltonian is defined by i∂tψ%Hψ. Isolating the time deri $ati$e in the =irac equation, ,q. (5.""), we read out, #» #» 0 i 0 m σ p H% iγ γ ∂i &γ m% #» #» · . ( .'1) − σ p m � · − � #» #» p H #» *elicity The helicity is the compenent of the spin in the direction of motion p :% p , and is defined by, | | #» #» ! #» #» ! σ pH( h% σ pH % · #» #» . ( .'") " · ⊗ " ( σ pH � · � By direct computation, one can check that IH,hJ % (, and thus there exist a set of eigen* functions diagonalizingH andh simultaneously. )he eig en$alues ofh are then constants of the motion and hence good quantum numbers to label the corresponding states. This quantum number can take two $alues, #» #» & 1 positi$e helicity ! p , % 2 � #» �� #» (5.'') 1 negati$e helicity ! p .
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