Chapter 5 Elements of Quantum Electrodynamics

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Chapter 5

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), (5.1) �−→ − ∇ E i∂ (energy). (5.2) �−→ t

For nonrelativistic quantum mechanics we get the celebrated Schr¨odinger equation,

1 #» i∂ ψ=Hψ, withH= +V( x), (5.3) t − 2m� #» whose free solution (V( x) 0) is, ≡ #» #» #» 2 #» i(Et p x) p ψ( x , t)=Ce − · withE= . 2m

The relativistic version of the energy-momentum relationship is however, #» E2 = p 2 +m 2, (5.4) from which we get, using again the correspondence principle Eq. (5.1),

∂2ψ=( +m 2)ψ. (5.5) − t −�

55 56 CHAPTER 5. ELEMENTS OF QUANTUM ELECTRODYNAMICS

At this point we deﬁne some important symbols which will follow us troughout the rest of this lecture,

∂ := (∂ , ), (5.6) µ t ∇ ∂µ :=g µν∂ = (∂ , ), (5.7) ν t −∇ :=∂ ∂µ =∂ 2 . (5.8) � µ t − � With this notation we can then reformulate Eq. (5.5) to get the Klein-Gordon equation,

2 �+m ψ=0 , (5.9)

with solutions, � � #» #» #» i(Et p x) #»2 2 ψ( x , t)=Ce − · withE= p +m . ± � We see that in this case it is possible to have negative energy eigenvalues, a fact not arising with the nonrelativistic case. As in the case of the Schr¨odinger equation (5.3) we can formulate a continuity equation. To do so we multiply the Klein-Gordon equation (5.9) by the left withψ ∗ and its conjugate, 2 (�+m )ψ ∗ = 0, byψ and then subtract both equations to get,

µ µ 0 =ψ ∗∂ ∂µψ ψ∂ ∂µψ∗ µ − =∂ (ψ∗∂ ψ ψ∂ ψ∗) µ − µ ∂ (ψ∗∂ ψ ψ∂ ψ∗) + (ψ ∗ ψ ψ ψ ∗) = 0. (5.10) ⇒ t t − t ∇· ∇ − ∇

We would like to interpretψ ∗∂tψ ψ∂ tψ∗ in Eq. (5.10) as a probability density, or more exactly, −

ρ=i(ψ ∗∂ ψ ψ∂ ψ∗), t − t which is not a positive definite quantity (as we can convince ourselves by computingρ for the plane wave solution), and hence cannot be interpreted as a probability density as it was the case in QM. When computing the continuity equation for the Schrödinger equation, where such a problem does not arise, we see that the problem lies essentially in the presence of a second order time derivative in the Klein-Gordon equation. We now make a big step, by imposing that our equation of motion only contains a first order time derivative. Since we want a Lorentz invariant equation of motion, we conclude that only a linear dependence on is allowed. Following D irac’s intuition, we make the ansatz, ∇ #» (iγµ∂ m)ψ=(iγ 0∂ +i γ m)ψ=0. (5.11) µ − t ·∇− 5.1. QUANTUM MECHANICAL EQUATIONS OF MOTION5 7

Turning back to the correspondence principle we remark that, #» #» (γ0E γ p m)ψ=0 (5.12) − · − #» #» (γ 0E γ p m) 2ψ=0, ⇒ − · − #» 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. × We now derive relations that theγ µ’s must fullﬁll, so that the mass-shell relation remains true. From Eq. (5.12), and again with the correspondence principle, we must have,

0 1 #» 0 1 i∂t =(γ )− γ ( i ) +(γ )− m · − ∇#» E p 3 ��2 1 � ��0 �1 i 0 1 j 0 1 j 0 1 i ∂ = (γ )− γ (γ )− γ + (γ )− γ (γ )− γ ∂ ∂ (5.13) ⇒ − t − 2 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 ∂ (5.14) − · i i=1 2 �0 1� 0 1 � +m (γ )− (γ )− (5.15) ! 2 =(∂i∂i +m ) (5.16) =( +m2). −�#» p 2 �� Comparing Eqs. (5.16) and (5.15) we conclude that,

0 1 0 1 ! 0 1 0 (γ )− (γ )− = (γ )− =γ . (5.17) ⇒

Deﬁning a, b := ab+ ba and comparing Eq. (5.1 6) and Eq. (5.14) we get { }

0 i 0 1 ! i 0 γ γ (γ )− = 0 γ , γ =0.( 5.18) ⇒{ }

Finally, comparing Eq. (5.16) and Eq. (5.13), we have,

1 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 = 2δ . (5.19) −2 ij ⇒ { } − ij � � 58 CHAPTER 5. ELEMENTS OF QUANTUM ELECTRODYNAMICS

We can summarize Eqs. (5.17), (5.18) and (5.19) in the Cliﬀord algebra of theγ- matrices,

γµ, γν =2g µν , (5.20) { } whereg 00 = 1,g ii = 1 and all the other elem ents vanish. − Important facts : The eigenvalues ofγ 0 can only be 1 and those ofγ i i and the γ-matrices have vanishing trace : ± ±

Trγ i = Tr (γ0γ0γi) = Tr (γ 0γiγ0) = Trγ i Trγ i = 0, − − ⇒ 0 0 i i 1 i 0 i 1 0 0 Trγ = Tr (γ γ (γ )− ) = Tr (γ γ (γ )− ) = Tr (γ ) Trγ = 0. − − ⇒ The eigenvalue property ofγ 0 implies with the last equation that the dimensionn of the γ-matrices must be even. Forn = 2 there are no matric es satisfying Eq. (5.20), as can be checked by direct computation. Forn = 4 there are many p ossibilities. The most common choice in textbooks is the Dirac-Pauli representation :

0 0σ i γ0 = σ 3 = , γi =σ i (iσ 2) = , (5.21) ⊗ 0 ⊗ σi 0 � − � � − � with the Pauli matrices,

1 0 0 1 0 i 1 0 σ0 == , σ1 = , σ2 = , σ3 = , 0 1 1 0 i0− 0 1 � � � � � � � − � and the Kronecker product of 2 2-matrices, × b A b A A B= 11 12 . ⊗ b21A b22A � � Looking at the Dirac equation

(iγµ∂ m)ψ=0 (5.22) µ − we see thatψ is no longer a function but a vector, called (4-)spinor,

ψ1 ψ ψ=  2  . ψ3  ψ   4    For 4-spinors, there are two types of adjoints, namely, 5.2. SOLUTIONS OF THE DIRAC EQUATION 59

the hermitian adjointψ † = (ψ∗ ψ∗ ψ∗, ψ∗), and • 1 2 3 4 0 the Dirac adjoint ψ¯ :=ψ †γ = (ψ∗ ψ∗ ψ ∗, ψ ∗). • 1 2 − 3 − 4

Note that ψ¯ satisﬁes a dirac equation of its own,

µ i∂µψγ¯ +m ψ¯=0. (5.23)

We now focus our attention on the continuity equation for the Dirac ﬁeld. From Eqs. (5.22) and (5.23),

0 i 0 iψ†(∂ ψ)= iψ†γ γ ∂ +ψ †γ m ψ, t − i � � and its hermitian conjugate,

0 i 0 i(∂ ψ†)ψ= i(∂ ψ†)γ γ +ψ †γ m ψ, − t i � � we get the diﬀerence,

0 i 0 i ∂ (ψ†ψ)= (∂ ψ†)γ γ ψ+ψ †γ γ i(∂ ψ) , t − i i ∂ (ψγ¯ 0ψ)= ∂ (ψγ¯ iψ). (5.24) t − �i �

We identify the components as, #» #» ρ= ψγ¯ 0ψ, j= ψ¯γ ψ,

or interpreting them as components of a 4-vector as in classical electrodynamics,

jµ = ψγ¯ µψ, (5.25)

we see that Eq. (5.24) can be reexpressed in the manifestly covariant form,

µ ∂µj = 0 . (5.26)

5.2 Solutions of the Dirac equation

Before we look at the solutions of the free Dirac equation, we introduce the slash notation µ for contraction with theγ-matrices : a/:=γ aµ. The Dirac equation then reads (i∂/ m)ψ= 0. − 60 CHAPTER 5. ELEMENTS OF QUANTUM ELECTRODYNAMICS

5.2.1 Free particle at rest

In the rest frame of a particle, the Dirac equation reduces to,

0 iγ ∂tψ= mψ, for which we ﬁnd four linearly indepedent solutions, namely,

1 0 imt 0 imt 1 ψ = e− , ψ = e− ,E=m (particles) 1  0  2  0   0   0       0   0  +imt 0 +imt 0 ψ3 = e   , ψ4 = e   ,E= m (antiparticles). 1 0 −  0   1          5.2.2 Free particle

In order to preserve the Lorentz invariance of a solution, it must only depend on Lorentz µ scalars – quantities which are invariant under Lorentz transformations – likep x=p µx . We make the ansatz, ·

ip x 0 ψ1,2 = e− · u (p), p >0 ± +ip x 0 ψ3,4 = e · v ( p), p <0. ∓ −

Pluging those ansatz in the Dirac equation, we get,

(p/ m)u (p)=¯u(p/ m) = 0,( 5.27) − ± ± − (p/+m)v (p)=¯v(p/+m) = 0,( 5.28) ± ∓ where we replaced p p in the second equation, having thusp 0 > 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± , (5.29) ± p0+· m χ � #» #» ± � � σ p 0· χ v (p)= p0 +m p +m ∓ , (5.30) ± χ � � ∓ � 5.2. SOLUTIONS OF THE DIRAC EQUATION 61

1 0 whereχ + = corresponds to a “spin up” state andχ = to a “spin down” 0 − 1 state. � � � � We note on the way that the application,

#» #» #» i p3 p1 ip2 p σ p=σ pi = − , �−→ · p1 + ip2 p3 � − � deﬁnes an isomorphism between the vector spaces of 3-vectors and hermitian 2 2- matrices. ×

5.2.4 Operators on spinor spaces

Hamiltonian The Hamiltonian is deﬁned by i∂tψ=Hψ. Isolating the time deri vative in the Dirac equation, Eq. (5.22), we read out, #» #» 0 i 0 m σ p H= iγ γ ∂i +γ m= #» #» · . (5.31) − σ p m � · − �

#» #» p ˆ #» Helicity The helicity is the compenent of the spin in the direction of motion p := p , and is deﬁned by, | | #» #» 1 #» #» 1 σ pˆ0 h= σ pˆ = · #» #» . (5.32) 2 · ⊗ 2 0 σ pˆ � · �

By direct computation, one can check that [H,h] = 0, and thus there exist a set of eigenfunctions diagonalizingH andh simultaneously. The eig envalues ofh are then constants of the motion and hence good quantum numbers to label the corresponding states. This quantum numberλ can take two values, #» #» + 1 positive helicity s p , λ= 2 � #» �� #» (5.33) 1 negative helicity s p . � − 2 � �� We stress here that helicity/handedness is not a Lorentz invariant quantity for massive particles. #» Consider p in thez-direction, then, 1 #» #» 1 1 σ pˆ χ = σ3χ = χ . 2 · ± 2 ± ± 2 ±

From the last argumentative steps, we are not surprised with the statement that the Dirac 1 equation describes spin- 2 particles. 62 CHAPTER 5. ELEMENTS OF QUANTUM ELECTRODYNAMICS

Chirality Consider the Dirac equation for the case of massless particles. This is a good approximation forE m, which is often the cas e in accelerator experiments. Setting m = 0 simpliﬁes Eq.� (5.22) leading to

µ iγ ∂µψ=0.

Eq. (5.29) and (5.30) change accordingly. We consider for now the particle solutionsu : ±

#» #» χ#» #» χ u (p)= p σ p± = p ± . (5.34) #» ± | | ·p χ | | χ � | | ±� �± ±� � �

It is convenient to deﬁne the so-called chirality matrixγ 5:

0 1 2 3 γ5 = iγ γ γ γ

which in the Dirac-Pauli representation reads

0 γ = . 5 0 � � Using that theγ-matrices fulﬁll γ µ, γν =2g µν (see Eq. (5.20)), one can show that { } γ , γµ =0and (5.35) { 5 } 2 γ5 = . (5.36)

These properties ofγ 5 imply that ifψ is a solution of the Dir ac equation then so isγ 5ψ. Furthermore, sinceγ 2 = the eigenvalues of the c hirality matrix are 1: 5 ±

γ5ψ = ψ ± ± ± which deﬁnes the chirality basisψ . ± Let us apply theγ 5 matrix to the spinor part of particle solutions of the free Dirac equation given in Eq. (5.34):

0 #» χ #» χ γ5u (p)= p ± = p ± ± (5.37) ± 0 | | χ | | χ � � �± ±� � ± � #» �χ � = p ± = u (p). (5.38) ± | | χ ± ± �± ±� � A similar calculation shows that for the antiparticle solutions

γ5v (p)= v (p). (5.39) ± ∓ ± Therefore, the helicity eigenstates form = 0 are equivalent to the chirality eigenstates. Results (5.38) and (5.39) lead to the notion of handedness (which is borrowed from chem- istry): 5.2. SOLUTIONS OF THE DIRAC EQUATION 63

#» u describes a right handed particle: −−→spin p and • + �� e− #» v describes a left handed antiparticle: ←−−spin p + • + �� e where the converse holds foru andv . − − Exploiting the eigenvalue equations (5.38) and (5.39), one can deﬁne the projectors

1 PR = ( γ 5). (5.40) L 2 ±

They project tou , v for arbitrary spinors. For example we have ± ± 1 1 0 PLu = ( γ 5)u = ( )u = . ± 2 − ± 2 ∓ ± u � − To show that Eq. (5.40) indeed deﬁnes projectors, we check (using Eq. 5.36) for idempo- tence,

2 1 1 2 1 PR = ( γ 5)( γ 5) = ( 2γ 5 +γ 5 ) = ( γ 5) =P R, L 4 ± ± 4 ± 2 ± L orthogonality, 1 1 P P = (+γ )( γ ) = ( γ 2) = 0, R L 4 5 − 5 4 − 5 and completeness,

PR +P L =.

Note that the projectorsP L andP R are often used to indicate the chirality basis:

uL,R =P L,Ru

vL,R =P L,Rv.

What has been derived so far rests on the assumption that the mass be zero. In this case, chirality is equivalent to helicity which is also Lorentz invariant. If, on the other handm= 0, chirality and helicity ar e not equivalent: In this case chirality, while Lorentz invariant,� is not a constant of the motion,

[γ ,H ]= 0, 5 Dirac � and therefore not a good quantum number. Helicity though is a constant of the motion, but, since spin is unaﬀected by boosts, it is not Lorentz invariant for non-vanishing mass: #» For every possible momentum p in one frame of reference there is another frame in 64 CHAPTER 5. ELEMENTS OF QUANTUM ELECTRODYNAMICS

L transformation

�p � p

Figure 5.1: Helicity for the case of non-vanishing mass.

Chirality Helicity #» #» #» γ = iγ0γ1γ2γ3 h( pˆ)= 1 σ pˆ 5 2 · ⊗ Constant of motion � � m=0 Lorentz invariant � � Constant of motion � � m= 0 � Lorentz invariant � � Table 5.1: Chirality and helicity.

#» #» which the particle moves in direction p / p (see Fig. 5.1). A comp arison of chirality and helicity is given in Tab. 5.1. − | | Although chirality is not a constant of the motion form= 0, it is still a useful conce pt (and becomes important when one considers weak interactions).� A solution of the Dirac equationψ can be decomposed:

ψ=ψ L +ψ R whereψ L andψ R are not solutions of the Dirac equation. TheW vector boson of the weak interaction only couples toψ L. As for the normalization of the orthogonal spinors (5.29) and (5.30), the most convenient choice is:

¯us(p)us� (p)=2mδss� ¯v(p)v (p)= 2mδ s s� − ss� where s, s� = . ± 0 Using ψ¯=ψ †γ , one can show that the following completeness relations (or polarization sum rules) hold:

us(p)¯us(p)= p/+m (5.41) s= �± vs(p)¯sv(p)= p/ m. (5.42) − s= �± Comparing these polarization sums with, for instance, the Dirac equation foru, Eq. (5.27), one sees that p/+m projects on the subspac e of particle solutions. 5.3. FIELD OPERATOR OF THE DIRAC FIELD 65 5.3 Field operator of the Dirac ﬁeld

The spinors

ipx #» 2 2 u (p)e− , eigenvaluesE = + p +m , and s p | | ipx #» 2 2 vs( p)e , eigenvaluesE p = � p +m , − − | | are eigenfunctions of the Dirac Hamiltonian and therefore� solutions of the Dirac equation. From these solutions we can deduce the field operator of the Dirac field (which fulfills the Dirac equation) 1:

3 d p 1 #» ipx #» ipx ψ(x)= as( p)u s(p)e− +b †( p)v s(p)e (5.43) (2π)3 0 s 2p s= � ± � � 3 � d p �1 #» ipx #» ipx ψ¯(x)= a†( p)¯us(p)e +b s( p)¯vs(p)e− (5.44) (2π)3 0 s 2p s= � �± � � where � #» #» as†( p ) : creation operator of particle with momentum p #» #» b†( p ) : creation operator of antiparticle with momentum p s #» #» as( p ) : annihilation operator of particle with momentum p #» #» bs( p ) : annihilation operator of antiparticle with momentum p .

In advanced quantum mechanics we have seen that field operators create or annihilate position eigenstates. The field operator in Eq. (5.44) does the same thing while furthermore consistently combining the equivalent possibilities for particle creation and antiparticle an- #» nihilation:a ( p ) creates individual particle momentum eigenstates from which a weighted s† #» superposition is formed, the integral overb s( p ) on the other hand, annihilates a weighted superposition of antiparticles. Since the creation of a particle at positionx is equivalent to the annihilation of its antiparticle at positionx, both terms have to app ear in the field operator ψ¯(x). Because we have to consider particles and antiparticles, here the energy spectrum is more complicated than in the pure particle case. The creation terms come ipx ipx with a positive-sign plane wave factore while the annihilation terms contributee − . The equivalence of particle creation and antiparticle annihilation is to be understood in the sense that they lead to the same change in a given field configuration. The Dirac field is a spin-1/2 field. Therefore, the Pauli exclusion principle must hold, imposing anti-commutation relations on the field operators: #» #» #» #» ψ( x , t),ψ( x �, t) = ψ¯( x , t), ψ¯( x �, t) =0 { #» #» } { 0 3 #» #» } ψ( x , t), ψ¯( x �, t) =γ δ ( x x �). { } − 1The normalization is chosen to avoid an explicit factor 2p0 in the anticommutators of the fields and of the creation and annihilation operators. 66 CHAPTER 5. ELEMENTS OF QUANTUM ELECTRODYNAMICS

Because of Eq. (5.43) and (5.44), this implies for the creation and annihilation operators #» #» #» #» ar†( p),a s†( p �) = a r( p),a s( p �) =0 { #» #» } { #» #» } b†( p),b †( p �) = b ( p),b ( p �) =0 { r s } { r s } #» #» 3 3 #» #» a ( p),a †( p �) =δ (2π) δ ( p p �) { r s } rs − #» #» 3 3 #» #» b ( p),b †( p �) =δ (2π) δ ( p p �). { r s } rs − As an example for the relation of ﬁeld operator and ladder operator anti-commutation #» #» relations, we calculate ψ( x , t), ψ¯( x �, t) , assuming anti-commutation relations for the creation and annihilatio{n operators: } #» #» ψ( x , t), ψ¯( y , t) { 3 3 #» } d pd q 1 ipx iqy #» #» = e e− vr(p)¯sv(q) b†( p),b s( q) (2π)6 0 0 { r } 2p 2q r,s � � � ipx iqy #» #» +e − e �u (p)¯u(q) a ( p),a †( q) r s { r s } 3 #» #» #» � #» #» #» d p 1 i p( x y) i p( x y) = e− − v (p)¯v(p)+e − u (p)¯u(p) (2π)3 2p0 s s s s s s � � � � � which, using the completeness relations, Eq. (5.41) and (5.42),

3 #» #» #» #» #» #» d p 1 i p( x y) 0 0 #» #» i p( x y) 0 0 #» #» = e− − (p γ p γ m)+e − (p γ p γ +m) (2π)3 2p0 − · − − · � � even odd even odd � 3 #» #» #» 0 d p i p( x y) 0 3 #» #» =γ e − =γ �δ��(�x � ��y).� �� (2π3) − � However, in the laboratory one prepares in general (to a ﬁrst approximation) momentum eigenstates, rather than position eigenstates. Therefore, we give the expression2 for the momentum operator: 3 #» #» #» #» µ d k µ P = k a†( k)a ( k)+b †( k)b ( k) (2π)3 s s s s s � � � � which is just the momentum weighted with the number operatorN=a †a+b †b. Using the anti-commutation relations for the ladder operators, one can show that the momentum operator fulﬁlls the following useful commutation relations: µ #» µ #» [P , as†( p)]=p as†( p) µ #» µ #» [P , bs†( p)]=p bs†( p) µ #» µ #» [P , as( p)]= p as( p) #» − #» [P µ, b ( p)]= p µb ( p). s − s

2This expression is obtained from Noether’s theorem using the technique of normal ordering. These topics are discussed in text books on quantum ﬁeld theory, e. g. by Peskin/Schroeder [14]. 5.3. FIELD OPERATOR OF THE DIRAC FIELD 67

Vacuum state The vacuum state is denoted by 0 and has the property 3, | � P µ 0 =0, (5.45) | � i.e. the vacuum has no momentum. Using the commutation relations stated above and the property (5.45), we conclude that,

µ #» µ #» P a†( p) 0 =p a†( p) 0 , (5.46) s | � s | � #» µ µ in other words, the statea †( p) 0 is an eigenstate ofP with momentump . s | � With this fact in mind, we deﬁne the following states, #» e−(p, s) = 2E #»a†( p) 0 , (5.47) p s | � + #» #» �e (p, s)� = �2E p bs†( p) 0 , (5.48) � | � � � � of a particle respectively antiparticle� with momentum eigenstatep and spins.

#» The factor 2E p is there in order to ensure a Lorentz invariant normalization,

The deﬁnition of states (5.47) and (5.48) corresponds to a continuum normalization in inﬁnite volume. From the above equation, it can be seen that the dimensionality of the one-particle norm e −(p, s) e−(p, s) is, � | � (energy) = (energy) (volume), (momentum)3 ·

meaning that we have a constant particle density of 2E particles per unit volume. To obtain single particle states in a given volumeV , one must therefore m ultiply e −(p, s) with a normalization factor 1/√2EV: | �

1 e−(p, s) = e−(p, s) (5.49) single-particle √2EV � � 1 � � �e+(p, s) = �e+(p, s) (5.50) single-particle √2EV � � � � 3After applying the nontrivial� concept of normal ordering, here� only motivated by the number interpretation in the operatorP µ. 68 CHAPTER 5. ELEMENTS OF QUANTUM ELECTRODYNAMICS

Figure 5.2: Integration paths for Dirac propagator.

5.4 Dirac propagator

In order to solve general Dirac equations, we want to apply a formalism similar to the one used in classical electrodynamics, namely Green’s functions. We introduce the scalar propagator,

3 1 d p ip x Δ±(x)= e∓ · ± i (2π)32p0 � 4 1 d p 2 2 ip x = δ(p m )e∓ · , (5.51) ± i (2π)3 − � which satisﬁes the Klein-Gordon equation,

2 (�+m )Δ±(x)=0.

Representation as a contour integral

4 ip x d p e− · Δ±(x)= , (5.52) − (2π)4 p2 m 2 C�± −

where the pathsC ± are depicted in Fig. 5.2. 5.4. DIRAC PROPAGATOR 69

Figure 5.3: Deformed integration paths and+iε convention.

5.4.1 Feynman propagator

To get a “true” Green’s function for the operator�+m 2, we need to introduce a discon- tinuity, and deﬁne the Feynman propagator

+ Δ (x)=θ(t)Δ (x) θ( t)Δ −(x), (5.53) F − − where we deform the paths of Fig. 5.3 according to the sign oft=x 0 to get convergent integrals over the real line (details can be found in a complex analysis book, see e.g. Freitag & Busam [15]) :

0 0 ip0x0 R + x >0, Imp <0 e − →∞0:C , • ⇒ −→ 0 0 ip0x0 R x <0, Imp >0 e − →∞0:C −. • ⇒ −→

+iε convention Instead of deforming the integration path, one can also shift the two poles and integrate over the whole realp 0-axis, without having to worry about the poles, #» #» p0 = p 2 +m 2 ( p 2 +m 2 iη), ± −→ ± − � � 70 CHAPTER 5. ELEMENTS OF QUANTUM ELECTRODYNAMICS

yielding

4 ip x d p e− · ΔF (x)= lim , (5.54) ε 0+ (2π)4 p2 m 2 + iε → � − the Green’s function of the Klein-Gordon equation,

4 2 2 2 d p ip x p +m (4) ( +m )Δ (x)= e− · − = δ (x). (5.55) � F (2π)4 p2 m 2 − � −

Propagator A propagator is the transition amplitude of a particle between creation µ µ atx and annihilation atx � (or vice-versa). It is a fundamental tool of quantum field theory. After getting the Feynman propagator for the Klein-Gordon field (spin 0), we want to focus on the propagator for fermions (spin 1/2). We compute the anticommutation relations for the field in this case getting,

d3pd3p � i(p x p� x�) ψ(x), ψ¯(x�) = e · − · vr(p)¯sv(p�) b†(p),bs(p�) { } 6 0 0 { r } (2π) 2p 2p� r,s � �� i(p x p x ) e− �· − �· �� u (p)¯u(p�) a (p),a†(p�) r s { r s } 3 � d p ip (x x ) ip (x x ) = e · − � (p/ m) + e − · − � (p/+m) (2π)32p0 − � � 3 � d p ip (x x ) ip (x x ) =(i∂/+m) e− · − � e · − � , (5.56) (2π)32p0 − � � � where we made use of the completeness relations (5.41) and (5.42) in going from the ﬁrst to the second line. We now deﬁne the Feynman fermion propagator,

+ iS(x x �) (i ∂/+m)(Δ (x x �) + Δ−(x x �)). (5.57) − ≡ − −

+ + Splittingψ and ψ¯ in their creationψ −, ψ¯ − and annihilationψ , ψ¯ parts (looking only

at the operatorsa s†, bs† anda s, bs respectively), we get the comutation relations,

+ + + ψ (x), ψ¯ −(x�) =(i ∂/+m)Δ (x x �) = iS (x x �), (5.58) { } − − + ψ−(x), ψ¯ (x�) =(i ∂/+m)Δ −(x x �) = iS−(x x �). (5.59) { } − −

S±(x x �) can as well be represented as contour integrals, − 4 4 d p ip x p/+m d p ip x 1 S±(x)= e− · = e− · , (5.60) (2π)4 p2 m 2 (2π)4 p/ m C�± − C�± − 5.4. DIRAC PROPAGATOR 71 which is well deﬁned because (p/+m)( p/ m) = (p 2 m 2). − − We take a look at the time ordered product of fermion operators,

ψ(x)ψ¯(x�), t>t� T(ψ(x) ψ¯(x�)) = ψ¯(x�)ψ(x), t� > t � − =θ(t t �)ψ(x)ψ¯(x�) θ(t � t) ψ¯(x�)ψ(x). − − − The Feynman fermion propagator is then the vacuum expectation value of this time ordered product,

iS (x x �) = 0 T(ψ(x) ψ¯(x�)) 0 . (5. 61) F − � | | � Remembering the destroying eﬀect of annihilation operators on the vacuum, we can skip some trivial steps of the calculation. We look separately at both time ordering cases, getting,

+ + + 0 ψ(x) ψ¯(x�) 0 = 0 ψ (x)ψ¯ −(x�) 0 = 0 ψ (x), ψ¯ −(x�) 0 = iS (x x �), � | | � � | + | � � |{ + }| � − 0 ψ¯(x�)ψ(x) 0 = 0 ψ¯ (x�)ψ−(x) 0 = 0 ψ¯ (x�),ψ−(x) 0 = iS −(x x �), � | | � � | | � � |{ }| � − yielding,

+ S (x)=θ(t)S (x) θ( t)S −(x)=(i∂/+m)Δ (x), (5.62) F − − F or, as a contour integral,

4 4 d p ip x 1 d p ip x p/+m SF (x)= e− · = lim e− · . (5.63) 4 ε 0+ 4 2 2 (2π) p/ m → (2π) p m + iε C�F − � −

We then see that the fermion propagator is nothing else than the Green’s function of the Dirac equation,

4 d p ip x (p/ m)( p/+m) (4) (i∂/ m)S (x)= e− · − =δ (x). (5.64) − F (2π)4 p2 m 2 � −

The interpretation ofS F is then similar to the one of the Green’s function in classical electrodynamics:

t > t� t� > t x� creation x� annihilation x annihilation x creation We can ask ourselves why the time ordering procedure is important. In scattering processes both orderings are not distinguishable (see Fig. 5.4) in experiments, so that we can understand as a sum over both time ordering possibilities. 72 CHAPTER 5. ELEMENTS OF QUANTUM ELECTRODYNAMICS

x x x + iS (x x �) − + = iSF (x x �) iS−(x x �) − − − x� �x� �x�

Figure 5.4: Sum of both time orderings 5.5 Photon ﬁeld operator

After being able to describe free scalar fields (Klein-Gordon, spin 0) and free fermion fields (Dirac, spin 1/2), we go on to vector fields (spin 1) like the one describing the photon. The photon field will be shown to have a fundamental importance in QED since it is the interaction field between fermions. To start, we recall the photon field operator of advanced quantum mechanics, which reads in Coulomb gauge, #» 3 #» #» #» #» d k 1 #» ik x #» ik x A(x)= a ( k) ε ( k)e − · +a † ( k) ε ∗ ( k)e · . (5.65) (2π)3 √2ω α α α α α=1,2 k � � � � #» #» #» In Eq. (5.65),a α† ( k ) creates a photon of momentum k and polarizationα, anda α( k) destroys the same. Since we are dealing with a bosonic field, we impose the commutation relations, #» #» #» #» 3 (3) [aα( k),a † ( k �)] = g αβ(2π) δ ( k k �), (5.66) #» β #» − #» #» − [aα( k),a β( k �)] = [aα† ( k),a β† ( k �)] = 0. (5.67)

µ Supposing that the photon propagates in thez-direction (k = (k,0,0,k) �), we have the following possibilites for the polarization vectors :

µ µ linear :ε = (0,1,0,0) �, ε = (0,0,1,0) �, • 1 2 µ 1 µ µ 1 µ 1 µ µ 1 circular :ε + = √ (ε1 + iε2 ) = √ (0,1, i,0) �, ε = √ (ε1 iε2 ) = √ (0,1, i,0) �. • 2 2 − 2 − 2 − These vector sets satisfy the completness relation, 0 µν µ ν 1 Π = ε∗ ε = . (5.68) λ λ  1  λ=�  0  ±   (orλ=1,2)   5.5. PHOTON FIELD OPERATOR 73

By applying a well chosen boost to Πµν we can easily check that it is in general not Lorentz invariant. We have to choose a specific gauge depending on the reference frame, parametrized by a real numbern. µ σ To do so we define a auxiliary vectorn =n(1,0,0, 1) � satisfyingn σk = 2kn and get the “axial gauge”, − µ ν µ ν µν µν n k +k n Π = g + σ . (5.69) − nσk Forn = 1, we recover the Co ulomb gauge, 1 1 0 − 1 0 1 Πµν = + = .  1   0   1   1   1   0     −          In physical processes, the photon field couples to an external current, µ µ ik x j (x)=j (k)e · , and we have the current conservation, µ ∂µj = 0, which yields in Fourier space, µ kµj = 0, and thus, µν µν jµΠ =j νΠ = 0, i.e. then µkν +k µnν term vanishes when contracted with external currents, such that we are left with an effective polarization sum, pµν = g µν. (5.70) eff − We now look at the time ordered product of photon field operators,

Aµ(x)Aν(x�), t>t� T(A µ(x)Aν(x�)) = . (5.71) A (x�)A (x), t� > t � ν µ Repeating the same steps as in the fermion case, we get the photon propagator,

iD (x x �) = 0 T(A (x)A (x�)) 0 (5 .72) F,µν − � | µ ν | � = ig Δ (x x �) (5.73) − µν F − 4 ik x d p e− · = ig µν lim . (5.74) − ε 0+ (2π)4 k2 + iε → � Finally, we see that the photon propagator is the Green’s function of the wave equation, (4) �DF,µν(x)=g µνδ (x). (5.75) 74 CHAPTER 5. ELEMENTS OF QUANTUM ELECTRODYNAMICS 5.6 Interaction representation

In the previous sections, we have gained an understanding of the free fields occurring in QED. The next step is to introduce a way to handle interactions between those fields. Idea: decompose the Hamiltonian in the Schrödinger representation,

HS =H 0,S +H S� , and deﬁne states and operators in the free Heisenberg representation,

iH0,S t ψI = e ψS

iH0,S t iH0,S t OI = e OSe− , and you get the interaction representation (also called Dirac representation). We have, in particular,

H0,I =H 0,S =H 0, (5.76) and the time evolution ofψ I respectivelyO I becomes,

i∂tψI =H I� ψI , (5.77) i∂ O = H O +O H = [O ,H ], (5.78) t I − 0 I I 0 I 0 i.e.ψ I is inﬂuenced only by the “true” interaction part; the “trivial” time evolution (free part) has been absorbed in the operatorsO I .

Comparison The Schr¨odinger, Heisenberg, and interaction representations diﬀer in the way they describe time evolution:

Schr¨odinger representation: states contain time evolution, operators are time inde- • pendent;

Heisenberg representation: states are time independent, operators contain time evo- • lution;

Interaction representation: time dependence of states only due to interactions, free • (also called “trivial”) time evolution for operators.

This comparison shows that the interaction representation is a mixture of both other representations. 5.6. INTERACTION REPRESENTATION 75

5.6.1 Time evolution operator

In preparation for time-dependent perturbation theory, we consider the time evolution operatorU(t, t 0) in the interaction representation:

ψI (t)=U(t, t 0)ψI (t0). (5.79)

The time evolution operator in Eq. (5.79) can be written in terms of the free and interaction Hamiltonians, Eq.(5.76), in the Schr¨odinger representation by using the time evolution properties:

iH0t iH0t iHS (t t0) iH0t iHS (t t0) iH0t0 ψI (t)=e ψS(t)=e e− − ψS(t0) =e e− − e− ψI (t0).

Comparing this result with Eq. (5.79) yields

iH0t iHS (t t0) iH0t0 U(t, t 0) =e e− − e− .( 5.80)

An interaction picture operator is related by

OH (t)=U †(t, t0)OI U(t, t 0) to its Heisenberg picture equivalent. Because of Eq. (5.80) the time evolution operator has the following properties:

U(t , t ) =, • 0 0 U(t , t )U(t , t ) =U(t , t ), • 2 1 1 0 2 0 1 U − (t , t ) =U(t , t ), and • 0 1 1 0 1 U †(t , t ) =U − (t , t ) =U(t , t ). • 1 0 1 0 0 1

5.6.2 Time ordering

To ﬁnd the time evolution operator, the time evolution (Schr¨odinger) equation

∂ i U(t, t ) =H � U(t, t ) (5.81) ∂t 0 I 0 has to be solved. This is equivalent to the integral equation

U(t, t ) =+( i) dt H� (t )U(t , t ) 0 − 1 I 1 1 0 t�0 76 CHAPTER 5. ELEMENTS OF QUANTUM ELECTRODYNAMICS which can be iterated to give the Neumann series

U(t, t ) =+( i) dt H� (t ) (5.82) 0 − 1 I 1 t�0

t t1 2 +( i) dt dt H� (t )H� (t ) (5.83) − 1 2 I 1 I 2 t�0 t�0 +... (5.84)

t t1 tn 1 − n +( i) dt dt ... dt H� (t )...H � (t ). (5.85) − 1 2 n I 1 I n t�0 t�0 t�0

This is not yet satisfactory since the boundary of every integral but the ﬁrst depends on the foregoing integration. To solve this problem, one uses time ordering. Let us ﬁrst consider the following identities:

t t1 t t

dt1 dt2HI� (t1)HI� (t2) = dt2 dt1HI� (t1)HI� (t2)

t�0 t�0 t�0 t�2 t t

= dt1 dt2HI� (t2)HI� (t1)

t�0 t�1 where in the ﬁrst line the integration domains are identical (see Fig. 5.5) and in going to the second line the variable labels are exchanged. We can combine these terms in a more compact expression:

t t1

2 dt1 dt2HI� (t1)HI� (t2)

t�0 t�0 t t t t

= dt2 dt1HI� (t1)HI� (t2) + dt1 dt2HI� (t2)HI� (t1)

t�0 t�2 t�0 t�1 t t

= dt dt H� (t )H� (t )θ(t t ) +H � (t )H� (t )θ(t t ) 1 2 I 1 I 2 1 − 2 I 2 I 1 2 − 1 t�0 t�0 � � t t

= dt1 dt2T HI� (t1)HI� (t2) .

t�0 t�0 � � 5.6. INTERACTION REPRESENTATION 77

t1 t0 t

Figure 5.5: Identical integration domains.

All terms of the Neumann series can be rewritten in this way. For then-th term in Eq. (5.85) we have

t tn 1 −

n! dt1 ... dtnHI� (t1)...H I� (tn)

t�0 t�0 t t

= dt1 ... dtnT HI� (t1)...H I� (tn) .

t�0 t�0 � �

We therefore obtain the following perturbation series4 for the time evolution operator:

t t ∞ 1 n U(t, t ) = ( i) dt ... dt T H�(t )...H �(t ) . (5.86) 0 n! − 1 n 1 n n=0 � t�0 t�0 � �

Deﬁning the time ordered exponential, Eq. (5.86) can be written as

U(t, t 0) =T exp i dt�H�(t�) . (5.87) � − � t�0

4We are working in the interaction picture and drop the indexI for simplicity. 78 CHAPTER 5. ELEMENTS OF QUANTUM ELECTRODYNAMICS

We check that this result indeed solves the time evolution equation (5.81):

t t ∞ ∂ 1 n i U(t, t 0) =i ( i) n dt1 ... dtn 1 ∂t n! − − n=1 � t�0 t�0

T H�(t0)...H �(tn 1)H�(t) − � t t � ∞ 1 n 1 =H �(t) ( i) − dt1 ... dtn 1 (n 1)! − − n=1 � − t�0 t�0

T H�(t0)...H �(tn 1) −

=H �(t)U(t, t 0). � �

5.7 Scattering matrix

Our overall aim is to develop a formalism to compute scattering matrix elements which describe the transition from initial states deﬁned att to ﬁnal states observe d att + . To this end, we split u p the Hamiltonian→−∞ into a solvable free part which determines→ ∞ the operators’ time evolution and an interaction part responsible for the time evolution of the states. Now we investigate how the time ordered exponential that is the time evolution operator, see Eq. (5.87), relates to the -matrix. S The scattering matrix element f i is the transition ampli tude for i f caused by interactions. The state of the� |S| system� is described by the time depen|dent�→| state� vector ψ(t) . The above statement about asymptotically large times can now be recast in a more| � explicit form: The initial state is given by

lim ψ(t) = φ i t →−∞ | � | � where φ i is an eigenstate of the free Hamilton operator andt is justiﬁed since | � 15 →−∞ the interaction timescale is about 10− s. The scattering matrix element fi is given by the projection of the state vector ψ(t) onto a ﬁnal state φ : S | � | f � fi = lim φf ψ(t) = φ f φi . t + S → ∞� | � � | S | � Using the time evolution operator (and its action on a state, see Eq. (5.79)), this can be expressed as

fi = lim lim φf U(t 2, t1) φ i . t2 + t1 S → ∞ →−∞ � | | � We can therefore conclude that

∞ ∞ ∞ 1 n =U(+ , ) = ( i) dt ... dt T H�(t )...H �(t ) . (5.88) S ∞ −∞ n! − 1 n 1 n n=0 � � � −∞ −∞ � � 5.7. SCATTERING MATRIX 79

As an instructive example, we consider 2 2 scattering: → k +k k +k . 1 2 → 3 4 The scattering matrix element is given by

= f i = 0 a(k )a(k ) a †(k )a†(k ) 0 . Sfi � |S| � � | 4 3 |S| 1 2 | � φf φi � | | � � �� The -operator itself consists of further creation and annihilation operators belonging to furtherS quantum ﬁelds. By evaluation of the creators and annihilators in (using commutation or anticommutation relations), it follows that there is only one singleS non- vanishing contribution to being of the (“normally ordered”) form S fi

f(k 1, k2, k3, k4)a†(k3)a†(k4)a(k2)a(k1).

Note that in the above expression, the annihilation operators stand on the right hand side, while the creation operators are on the left. Such expressions are said to be in normal order and are denoted by colons, :ABC:. Since the aim is to ﬁ nd the non- vanishing contributions, a way has to be found how time ordered products can be related to products in normal order. For instance, consider the time ordered product of two Boson + + 5 ﬁeld operators (whereA ,B are annihilators andA −,B− creators)

T A(x1)B(x2) =A(x 1)B(x2) t1>t2 � �� + + + � =A (x1)B (x2) +A −(x1)B (x2) � + +A (x1)B−(x2) +A−(x1)B−(x2).

not in normal order

5The sign is motivated by t he decomposition� of ﬁe��ld operators� in positive and negative frequency parts: ±

+ φ(x) =φ (x)+φ −(x).

Consider for example the Klein-Gordon ﬁeld where

3 #» #» #» #» d p 1 +i p x i p x φ(x) = a(p)e · +a †(p)e− · (2π)3 2p0 � � � and therefore �

3 #» #» + d p 1 +i p x φ (x)= a(p)e · (2π)3 2p0 � 3 #» #» d p 1 i p x φ−(x)= � a†(p)e− · . (2π)3 2p0 � � 80 CHAPTER 5. ELEMENTS OF QUANTUM ELECTRODYNAMICS

One can observe that only one of the above terms is not in normal order while the other three would vanish upon evaluation in 0 0 . Using � |·| � + + + A (x1)B−(x2) =B −(x2)A (x1) + [A (x1),B−(x2)],

not in normal order in normal order c-number

we rewrite � ��

+ + [A (x1),B−(x2)] = 0 [A (x1),B−(x2)] 0 � | + | � = 0 A (x )B−(x ) 0 � | 1 2 | � = 0 T(A(x )B(x )) 0 . � | 1 2 | �

Since the same holds fort 1 < t2, we draw the conclusion

T A(x )B(x ) = :A(x )B(x ):+ 0 T(A(x )B(x )) 0 . 1 2 1 2 � | 1 2 | � � � An analogous calculation for fermion operators yields the same result. The next step towards Feynman diagrams is to formalize this connection between time and normal ordered products. We ﬁrst deﬁne the following shorthand

φ (x )φ (x ) = 0 T(φ (x )φ (x )) 0 A 1 B 2 � | A 1 B 2 | �

which is called contraction of operators. This allows to state the following in compact notation.

Wick’s theorem: The time ordered product of a set of operators can be decomposed into the sum of all corresponding contracted products in normal order. All combinatorially allowed contributions appear:

T(ABC . . . XY Z)=:ABC...XYZ: +:AB C...XYZ:+ +:ABC...XYZ :+ +:ABC...XYZ : ··· ··· +:AB CD ...XYZ:+:ABCD ...XYZ:+...

+:threefold contractions:+....

5.8 Feynman rules of quantum electrodynamics

The Lagrangian density of QED is given by

Dirac photon = + + � L L 0 L 0 L 5.8. FEYNMAN RULES OF QUANTUM ELECTRODYNAMICS 81

where the subscript 0 denotes the free Lagrangian densities and � denotes the interaction part. In particular, we have

Dirac = ψ¯(i∂/ m)ψ (5.89) L0 − 1 photon = F F µν (5.90) L0 − 4 µν µ µ � = e ψγ¯ ψA = j A (5.91) L − µ − µ µν µ ν ν µ photon whereF =∂ A ∂ A . Note that from 0 the free Maxwell’s equations can be − L photon derived using the Euler-Lagrange equations. Using 0 + � yields Maxwell’s equations Dirac L L in the presence of sources and + � does the same for the Dirac equation. The L 0 L interaction term � describes current-ﬁeld interactions and therefore couples the fermions described by theL Dirac equation to photons described by Maxwell’s equations. Using Eq. (5.91), one ﬁnds the quantized interaction Hamiltonian density

µ � = � =e ψγ¯ ψA . H −L µ Integrating the interaction Hamiltonian density over all space yields the interaction Hamil- tonian,

3 #» H� = d x �, H � and, in the integral representation of given in Eq. (5.88), this leads to integrations over space-time: S

∞ 1 = ( ie)n d4x . . . d4x T ψ¯(x )γ ψ(x )Aµ1 ... ψ¯(x )γ ψ(x )Aµn . (5.92) S n! − 1 n 1 µ1 1 n µn n n=0 � � � � Sincee= √4πα (see Eq. (1.9)), the coupling constant appears in the interaction term and n-th order terms are suppressed withe n. This means that we found an expansion of in the small parametere which is the starting p oint for perturbation theory. The structureS of then-th term in the perturba tion series in Eq. (5.92) is 1 (n) = d4x . . . d4x (5.93) S n! 1 nSn � where

= K(x , . . . , x ):... ψ¯(x )...ψ(x )...A(x ):. (5.94) Sn 1 n i j n contractions � For a speciﬁc scattering process, the relevant matrix element is

= f i Sfi � | S | � a a ∼ ∼ † �� 82 CHAPTER 5. ELEMENTS OF QUANTUM ELECTRODYNAMICS

Figure 5.6: First order contributions (1). These processes violate energy-momentum conservation and are therefore unphysical.S

which means that only terms in matching f i yield contributions to t he transition amplitude. The following ﬁeld operators,S which� |·|constitute� the Feynman rules in position space, are contained in ( −−−→time ). S + x ψ (x) absorption of electron atx ¯+ x ψ (x) absorption of positron atx ¯ x ψ −(x) emission of electron atx x ψ−(x) emission of positron atx + x A (x) absorption of photon atx x A−(x) emission of photon atx ψ(x )ψ(x ) 2 1 x1 x2 Fermion propagator = iSF (x2 x 1) µ ν− A (x2)A (x1) x1 x2 photon propagator = iDµν(x x ) F 2 − 1 ¯ µ ieψ(x)γµψ(x)A (x) vertex atx −= ieγ vertex atx x − µ · The -operator at ordern is examined using Wic k’s theorem. At ﬁst order, this yields (rememberingS Eq. (5.92) while ignoring disconnected contributions from Wick’s theorem) the following 23 = 8 contributions:

(1) = ie d4xT( ψ¯(x)γ ψ(x)Aµ(x)) = ie d4x: ψ¯(x)γ ψ(x)Aµ(x):. S − µ − µ � � There is a total of 8 possible combinations, sinceA µ creates or annihilates a photon, ψ¯ creates an electron or annihilates a positron, andψ creates a position or a nnihilates an electron. Fig. 5.6 shows the corresponding Feynman diagrams. However, all these processes are unphysical because they violate energy-momentum con- 5.8. FEYNMAN RULES OF QUANTUM ELECTRODYNAMICS 83

servation:

p + p p = 0 ± e ± e− ± γ �

which is because free particles fulﬁll

2 2 2 2 2 p + =m p =m p = 0. e e e− e γ

To ﬁnd physical contributions to the interaction Hamiltonian, we turn to the second order contributions to (see Eq. (5.92)): S

1 (2) = ( ie)2 d4x d4x T ψ¯(x )γ ψ(x )Aµ1 (x )ψ¯(x )γ ψ(x )Aµ2 (x ) . S 2! − 1 2 1 µ1 1 1 2 µ2 2 2 � � �

Application of Wick’s theorem yields contraction terms. We ﬁrst note that contractions of the form

ψ(x1)ψ(x2) ψ(x1)ψ(x2)

vanish because they contain creators and annihilators, respectively, for diﬀerent particles and thus

0 T(ψ(x )ψ(x )) 0 =0. � | 1 2 | �

The remaining terms read, using shorthands like ψ¯(x1) = ψ¯1, 84 CHAPTER 5. ELEMENTS OF QUANTUM ELECTRODYNAMICS

(2) ( ie)2 4 4 = − d x d x S 2! 1 2 � �

¯ µ1 µ2 :ψ1γµ1 ψ1ψ2γµ2 ψ2A1 A2 : (a) x1

¯ µ1 µ2 x2 +:ψ1γµ1 ψ1ψ2 γµ2 ψ2A1 A2 : (b)

µ1 µ2 x1 +:ψ1γµ1 ψ1ψ2γµ2 ψ2 A1 A2 : (c)

x1 x2 ¯ µ1 µ2 +:ψ1γµ1 ψ1ψ2γµ2 ψ2 A1 A2 : (d)

x2 ¯ µ1 µ2 +:ψ1γµ1 ψ1ψ2 γµ2 ψ2 A1 A2 : (e)

x1 µ1 µ2 +:ψ1γµ1 ψ1ψ2γµ2 ψ2 A1 A2 : (f)

µ1 µ2 +:ψ1γµ1 ψ1ψ2γµ2 ψ2 A1 A2 : (g)

µ1 µ2 +:ψ1γµ1 ψ1 ψ2γµ2 ψ2 A1 A2 : (h)

µ1 µ2 +:ψ1γµ1 ψ1 ψ2γµ2 ψ2 A1 A2 : (i)

µ1 µ2 +:ψ1γµ1 ψ1ψ2γµ2 ψ2 A1 A2 : (j)

µ1 µ2 +:ψ1γµ1 ψ1 ψ2γµ2 ψ2 A1 A2 : . (k) � 5.8. FEYNMAN RULES OF QUANTUM ELECTRODYNAMICS 85

It follows a discussion of the contributions (a) through (k).

(a) Independent emission or absorption. These diagrams violate energy-momentum conservation.

(b)&(c) Processes involving two electrons or positrons and two photons.

+ + 1. Compton scattering:γe − γe −, γe γe →→

+ 2. Electron-positron pair annihilation:e e− γγ µ1 → µ1 e−A e−A e+ Aµ2 e+ Aµ2 + 3. Electron-positron pair creation:γγ e e− µ1 µ→1 A e− A e− Aµ2 e+ Aµ2 e+

(d) Processes involving four electrons or positions.

+ + + + 1. Møller scattering:e −e− e −e−, e e e e → → e−e− e−e− e− e− e− e− + + 2. Bhabha scattering:e e− e e− + → + + ee− ee + e− e e− e− 86 CHAPTER 5. ELEMENTS OF QUANTUM ELECTRODYNAMICS

(e)&(f) No interaction between external particles. No scattering takes place, these terms are corrections to the fermion propagator.

(g) Correction to photon propagator.

(h)&(i) Corrections to fermion propagator, vanishing.

(j)&(k) Vacuum vacuum transitions, dis connected graphs. →

This constitutes a list of all known processes (for practical purposes) in (2); in general, we can find all processes by examining all orders of the scattering matrixS operator . S The -matrix elements are d efined as matrix elements between single-particle states. Consequently,S we need to apply the norm (5.49) repectively (5.50) to external states. The invariant amplitudes fi, which are derived from the -matrix elements accord ing to Eq. (3.11) properly accountM for this normalization factor,S and are evaluated for continuum states as defined in Eq. (5.47) and Eq. (5.48). The contractions of the field operators (see Eq. (5.43) and (5.44)) with external momentum eigenstates (as given in Eq. (5.49) and (5.50)) are for electrons

1 d3k 1 ψ(x) e−(p, s) = ar(k)ur(k) 2Epa†(p) 0 single-particle 2E V (2π)3 √2E s | � p � k r � � � � � 1 ipx = � e− us(p) 0 2EpV | �

1 +ipx − ¯ � e (p, s) single-particle ψ(x)= e 0 ¯us(p), 2EpV � | � � � � for positrons

+ 1 ipx ¯ − ψ(x) e (p, s) single-particle = e ¯sv(p) 0 2EpV | � � � 1 + � � +ipx e (p, s) single-particle ψ(x)= e 0 v s(p), 2EpV � | � � � � and for photons

1 ikx λ Aµ(x) γ(k,λ) = e− εµ(k) | � √2EkV 1 +ikx λ γ(k,λ) A µ(x)= e εµ∗ (k). � | √2EkV 5.8. FEYNMAN RULES OF QUANTUM ELECTRODYNAMICS 87

Example We treat the case of Møller scatteringe −e− e −e− as a typical example for the application of the Feynman rules. → We first define our initial and final states,

1 i = e−(p , s ) e−(p , s ) = 2E 2E a† (p )a† (p ) 0 , 1 1 single-particle 2 2 single-particle 1 2 s1 1 s2 2 | � ⊗ √2E1V2E 2V | � � � � � � 1 � − � − f = e (p3, s3) single-particle e (p4, s4) single-particle = 2E32E4 0 a s4 (p4)as3 (p3). � | ⊗ √2E3V2E 4V � | � � � � � � � The transition matrix element is then, S fi ( ie)2 = f i = − d4x d4x 16E E E E 0 a (p ) a (p ) Sfi � |S| � 2! 1 2 1 2 3 4 � | s4 4 s3 3 � � E D µ ν :ψ¯(x ) γ ψ(x ) ψ¯(x ) γ ψ(x ):A (x )A (x ) a† (p ) a† (p ) 0 , (5.95) 1 µ 1 2 ν 2 1 2 s1 � 1�� s�2 � 2��| � D C E A A C

B � �� =��ψ¯(�x2�)ψ��(x1�) � �� − � �� yielding 2 2 = 4 Feynman graphs in position space (of which 2! are topologically identical). In Fig.× 5.7, we labeled the last Feynman graph according to Eq. (5.95).

p1 p4 p1 p4 p1 p3

x2 x1 x2

+ − x1 �x2 �x1 p2 p3 p2 p3 p2 p4 p1 p3

x1 C D

−B A E �x2 p2 p4

Figure 5.7: Feynman graphs associated with the Møller scattering.

We recall that each ordering ofψ, ψ¯ corresponds to a Feynman diagram. The anticommutation relations are responsible for the relative sign changes. 88 CHAPTER 5. ELEMENTS OF QUANTUM ELECTRODYNAMICS

With the photon propagator in momentum space, igµν iDµν(q)= , (5.96) F − q2 we get, 1 =( ie)2(2π)4δ(4)(p +p p p ) fi 3 4 1 2 2 S − − − √16E1E2E3E4V ¯u (p )γ u (p )iDµν(p p )¯u (p )γ u (p ) s4 4 µ s2 2 F 3 − 1 s3 3 µ s1 1 � µν ¯u (p )γ u (p )iD (p p )¯u (p )γ u (p ) . (5.97) − s4 4 µ s1 1 F 3 − 2 s3 3 µ s2 2 � We now deﬁne the invariant amplitude (see Eq. (3.11)) via, M fi 1 =δ +i(2π) 4δ(4)(p +p p p ) . (5.98) fi fi 3 4 1 2 2 fi S − − √16E1E2E3E4V M can then be computed using the Feynman rules in momentum space. Mfi

Application of the Feynman rules

Momentum conservation at each vertex • Fermion number conservation at each vertex (indicated by the direction of the ar- • rows) All topologically allowed graphs contribute • Exchange factor ( 1) when interchanging two external fermions with each other • − Each closed fermion loop yields a factor ( 1), e.g. x1 x2 coming from the • − contraction :ψ(x1)ψ(x1)ψ(x2)ψ(x2):

graphs in which the ordering of the vertices along a fermion line is diﬀerent are not • topologically equivalent, and must be summed, eg.

(2) : = S � �

(4) : = S � � 5.8. FEYNMAN RULES OF QUANTUM ELECTRODYNAMICS 89

External states

p, s incoming electron us(p)

p, s outgoing electron ¯us(p)

p, s incoming positron ¯sv(p)

p, s outgoing positron vs(p)

µ incoming photon λ, µ, p (ελ) (p)

µ outgoing photon λ, µ, p (ελ∗ ) (p)

Propagators

i(p/+m) p electron p2 m2+iε − µ ν igµν photon k −k2+iε

Vertex µ

µ electron-photon-electron q ieqeγ e −

Table 5.2: Feynman rules in momentum space. 90 CHAPTER 5. ELEMENTS OF QUANTUM ELECTRODYNAMICS 5.9 Trace techniques forγ-matrices

2 2 Cross sections are proportinal to fi ¯usf (pf )Γusi (pi) , where Γ denotes an arbitrary product ofγ-matrices. |M | ∝ | | In many experiments – but not all! –, the spin states of the initial and ﬁnal states are not observed. This is for example the case at the CMS and ATLAS experiments of LHC. We then need to follow the following procedure :

If the spin state of the ﬁnal state particles cannot be measured, one must sum over • the ﬁnal state spins : 2, sf | · · · | If the initial states par�ticles are unpolarized, one must average over the initial state • spins : 1 2. 2 si | · · · | � 0 Then, remembering that ¯u=u†γ , we can write,

1 2 1 0 0 0 ¯u (p )Γu (p ) = ¯u (p )Γu (p )u† (p )γ γ Γ†γ u (p ) 2 | sf f si i | 2 sf f si i si i sf f s ,s s ,s �i f �i f 1 = ¯u (p )Γu (p )¯u(p )Γ¯u (p ) 2 sf f si i si i sf f s ,s �i f 1 = (¯u (p )) Γ (u (p )) (¯u(p )) Γ¯ (u (p )) 2 sf f α αβ si i β si i γ γδ sf f δ s ,s �i f (5.41) 1 ¯ = Γαβ(p/ +m) βγΓγδ(p/ +m) δα 2 i f 1 = Γ(p/ +m) Γ(¯ p/ +m) 2 i f αα 1 � � = Tr Γ(p/ +m) Γ(¯ p/ +m) , 2 i f � � 0 0 where the indices α,β,γ andδ label the matrix elemen t, and Γ¯ :=γ Γ†γ . We thus get the important result,

1 2 1 ¯ ¯us (pf )Γusi (pi) = Tr Γ(p/ +m) Γ(p/ +m) , (5.99) 2 | f | 2 i f s ,s �i f � � and its analogon for antiparticles,

1 2 1 ¯ ¯sv (pf )Γvsi (pi) = Tr Γ(p/ m) Γ(p/ m) , (5.100) 2 | f | 2 i − f − s ,s �i f � � i.e. the Cliﬀord algebra ofγ-matrices is taking care of the spin summation for us. We now compute Γ¯ for an arbitrary number ofγ-matrices. 5.9. TRACE TECHNIQUES FORγ-MATRICES 91

µ 0 0 i µ 0 µ 0 µ µ For Γ =γ ,(γ )† =γ ,(γ )† = γ i hence (γ )† =γ γ γ γ¯ =γ . For later use, • note that γ¯5 = γ 5. − ⇒ −

µ1 µn µ1 µn 0 µn µ1 0 µn µ1 For Γ =γ γ ,Γ† = (γ γ )† =γ γ γ γ Γ¯ =γ γ . In • other words, to··· get Γ,¯ we just need··· to read Γ in the··· inverse orderin⇒ g. ···

We ﬁnally want to compute some traces for products ofγ-matrices, since they ap pear ¯ µ explicitly (Γ, Γ) and implicitely (p/=γ pµ) in the formulas (5.99) and (5.100). In doing this, one should remember that the trace is cyclic (Tr(ABC) = Tr(BCA)) and the Cliﬀord algebra ofγ-matrices.

0γ-matrix : Tr = 4. • 1γ-matrix : Trγ µ = 0, Trγ 5 = 0. The last one is shown using the fact that • γ5, γµ =0. { } µ ν 1 µ ν ν µ µν / 2γ-matrices : Tr(γ γ ) = 2 Tr(γ γ +γ γ ) = 4g Tr(a/b)=4a b, where is • the scalar product of 4-vectors. ⇒ · ·

4γ-matrices : • Tr(γµγνγργσ) = Tr(γνγργσγµ) = Tr(γ νγργµγσ) + 2gµσTr(γνγρ) − = Tr(γνγµγργσ) + 8gµσgνρ 8g µρgνσ − = Tr(γ µγνγργσ) + 8gµσgνρ 8g µρgνσ + 8gµνgρσ − − Tr(γµγνγργσ) = 4(gµνgρσ +g µσgνρ g µρgνσ) ⇒ − Tr(a/ a/ a/ a/ ) = 4 [(a a )(a a ) + (a a )(a a ) (a a )(a a )]. ⇒ 1 2 3 4 1 · 2 3 · 4 1 · 4 2 · 3 − 1 · 3 2 · 4 and in general,

Tr(a/ a/ ) =(a a )Tr(a/ a/ ) (a a )Tr(a/ a/ a/ ) 1 ··· n 1 · 2 3 ··· n − 1 · 3 3 4 ··· n + (a 1 a n)Tr(a/2 a/n 1), ···± · ··· − which implies inductivly that the trace of a string ofγ-matrices is a real numb er.

nγ-matrices (n odd) : • Tr(γµ1 γ µn ) = Tr(γµ1 γ µn γ5γ5) = Tr(γ5γµ1 γ µn γ5) ··· ··· ··· = = ( 1)nTr(γµ1 γ µn ) Tr(γ µ1 γ µn ) = 0. − ···�� ⇒ ···

nγ-matrices (n even) : •

µ1 µn µ1 µn 0 µn µ1 0 µn µ1 Tr(γ γ ) = Tr((γ γ )†) = Tr(γ γ γ γ ) = Tr(γ γ ). ··· ··· ··· ··· 92 CHAPTER 5. ELEMENTS OF QUANTUM ELECTRODYNAMICS

γ 5 and 2γ-matrices : Tr(γ 5γµγν) = 0. To show this identity, we remark that • γ5γµγν is a rank-2 tensor, which does not depend on any 4-momenta. Therefore, 5 µ ν µν 5 µ µν Tr(γ γ γ ) = cg . We contract withg µν to get Tr(γ γ γµ) = cg gµν = 4c, but µ 5 sinceγ γµ = 4 we getc = Trγ = 0. γ 5 and 4γ-matrices : Tr(γ 5γµγνγργσ) = 4iε µνρσ. • − Contractions : • µ γ γµ = 4 (5.101) γµaγ/ = 2 a/ (5.102) µ − γµa//bγ = 4(a b) (5.103) µ · γµa//b/cγ = 2 /c/ba/ (5.104) µ −

+ + 5.10 Annihilation process :e e− µ µ− → In this section, we compute the diﬀerential cross section of the simplest of all QED process, the reaction

+ + e−(p )e (p ) µ −(p )µ (p ), 1 2 → 3 4 illustrated on Fig. 5.8. The simplicity arises frome the fact thate − =µ −, and hence only + � one diagram contributes (thee e−-pair must be annihilated).

e− µ−

p1 p3

p1 +p 2 p2 p4 e+ µ+

+ + Figure 5.8: Annihilation processe e− µ µ− → We recall the Mandelstam variables for this process,

2 s=(p 1 +p 2) t=(p p )2 1 − 3 u=(p p )2. 1 − 4 We make the following assumptions (very common for QED processes), + + 5.10. ANNIHILATION PROCESS :E E− µ µ− 93 → Unpolarized leptons : 1 , • 2 si,sf � High energy limit :m , m = 0 √s m , m . • e µ ⇔ � e µ

Using the Feynman rules of Table 5.2 for the diagram depicted in Fig. 5.8, we get,

µ igµν ν i fi = ¯us3 (p3)ieγ vs4 (p4) − 2 ¯sv2 (p2)ieγ us1 (p1) − M (p1 +p 2) 1 1 2 = 2 |Mfi| 2 2 |Mfi| s1 s2 s3 s4 �4 � � � 1 e µ ν = Tr(γ p/ γ p/ )Tr(γµp/ γνp/ ) 4 s2 4 3 1 2 1 e4 = 16 [2(p p )(p p ) + 2(p p )(p p )]. 4 s2 1 · 3 2 · 4 1 · 4 2 · 3

2 Since we are working in the high energy limit, we havep i = 0 and hencet= 2p 1 p 3 = 2 − 2 · 2p2 p 4 andu= 2p 1 p 4 = 2p 2 p 3. Using the identitys+t+u=2m e + 2mµ = 0 to − · − · − · e2 get rid of the Mandelstamu-variable and withα= 4π we have,

t2 + (s+t) 2 2 = 32π2α2 . (5.105) |Mfi| s2

2 s Considering the center of mass frame, we haves=4(E ∗) , t= 2 (1 cos Θ ∗) and with the help of Eq. (3.34), this yields, − −

2 dσ 1 2 πα 2 = = (1 + cos Θ∗), (5.106) dt 16πs2 |Mfi| s2

or using,

dσ dΩ dσ 4π dσ = ∗ = , dt dt dΩ∗ s dΩ∗

+ + we get the diﬀerential cross section fore e− µ µ− in the center of mass frame, →

+ + e e− µ µ− 2 dσ → α 2 = (1 + cos Θ∗) . (5.107) dΩ∗ 4s

This diﬀerential cross section (see Fig. 5.9) has been very well measured and is one of the best tests of QED at high energies. 94 CHAPTER 5. ELEMENTS OF QUANTUM ELECTRODYNAMICS

dΣ d��

�� Π Π 2

+ + Figure 5.9: Diﬀerential cross section fore e− µ µ− in the center of mass frame. →

Using this result, we can calculate the total cross-section by integration over the solid angle:

π 2π 2 dσ α 2 σ= dΩ∗ = (1 + cos Θ∗) sin Θ∗dΘ∗ dφ (5.108) dΩ∗ 4s � �0 d cos Θ∗ �0 � �� 2π α2 8 = 2π � �� (5.109) 4s 3 2 e+e µ+µ 4πα 86.9 nb σ −→ − = = (5.110) ⇒ 3s s [GeV2]

33 2 2 2 where 1 nb = 10− cm . If one considers non-asymptotic energies,s m µ (buts m e), 2 � � one ﬁnds a result which reduces to Eq. (5.110) form µ = 0 :

2 2 2 e+e µ+µ 4πα mµ 4mµ σ −→ − = 1 + 2 1 . 3s s − s � � �

5.11 Compton scattering

Let us now consider Compton scattering:

γ(k)+e −(p) γ(k �) +e −(p�). → 5.11. COMPTON SCATTERING 95

The diagrams corresponding to this process have been introduced in Sect. 5.8. k, λ, ν k�, λ�, µ k, λ, ν k�, λ�, µ + p p� p p� k+p= p k � = − k� +p � p� k s-channel u-channel− This yields the amplitude

 µ i ν ν i µ i fi =ε µ∗ (k�, λ�)εν(k,λ)¯u(�p) ieγ ieγ + ieγ ieγ u(p) − M p/+ k/ m p/ k/� m  − − −     LHS diagram RHS diagram    where the on-shell conditions read � ��

2 2 2 2 2 k =k � = 0 p =p � =m and the photons are transversal:

k ε(k)=k � ε(k �) = 0. · · It is instructive to check that the invariant amplitude is indeed also gauge invariant. Consider the gauge transformation

A (x) A (x)+∂ Λ(x) ν → ν ν which leaves Maxwell’s equations unaltered. In the photon ﬁeld operator this can be implemented by

εν(k,λ) ε ν(k,λ)+βk ν, β R arbitrary. → ∈ We observe the change of the matrix element for transformation of one of the photons:

2 µ 1 1 µ i fi(εν k ν) = ie εµ∗ (k�, λ�)¯u(�p) γ k/+ k/ γ u(p). − M → − � p/+ k/ m p/ k/� m � − − − In simplifying this expression, we use 1 1 ku/ (p)= (k/+ p/ m)u(p)=u(p) p/+ k/ m p/+ k/ m − − − 96 CHAPTER 5. ELEMENTS OF QUANTUM ELECTRODYNAMICS where we added a zero since (p/ m)u(p) = 0 and analogously − 1 1 ¯u(�p)k/ = ¯u(�p)(k/ p/� +m) = ¯u(p�). p/ k/� m − p/� k/ m − − − − − Putting the terms together, we therefore ﬁnd

2 µ µ i (ε k ) = ie ε∗ (k�, λ�)¯u(�p)(γ γ )u(p)=0. − Mfi ν → ν − µ −

The result is the same for the transformationε ∗ ε ∗ +βk � . µ → µ µ It is generally true that only the sum of the contributing diagrams is gauge invariant. Individual diagrams are not gauge invariant and thus without physical meaning. Recall that the aim is to ﬁnd the diﬀerential cross section and therefore the squared matrix element. Since there are two contributing diagrams, one has to watch out for interference terms. Applying the trace technology developed in Sect. 5.9 yields 1 1 2 = 2 |Mfi| 2 2 |Mfi| s �λ � �λ� �s� m2 u m2 s m2 m2 m2 m2 2 = 2e4 − + − + 4 + + 4 + . s m 2 u m 2 s m 2 u m 2 s m 2 u m 2 � − − � − − � � − − � � (5.111)

Bearing in mind thats+t+u=2m 2, this yields the unpolarized Compton cross-section dσ 1 = 2 (5.112) dt 16π(s m 2)2 |Mfi| − which is a frame independent statement. Head-on electron-photon collision is rather uncommon; usually photons are hitting on a target. Therefore it is useful to consider the electron’s rest frame (laboratory frame):

k�

ΘL k p� #» #» #» L L T Withω= k =E , ω� = k � =E � , andp=(m, 0 ) one ﬁnds | | γ | | γ s m 2 = 2mω (5.113) − 2 u m = 2p k � = 2mω � (5.114) − − · − t= 2ωω �(1 cos Θ ).( 5.115) − − L 5.11. COMPTON SCATTERING 97

One of the three variables can be eliminated usings+t+u=2m 2:

1 2 ωω� ω� = (s+t m ) =ω (1 cos Θ ) (5.116) 2m − − m − L 1 1 1 = (1 cos Θ ) (5.117) ⇒ ω − ω m − L � ω ω � = . (5.118) ⇒ 1 + ω (1 cos Θ ) m − L We continue calculating the diﬀerential cross-section. Eq. (5.115) yields

ω 2 ω 2 dt= � 2πd cos Θ = � dΩ . π L π L Furthermore, we can use Eq. (5.117) to simplify Eq. (5.111):

m2 m2 m2 m2 m 1 1 1 + = + = = (1 cos Θ ). s m 2 u m 2 2mω 2mω 2 ω − ω − 2 − L − − − � � � � Using these results and remembering Eq. (5.112), we obtain

γe γe 2 dσ → dt dσ ω� 1 2 2mω� 2mω 2 = = 2 2e + − sin ΘL (5.119) dΩL dΩL dt π 16π(2mω) 2mω 2mω� − 2 2 2 � − � ω� 2 16π α ω� ω = · + sin2 Θ (5.120) π 16π4m2ω2 ω ω − L � � � dσγe γe α2 ω 2 ω ω → = � � + sin2 Θ (5.121) ⇒ dΩ 2m2 ω ω ω − L L � � � � � which is called the Klein-Nishima formula. It follows a discussion of important limiting cases.

Classical limit:ω m ω � ω • In the classical limit,� ⇒ Eq. (5.121)� simpliﬁes to the classical Thomson cross-section (which was used to measureα)

γe γe 2 dσ → α 2 = 2 1 + cos ΘL , dΩL 2m � � yielding the total cross-section

2 γe γe α 16π σ → = . 2m2 3 98 CHAPTER 5. ELEMENTS OF QUANTUM ELECTRODYNAMICS

Asymptotic limit:s m 2 ω m • In this case, the so-calle� d leading⇒ � log approximation holds:

2 2 2 2 γe γe 2πα m s 1 m 2πα s σ → = ln + + ln . m2 s m2 2 O s � s m2 � � �� In general we can conclude that • 2 γe γe α 25 2 σ → 10− cm ∼ m2 � from which one can infer the “classical electron radius”

classical α 13 r √σ = 2.8 10 − cm. e ∼ Thomson ∼ m ·

5.12 QED as a gauge theory

Recall the QED Lagrangian 1 QED = ψ¯(iγ µ∂ m)ψ eq ψγ¯ µψA F F µν L µ − − e µ − 4 µν Dirac photon = + � + L 0 L L 0 introduced in Sect. 5.8 which includes the following observables:

Fermions: components of ψγ¯ µψ=j µ • #» #» Photons: components ofF µν: E and B ﬁeld. •

Neitherψ norA µ as such are observables. In particular, the phase ofψ cannot be observed. This means that QED must be invariant under phase transformations ofψ:

ieqeχ(x) ψ(x) ψ �(x)=e ψ(x) → which is a unitary one-dimensional i. e.U(1) transformation. Obs erve ﬁrst the action on the Dirac Lagrangian:

Dirac µ ψ¯�(iγ ∂ m)ψ � L0 → µ − ieqeχ(x) ieqeχ(x) µ = ψe¯ − e (i∂/ m)ψ ψγ¯ (∂ eq χ(x))ψ − − µ e = Dirac eq ψγ¯ µψ(∂ χ(x)). L 0 − e µ Therefore, the free Dirac ﬁeld Lagrangian alone is not invariant under this transformation. In order for the extra term to vanish,A µ has to be transformed, too:

A A � =A ∂ χ(x) µ → µ µ − µ 5.12. QED AS A GAUGE THEORY 99

QED photon U(1) 0 Weak interaction W ±,Z SU(2) QCD gluon SU(3)

Table 5.3: Summary of gauge theories.

µν µν µν µ ν ν µ such thatF =F � sinceF =∂ A ∂ A . This means that we are dealing with the gauge transformation known from classical− electrodynamics. Because we have

eq ψγ¯ µψA eq ψγ¯ µψA +eq ψγ¯ µψ(∂ χ) − e µ → − e µ e µ the complete Lagrangian QED is invariant underU(1) gauge transformatio ns. This mo- tivates the deﬁnition of theL gauge covariant derivative

Dµ =∂ µ + ieqeAµ which contains the photon-electron interaction. In summary, the requirement of gauge invariance uniquely determines the photon-electron interaction and QED is aU(1) gauge theory. This suggests a new approach on theory building: start from symmetries instead of ﬁnding them in the ﬁnal Lagrangian:

local symmeties existence of vector ﬁelds gauge interactions . (gauge invariance) → (gauge ﬁelds) →

A summary of gauge theories with the corresponding gauge ﬁelds and gauge groups is given in Tab. 5.3. 100 CHAPTER 5. ELEMENTS OF QUANTUM ELECTRODYNAMICS