Advanced Quantum Field Theory Chapter 2 Physical States. S Matrix

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Advanced Quantum Field Theory Chapter 2 Physical States. S Matrix. LSZ Reduction

Jorge C. Rom˜ao Instituto Superior T´ecnico, Departamento de F´ısica & CFTP A. Rovisco Pais 1, 1049-001 Lisboa, Portugal

Fall 2013 Lecture 3

Physical states

LSZ Reduction

Fermions

Photons

Cross sections

Lecture 3

Jorge C. Rom˜ao TCA-2012 – 2 Physical states

❐ In the previous chapter we saw, for the case of free ﬁelds, how to construct Lecture 3 the space of states, the so-called Fock space of the theory. When we Physical states • In states consider the real physical case, with interactions, we are no longer able to • Spectral rep • Out States solve the problem exactly. For instance, the interaction between electrons

LSZ Reduction and photons is given by a set of nonlinear coupled equations,

Fermions Photons (i∂/ m)ψ = eA/ψ − Cross sections µν ν ∂µF = eψγ ψ

that do not have an exact solution. ❐ In practice we have to resort to approximation methods. In the following chapter we will learn how to develop a covariant perturbation theory. Here we are going just to study the general properties of the theory. ❐ Let us start by the physical states. As we do not know how to solve the problem exactly, we can not prove the assumptions we are going to make about these states. However, these are reasonable assumptions, based on Lorentz covariance. We choose our states to be eigenstates of energy and momentum, and of all the other observables that commute with P µ.

Jorge C. Rom˜ao TCA-2012 – 3 Physical states

Lecture 3 ❐ Besides that, we will also assume the following properties: Physical states • In states 2 0 • Spectral rep ◆ The eigenvalues of p are non-negative and p > 0. • Out States

LSZ Reduction ◆ There exists one non-degenerate base state, with the minimum of Fermions energy, which is Lorentz invariant. This state is called the vacuum state Photons 0 and by convention Cross sections | i

pµ 0 = 0 | i ◆ There exist one particle states p(i) , such that,

(i) (i)µ 2 pµ p = mi

for each stable particle with mass mi. ◆ The vacuum and the one-particle states constitute the discrete spectrum of pν .

Jorge C. Rom˜ao TCA-2012 – 4 In states

❐ As we are mainly interested in scattering problems, we should construct Lecture 3 states that have a simple interpretation in the limit t . At that time, Physical states → −∞ • In states the particles that are going to participate in the scattering process have not • Spectral rep • Out States interacted yet (we assume that the interactions are adiabatically switched off when t which is appropriate for scattering problems). LSZ Reduction | | → ∞ Fermions ❐ Photons We look for operators that create one particle states with the physical mass. Cross sections To be explicit, we start by an hermitian scalar field given by the Lagrangian 1 1 = ∂µϕ∂ ϕ m2ϕ2 V (x) L 2 µ − 2 − where V (x) is an operator made of more than two interacting fields ϕ at point x. For instance, those interactions can be self-interactions of the type λ V (x) = ϕ4(x) 4!

❐ The field ϕ satisfies the following equation of motion ∂V ( + m2)ϕ(x) = j(x) ⊔⊓ −∂ϕ(x) ≡ Jorge C. Romão TCA-2012 – 5 In states

❐ The equal time canonical commutation relations are, Lecture 3

Physical states • In states [ϕ(~x, t)ϕ(~y, t)]=[π(~x, t)π(~y, t)]=0 • Spectral rep 3 • Out States [π(~x, t),ϕ(~y, t)] = iδ (~x ~y) , where π(x)=ϕ ˙(x) − − LSZ Reduction Fermions if we assume that V (x) has no derivatives. Photons ❐ Cross sections We designate by ϕin(x) the operator that creates one-particle states. It will be a functional of the ﬁelds ϕ(x). Its existence will be shown by explicit construction. We require that ϕin(x) must satisfy the conditions:

i) ϕin(x) and ϕ(x) transform in the same way for translations and Lorentz transformations. For translations we have then

µ µ i [P ,ϕin(x)] = ∂ ϕin(x)

ii) The spacetime evolution of ϕin(x) corresponds to that of a free particle of mass m, that is

( + m2)ϕ (x) = 0 ⊔⊓ in

Jorge C. Rom˜ao TCA-2012 – 6 In states

❐ From these deﬁnitions it follows that ϕin(x) creates one-particle states from Lecture 3 the vacuum. In fact, let us consider a state n , such that, Physical states | i • In states • Spectral rep P µ n = pµ n • Out States | i n | i LSZ Reduction ❐ Then Fermions Photons µ µ µ ∂ n ϕin(x) 0 = i n [P ,ϕin(x)] 0 = ipn n ϕin(x) 0 Cross sections h | | i h | | i h | | i and therefore

n ϕ (x) 0 = p2 n ϕ (x) 0 ⊔⊓h | in | i − n h | in | i ❐ Then

( + m2) n ϕ (x) 0 =(m2 p2 ) n ϕ (x) 0 = 0 ⊔⊓ h | in | i − n h | in | i

where we have used the fact that ϕin(x) is a free ﬁeld

❐ Therefore the states created from the vacuum by ϕin are those for which 2 2 pn = m , that is, the one-particle states of mass m

Jorge C. Rom˜ao TCA-2012 – 7 In states

❐ The Fourier decomposition of ϕin(x) is then the same as for free ﬁelds, Lecture 3

Physical states • In states ϕ (x) = dk a (k)e−ik·x + a† (k)eik·x • Spectral rep in in in • Out States Z h i LSZ Reduction f† Fermions where ain(k) and ain(k) satisfy the usual algebra for creation and † Photons annihilation operators. In particular, by repeated use of ain(k) we can create Cross sections one state of n particles.

❐ To express ϕin(x) in terms of ϕ(x) we start by introducing the retarded Green’s function of the Klein-Gordon operator,

( + m2)∆ (x y; m) = δ4(x y) ⊔⊓x ret − − where

∆ (x y; m) = 0 if x0 < y0 ret − ❐ We can then write

√Zϕ (x) = ϕ(x) d4y∆ (x y; m)j(y) in − ret − Z Jorge C. Rom˜ao TCA-2012 – 8 In states

❐ The field ϕin(x), satisfies the two initial conditions. Lecture 3 Physical states ❐ √ • In states The constant Z was introduced to normalize ϕin in such a way that it has • Spectral rep amplitude 1 to create one-particle states from the vacuum. The fact that • Out States ∆ret = 0 for x0 , suggests that √Zϕin(x) is, in some way, the limit LSZ Reduction of ϕ(x) when x → −∞ . Fermions 0 → −∞ Photons ❐ In fact, as ϕ and ϕ are operators, the correct asymptotic condition must Cross sections in be set on the matrix elements of the operators. Let α and β be two normalized states. We define the operators | i | i

↔ ↔ f 3 ∗ f 3 ∗ ϕ (t) = i d xf (x) ∂ 0 ϕ(x) , ϕin = i d xf (x) ∂ 0 ϕin(x) Z Z f where f(x) is a normalized solution of the Klein-Gordon equation. ϕin does −ik·x f not depend on time (for plane waves f = e and ϕin = ain). ❐ Then the asymptotic condition of Lehmann, Symanzik e Zimmermann (LSZ), is

lim α ϕf (t) β = √Z α ϕf β t→−∞ h | | i h | in | i Jorge C. Rom˜ao TCA-2012 – 9 Spectral representation for scalar ﬁelds

❐ We saw that Z had a physical meaning as the square of the amplitude for Lecture 3 the field ϕ(x) to create one-particle states from the vacuum. Let us now find Physical states • In states a formal expression for Z and show that 0 Z 1. • Spectral rep ≤ ≤ • Out States ❐ We start by calculating the expectation value in the vacuum of the LSZ Reduction commutator of two fields, Fermions

Photons i∆′(x, y) 0 [ϕ(x),ϕ(y)] 0 Cross sections ≡h | | i ❐ As we do not know how to solve the equations for the interacting fields ϕ, we can not solve exactly the problem of finding the ∆′, in contrast with the free field case. We can, however, determine its form using general arguments of Lorentz invariance and the assumed spectra for the physical states.

❐ We introduce a complete set of states between the two operators and we use the invariance under translations in order to obtain,

n ϕ(y) m = n eiP ·yϕ(0)e−iP ·y m h | | i h | | i =ei(pn−pm)·y n ϕ(0) m h | | i

Jorge C. Rom˜ao TCA-2012 – 10 Spectral representation for scalar ﬁelds

❐ Therefore we get Lecture 3

Physical states ′ −ipn·(x−y) ipn·(x−y) • In states ∆ (x, y) = i 0 ϕ(0) n n ϕ(0) 0 (e e ) • Spectral rep − h | | ih | | i − n • Out States ′ X LSZ Reduction ∆ (x y) ≡ − Fermions ′ Photons that is, like in the free ﬁeld case, ∆ is only a function of the diﬀerence x y. Cross sections − ❐ Introducing now

1 = d4q δ4(q p ) − n Z we get

d4q ∆′(x y) = i (2π)3 δ4(p q) 0 ϕ(0) n 2 (e−iq·(x−y) eiq·(x−y)) − − (2π)3 n − |h | | i| − " n # Z X d4q = i ρ(q)(e−iq·(x−y) eiq·(x−y)) − (2π)3 − Z

Jorge C. Rom˜ao TCA-2012 – 11 Spectral representation for scalar ﬁelds

❐ We have deﬁned the density ρ(q) (spectral amplitude), Lecture 3 Physical states ρ(q)=(2π)3 δ4(p q) 0 ϕ(0) n 2 • In states n − |h | | i| • Spectral rep n • Out States ❐ This spectral amplitudeX measures the contribution to ∆′ of the states with LSZ Reduction 4-momentum qµ. ρ(q) is Lorentz invariant (as can be shown using the Fermions invariance of ϕ(x) and the properties of the vacuum and of the states n ) Photons | i Cross sections and vanishes when q is not in future light cone, due the assumed properties of the physical states. ❐ Then we can write

ρ(q) = ρ(q2)θ(q0)

and we get d4q ∆′(x y) = i ρ(q2)θ(q0)(e−iq·(x−y) eiq·(x−y)) − − (2π)3 − Z d4q = i dσ2δ(q2 σ2)ρ(σ2)θ(q0) e−iq·(x−y) eiq·(x−y) − (2π)3 − − ∞Z Z h i = dσ2ρ(σ2)∆(x y; σ) − Z0 Jorge C. Rom˜ao TCA-2012 – 12 Spectral representation for scalar ﬁelds

❐ Where Lecture 3

Physical states 4 d q 2 2 0 −iq·(x−y) iq·(x−y) • In states ∆(x y; σ) = i δ(q σ )θ(q )(e e ) • Spectral rep − − (2π)3 − − • Out States Z LSZ Reduction is the invariant function for the commutator of free ﬁelds with mass σ.

Fermions ❐ Photons The above relation is known as the spectral decomposition of the

Cross sections commutator of two ﬁelds. This expression will allow us to show that 0 Z < 1. ≤ ❐ To show that, we separate the states of one-particle from the sum. Let p be a one-particle state with momentum p. Then | i

0 ϕ(x) p =√Z 0 ϕ (x) p + d4y∆ (x y; m) 0 j(y) p h | | i h | in | i ret − h | | i Z =√Z 0 ϕ (x) p h | in | i where we have used

0 j(y) p = 0 ( + m2)ϕ(y) p =( + m2)e−ip·y 0 ϕ(0) p h | | i | ⊔⊓ | ⊔⊓ h | | i =(m2 p2)e−ip·y 0 ϕ(0) p = 0 − h | | i Jorge C. Rom˜ao TCA-2012 – 13 Spectral representation for scalar ﬁelds

❐ On the other hand Lecture 3 Physical states d3k • In states −ik·x 0 ϕin(x) p = 3 e 0 ain(k) p • Spectral rep h | | i (2π) 2ωk h | | i • Out States Z −ip·x LSZ Reduction =e Fermions Photons and then Cross sections

ρ(q) =(2π)3 dp δ4(p q)Z + contributions from more than one particle − Z =Zδ(q2 fm2)θ(q0) + − ··· ❐ Therefore ∞ ∆′(x y) = Z∆(x y; m) + dσ2ρ(σ2)∆(x y; σ) − − 2 − Zm1

where m1 is the mass of the lightest state of two or more particles.

Jorge C. Rom˜ao TCA-2012 – 14 Spectral representation for scalar ﬁelds

❐ Finally noticing that Lecture 3 Physical states ∂ ∂ • In states ′ 3 ∆ (x y) x0=y0 = ∆(x y; σ) x0=y0 = δ (~x ~y) • Spectral rep ∂x0 − | ∂x0 − | − − • Out States LSZ Reduction we get the relation Fermions ∞ Photons 2 2 Cross sections 1 = Z + dσ ρ(σ ) 2 Zm1 ❐ This means

0 Z < 1 ≤ where this last step results from the assumed positivity of ρ(σ2).

Jorge C. Rom˜ao TCA-2012 – 15 Out states

❐ In the same way as we reduced the dynamics of t to the free fields Lecture 3 → −∞ ϕin, it is also possible to define in the limit t + the corresponding free Physical states → ∞ • In states fields, ϕout(x). • Spectral rep • Out States ❐ These free fields will be the final state of a scattering problem. LSZ Reduction

Fermions ❐ The formalism is copied from the case of ϕin, and therefore we will present Photons the results without going into the details of the derivations. ϕout(x) obey Cross sections the following relations:

µ µ i [P ,ϕout] = ∂ ϕout ( + m2)ϕ = 0 ⊔⊓ out and has the expansion

−ik·x † ik·x ϕout(x) = dk aout(k)e + aout(k)e Z ❐ The asymptotic conditionh is now i f f √ f lim α ϕ (t) β = Z α ϕout β t→∞ h | | i h | | i

Jorge C. Rom˜ao TCA-2012 – 16 Out states

❐ Also Lecture 3

Physical states 4 • In states √Zϕout(x) = ϕ(x) d y∆adv(x y; m)j(y) • Spectral rep − − • Out States Z LSZ Reduction where the Green’s functions ∆adv satisfy Fermions

Photons 2 4 ( x + m )∆adv(x y; m) = δ (x y) Cross sections ⊔⊓ − − ∆ (x y; m)=0 ; x0 > y0 . adv − ❐ For one-particle states we get

0 ϕ(x) p =√Z 0 ϕ (x) p h | | i h | out | i =√Z 0 ϕ (x) p h | in | i =√Ze−ip·x

Jorge C. Rom˜ao TCA-2012 – 17 S matrix

❐ We have now all the formalism needed to study the transition amplitudes Lecture 3 from one initial state to a given ﬁnal state, the so-called S matrix elements. Physical states • In states Let us start by an initial state with n non-interacting particles • Spectral rep • Out States p p ; in α ; in LSZ Reduction | 1 ··· n i≡| i Fermions where p p are the 4-momenta of the n particles. Other quantum Photons 1 ··· n Cross sections numbers are assumed but not explicitly written. ❐ The ﬁnal state will be, in general, a state with m particles

p′ p′ ; out β ; out | 1 ··· m i≡| i ❐ The S matrix element Sβα is deﬁned by the amplitude

S β ; out α ; in βα ≡h | i ❐ The S matrix is an operator that induces an isomorphism between the in and out states, that by assumption are a complete set of states,

β ; out = β ; in S, β ; in = β ; out S−1 h | h | h | h | β ; out α ; in = β ; in S α ; in = β ; out S α; out h | i h | | i h | | i Jorge C. Rom˜ao TCA-2012 – 18 S matrix

❐ From the assumed properties for the states we can show the following results Lecture 3 for the S matrix. Physical states • In states • Spectral rep i) 0 S 0 = 0 0 = 1 (stability and unicity of the vacuum) • Out States h | | i h | i LSZ Reduction ii) The stability of the one-particle states gives Fermions

Photons Cross sections p ; in S p ; in = p ; out p ; in = p ; in p ; out = 1 h | | i h | i h | i because p ; in = p ; out . | i | i −1 iii) ϕin(x) = Sϕout(x)S iv) The S matrix is unitary. To show this we have

δ = β ; out α ; out = β ; in SS† α ; in βα h | i | | and therefore

SS† = 1

Jorge C. Rom˜ao TCA-2012 – 19 S matrix

v) The S matrix is Lorentz invariant. In fact we have Lecture 3

Physical states • In states • Spectral rep −1 −1 −1 • Out States ϕin(ax + b) =U(a, b)ϕin(x)U (a, b) = USϕout(x)S U LSZ Reduction −1 −1 −1 Fermions =USU ϕout(ax + b)US U . Photons Cross sections But

−1 ϕin(ax + b) = Sϕout(ax + b)S ,

and therefore we get ﬁnally

S = U(a, b)SU −1(a, b) .

Jorge C. Rom˜ao TCA-2012 – 20 LSZ reduction formula for scalar ﬁelds

❐ The S matrix elements are the quantities that are directly connected to the Lecture 3 2 experiment. In fact, Sβα represents the transition probability from the Physical states initial state α ; in to| the| ﬁnal β ; out . LSZ Reduction | i | i Fermions ❐ We are going in this section to use the previous formalism to express these Photons

Cross sections matrix elements in terms of the so-called Green functions for the interacting ﬁelds. In this way the problem of the calculation of these probabilities is transferred to the problem of calculating these Green functions. These, of course, can not be evaluated exactly, but we will learn in the next chapter how to develop a covariant perturbation theory for that purpose. ❐ Let us then proceed to the derivation of the relation between the S matrix elements and the the Green functions of the theory. This technique is known as the LSZ reduction from the names of Lehmann, Symanzik e Zimmermann that have introduced it. The starting point is, by deﬁnition

p ; out q ; in = p , ; out a† (q ) q , ; in h 1 ··· | 1 ··· i 1 ··· | in 1 | 2 ··· Using D E ↔ a† (q ) = i d3xe−iq1·x ∂ ϕ (x) in 1 − 0 in Zt Jorge C. Rom˜ao TCA-2012 – 21 LSZ reduction formula for scalar ﬁelds

❐ The last integral is time-independent, and therefore can be calculated for an Lecture 3 arbitrary time t. If we take t and use the asymptotic condition for Physical states → −∞ LSZ Reduction the in ﬁelds, we get Fermions ↔ −1/2 3 −iq1·x Photons p1 ; out q1 ; in = lim iZ d xe ∂ 0 p1 ; out ϕ(x) q2 ; in Cross sections h ··· | ··· i −t→−∞ h ··· | | ··· i Zt ❐ In a similar way one can show that

p ; out a† (q ) q ; in = 1 ··· | out 1 | 2 ··· D ↔E −1/2 3 −iq1·x = lim iZ d xe ∂ 0 p1 ; out ϕ(x) q2 ; in . − t→∞ h ··· | | ··· i Zt ❐ Then, using the result,

tf ∂ lim lim d3xf(~x, t)= lim dt d3xf(~x, t) t→∞ − t→−∞ tf →∞,ti→−∞ ∂t Z Zti Z 4 = d x∂0f(~x, t) Z

Jorge C. Rom˜ao TCA-2012 – 22 LSZ reduction formula for scalar ﬁelds

❐ Subtracting the last two equations we get Lecture 3

Physical states p ; out q ; in = p ; out a† (q ) q ; in LSZ Reduction h 1 ··· | 1 ··· i 1 ··· | out 1 | 2 ··· Fermions D ↔ E −1/2 4 −iq1·x Photons + iZ d x ∂ e ∂ p ; out ϕ(x) q ; in 0 0 h 1 ··· | | 2 ··· i Cross sections Z h i ❐ The first term on the right-hand side of corresponds to a sum of disconnected terms, in which at least one of the particles is not affected by the interaction (it will vanish if none of the initial momenta coincides with one of the final momenta). Its form is

p ; out a† (q ) q ; in = 1 ··· | out 1 | 2 ··· D n E = (2π)32p0 δ3(~p ~q ) p , , p , ; out q , ; in k k − 1 h 1 ··· k ··· | 2 ··· i Xk=1 b where pk means that this momentum was taken out from that state.

Jorge C. Rom˜ao TCA-2012 – 23 LSZ reduction formula for scalar ﬁelds

❐ For the second term we write, Lecture 3

Physical states ↔ d4x ∂ e−iqix∂ p ; out ϕ(x) q ; in LSZ Reduction 0 0 h 1 ··· | | 2 ··· i Fermions Z h i Photons 4 2 −iq1·x −iq1·x 2 Cross sections = d x ∂ e + e ∂ − 0 h···i 0 h···i Z = d4x ( ∆2 + m2)e−iq1·x + e−iq1·x∂2 − h···i 0 h···i Z = d4xe−iq1·x( + m2) p ; out ϕ(x) q ; in ⊔⊓ h 1 ··· | | 2 ··· i Z where we have used ( + m2)e−iq1·x = 0, and have performed an integration by parts (whose justiﬁcation⊔⊓ would imply the substitution of plane waves by wave packets).

Jorge C. Rom˜ao TCA-2012 – 24 LSZ reduction formula for scalar ﬁelds

❐ Therefore, after this ﬁrst step in the reduction we get, Lecture 3

Physical states p1, pn; out q1 qℓ; in = LSZ Reduction h ··· | ··· i Fermions n Photons 0 3 3 = 2pk(2π) δ (~pk ~q1) p1, pk; pn; out q2 q2 qℓ; in Cross sections − h ··· ··· | ··· ··· i Xk=1 b + iZ−1/2 d4xe−iq1x( + m2) p p ; out ϕ(x) q q ; in ⊔⊓ h 1 ··· n | | 2 ··· ℓ i Z ❐ We will proceed with the process until all the momenta in the initial and final state are exchanged by the field operators. To be specific, let us now remove one momentum in the final state. ❐ From now on we will no longer consider the disconnected terms, because in practice we are only interested in the cases where all the particles interact. Once we know the cases where all the particles interact, we can always calculate situations where some of the particles do not participate in the scattering

Jorge C. Rom˜ao TCA-2012 – 25 LSZ reduction formula for scalar ﬁelds

❐ We have then Lecture 3

Physical states p1 pn; out ϕ(x1) q2 qℓ; in = p2 pn; out aout(p1)ϕ(x1) q2 qℓ; in LSZ Reduction h ··· | | ··· i h ··· | | ··· i Fermions ↔ −1/2 3 ip1·y1 Photons 0 =0 lim iZ d y1 e ∂ y1 p2 pn; out ϕ(y1)ϕ(x1) q2 qℓ; in y1 →∞ h ··· | | ··· i Cross sections Z = p p ; out ϕ(x )a (p ) q q ; in h 2 ··· n | 1 in 1 | 2 ··· ℓ i ↔ − 1· 1 1/2 3 ip y 0 +0 lim iZ d y1 e ∂ y1 p2 pn; out ϕ(y1)ϕ(x1) q2 qℓ; in y1 →∞ h ··· | | ··· i Z ↔ − 1· 1 1/2 3 ip y 0 0lim iZ d y1 e ∂ y1 p2 pn; out ϕ(x1)ϕ(y1) q2 qℓ; in − y1 →−∞ h ··· | | ··· i Z = p p ; out ϕ(x )a (p ) q q ; in h 2 ··· n | 1 in 1 | 2 ··· ℓ i ↔ − 1· 1 1/2 3 ip y 0 +iZ 0lim 0lim d y1e ∂ y1 p2 pn; out Tϕ(y1)ϕ(x1) q2 qℓ; in y1 →∞−y1 →−∞ h ··· | | ··· i Z

Jorge C. Rom˜ao TCA-2012 – 26 LSZ reduction formula for scalar ﬁelds

❐ Applying the same procedure as before we obtain, Lecture 3

Physical states p1 pn, ; out ϕ(x1) q2 qℓ; in = disconnected terms LSZ Reduction h ··· | | ··· i Fermions −1/2 4 ip1·y1 2 Photons + iZ d y e ( + m ) p p ; out Tϕ(y )ϕ(x ) q q ; in 1 ⊔⊓y1 h 2 ··· n | 1 1 | 2 ··· ℓ i Cross sections Z ❐ It is not very difficult to generalize this method to obtain the final reduction formula for scalar fields,

p p ; out q q ; in = disconnected terms h 1 ··· n | 1 ··· ℓ i n+ℓ i n ℓ 4 4 4 4 [i P1 pk·yk −i P1 qr ·xr ] + d y1 d ynd x1 d xℓe √Z ··· ··· Z ( + m2) ( + m2) 0 Tϕ(y ) ϕ(y )ϕ(x ) ϕ(x ) 0 ⊔⊓y1 ··· ⊔⊓xℓ h | 1 ··· n 1 ··· ℓ | i ❐ This last equation is the fundamental equation in quantum ﬁeld theory. It allows us to relate the transition amplitudes to the Green functions of the theory.

Jorge C. Rom˜ao TCA-2012 – 27 LSZ reduction formula for scalar ﬁelds

❐ The quantity Lecture 3

Physical states 0 Tϕ(x1) ϕ(xn) 0 G(x1 xn) LSZ Reduction h | ··· | i ≡ ··· Fermions is known as the complete green function for n = m + ℓ particles Photons Cross sections ❐ We will introduce the following diagrammatic representation for it xn xl

G(x x ) = 1 ··· n

x1 xi ❐ The factors ( + m2) in force the external particles to be on-shell. In fact, in momentum space⊔⊓ ( + m2) ( p2 + m2). Therefore the amplitude will vanish unless the propagators⊔⊓ → of− the external legs are on-shell, as in that case 1 they will have a pole, p2−m2 . We conclude that for the transition amplitudes only the truncated Green functions will contribute, that is the ones with the external legs removed. In the next chapter we will learn how to evaluate these Green functions in perturbation theory.

Jorge C. Rom˜ao TCA-2012 – 28 Reduction Formula for fermions: States in and out

❐ The deﬁnition of the in and out follows exactly the same steps as in the case Lecture 3 of the scalar ﬁelds. We will therefore, for simplicity, just state the results Physical states

LSZ Reduction with the details. Fermions ❐ • States In & Out The states ψin(x) satisfy the conditions, • Spectral rep • LSZ for Fermions (i∂/ m)ψin(x) = 0 Photons − Cross sections [P , ψ (x)] = i∂ ψ (x) . µ in − µ in

❐ The states ψin(x) will create one-particle states and they are related with the ﬁelds ψ(x) by,

Z ψ (x) = ψ(x) d4yS (x y,m)j(y) 2 in − ret − p Z where ψ(x) satisﬁes the Dirac equation,

(i∂/ m)ψ(x) = j(x) −

and Sret is the retarded Green function for the Dirac equation,

Jorge C. Rom˜ao TCA-2012 – 29 Reduction Formula for fermions: States in and out

❐ Where the retarded Green function is Lecture 3 Physical states 4 (i∂/x m)Sret(x y,m) = δ (x y) LSZ Reduction − − − Fermions 0 0 • States In & Out Sret(x y)=0; x < y • Spectral rep − • LSZ for Fermions Photons ❐ The fields ψin(x), as free fields, have the Fourier expansion, Cross sections −ip·x † ip·x ψin(x) = dp bin(p,s)u(p,s)e + din(p,s)v(p,s)e s Z X h i f where the operators bin, din satisfy exactly the same algebra as in the free field case. The asymptotic condition is now,

f f lim α ψ (t) β = Z2 α ψ β t→−∞ h | | i h | in | i p f f where ψ (t) and ψin have a meaning similar to the scalar case

❐ For the ψout ﬁelds we get essentially the same expressions with ψin substituted by ψout.

Jorge C. Rom˜ao TCA-2012 – 30 Reduction Formula for fermions: States in and out

❐ The main difference with respect to the in States is in the asymptotic Lecture 3 condition that now reads, Physical states LSZ Reduction f f lim α ψ (t) β = Z2 α ψout β Fermions t→∞ h | | i h | | i • States In & Out • Spectral rep p • LSZ for Fermions implying the following relation between the fields ψout and ψ, Photons Cross sections Z ψ = ψ(x) d4yS (x y; m)j(y) 2 out − adv − p Z ❐ The advanced Green function is defined by

(i∂/ m)S (x y; m) = δ4(x y) x − adv − − S (x y; m) = 0 x0 > y0 . adv −

Jorge C. Rom˜ao TCA-2012 – 31 Spectral representation fermions

❐ Let us consider the vacuum expectation value of the anti-commutator of two Lecture 3 Dirac ﬁelds, Physical states LSZ Reduction ′ S (x, y) i 0 ψα(x), ψ (y) 0 Fermions αβ ≡ h | { β }| i • States In & Out • Spectral rep • LSZ for Fermions −ipn(x−y) =i 0 ψα(0) n n ψβ(0) 0 e Photons h | | ih | | i n " Cross sections X

ipn·(x−y) + 0 ψβ(0) n n ψα(0) 0 e h | | ih | | i # S′ (x y) ≡ αβ − where we have introduced a complete set of eigen-states of the 4-momentum.

❐ As before we introduce the spectral amplitude ραβ(q),

ρ (q) (2π)3 δ4(p q) 0 ψ (0) n n ψ (0) 0 αβ ≡ n − h | α | ih | β | i n X

Jorge C. Rom˜ao TCA-2012 – 32 Spectral representation fermions

❐ We will now find the most general form for ραβ(q) using Lorentz invariance Lecture 3 arguments. ραβ(q) is a 4 4 matrix in Dirac space, and can be written as Physical states × LSZ Reduction µ µν 5 µ 5 Fermions ραβ(q) = ρ(q)δαβ + ρµ(q)γαβ + ρµν(q)σαβ +ρ ˜(q)γαβ +ρ ˜µ(q)(γ γ )αβ • States In & Out • Spectral rep • LSZ for Fermions ❐ Lorentz invariance arguments restrict the form of the coefficients ρ(q), Photons ρµ(q), ρµν(q), ρ˜(q) and ρ˜µ(q). Under Lorentz transformations the fields Cross sections transform as

−1 −1 U(a)ψα(0)U (a) = Sαλ (a)ψλ(0) −1 U(a)ψα(0)U (a) = ψλ(0)Sλα(a) −1 µ µ ν S γ S = a ν γ

❐ Then we can show that the matrix (in Dirac space), ραβ must obey the relation,

ρ(q) = S−1(a)ρ(qa−1)S(a)

where we have used a matrix notation. This relation gives the properties of the diﬀerent coeﬃcients

Jorge C. Rom˜ao TCA-2012 – 33 Spectral representation fermions

❐ For instance, Lecture 3 Physical states µ µ ν −1 ρ (q) = a ν ρ (qa ) LSZ Reduction Fermions µ • States In & Out which means that ρ transform as a 4 vector. • Spectral rep − • LSZ for Fermions ❐ Using the fact that ραβ is a function of q and vanishes outside the future Photons light cone, we can ﬁnally write Cross sections 2 2 2 5 2 5 ραβ(q) = ρ1(q )q/αβ + ρ2(q )δαβ +ρ ˜1(q )(q/γ )αβ +ρ ˜2(q )γαβ

2 that is, ραβ(q) is determined up to four scalar functions of q . ❐ Requiring invariance under parity transformations we get

ρ (~q, q ) = γ0 ρ ( ~q, q0)γ0 αβ 0 αλ λδ − δβ and we obtain,

ρ1 = ρ2 = 0

e e Jorge C. Rom˜ao TCA-2012 – 34 Spectral representation fermions

❐ Therefore for the Dirac theory, that preserves parity, we get, Lecture 3 Physical states 2 2 ραβ(q) = ρ1(q )q/αβ + ρ2(q )δαβ LSZ Reduction

Fermions • States In & Out ❐ Repeating the steps of the scalar case we write, • Spectral rep • LSZ for Fermions ∞ Photons ′ 2 2 Sαβ(x y) = dσ ρ1(σ )Sαβ(x y; σ)+ Cross sections − − Z0 + σρ (σ2) ρ (σ2) δ ∆(x y; σ) 1 − 2 αβ − where ∆ and Sαβ are the functions deﬁned for free ﬁelds. ❐ We can then show that

i) ρ1 e ρ2 are real ii) ρ (σ2) 0 1 ≥ iii) σρ (σ2) ρ (σ2) 0 1 − 2 ≥ ❐ Using the previous relations we can extract of the one-particle states

Jorge C. Rom˜ao TCA-2012 – 35 Spectral representation fermions

❐ We get, Lecture 3 Physical states ′ Sαβ(x y) =Z2Sαβ(x y; m) LSZ Reduction − − Fermions ∞ • States In & Out 2 2 • Spectral rep + dσ ρ1(σ )Sαβ(x y; σ) 2 − • LSZ for Fermions Zm1 ( Photons Cross sections 2 2 + σρ1(σ ) ρ2(σ ) δαβ∆(x y; σ) − − ) where m1 is the threshold for the production of two or more particles. ❐ Evaluating at equal times we can obtain

2 2 1 = Z2 + dσ ρ1(σ ) 2 Zm1 that is

0 Z < 1 ≤ 2

Jorge C. Rom˜ao TCA-2012 – 36 Reduction formula for fermions

❐ To get the reduction formula for fermions we will proceed as in the scalar Lecture 3 case. Physical states LSZ Reduction ❐ The only diﬃculty has to do with the spinor indices. The creation and Fermions • States In & Out annihilation operators can be expressed in terms of the ﬁelds ψin by the • Spectral rep relations, • LSZ for Fermions

Photons 3 ip·x 0 Cross sections bin(p,s) = d xu(p,s)e γ ψin(x) Z † 3 −ip·x 0 din(p,s) = d xv(p,s)e γ ψin(x) Z † 3 0 −ip·x bin(p,s) = d xψin(x)γ e u(p,s) Z 3 0 ip·x din(p,s) = d xψin(x)γ e v(p,s) Z with the integrals being time independent. ❐ In fact, to be more rigorous we should substitute the plane wave solutions by wave packets, but as in the scalar case, to simplify matter we will not do it.

Jorge C. Rom˜ao TCA-2012 – 37 Reduction formula for fermions

❐ To establish the reduction formula we start by extracting one electron from Lecture 3 the initial state, Physical states

LSZ Reduction † Fermions β; out (ps)α; in = β; out bin(p,s) α,in • States In & Out h | i | | • Spectral rep D E† † • LSZ for Fermions = β (p,s); out α; in + β; out bin(p,s) bout(p,s) α; in Photons h − | i | − | D E Cross sections =disconnected terms

+ d3x β; out ψ (x) ψ (x) α; in γ0e−ip·xu(p,s) | in − out | Z =disconnected terms 1 lim lim d3x β; out ψ(x) α; in γ0e−ip·xu(p,s) − t→+∞ − t→−∞ √Z | | 2 Z

=disconnected terms

Z−1/2 d4x β; out ∂ ψ(x) α; in γ0e−ip·xu(p,s) − 2 | 0 | Z + β; out ψ(x) α; in γ0∂ (e−ip·xu(p,s)) | | 0 Jorge C. Rom˜ao TCA-2012 – 38 Reduction formula for fermions

❐ Using now Lecture 3 Physical states 0 i −ip·x (iγ ∂0 + iγ ∂i m)(e u(p,s))=0 LSZ Reduction − Fermions • States In & Out we get, after an integration by parts, • Spectral rep • LSZ for Fermions β; out b† (ρ,s) α; in = disconnected terms Photons | in | Cross sections D E ← iZ−1/2 d4x β; out ψ(x) α; in ( i∂/ m)e−ip·xu(p,s) − 2 | | − x − Z

❐ In a similar way the reduction of an anti-particle from the initial state gives,

β; out d† (p,s) α; in = disconnected terms | in | D E + iZ−1/2 d4xe−ip·xv(p,s)(i∂/ m) β; out ψ(x) α; in 2 x − h | | i Z

Jorge C. Rom˜ao TCA-2012 – 39 Reduction formula for fermions

❐ The reduction of a particle or of an anti-particle from the ﬁnal state give, Lecture 3 respectively, Physical states

LSZ Reduction β; out bout(p,s) α; in = disconnected terms Fermions h | | i • States In & Out • Spectral rep −1/2 4 ip·x • LSZ for Fermions iZ d xe u(p,s)(i∂/ m) β; out ψ(x) α; in − 2 x − h | | i Photons Z Cross sections and

β; out d (p,s) α; in = disconnected terms h | out | i ← + iZ−1/2 d4x β; out ψ(x) α; in ( i∂/ m)v(p,s)eip·x 2 | | − x − Z

❐ Notice the formal relation between one electron in the initial state and a positron in the ﬁnal state. To go from one to the other one just has to do,

u(p,s)e−ip·x v(p,s)eip·x → −

Jorge C. Rom˜ao TCA-2012 – 40 Reduction formula for fermions

❐ To write the ﬁnal expression we denote the momenta in the state in by pi Lecture 3 h | or pi, respectively for particles or anti-particles, and those in the state out Physical states ′ ′ h | LSZ Reduction by pi, pi.

Fermions • States In & Out ❐ We also make the following conventions (needed to deﬁne the global sign), • Spectral rep • LSZ for Fermions † † Photons (p1,s1), , (p1, s1); ; in = bin(p1,s1) din(p1, s1) 0 | ′ ···′ ′ ···′ i ···′ ′ ···|′ i′ Cross sections out;(p ,s ) , (p , s ) = 0 d (p , s ), b (p ,s ) h 1 1 ··· 1 1 ···| h |··· out 1 1 ··· out 1 1 ❐ Then, if n(n′) denotes the total number of particles (anti-particles), we get

out;(p′ ,s′ ) , (p′ , s′ ) (p ,s ), (p , s ), ; in = disc. terms h 1 1 ··· 1 1 ···| 1 1 ··· 1 1 ··· i ′ +( iZ−1/2)n(iZ−1/2)n d4x d4y d4x′ d4y′ − 2 2 1 ··· 1 ··· 1 ··· 1 ··· Z ′ ′ ′ ′ − · − · · · e i P(pi xi) i P(pi yi)e+i P(pi xi)+i P(pi yi) ′ ′ u(p ,s )(i∂/ ′ m) v(p , s )(i∂/ m) 1 1 x1 − ··· 1 1 y1 − ′ ′ 0 T ( ψ(y1) ψ(x1)ψ(x1) ψ(y1) 0 h | ← ··· ··· ← ··· ···| i ′ ′ ( i∂/ m)u(p ,s ) ( i∂/ ′ m)v(p , s ) − x1 − 1 1 ··· − y1 − 1 1 Jorge C. Rom˜ao TCA-2012 – 41 Reduction formula for fermions

❐ Last equation is the fundamental expression that allows to relate the Lecture 3 elements of the S matrix with the Green functions of the theory. Physical states LSZ Reduction ❐ The sign and ordering shown correspond to the previous conventions Fermions • States In & Out • Spectral rep ❐ In terms of diagrams, we represent the Green function, • LSZ for Fermions

Photons ′ ′ ′ ′ 0 T ψ(ym′ ) ψ(y1)ψ(xℓ′ ) ψ(x1)ψ(x1) ψ(xℓ)ψ(y1) ψ(ym) 0 Cross sections h | ··· ··· ··· ··· | i by the diagram of the ﬁgure ′ y′ ❐ ′ xl 1 ′ With lepton number conservation, the number of x1 ym particles minus anti-particles is conserved, that is

ℓ m = ℓ′ m′ − − ← ❐ The operators (i∂/ m) e ( i∂/ m) force the particles to be on-shell− and remove− the− propagators from the external lines (truncated Green functions). In the x1 ym next chapter we will learn how to determine these xl y1 functions in perturbation theory.

Jorge C. Rom˜ao TCA-2012 – 42 Reduction formula for photons

❐ The LSZ formalism for photons, has some difficulties connected with the Lecture 3 problems in quantizing the electromagnetic field. When one adopts a Physical states µ LSZ Reduction formalism (radiation gauge) where the only components of the field A are

Fermions transverse, the problems arise in showing the Lorentz and gauge invariance Photons of the S matrix. In the formalism of the undefined metric, the difficulties are • LSZ for photons connected with the states of negative norm, besides the gauge invariance. Cross sections ❐ We will see later a more satisfactory procedure to quantize all gauge theories, including Maxwell theory of the electromagnetic field and the resulting perturbation theory coincides with the one we get here. This is our justification to assume that we can define the in fields by the relation,

Z Aµ (x) = Aµ(x) d4yDµν (x y)j (y) 3 in − ret − ν and inp the same way for the outZ ﬁelds,

Z Aµ (x) = Aµ(x) d4yDµν (x y)j (y) 3 out − adv − ν wherep Z Aµ = Aµ = 0 , Aµ = jµ and Dµν = δµν δµ(x y) ⊔⊓ in ⊔⊓ out ⊔⊓ ⊔⊓ adv, ret − Jorge C. Rom˜ao TCA-2012 – 43 Reduction formula for photons

❐ The ﬁelds in and out are free ﬁelds, and therefore they have a Fourier Lecture 3 expansion in plane waves and creation and annihilation operators of the form Physical states

LSZ Reduction 3 Fermions µ µ −ik·x † µ∗ ik·x A (x) = dk ain(k,λ)ε (k,λ)e + a (k,λ)ε (k,λ)e Photons in in • LSZ for photons Z λ=0 ❐ X h i Cross sections Where f ↔ a (k,λ) = i d3xeik·x∂ εµ(k,λ)Ain(x) in − 0 µ Z ↔ † 3 −ik·x µ∗ in ain(k,λ) =i d xe ∂ 0ε (k,λ)Aµ (x) Z † and, as usual, ain(k,λ) and ain(k,λ) are time independent. ❐ Above, all the polarizations appear, but as the elements of the S matrix are between physical states, we are sure that the longitudinal and scalar polarizations do not contribute. ❐ In this formalism what is difficult to show is the spectral decomposition. We are not going to enter those details, just state that we can show that Z3 is gauge independent and satisfies 0 Z < 1. ≤ 3 Jorge C. Romão TCA-2012 – 44 Reduction formula for photons

❐ The reduction formula is easily obtained. We get Lecture 3

Physical states β; out (kλ)α; in = β (k,λ); out α; in + β; out a† (k,λ) a† (k,λ) α; in LSZ Reduction h | i h − | i | in − out | Fermions D E Photons = disconnected terms • LSZ for photons ↔ Cross sections + i d3xe−ik·x∂ ε∗ (k,λ) β; out Aµ (x) Aµ (x) α; in 0 µ h | in − out | i Z = disconnected terms ↔ −1/2 3 −ik·x µ ∗ i( lim lim )Z d xe ∂ 0 β; out A (x) α; in ε (k,λ) − t→+∞ − t→−∞ 3 h | | i µ Z = disconnected terms ↔ iZ−1/2 d4xe−ik·x∂ β; out Aµ(x) α; in ε∗ (k,λ) − 3 0 h | | i µ Z = disconnected terms

iZ−1/2 d4xe−ik·x~ β; out Aµ(x) α; in ε∗ (k,λ) − 3 ⊔⊓x h | | i µ Z

Jorge C. Rom˜ao TCA-2012 – 45 Reduction formula for photons

❐ The ﬁnal formula for photons is then Lecture 3 Physical states ′ ′ k1 kn; out k1 kℓ; in = disconnected terms LSZ Reduction h ··· | ··· i Fermions n+ℓ Photons i n ′ ℓ 4 4 4 4 [i P ki·yi−i P ki·xi] • LSZ for photons + − d y1 d ynd x1 d xℓ e √Z ··· ··· Cross sections 3 Z ′ ′ 1 ∗ ′ ′ ∗ ′ ′ εµ (k ,λ ) εµℓ (k ,λ )ε µ1 (k ,λ ) ε µn (k ,λ ) 1 1 ··· ℓ ℓ 1 1 ··· n n 0 T (A ′ (y ) A ′ (y )A (x ) A (x ) 0 ⊔⊓y1 · · · ⊔⊓xℓ h | µ1 1 ··· µn n µ1 1 ··· µℓ ℓ | i ❐ This corresponds to the diagram

xn xl

x1 xi

Jorge C. Rom˜ao TCA-2012 – 46 Cross sections

❐ The reduction formulas are the fundamental results of this chapter. They Lecture 3 relate the transition amplitudes from the initial to the ﬁnal state with the Physical states

LSZ Reduction Green functions of the theory. In the next chapter we will show how to

Fermions evaluate these Green functions setting up the so-called covariant Photons perturbation theory. Before we close this chapter, let us indicate how these Cross sections transition amplitudes

S f; out i; in fi ≡h | i are related with the quantities that are experimentally accessible, the cross sections. Then the path between experiment (cross sections) and theory (Green functions) will be established. ❐ As we have seen in the reduction formulas there is always a trivial contribution to the S matrix, that corresponds to the so-called disconnected terms, when the system goes from the initial to the ﬁnal state without interaction. The subtraction of this trivial contribution leads to the T matrix

S = 1 i(2π)4δ4(P P )T fi fi − f − i fi where we have factorized explicitly the delta function expressing the 4-momentum conservation. Jorge C. Rom˜ao TCA-2012 – 47 Cross sections

❐ If we neglect the trivial contribution, the transition probability from the Lecture 3 initial to the ﬁnal state will be given by Physical states LSZ Reduction 4 4 2 Wf←i = (2π) δ (Pf Pi)Tfi Fermions − Photons ❐ To proceed we have to deal with the meaning of a square of a delta function.

Cross sections This appears because we are using plane waves. To solve this problem we can normalize in a box of volume V and consider that the interaction has a duration of T . Then

T/2 (2π)4δ4(P P )= lim d3x dx0ei(Pf −Pi)·x . f − i V ZV Z−T/2 T → ∞ ❐ However → ∞

T/2 3 0 i(Pf −Pi)·x 2 T F d x dx e = Vδ ~ ~ sin (Ef Ei) ≡ Pf Pj E E 2 − ZV Z−T/2 f i | − |

and the square of the last expression can be done, giving,

2 2 4 2 T F = V δ ~ ~ sin (Ef Ei) | | Pf ,Pi E E 2 2 − f i | − | Jorge C. Rom˜ao TCA-2012 – 48

Cross sections

❐ If we want the transition rate by unit of volume (and unit of time) we divide Lecture 3 by VT . Then Physical states

LSZ Reduction sin2 T (E E ) Fermions 2 f i 2 Γfi = lim Vδ ~ ~ 2 − Tfi Pf ,Pi T 2 Photons (E E ) | | V 2 f − i Cross sections T → ∞ → ∞ ❐ Using now the results

3 3 lim Vδ ~ ~ = (2π) δ (P~f P~i) V →∞ Pf Pj −

2 T sin 2 (Ef Ei) lim 2 − = (2π) δ(Ef Ei) T →∞ T (E E )2 − 2 f − i we get for the transition rate by unit volume and unit time,

Γ (2π)4δ4(P P ) T 2 fi ≡ f − i | fi| ❐ To get the cross section we have to further divide by the incident ﬂux, and normalize the particle densities to one particle per unit volume. Finally, we sum (integrate) over all ﬁnal states in a certain energy-momentum range.

Jorge C. Rom˜ao TCA-2012 – 49 Cross sections

❐ We get, Lecture 3 Physical states n 3 1 1 d pj LSZ Reduction dσ = Γ ρ ρ ~v fi 2p0(2π)3 Fermions 1 2 | 12| j=3 j Photons Y

Cross sections where ρ1 = 2E1 and ρ2 = 2E2 ❐ An equivalent way of writing this equation is

n 1 4 4 2 dσ = (2π) δ (Pf Pi) Tfi dpj 2 2 2 1/2 − | | 4[(pi p2) m m ] j=3 · − 1 2 Y that exhibits well the Lorentz invariance of each part that enters the cross section along the direction of collision. The incident ﬂux and phase space factors are purely kinematics. The physics, with its interactions, is in the matrix element Tfi. ❐ We note that with our conventions, fermion and boson ﬁelds have the same normalization, that is, the one-particle states obey

p p′ = 2p0(2π)3δ3(~p ~p′) h | i − Jorge C. Rom˜ao TCA-2012 – 50