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Advanced Quantum Field Theory Chapter 2 Physical States. S Matrix

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Advanced Quantum Field Theory Chapter 2 Physical States. S Matrix 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. Rom˜ao 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.
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