Old-Fashioned Perturbation Theory

2012 Matthew Schwartz I-4: Old-fashioned perturbation theory 1 Perturbative quantum field theory The slickest way to perform a perturbation expansion in quantum field theory is with Feynman diagrams. These diagrams will be the main tool we’ll use in this course, and we’ll derive the diagrammatic expansion very soon. Feynman diagrams, while having advantages like producing manifestly Lorentz invariant results, give a very unintuitive picture of what is going on, with particles that can’t exist appearing from nowhere. Technically, Feynman diagrams introduce the idea that a particle can be off-shell, meaning not satisfying its classical equations of motion, for 2 2 example, with p m . They trade on-shellness for exact 4-momentum conservation. This con- ceptual shift was critical in allowing the efficient calculation of amplitudes in quantum field theory. In this lecture, we explain where off-shellness comes from, why you don’t need it, but why you want it anyway. To motivate the introduction of the concept of off-shellness, we begin by using our second quantized formalism to compute amplitudes in perturbation theory, just like in quantum mechanics. Since we have seen that quantum field theory is just quantum mechanics with an infinite number of harmonic oscillators, the tools of quantum mechanics such as perturbation theory (time-dependent or time-independent) should not have changed. We’ll just have to be careful about using integrals instead of sums and the Dirac δ-function instead of the Kronecker δ. So, we will begin by reviewing these tools and applying them to our second quantized photon. This is called old-fashioned perturbation theory (OFPT). As a historical note, OFPT was still a popular way of doing calculations through at least the 1960s. Some physicists, such as Schwinger, never embraced Feynman diagrams and continued to use OFPT. It was not until the 1950s though the work of Dyson and others that it was shown that OFPT and Feynman diagrams gave the same results. Despite the prevalence of Feynman’s approach in modern calculations, and the efficient encapsulation by the path integral formalism (Lectures II-7 and onward), OFPT is still worth understanding. It provides complementary physical insight into quantum field theory. For example, a souped-up version of OFPT is given by Schwinger’s proper time formalism (Lecture IV-9), which is still the best way to do certain effective action calculations. This lecture can be skipped without losing continuity with the rest of the text. 2 Lippmann-Schwinger equation Just like in quantum mechanics, perturbation theory in quantum field theory works by splitting the Hamiltonian up into two parts H = H0 + V (1) where the eigenstates of H0 are known exactly, and the potential V gives corrections which are small in some sense. The difference from quantum mechanics is that in quantum field theory the states often have a continuous range of energies. For example, in a Hydrogen atom coupled to an electromagnetic field the associated photon energies E = ωk = kQ can take any values. Because of the infinite number of states, the methods look a little different, but we will just be doing the natural continuum generalization of perturbation theory in quantum mechanics. 1 2 Section 2 We are often interested in a situation where we know the state of a system at early times and would like to know the state at late times. Say the state has a fixed energy E at early and late times (of course, it’s the same E). There will be some eigenstate of H0 with energy E, call it | φ i . So, H0| φ i = E | φ i (2) If the energies E are continuous then we should be able to find an eigenstate | ψ i of the full Hamiltonian with the same eigenvalue H | ψ i = E| ψ i (3) and we can formally write 1 | ψ i = | φ i + V | ψ i (4) E − H0 which is trivial to verify. This is called the Lippmann-Schwinger equation. 1 The inverted object appearing in the Lippmann-Schwinger equation is a kind of Green’s function known as the Lippmann-Schwinger kernel 1 ΠLS = (6) E − H0 The Lippmann-Schwinger equation is useful in scattering theory where | φ i are free states, such as momentum eigenstates with continuous energies, at early and late times, and the potential acts at intermediate times to induce transitions among these states. It says the full wavefunction | ψ i is given by the free wavefunction | φ i plus a scattering term. What we would really like to do is express | ψ i entirely in terms of | φ i . Thus we define an operator T by V | ψ i = T| φ i (7) T is known as the transfer matrix. Inserting this definition turns the Lippmann-Schwinger equation into 1 | ψ i = | φ i + T| φ i (8) E − H0 which formally gives | ψ i in terms of | φ i . Multiplying this by V and demanding the two sides be equal when contracted with any state h φ j | gives an operator equation for T 1 T = V + V T (9) E − H0 We can then solve perturbatively in V to get 1 1 1 T =V + V V + V V V + E − H0 E − H0 E − H0 (10) =V + V ΠLS V +V ΠLS V ΠLS V + If we insert complete set j | φ j ih φ j | of eigenstates | φ j i of H0 the matrix elements P 1 h φi | T| φ f i = h φi | V | φ f i + h φ f | V | φ j ih φ j | V | φi i + (11) E − H0 Writing Tfi = h φ f | T| φi i and Vij = h φi | V | φ j i this becomes Tfi = Vfi + VfjΠLS ( j) Vji + VfjΠLS ( j) Vjk ΠLS ( k) Vki + (12) 1 where ΠLS ( k) = , where again, E = Ei = Ej is the energy of the initial and final state we E − Ek are interested in. This expansion is old-fashioned perturbation theory. 1. Formally, the inverse of E − H0 is not well-defined. Since E is an eigenvalue of H0, det ( E − H0 ) = 0 and − 1 ( E − H0 ) is singular. To regulate this singularity, we can add an infinitesimal imaginary factor iε, leading to 1 | ψ i = | φi + V | ψ i (5) E − H0 + iε with the understanding that ε should be taken to zero at the end of the calculation. Lippmann-Schwinger equation 3 Eq. (12) says that to calculate transitions in perturbation theory there is a sum of terms. In each term the potential creates an intermediate state | φ j i which propagates with the propagator ΠLS ( j) until it hits another potential, where it creates a new field | φk i which then propagates and so on, until they hit the final potential factor, which transitions it to the final state. There is a nice diagrammatic way of drawing this series, called Feynman graphs, which we will see through an example in a moment and in more detail in upcoming lectures. For some termi- nology, the first term, Vij gives the Born approximation (or first Born approximation), the second term the second Born approximation and so on. See for example [Sakuraii chapter 7] for applications of the Lippmann-Schwinger approximation in non-relativistic quantum mechanics. 2.1 Coulomb’s law revisited The example we will compute is one we will revisit many times: an electron scattering off another electron in QED. The transition matrix element for this process is given by the Lipp- mann-Schwinger equation as 1 Tfi = Vfi + Vfn Vni + (13) Ei − En Xn Here Ei are the initial energy (which is the same as the final energy), and En is the energy of 1 2 the intermediate state. The initial and final states each have two electrons | i i = | ψe ψe i and 3 4 1 1 h f | = h ψe ψe | where the superscripts label the momenta, i.e. ψe has Qp , etc. The intermediate state can be anything in the whole Fock space, but only certain intermediate states will have non-vanishing matrix elements with V. In relativistic field theory, the instantaneous action-at-a-distance of Coulomb’s law is replaced by a process where two electrons exchange a photon which travels only at the speed of light. Thus there should be photons in the intermediate state. Ignoring the spin of the electron and the photon, the interaction of the electron with the photon field can be written as 1 3 V = e d xψe( x) φ( x) ψe ( x) (14) 2 Z This interaction is local, since the fields ψe ( x) and φ( x) , corresponding to the electrons and 1 photon are all evaluated at the same point. The factor of 2 comes from ignoring spin and treating all fields as representing real scalar particles. This interaction can turn a state with an electron of momentum Qp1 into a state with an elec- Q tron with momentum pQ 3 and a photon with momentum pγ. Since initial and final states both have two electrons and no photons, the leading term in Eq. (13) vanishes, Vfi = 0. To get a non-zero matrix element, we need an intermediate state | n i with a photon. There are two intermediate states which can contribute. In the first, the photon is emitted from the first electron and the intermediate state is before that photon hits the second electron. We can draw a picture representing this process: − − e1 time→ e3 ~p3 ~p1 (15) ~p2 ~p4 ¡ − − e2 e4 The vertical dashed line indicates the time at which the intermediate state is evaluated.

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