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Renormalization Renormalization In this chapter we face the ultraviolet divergences that we have found in perturbative quantum field theory. These divergences are not simply a technical nuicance to be disposed of and forgotten. As we will explain, they parameterize the dependence on quantum fluctuations at short distance scales (or equivalently, high momenta). Historically, it took a long time to reach this understanding. In the 1930’s, when the ultraviolet divergences were first discovered in qunatum electrodynamics, many physicists believed that fundamental principles of physics had to be changed to elim- inate the divergences. In the late 1940’s Bethe, Feynman, Schwinger, Tomonaga, and Dyson, and others proposed a program of ‘renormalization’ that gave finite and physically sensible results by absorbing the divergences into redefinitions of physi- cal quantities. This leads to calculations that agree with experiment to 8 significant digits in QED, the most accurate calculations in all of science. Even after the technical aspects of renormalization were understood, conceptual difficulties remained. It was widely believed that only a limited class of ‘renormaliz- able’ theories made physical sense. (The fact that general relativity is not renormaliz- able in this sense was therefore considered a deep problem.) Also, the renormalization program was viewed by many physicists as an ad hoc procedure justified only by the fact that it yields physically sensible results. This was changed by the profound work of K. Wilson in the 1970’s, which laid the foundation for the modern understand- ing of renormalization. According to the present view, renormalization is nothing more than parameterizing the sensitivity of low-energy physics to high-energy phy- sics. This viewpoint allows one to make sense out of ‘non-renormalizable’ theories as effective field theories describing physics at low energies. We now understand that even ‘renormalizable’ theories are effective field theories in this sense, and this view- point explains why nature is (approximately) described by renormalizable theories. This modern point of view is the one we will take in this chapter. 1 Renormalization in Quantum Mechanics Ultraviolet divergences and the need for renormalization appear not only in field theory, but also in simple quantum mechanical models. We will study these first to understand these phenomena in a simpler setting, and hopefully dispell the air of mystery that often surrounds the subject of renormalization. 1 1.1 1 Dimension We begin with 1-dimensional quantum mechanics, described by the Hamiltonian ˆ 1 2 ˆ H = 2 pˆ + V . (1.1) We are using units whereh ¯ = 1, m = 1. In these units, all quantities have dimensions of length to some power. Sincep ˆ = id/dx acting on position space wavefunctions, − we have dimensions 1 1 [p]= , [E]= . (1.2) L L2 Suppose that the potential V (x) is centered at the origin and has a range of order a and a height of order V0. We will focus on scattering, so we consider an incident plane wave to the left of x = 0 with momentum p > 0. That is, we assume that the position-space wavefunction has the asymptotic forms Aeipx + Be−ipx x Ψ(x) → −∞ (1.3) → ( Ceipx x + , → ∞ where the contributions proportional to A (B) [C] represent the incoming (reflected) [transmitted] waves (See Fig. 2.1). V (x) V0 incoming transmitted a x re ected Fig. 1. Scattering from a local potential in one-dimensional quantum mechanics. Suppose that the range of the potential a is small compared to the de Broglie wavelength λ =2π/p. This means that the incoming wavefunction is approximately 2 constant over the range of the potential, and we expect the details of the potential to be unimportant. We can then obtain a good approximation by approximating the potential by a delta function: V (x) c δ(x), (1.4) ≃ where c is a phenomenological parameter (‘coupling constant’) chosen to reproduce the results of the true theory. Note that δ(x) has units of 1/L (since dx δ(x) = 1), so the dimension of c is R 1 [c]= . (1.5) L Approximating the potential by a delta function can be justified by considering trial wavefunctions ψ(x) and χ(x) that vary on a length scale λ a. Consider matrix ≫ elements of the potential between such states: χ Vˆ ψ = dx χ∗(x)ψ(x)V (x). (1.6) h | | i Z The wavefunctions χ(x) and ψ(x) are approximately constant in the region where the potential is nonvanishing, so we can write χ Vˆ ψ χ∗(0)ψ(0) dxV (x). (1.7) h | | i ≃ Z This is equivalent to the approximation Eq. (1.4) with c = dxV (x). (1.8) Z We can systematically correct this approximation by expanding the wavefunctions in a Taylor series around x = 0: def f(x) = χ∗(x)ψ(x)= f(0) + f ′(0)x + (1/λ2). (1.9) O Substituting into Eq. (1.6), we obtain χ Vˆ ψ = f(0) dxV (x)+ f ′(0) dxxV (x)+ . (1.10) h | | i Z Z ··· The first few terms of this series can be reproduced by approximating the potential as V (x)= c δ(x)+ c δ′(x)+ (V a2/λ2), (1.11) 0 1 O 0 3 where c0 = dxV (x), c1 = dxxV (x), (1.12) Z − Z etc. Note that this expansion is closely analogous to the multipole expansion in electrostatics. In that case, a complicated charge distribution can be replaced by a simpler one (monopole, dipole, . .) for purposes of finding the potential far away. In the present case, we see that multipole moments of the potential are sufficient to approximate the matrix elements of the potential for low-momentum states. This is true no matter how complicated the potential V (x) is, as long as it has short range. The solution of the scattering problem for a delta function potential is an el- ementary exercise done in many quantum mechanics books. The solution of the time-independent Schr¨odinger equation 1 Ψ′′(x)+ c δ(x)Ψ(x)= EΨ(x) (1.13) − 2 0 has the form Aeipx + Be−ipx x< 0 Ψ(x)= (1.14) ( Ceipx x> 0, with p = √2E. Integrating the equation over a small interval ( ǫ, ǫ) containing the − origin, we obtain 1 [Ψ′(ǫ) Ψ′( ǫ)] + c Ψ(0) = (ǫ). (1.15) − 2 − − 0 O Taking ǫ 0 gives the condition → ip [C A + B]+ c C =0. (1.16) 2 − 0 In order for Ψ(0) to be well-defined, we require that the solution is continuous at x = 0, which gives A + B = C. (1.17) We can solve these equations up to the overall normalization of the wavefunction, which has no physical meaning. We obtain def C T = = transmission amplitude A p = , (1.18) p + ic0 def B R = = reflection amplitude A ic = 0 . (1.19) −p + ic0 4 Note that T 2 + R 2 =1, (1.20) | | | | as required by unitarity (conservation of probability). From the discussion above, we expect this to be an accurate result for any short- range potential as long as p 1/a. The leading behavior in this limit is ≪ ip T . (1.21) ≃ −c0 Note that T is dimensionless, so this answer is consistent with dimensional analysis. Eq. (1.21) is the ‘low-energy theorem’ for scattering from a short-range potential in one-dimensional quantum mechanics. Suppose, however, that the microscopic potential is an odd function of x: V ( x)= V (x). (1.22) − − Then the first nonzero term in Eq. (1.11) is V (x) c δ′(x). (1.23) ≃ 1 Note that c1 is dimensionless. We must then solve the Schr¨odinger equation 1 Ψ′′(x)+ c δ′(x)Ψ(x)= EΨ(x). (1.24) − 2 1 This is not a textbook exercise, for the very good reason that no solution exists! To see this, look at the jump condition: 1 [Ψ′(ǫ) Ψ′( ǫ)] c Ψ′(0) = (ǫ). (1.25) 2 − − − 1 O The problem is that Ψ′(0) is not well-defined, because the jump condition tells us that Ψ′ is discontinuous at x = 0. In fact, this inconsistency is a symptom of an ultraviolet divergence precisely anal- ogous to the ones encountered in quantum field theory. To see this, let us formulate this problem perturbatively, as we do in quantum field theory. We write the Dyson series for the interaction-picture time-evolution operator tf UˆI (tf , ti) = Texp i dt HˆI(t) , (1.26) − Zti where +iHˆ0t −iHˆ0t 2 HˆI (t)= e Vˆ e , Hˆ0 =p ˆ (1.27) 5 defines the interaction-picture Hamiltonian. The S-matrix is given by Sˆ = lim UˆI (+T, T ), (1.28) T →∞ − so the Dyson series directly gives an expansion of the S-matrix. The first few terms in the expansion are pf Sˆ pi = δ(pf pi)+2πδ(Ef Ei) pf Vˆ pf h | | i − − "h | | i (1.29) ˆ i ˆ + dp pf V p 1 p V pi + . h | | iE p2 + iǫh | | i ···# Z − 2 This perturbation series is precisely analogous to the perturbation series used in quantum field theory. This series can be interpreted (`ala Feynman) as describing the amplitude as a sum of terms where the particle goes from its initial state p to the | ii final state p in ‘all possible ways.’ The first term represents the possibility that | f i the particle does not interact at all; the higher terms represent the possibility that the particle interacts once, twice, . with the potential. The interaction with the potential does not conserve the particle momentum (the potential does not ‘recoil’), so the momentum of the particle between interactions with the potential takes on all possible values, as evidenced by the momentum integral in the third term.
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