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Quantum Field Theory II

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Quantum Field Theory II Quantum Field Theory II GEORGE SIOPSIS Department of Physics and Astronomy The University of Tennessee Knoxville, TN 37996-1200 U.S.A. e-mail: [email protected] Spring 2018 ii Contents 1 Interactions 1 1.1 Scattering theory . 1 1.2 Wick’s theorem . 5 1.3 Feynman diagrams . 15 1.4 Reaction rates . 19 1.5 Unitarity . 22 1.6 Path Integrals . 24 iii iv CONTENTS UNIT 1 Interactions When you include interactions, the plot thickens. Until now, we have only considered free fields, described by a Hamiltonian H = H0, which we were able to diagonalize (find all eigenvalues and corresponding eigenstates). But that is no physics! It is time to do some serious physics. When you include interactions, the Hamiltonian is modified to H = H0 + HI (1.0.1) where HI describes interactions. It is no longer possible to solve the eigenvalue problem for H, except in very few special cases (mostly in two dimensions). The only thing we can do is perturbation theory, assuming HI is small. This does not answer deep questions, such as “what is a proton?”, but it provides a method for very accurate calculations (e.g., the magnetic moment of the electron is known to about 10 significant digits both theoretically and experimentally, and they agree with each other!). Experimental results are obtained primarily through scattering. To compare with them, we need to develop scattering theory. It turns out that this type of processes (scattering) holds all the information of a quantum field theory. 1.1 Scattering theory Recall in quantum mechanics, H = H0 + V (1.1.1) where H0 is the kinetic energy and V the potential. If V has compact support, then at times t ! ±∞, H = H0, because V = 0. Thus, we may define incoming and outgoing states, jini (at t ! −∞) and jouti (at t ! +1), respectively, which are eigenstates of H. We shall attempt to do the same in quantum field theory. Consider a state jini. It evolves in time as e−iHtjini (1.1.2) As t ! −∞, H ! H0 (no interactions), so asymptotically, our state approaches a state in the Hilbert space constructed from H0. Call that state jin,0i. It evolves in time as e−iH0tjin,0i (1.1.3) Then the statement that this is asymptotic to the state jini amounts to e−iHtjini −! e−iH0tjin,0i ; as t ! −∞ (1.1.4) Therefore, jini = lim eiHte−iH0tjin,0i (1.1.5) t→−∞ 2 UNIT 1. INTERACTIONS The operator U(t) ≡ eiH0te−iHt (1.1.6) y (a unitary map, U U = I) maps a state in the Hilbert space of H0 (jin,0i) to a state in the Hilbert space of H (jini) by jini = lim U y(t)jin,0i (1.1.7) t→−∞ −iHI t Notice that if H and H0 commute ([H; H0] = 0), then we may write U(t) = e , where we used (1.0.1). But this rarely happens. Thus, in general, U(t) is a very complicated object. Similarly, in the infinite future, we may map jouti = lim U y(t)jout,0i (1.1.8) t!+1 where jouti (jout,0i) is in the Hilbert space of H (H0). We wish to calculate the amplitude of the process jini −! jouti (1.1.9) i.e., y houtjini = lim hout,0jU(t+)U (t−)jin,0i = hout,0jSjin,0i (1.1.10) t±→±∞ where y S ≡ lim U(t+)U (t−) (1.1.11) t±→±∞ is the S-matrix. S maps in-asymptotes to out-asymptotes (these two Hilbert spaces may, in general, be distinct, but not here). In fact, S contains all the information of the quantum field theory. It is an important object to study. Let us bring it into a more convenient form. To this end, we need to establish some properties of U(t) first. We have dU(t) = iH eiH0te−iHt + eiH0t(−iH)e−iHt dt 0 iH0t −iHt = −ie (H − H0)e iH0t −iHt = −ie HI e iH0t −iH0t iH0t −iHt = −ie HI e e e = −iHI (t)U(t) (1.1.12) Thus U(t) is a true evolution operator with (time-dependent) Hamiltonian HI (t). The latter is ob- tained from HI by evolving with H0. The first-order ODE (1.1.12) together with the initial condition U(0) = I uniquely determine U(t). To solve (1.1.12), convert it into an integral equation, Z t 0 Z t dU(t ) 0 0 0 0 0 dt = −i dt HI (t )U(t ) 0 dt 0 Z t 0 0 0 U(t) = I − i dt HI (t )U(t ) (1.1.13) 0 and then use iteration, Z t 0 0 0 U0(t) = I ;Un(t) = I − i dt HI (t )Un−1(t )(n ≥ 1) (1.1.14) 0 1.1 Scattering theory 3 We obtain U0(t) = I Z t 0 0 U1(t) = I − i dt HI (t ) 0 Z t Z t Z t1 0 0 2 U2(t) = I − i dt HI (t ) + (−i) dt1 dt2HI (t1)HI (t2) (1.1.15) 0 0 0 etc.. This may be brought into a more compact form. Look at the second-order term. In it, t ≥ t1 ≥ t2 ≥ 0. The two-dimensional integral is over a triangle in the (t1; t2) plane, which is half of the square 0 ≤ t1; t2 ≤ t. The other half gives the same answer, but with t1 and t2 interchanged. It follows that the two-dimensional integral can be written in terms of a time-ordered product as Z t Z t1 1 Z t Z t dt1 dt2HI (t1)HI (t2) = dt1 dt2T [HI (t1)HI (t2)] (1.1.16) 0 0 2 0 0 For the nth term, we similarly obtain Z t Z tn−1 1 Z t Z t dt1 ··· dtnHI (t1) ··· HI (tn) = dt1 ··· dtnT [HI (t1) ··· HI (tn)] (1.1.17) 0 0 n! 0 0 It follows that 1 X (−i)n Z t Z t U(t) = T dt ··· dt H (t ) ··· H (t ) n! 1 n I 1 I n n=0 0 0 h −i R t dt0H (t0)i = T e 0 I (1.1.18) which is Dyson’s formula. Next, define the generalized evolution operator R t2 0 0 y h −i dt HI (t )i U(t1; t2) ≡ U (t1)U(t2) = T e t1 (1.1.19) It is easily seen from its definition that it has the following properties 0 0 00 00 y 0 0 y 0 0 U(t; t )U(t ; t ) = U(t; t ) ;U (t; t ) = U(t ; t) ;U (t; t )U(t; t ) = I (1.1.20) The S-matrix can be written as h R +1 i −i dtHI (t) S = lim U(t+; t−) = T e −∞ (1.1.21) t±→±∞ Various important properties of the S-matrix follow. • The S-matrix is unitary, y S S = I (1.1.22) This may look like a trivial statement (direct consequence of the unitarity of U(t+; t−)), but we need to take limits t± ! ±∞, and that may spoil unitarity, unless the range of S is the entire Hilbert space. The latter is a consequence of the requirement that probability be conserved: everything that comes in should go out. There are cases where this is not true and particles may be trapped into bound states. The S-matrix has a way to address these issues. For the most part, we shall assume that no trapping occurs and the S-matrix is unitary. • S commutes with the free Hamiltonian H0, [S; H0] = 0 (1.1.23) i.e., a scattering process conserves unperturbed energy. 4 UNIT 1. INTERACTIONS Proof. eiH0 Se−iH0 = lim eiH0 eit−H0 e−it−H eit+H e−it+H0 e−iH0 t±→±∞ = lim eiH0(t−+)e−iH(t−+)eiH(t++)eiH0(t−−) t±→±∞ y = lim U(t− + )U (t+ + ) t±→±∞ But as t± ! ±∞, adding an makes no difference, so eiH0 Se−iH0 = S (1.1.24) Expanding in , at first order we obtain the desired result (1.1.23). • S is Lorentz-invariant. Proof. The Lagrangian density (a Lorentz-invariant quantity) can be split as L = L0 + LI (1.1.25) If LI contains no time derivatives, then Z 3 HI = − d xLI (1.1.26) Therefore, using (1.1.21), h −i R 1 dtH (t)i h i R d4xL i S = T e −∞ I = T e I (1.1.27) which is manifestly Lorentz-invariant. The integral is over the entire space-time. It should be noted that time-ordering does not spoil Lorentz invariance. Indeed, time-ordering is frame-dependent only for spacelike separations, and fields commute at spacelike separations (causality). • Corollary: By Lorentz invariance, it follows from (1.1.23) that S commutes with the unper- turbed four-momentum, µ [S;P0 ] = 0 (1.1.28) This implies momentum conservation in scattering. 0 0 Indeed, consider an incoming (outgoing) state of free particles with momenta p1; p2;::: (p1; p2;::: ), i.e., 0 0 jin,0i = j~p1; ~p2;::: i ; jout,0i = j~p1; ~p2;::: i (1.1.29) (notation as in eqs. (1.1.7) and (1.1.8)). These are eigenstates of unperturbed momentum µ µ µ µ 0µ 0µ P0 jin,0i = (p1 + p2 + ::: )jin,0i ;P0 jout,0i = (p1 + p2 + ::: )jout,0i (1.1.30) It follows that µ X µ X 0µ 0 = hout,0j[S;P0 ]jin,0i = pi − pi houtjini and so X µ X 0µ pi = pi (1.1.31) i.e., momentum is conserved.
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