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Time Ordered Perturbation Theory Time Ordered Perturbation Theory Physics 217 2013, Quantum Field Theory Michael Dine Department of Physics University of California, Santa Cruz October 2013 Physics 217 2013, Quantum Field Theory Time Ordered Perturbation Theory Quantization of the Free Electromagnetic Field We have so far quantized the free scalar field and the free Dirac field. We also would like to understand the free electromagnetic field, before proceeding to interacting theories. Your text deals with this very briefly, simply guessing a form for the propagator. We can be a little more systematic, in a fashion which is instructive. Physics 217 2013, Quantum Field Theory Time Ordered Perturbation Theory A Tension In a gauge theory like electromagnetism, there is a tension between two basic principles: Lorentz invariance and unitarity (unitarity is the statement that in quantum mechanics, time evolution is described by a unitary operator, as a result of which probability is conserved). If we choose Lorentz gauge: µ @µA = 0 (1) and follow our usual quantization procedure, we will be lead to write: Z d 3p Aµ(x) = (aµ(p)eip·x + ay µe−ip·x ) (2) 3p (2π) 2Ep [aµ(p); aν y(p)] = gµν(2π)3δ(~p − ~p0): (3) Physics 217 2013, Quantum Field Theory Time Ordered Perturbation Theory The problem is with the ao commutator. The states aoyj0 > have negative norm (check!). Doesn’t sound good for quantum mechanics. It turns out that the states with negative norm are never produced in scattering processes, but proving this is a bit involved. An alternative approach gives up manifest Lorentz invariance. One chooses the Coulomb (or “transverse" or “radiation") gauge: r~ · A~ = 0 (4) Physics 217 2013, Quantum Field Theory Time Ordered Perturbation Theory Note that in writing this condition, we are making a choice of Lorentz frame. The expansion of the gauge field is now: Z d 3p Ai (x) = (a(p; λ)i (p; λ)eip·x +ay(p; λ)i ∗(p; λ)e−ip·x ): 3p (2π) 2Ep (5) Physics 217 2013, Quantum Field Theory Time Ordered Perturbation Theory From the gauge condition, ~p · ~(p; λ) = 0: (6) The commutation relations of the a’s are just what you might expect: y 0 0 3 0 [a(p; λ); a (p ; λ )] = δλ,λ0 (2π) δ(~p − ~p ): (7) Physics 217 2013, Quantum Field Theory Time Ordered Perturbation Theory From these expressions we can work out the propagator. In this computation, analogous to what we saw in the Dirac case, one encounters: X pi pj i (p; λ)∗j (p; λ) = P (p) = (δ − ): (8) ij ij ~p2 λ Then (Exercise: check): Z d 4p iP (~p) T < Ai (x)Aj (y) >= eip·x ij (9) (2π)4 p2 + i Physics 217 2013, Quantum Field Theory Time Ordered Perturbation Theory It is also natural to define a propagator for the scalar potential, remembering that propagators are just Green’s functions. In momentum space, this is just 1 < A A >= i : (10) o o ~p2 Physics 217 2013, Quantum Field Theory Time Ordered Perturbation Theory Not surprisingly, these propagators don’t look very Lorentz invariant. But we can fix this by noting that the full propagator can be written (in momentum space, using the "west coast metric"): gµν pµpν ηµpν + ηνpµ Dµν = − − + (11) p2 + i ~p2(p2 + i) (p2 + i) o η = ( p ; ; ; ) where ~p2 0 0 0 is a fixed four vector. Exercise: Check this. Don’t worry about the i’s. Physics 217 2013, Quantum Field Theory Time Ordered Perturbation Theory Now in electrodynamics, Aµ couples to jµ, a conserved current. So pµ always multiplies jµ(p), and thus these terms vanish by current conservation. We will actually see how this works in scattering amplitudes later. As a result, we can use the covariant propagator. Note that this is the propagator one might have written in Lorentz gauge by analogy with the propagator for a scalar field. The fact that in the end one can write manifestly Lorentz invariant Feynman rules means that the non-Lorentz invariant gauge choice doesn’t matter in the end. It is possible to prove that, in Coulomb gauge, there are a nice set of operators which generate Lorentz invariance. But this is rather involved and, for the moment, not particularly instructive. Physics 217 2013, Quantum Field Theory Time Ordered Perturbation Theory Interacting Field Theories Consider, first, our scalar field. Lorentz invariance allows many other terms: 1 λ 1 L = ((@ φ)2 − m2φ2 − Γ3φ3 − φ4 − φ5 + :::: (12) 2 µ 4 M We will stop at φ4; generally we won’t include operators of negative mass dimension (“non-renormalizable"). Let’s look at φ3, in the language of old fashioned perturbation theory. We can have processes which change the number of particles. Physics 217 2013, Quantum Field Theory Time Ordered Perturbation Theory Quantum Electrodynamics So far we have free fields: 1 L = − F 2 + ¯(6@ − m) : (13) 4 µν To decide how to couple the gauge field, Aµ, to fermions, we will focus on the principle of gauge invariance. We take i!(x) Aµ ! Aµ + @µ!(x) ! e (x): (14) Then the covariant derivative transforms like : i!(x) Dµ = (@µ − iAµ) ; Dµ ! e Dµ (15) We also introduce a constant e, the electric charge: 1 L = − F 2 + ¯(D6 − m) : (16) 4e2 µν This lagrangian is Lorentz and gauge invariant. Physics 217 2013, Quantum Field Theory Time Ordered Perturbation Theory One often rescales the fields, Aµ ! eAµ (17) in which case 1 A ! A + @ !(x) ! ei!(x) (x): (18) µ µ e µ Dµ = (@µ − ieAµ) (19) In this form, the role of e as a coupling is clearer, but gauge invariance involves e in an odd way (it will be more odd when we confront renormalization). The alternative form gives gauge invariance a more “geometric" character. Physics 217 2013, Quantum Field Theory Time Ordered Perturbation Theory Now, adopting the methods of old fashioned (time-ordered) perturbation theory, we have, for the interaction Hamiltonian: ¯ µ HI = e A γµ (20) and we have the possibility of creating and destroying electrons, positrons and photons in physical processes. Our goal is to develop a more covariant treatment of these processes. Physics 217 2013, Quantum Field Theory Time Ordered Perturbation Theory Pictures in Quantum Mechanics We usually work with quantum mechanics in the Schrodinger picture, in which states depend on time, but operators like x and p are independent of time. In the Schrodinger picture: jΨS(t) >= U(t; to)jΨ(to) > : (21) The time-development operator is just: −iH(t−to) U(t; to) = e (22) and it satisfies: @ i U = HU: (23) @t Physics 217 2013, Quantum Field Theory Time Ordered Perturbation Theory U has the important properties that: y U (t; to) = U(to; t); U(t; t1)U(t1; to) = U(t; to) (24) Another picture which is convenient for certain purposes is the Heisenberg picture. In this picture, the states are independent of time, but operators are now time-dependent: y jΨH (t) >= U (t; to)jΨS(t) >= jΨS(to) > (25) while y OH (t) = U (t; to)OS U(t; to): (26) Physics 217 2013, Quantum Field Theory Time Ordered Perturbation Theory Note that with this definition: < ΨH jOH jΨH >=< ΨSjOSjΨS > (27) and @ i O = −UyHO U + UyO HU = −[H; O ] (28) @t H S S H This is actually an operator version of Ehrnfest’s theorem. You can check, for example, that for a particle in a potential, the operators obey the classical equations of motion. Physics 217 2013, Quantum Field Theory Time Ordered Perturbation Theory As an example, consider the raising and lowering operators for a harmonic oscillator. 1 H = !( + aya)[a; ay] = 1: (29) 2 [H; a] = −!a (30) So d a = −i!a : (31) dt H H So −i!t aH (t) = e aH (0) (32) which is the form that we have become familiar with in our field expansions. Physics 217 2013, Quantum Field Theory Time Ordered Perturbation Theory Probably the most useful picture is the “interaction picture." Here we suppose that the Hamiltonian is of the form: H = Ho + HI (33) where Ho is a Hamiltonian which we know how to diagonalize (in the field theory case, this will be the free Hamiltonian). The basic assumption of perturbation theory will be that HI is in some sense small. Physics 217 2013, Quantum Field Theory Time Ordered Perturbation Theory Then we remove the part of the time dependence we understand and write iHo(t−to) jΨI >= e jΨS(t) > (34) iHo(t−to) −iH(t−to) = e e jΨS(t0) > So we define the time-development operator in the interaction picture: iHo(t−to) −iH(t−to) UI(t; to) = e e (35) and define the time-dependent operators: iHo(t−to) −iHo(t−to) OI(t) = e OSe (36) Physics 217 2013, Quantum Field Theory Time Ordered Perturbation Theory It is a simple exercise to show that UI obeys: @ i U = H (t)U ; U (t; t) = 1: (37) @t I I I I where iHo(t−to) −iHo(t−to) HI(t) = e HIe (38) UI has composition properties similar to those of U. These are most easily proven by writing U in a slightly different form: 0 0 0 iHo(t−to) −iH(t−t ) −iHo(t −to) UI(t; t ) = e e e (39) This obeys the correct equation and boundary condition (check!). It also manifestly satisfies: y UI (t; to) = UI(to; t); UI(t; t1)UI(t1; t2) = U(t; t2) (40) Physics 217 2013, Quantum Field Theory Time Ordered Perturbation Theory Solving this equation iteratively (see your favorite quantum mechanics book, e.g.
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