8.323 Relativistic Quantum Field Theory I

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1 MIT OpenCourseWare Relativistic Quantum Field Theory I Spring 2008 For information about citing these materials or our Terms of Use, visit:

2 8.323 Lecture Notes 2: Particle Production by a Classical urce, Part II (incomplete), p. 1. MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department 8.323: Relativistic Quantum Field Theory I Particle Creation by a Classical urce (Part II, and incomplete) March 18, , March 18, 2008 lution to Differential Equation Equationof motion: Initial condition: ( + m 2 )φ(x) = j(x). (2.1) φ(x) = φ in (x). (2.2) Eqs. (2.1) and (2.2) = unique solution for Heisenberg operator φ(x). lution: φ(x) = φ in (x)+ i d 4 yd R (x y)j(y), (2.3) where D R (x y) is the retarded propagator: ( x + m 2 )D R (x y) = iδ (4) (x y) where D R (x y) =0 if x 0 <y 0 (retarded). (2.4) 8.323, March 18,

3 8.323 Lecture Notes 2: Particle Production by a Classical urce, Part II (incomplete), p. 2. We know that D R (x y) = θ(x 0 y 0 ) 0 [φ in (x), φ in (y)] 0 [ ] = θ(x 0 y 0 d 3 p 1 ) e ip (x y) e ip (x y) p 0=E 2 p = p 2 +m (2.5) Note that D R (x y) is defined by the free wave equation. It canbe writtenin terms of [φ in (x), φ in (y)] as above, or interms of [φ out (x), φ out (y)], but not interms of [φ(x), φ(y)]. θ(x 0 y 0 )in D R is hard to deal with, but for x 0 t >t 2 we canset θ(x 0 y 0 )=1. Then d 3 [ ] φ(x) = φ in (x)+ i d 4 p 1 yj(y) e ip (x y) e ip (x y). (2.6) Guth Alan Massachusetts Institute Technology of 8.323, March , 2 Repeating, φ(x) = φ in (x)+ i d 4 yj(y) d 3 p 1 [ ] e ip (x y) e ip (x y). (2.6) Define so d 3 p 1 [ ip x ip x φ(x) = φ ] in (x)+ i j(p)e j( p)e {[ ] d 3 p 1 i ip x = a in ( p)+ j(p) e 2E p [ ] } (2.8) = j(p) d 4 ye ip y j(y), (2.7) + a ( p) i j( p) e ip x in 2E p d 3 p 1 [ aout ( p)e ipx +h.c. ] , March 18,

4 8.323 Lecture Notes 2: Particle Production by a Classical urce, Part II (incomplete), p. 3. a out ( p) = a in ( p)+ a out( p) = a in ( p) i j(p) i j( p), (2.9) where since j(x) is real, and j( p) = j (p), (2.10) p 0 = p 2 + m 2. (2.11) Thus, only the mass shell component (p 0 = p 2 + m 2 )of j(p) results inparticle creation. This is just the classical phenomenon of resonance occurring in the quantum field theory setting , March 18, Unitary Transformation Between In and Out It is useful to construct a unitary transformation that relates in and out quantities. Remembering that D R (x y) = θ(x 0 y 0 ) 0 [φ in (x), φ in (y)] 0, recall also that [φ in (x), φ in (y)] is a c-number, so 0 [φ in (x), φ in (y)] 0 = [φ in (x), φ in (y)]. for x 0 t >t 2, φ(x) = φ out (x) = φ in (x)+ i d 4 y [φ in (x), φ in (y)] j(y). (2.12) If we define B d 4 yj(y) φ in (y), (2.13) then φ out (x) = φ in (x)+ i [φ in (x), B]. (2.14) But [φ in (x), B] is also a c-number, so we can write φ out (x) = e ib φ in (x) e ib. (2.15) 8.323, March 18,

5 8.323 Lecture Notes 2: Particle Production by a Classical urce, Part II (incomplete), p. 4. Since φ out (x) = e ib φ in (x) e ib, (2.15) we know from the uniqueness of the Fourier expansion that a out ( p) = e ib a in ( p)e ib. (2.16) We canalso verify that this equationis true by using a out ( p) = a in ( p)+ i j(p) (2.9a) with [ ] a in ( p), a in ( q ) = (2π) 3 δ (3) ( p q ). (2.17) 8.323, March 18, The S-Matrix Define S e ib (the FAMOUS S-Matrix) (2.18) Mapping of states: a out ( p) 0 out = 0 S 1 a in ( p)s 0 out = 0 (2.19) = a in ( p) S 0 out = 0 This implies, up to a phase, the S 0 out = 0 in. We canredefine the phase of 0 out (or 0 in )sothat S 0 out = 0 in. (2.20) 8.323, March 18,

6 8.323 Lecture Notes 2: Particle Production by a Classical urce, Part II (incomplete), p. 5. Onone particle states, S p out = Sa out( p) 0 out = Sa out( p)s 1 S 0 out }{{}}{{} a in ( p) 0 in = p in (2.21) Ingeneral, we could show that S p 1... p N,out = p 1... p N,in. (2.22) 8.323, March 18, Normal Ordering of S We know that i d 4 yj(y) φ in (y) S = e ib = e. (2.23) It is useful to write S so that all the annihilation operators are on the right. Let [ ] ib = i d 4 d 3 p yj(y) (2π) 1 a 3 in ( p)e ip y + a ( p)eip y in = G + F, 2E p (2.24) where d 3 p F = i 1 j(p)a d 3 p ( p), G = i 1 p), (2.25) (2π) 3 in j( p)a 2E p (2π) 3 in ( 2E p where we recall that j(p) d 4 ye ip y j(y). (2.7) 8.323, March 18,

7 8.323 Lecture Notes 2: Particle Production by a Classical urce, Part II (incomplete), p. 6. S = e ib = e F +G. (2.26) F and G do not commute, but [F,G] is a c-number and therefore commutes with both F and G. Whenever F and G commute with [F,G], e F +G = e F e G e 1 2 [F, G]. (2.27) 8.323, March 18, Aside about e F +G = e F e G e 2 1 [F,G] To prove this identity, define H 1 (λ) e λ(f +G), H 2 (λ) = e λf e λg e 1 λ 2 [F,G] 2. (2.28) Clearly H 1 (0) = H 2 (0) = I (identity operator), and dh 1 (λ) =(F + G) H1 (λ). (2.29) dλ if we canshow that H 2 (λ) obeys the same differential equation as above, then it follows that H 2 (λ) = H 1 (λ). You ll get to show this onyour next problem set. This is actually a special case of the Baker-Campbell-Hausdorff formula, which has the general form e F e G = e F +G+ 1 [F,G]+...(iterated commutators) 2. (2.30) We ll prove this, too, ona problem set soon , March 18,

8 8.323 Lecture Notes 2: Particle Production by a Classical urce, Part II (incomplete), p. 7. Returning to the main argument: S = e ib = e F +G. (2.26) F and G do not commute, but [F,G] is a c-number and therefore commutes with both F and G. Whenever F and G commute with [F,G], Recalling F = i one sees that [F,G] = d 3 p 1 j(p)a (2π) 3 in ( p), = e F +G = e F e G e 1 2 [F, G]. (2.27) G = i d 3 p 1 j( p)a (2π) 3 in ( p), (2.25) d 3 p 1 d 3 q 1 j(p) j( q)[a (2π) 3 (2π) 3 in ( p),a in( q )] 2Eq }{{} (2π) 3 δ (3) ( p q ) d 3 p (2π) 3 1 2E p j(p) 2. (2.31) 8.323, March 18, { S = e ib =exp 1 2 d 3 } p 1 j(p) 2 e F e G. (2.32) 8.323, March 18,

9 8.323 Lecture Notes 2: Particle Production by a Classical urce, Part II (incomplete), p. 8. Vacuum to Vacuum Probability The probability that no particles are produced by the source is given by P (no particle production) = 0 out 0 in 2. (2.33) Logic: physical state is 0 in, independent of time (in the Heisenberg picture). 0 out = state with no particles for t >t 2., 0 out 0 in 2 is the probability that the physical state of the system would be measured to have 0 particles at t >t 2. To express the answer in terms of the S-matrix, recall 0 out = S 1 0 in = 0 out = 0 in S. (2.34) 0 out 0 in = 0 in S 0 in { } 1 d 3 p 1 =exp j(p) 2 0in e F e G (2.35) 0 2 (2π) 3 in. 2E p 8.323, March 18,