MIT OpenCourseWare http://ocw.mit.edu 8.323 Relativistic Quantum Field Theory I Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.8.323 Lecture Notes 2: Particle Production by a C lassical Source, Part II (incomplete),p. 1. MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department ������ ����������� ������� ����� ������ � ������� �������� �� � �������� ����� ����� ��� ��� ���������� ���� ��� ���� — ���� ���� ���� ���� ������������ ��������� �� ��������� ������ ���� ��� ���� �������� �� ������������ �������� Equation of motion: ( + m 2)φ(x)= j(x) . (2.1) Initial condition: φ(x)= φin(x) . (2.2) Eqs. (2.1) and (2.2) =⇒ unique solution for Heisenberg operator φ(x). Solution: � φ(x)= φin(x)+ i d4yDR(x − y)j(y) , (2.3) where DR(x − y) is the retarded propagator: x + m 2)DR(x − y)= −iδ(4)(x − y)( (2.4) where DR(x − y)=0 if x 0 <y0 (retarded) . ���� ���� ������������ ��������� �� ��������� ������ ���� ��� ���� –1–� � � � � � � � �� � � 8.323 Lecture Notes 2: Particle Production by a C lassical Source, Part II (incomplete),p. 2. We know that DR(x − y)= θ(x 0 − y 0) 0 |[φin(x) ,φin(y)]| 0 = θ(x 0 − y 0) � d3p 1 e −ip·(x−y) − eip·(x−y) √ 0 2(2π)3 2Epp =Ep = p 2+m(2.5) Note that DR(x − y) is defined by the free wave equation. It can be written in terms of [φin(x) ,φin(y)] as above, or in terms of [φout(x) ,φout(y)], but not in terms of [φ(x) ,φ(y)]. θ(x0 −y0)in DR is hard to deal with, but for x0 ≡ t>t2 we can set θ(x0 −y0)=1. Then d3� � φ(x)= φin(x)+ i d4yj(y) p 1 e −ip·(x−y) − eip·(x−y) . (2.6)(2π)3 2Ep ���� ���� ������������ ��������� �� ��������� ������ ���� ��� ���� –2– Repeating, d3� � φ(x)= φin(x)+ i d4yj(y) p 1 e −ip·(x−y) − eip·(x−y) . (2.6)(2π)3 2Ep Define � ˜(p) ≡ d4 yeip·y j(y) , (2.7) so d3p 1 � −ip·x − ˜ip·x � φ(x)= φin(x)+ i ˜(p)e (−p)e(2π)3 2Ep d3p 1 i −ip·x = � ain(p)+ � ˜(p) e (2π)3 2Ep 2Ep � � � (2.8) + a † (p) − � i ˜(−p) eip·x in2Ep = d3p � 1 � aout(p)e −ipx +h.c. � . (2π)3 2Ep ���� ���� ������������ ��������� �� ��������� ������ ���� ��� ���� –3–� 8.323 Lecture Notes 2: Particle Production by a Classical Source, Part II (incomplete),p. 3. So aout(p)= ain(p)+ i � 2Ep ˜(p) a † out(p)= a † in(p) − i � 2Ep ˜(−p) , (2.9) where ˜(−p)= ˜ ∗ (p) , (2.10) since j(x)is real, and p 0 = � p2 + m2 . (2.11) Thus, only the mass shell component (p0 = � p2 + m2)of ˜(p) results in particle creation. This is just the classical phenomenon of resonance occurring in the quantum field theory setting. ���� ���� ������������ ��������� �� ��������� ������ ���� ��� ���� –4– ������� �������������� ������� �� ��� ��� It is useful to construct a unitary transformation that relates in and out quantities. Remembering that DR(x − y)= θ(x0 − y0) 0 |[φin(x) ,φin(y)]| 0, recall also that [φin(x) ,φin(y)] is a c-number, so 0 |[φin(x) ,φin(y)]| 0 =[φin(x) ,φin(y)].So for x0 ≡ t>t2, φ(x)= φout(x)= φin(x)+ i d4 y[φin(x) ,φin(y)] j(y) . (2.12) If we define � B ≡ d4yj(y) φin(y) , (2.13) then φout(x)= φin(x)+ i[φin(x) ,B] . (2.14) But [φin(x) ,B] is also a c-number, so we can write φout(x)= e −iB φin(x) e iB . (2.15) ���� ���� ������������ ��������� �� ��������� ������ ���� ��� ���� –5–8.323 Lecture Notes 2: Particle Production by a Classical Source, Part II (incomplete),p. 4. Since φout(x)= e −iB φin(x) e iB , (2.15) we know from the uniqueness of the Fourier expansion that aout(p)= e −iB ain(p)e iB . (2.16) We can also verify that this equation is true by using aout(p)= ain(p)+ i � 2Ep ˜(p) (2.9a) with � ain(p) ,a † in(q ) � =(2π)3δ(3)(p − q ) . (2.17) ���� ���� ������������ ��������� �� ��������� ������ ���� ��� ���� –6– ��� S������� Define S ≡ e iB (the ������ S-Matrix) (2.18) Mapping of states: aout(p) |0out =0 S−1 ain(p)S |0out =0 (2.19) =⇒ ain(p) S |0out =0 This implies, up to a phase, the S |0out = |0in. We can redefine the phase of |0out (or |0in)sothat S |0out = |0in . (2.20) ���� ���� ������������ ��������� �� ��������� ������ ���� ��� ���� –7–� � 8.323 Lecture Notes 2: Particle Production by a Classical Source, Part II (incomplete),p. 5. On one particle states, S |pout = Sa† out(p) |0out = Sa† out(p)S−1 � �� � a † in(p) S |0out � �� � |0in = |pin (2.21) In general, we could show that S � �p1 ...pN,out � = � �p1 ...pN,in � . (2.22) ���� ���� ������������ ��������� �� ��������� ������ ���� ��� ���� –8– ������ �������� �� S We know that � i d4yj(y) φin(y) S = e iB