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Unformatted text preview: 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. Alan Guth, 8.323 Lecture, May 15, 2008: Interacting Field Theories (Summary), p. 1. MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department
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���� ����� � ������ ���� ���� �� Calculate Green’s functions: expectation values of timeordered Heisenberg ﬁelds in the true vacuum state. Deﬁne S matrix, with S ≡ 1 + iT , and T = (2π )4 δ (4) (∆ptot )iM . Express cross sections and lifetimes in terms of M. Express S matrix in terms of Green’s functions. �� �� ���� ���� �����
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���� ����� ��������� ������� –1– Alan Guth, 8.323 Lecture, May 15, 2008: Interacting Field Theories (Summary), p. 2. ������ ������� λφ4 � �
Then where = 1 1 λ (∂µ φ)2 − m2 φ2 − φ4 . 2 2 4! H = H0 + Hint , Hint = d3 x λ4 φ (x ) . 4! Goal: to perturbatively calculate matrix elements of the Heisenberg ﬁeld φ(x , t) = eiH t φ(x , 0)e−iHt in the state Ω , the true ground state of the interacting theory. ������ ��� ��� ����� ��
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����� Fields evolve, states are timeindependent. ������
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����� Fields evolve according to free ﬁeld theory. Matrix elements, propagators, etc. can be calculated as in free ﬁeld theory. States evolve as necessary, with the evolution driven by the interaction Hamiltonian. ���� ���� �����
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������� ���� ���� �������������� ������������ ������ –2– –3– Alan Guth, 8.323 Lecture, May 15, 2008: Interacting Field Theories (Summary), p. 3. Cho ose any reference time t0 , at which the interaction picture operators and Heisenberg operators will coincide. Deﬁne φI (x , t) ≡ eiH0 (t−t0 ) φ(x , t0 )e−iH0 (t−t0 ) . At t = t0 , can expand Heisenberg φ and π in creation and annihilation operators: φ(x , t0 ) = ˙ π (x , t0 ) = φ(x , t0 ) = d3 p ( 2π )2 d3 p ( 2π )2 1 ap eip ·x + a† e−ip ·x p 2E p . † ap creates state of momentum p , but not energy Ep — not single particle. But [φ(x , t0 ) , π (y , t0 )] = iδ 3 (x − y ) Can write φI (x , t) for all t: d3 p φI (x , t) = ( 2π )2 How to express φ(x , t): φ(x , t) = eiH (t−t0 ) e−iH0 (t−t0 ) φI (x , t)eiH0 (t−t0 ) e−iH (t−t0 ) ≡ U † (t, t0 )φI (x , t)U (t, t0) , U (t, t0 ) = eiH0 (t−t0 ) e−iH (t−t0 ) . Diﬀerential equation for U : ∂ i U (t, t0 ) = eiH0 (t−t0 ) (H − H0 )e−iH (t−t0 ) ∂t = eiH0 (t−t0 ) Hint e−iH (t−t0 ) = eiH0 (t−t0 ) Hint e−iH0 (t−t0 ) eiH0 (t−t0 ) e−iH (t−t0 ) = HI (t)U (t, t0 ) , where HI (t) = eiH0 (t−t0 ) Hint e−iH0 (t−t0 ) = d3 x λ4 φ (x , t) . 4! I –5– where ������
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����� 1 −iEp ap eip ·x + iEp a† e−ip ·x p 2E p . =⇒ † ap , aq = (2π )3 δ (3) (p − q ) . –4– 1 2E p ap e−ip·x + a† eip·x p x0 =t−t0 . Alan Guth, 8.323 Lecture, May 15, 2008: Interacting Field Theories (Summary), p. 4. �������� �� ������������ ��������� ∂ U (t, t0 ) = HI (t)U (t, t0 ) ∂t implies the integral equation i U (t, t0 ) = I − i To ﬁrst order in HI , U (t, t0 ) = I − i To second order in HI , U (t, t0 ) = I − i To third order, U (t, t0 ) = . . . + (−i)3
t t0 t1 t t0 t t0 with U (t0 , t0 ) = I dt HI (t )U (t , t0 ) .
t t0 dt1 HI (t1 ) . dt1 HI (t1 ) + (−i)2 t t0 t1 dt1 t0 dt2 HI (t1 )HI (t2 ) . t2 dt1 t0 dt2 t0 dt3 HI (t1 )HI (t2 )HI (t3 ) . –6– Note that t1 ≥ t2 ≥ t3 . Can rewrite 3rd order term as U (t, t0 ) = . . . + (−i) = ...+
3 t t0 t1 t2 dt1
t t0 t dt2 dt2 t0 t t0 dt3 HI (t1 )HI (t2 )HI (t3 ) dt3 T {HI (t1 )HI (t2 )HI (t3 )} , (−i)3 3! t0 dt1 t0 where T {} is timeordered pro duct (earliest time to right). Finally, U (t, t0 ) = I + (−i) ≡T
t t0 (−i)2 dt1 HI (t1 ) + 2!
t t0 t t0 t dt1 t0 dt2 T {HI (t1 )HI (t2 )} + . . . exp −i dt HI (t ) . –7– Alan Guth, 8.323 Lecture, May 15, 2008: Interacting Field Theories (Summary), p. 5. Generalize to arbitrary t0 :
t2 t1 U ( t2 , t1 ) ≡ T exp −i dt HI (t ) , where (for t1 < t0 < t2 ) U ( t 2 , t1 ) = T exp −i
t2 t0 dt HI (t ) T exp −i t0 t1 dt HI (t ) = U (t2 , t0 )U −1 (t1 , t0 ) . Given have U (t, t0 ) = eiH0 (t−t0 ) e−iH (t−t0 ) , U (t2 , t1 ) = eiH0 (t2 −t0 ) e−iH (t2 −t0 ) eiH (t1 −t0 ) e−iH0 (t1 −t0 ) = eiH0 (t2 −t0 ) e−iH (t2 −t1 ) e−iH0 (t1 −t0 ) . –8– U ( t2 , t1 ) ≡ T Properties: • U is unitary. exp −i t2 t1 dt HI (t ) . • U (t3 , t2 )U (t2 , t1 ) = U (t3 , t1 ) . • U (t2 , t1 )−1 = U (t1 , t2 ) . –9– Alan Guth, 8.323 Lecture, May 15, 2008: Interacting Field Theories (Summary), p. 6. Ω �������
���� �� ���� ������ ����� � Assume that 0 has nonzero overlap with Ω : e−iHT 0 =
n e−iEn T n n 0 . If T had large negative imaginary part, all other states would be suppressed relative to  Ω . Ω = = Recall so Ω =
T →∞(1−i ) T →∞(1−i ) lim e−iH (T +t0 ) 0 e−iE0 (T +t0 ) Ω 0 −1 T →∞(1−i ) lim e−iH (T +t0 ) eiH0 (T +t0 ) 0 e−iE0 (T +t0 ) Ω 0 −1 . U (t2 , t1 ) = eiH0 (t2 −t0 ) e−iH (t2 −t1 ) e−iH0 (t1 −t0 ) , lim U (t0 , −T ) 0 e−iE0 (T +t0 ) Ω 0
−1 . –10– Similarly, Ω = Recall So, for x0 > y 0 , Ω φ(x)φ(y ) Ω =
T →∞(1−i ) T →∞(1−i ) lim e−iE0 (T −t0 ) Ω 0 −1 0 U (T, t0 ) . φ(x , x0 ) = U † (x0 , t0 )φI (x , x0 )U (x0 , t0 ) . lim 0 U (T, t0 )U (t0 , x0 )φI (x , x0 )U (x0 , t0 ) × U (t0 , y 0 )φI (y , y 0 )U (y 0 , t0 )U (t0 , −T ) 0 × Normalization factor =
T →∞(1−i ) lim 0 U (T, x0 )φI (x , x0 )U (x0 , y 0 ) × φI (y , y 0 )U (y 0 , −T ) 0 × Normalization factor . But Normalization factor = Ω Ω
−1 , –11– Alan Guth, 8.323 Lecture, May 15, 2008: Interacting Field Theories (Summary), p. 7. so 0T Ω φ(x)φ(y ) Ω =
T →∞(1−i ) φI (x)φI (y ) exp −i 0T exp −i
T −T T −T dt HI (t) 0 0 . lim dt HI (t) ���� ���� �����
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���� ����� � ������ –12– Recall from last screen: 0T Ω φ(x)φ(y ) Ω =
T →∞(1−i ) φI (x)φI (y ) exp −i 0T exp −i
T −T T −T lim dt HI (t) Recall from Lecture Notes on Path Integrals and Green’s Functions: x(T )=x0 G(tN , . . . , t1 ) =
T →∞(1−i )
¯ D x(t) e h S [x(t)] x(tN ) . . . x(t1 ) i lim x(−T )=x0 x(T )=x0 x(−T )=x0 D x(t) e i h S [x(t)] ¯ with an obvious generalization to ﬁeld theories.
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���� dt HI (t) 0 0 . , (6.20) –13– Alan Guth, 8.323 Lecture, May 15, 2008: Interacting Field Theories (Summary), p. 8. Generalizing path integrals to ﬁeld theories,
φ(x,T )=φ0
¯ D φ(x) e h S [φ(t)] φ(x) φ(y ) i Ω φ(x)φ(y ) Ω = T →∞(1−i ) lim φ(x,−T )=φ0 φ(x,T )=φ0 , D φ(x) e
i h S [φ(t)] ¯ φ(x,−T )=φ0 To calculate perturbatively, just write S [φ(t)] = S0 [φ(t)] + and expand in powers of λ. 1 4! d4 xλφ4 (t) , ���� ���� �����
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���� ����� � ������ –14– ���������� HI (t) = so 0T Ω φ(x)φ(y ) Ω = d3 x � (x , t) , φI (x)φI (y ) exp −i 0T exp −i dz
4 � d4 z � I (z ) 0 . I (z ) 0 If z1 and z2 are spacelikeseparated, their time ordering is framedependent. Need I (z1 ) , I (z2 ) = 0 to get same answer in all frames. � � –15– Alan Guth, 8.323 Lecture, May 15, 2008: Interacting Field Theories (Summary), p. 9. 0T Ω φ(x)φ(y ) Ω = φI (x)φI (y ) exp −i 0T exp −i dz
4 GianCarlo Wick October 15, 1909 – April 20, 1992 For more information see The National Academies Press Biographical Memoir http://www.nap.edu/readingroom.php?book=biomems&page=gwick.html ��
��� ������� �������������� ������������ ������� � d4 z � I (z ) 0 . I (z ) 0 –16– –17– Alan Guth, 8.323 Lecture, May 15, 2008: Interacting Field Theories (Summary), p. 10. T {φ(x1 )φ(x2 ) . . . φ(xm )} = N {φ(x1 )φ(x2 ) . . . φ(xm ) + all possible contractions . Example: Corollary: 0 T {φ(x1 )φ(x2 ) . . . φ(xm )} 0 = all possible FULL contractions . Example: 0 T {φ(x1 )φ(x2 )φ(x3 )φ(x4 )} 0 = ∆F (x1 − x2 )∆F (x3 − x4 ) + ∆F (x1 − x3 )∆F (x2 − x4 ) + ∆F (x1 − x4 )∆F (x2 − x3 ) . –18– ������� �������� Example: 0 T {φ(x1 )φ(x2 )φ(x3 )φ(x4 )} 0 = ∆F (x1 − x2 )∆F (x3 − x4 ) + ∆F (x1 − x3 )∆F (x2 − x4 ) + ∆F (x1 − x4 )∆F (x2 − x3 ) . Feynman diagrams: [Note: the diagrams and some equations on this and the next 12 pages were taken from An Introduction to Quantum Field Theory, by Michael Peskin and Daniel Schroeder.] –19– Alan Guth, 8.323 Lecture, May 15, 2008: Interacting Field Theories (Summary), p. 11. ���������� �������� �
0T I (z ) = λ4 φ (z ) 4! Ω T {φ(x)φ(y )} Ω = φ(x)φ(y ) + φ(x)φ(y ) −i d4 z d4 z � I (z ) +... 0 . 0T φ(x)φ(y ) −i −iλ 4! � I (z ) 0 =3· DF (x − y ) −iλ 4! d4 zDF (z − z )DF (z − z ) + 12 · d4 zDF (x − z )DF (y − z )DF (z − z ) = –20– How many identical contractions are there? Overall factor: 1 3! × 13 4! × 3! × 4 · 3 × 4! × 4 · 3 × ���� ���������� �������� 1 2 = 1 8 ≡ 1 symmetry factor . –21– Alan Guth, 8.323 Lecture, May 15, 2008: Interacting Field Theories (Summary), p. 12. �������� ��
����� –22– ������� ����� ��� λφ4 ������ 0T Rules: φ(x)φ(y ) exp −i d4 z � I (z ) 0 = sum of all possible diagrams with two external points . –23– Alan Guth, 8.323 Lecture, May 15, 2008: Interacting Field Theories (Summary), p. 13. �������� ���
� ������� ������ DF (x − y ) = d4 p i e−ip·(x−y) . (2π )4 p2 − m2 + i Vertex: d4 z e−ip1 z e−ip2 z e−ip3 z e−ip4 z = (2π )4 δ (4) (p1 + p2 + p3 + p4 ) . –24– –25– Alan Guth, 8.323 Lecture, May 15, 2008: Interacting Field Theories (Summary), p. 14. Consider Momentum conservation at one vertex implies conservation at the other. Graph is proportional to (2π )4 δ (4) (0) Use (2π )4 δ (4) (0) = (volume of space) × 2T . Disconnected diagrams: Give each disconnected piece a name: Then Diagram = (value of connected piece) ·
i 1 (Vi )ni . ni ! So, the sum of all diagrams is: Factoring, ���
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��� �������� –26– –27– Alan Guth, 8.323 Lecture, May 15, 2008: Interacting Field Theories (Summary), p. 15. Factoring even more: –28– For our example, Now lo ok at denominator of matrix element: –29– Alan Guth, 8.323 Lecture, May 15, 2008: Interacting Field Theories (Summary), p. 16. Finally, Summarizing, Vacuum energy density: –30– Final sum for fourpoint function: –31– Alan Guth, 8.323 Lecture, May 15, 2008: Interacting Field Theories (Summary), p. 17. ������� �� �������������� ������������ ������ Example: The fourpoint function: –32– ����� ��
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�� ����� where ρA and ρB = number density of particles A = cross sectional area of beams A and B = lengths of particle packets σ≡ Number of scattering events of speciﬁed type . ρA A ρB B A
AA Then Note that σ depends on frame. Special case: 1 particle in each beam, so ρA = 1 and ρB
BA Number of events = σ/A . –33– ���������� �� ����� ��
����� = 1. Then Alan Guth, 8.323 Lecture, May 15, 2008: Interacting Field Theories (Summary), p. 18. Γ≡ Number of decays per unit time . Number of particles present Number of surviving particles at time t: N (t) = N0 e−Γt . Mean lifetime: τ= Halflife: e−Γt = 1 N0
∞ Unstable particles are not eigenstates of H ; they are resonances in scattering experiments. In nonrelativistic quantum mechanics, the BreitWigner formula f (E ) ∝ 1 E − E0 + iΓ/2 =⇒ σ∝ 1 . (E − E0 )2 + Γ2 /4 The “full width at half max” of the resonance = Γ. In the relativistic theory, the BreitWigner formula is replaced by a mo diﬁed (Lorentzinvariant) propagator: 1 1 ≈ , 2 + imΓ 0 − E + i(m/E )Γ/2) −m 2E p ( p p p p2 which can be seen using
2 (p0 )2 − p 2 − m2 = (p0 )2 − Ep = p0 + Ep �������� �����
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0 − dN dt t = 1/Γ . 1 2 =⇒ t1 / 2 = τ l n 2 . –34– p0 − Ep ≈ 2Ep p0 − Ep . –35– Alan Guth, 8.323 Lecture, May 15, 2008: Interacting Field Theories (Summary), p. 19. Recall our discussion of particle creation by an external source, ( + m2 )φ(x) = j (x) , where j was assumed to be nonzero only during a ﬁnite interval t1 < t < t2 . • In that case, the Fock space of the free theory for t < t1 deﬁned the instates, the Fock space of the free theory for t > t2 deﬁned the outstates, and we could calculate exactly the relationship between the two. • We started in the invacuum and stayed there. The amplitude p 1 p 2 . . . p N , out 0, in was then interpreted as the amplitude for pro ducing a set of ﬁnal particles with momenta p 1 . . . p N . –36– For interacting QFT’s, it is more complicated. The interactions do not turn oﬀ, and aﬀect even the 1particle states. It is still possible to deﬁne in and outstates p 1 . . . p N , in and p 1 . . . p N , out with the following properties: • They are exact eigenstates of the full Hamiltonian. • At asymptotically early times, wavepackets constructed from p 1 . . . p N , in evolve as free wavepackets. (The pieces of this ket that describe the scattering vanish in stationary phase approximation at early times.) These states are used to describe the initial state of the scattering. • At asymptotically late times, wavepackets constructed from p 1 . . . p N , out evolve as free wavepackets. These states are used to describe the ﬁnal state. ������� ��� ����� ������� ��� ��� ����������� ����� ��
����� ��� ��� �������� –37– Alan Guth, 8.323 Lecture, May 15, 2008: Interacting Field Theories (Summary), p. 20. Oneparticle incoming wave packet: φ = where φ φ = 1 Twoparticle initial state: φA φB , b , in = d3 kA d3 kB φA (k A ) φB (k B )e−ib ·k B k A k B , in (2π )3 (2π )3 (2EA )(2EB ) , =⇒ d3 k ( 2π )3 1 φ(k ) k , in 2E k d3 k φ(k ) (2π )3
2 where b is a vector which translates particle B orthogonal to the beam, so that we can construct collisions with diﬀerent impact parameters. Multiparticle ﬁnal state: n φ1 . . . φn , out = Deﬁnition: S Ψ, out = Ψ, in . Therefore Ψ, out Ψ, in = Ψ, out S  Ψ, out . But S maps a complete set of orthonormal states onto a complete set of orthonormal states, so S is unitary. Therefore Ψ, out S  Ψ, out = Ψ, out S † S S Ψ, out = Ψ, in S  Ψ, in , so P&S often do not label the states as in or out. ��� ��������� ������
��� ������� , =1. d pf φf (p f ) p 1 . . . p n , out . ( 2π )3 2E f –38–
3 f =1 –39– Alan Guth, 8.323 Lecture, May 15, 2008: Interacting Field Theories (Summary), p. 21. No scattering =⇒ But T must contain a momentumconserving δ function, so deﬁne p 1 . . . p n iT  kA kB ≡ (2π )4 δ (4) kA + kB − pf · iM(kA kB → {pf }) . The probability of scattering into the speciﬁed ﬁnal states is just the square of the S matrix element, summed over the ﬁnal states: � AB, b → p1 . . . pn = To relate to the cross section, think of a single particle B scattering oﬀ of a particle A, with impact parameter vector b : Remembering that the cross section can be viewed as the cross sectional area blo cked oﬀ by the target particle, dσ = d2 b � AB, b → p1 . . . pn . –41– ���������� �� ��� ����� ��
����� S , T , and M � ﬁnal = initial, so separate this part of S : S ≡ 1 + iT . –40– n d pf 1 (2π )3 2Ef
3 p 1 . . . p n S  φA φB , b 2 . f =1 Alan Guth, 8.323 Lecture, May 15, 2008: Interacting Field Theories (Summary), p. 22. Substituting the expression for � and writing out the wavepacket integrals describing the initial state, n 3 d3 kA d3 kB φA (k A ) φB (k B ) d pf 1 2 dσ = d b ( 2π )3 2E f ( 2π )3 ( 2π )3 (2EA )(2EB )
f =1 × ¯ d3 k A (2π )3 ¯ ¯ ¯ d3 k B φ∗ (kA ) φ∗ (k B ) ib ·(kB −k B ) ¯ A B e 3 ¯ A )(2E B ) ¯ ( 2π ) ( 2E ¯¯ p 1 . . . p n S  k A k B
∗ × p 1 . . . p n S  k A k B This can be simpliﬁed by using . ¯ ¯ d2 b eib ·(kB −k B ) = (2π )2 δ (2) k ⊥ − k ⊥ B B , pf pf –42– , . p 1 . . . p n S  k A k B = iM k A k B → {p f } (2π )4 δ (4) kA + kB − ¯¯ p 1 . . . p n S  k A k B
∗ ¯¯ ¯ ¯ = −iM∗ k A k B → {p f } (2π )4 δ (4) k A + k B − ⊥ ¯ ¯ ¯ We ﬁrst integrate over k A and k B using δ (2) k ⊥ − k B and B ¯ ¯ δ (4) k A + k B − pf ¯ ¯ = δ (2) k ⊥ + k ⊥ − A B ¯ ¯ × δ EA + EB − ⊥ ¯ ¯ pf δ k z + k z − A B pz f Ef , ¯ ¯ where the beam is taken along the z axis. After integrating over k ⊥ and k ⊥ , we A B are left with ¯ ¯ ¯¯ dk z dk z δ k z + k z − A B A B
z ¯ ¯ pf δ E A + E B − Ef = ¯ ¯ dk z δ F (k z ) , A A ¯ where the ﬁrst δ function was used to integrate k z , and B ¯ F (k z ) = A Then ¯ ¯z dk z δ F (k A ) = A ¯ k⊥ A
2 ¯ + kz A 2 + m2 + ¯ k⊥ B 2 + ¯ pz − k z A f 2 + m2 − Ef . 1 ¯ , evaluated where F (kz ) = 0 . A dF ¯ dk z A –43– Alan Guth, 8.323 Lecture, May 15, 2008: Interacting Field Theories (Summary), p. 23. Rewriting ¯ F (k z ) = A one ﬁnds ¯ k⊥ A
2 ¯ + kz A 2 + m2 + ¯ k⊥ B 2 + ¯ pz − k z A f 2 + m2 − Ef , ¯ kz dF = ¯A − ¯ EA dk z A ¯ pz − k z A f ¯B E from the previous ¯ ¯ Remembering the δ function constraint δ k z + k z − pz A B f slide, one has ¯ ¯ dF kz kz = ¯A − ¯B = v z − v z  . ¯A ¯B ¯ EA EB dk z
A ¯ What values of satisfy the constraint F (kz ) = 0? There are two solutions, since A ¯z ) = 0 can be manipulated into a simple quadratic equation. (To see this, move F (k A ¯ one square ro ot to the RHS of the equation and square both sides. The (k z )2 term A on each side cancels, leaving only linear terms and a square root on the LHS. Isolate the square ro ot and square both sides again, obtaining a quadratic equation.) One ¯ solution gives k A = k A , and the other corresponds to A and B approaching each other from opposite directions. Assume that the initial wavepacket is too narrow to overlap the 2nd solution. –44– ¯ kz A Then dσ = n d pf 1 ( 2π )3 2E f
3 f =1 d3 kA ( 2π )3
2 d3 kB φA (k A )2 φB (k B )2 z z (2π )3 (2EA )(2EB ) vA − vB  (2π )4 δ (4) kA + kB − pf . × M k A k B → {p f } dΠn (P ) ≡ Deﬁne the relativistically invariant nbo dy phase space measure
n d pf 1 (2π )4 δ (4) P − 3 2E ( 2π ) f
3 pf , f =1 z z and assume that EA (k A ), EB (k B ), vA − vB , M k A k B → {p f } 2 , and dΠn (kA + kB ) are all suﬃciently slowly varying that they can be evaluated at the central momenta of the two intial wavepackets, k A = p A and k B = p B . Then the normalization of the wavepackets implies that d3 kA ( 2π )3 d3 kB φA (k A )2 φB (k B )2 = 1 , 3 ( 2π ) –45– Alan Guth, 8.323 Lecture, May 15, 2008: Interacting Field Theories (Summary), p. 24. so ﬁnally dσ = M (p A p B → {p f }) z z dΠn (pA + pB ) . (2EA )(2EB ) vA − vB 
2 This formula holds whether the ﬁnal state particles are distinguishable or not. In calculating a total cross section, however, one must not doublecount ﬁnal states. If the ﬁnal state contains n identical particles, one must either restrict the integration or divide the answer by n!. –46– In the center of mass (CM) frame, p A = −p B and Ecm = EA + EB , so dΠ2 (pA + pB ) = 2 d pf 1 (2π )4 δ (4) pA + pB − (2π )3 2Ef
3 f =1 = 1 d3 p1 d3 p2 (2π )4 δ (4) (pA + pB − p1 − p2 ) 3 (2π )3 (2E )(2E ) ( 2π ) 1 2 1 d3 p1 = (2π )δ (Ecm − E1 − E2 ) 3 (2E )(2E ) (2π ) 1 2 = dΩ 1 p2 dp1 1 (2π )δ Ecm − 3 (2E )(2E ) ( 2π ) 1 2
−1 p2 p1 1 p1 = dΩ 1 2 + (2π ) (2E1 )(2E2 ) E1 E2 p1 = dΩ . 16π 2 Ecm ���
��� ����� ���������
�� ����� ������ pf p2 + m2 − 1 1 p2 + m2 1 2 –47– Alan Guth, 8.323 Lecture, May 15, 2008: Interacting Field Theories (Summary), p. 25. The twoparticle ﬁnal state, centerofmass cross section is then p A  M (p A p B → p 1 p 2 ) = z z. 64π 2 EA EB (EA + EB ) vA − vB 
2 dσ dΩ cm If all four masses are equal, then EA = EB = and
z z vA − vB  = 1 Ecm 2 2 p A  4 p A  = , EA Ecm so dσ dΩ M2 = 2 64π 2 Ecm (all masses equal) . cm –48– The formula for decay rates is more diﬃcult to justify, since decaying particles have to be viewed as resonances in a scattering experiment. For now we just state the result. By analogy with the formula for cross sections, M (p A p B → {p f }) dσ = z z dΠn (pA + pB ) , (2EA )(2EB ) vA − vB  we write M (p A → {p f }) dΓ = dΠn (pA ) . 2E A Here M cannot be deﬁned in terms of an S matrix, since decaying particles cannot be described by wavepackets constructed in the asymptotic past. M can be calculated, however, by the Feynman rules that Peskin & Schroeder describe in Section 4.6. If some or all of the ﬁnal state particles are identical, then the same comments that were made about cross sections apply here. –49–
2 2 ���������� �� ��� ��
�� ����� Alan Guth, 8.323 Lecture, May 15, 2008: Interacting Field Theories (Summary), p. 26. Timedependent p erturbation theory: Ω T {φ(x1 ) . . . φ(xn )} Ω = 0T φI (x1 ) . . . φI (xn ) exp −i d4 z � I (z ) 0
connected = Sum of all connected diagrams with external points x1 . . . xn . Status: derivation was more or less rigorous, except for ignoring problems connected with renormalization: evaluation of integrals in this expression will lead to divergences. These questions will be dealt with next term. If the theory is regulated, for example by deﬁning it on a lattice of ﬁnite size, the formula above would be exactly true for the regulated theory. One ﬁnds, however, that the limit as the lattice spacing goes to zero cannot be taken unless the parameters m, λ, etc., are allowed to vary as the limit is taken, and in addition the ﬁeld operators must be rescaled. –50– Cross sections from S matrix elements: S = 1 + iT , where p 1 . . . p n iT  kA kB ≡ (2π )4 δ (4) kA + kB − pf · iM(kA kB → {pf }) . The relativistically invariant nbo dy phase space measure is n 3 d pf 1 dΠn (P ) ≡ (2π )4 δ (4) P − ( 2π )3 2E f
f =1 pf , and the diﬀerential cross section is dσ = M (p A p B → {p f }) z z dΠn (pA + pB ) . (2EA )(2EB ) vA − vB 
2 Status: this derivation was more or less rigorous, making mild assumptions about in and outstates. These assumptions, and the formula above, will need to be mo diﬁed slightly when massless particles are present, since the resulting longrange forces mo dify particle trajectories even in the asymptotic past. These mo diﬁcations arise only in higher order perturbation theory, and are part of the renormalization issue. –51– ������� �� ��� ������� �� ��� Alan Guth, 8.323 Lecture, May 15, 2008: Interacting Field Theories (Summary), p. 27. Special case— twoparticle ﬁnal states, in the centerofmass frame: dσ dΩ =
cm p A  M (p A p B → p 1 p 2 ) z z 64π 2 EA EB (EA + EB ) vA − vB  M2 2 64π 2 Ecm (if all masses are equal) . 2 = Decay rate from S matrix elements: M (p A → {p f }) dΠn (pA ) . dΓ = 2E A Status: completely nonrigorous at this point. Unstable particles should be treated as resonances, an issue which is discussed in Peskin & Schroeder in Chapter 7.
2 –52– ��������� ��� �������� ���� ������� �������� Complication: p 1 . . . p n S  p A p B ≡ p 1 . . . p n , out p A p B , in , but the in and outstates are hard to construct: even singleparticle states are mo diﬁed by interactions. The solution will make use of the fact that Ω φ(x) p = Ω eiP ·x φ(0)e−iP ·x p = e−ip·x Ω φ(0) p is an exact expression for the interacting ﬁelds, with the full operator P µ and the exact eigenstate p . By generalizing this to in and outstates, it will be possible to manipulate the correlation functions Ω T (φ1 . . . φn ) Ω by inserting complete sets of in and outstates at various places. When the correlation function is Fouriertransformed in its variables x1 . . . xn to produce a function of p1 . . . pn , one can show that it contains poles when any pi is on its mass shell, p2 = m2 , and i i that the residue when all the pi are on mass shell is the S matrix element. –53– Alan Guth, 8.323 Lecture, May 15, 2008: Interacting Field Theories (Summary), p. 28. A derivation will be given in Chapter 7, but for now we accept the intuitive notion that U (t2 , t1 ) describes time evolution in the interaction picture, and that the S matrix describes time evolution from minus inﬁnity to inﬁnity. So we write p 1 . . . p n S  p A p B =
I p1 . . . pn T exp −i d4 x � I (x) p Ap B I connected, amputated , where “connected” means that the disconnected diagrams will cancel out as before, and the meaning of “amputated” will be discussed below. It will be shown in Chapter 7 that this formula is valid, up to an overall multiplicative factor that arises only in higherorder perturbation theory, and is asso ciated with the rescaling of ﬁeld operators required by renormalization. –54– ���
������� �� 2 → 2 �
��������� 2Ep a† (p ) 0 , a(q ) , a† p = (2π )3 δ (3) (q − p ) . Normalization conventions: p = To zeroth order in p 1 p 2 S  p A p B = I p 1 p 2 p A p B = � I, I (2E1 )(2E2 )(2EA)(2EB ) 0 a1 a2 a† a† 0 AB = (2EA )(2EB )(2π )6 δ (3) (p A − p 1 )δ (3) (p B − p 2 ) + δ (3) (p A − p 2 )δ (3) (p B − p 1 ) . Graphically, Contributes only to “1” of S = 1 + iT . –55– Alan Guth, 8.323 Lecture, May 15, 2008: Interacting Field Theories (Summary), p. 29. To ﬁrst order in � I: I p 1p 2 T = λ −i 4! p 1p 2 N d4 x φ4 (x) I −i λ 4! p A p B
I I d4 x φ4 (x) + contractions I p Ap B I , where contractions = −i λ 6 4! φ(x)φ(x) + 3 . Uncontracted ﬁelds can destroy particles in initial state or create them in the ﬁnal state: φI (x) = φ+ (x) + φ− (x) = I I so φ+ (x) p I
I d3 k ( 2π )3 1 aI (k )e−ik·x + a† (k )eik·x I 2E k
I , = e−ip·x 0 , where φ+ and φ− refer to the parts of φI (x) containing annihilation and creation I I operators, respectively. –56– This leads to a new type of contraction: We show this kind of contraction in a Feynman diagram as an external line. Lo oking at the contracted terms from the Wick expansion, the fully contracted term pro duces a multiple of the identity matrix element, −i λ 4! d4 xI p 1p 2 p Ap B = −i λ 4! d4 x × I p 1 p 2 p A p B
I I , so this term also contributes only to the uninteresting 1 part of S = 1 + iT . –57– Alan Guth, 8.323 Lecture, May 15, 2008: Interacting Field Theories (Summary), p. 30. The singly contracted term −i 6λ 4! d4 xI p 1p 2 φ(x)φ(x) p A p B I contains terms where one φ(x) contracts with an incoming particle and the other contracts with an outgoing particle, giving the Feynman diagrams The integration over x gives an energymomentum conserving δ function, and the uncontracted inner product produces another, so these diagrams are again a contribution to the 1 of S = 1 + iT . The contributions to T come from fully connected diagrams, where all external lines are connected to each other. –58– Nontrivial contribution to T: There are 4! ways of contracting the 4 ﬁelds with the 4 external lines, so the contribution is λ d4 x e−i(pA +pB −p1 −p2 )·x 4! · −i 4! = −iλ(2π )4 δ (4) (pA + pB − p1 − p2 ) ≡ iM(2π )4 δ (4) (pA + pB − p1 − p2 ) , so M = −λ . –59– Alan Guth, 8.323 Lecture, May 15, 2008: Interacting Field Theories (Summary), p. 31. Repeating, M = −λ . By our previous rules, this implies dσ dΩ λ2 . 2 64π 2 Ecm =
cm For σtotal one uses the fact that the two ﬁnal particles are identical. If we integrate over all ﬁnal angles we have doublecounted, so we divide the answer by 2!. λ2 1 = × 4π × 2E 2 64π cm 2! λ2 . 2 32πEcm σtotal = –60– ����������� Consider the 2nd order diagram Contribution is 1 2 d4 p i 4 p 2 − m2 ( 2π ) i d4 k 4 k 2 − m2 ( 2π ) × (−iλ)(2π )4 δ (4) (pA + pB − p1 − p2 ) × (−iλ)(2π )4 δ (4) (pB − p ) . Note that δ (4) (pB − p ) =⇒ p 2 = m2 , so 1 1 =, 2 p −m 0
2 which is inﬁnite. Any diagram in which all the momentum from one external line is channeled through a single internal line will pro duce an inﬁnite propagator. –61– Alan Guth, 8.323 Lecture, May 15, 2008: Interacting Field Theories (Summary), p. 32. Note that δ (4) (pB − p ) =⇒ p 2 = m2 , so 1 1 =, 2 p −m 0
2 which is inﬁnite. Any diagram in which all the momentum from one external line is channeled through a single internal line will pro duce an inﬁnite propagator. Amputation: Eliminate all diagrams for which cutting a single line results in separating a single leg from the rest of the diagram. For example, –62– ������� ����� ��� λφ4 �� �������� ���
�� iM · (2π )4 δ (4) (pA + pB − pf ) = (sum of all connected, amputated diagrams) , where the diagrams are constructed by the following rules: –63– Alan Guth, 8.323 Lecture, May 15, 2008: Interacting Field Theories (Summary), p. 33.
������� ����� ��� λφ4 �� �������� ���
�� iM = (sum of all connected, amputated diagrams) , where the diagrams are constructed by the following rules: –64– Timedependent p erturbation theory: Generalizes easily, since relations: ¯ Ω T φ...ψ ...ψ ... =
I I � is bilinear in Fermi ﬁelds, so it obeys commutation Ω d4 z 0T ¯ φI . . . ψI . . . ψ I . . . exp −i � I (z ) = Sum of all connected diagrams with speciﬁed external points . But, to use Wick’s theorem, we must deﬁne timeordering and normal ordering for fermion operators. ������� ����� ��� �������� 0 I ,connected –65– Alan Guth, 8.323 Lecture, May 15, 2008: Interacting Field Theories (Summary), p. 34.
������������ �����
�� �� ����� ������� Suppose x and y are spacelike separated, with y 0 > x0 . Then ψ (x)ψ (y ) = −ψ (y )ψ (x) . The RHS is already timeordered, so T {−ψ (y )ψ (x)} = −ψ (y )ψ (x). If T is to act consistently on both sides, then T {ψ (x)ψ (y )} = −ψ (y )ψ (x) . Generalizing, T {ψ1 ψ2 . . . ψn } = (product of ψ ’s ordered by time, earliest to right) × (−1)N , where N is the number of interchanges necessary to bring the ordering on the LHS to the ordering on the RHS. (Here ψ represents a general Fermi ﬁeld, ψ ¯ or ψ .) –66– By this deﬁnition ¯ T {ψ (x)ψ(y )} ≡ ¯ ψ (x)ψ(y ) for x0 > y 0 ¯ −ψ (y )ψ (x) for y 0 > x0 . In free ﬁeld theory we have already learned that d4 p i( p + m) −ip·(x−y) e ≡ SF (x − y ) . (2π )4 p2 − m2 + i ¯ 0 T { ψ ( x) ψ ( y ) } 0 = ��� ������� ����������� –67– Alan Guth, 8.323 Lecture, May 15, 2008: Interacting Field Theories (Summary), p. 35. For p = q , as (q )as † (p ) = −as † (p )as (q ) . The RHS is normalordered, so one presumably deﬁnes N {−as † (p )as (q )} = −as † (p )as (q ). If N is to act consistently on both sides, then N {as (q )as † (p )} = −as † (p )as (q ) . Generalizing, N {product of a’s and a† ’s} = (pro duct with all a’s to the right) × (−1)N , where N is the number of interchanges necessary to bring the ordering on the LHS to the ordering on the RHS. –68– ¯ ¯ T {ψ1 ψ 2 ψ3 . . .} = N {ψ1 ψ 2 ψ3 . . . + (all possible contractions)} . A sample contraction would be ��
��� �������� �������������� �����
�� �� ����� ���������� –69– ...
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This note was uploaded on 11/08/2011 for the course PHY 8.323 taught by Professor Staff during the Spring '08 term at MIT.
 Spring '08
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 Quantum Field Theory

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