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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 1, 2008, p. 1. MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department ������ ����������� ������� ����� ������ � ��� ���� ����� ���� �� ��������� ��� �� ���� — ���� ���� ���� ���� ����� ������� ��������� �� �� ������� ������ ��� �� ���� ������ ������ Corrections to Problem Set 9: Problem 2: Notational improvement. The two equations should read: 3 Λ 1 B3 (η ) ≡ e−iηK = e−iηS 2 3 Λ 1 R3 (θ ) ≡ e−iθJ = e−iθS 2 03 12 . . i Problem 4: Should read S µν = 4 [γ µ , γ ν ] in the preamble, and identity (vi) should read /q p / = 2p · q − q / = p · q − 2iS µν pµ qν . /p ���� ���� ����� ������� ��������� �� �� ������� ������ ��� �� ���� –1– Alan Guth, 8.323 Lecture, May 1, 2008, p. 2. ������� �� ����� �� ��� Poincar´ symmetry implies e µ µ eiPµ x ψa (y ) e−iPµ x = ψa (x + y ) , U −1 (Λ)ψa (x)U (Λ) = Mab ψb (Λ−1 x) , (68) where M (Λ) is a representation of the Lorentz group. Following Dirac we have chosen µν i i (69) Mab ≡ Λ 1 ,ab = e− 2 ωµν S , with S µν = [γ µ , γ ν ] , 2 4 and following P&S we have made a particular choice of the γ -matrices. P 2 is a Casimir operator, which we take to equal m2 , the square of the particle mass, and we assume that P 0 > 0. ���� ���� –2– ����� ������� ��������� �� �� ������� ������ ��� �� ���� Creation operators transform as U (Λ) a† (p ) U −1 (Λ) = s EΛp Ep s� a†� (Λp ) Ds� s RW (Λ, p ) . s (70) Taking the adjoint: U (Λ) as (p ) U −1 (Λ) = = EΛp Ep EΛp Ep ∗ as� (Λp ) Ds� s RW (Λ, p ) s� s� (71) − Dss1 RW (Λ, p ) as� (Λp ) , � ���� ���� ����� ������� ��������� �� �� ������� ������ ��� �� ���� –3– Alan Guth, 8.323 Lecture, May 1, 2008, p. 3. Expand the field as d3 p ( 2π )3 ψa (x) = 1 2E p µ s µ s as (p ) us (p ) e−ipµ x + b† (p ) va (p ) eipµ x a s , (72) where p0 = + |p |2 + m2 . The sign of p0 , which follows from our assumption that the operator P 0 was positive, will be important. The above equations imply that � − us (Λp ) = Λ 1 ,ab us (p )Ds� 1 RW (Λ, p ) . b a s 2 (73) Writing ψa (x) as ψ1 (x) ψ (x) ψa (x) = 2 = ψ3 (x) ψ4 (x) ψL (x) ψR (x) , (74) ���� ���� ����� ������� ��������� �� �� ������� –4– ������ ��� �� ���� we found that rotations in the rest frame implied that us ,a (p = 0) = L √ s m δa , (75) up to an arbitrary normalization. This is often written as us (p = 0) = L √ m ξ s , where ξ + = 1 0 , ξ− = 0 1 , (76) where ξ + corresponds to spin-up, and ξ − corresponds to spin-down, along the z -axis. The constraints on uR are identical, so we similarly choose us (p = 0) = R √ m ξs . (77) In both cases the normalization is an arbitrary convention, and we can cho ose them to have the same normalization without any loss of generality. ���� ���� ����� ������� ��������� �� �� ������� ������ ��� �� ���� –5– Alan Guth, 8.323 Lecture, May 1, 2008, p. 4. To discuss nonzero momenta, we bo ost from the rest frame: us ( q ) = √ ξs ξs m Λ 1 (B q ) 2 . (78) We will calculate this explicitly in the canonical basis: ˆ Λ 1 (Bp ) = e−iηp·K , 2 (79) where η is the rapidity, with cosh η = E /m, sinh η = |p |/m, and tanh η = v . (Earlier I called the rapidity ξ , but ξ is now being used for the basis spinors.) In our basis i σj 0 . K j = J 0j = − 0 −σ j 2 Λ 1 (Bp ) = exp 2 1 η 2 −p · σ ˆ 0 0 p·σ ˆ . Note that (p · σ )2 = 1, so ˆ ���� ���� ����� ������� ��������� �� �� ������� –6– ������ ��� �� ���� 1 η 2 −p · σ ˆ 0 0 p·σ ˆ =1+ η 2 −p · σ ˆ 0 0 p·σ ˆ + 1 3! 1 0 0 1 Λ 1 (Bp ) = exp 2 = η 2 3 cosh + 1 2! −p · σ ˆ 0 0 p·σ ˆ η + 2 −p · σ ˆ 0 η 2 2 1 0 0 1 + +... 0 p·σ ˆ sinh η 2 . To simplify, look at 2 × 2 blo cks: ΛL 0 Λ 1 (B p ) = 2 0 ΛR , where Λ L = cosh R η η ∓ p · σ sinh ˆ 2 2 . Then remember that e±η = (E ± |p |)/m, so ΛL = 1 2 E + |p | ˆ [1 − p · σ ] + m E − |p | ˆ [1 + p · σ ] m . ���� ���� ����� ������� ��������� �� �� ������� ������ ��� �� ���� –7– Alan Guth, 8.323 Lecture, May 1, 2008, p. 5. ΛL = 1 2 E + |p | [1 − p · σ ] + ˆ m The eigenvalues of p · σ are ±1, so ˆ ΛL = = E − |p | [1 + p · σ ] ˆ m E −|p | m if p · σ = 1 ˆ E +|p | m . if p · σ = −1 ˆ E−p ·σ . m If the eigenvalues of a matrix are positive, we define its square ro ot by diagonalizing the matrix and taking the positive square ro ot of its eigenvalues. ���� ���� ����� ������� ��������� �� �� ������� –8– ������ ��� �� ���� Similarly, ΛR = E +p ·σ . m The notation can be simplified further by defining 4-vectors σ µ = (1, σ i ) and σ µ = (1, −σ i ), so ¯ p·σ p·σ ¯ , ΛR = , ΛL = m m and finally us ( p ) = √ p · σ ξs √ p · σ ξs ¯ . (80) ���� ���� ����� ������� ��������� �� �� ������� ������ ��� �� ���� –9– Alan Guth, 8.323 Lecture, May 1, 2008, p. 6. ������ ���������� ��� us(p ) (p · σ )(p · σ ) = p2 = m2 . ¯ (81) Normalization: † u ( p ) u( p ) = ξ †√ p·σ ξ †√ p·σ ¯ √ p · σ ξs √ p · σ ξs ¯ . = ξ † (p · σ + p · σ )ξ ¯ (82) = 2E p ξ † ξ . ¯ u† ( p ) γ 0 u( p ) ≡ u( p ) u( p ) √ √ √ √ ¯ ¯ = ξ † p · σ p · σξ + ξ † p · σ p · σξ (83) = 2m ξ † ξ . ����� ������� ��������� �� �� ������� –10– ���� ���� ������ ��� �� ���� √ s √p · σ ξ u (p ) = p · σ ξs ¯ s u s ( p ) us ( p ) = γ · p + m , ¯ =⇒ (84) s ¯ where us (p ) ≡ us† (p )γ 0 . For the v s, � − us (Λp ) = Λ 1 ,ab us (p )Ds� 1 RW (Λ, p ) . b a s 2 (73) is replaced by � − s s va (Λp ) = Λ 1 ,ab vb (p )Ds� 1 ∗ RW (Λ, p ) . s 2 (85) ���� ���� ����� ������� ��������� �� �� ������� ������ ��� �� ���� –11– Alan Guth, 8.323 Lecture, May 1, 2008, p. 7. From lo oking at the rest frame, we use D ∗ (R ) = σ 2 D (R )σ 2 , and following the same logic that we used to show that ua s (p ) must be a multiple of the identity, here we show that (vσ 2 )a s must be a multiple of the identity. The normalization is arbitrary, so we can write √ √ s 2 s 2 (86) vL,a (p = 0) = −i m σas βL , vR,a (p = 0) = i m σas βR , where βL and βR are used here to indicate our freedom to cho ose the normalization of these terms arbitrarily, as far as the Lorentz transformations of the field are concerned. We will so on find, however, that these constants are constrained. Bo osting to an arbitrary momentum, √ 2s p √ · σ (−iσ ξ )βL , v (p ) = − p · σ (−iσ 2 ξ s )βR ¯ s (87) ���� ���� ����� ������� ��������� �� �� ������� –12– ������ ��� �� ���� 1 and ξ − = 0 ξ= 0 1 describe spin-up and spin-down particles. We can also define where as before + η s ≡ −iσ 2 ξ s , (88) and the two spinors η s , for s = + and s = −, will form a pair of orthonormal vectors in the space of 2-component spinors. Then one has √ s p √ · σ η βL , v (p ) = − p · σ η s βR ¯ s (89) The spin sum for the v ’s then becomes ∗ −mβL βR v (p ) v (p ) = ¯ p · σ |βR |2 ¯ s s s p · σ |βL |2 ∗ . −mβR βL (90) ���� ���� ����� ������� ��������� �� �� ������� ������ ��� �� ���� –13– Alan Guth, 8.323 Lecture, May 1, 2008, p. 8. ���� ��� �������� � A key result of quantum field theory is the Spin-Statistics Theorem, which says that particles of integer spin must be bosons, while particles of half-integer spin must be fermions. (Note that there is no such theorem in nonrelativistic quantum theory.) A full pro of is beyond this course. See, for example, PCT, Spin & Statistics, and All That, by R.F. Streater and Arthur S. Wightman, 1964. Despite the informal title and mo dest length, this bo ok is not easy reading for most of us. We will, however, demonstrate a limited version of this theorem. We will consider the possibility that the Dirac particles are either bosons or fermions, and show that only the second choice gives a consistent theory. ����� ������� ��������� �� �� ������� –14– ���� ���� ������ ��� �� ���� ��� ����� ����������� ���� ����� ����������� Consider first the boson possibility. Bosons are like the scalar particles that we have studied, with creation/annihilation operator commutators as (p ) , ar† (q ) = (2π )3 δ rs δ (3) (p − q ) . bs (p ) , br† (q ) = (2π )3 δ rs δ (3) (p − q ) , (91h) with other commutators ([a , a], [b , b], and their adjoints) vanishing. The “h” in the equation number stands for “hypothesis”. It will turn out NOT to be consistent. We will now compute the equal-time commutator ψ (x ,0) , ψ † (y ,0) which is clearly a spacelike separation. There are two reasons why this should vanish. ���� ���� ����� ������� ��������� �� �� ������� ������ ��� �� ���� –15– Alan Guth, 8.323 Lecture, May 1, 2008, p. 9. ��� ��� ���������� ������ ������ �� ��� ����� ����������� �� The fields should represent measurable quantities, so a nonzero commutator implies an uncertainty principle — a precise measurement of one causes the other to become completely uncertain. But if this happens at spacelike separations, then information can be transmitted faster than light. �� Later we will use these fields to describe terms in an interaction Hamiltonian, and perturbation theory will allow us to express evolution (and hence scattering cross sections) in terms of time-ordered products. However, the time-ordering between 2 spacetime points is ambiguous if they are spacelike separated, so a unique answer is found only if the operators commute. So, the Lorentz­ invariance of interactions calculated in QFT would be violated if fields did not commute at spacelike separations. The requirement that commutators vanish at spacelike separations is called ���������� ���� ���� ����� ������� ��������� �� �� ������� –16– ������ ��� �� ���� ��� ������� �� ��� ���������� From Eq. (72), ψa (x , 0) = = d3 p ( 2π )3 1 2E p d3 p ( 2π )3 1 2E p µ µ s as (p ) us (p ) e−ipµ x + b† (p ) va (p ) eipµ x a s s s as (p ) us (p ) + b† (−p ) va (−p ) eip ·x . a s s (92) So ¯ ψ (x ,0) , ψ (y ,0) = × ei(p ·x −q ·y ) d3 p d3 q ( 2π )6 1 ( 2E p ) ( 2E q ) a r ( p ) , a s † ( q ) u r ( p ) us ( q ) ¯ (93) r,s + br† (−p ) , bs (−q ) v r (−p ) v s (−q ) . ¯ –17– Alan Guth, 8.323 Lecture, May 1, 2008, p. 10. Using the creation/annihilation operator commutators (91h) and the spin sums (84) and (90), this becomes ¯ ψ (x ,0) , ψ (y ,0) = Ep d3 p 1 ip ·(x −y ) e × ( 2π )3 2E p ∗ m (1 + βL βR ) 1 − |βR |2 + p · σ 1 + |βR |2 Ep 1 − |βL |2 − p · σ 1 + |βL |2 . ∗ m (1 + βR βL ) (94) The expression has particle contributions, with no β ’s, and antiparticle contribu­ tions, with 2 β ’s. The killer terms are the ones in p · σ , since the co efficients of particles and antiparticles add. Lo oking back to Eq. (93), one sees that the antiparticle contribution has 2 sign changes relative to the particle contribution: the signs of p and q are reversed, and the commutator has the creation operator first. ���� ���� ����� ������� ��������� �� �� ������� –18– ������ ��� �� ���� The term proportional to p · σ does not vanish when integrated, as it is proportional to −iσ · ∇ d3 p 1 ip ·(x −y ) e , ( 2π )3 2E p (95) where the integral does not vanish. ���� ���� ����� ������� ��������� �� �� ������� ������ ��� �� ���� –19– Alan Guth, 8.323 Lecture, May 1, 2008, p. 11. ��� ������� ����������� ���� ����� ����� ���������� This time let us assume that the particles are fermions, which means that only one particle can have a specified momentum and spin, and that the multi-particle basis state |p 1 s1 , . . . , p N sN is antisymmetric in its particle labels. For example, |p 1 s1 , p 2 s2 = − |p 2 s2 , p 1 s1 . (96) For fermions the creation and annihilation operators satisfy anticommutation relations as (p ) , ar† (q ) = (2π )3 δ rs δ (3) (p − q ) . (97) bs (p ) , br† (q ) = (2π )3 δ rs δ (3) (p − q ) , ���� ���� ����� ������� ��������� �� �� ������� ������ ��� �� ���� –20– Note that the antisymmetry of states, as in Eq. (96), implies that as1 † (p 1 ) as2 † (p 2 ) = −as2 † (p 2 ) as1 † (p 1 ) , (98) since the creation operators create multiparticle states with the particle labels in the same order as the operators. Thus there is no possibility that fermion creation/annihilation operators could obey commutation rather than anticommutation relations. Since the fields are linear in creation and annihilation operators, if the creation and annihilation operators satisfy anticommutation relations, then so must the fields. If the fields anticommute at spacelike separations, that is okay. It means that any measurable quantity — any quantity that can appear in the Hamiltonian — must be bilinear in the fermi fields, and then they will commute. We might have expected this anyway for a fermi field, since under a 360◦ rotation it turns into minus itself! If such a field were measurable, what would it look like? ���� ���� ����� ������� ��������� �� �� ������� ������ ��� �� ���� –21– Alan Guth, 8.323 Lecture, May 1, 2008, p. 12. ��� ������� �� ��� ���� ��������� The calculation lo oks much like the previous one, with one key difference. In the boson calculation we found that the antiparticle contribution was proportional to bs† (−p ) , bs (−q ) , which produced a minus sign because the creation and annihilation operators were not in their usual order. This time, however, we find an anticommutator instead, so there is no change in sign. Our final answer will be identical to the previous one, except that the entire antiparticle contribution will change sign: ¯ ψ (x ,0) , ψ (y ,0) = Ep d3 p 1 ip ·(x −y ) e × ( 2π )3 2E p ∗ m (1 − βL βR ) 1 + |βR |2 + p · σ 1 − |βR |2 Ep 1 + |βL |2 − p · σ 1 − |βL |2 . ∗ m (1 − βR βL ) (99) The change in sign makes all the difference! ���� ���� ����� ������� ��������� �� �� ������� –22– ������ ��� �� ���� ��������� �� �� ��� ��� ������� ¯ ψ (x ,0) , ψ (y ,0) = Ep d3 p 1 ip ·(x −y ) e × ( 2π )3 2E p ∗ m (1 − βL βR ) 1 + |βR |2 + p · σ 1 − |βR |2 Ep 1 + |βL |2 − p · σ 1 − |βL |2 . ∗ m (1 − βR βL ) (99) The previously fatal p · σ terms will cancel if |βL |2 = |βR |2 = 1 . (100) The terms in m will cancel as long as βL and βR have the same phase. Thus, we require βL = βR . (101) From now on we will adopt the standard phase convention, (102) βL = βR = 1 . ���� ���� ����� ������� ��������� �� �� ������� ������ ��� �� ���� –23– Alan Guth, 8.323 Lecture, May 1, 2008, p. 13. d3 p 1 ip ·(x −y ) e × ( 2π )3 2E p ¯ ψ (x ,0) , ψ (y ,0) = Ep ∗ m (1 − βL βR ) 1 + |βR |2 + p · σ 1 − |βR |2 Ep 1 + |βL |2 − p · σ 1 − |βL |2 . ∗ m (1 − βR βL ) (99) The terms in Ep do not cancel, but notice that the Ep cancels the same factor in the denominator, pro ducing a Dirac δ -function which vanishes when x = y . ����� ������� ��������� �� �� ������� ���� ���� ������ ��� �� ���� –24– ...
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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.

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