8.323 Relativistic Quantum Field Theory I

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1 MIT OpenCoureWare Relativitic Quantum Field Theory I Spring 008 For information about citing thee material or our Term Ue, viit:

2 , 8.33 Lecture, April 4 & 9, 008, p.. MASSACHUSETTS ISTITUTE OF TECHOLOGY Phyic Department 8.33: Relativitic Quantum Field Theory I The Dirac Field April 4 & 9, 008 Eq. (55) wa corrected on 5/6/08 Maachuett Intitute Technology 8.33, April 4 & 9, 008 Tranformation the Dirac Field Let u aumethat wearetrying to contruct a freefield ψ a (x) for electron which, like the free field φ(x) for calar particle, i linear in creation and annihilation operator. Then we expect a nonzero value for 0 ψ ( x) electron, p =0. a Under rotation the tate() on the right tranform under the pin- repreentation, o ψ a (x) mut contain thi repreentation, or ele the matrix element vanihe (i.e. the only pin that can be added to pin- to get pin-0 i pin- ). But ψ a (x) mut tranform under ome repreentation the Lorentz group, generated by J + = (J + ik ) J = J + + J (4) J = (J ik ) K = i J J +. The implet allowed repreentation are (j +,j )=, 0 or 0,. Wewill ue both. Maachuett Intitute Technology 8.33, April 4 & 9, 008

3 , 8.33 Lecture, April 4 & 9, 008, p.. Mixing (, 0 ) and 0, otethat under a parity tranformation, K K J J, o J = ( J + ik ), J = ( J ik ), implie that + J + J. In addition, recall that the 4-vector rep i, (. Soif χ tranform under, ay, the ), 0 ( rep, ) then µ χ mut ( belong ) to the,, 0 rep, which contain the 0, rep but not the, 0. Thu, µ χ mut interchange thee two rep. Technology Maachuett Intitute 8.33, April & 9, The Lorentz Group and SL(,C) From the, 0 or 0, repreentation, one can ee the relation between the proper orthochronou Lorentz group L + and SL(,C). For the, 0 rep, i J + = σ, J =0 = J = σ, K = σ, where the σ i are the (tracele) Pauli pin matrice. Exponentiation the σ i with imaginary coefficient produce all unitary determinant one matrice, or SU(). Since e πij z =, there are matrice in SU() for every rotation group element, o the rotation group i SU()/Z. The Lorentz group include the exponentiation the σ i matrice with arbitrary complex coefficient, o unitarity i lot. Exponentiation generate SL(,C), thegroup complex matrice with determinant. One till ha e πij z = and thereulting : relationhip, o L + = SL(,C)/Z. (4) Maachuett Intitute Technology 8.33, April 4 & 9, 008 3

4 , 8.33 Lecture, April 4 & 9, 008, p. 3. The Dirac Field Let ψ L (x) be a -component field, ( repreented ) a a column vector, tranforming according to the, 0 rep. Let ψ R (x) be a -component field, ( repreented ) a a column vector, tranforming according to the 0, rep. TheDirac field ψ a (x) i 4-component field, repreented a a 4 column vector, contructed by placing ψ L (x) on top ψ R (x): ψ (x) ψ (x) ψ L (x) ψ a (x) = ψ3 (x) = ψ R (x) ψ 4 (x). (43) Guth Alan Maachuett Intitute Technology 8.33, April 4 & 008 9, 4 The Dirac Matrice Dirac dicovered that one can contruct the, 0 + 0, repreentation the Lorentz group by tarting with four 4 4 Dirac matrice γ µ,choen to atify {γ µ,γ ν } = g µν, (44) wherethecurly bracket denotetheanticommutator, and theright-hand ide i multiplied by an implicit 4 4 identity matrix. Given Eq. (44), the matrice S µν = i [γ µ,γ ν ] (45) 4 automatically havethelorentz group commutation relation, [S µν,s ρσ ]=4i {g ρν S µσ } µν, ρσ where the pair ubcript denote antiymmetrization a defined by Eq. (), 4/3/06. Maachuett Intitute Technology 8.33, April 4 & 9, 008 5

5 , 8.33 Lecture, April 4 & 9, 008, p. 4. Choice Dirac Matrix Convention Following Pekin and Schroeder, we will ue γ 0 0 =, γ i 0 σ = i, (46) 0 σ i 0 where the entrie in γ 0 repreent block zero or the identity matrix, and the σ i repreent the Pauli pin matrice, 0 0 i 0 σ =, σ =, σ 3 =. (47) 0 i 0 0 Thi i called the Weyl or the chiral repreentation. Another popular choice i to diagonalize γ 0. Guth Alan Maachuett Intitute Technology 8.33, April 4 & 008 9, 6 Propertie thi Choice Dirac Matrice With omecalculation, onefind that ( σ k 0 ) i σ k 0 J k = ɛ klm S lm =, K k = S 0k =. 0 σ k 0 σ k (48) Thi give J k = (J k + ik k )= ( σ 0 k ), J k = (J k ik k )= 0 0 k. 0 σ (49) Maachuett Intitute Technology 8.33, April 4 & 9, 008 7

6 , 8.33 Lecture, April 4 & 9, 008, p. 5. One-Particle State and the Poincaré Group A one-particle tate mut remain a one-particle tate under a Poincarétranformation, o the pace one-particle tate form a repreentation the Poincaré group. P P µ P µ i a Caimir operator the Poincarégroup, o a ingleparticleha a uniquevalue P, P = m,where m i called the ma theparticle. Particle eem to alo have a definite pin. I thi a Caimir operator? Anwer: Ye. The operator i W W µ W µ, where W µ = ɛ µνλσ J νλ P σ. (50) W µ i called the Pauli-Lubanki peudovector. In the ret frame P µ oneha [ ] W = m J. Oneknow that W,J µν =0,ince W µ tranform a a vector under Lorentz tranformation. It i alo tranlation-invariant, ince [ ] J νλ,p ρ = i g ρλ P ν g ρν P λ, and theum with ɛ µνλσ will vanih when two indice are contracted with P. Guth Alan Maachuett Intitute Technology 8.33, April 4 & 008 9, 8 One-Particle the Poincaré Group Repreentation Conider firt P > 0, i.e., maive particle. Diagonalize P µ and conider a particle in it ret frame, P µ =(m, 0, 0, 0). The ubgroup the Lorentz group that leave P µ invariant i called the little group. In thi cae, thelittlegroup i therotation group. But weknow about therotation group! Theparticlemut havepin j, an integer or half-integer, with j + pin tate decribed by J z,with = j, j +,...,j. (The eigenvalue J z i ten called m, but m = ma.) [ ] iθnˆ J p =0, U R(ˆn, θ) p =0, = e D R(ˆn, θ). (5) 9

7 , 8.33 Lecture, April 4 & 9, 008, p. 6. State with onzero P By Wigner theorem, we know that we hould expect to contruct a unitary repreentation the Poincare group in the Hilbert pace. (Anti-unitary i excluded here, ince the group i continuou.) We can therefore decribe tate with p 0 by booting the p =0 tate. So we can define p, U(B p ) p =0,, (5) where B p i a boot that tranform the ret vector to the pecified momentum p. B p i not uniquely defined by thi criterion. For Eq. (5) to beunitary, wemut beuing thecovariant normalization: p, p, =E p (π) 3 δ δ (3) ( p p). Guth Alan Maachuett Intitute Technology 8.33, April 4 & 008 9, 0 Canonical v. Helicity Bai Therearetwo tandard way to chooeb p, and therefore to define a bai for the Hilbert pace -particle tate: Canonical: boot in direction p: B (c) = B p, ˆ ξ( p ) = e p iξˆ K p, (53) where ξ( p ) i therapidity aociated with p, tanh ξ = p /E. Helicity: boot in poitive z-direction, then rotate in the z-pˆ plane(along thehorter thetwo option): B (h) p = R(ˆp) B z ξ( p ). (54) Thi proce preerve the helicity, thecomponent thepin in the direction the momentum. Thu, = helicity.

8 , 8.33 Lecture, April 4 & 9, 008, p. 7. Lorentz Tranformation Bai State Tranlation are no problem, ince thee are eigentate momentum. To apply a Lorentz tranformation, ue U(Λ) p, = U(Λ) U(B p ) p =0, = U(B Λ p ) U (B Λ p ) U(Λ) U(B p ) p =0,. ow define the Wigner rotation R W (Λ, p) B Λ p Λ B p. (55) otethat R W map a ret vector to a ret vector, o it i a rotation. But we know about rotation! U(Λ) p, = U(B Λ p ) p =0, ( p = 0, U RW (Λ, p) ) p =0,. Guth Alan Maachuett Intitute Technology 8.33, April 4 & 008 9, U(Λ) p, = U(B Λ p ) p =0, p = 0, ( U RW (Λ, p) ) p =0,. U(Λ) p, = Λ p, ( D RW (Λ, p) ). (56) Maachuett Intitute Technology 8.33, April 4 & 9, 008 3

9 , 8.33 Lecture, April 4 & 9, 008, p. 8. Bai State for -Particle Hilbert Space (Review, Maive Particle) Bai State in Ret Frame: [ ] p =0, U R(ˆn, θ) p =0, = e iθnˆ J D R(ˆn, θ). Bai State in Arbitrary Frame: (5) p, U(B p ) p =0,, (5) where B p i a boot (canonical or helicity) that boot a ret vector to p. Guth Alan Maachuett Intitute Technology 8.33, April 4 & 008 9, 4 Lorentz Tranformation Bai State where i called the Wigner rotation. U(Λ) p, = ( Λ p, D RW (Λ, p) ), (56) R W (Λ, p) B Λ p Λ B p. (55) Recall: B p i the tandard boot, in either the canonical or helicity bai. Maachuett Intitute Technology 8.33, April 4 & 9, 008 5

10 , 8.33 Lecture, April 4 & 9, 008, p. 9. Multiparticle State FreeparticleFock pacebai vector: p,..., p. If two ket ψ and ψ are identical except for the ordering the particle, they repreent the ame phyical tate. For calar particle, ψ = ψ. We will learn (oon!) that for pin- particle, ψ = ± ψ, depending on whether the permutation i even (+) or odd (-). Lorentz Tranformation: each particle tranform independently. U(Λ) p,..., p = Λ p,...,λ p j { } D R W (Λ, p )...D R W (Λ, p ). (57) Maachuett Intitute Technology 8.33, April 4 & 9, Tranformation Creation and Annihilation Operator U(Λ)a ( p ) E p p,..., p = a (Λ p ) E Λ p Λ p,...,λ p o j { } D R W (Λ, p )...D R W (Λ, p ) = a (Λ p ) E Λ p U(Λ) p,..., p D R W (Λ, p ), U(Λ)a ( p ) E p ψ = a (Λ p ) E Λ p D R W (Λ, p ) U(Λ) ψ. Maachuett Intitute Technology 8.33, April 4 & 9, 008 7

11 , 8.33 Lecture, April 4 & 9, 008, p. 0. So, U(Λ)a ( p ) E p ψ = a (Λ p ) ( E Λ p D RW (Λ, p ) ) U(Λ) ψ. U(Λ) a ( p) U (Λ) = E Λ p E p a ( (Λ p) D RW (Λ, p) ). (58) Taking theadjoint: U(Λ) a ( p) U (Λ) = E Λ p E p D ( RW (Λ, p) ) a (Λ p), (59) where I ued the unitarity D : D ( RW (Λ, p) ) = D ( RW (Λ, p) ). Maachuett Intitute Technology 8.33, April 4 & 9, Male Particle What i the littlegroup when P =0, o there i no ret frame? Let p µ =(ω, 0, 0,ω) Little group: generated by J 3, M,and M,where J 3 = J M K J = J 0 + J 3 (60) M K + J = J 0 + J 3. Algebra little group: [M,M ]=0 [ ] J 3,M = im (6) [ ] J 3,M = im. Thelittlegroup i E(), theeuclidean group (rotation and tranlation in R ) Maachuett Intitute Technology 8.33, April 4 & 9, 008 9

12 , 8.33 Lecture, April 4 & 9, 008, p.. Repreentation E() Caimir Operator: M =(M ) +(M ) Rep with M =0. M could then be rotated by any amount, o the rep i infinite dimenional. Thi would involve an infinite number tate with the ame momentum, and doe not correpond to anything known phyically. Finite-dimenional (D) repreentation [ with ] M =0. Whe n M =0, J 3 become a Caimir operator, ince J 3,M i i proportional to M j.so in thi cae M = M =0 J 3 = contant Z/. (ince e 4πiJ3 = in SL(, C)) So, for m =0 thelittlegroup doe not mix different value. The helicity h a male particle i Lorentz-invariant, o thephoton can haveh = ±, but doe not need an h =0 tate. The h = ± tate are related by parity, but not by Lorentz tranformation. If neutrino were male, they could have only h = ( left-handed only). Maachuett Intitute Technology 8.33, April 4 & 9, Dirac Field: Spacetime Dependence Weaumethat ψ a (x) i linear in creation and annihilation operator. Electron are charged. The field hould have a definite charge, to contruct charge-conerving interaction. So let ψ a (x) contain annihilation operator for electron. Then it mut contain creation operator for antiparticle, i.e. poitron. (Thi i like the charged calar field.) Then ψ a (x) will create electron and annihilate poitron. Thefield mut atify: e ip µx µ ψ a (y) e ip µx µ = ψ a (x + y), U (Λ)ψ a (x)u(λ) = Λ,ab ψ b (Λ x), (6) where Λ,ab M ab = e i ω µν S µν, with S µν i = [γ µ,γ ν ]. (63) 4 Maachuett Intitute Technology 8.33, April 4 & 9, 008

13 , 8.33 Lecture, April 4 & 9, 008, p.. ψ a (x) = e ip µx µ ψ a (0) e ip µx µ. But ψ a (0) i linear in electron annihilation operator a ( p) andpoitroncreation operator b ( p), owrite d 3 p { ψ a (0) = a (π) 3 ( p) u a( p)+ b ( p) v a( p) }. E p But e ip µx µ a ( p) e ip µ x µ = e ip µ x µ a ( p), o e ip µx µ b ( p) e ip µ x µ = e ip µ x µ b ( p), ψ a (x) = d 3 p (π) 3 E p { a ( p) u a( p) e ip µ x µ + b ( p) v a( p) e ip µx µ }, where p 0 =+ p + m. Thechoicep 0 > 0 will beimportant. (64) Maachuett Intitute Technology 8.33, April 4 & 9, 008 Lorentz Tranformation ψ (x) a o U(Λ)ψ (x)u (Λ) = Λ ψ b (Λ x) a = { } d 3 p (π) EΛ p D ( 3 E p E RW (Λ, p) ) a (Λ p)u a ( p)e ip x ow let q Λ p, and ue U (Λ)ψ a (x)u(λ) = Λ,ab ψ b (Λ x),,,ab p d 3 p d 3 q (π) 3 = E p (π) 3, o E q U(Λ)ψ a (x)u (Λ) = d 3 q { } D i(λ (π) q) x 3 R W (Λ, p) a ( q )u a ( p)e E q, Maachuett Intitute Technology 8.33, April 4 & 9, 008 3

14 , 8.33 Lecture, April 4 & 9, 008, p. 3. U(Λ)ψ a (x)u (Λ) = d 3 q (π) 3 E q But thi mut equal, Λ ψ b(λ x) =,ab d 3 q (π) 3 E q Thee two expreion match if or equivalently, Λ u,ab { D ( RW (Λ, p) ) } a ( q )u a ( p)e i(λ q) x +... { b (Λ p) = } a ( q )Λ u,ab b ( q )e iq (Λx) +... u a ( p)d ( RW (Λ, p) ), u a (Λ p) =Λ,ab u ( b ( p)d RW (Λ, p) ). (65).. Maachuett Intitute Technology 8.33, April 4 & 9, Rotation in Ret Frame ( p =0): u a (Λ p) =Λ,ab u b ( p)d R W (Λ, p). (65) In theret frame p =0, Λ = R (rotation matrix), o Λ p =0,and B p = B Λ p = I, o R W = B Λ p ΛB p = R. So u a ( p = 0)= Λ,ab (R)u b Look eparately at upper and lower pinor ψ: ( p = 0) D (R). ψ (x) ψ (x) ψ L (x) ψ a (x) = ψ3 (x) = ψ R (x) ψ 4 (x). (43) Guth Alan Maachuett Intitute Technology 8.33, April 4 & 008 9, 5

15 , 8.33 Lecture, April 4 & 9, 008, p. 4. Weknow that for rotation, Λ,ab are jut the matrice generated by J i = σ i 0 0 σ i, and the D (R) matrice are the ame! So the above equation can be rewritten a u L,a = D ab (R) u L,b D (R), or u L D(R) =D(R) u L for every R. Sincethepin- repreentation i irreducible, the only matrice that commutewith all D(R) aremultiple theidentity. Chooe u L,a( p =0)= mδ a. (66) Booting from the ret frame: For p =0,let Λ=B q,o Λ p = q.then So R W (Λ, p) =B Λ p ΛB p = B q B q I = I, u a ( q )=Λ,ab(B q ) u b ( p = 0). (67) Maachuett Intitute Technology 8.33, April 4 & 9, 008 6