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Lecture 8: Spinors & Dirac Equation

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Lecture 8: Spinors & Dirac Equation QFT Dr Tasos Avgoustidis (Notes based on Dr A. Moss’ lectures) Lecture 8: Spinors & Dirac Equation !1 QFT Lorentz Group (Recap) µ µ µ • Consider transformation ⇤ ⌫ = δ ⌫ + ✏w ⌫ µ ⌧ ⌫ µ⌫ • Using definition of LT ⇤ σ⌘ ⇤ ⌧ = ⌘ • For terms linear in ✏ then wµ⌫ + w⌫µ =0 • For infinitesimal LT the matrix needs to be anti- symmetric. This has 6 degrees of freedom, corresponding to the 6 transformations of the Lorentz group • Introduce basis of 6 anti-symmetric 4x4 matrices ( ⇢)µ⌫ = ⌘⇢µ⌘⌫ ⌘σµ⌘⇢⌫ M − • ⇢ , σ label which matrix, µ, ⌫ the row/column of each matrix !2 Lecture equations Adam Moss (Dated: February 28, 2012) 1 d3xd3kd3q Hˆ = × (1) 8 (2π)6E(k)E(q) ! LectureE equations(k)E(q) −(ˆa†(k)eik·x − aˆ(k)e−ik·x)(ˆa†(q)eiq·x − aˆ(q)e−iq·x) (2) † ik·x −ik·x † iq·x −iq·x (−ik aˆ (k")e + ik aˆ(k)e )(−iq aˆ (q)e + iq aˆ(q)e #) (3) Adam2 Moss† ik·x −ik·x † iq·x −iq·x (Dated: February"m (ˆa 28,(k 2012))e +ˆa(k)e )(ˆa (q)e +ˆa(q)e ) # (4) " # 1 d3k Hˆ = × (5) 1 d3xd3kd3q 3 2 Hˆ × 8 (2π) E(k) = 6 ! (1) 8 (2π) E(k)E(q) k k ! (−E(k)2 + k2 + m2)(ˆa†(k)ˆa†(−k)e2iE( )t +ˆa(k)ˆa(−k)e−2iE( )t) (6) E(k)E(q) −(ˆa†(k)eik·x − aˆ(k)e−ik·x)(ˆa†(q)eiq·x − aˆ(q)e−iq·x) (2) 2 2 2 † † † ik·x −$(ikE·x(k) + k† + miq·x)(ˆa (k)ˆa(k)+ˆ−iq·xa(k)ˆa (k)) % (7) (−ik aˆ (k")e + ik aˆ(k)e )(−iq aˆ (q)e + iq aˆ(q)e #) (3) 2 † ik·x −ik·x † iq·x −iq·x "m (ˆa (k)e +ˆa(k)e ")(ˆa (q)e +ˆa(q)e ) # # (4) " # 1 d3k Hˆ = (ˆa†(k)ˆa(k)+ˆa(k)ˆa†(k)) (8) π 3 1 d3k 4 (2 ) Hˆ = × ! (5) 8 (2π)3E(k)2 ! k k (−E(k)2 + k2 + m2)(ˆa†(k)ˆa†(−k)e2iE( )t +ˆa(k)ˆa(−k)e−2iE( )t) (6) φˆ(x)φˆ(y), if x0 >y0 T φˆ(x)φˆ(y)= (9) $ 2 2 2 † † ˆ ˆ % 0 0 2 (E(k) + k + m )(ˆa (k)ˆa(k)+ˆa(k)ˆa (k)) &φ(y)φ(x), if y >x (7) " # T [φ φ 1φ φ ]=:d3k φ φ φ φ :+ (10) Hˆ 1= 2 3 4 (ˆa1†(k2)ˆa3(k4)+ˆa(k)ˆa†(k)) (8) 0001 4 (2π)3 − − − ! ∆F (x1 x2):φ3φ4 :+∆F (x1 x3):φ2φ4 :+∆F (x1 x4):φ2φ3 :+ (11) ⎛ 0000⎞ ∆F (x2 − x3):φ1φ4 :+∆F (x2 − x4):φ1φ3 :+∆F (x3M−03x4):µ φ1φ2 :+ (12) QFT ( ) ν = (16) − − Lorentz− Group− −⎜ ⎟− ∆F (x1 x2)∆F (x3 x4)+∆F (x1 x3)∆F (x2 x4)+∆F (x1 ⎜x00004)∆F (x2 ⎟x3)(13) φˆ(x)φˆ(y), if x0 >y0 ⎜ ⎟ T φˆ(x)φˆ(y)= (9) ⎜ ⎟ φˆ(y)φˆ(x), if y0 >x0 ⎜ 1000⎟ & ⎜ ⎟ 2 ⎝ ⎠ ⇢0100µ ⇢µ σ σµ ⇢ T [φ1φ2φ3φ4]=:φ1φ2φ3φ4 :+ (10) 00 0 0 • Lower one index ( ⎛ 1000) ⌫ =⎞ ⌘ δ ⌫ ⌘ δ ⌫ ∆ (x − x ):φ φ :+∆ (x − x ):φ φ :+∆ (Mx 01− xµ ):φ φ 0001:+ (11) F 1 2 3 4 F 1 3 2 4 F( 1 ) 4Mν = 2 3 −⎛ 00−10⎞ (14) ⎜ ⎟ M12 µ ∆F (x2 − x3):φ1φ4 :+∆F (x2 − x4):φ1φ3 :+∆F (x3 − x4):φ⎜1⎛φ00002 :+⎟⎞ ( ) (12)ν = (17) ⎜ 0000⎟ ⎜ ⎟ − • −Matrices− are now− (M 03no-longer)µ =⎜− antisymmetric⎟− ⎜ 01 on 0 0 µ,⎟ ⌫ (16) 2 ∆F (x1 x2)∆F (x3 x4)+∆F (x1 x3)∆F (x2 x4)+∆νF (x⎜1⎜0000x4)∆F (x⎟2⎟ x3)(13)⎜ ⎟ ⎜⎜ 0000⎟⎟ ⎜ ⎟ ⎝⎜ ⎠⎟ ⎜ 00 0 0⎟ • Infinitesimal boosts: ⎜ ⎟ ⎜ ⎟ ⎜ 1000⎟ ⎜ ⎟ 2 ⎝ ⎠ ⎝ ⎠ 0100 0010 0001 000 0 ⎛ 1000⎞ ⎛ 000000 0 0⎞ ⎛ 0000⎞ M01 µ M02 µ M03 µ ( ) ν = 0001 ( ) ν = ( ) ν(14)= ⎛ 000−1 ⎞ (15) (16) ⎜ 0000⎟ ⎜⎛ 001000−10⎟⎞ (M13)µ =⎜ 0000⎟ (18) ⎜⎛ ⎞⎟ M12 µ ⎜ ⎟ ν ⎜ ⎟ ⎜ 0000⎟ ( ) ν = ⎜ ⎟ ⎜⎜ 000⎟ 0⎟ (17) (M03)µ = ⎜ ⎟ ⎜ ⎟⎟ (16) ⎜⎜ ⎟ ⎟ ν ⎜⎜ 0000⎟ ⎜ 010000 0 0⎟⎟ ⎜⎜1000⎟ ⎟ ⎜⎜ 0000⎟ ⎜ ⎟⎟ ⎜⎜ ⎟ ⎟ ⎝⎜ ⎟⎠ ⎝⎜ ⎠⎟ ⎜ 010 0⎟ ⎜ ⎟ ⎜ 00 0 0⎟ ⎝⎜ ⎠ ⎟ ⎜ 1000⎟ ⎜ ⎟ ⎝ ⎠ ⎜ ⎟ ⎝ ⎠ • Infinitesimal⎝ ⎠rotations: 0010 00 0 0 000 0 ⎛ 0000⎞ 000 0 M02 µ 00 0 0 ⎛ 00−10⎞ ( ) ν = M12 µ (15) ⎛ ⎞ ⎜ ⎟ ⎛ 000−1 ⎞( ) ν = 000 0 (17) ⎛⎜ 1000− ⎟⎞ 13 µ M23 µ ⎜ ⎟ 12 µ 00 10 M ( ) = 01 0 0 (19) (M ) = ⎜ ⎟ ( ) ν = (17)ν ⎜ ⎜ ⎟ ⎟ (18) ν ⎜⎜ ⎟⎟ ⎜ 000 0⎟ ⎜ ⎜ 000−⎟1 ⎟ ⎜⎜010000 0 0⎟⎟ ⎜ ⎟ ⎜ ⎟ ⎜⎜ ⎟⎟ ⎜ ⎟ ⎜ ⎜00 0 0⎟ ⎟ ⎜⎝ ⎠⎟ ⎜ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ 00 0 0⎟ ⎜ 010 0⎟ ⎜ 001 0⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎜ ⎠ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ !3 000 0 000 0 000 0 ⎛ 000−1 ⎞ ⎛ 000−1 ⎞ ⎛ 000 0⎞ M13 µ (M13)µ = (M23)µ = ( ) (18)ν = (19) (18) ν ⎜ ⎟ ν ⎜ ⎟ ⎜ 000 0⎟ ⎜ 000 0⎟ ⎜ 000−1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 010 0⎟ ⎜ 001 0⎟ ⎜ 010 0⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 000 0 000 0 ⎛ ⎞ 23 µ 000 0 ⎛ ⎞ (M ) = 23 µ(19) 000 0 ν ⎜ − ⎟ (M ) = (19) ⎜ 000 1 ⎟ ν ⎜ ⎟ ⎜ ⎟ ⎜ 000−1 ⎟ ⎜ 001 0⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 001 0⎟ ⎝ ⎠ ⎜ ⎟ ⎝ ⎠ QFT Lorentz Group • Can write any infinitesimal LT in terms of this basis 1 wµ = ⌦ ( ⇢)µ ⌫ 2 ⇢ M ⌫ • Here ⌦ ⇢ are six real numbers specifying the LT 1 • Any finite LT can be written as ⇤µ =exp ⌦ ( ⇢)µ ⌫ 2 ⇢ M ⌫ ✓ ◆ • The six basis matrices obey the Lie algebra [ ⇢, ⌧⌫]=⌘⌧ ⇢⌫ ⌘⇢⌧ ⌫ + ⌘⇢⌫ ⌧ ⌘⌫ ⇢⌧ M M M − M M − M • Here the row/column index is suppressed. This equation encapsulates the properties of the Lorentz group. We are interested in other matrices which satisfy this algebra !4 2 2 0001 0 00001 ( 03)µ = (16) 0001M ⌫ B C B 0000C B C 0 1 B C 03 µ 0000 B 1000C ( ) = B C (16) M ⌫ B C B 0000C @ A B C B C B 1000C B C 00 0 0 @ A 0 00 101 ( 12)µ = − (17) M ⌫ B C 00 0 0 B 01 0 0C B C B C 0 1 B 00 0 0C 12 µ 00 10 B C ( ) = − @ A (17) M ⌫ B C B 01 0 0C B C B C B 00 0 0C 000 0 B C @ A 0 000 1 1 ( 13)µ = − (18) QFT M ⌫ B C Spinor RepresentationB 000 0C 000 0 B C B C B 010 0C 0 000 1 1 B C ( 13)µ = − @ A (18) M ⌫ B C B 000 0C B C • Interested in findingB otherC representations000 0 of the Lorentz B 010 0C group B C 0 000 01 @ ( A23)µ = (19) M ⌫ B C B 000 1 µC ⌫ µ⌫ • The Clifford algebra is definedB as −γC,γ =2⌘ 1 000 0 B C B 001{ 0C } µ B C • γ with µ =0 0 , 000 1 , 2 , 3 0 are1 a set@ of 4 matrices,A so ( 23)µ = (19) M ⌫ B C 0 2 B 000i 2 1 C 01 µ ⌫ ⌫ µ B − C 0 ⌫ = µ (γ ) =1 (Bγ ) = C1 γ = γ γ = γ γ (20) B 001 0−C 0 1 6 B C 10 − • Simplest representation@ A is 4x4@ matricesA 01 0 σi γ0 = γi = (20) (21) 0 101 0 σi 0 1 − @ A @ A • Where σ i are the Pauli matrices 0 σi γi = (21) 0 σi 0 1 !5 − @ A QFT Spinor Representation • There is a “unique” (up to a similarity transformation) irreducible representation of the Clifford algebra. These γ µ matrices define the chiral (or Weyl) rep • Consider the commutator of two γµ 1 1 1 S⇢ = [γ⇢,γσ]= γ⇢γσ ⌘⇢ 4 2 − 2 • Can show these form a representation of the Lorentz group such that [S⇢,S⌧⌫]=⌘⌧ S⇢⌫ ⌘⇢⌧ S⌫ + ⌘⇢⌫ S⌧ ⌘⌫S⇢⌧ − − !6 QFT Dirac Spinor • Introduce a Dirac spinor, a complex valued object with 4 components which transforms as ↵ ↵ β 1 (x) S[⇤] (⇤− x) ! β • Here ↵ =1 , 2 , 3 , 4 labels the row/column of the S µ ⌫ matrices and 1 ⇢ 1 ⇢ ⇤=exp ⌦ S[⇤] = exp ⌦⇢S 2 ⇢M 2 ✓ ◆ ✓ ◆ • Particular LT specified by ⌦ ⇢ - these are the same for both ⇤ and S[⇤] • Lets look at S [ ⇤ ] in the chiral representation !7 2 2 0001 0001 2 0 00001 0 1 03 µ 03 µ 0000 ( ) ⌫ = ( ) = (16) (16) M B CM ⌫ B C B 0000C B 0000C B C B C B C B C B 1000C B 1000C B C 0001B C @ A @ A 0 00001 ( 03)µ = (16) M ⌫ B C 00 0 0 B 000000 0C 0 B C 0 1 B 01000C 1 00 10 12 µ B 00 C10 ( 12)µ = − ( ) B= −C (17) (17) ⌫ M ⌫ @ B A C M B 01 0 0C B 01 0 0C B C B C B C B C B C 00B 00 0 0 0 0C B 00 0 0C B C B C @ A @ A 0 00 101 ( 12)µ = − (17) M ⌫ B C B 01000 0 0C 0 000 0 B C B 000 0 0C 1 0 1 B 000C1 000 (1 13)µ B= −C (18) ( 13)µ = − M ⌫@ B A C (18) M ⌫ B C B 000 0C B 000 0C B C B C B C B C 000B 010 0 0C B 010 0C B C B C 0 000@ 1 1 A @ ( 13A)µ = − (18) M ⌫ B C B 000 0C B 000C 0 000 0 B C B 0100 0C 1 B 000C 0 0 000 0( 123)µ @= A (19) 23 µ M ⌫ B C ( ) ⌫ = B 000 1 C (19) M B C B − C B 000 1 C B C B − C 000B 001 0 0C B C B C B 001 0C 0 000@ 01 A B ( 23C)µ = (19) @ M A⌫ B C B 000 1 C B 01− C γ0B= C (20) 01 B 001 0C 0 B 0 101 C γ = @ A (20) 0 101 @ A @ A 0 01 i γ i= 0 σ (20) i γ =0 101 (21) i i 0 σ 0 σ 0 1 QFT γ = @ − A (21) Dirac Spinor0 σi 0 1 @ A − @ A i i 0 σ γ1 = i σk 0 (21) Sij = [γi,0γj]=σi 0 ✏1ijk (22) 1 i σk4 0 − −2 0 0 σk 1 • For rotations Sij = [γi, γj]= ✏ijk @ A (22) 4 −2 0 0 σk 1 @ A k ij @1 i j A i ijk σ 0 S = [γ , γ ]= ei'✏σ/2 0 (22) k 4 −2 · 0 k 1 • Writing rotation as ⌦ ij = ✏ ijk ' S[⇤]= 0 σ (23) 0 i' σ/2 1 − 0 @e · A @ A 3 ei⇡ 0 • For a rotation of ' =(0, 0, 2⇡) S[⇤]= 3 = 1 (23) 0 0 ei⇡ 1 − @ A • This means that under 2 ⇡ rotations ↵ ( x ) ↵ ( x ) which is not what happens to a vector - different! rep • For rotations in the chiral representation S [ ⇤ ] is unitary, i.e.
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