Lecture 9/10 (February 19/24, 2014) DIRAC EQUATION(III) i 2. ( x) σ = = Equation 66 is similar to the rotation of two-component Pauli spinor ( ) ( )

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1 P47 For a Lorentz boost along the x-axis, Lecture 9/ (February 9/4, 4) DIRAC EQUATION(III) i ψ ωσ ψ ω exp α ψ ( x) ( x ) exp ( x) (65) where tanh ω β, cosh ω γ, sinh ω βγ β imilarly, for a rotation around the z-axis of an angle ω: i ψ ( x ) exp ωσ ψ( x) σ 3 where σ σ 3 Equation 66 is similar to the rotation of two-component Pauli spinor ( i ) ψ x e ωσ ψ x (66) (67) Are the spinor transformation unitary? For spatial rotation, Rot, is unitary, since but for Lorentz boost ij + i i ij + ωσ ωσ e e (68) Rot Rot ( ω ) i + ω + α αi e e (69) Lor Lor Lor

2 P47 Nevertheless, both Rot and Lor have the property γ γ (7) + o which can be verified by expanding Rot, Lor in power series. o Here is proof for β tanh ω Consider a Lorentz boost along the x-axis I I I ν I 3 I ν ν ω χ lim + I x N N I ( e ) ω ν x ( coshω sinhω ) I + I x ( coshω sinhω) I + I + I x ν coshω sinhω sinhω coshω x x t (cosh ω) t (sinh ω) x x -(sinh ω) t + (cosh ω) x ν

3 P47 3 We also have Therefore t γ (t βx) x γ (-βt + x) γ cosh ω γβ sinh ω β tanh ω The fact that Lor is not unitary (Equation 69) should not be too surprising. This simply reflects the situation that ψ + ψ is not a conserved quantity in a Lorentz boost. We now consider space inversion: ν Λ This discrete transformation cannot be constructed out of infinitesimal transformation. Rather, we rely on the definition of (Equation 53) to find the parity transformation p for the Dirac spinor ψ. Equation 53 and Equation 7 imply This can be satisfied with p p o γ i γ p p o γ i γ (7) p o γ (73) Note that p also satisfies Equation 7 since γ γ o + p p o

4 P47 4 For a Dirac particle at rest, Equation 73 shows that U and U have positive parity while U 3 and U 4 have negative parity (Equation 46)! Having established the properties of, we can show that ψψ transforms like a scalar, while ψγ ψ transforms like a 4-vector under Lorentz transformation. + o o o o ψ x ψ x γ ψ x γ ψ x γ ψ x γ ψ x (74) Therefore ( x ) ( x ) ( x) ( x) ( x) ( x) ψ ψ ψ ψ ψ ψ (75) ψψ is a scalar ψ x γψ x ψ x γ ψ x ν ( x) ν γψ( x) ν ψ( x) γψ( x) ψ Λ Λ ν ψγ ψ is a 4-vector To obtain the plane-wave solution to the Dirac equation for a particle moving along the x-axis with velocity β, we boost along x with a velocity of β. ( i ) ωσ e ; σ iα e ω ω cosh αsinh where tanh ω -β ω α (76) The spinors u ν (p) are ν ω ω u p U ν cosh α sinh

5 P47 5 where and ω tanh ω tanh ω cosh u ν ω tanh ω tanh ω tanhω β tanh + tanh ω + ( β ) β βγ βγ m P + + γ ( + γ) m E + m γ ω m + E m tanh + γ cosh ω (77) (78) using Equation 78, Equation 77 becomes Px E + m E Px ν m + E + m u p u m Px E + m Px E + m For a boost along an arbitrary direction ν (79)

6 P47 6 I ν Px Py Pz p p p Px p Py p Pz p (8) and Pˆ e ω α (8) Pz Px ipy E+ m E+ m Px + ipy Pz ν m + E E+ m E+ m u p u m Pz Px ipy E + m E+ m Px + ipy Pz E+ m E+ m ν (8) The four solutions of the spinors are

7 P47 7 m+ E Pz m+ E Px ipy m E m m E m + + Px + ipy Pz E+ m E+ m Pz Px ipy E+ m E+ m m+ E Px + ipy m+ E Pz m E+ m m E+ m ; u u p p ; u 3 4 u p p (83) Note that Equation 83 is obtained by using a normalization: 3 4 u ; u ; u ; u (84) We can also solve the Dirac equation directly: m σ P Eu ( α P+ βmu ) u (85) σ P m u u A u B, where u A, u B are two-component wave functions u m σ P u E A A u B σ P m u B

8 P47 8 We obtain the coupled equations σ PuB ( E m) ua and (86) σ Pu E + m u A A possible solution to Equation 86 is x x For E >, u N σ P, ( x ) E+ m x, σ P ( ) ( + ) x For E <, u N E m ( ) x Explicitly: B Pz ; Px ipy ; E > E+ m E+ m Px + ipy Pz E+ m E+ m u p N u p N Pz Px ipy E m E m Px + ipy Pz ; ; E< E m E m 3 4 u p N u p N (87)

9 P47 9 Note that u 3 (p), u 4 (p) from Equation 87 are different from those obtained by the boost method (Equation 83). Apart from a difference in the normalization factor N, ( 3,4 ) ( 3,4 ) ( ), where p ( E, p) u p u p boost direct Note that E > for u ( 3,4 ) boost p, while E < for ( 3,4 ) u p. direct In order to interpret the E < solutions for u (3,4)( p), we examine the chargeconjugation transformation on the Dirac equation. The Dirac equation for an electron in an EM field can be obtained with the gauge substitution For an electron with charge q -e (e > ), we have The Dirac equation for an electron becomes D + iqa (89) i i + ea (9) γ ( i + ea ) m ψ (9) The Dirac equation for a positron, ψ c, would be γ ( i ea ) m ψc (9) Our task is to find the transformation linking ψ to ψ c. To change the relative sign between the and A μ terms in Equation 9, one can take a complex conjugate of Equation 9 * ( γ ) ( ) i ea m ψ * + (93) To find a 4 x 4 matrix c, which satisfies

10 P47 c ψ* ψ c (94) We insert c + c term in front of ψ* in Equation 93 and multiply c from the left: * * ( γ ) ( ) ( ψ ) c c i n + ea m c (95) From Equations 94 and 95, it follows that c satisfies the following equation: c * γ c γ (96) Now, the Pauli-Dirac representation of γ μ gives * γ γ for,, 3 * γ γ for (97) Therefore, Equation 96 implies that c commutes with γ and anticommutes with γ, γ, γ 3. This can be satisfied if c is proportional to γ : c iγ (98) The factor i is chosen by convention. To obtain the positron wave function, one takes a complex conjugate of the electron wave function, followed by a multiplication of iγ : ψ c * iγψ (99) We can apply Equation 99 to ψ () (p):

11 P47 ψ ( ) Pz ip x o p N e, P E> E + m Px + ipy E + m γψ Pz E + m Px ipy E + m Px ipy E + m Pz ip x o N e ; P E > E + m * ip x i p N e () () We can also write ψ (4) (p ) as ψ Px ipy E-m Pz p N e where P E E-m < ( 4 ) ip x () ubstituting p -p into Equation, we obtain

12 P47 ψ Px ipy E m Pz p N e ; P E > E m 4 ip x Equation 3 is identical to Equation. ( * ) 4 Hence, ψ iγψ p ψ p imilarly, c * 3 c ψ iγψ p ψ p (3) (4) It also follows 4 3 u p v p u p v p (5) where v (), v () are the spinors for positron wave functions.

13 P47 3 It is instructive to consider time-reversal operation on the Dirac equation. In a procedure analogous to that for the charge-conjugation operation, we can write down the Dirac equation for a time-reversed electron state, ψ t, as 3 iγ + iγ + iγ + iγ mψt t x y z (6) and the ordinary Dirac equation as 3 iγ + iγ + iγ + iγ m ψ t x y z (7) Note that Equation 6 is obtained by t -t transformation applied to the Dirac equation. Taking the complex conjugate of Equation 7 gives * * * 3 * * i( γ ) i( γ ) i( γ ) i( γ ) m ψ t x y z (8) To find a 4 x 4 matrix which transforms ψ to ψ t t ψ* ψ t (9) When we multiply t from the left, we get t in front of ψ* in Equation 8 followed by multiplying t * * * 3 * * t i( γ ) i( γ ) i( γ ) i( γ ) t m tψ t x y z () Equations 9 and imply that t should have the following properties:

14 P47 4 t ( γ ) i ( γ ) * t γ * i γ ( i 3) t t () ince (γ )* γ, (γ )* γ, (γ )* -γ, (γ 3 )* γ 3, Equation implies t commutes with γ, γ and anticommutes with γ, γ 3. t is therefore proportional to γ, γ 3 and, by convention, one chooses t 3 iγγ σ i σ () Applying t ψ*(p) for ψ (), one obtains ( * ) tψ ( p) i N Pz e E + m Px ipy E + m Px ipy ip x in e E + m Pz E + m ip x (3) ( Hence, the time-reversal operation transforms ) ( ) ψ pt, to ψ ( p, t), as one would expect (since time-reversal flips pt, and spin, and ψ and ψ have opposite spin).

15 P47 5 The combined operation of CPT can be obtained too. ince we have Pψ γ ψ Cψ iγ ψ* Tψ iγ γ 3 ψ* ( 3 *) 3 * ( γ iγγψ ) ψcpt CPTψ CP iγγψ C i i 3 γ γ iγγψ 3 γ γγ γψ 5 γψ We can also examine the operation of CPT on solutions for the Klein-Gordon equation + m x ( ) φ In the presence of an EM field: and the K-G equation becomes i i ea or ( i ea )( i ea ) φ( x) m φ( x) φ + m + ie A + A e A x It is straight forward to show that The spin operator is defined as P: φ ( x P ) φ ( x ) C: φ ( x * C ) φ ( x ) * T: φ T ( x ) φ ( x) CPT: φ ( ) φ CPT x x

16 P47 6 σ σ k k σ ij ( i, j, k,, 3 cyclic) (4) k It can be shown that the Dirac Hamiltonian H α P+ βm does not commute with Σ, namely H, Σ i( α P) (5) It does not commute with the orbital angular momentum L either H, L i( α P) (6) However, Equations 5 and 6 show that H commutes with the total angular momentum J L+ Σ H, J H, L + H, Σ (7) Although spin itself is not conserved, the spin projection along the momentum vector is conserved. This spin projection is also called helicity and is defined as σ Pˆ Σ P (8) σ Pˆ One can readily show that H, Pˆ Σ. The reason why Σ Pˆ is conserved can be understood if one recognizes that there is no orbital angular momentum along the direction of momentum. Hence the particle s spin is the same as its total angular momentum in this case, and it is conserved. k 5 k Note that Σ γγγ ( k,, 3) The Dirac spinors given in Equation 87 are not eigenstates of the helicity operator:

17 P47 7 Pz Px ipy ˆ ( ) Px ipy Pz P u ( P) + Σ N Pz P Pz Px ipy E+ m Px + ipy Pz Px + ipy E+ m Pz N Px ipy + P P (9) E+ m Equation 9 shows that in general, U () (P) is not an eigenstate of Σ Pˆ. In the special case when Px Py, U () (P) is an eigenstate of Σ Pˆ with an eigenvalue of +½. We summarize here some of the pertinent properties of the Dirac spinors u and v: First, u and v satisfy the following equations: ( P mu ), ( P+ mv ) u( P m), v( P + m) () The orthogonality relations are + + ( r) ( s) ( r) ( s) u u E δ, v v E δ ( s) ( s) ( s) ( s) u u m, v v m rs rs () Note that different normalizations have been adopted in the literature. For example ( s) ( s) ( s) ( s) u u, v v ()

18 P47 8 have been chosen by many textbooks The completeness relations are s, s, ( s ) ( s ) u Pu P P ( s ) ( s ) v Pv P P + m m (3) Why are these relations called completeness relations? One can show easily that ( s ) ( s ) ( s ) ( s u Pu P v ( Pv ) ) ( P) m (4) s, Using the normalization scheme of Equation, one obtains s, s s s s u Pu P v Pv P Equations 3 are very important for evaluating cross sections later on. Explicitly u E+ m Pz ; u E+ mpx ipy E+ m E+ m Px ipy Pz + E+ m E+ m Px ipy Pz E+ m E + m Pz Px + ipy v E+ m E+ m ; v E+ m E+ m

19 P47 9 In addition to ψψ and ψγ ψ discussed earlier, there are other bilinear covariants. One can show that the following 6 bilinear covariants are independent and complete: ψψ ψγ ψ ψσ ν ψ 5 ψγ ψ 5 ψγ γ ψ scalar 4 vector 6 tensor pseudoscalar 4 axial vector (5) where i, (6) γ γγγγ γ γ + γ 5 anticommutes with all γ μ { } γ 5 commutes with proper Lorentz transformation 5 γ, γ (7) 5 5 γ γ (8) but anticommutes with space inversion P γ γ (9) 5 5 P P The transformation of the 6 bilinear covariants under proper Lorentz transformations and space inversion, as given by Equation 5, can be readily proved using Equations 53, 74, 8 and 9. This concludes our discussion on the DIRAC Equation.