11 Spinor solutions and CPT

Transcription

1 11 Spinor solutions and CPT 184 In the previous chapter, we cataloged the irreducible representations of the Lorentz group O(1, 3. We found that in addition to the obvious tensor representations, φ, A µ,h µν etc., there are a whole set of spinor representations, such as Weyl spinors ψ L,ψ R.ADirac spinor ψ transforms in the reducible ( 1 2, (, 2 1 representation. We also found Lorentzinvariant Lagrangians for spinor fields, ψ(x. Thenextsteptowardsquantizingatheory with spinors is to use these Lorentz group representations to generate irreducible unitary representations of the Poincaré group. We discussed how unitary representations of the Poincarégroupareinducedfromrep- resentations of its little group. The little group is the group that leaves a given momentum 4-vector p µ invariant. When p µ is massive, the little group is SO(3; whenp µ is massless, the little group is ISO(2. Asaconsequence, massiveparticlesofspinj should have 2j +1degrees of freedom and massless particles of any spin > have two degrees of freedom. In the spin-1 case, we found that there were ambiguities in what the free Lagrangian was (it could have been aa µ A µ + ba µ µ ν A ν for any a and b, but we found that there was a unique Lagrangian that propagated the correct degrees of freedom. We then solved the free equations of motion for a fixed momentum p µ generating two or three polarizations ϵ i µ(p. Thesesolutions,whichwererepresentationsofthelittle group, if known for every value of p µ,inducerepresentationsofthefullpoincaré group. For the spin- 1 2 case, there is a unique free Lagrangian (up to Majorana masses that automatically propagates the right degrees of freedom. In this sense, spin 1 2 is easier than spin 1, since there are no unphysical degrees of freedom. The mass term couples left- and right-handed spinors, so it is natural to use the Dirac representation. As in the spin-1 case, we will solve the free equations of motion to find basis spinors, u s (p and v s (p (analogs of ϵ i µ, which we will use to define our quantum fields. As with complex scalars, we will naturally find both particles and antiparticles in the spectrum with the same mass and opposite charge: these properties fall out of the unique Lagrangian we can write down. Aspinorcanalsobeitsownantiparticle,inwhichcasewecallitaMajoranaspinor. As we saw, since particles and antiparticles have opposite charges, Majorana spinors must be neutral. We will define the operation of charge conjugation C as taking particles to antiparticles, so Majorana spinors are invariant under C. AfterintroducingC, itisnatural to continue to discuss how the discrete symmetries parity, P,andtimereversal,T,acton spinors.

2 11.1 Chirality, helicity and spin Chirality, helicity and spin In a relativistic theory, spin can be a confusing subject. There are actually three concepts associated with spin: spin, helicity and chirality. In this section we define and distinguish these different quantities. Recall from Eq. (1.15 thatthediracequation(i /D mψ =implies ( (i µ ea µ 2 e 2 F µνσ µν m 2 ψ =, (11.1 and for the conjugate field ψ = ψ γ, ( ψ (i µ + ea µ 2 + e 2 F µνσ µν m 2 =. (11.2 Thus, ψ is a particle with mass m and charge opposite to ψ; thatis, ψ is the antiparticle of ψ. We will often call ψ an electron and ψ apositron,althoughtherearemanyother particle antiparticle pairs described by the Dirac equation besides these. When we constructed the Dirac representation, we saw that it was the direct sum of two irreducible representations of the Lorentz group: ( 1 2, (, 2 1.Nowweseethatit describes two physically distinguishable particles: the electron and the positron. Irreducible unitary spin- 1 2 representations of the Poincaré group,weylspinors,havetwodegreesof freedom. Dirac spinors have four. These are two spin states for the electron and two spin states for the positron. For charged spinors, there is no other way. Uncharged spinors can be their own antiparticles if they are Majorana spinors, as discussed in Section 11.3 below. To understand the degrees of freedom within a four-component Dirac spinor, first recall that in the Weyl basis the γ-matrices have the form γ µ = ( σµ σ µ, (11.3 and the Lorentz generators S µν = i 4 [γµ,γ ν ] are block diagonal. Under an infinitesimal Lorentz transformation, ψ ψ + 1 ( (iθi β i σ i ψ. ( (iθ i + β i σ i In this basis, a Dirac spinor is a doublet of a left- and a right-handed Weyl spinor: ( ψl ψ =. (11.5 Here left-handed and right-handed refer to the ( 1 2, or (, 1 2 representations of the Lorentz group. The handedness of a spinor is also known as its chirality. It is helpful to be able to project out the left- or right-handed Weyl spinors from a Dirac spinor. We can do that with the γ 5 -matrix: ψ R γ 5 iγ γ 1 γ 2 γ 3. (11.6

3 186 Spinor solutions and CPT In the Weyl representation ( 1 γ 5 =, ( so left- and right-handed spinors are eigenstates of γ 5 with eigenvalues 1. Wecanalso define projection operators, ( P R = 1+γ5 =, P L = 1 ( γ5 1 =, ( which satisfy PR 2 = P R and PL 2 = P L and ( ( ( ψl ψl P R =, P L ψ R ψ R ψ R = ( ψl. (11.9 Writing projectors as 1±γ5 2 is basis independent. It is easy to check that ( γ 5 2 { = 1 and γ 5,γ µ} =.Thusγ 5 is like another γ- matrix, which is why we call it γ 5. This lets us formally extend the Clifford algebra to five generators, γ M = γ,γ 1,γ 2,γ 3,iγ 5 so that { γ M,γ } N = 2g MN with g MN = diag(1, 1, 1, 1, 1. Ifwewerelookingatrepresentationsofthefive-dimensional Lorentz group, we would use this extended Clifford algebra. See [Polchinski, 1998] fora discussion of spinors in various dimensions. To understand the degrees of freedom in the spinor, let us focus on the free theory. In the Weyl basis, the Dirac equation is ( ( m iσ µ µ ψl i σ µ =. (11.1 µ m ψ R In Fourier space, this implies σ µ p µ ψ R =(E σ pψ R = mψ L, (11.11 σ µ p µ ψ L =(E + σ pψ L = mψ R. (11.12 So the mass mixes the left- and right-handed states. In the absence of a mass, left- and right-handed states are eigenstates of the operator ĥ = σ p p with opposite eigenvalue, since E = p for massless particles. This operator projects the spin on the momentum direction. Spin projected on the direction of motion is called the helicity,sotheleft-andright-handedstateshaveoppositehelicityinthemassless theory. When there is a mass, the left- and right-handed fields mix due to the equations of motion. However, since momentum and spin are good quantum numbers in the free theory, even with a mass, helicity is conserved as well. Therefore, helicity can still be a useful concept for the massive theory. The distinction is that, when there is a mass, helicity eigenstates are no longer the same as the chirality eigenstates ψ L and ψ R. By the way, the independent solutions to the free equations of motion for massless particles of any spin are the helicity eigenstates. For any spin, we will always find S pψ s = ±s p Ψ s,wheres = J are the rotation generators in the Lorentz group for

4 11.1 Chirality, helicity and spin 187 spin s. Forspin 1 2, S = 1 2 σ. Forspin1,therotationgeneratorsaregiveninEqs.(1.14. For example, J 3 has eigenvalues ±1 with eigenstates (,i,1, and (, i, 1,. These are the states of circularly polarized light in the z direction, which are helicity eigenstates. In general, the polarizations of massless particles with spin > can always be taken to be helicity eigenstates. This is true for spin 1 2 and spin 1, as we have seen; it is also true for gravitons (spin 2, Rarita Schwinger fields (spin 3 2 andspinss>2(although, as we proved in Section 9.5.1,itisimpossibletohaveinteractingtheorieswithmasslessfieldsof spin s>2. We have seen that the left- and right-handed chirality states ψ L and ψ R do not mix under Lorentz transformations they transform in separate irreducible representations. each have two components on which the σ-matrices act. These are the two spin states of the electron; both left- and right-handed spinors have two spin states. are eigenstates of helicity in the massless limit. We have now seen three different spin-related quantities: Spin is a vector quantity. We say spin up, or spin down, spin left, etc. It is the eigenvalue of S = σ 2 for a fermion. If there is no angular momentum, for example for a single particle, the spin and the rotation operators are identical S = J.Wealsotalkaboutspins as ascalar,whichistheeigenvalues(s +1of the operator S 2. When we say spin 1 2 we mean s = 1 2. Helicity refers to the projection of spin on the direction of motion. Helicity eigenstates S p satisfy p Ψ=±Ψ. Helicityeigenstatesexistforanyspin.Thehelicityeigenstatesofthe photon correspond to what we normally call circularly polarized light. Chirality is a concept that only exists for spinors, or more precisely for (A, B representations of the Lorentz group with A B. Youmayrememberthewordchiral from chemistry: DNA is chiral, so is glucose and many organic molecules. These are not symmetric under reflection in a mirror. In field theory, a chiral theory is one that is not symmetric on interchange of the (A, B representations with the (B, A representations. Almost always, chirality means that a theory is not symmetric between left-handed Weyl spinors ψ L and right-handed spinors ψ R.ThesechiralspinorscanalsobewrittenasDirac spinors that are eigenstates of γ 5. By abuse of notation we also write ψ L and ψ R for Dirac spinors, with γ 5 ψ L = ψ L and γ 5 ψ R = ψ R. Whether a Weyl or Dirac spinor is meant by ψ L and ψ R will be clear from context. Chirality works for higher half-integer spins too. For example, a spin- 3 2 field can be put in a Dirac spinor with a µ index, ψ µ.thenγ 5 ψ µ = ±ψ µ are the chirality eigenstates. Whether spin, helicity or chirality is important depends on the physical question you are interested in. For free massless spinors, the spin eigenstates are also helicity eigenstates and chirality eigenstates. In other words, the Hamiltonian for the massless Dirac equation S p commutes with the operators for chirality, γ 5, helicity, E,andthespinoperators, S.The QED interaction ψ /Aψ = ψ L /Aψ L + ψ R /Aψ R is non-chiral, that is, it preserves chirality. Helicity, on the other hand, is not necessarily preserved by QED: if a left-handed spinor has its direction reversed by an electric field, its helicity flips. When particles are massless

5 188 Spinor solutions and CPT (or ultra-relativistic they do not change direction so easily, but the helicity can flip due to an interaction. In the massive case, it is also possible to take the non-relativistic limit. Then it is often better to talk about spin, the vector. Projecting on the direction of motion does not make so much sense when the particle is nearly at rest, or in a gas, say, when its direction of motion is constantly changing. The QED interactions do not preserve spin, however; only a strong magnetic field can flip an electron s spin. So, as long as magnetic fields are weak, spin is a good quantum number. That is why spin is used in quantum mechanics. In QED, we hardly ever talk about chirality. The word is basically reserved for chiral theories, which are theories that are not symmetric under L R, suchasthetheoryof the weak interactions. We talk very often about helicity. In the high-energy limit, helicity is often used interchangeably with chirality. As a slight abuse of terminology, we say ψ L and ψ R are helicity eigenstates. In the non-relativistic limit, we use helicity for photons and spin (the vector for spinors. Helicity eigenstates for photons are circularly polarized light Solving the Dirac equation Now let us solve the free Dirac equation. Since spinors satisfy the Klein Gordon equation, ( + m 2 ψ =(in addition to the Dirac equation they have plane-wave solutions: ψ s (x = d 3 p (2π 3 u s(pe ipx, (11.13 with p = p 2 + m 2 >. ThesearelikethesolutionsA µ (x = d 4 p (2π ϵ 4 µ (pe ipx for spin-1 plane waves. There are of course also solutions to ( + m 2 ψ =with p <.We will give these antiparticle interpretations, as in the complex scalar case (Chapter 9, and write d 3 p χ s (x = (2π 3 v s(pe ipx, (11.14 also with p = p 2 + m 2 >. Theseareclassicalsolutions,butthequantumversions will annihilate particles and create the appropriate positive-energy antiparticles. The spinors u s (p and v s (p are the polarizations for particles and antiparticles, respectively. They transform under the Poincaré groupthroughthetransformationofp µ and the little group that stabilizes p µ.thus,weonlyneedtofindexplicitsolutionsforfixedp µ,aswe did for the spin-1 polarizations. To find the spinor solutions, we use the Dirac equation in the Weyl basis: ( m p σ p σ µ m u s (p = ( m p σ p σ m v s (p =. (11.15

6 11.2 Solving the Dirac equation 189 In the rest frame, p µ =(m,,, and the equations of motion reduce to ( 1 1 u s = 1 1 ( v s =. (11.16 So, solutions are constants: u s = ( ξs ξ s ( ηs, v s =, (11.17 η s for any two-component spinors ξ s and η s.forexample,fourlinearlyindependentsolutions are 1 1 u = 1, u = 1, v = 1, v = 1. ( The Dirac spinor is a complex four-component object, with eight real degrees of freedom. The equations of motion reduce it to four degrees of freedom, which, as we will see, can be interpreted as spin up and spin down for particle and antiparticle. To derive a more general expression, we can solve the equations again in the boosted frame and match the normalization. If p µ =(E,,,p z then ( E pz p σ = E + p z ( E + pz, p σ =. (11.19 E p z Let a = E p z and b = E + p z,thenm 2 =(E p z (E + p z =a 2 b 2 and Eq. (11.15 becomes ab a 2 ab b 2 b 2 ab u s(p =. (11.2 a 2 ab The solutions are ( a u s = b ( b a ξ s ξ s (11.21 for any two-component spinor ξ s.notethatintherestframep z =, a 2 = b 2 = m, and these solutions reduce to Eq. (11.17 above.thesolutionsinthep z frame are ( E pz ξ s u s (p = E + pz ( E + pz. (11.22 ξ s E pz

7 19 Spinor solutions and CPT Similarly, ( E pz η s v s (p = E + pz ( E + pz E p z η s. (11.23 Using ( E pz p σ =, E + pz ( E + pz p σ = E pz (11.24 we can write more generally u s (p = ( ( p σξs p σηs, v s (p = p σξs, (11.25 p ση s where the square root of a matrix can be defined by changing to the diagonal basis, taking the square root of the eigenvalues, then changing back to the original basis. In practice, we will usually pick p µ along the z axis, so we do not need to know how to make sense of p σ.thenthefoursolutionsare u 1 p = E pz E + pz, u 2 E pz p =, v 1 p = E + pz E pz E + p z, vp 2 = E pz E + p z. (11.26 In any frame u s are the positive frequency electrons, and the v s are negative frequency electrons, or equivalently, positive frequency positrons. For massless spinors, p z = ±E and the explicit solutions in Eq. (11.26 are 4-vectors with one non-zero component describing spinors with fixed helicity. The spinor solutions for massless electrons are sometimes called polarizations, and are useful for computing polarized electron scattering amplitudes. For Weyl spinors, there are only four real degrees of freedom off-shell and two real degrees of freedom on-shell. Recalling that the Dirac equation splits up into separate equations for ψ L and ψ R,theDiracspinorswithzerosinthebottomtworowswillbeψ L and those with zeros in the top two rows will be ψ R.Sinceψ L and ψ R have two degrees of freedom each, these must be particle and antiparticle for the same helicity. The embedding of Weyl spinors into fields this way induces irreducible unitary representations of the Poincaré groupform = Normalization and spin sums To figure out what the normalization is that we have implicitly chosen, let us compute the inner product:

8 11.2 Solving the Dirac equation 191 ( p ū s (pu s (p =u σξs s(pγ u s (p = = ( ξs ξ s p σξs ( 1 1 ( p σξs p σξs ( ( (p σ(p σ ξs (p σ(p σ ξ s =2mδ ss. (11.27 Similarly, v s (pv s (p = 2mδ ss.thisisthe(conventionalnormalizationforthespinor inner product for massive Dirac spinors. It is also easy to check that v s (pu s (p = ū s (pv s (p =. We can also calculate u s(pu s (p = ( ( p σξs p σξs =2Eξ p σξs p σξs sξ s =2Eδ ss, (11.28 and similarly, v s(pv s (p =2Eδ ss.thisistheconventionalnormalizationformassless Dirac spinors. Another useful relation is that, if we define p µ =(E p, p as a momentum backwards to p µ,thenv s(pu s ( p =u s(pv s ( p =. We can also compute the spinor outer product: 2 u s (pū s (p =/p + m, (11.29 s=1 where the sum is over the spins. Both sides of this equation are matrices. It may help to think of this equation as s s s. Fortheantiparticles, 2 v s (p v s (p =/p m. (11.3 s=1 You should verify these relations on your own (see Problem To keep straight the inner and outer products, it may be helpful to compare to spin-1 particles. We have found s s : s s : s ϵ i µ pϵ j µ(p = δ ij ū s (pu s (p =2mδ ss, ( ϵ µ i (p ϵ ν i (p = g µν + pµ p ν 2 m 2 u s (pū s (p =/p + m. i=1 s=1 (11.32 So, when we sum over internal spin indices, we use an inner product and get a number. When we sum over polarizations/spins, we get a matrix.