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Home , Charge (physics), Dirac spinor, Majorana fermion, Spinor, Transpose

2012 Matthew Schwartz II-4: solutions and CPT

1 Introduction

In the previous lecture, we characterized the irreducible representations of the Lorentz O(1,3). We found that in addition to the obvious representations φ, A µ , h µν etc., there are a whole set of spinor representations, such as Weyl ψL , ψR. A ψ trans- forms in the reducible 1 , 0 0, 1 representation. We also found Lorentz invariant 2 ⊕ 2 Lagrangians for spinor fields , ψ( ) . The next step towards quantizing a theory with spinors is to   use these representations to generate irreducible unitary representations of the Poincaré group. Recall that unitary representations of the Poincaré group are induced from representations of its little group which stabilizes pµ, SO(3) or ISO(2), for the massive or massless cases respec- tively. For massive , we expect 2 J + 1 degrees of freedom and for massless particles, 2 degrees of freedom. In the vector case, we found that there were ambiguities in what the free Lagrangian was (it could have been aA µ A µ + bA µ∂µ∂νAν for any a and b), but we found that there was a unique Lagrangian which propagated the correct degrees of freedom. We then solved i the free equations of motion for a fixed momentum pµ generating 2 or 3 polarizations ǫ µ( p) . These solutions, which were representations of the little group, if known for every value of pµ, induce representations of the full Poincaré group. For the 1 case, there is a unique free Lagrangian (up to Majorana ) which auto- 2 matically propagates the right degrees of freedom. In this sense, spin 1 is easier than spin 1 , 2 since there are no unphysical degrees of freedom. The 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 i free equations of motion to find spinors, us ( p) and vs ( p) (analogs of ǫ µ) which we will use to define our quantum fields. As with complex scalars, we will naturally find both particles and in the spectrum with the same mass and opposite : these properties fall out of the unique Lagrangian we can write down. A spinor can also be its own , in which case we call it a Majorana spinor. As we saw, since particles and anti-particles have opposite charges Majorana spinors must be neutral. We will define the of charge conjugation C as taking particles to antiparticles, so Majorana spinors are invariant under C. After introducing C, it is natural to continue to discuss how the discrete symmetries , P, and time-reversal, T, act on spinors.

2 , helicity and spin

In a relativistic theory, spin can be a confusing subject. There are actually three concepts asso- ciated with spin: spin, helicity and chirality. In this section we define and distinguish these dif- ferent quantities. Recall from the previous lecture that the ( iD m) ψ = 0 implies − 2 e µν 2 ( i∂µ eA µ) Fµν σ m ψ = 0 (1) − − 2 −   and for the conjugate field ψ¯ = ψ † γ0

2 e µν 2 ψ¯ ( i∂ µ + eA µ) + Fµν σ m = 0 (2) 2 −   Thus ψ¯ , is a with mass m and charge opposite to ψ; that is, ψ¯ is the antiparticle of ψ. We will often call ψ an and ψ¯ a , although there are many other particle antiparticle pairs described by the Dirac equation besides these.

1 2 Section 2

When we constructed the Dirac representation, we saw that it was the direct sum of two irreducible representations of the Lorentz group: 1 , 0 0, 1 . Now we see that it describes 2 ⊕ 2 two physically distinguishable particles: the electron and the positron. Irreducible unitary spin 1   2 representations of the Poincaré group, Weyl spinors, have 2 degrees of freedom. Dirac spinors have 4. These are 2 spin states for the electron and 2 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 4 below. To understand the degrees of freedom within a 4-component Dirac spinor, first recall that in the Weyl basis, the γ matrices have the form 0 σ γ = µ (3) µ σ¯ 0  µ  and the Lorentz generators S µν = i [ γµ , γν ] are block diagonal. Under an infinitesimal Lorentz 4 transformation 1 ( iθi βi) σi ψ ψ + − ψ (4) → 2 ( iθi + βi) σi   In this basis, a Dirac spinor is a doublet of a left and a right-handed Weyl spinor

ψ ψ = L (5) ψ  R  1 , , 1 Here left-handed and right-handed refer to the 2 0 or 0 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 γ5 γ5 = iγ0 γ1 γ2 γ3 (6) In the Weyl representation 5 1 γ = − 1 , (7)   so left- and right-handed spinors are eigenstates of γ5 with eigenvalues 1 . We can also define projection operators ±

1 + γ5 0 1 γ5 1 PR = = ,PL = − = (8) 2 1 2 0     2 2 which satisfy PR = PR and PL = PL and ψ 0 ψ ψ P L = ,P L = L (9) R ψ ψ L ψ 0  R   R   R    1 γ 5 Writing projectors as ± is useful because it is basis-independent. 2 It is easy to check that that ( γ5 ) 2 = 1 and γ5 , γµ = 0. Thus γ5 is like another gamma { } matrix, which is why we call it γ5 . This lets us formally extend the Clifford algebra to 5 genera- tors, γM = γ0,γ1 ,γ2 ,γ3,iγ5 so that γM ,γN = 2 gMN with gMN = diag(1 , 1 , 1 , 1 , 1) . If we were looking at representations{ of the 5-dimensional} Lorentz group, we− would− − use− this extended Clifford algebra. See [Polchinski] for a discussion of spinors in various . To understand the degrees of freedom in the spinor, let us focus on the free theory. In the Weyl basis, the Dirac equation is

µ m iσ ∂µ ψL −µ = 0 (10) i σ¯ ∂µ m ψR  −   In Fourier space, this implies

µ Q σ pµ ψR = ( E Qσ p ) ψR = mψL (11) − ·

µ Q σ¯ pµ ψL = ( E + Qσ p ) ψL = mψR (12) · Chirality, helicity and spin 3

So the mass mixes the left and right handed states. Q Qσ p In the absence of a mass, left and right handed states are eigenstates of the operator hˆ = ·

Qp | | with opposite eigenvalue, since E = Qp for massless particles. This operator projects the spin on the momentum direction. Spin projected| | on the direction of motion is called the helicity, so the left and right handed states have opposite helicity in the massless 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 mas- sive theory. The distinction is that when there is a mass, helicity eigenstates are no longer the same as the chirality eigenstates ψL and ψR.

By the way, the independent solutions to the free equations of motion for massless particles

Q Q of any spin are the helicity eigenstates. For any spin, we will always find S Qp Ψ s = s p Ψ s ,

· 1 ± | | 1

Q Q Q where S = J are the generators in the Lorentz group for spin s. For spin , S = σQ . 2 2 For spin 1, the rotation generators are given in Eqs. (15??) of the previous Lecture. For example, J3 has eigenvalues 1 with eigenstates (0, i, 1 , 0) and (0, i, 1 , 0) . 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 > 0 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 (spin 2), Rarita-Schwinger fields (spin 3/2) and spins s > 2 (although, as we proved in Lecture II-2, it is impossible to have interacting theories with massless fields of spin s > 2). We have seen that the left- and right-handed chirality states ψL and ψR do not mix under Lorentz transformations – they transform in separate irreducible repre- • sentations.

each have two components on which the σQ matrices act. These are the two spin states of • the electron; both left- and right-handed spinors have 2 spin states. are eigenstates of helicity in the massless limit. • We have now seen 3 different spin-related quantities: Spin is a vector quantity. We say spin up, or spin down, spin left, etc. It is the eigenvalue of

Qσ SQ = for a . If there is no , for example for a single particle, the spin

2 Q and the rotation operators are identical SQ = J . We also talk about spin s, as a , which is 2 1 1 s s SQ s the eigenvalue ( + 1) of the operator . When we say spin 2 we mean = 2 .

Helicity refers to the projection of spin on the direction of motion. Helicity eigenstates sat- Q

S Qp isfy · Ψ = Ψ. Helicity eigenstates exist for any spin. For spin 1, circularly polarized light are

Qp | | ± the helicity eigenstates. Chirality is a concept that only exists for spinors, or more precisely for ( A, B) representa-

tions of the Lorentz group with A  B. You may remember the word chiral from chemistry: DNA is chiral, so is glucose and many organic . These are not symmetric under reflec- tion 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 ψL and right-handed spinors ψR. These chiral spinors can also be written as Dirac spinors which are eigenstates of γ5 . By we also write ψL and ψR for Dirac spinors, with γ ψL = ψL and γ ψR = ψR. 5 − 5 Whether a Weyl or Dirac spinor is meant by ψL and ψR will be clear from context. Chirality 3 works for higher half-integer spins too. For example, a spin 2 field can be put in a Dirac spinor with a µ index, ψµ. Then γ5 ψµ = ψµ 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 chi-

rality eigenstates. In other words, the Hamiltonian for the massless Dirac equation commutes Q

S Qp γ · SQ with the operators for chirality, 5 , helicity, E , and the spin operators, . The QED interac- ¯ ¯ ¯ tion ψ Aψ = ψL AψL + ψRAψR 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 (or ultrarelativistic) they don’t change direction so easily, but the helicity can flip due to an interaction. 4 Section 3

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 mag- netic field can flip an electron’s spin. So as long as magnetic fields are weak, spin is a good . That’s why spin is used in . In QED, we hardly ever talk about chirality. The word is basically reserved for chiral theo- ries, which are theories that are not symmetric under L R, such as the theory of 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 ψL and ψR are helicity eigenstates. In the non-relativistic limit, we use helicity for and spin (the vector) for spinors. Helicity eigenstates for photons are circularly polarized light.

3 Solving the Dirac equation

Now let’s solve the free Dirac equation. We expect four solutions: two for particles and two for antiparticles, each with two spins. Since spinors satisfy the Klein-Gordon equation (  + m2 ) ψ = 0 in addition to the Dirac equation, they have plane wave solutions d3 p ψ ( x) = u ( p) e ipx (13) s (2 π) 3 s − Z 4 2 2 d p ipx with p = m and p > 0 for any spin s. These are like the solutions A ( x) = 4 ǫ ( p) e 0 µ (2 π) µ  2 0 for spin-1 plane waves. There are of course also solutions to ( + m ) ψ = 0 withR p < 0. We will give these antiparticle interpretations, as in the complex scalar case (Lecture II-2), and write d3 p χ ( x) = v ( p) eipx (14) s (2 π) 3 s Z 2 2 with p0 Qp + m > 0. This is a classical solution, but the quantum version will annihilate particles≡ and create the appropriate positive-energy antiparticle. The spinors u ( p) and v¯ ( p) are p s s the polarizations for particles and anti-particles, respectively. These transform under the Poincaré group through the transformation of pµ and the little group which stabilizes pµ. Thus we only need to find explicit solutions for fixed pµ, as we did for the spin 1 polarizations. To find the spinor solutions, we use the Dirac equation in the Weyl basis: m p σ m p σ u ( p) = v ( p) = 0 (15) p− σ¯ µ ·m s −p σ¯ − m· s  · −   − · −  In the rest frame, pµ = ( m, 0, 0, 0) and the equations of motion reduce to

1 1 1 1 u = v = 0 (16) −1 1 s − 1 − 1 s So solutions are constants  −   − −  ξs ηs us = , vs = (17) ξs ηs    −  For any two-component spinors ξs and ηs . For example, 4 linearly independent solutions are 1 0 1 0 0 1 −0 1 u =  , u =  , v =  , v =  (18) ↑ 1 ↓ 0 ↑ 1 ↓ 0  0   1   0   1         −          The Dirac spinor is a complex 4-component object, with 8 real degrees of freedom. The equa- tions of motion reduce it to 4 degrees of freedom which, as we will see, can be interpreted as spin up and spin down for particle and anti-particle. Solving the Dirac equation 5

Now let’s boost in the z-direction. Peskin and Schroeder do the actual boost, but we’ll just µ solve the equations again in the boosted frame and match the normalization. If p = ( E, 0, 0,pz) then E pz 0 E + pz 0 p σ = − , p σ¯ = (19) · 0 E + pz · 0 E pz    −  2 2 2 Let a = √E pz and b = √E + pz , then m = ( E pz)( E + pz) =a b and Eq. (15) becomes − − ab 0 a2 0 − 2  0 ab 0 b  − us ( p) = 0 (20) b2 0 ab 0  2 −   0 a 0 ab   −  The solutions are   a 0 ξ 0 b s us =    (21) b 0 ξ 0 a s       2 2 for any two-component spinor ξs . Note that in the rest frame pz = 0, a = b = m, and these solu- tions reduce to Eq. (17) above. The solutions in the pz frame are

√E pz 0 − ξs 0 √E + pz ! us ( p) =  (22) √E + pz 0 ξs  0 √E pz !   −  Similarly,   √E pz 0 − ηs 0 √E + pz ! vs ( p) =  (23) √E + pz 0 − ηs  0 √E pz !   − −  Using   √E pz 0 √E + pz 0 √p σ = − ,√ p σ¯ = (24) · 0 √E + pz · 0 √E pz ! − ! we can write more generally

√p σ ξs √p σ ηs us ( p) = · , vs ( p) = · (25) √p σ¯ ξs √p σ¯ ηs · ! − · ! where the 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 usu- ally pick pµ along the z axis, so we don’t need to know how to make sense of √p σ . Neverthe- less, relations like ( p σ)( p σ¯) = m2 . Then the 4 solutions are · · ·

√E pz 0 √E pz 0 − − 1  0  2  √E pz  1  0  2  √E pz  up = , u p = − , vp = , vp = − (26) √E + pz 0 √E + pz 0      −     0   √E + pz   0   √E + pz         −          In any frame us are the positive frequency , and the vs are negative frequency electrons, or equivalently, positive frequency . For Weyl spinors, there are only 4 real degrees of freedom off-shell and 2 real degrees of freedom on-shell. Recalling that the Dirac equation splits up into separate equations for ψL and ψR, the Dirac spinors with zeros in the bottom two rows will be ψL and those with zeros in the top two rows will be ψR. Since ψL and ψR have 2 degrees of freedom each, these must be par- ticle and antiparticle for the same helicity. The embedding of Weyl spinors into fields this way induces irreducible unitary representations of the Poincaré group for m = 0. 6 Section 4

3.1 Normalization and spin sums To figure out what the normalization is which we have implicitly chosen, let’s compute the inner product

√p σ ξs † 0 1 √p σ ξs ′ ′ ′ u¯s ( p) us ( p) =us† ( p) γ0 us ( p) = · · √p σ¯ ξs 1 0 √p σ¯ ξs ′ · !   · ! (27) ξ ( p σ)( p σ¯) ξ ′ = s † · · s ξs ( p σ)( p σ¯) ! ξs ′   p · ·   ′ =2 mδss p

Similarly v¯s ( p) vs ′ ( p) = 2 mδss ′ . This is the (conventional) normalization for the spinor inner product. For m = 0 this− way of writing the normalization convention is not useful. Instead we can calculate a different inner product

√p σ ξs † √p σ ξs ′ ′ ′ ′ us† ( p) us ( p) = · · = 2 E ξs† ξs = 2 E δss (28) √p σ¯ ξs √p σ¯ ξs ′ · ! · ! These are the same √2 E factors which go between the non-relativistic and relativistic normal- ization as discussed in Lectures I-2 through I-4. Other useful relations which you can check are

′ ′ ′ ′

Q Q Q that v¯s ( p) us ( p) = u¯s ( p) vs ( p) = 0 and us† ( Qp ) vs ( p ) = vs† ( p ) us ( p ) = 0 as well. We can also compute the spinor − −

2 us ( p) u¯s ( p) = p + m (29) s =1 X where the sum is over the spins. Both sides of this equation are matrices. It may help to think of this equation like s s . For the antiparticles, s | ih | P 2 vs ( p) v¯s ( p) = p m (30) − s =1 X 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 parti- cles. We have found

i⋆ j ij s s ′ : ǫ ( p) ǫ ( p) = δ u¯s ( p) us ′ ( p) = 2 mδss ′ (31) h | i µ µ − ↔ 3 µ ν 2 µ ⋆ ⋆ν µν p p s s : ǫ ( p) ǫ ( p) = g + us ( p) u¯s ( p) = p + m (32) | ih | i i − m2 ↔ s i =1 s =1 X X X 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.

4 Majorana spinors

Recall from the previous lecture that if we allow to be anti-commuting Grasmann num- bers (these “numbers” be discussed more formally in Lecture II-7), then we can write down a Lagrangian for a single Weyl spinor with a mass term

m ⋆ T = iψ † σµ∂µψL + i ( ψ † σ ψ ψ σ ψL ) (33) L L 2 L 2 L − L 2 The mass terms in this Lagrangian are called Majorana masses, and the Lagrangian is said to describe Majorana fermions. Majorana fermions transform under the same representations of the Lorentz group as Weyl fermions. The distinction comes in the quantum theory in which Majorana fermions are their own antiparticles. We will make this more precise through the notion of charge conjugation defined below. Majorana spinors 7

It is sometimes useful to use the to represent Majorana fermions, like how we P 1 γ use it to describe Weyl fermions with the R/ L = 2 (1 5 ) projection operators. Majorana fermions can be put in 4-component Dirac spinors as ± ψ ψ = L (34) iσ ψ⋆  2 L  ⋆ This transforms like a Dirac spinor because σ2 ψL transforms like ψR. Then the Majorana mass can be written as m m ⋆ T ψ¯ ψ = i ( ψ † σ ψ ψ σ ψL ) (35) 2 2 L 2 L − L 2 which agrees with Eq. (33). 2 Note that (in the Weyl basis), using σ2 = 1 ,

⋆ ⋆ ⋆ 0 σ2 ψL ( i)( i) σ2 σ2 ψL ψL iγ2 ψ = i ⋆ = − − ⋆ = ⋆ = ψ (36) − − σ2 0 iσ2 ψL ( i)( 1) σ2 ψL iσ2 ψL  −    − −    Let us then define the operation of charge conjugation C by ⋆ C: ψ iγ ψ ψc (37) → − 2 ≡ ⋆ where ψc iγ2 ψ means the charge conjugate of the fermion ψ. Thus a Majorana fermion is its own charge≡ − conjugate. To understand why C is called charge conjugation, take the of the Dirac equation ( i∂ e A m) ψ = 0 to get − − ⋆ ⋆ ⋆ ( iγ ∂µ eγ A µ m) ψ = 0 (38) − µ − µ − which implies ⋆ ⋆ γ ( iγ ∂µ eγ A µ m) γ ψc = 0 (39) 2 − µ − µ − 2 Now recall that in the Weyl basis γ2 is imaginary and γ0 , γ1 , and γ3 are real. (Of course, we could just as well have taken γ3 or γ1 imaginary and γ2 real, but it’s conventional to pick out ⋆ γ2 .). So we can define a new representation of the γ-matrices by γˆ µ = γ2 γµ γ2 . This satisfies the Dirac Algebra because γ2 = 1 . So we get 2 − ( iγˆ ∂µ + eγˆ A µ m) ψc = 0 (40) µ µ − which shows that ψc satisfies the Dirac equation, albeit in a different γ basis. Since the physics is basis independent, we can read off that ψc has the opposite charge from ψ, justifying why we call this charge conjugation. ⋆ Because ψ = ψc = iγ2 ψ for Majorana fermions, they cannot be charged under any U(1) − iα iα gauged or global symmetry of a theory. Under such a symmetry ψ e ψ and ψc e − ψc so → → ψ = ψc cannot hold. We can also see this through the mass term, which is not invariant under the U(1) transformation T T iα iα 2 iα T ψ σ ψL ψ e σ e ψL = e ψ σ ψL (41) L 2 → L 2 L 2 This is true for gauge charges, that is those with a corresponding gauge , like the , and also for global charges like number (which counts the number of electrons and neu- trinos minus the number of positrons and antineutrinos) which have no associated . If there are multiple Majorana fermions, they can transform together under a real representa- tions of an internal non-Abelian symmetry group. For example, in can be Majorana, transforming under the of SU(3) . Non-Abelian gauge groups are introduced in Lecture IV-1. In summary, we have seen three types of spinors Dirac Spinors: massive, left and right handed. • Weyl Spinors: massless, left or right handed. Embedded in Dirac spinors with γ ψ = ψ. • 5 ± Majorana spinors: left or right handed. Embedded in Dirac spinors with ψ = ψc = • ⋆ − iγ2 ψ 8 Section 5

As an important application there are particles in called which apparently carry no charges. Thus they may be Majorana or Dirac fermions. In fact, a number of experi- ments are trying hard to find out if neutrinos are Majorana (see Problem ??). Weyl spinors do exist in nature, in an obvious way, since Dirac spinors are just two Weyl spinors put together. But Weyl spinors are also integral to the theory of Weak interactions which is chiral.

5 Charge conjugation

The notion of charge conjugation, under which Majorana fermions are invariant, can be applied to any 4-dimensional spinor. For example, we can see how it affects the different spins of a Dirac spinor. Recall from Eq. (18) a basis for free Dirac spinor in its rest frame is given by

1 0 1 0 0 1 −0 1 u =  , u =  , v =  , v =  (42) ↑ 1 ↓ 0 ↑ 1 ↓ 0  0   1   0   1  Then        −    ⋆       1 0 0 0 i 1 0 − c 0 0 0 i 0 0 1 ( u ) = iγ2  = i   =  = v (43) ↑ − 1 − 0 i 0 0 1 0 ↓  0   i 0 0 0   0   1     −     −  and so on, giving       

( u ) c = v , ( u ) c = v , ( v ) c = u , ( v ) c = u (44) ↑ ↓ ↓ ↑ ↑ ↓ ↓ ↑ Thus charge conjugation takes particles to anti-particles and flips the spin. In particular, invari- ance under C of a theory constrains how different spin states interact. Charge conjugation may or may not be a symmetry of a particular Lagrangian. The opera- tion of charge conjugation acts on spinors and their conjugates by C: ψ iγ ψ⋆ (45) → − 2 In the Weyl basis, γ⋆ = γ and γT = γ so, 2 − 2 2 2 C: ψ⋆ iγ ψ (46) → − 2 and in particular C2 = 1 which is why C is called a conjugation operator. Then C: ψ¯ ψ ( iγ ψ) Tγ ( iγ ψ⋆) = ψT γT γ γ ψ⋆ = ψT γ ψ⋆ (47) → − 2 0 − 2 − 2 0 2 − 0 The on a spinor isn’t really necessary. This last expression just means

T ⋆ ⋆ ψ γ ψ = ( γ ) αβ ψα ψ (48) − 0 − 0 β Now, anticommuting the spinors, relabeling α β and combining shows that ↔ ⋆ ⋆ ⋆ T ( γ ) αβ ψα ψ = ( γ ) αβ ψ ψα = ( γ ) βα ψ ψβ = ψ † γ ψ = ψ¯ ψ (49) − 0 β 0 β 0 α 0 Thus C: ψ¯ ψ ψ¯ ψ (50) → Similarly, C: ψ¯ ∂ψ ψ¯ ∂ψ (51) → So the free Dirac Lagrangian is C invariant. We can also check that C: ψ¯ γµ ψ ψ¯ γµ ψ (52) → − µ This implies that the interaction eA µ ψ¯ γ ψ will only be C invariant if

C: A µ A µ (53) → − Parity 9

2 Since the kinetic term Fµν is invariant under A µ A µ the whole QED Lagrangian is therefore C invariant. → ± The transformation A µ A µ under C may seem strange, since a vector field is real, so it shouldn’t transform under an→ operation− which switches particles with anti-particles. Since parti- cles and anti-particles have opposite charge and A µ couples proportionally to charge this trans- formation is needed to compensate for the transformation of the charged fields. There is an important lesson here: you could take C: A µ A µ, but then the Lagrangian wouldn’t be invariant. Thus, rather than trying to figure out how→ C acts, the right question is: how can we enlarge the action of the transformation C, which we know for Dirac spinors, to a full interacting theory so that the symmetry is preserved? Whether we interpret C with the words “takes particles to antiparticles,” has no physical implications. In contrast, a symmetry of a theory does have physical implications: preservation of the symmetry gives a superselection rule – certain transitions cannot happen. An important example is that C invariance forces matrix elements involving an odd number of photons to vanish, a result known as Furry’s the- orem (see problem ??). Thus, cataloging the symmetries of a theory is important, whether or not we have interesting names or simple physical interpretations of those symmetries. For future reference, it is also true that C: i ψ¯ γ5 ψ i ψ¯ γ5 ψ (54) → C: i ψ¯ γ5 γµ ψ i ψ¯ γ5 γµ ψ (55) → C: ψ¯ σ µν ψ ψ¯ σ µν ψ (56) → − which you can prove in Problem 4.

6 Parity Recall that the full Lorentz group, O(1,3) is the group of 4 4 matrices Λ with ΛT g Λ = g. In addition to the transformations smoothly connected to 1 , this× group also contains the transfor-

mations of parity and time-reversal Q P: ( t, xQ ) ( t, x ) (57)

→ − Q T: ( t,Q x ) ( t, x ) (58) → − Just as with charge conjugation, we would like to know how to define these transformations acting on spinors, and other fields, so that they are symmetries of QED or whatever theory we are studying. You might expect that the action of P and T should be determined from . However, recall that technically, spinors do not actually transform under the Lorentz group, O(1,3) only its universal cover, SL(2, C) , so we are not guaranteed that T and P will act in any nice way on irreducible spinor representations. In fact they don’t. Although we can define an action of T and P on spinors (and other fields) these definitions are only useful to the extent that they are symmetries of the theory we are interested in. For example, we will define P so that it is a good symmetry of QED, but there is no way to define it so that it is preserved under the weak interactions. In any representation, we should have P2 = T2 = 1 .

6.1 Scalars and vectors Before discussing vectors and spinors, let us begin with real scalar fields. For real scalars, parity should be a symmetry of the kinetic terms = 1 φ φ 1 m2 φ2 or we are dead in the water. L − 2 − 2 Thus P2 = 1 (we do not need to use P2 = 1 in the Lorentz group for this argument). Thus there

are two choices Q P: φ( t,Q x ) φ( t, x ) (59) → ± − The sign is known as the intrinsic parity of a particle. In nature, there are particles with even

parity (scalars, like the ) and particles with odd parity (, like the π0).

Q Q Since the action integrates over all Qx , we can change x x and the action will be invariant. → − 10 Section 6

For complex scalars, the free theory has Lagrangian = φ⋆ φ m2 φ⋆ φ, so the most gen-

eral possibility is L − − Q P: φ( t,Q x ) η φ( t, x ) (60) → − where η is a pure phase. Recall that charged scalars always have a global symmetry under φ eiα φ for a constant α, which is why they can couple to the photon. So η is not even well-→ defined, since we can always combine this transformation with a phase rotation and still have a symmetry. However, all charged particles must rotate the same way under the global symmetry of QED, so if we pick a convention for the phase of one charged particle, the phase of the others then becomes physical. We can go further, and redefine P so that all the parity phases for all particles are 1 . To iαQ ± see that, suppose η = e where Q is the charge of φ and α R. Then the operator P ′ = α ∈ i 2 Q 2 2 Pe − is also a legitimate discrete symmetry, which satisfies ( P ′) : ψ ψ so ( P ′) = 1 . Thus we might as well call this parity, P, and P: ψ ψ. We actually have 3→ global continuous sym- metries in the : → ( ± only), number ( only) and charge. Thus we can pick three phases, which conventionally are taken so that the , neu- tron and electron all have parity +1 . Then every other particle has parity 1 . ± From measurements, it was deduced that the , π0, and its charged sib- + lings π and π− all have parity 1 . Then it was very strange to find that a particle called the decayed to both 2 and− 3 pions. People thought for a while that the kaon was two particles, the θ (with parity +1 which decayed to two pions) and the τ (with parity 1 which decayed to three pions). Lee and Yang finally figured out, in 1956, that these were the same− par- ticle, and that parity was not conserved in kaon decays. For vector fields, P acts as it does on four-vectors. However, for the free vector theory to be

invariant, we only require that

Q Q Q P: V ( t,Q x ) V ( t, x ) Vi( t, x ) Vi( t, x ) (61) 0 → ± 0 − → ∓ −

The notation is that if P: Vi Vi, like Qx , we say Vµ has parity 1 and call it a vector. If P: → − − Vi Vi, we call it a pseudovector, with parity +1 . For example, the ρ is a vector and the → a1 meson is a pseudovector. You have already seen pseudovectors in three dimensions: the elec- tric field is a vector, which flips sign under under parity, while the magnetic field is a pseu- dovector which remains invariant under parity. Massless vectors like the photon have to have parity 1 . To see this, just look at the cou- pling to a charged scalar. Under parity we would like − ⋆ ⋆ ⋆ ⋆ P: A µ( φ ∂µφ φ∂µφ ) A µ( φ ∂µφ φ∂µφ ) (62) − → −

which is only possible if A µ transforms like ∂µ. That is, A µ is a vector:

Q Q Q P: A ( t,Q x ) A ( t, x ) Ai( t, x ) Ai( t, x ) (63) 0 → 0 − → − − 6.2 Spinors Now let’s turn to spinors. In the Lorentz group, P commutes with the rotations. Thus, P does not change the spin of a state embedded in a vector field. This should be true for spinors too. For massless spinors, recall that left- and right-handed spinors are eigenstates of the helicity

operator, which projects spin onto the momentum axis:

Q Q Q σQ p σ p

· ψR = ψR , · ψL = ψL (64) Q Qp p − | | | |

Since parity commutes with spin Qσ and energy but flips the momentum, it will take left-handed 1 , , 1 spinors to right-handed spinors. That is, it will map 2 0 representations to 0 2 . Therefore P cannot be appended to either of the spin 1 irreducible representations alone. 2   For Dirac spinors, which comprise left and right-handed spinors, we can see that parity just swaps left and right, keeping the spin invariant. In the Weyl basis, this transformation can be written in the simple form P: ψ γ ψ (65) → 0 Time reversal 11

There is in principle a phase ambiguity here, like for charged scalars. But as in that case, we can use invariance under global phase rotations, associated with , to simply choose this phase to be 1, as we have done here. Despite this phase, a chiral theory (one with no symmetry under L R), such as the theory of weak interactions, cannot be invariant under parity. ↔ Note that

¯ ¯

Q Q P: ψ ψ( t,Q x ) ψ † γ γ γ ψ( t, x ) = ψ ψ( t, x ) (66) → 0 0 0 − −

¯ ¯

Q Q P: ψ γµ ψ( t, xQ ) ψ † γ γ γµ γ ψ( t, x ) = ψ γ † ψ( t, x ) (67) → 0 0 0 − µ −

Recalling that γ † = γ and γ † = γi, we see that 0 0 i −

¯ ¯ ¯ ¯

Q Q Q P: ψ γ ψ( t,Q x ) ψ γ ψ( t, x ) ψ γi ψ( t, x ) ψ γi ψ( t, x ) (68) 0 → 0 − → − − so that ψ¯ γµ ψ transforms exactly as a 4-vector and hence the Dirac Lagrangian is parity invariant. The parity transformations are opposite for bilinears with γ5 :

¯ 5 ¯ 5 ¯ 5 ¯ 5 Q P: ψ γ γ ψ ψ γ γ ψ( t, Qx ) ψ γi γ ψ ψ γi γ ψ( t, x ) (69) 0 → − 0 − → − So that ¯ ¯ P: ψ Aψ ψ Aψ( t, xQ ) , (70) → − ¯ 5 ¯ 5 P: ψ Aγ ψ ψ Aγ ψ( t, xQ ) , (71) → − − µ µ ¯ µ The currents contracted with A in these terms are known as the vector current, JV = ψ γ ψ µ ¯ µ 5 and the axial vector current, JA = ψ γ γ ψ. These currents play a crucial role in theory of weak interactions, which involves J µ J µ, or the V-A current. V − A

7 Time reversal

Finally, let us turn to the most confusing of the discrete symmetries, time reversal. As a Lorentz

transformation, Q T: ( t,Q x ) ( t, x ) (72) → − We are going to need a transformation of our spinor fields ψ such that (at least) the kinetic Lagrangian is invariant. To do this, we need ψ¯ γµ ψ to transform like a 4-vector under T, so that ¯ i ψ ∂ψ( t,Q x ) iψ ∂ψ( t, x) and the action will be invariant. In particular we need the 0-compo- 0 → 0− nent, ψ¯ γ ψ ψ¯ γ ψ which implies ψ † ψ ψ † ψ. But this last form of the requirement is very odd – it→ says − we need to turn a positive→ definite − quantity into a negative definite quantity. This is impossible for any linear transformation ψ Γ ψ. Thus we need to think harder. → I’ll describe two possibilities, one which I call “simple Tˆ ,” and denote Tˆ . It is the obvious parallel to parity. The other is the T symmetry which is normally what is meant by T in the lit- erature. This second T was invented by Wigner in 1932 and requires T to take i i in the → − whole Lagrangian in addition to acting on fields. While the simple Tˆ is the more natural gener- alization of the action of T on 4-vectors, it is also kind of trivial. Wigner’s T has important physical implications.

7.1 The simple Tˆ Before doing anything drastic, the simplest thing besides T: ψ Γ ψ would be T: ψ Γ ψ⋆, like → → with charge-conjugation. We’ll call this transformation Tˆ to distinguish it from what is conven- tionally called T in the literature. So,

⋆ ⋆ T Tˆ : ψ Γ ψ , ψ † (Γ ψ ) † = ψ Γ † (73) → → That Tˆ should take particles to anti-particles is also understandable from the picture of antipar- ticles as particles moving backwards in time. 12 Section 7

Then

T ⋆ ⋆ ⋆ T T ψ ψ ψ Γ Γ ψ = Γ † Γ βγ ψα ψ = ψ (Γ † Γ βγ) ψα = ψ (Γ Γ) ψ (74) † → † αβ γ − γ αβ − † †

So we need Γ † Γ = 1 which says that Γ is a . That’s fine. But we also need ψ¯ γi ψ and the mass term ψ¯ ψ to be preserved. For the mass term, we need

T ⋆ T ψ¯ ψ ψ Γ † γ Γ ψ = ψ¯ (Γ † γ Γ γ ) ψ (75) → 0 − 0 0 to equal ψ¯ ψ. Then Γ,γ = 0. Next, { 0} T ⋆ T T ψ¯ γi ψ ψ Γ † γ γi Γ ψ = ψ¯ (Γ † γ γi Γ γ ) ψ = ψ¯ (Γ † γi Γ) ψ (76) → 0 − 0 0 − T which should be equal to ψ¯ γi ψ for i = 1 , 2, 3. So γi Γ + Γ γi = 0, which implies [Γ,γ1 ] = 0, [Γ, γ ] = 0 and Γ,γ = 0. The unique (up to a constant) matrix which commutes with γ and γ 3 { 2 } 1 3 and anticommutes with γ2 and γ0 is Γ = γ0 γ2 . Thus

⋆ T

Q Q Q ψ( t,Q x ) γ γ ψ ( t, x ) , ψ † ( t, x ) ψ γ γ ( t, x ) (77) → 0 2 − → − 2 0 − Note that this is very similar to P C. On vectors, we should have ·

ˆ Q Q Q T: A ( t,Q x ) A ( t, x ) , Ai( t, x ) Ai( t, x ) (78) 0 → − 0 − → − So that the interaction ψ¯ Aψ in the action is invariant. Now consider the action of C P Tˆ . This sends · ·

ˆ ⋆ ⋆ Q Q CPT: ψ( t,Q x ) iγ ( γ [ γ γ ψ ]) ( t, x ) = iψ( t, x ) (79) → − 2 0 0 2 − − − − − and so

ˆ ¯ µ ¯ µ Q Q Q CPT: ψ ( t,Q x ) γ ψ( t, x ) ψ ( t, x ) γ ψ( t, x ) (80) → − − − −

ˆ Q CPT: A µ( t,Q x ) A µ( t, x ) (81) → − − µ µ CPTˆ also sends ∂µ ∂µ. Thus ψ¯ ψ, i ψ¯ γ ∂µψ and ψ¯ A µ γ ψ are all invariant in the action. → − 1 This time reversal is essentially defined to be Tˆ ( C P) − , which makes CPTˆ invariance trivial. The actual CPT theorem concerns a different∼T symmetry,· which we will now discuss.

7.2 Wigner’s T (i.e. what is normally called T) What is normally called time-reversal is a symmetry T that was described in a 1932 paper by Wigner, and shown to be an explanation of Kramer’s degeneracy. To understand Kramer’s

degeneracy, consider the Schrödinger equation Q i∂tψ( t,Q x ) = H ψ( t, x ) (82)

2 H p V x where for simplicity, let’s say = 2 m + ( ) which is real and time-independent. If we take the complex conjugate of this equation and also t t, we find → − ⋆ ⋆ i∂tψ ( t, x¯) = H ψ ( t, xQ ) (83) − − ⋆ Thus ψ ′( t, x) = ψ ( t, x) is another solution to the Schrödinger equation. If ψ is an energy − ⋆ eigenstate, then as long as ψ  ξ ψ for any ξ, ψ ′ will be another state with the same energy. This degeneracy of states at the same energy is known as Kramer’s degeneracy.

In particular, for the Hydrogen , ψnlm( Qx ) = Rn( r) Ylm( θ, φ) are the energy eigenstates, so Kramer’s degeneracy says that the states with m and m will be degenerate (which they are). The importance of this theorem is that it also holds for− more complicated systems, and for sys- tems in external electric fields, for which the exact eigenstates may not be known.

⋆ Q As we will soon see, this mapping ψ( t, xQ ) ψ ( t, x ) sends particles to particles (not → − antiparticles), unlike the simple Tˆ operator above. It has a nice interpretation: suppose you made a movie of some physics process, then watched the movie backwards; time-reversal implies you shouldn’t be able to tell which was “play” and which was “reverse”. Time reversal 13

⋆ The trick to Wigner’s T is that we had to complex conjugate and then take ψ ′ = ψ . This means in particular that the i in the Schrödinger equation goes to i as well as the field trans- − forming. This is the key to finding a way out of the problem that ψ † ψ needed to flip sign under 0 T which we discussed at the beginning of the section. The kinetic term for ψ is i ψ¯ γ ∂0 ψ so if i i then since ∂ ∂ , ψ † ψ can be invariant. Thus we need → − 0 → − 0 T: i i (84) → − This makes T an anti-linear operator. What that means is that if we write any object on which T acts as a real plus an imaginary part ψ = ψ1 + iψ2 , with ψ1 and ψ2 real, then T( ψ1 + iψ2 ) = Tψ1 iTψ2 . Since T−changes all the factors of i in the Lagrangian to i, it also affects the γ matrices. In − the Weyl basis, only γ2 is imaginary, so

T: γ , , γ , , , γ γ (85) 0 1 3 → 0 1 3 2 → − 2 For a real spinor, T is simply linear, so we can write its action as

˜ Q T: ψ( t,Q x ) Γ ψ( t, x ) (86) → − µ 0 with Γ~ a Dirac matrix. Then, for i ψ¯ γ ∂µψ to be invariant, we need ψ¯ γ ψ to be invariant and ψ¯ γi ψ ψ¯ γi ψ. Thus, → − [Γ˜ ,γ0] = Γ˜ ,γ1 = [Γ˜ ,γ2 ] = Γ˜ ,γ3 = 0 (87)

The only element of the Dirac algebra which satisfies these constraints is Γ˜ = γ1 γ3, up to a con- stant. Thus we take 0 1

1 0

Q Q T: ψ( t, xQ ) γ1 γ3 ψ( t, x ) = −  ψ( t, x ) (88) → − 0 1 −  1 0   −    Thus T flips the spins of particles, but does not turn particles into antiparticles, as expected. T

does not have a well-defined action on Weyl spinors, which have one spin state. T also reverses Q the momenta, Qp = i because of the i. Thus, T makes it look like things are going forward in time, but with their∇ momenta and spins flipped.

Similarly, for ψ¯ D ψ to be invariant, we need A µ to transform like i∂µ which is

Q Q Q T: A ( t, xQ ) A ( t, x ) , Ai( t, x ) Ai( t, x ) (89) 0 → 0 − → − − It is straightforward to check now that the Dirac Lagrangian is invariant under T. Next, consider the combined operation of CPT. This sends particles into anti-particles moving as if you watched them in reverse in a mirror. On Dirac spinors, it acts as C P T: ψ( x) iγ γ γ γ ψ⋆( x) = γ ψ⋆( x) (90) · · → − 2 0 1 3 − − 5 − ⋆ It also sends A µ( x) A µ( x) , φ φ ( x) and of course i i. You can check (Problem→ − 6) that→ any− term you could possibly→ − write down, for example,

µ 5 µν = ψ¯ ψ, iψ¯ ∂ψ, iψ¯ Aψ, ψ¯ γ γ ψWµ , i ψ¯ σ ψFµν (91) L and so on, are all CPT invariant. The CPT theorem, which requires some fancy to prove, says that this is a consequence of Lorentz invariance and unitarity (See Streater and Wightman). The physicist’s proof is that any Lorentz invariant term you can write down in the Lagrangian is CPT invariant.