where k was changed to -k in the sum in the last expression. Hence, equating Fourier coefficients gives

On single particle states, this means that

which shows that parity changes the direction of momentum of a state (as expected) and that particles and antiparticles have the same intrinsic parity. For spinors, a similar argument gives

= ^TlfP s) e~<fc"+"(-*>s)dl^ e'k'x} •

The Dirac spinors satisfy the relations

7°u(— k, s) — u{k,s) ■y°v(-k,s) =-v{k,s) , and therefore

Hence Fermi particles and antiparticles have opposite parity, and the direction of their spin is unchanged (as we would expect from its cross product nature

For the transverse EM field we have a similar result. Using the circular polarization basis,

fc,a where the two helicity states are [recall Eq. (2.61)]

If the transformation from z to -z is achieved by rotating about the y-axis by it, then c+ -

Hence the photon has odd parity, and its momentum and helicity change sign under parity.

Now, if parity is a symmetry of the Lagrangian, then

implying that £ is a scalar. This places restrictions on the types of interactions permitted in the Lagrangian. For example, for nNN interactions with no derivatives, we must have

£p = ,ip(x)')5ip(x) <j>(x) pseudoscalar (8.53)

if the pion field is pseudoscalar and

if it is a scalar. To prove this formally, use the transformation laws (8.41),

= -vPM-r, i)7°757V(-r, t) 0(-r, t) = Cp(-r,t) , (8.55)

Parity Transformation of Spin and Angular Momentum

Recall that angular momenta (and hence spins) have even parity and are therefore axial vectors (since normal vectors change sign under parity). To show this, recall that the angular momentum vector is a cross product, fi = r x p .

In terms of its components,

8.5 CHARGE CONJUGATION 219

and under any transformation r'3 = AJere,

But fi« = eljkr'Jp'k = eljkVtA kmrepm eijkA1nA3iAkm = (det A)engm

and hence, multiplying this expression by A*' „ and summing over n (for orthogonal transformations A where A* nA'n = <5^) give

€i'jkAJiAkm = (detA)Ai'n€nim . Hence the transformation (8.56) can be written

The extra factor of det A shows that ft does not change sign under the parity transformation and hence is an axial vector.

The charge conjugation transformation, as we saw in Chapters 4 and 5, is associated with a symmetry of a relativistic wave equation which allows us to transform ne negative energy solutions into positive energy solutions which satisfy the same wave equation but with the sign of the charge reversed. Such a transformation can also be defined when a particle has no charge but has some other quantum number which changes sign under the transformation (for example, the Ko, Ko system). Hence charge conjugation might be more appropriately referred to as "field conjugation."

To extend this idea to field theory, we postulate the existence of a unitary operator C which transforms the fields according to the relations

C4>{x)& =t)$4>Hx) Cl>(x)tf = rftCpix) CA(x)Cf = -A(x) ,

where again the 77's are phases which can be ±1 and C (to be distinguished from C) is the Dirac conjugation matrix introduced in Eq. (5.32) with the following properties:

8.5 CHARGE CONJUGATION 221 Since the exponentials are independent, and since C is unitary, we have

Thus C interchanges particles and antiparticles, with the same phase. For single particle states, assuming C|0) = |0), we have

where |k) is the state of a single antiparticle with momentum k. The same relation holds for antiparticles,

For the Dirac field, the derivation requires use of Eq. (5.38), which can be written

Using these relations,

CiP(x)Ct =r)$CY, -?=f ibl>J(P>eiP'X + dP>°vT(P>s) e"'P X}

p,s VZtjVL

Hence

C6P, |
,ct |
= |

Cdp, |
,ct |
= V$bp,s |

and the effect of C on fermions is the same as it is on bosons. Finally, for photons

and this requires

Photons are odd under charge conjugation.

222 SYMMETRIES I Positronium Decay

To illustrate the usefulness of charge conjugation invariance, we consider positronium decay. Positronium is a bound state of the e+e~ system, which can decay through the annihilation of the e+e~ pair into photons. This decay is most likely to happen when the e+ and e~ are very close to each other, and this in turn is only probable in S-states, where the wave function at the origin is not zero. There are two types of S-states: the spins of the particles may be aligned, so that the total spin is one (a spin triplet state), or they may be antialigned in a spin zero (a spin singlet state). The standard spectroscopic notation for these states is 2S+1Lj, where J, L, and S are the total angular momentum, orbital angular momentum (in the S, P, D, ■ ■ • notation), and total spin, respectively. Now, experimentally, it is observed that the 3 Si-states of positronium decay only into 37's (2-y decay is not observed and I7 decay is forbidden by energy momentum conservation) while the xS0-states decay only into 27's. Since 37 decay is much less probable that 27 decay (because the decay rate is smaller by an extra factor of a and is further suppressed by the small size of the three-body phase space), the 3Si-state is metastable.

To understand these results, we will represent the positronium states by the vector where p is the relative momentum of the e+e~ pair (the total momentum being zero by assumption) and s and s' are the possible spin states of the electron and positron, respectively. The wave function of the state is related to /, which for c states depends only on the magnitude of p, so that f(p; s, s') = f(-p\ s, s') = f(p; s, s'). Angular momentum conservation insures that / separates into a triplet part, symmetric in s and s', and a single part, antisymmetric in s and s'. We can therefore distinguish two different functions /, one symmetric under s ♦-» s' and the other antisymmetric, s,s'

fc(p;s,s') = ef((p;s',s) , where e = +1 for 3Si-states and -1 for 'So-states. Hence,

fc(p;s,s') = ef((p;s',s) , where e = +1 for 3Si-states and -1 for 'So-states. Hence,

Now note that s,s'

8.6 TIME REVERSAL 225 The transformed O, denoted by O', is defined by the requirement

Hence

This equality must hold for any |x), which gives the relation

Furthermore, under UA the matrix element (y\z) transforms as

(vV) = (v\*r , from which we obtain the general result for the transformation of operators under antiunitary transformations,

Now we will discuss why the physics requires that the operation of time inversion, T, must be an antiunitary operator. If O is a scalar operator which depends on time, 0'(t) = O(-t). Since the Hamiltonian is the time component of a four-vector, we would expect the transformation law to be H'(t) = —H(—t), but this would have the undesirable effect of changing the sign of all energies. We thus require that the correct transformation for H be H'(t) — H(-t). If this is the correct transformation law for the Hamiltonian, the effect of time inversion on the interaction time translation operator can be found by examining its effect on the nth term in its expansion, Eq. (3.24), which is

= LJr-/ •••/ dt1-.-dtnT(H,(t1)---Hi(tn)) . (8.85)

Hence the operation of time inversion gives ruj

nl 7-oo

where in the first line we used the assumed antiunitarity of T when we changed the factor of H)n to (+i)n, in the second line noted that when the action of T on H changed the sign of the time, terms which were time ordered became "anti-time-ordered" (denoted by Ta), in the third line changed t —> -t in each of the time integrals, and finally in the last line replaced the anti-time-ordering with the Hermitian conjugate. Hence we conclude from (8.86) that

This is a very elegant result. Taking matrix elements and using Eq. (8.84)

give v here |/) and \i) are the final and initial state, respectively, and \it) and |/t) are time-reversed initial and final states. As illustrated in Fig. 8.3, this gives a beautiful interpretation of the T operation as the reversal of direction of motion. This interpretation would not emerge without the mapping |j) —► |/t) and |/) —> \it) and without the change in the sign of i which insures that it is the same (correct) time translation operator Ui which is involved in both cases. And both of these properties are a consequence of the assumption that T is antiunitary.

The necessity for time reversal to be antiunitary can also be demonstrated with two simple arguments:

• If the energy E is to be kept positive under time inversion, then the exponent in the factor e~tEl must not be allowed to change sign. If t —* —t, the only way to insure this is to change the sign of i, which requires that T be antiunitary.

• If we wish to preserve the meaning of H as a time translation operator, we must preserve the relation

and this requires that T be antiunitary.

where C is the charge conjugation matrix. We turn now to a discussion of the implications of these transformations.

We will now show that these two equations are consistent with, and required by, the normal phase convention used to define angular momentum states. To see this, work with a particle at rest, p = 0, and recall that the states are related to each other by the raising and lowering operators.

J+\0,-i) = (JX+iJy)\0,-l) = \0, i) J_|0, = (Jx - Uy) |0, i) = |0,-l) .

If we choose the phase so that m-i> = io,4> , and use the fact that TJl = —Ji T, we obtain

T|0, i) = T (Jx + iJy) 10, -I) = (-Jx + Uy) T|0, -I)

Hence, the negative phase for the spin "up" state is required by all of the definitions and choices previously imposed!

We conclude this section by discussing time inversion for the photon. Using the helicity representation, Eq. (8.50),

4 = -^(fi-w), the photon polarization vectors satisfy the following relations:

Therefore, under time inversion

k,a and the transformation law for the annihilation operators is

Note that the phase is positive, and the helicity does not change sign, because both the momentum and the spin do.

8.7 THE PCT THEOREM

We conclude this chapter with a discussion of one of the more interesting theorems in the subject of symmetry.

Theorem: If the Lagrangian density is a Hermitian, normal-ordered, Lorentz invariant operator constructed from fields quantized with the usual connection between spin and statistics, then the product of the VCT transformations is always a symmetry of the theory.

Interactions can readily be constructed which violate V, C, or T individually (often it is only necessary to change the phase of a coupling constant to achieve this), but the PCT theorem says that it is possible to choose the phases r) [which enter into the transformation laws Eqs. (8.41), (8.58), and (8.94)] so that the product of all of these symmetries, O = VCT, is always a symmetry of the theory, regardless of how the interactions are constructed (provided only that they conform to the restrictions stated in the theorem) and regardless of the phases of the coupling constants. In this section, a proof of this theorem will be sketched and implications discussed.

If O = VCT, then from the previous sections, where, as in the previous cases, the phase rty is free to be chosen, subject to the condition rfy = 1. However, as we will discuss below, the PCT symmetry will emerge only if the phase r]^ accompaning the transformation of spin zero fields is chosen to be +1. Since the Lagrangian density is a scalar, invariance under this transformation implies because £ is also Hermitian.

Now consider the most general Lagrangian density imaginable. Lorentz invariance requires that all terms involving the Dirac ip fields be constructed from the bilinear covariants discussed in Sec. 5.10, which must in turn be1 contracted with other tensors constructed from other fields which have the same Lorentz invariance properties. We have denoted these bilinear covariant matrices by T, and without loss of generality we may assume that the T's are Hermitian (any non-Hermitian terms we might want to consider can be constructed from Hermitian T's multiplied by complex coefficients). The ip fields will therefore enter the

0<t>(x)0~l = ??0 0t(-x) OA"(x)0~l = -A"(-z) ,

Lagrangian through combinations like ip(x)Tip(x). Consider the transformation of this quantity under O. Because O is antiunitary,

Now, we can remove the transpose in the Dirac space by re-ordering the Dirac fields (as we did in Sec. 8.5), remembering that this will give a minus sign. Hence, since r)l - 1,

07p(x)rtp(x)0~1 = —xp(—x)y°f5r^y°-y5ip(—x) . (8.115)

[The normal ordering which is implied means that we can drop all anticommutators which would otherwise emerge when the btf terms and dtf terms were exchanged under interchange of 1p and ip.] Now, the implied hermiticity of the T's gives the requirement

so that

Using 7°7Mt7° = 7M, the specific (Hermitian) forms of the 16 F's are r={l, 7", 7V, ¿75} • (8.117)

rlence, recalling that 75 anticommutes with all 7M, under O the T's transform to r' = 75r75 . (8.118)

Under PCT the Fs therefore fall into two classes:

r's = rs for rs = {i)Cr^,n5} T'a = -Ta for = {7m,757'j} •

Note that F's with one Lorentz vector index change sign; those with an even number (0 or 2) do not. However, T's with one vector index must necessarily be multiplied by

All of these also change sign under Ö, so that their product with does not. If a T has an even number of vector indices, it must be contracted with other terms which have the same (even) number of indices, and there will again be no sign change. This result can be immediately extended to include interaction terms with complex coupling constants; if the Lagrangian includes the term Ar x r, where A is complex, the hermiticity of the Lagrangian density guarantees that the Hermitian conjugate of this term is also present, and the combined term Ar x T + AT x T does not change sign under PCT. Hence, while individual factors involving the fermion fields may change sign (or phase) under O, this change of sign (or phase) is always balanced by other terms with a compensating change of sign (or phase) and therefore all terms involving fermions are invariant under O.

It now remains to examine interactions involving <f>. The Lagrangian density could contain (for example) an interaction term of the form

£ = -\<f)3(x) — X'^3(x) , which is both Lorentz invariant and Hermitian. Under PCT, <p(x) is transformed to r}^{—x) and to rj^4>(—x), and remembering that O is antilinear, these terms transform into each other, with an overall phase change of rfy. If we chose the phase 77^, = — 1, these terms would be odd under PCT, and the theorem would not hold. However, choosing 77^ = +1 guarantees that all such terms are even. This choice also insures that any terms involving products of Dirac and scalar fields are even. We conclude that C is invariant under O [i.e., it transforms like Eq. (8.113)].

As we have seen, the freedom to choose 77^ = +1 is central to the proof of the PCT theorem, and it must be demonstrated that this choice is always permitted by the physics. A full discussion of this point requires techniques which we have not developed in this book. Briefly, it can be shown that a field theory can be fully defined by the vacuum expectation values of all products of its field operators, and that these matrix elements, initially defined for real space-time points, can be analytically continued into complex space-time. The concept of analytic continuation is familiar from elementary studies of complex functions. It is a remarkable fact that a smooth function which is initially known only over an interval of the real axis can be continued into the entire complex plane and that this continuation is unique. In a similar fashion, the matrix elements of a Hermitian scalar field (a simple example), initially defined for real space-time points, can be defined uniquely in complex space-time. Now, for points in real space-time, the matrix elements of a scalar field are invariant under the real Lorentz group, and their unique extension into complex space-time is necessarily invariant under the group of complex Lorentz transformations. However, as we discussed briefly in Sec. 5.8, for the group of complex Lorentz transformations, TP is continuously connected to the identity, so that any function must transform under TP with the same phase as it does under the restricted group L\_ (recall Fig. 5.3 and Table 5.2). A scalar field is invariant under L\, and because TP is connected to the identity through the process of analytic continuation, the phase 77^, for a Hermitian scalar field must therefore equal +1. Hence there can be no physical impediment to this choice, which is, in this case, required by the interpretation of TV as space-time inversion. For further discussion, the interested reader is referred to Streater and Wightman (1964) and to the literature (see, for example, [Lu 57]) .

We begin our examination of the implications of PCT invariance by considering the effect of VCT on meson annihilation and creation operators. Since 7]$ = 1, we have

Hence

where, as before, |fc) is the state of an antiparticle with momentum k. The operator O turns particles into antiparticles. We can therefore use PCT invariance to prove the following theorem:

Theorem: The masses of particles and antiparticles are equal provided they are stable or cannot decay into any other single particle state.

Proof: For stable particles, the proof is simple because they are eigenstates of the total Hamiltonian, and therefore

where Ek = yjm2 + k2 and Ek = \/fh2 + k2 are the energies of the particle and antiparticle states of momentum k. But because of the invariance under PCT, O commutes with H, and

Hence Ek — Ek, which implies m — m. An example is the proton; mp = fhp.

For unstable particles the energy is, by definition, the expectation value of the Hamiltonian,

Therefore, under PCT

236 SYMMETRIES I corresponding to eigenstates m_ m+

Note that the masses are no longer equal. If CP is conserved, it turns out that B = B*, and Kg decays into an even number of rr's (mainly 2ir), while Ki decays into an odd number (only 37r's since single pion decay is forbidden by energy-momentum conservation). It was the observation that Kl sometime decays into 2tt's, which led to the discovery of CP violation.

PROBLEMS

8.1 Find the momentum operator Pl for the Klein-Gordon field (using Noether's theorem) and prove that

8.2 (a) Show explicitly that i: ■ip(x)-y5'ip(x): <j>(x) + i: tp{x)j5ip(x):

is Hermitian. (Here ip is a Dirac field, (j> a scalar field.) Also show (assuming all phases are unity) that

(i) this interaction does not conserve parity, but

(ii) the interaction does conserve VCT.

(b) Assuming all phases are unity, what other transformation (T or C) is not conserved by this interaction? Compute the effect of T and C on the interaction.

8.3 Construct a field theory with spin zero particles only, in which

(i) 3 particles interact at a point

(ii) 4 particles interact at a point

Assume that parity is conserved. What is the parity of the particles in each case? [To "construct a field theory," it is sufficient to write down £ and H.]

8.4 Show that the following Lagrangian density is invariant under charge conjugation:

C =ip(:r) ^ $ - rnj tp{x) - igi>(x)'y5ip(x)(j)(x)

+ (d^(x)d^(x) - ml^ix)) , where 0 is a Hermitian scalar field with a positive phase under charge conjugation and ip is a Dirac field.

8.5 Consider a field theory with the following interaction:

Ant = ~9 [i>{x)^lP(x)} [tP(x)f^^P(x)} , where g is a real constant.

(a) Prove that £int is Hermitian.

(b) Prove that £int does not conserve parity.

(c) What are the simplest physical interactions described by Cmt?

8.6 Suppose the electron had a static electric dipole moment analogous to the magnetic moment. Write a Hamiltonian density that represents the interaction of the electric dipole moment with the electromagnetic field and prove that it is not invariant under parity.

8.7 Show that

Tu(-p,-s) = (-l)t+3u*(p,s) Tt;(-p1-s) = (-l)i+V(p,a) , where T = Cj5.

Relativistic Quantum Mechanics and Field Theory

FRANZ GROSS Copyright© 2004 WILEY-VCH Verlag GmbH

CHAPTER 9

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