Symmetries I

This is the first of two chapters devoted to the discussion of symmetries in field theory. Symmetries are important for two reasons. First, if a symmetry is observed in nature, then the Lagrangian must be invariant under the transformations which describe the symmetry, and this imposes a constraint on the form of the interaction Lagrangian. Second, any properties which can be shown to be the consequences of an exact symmetry must be exact results, regardless of the details of the interactions. It is very difficult to obtain exact results in any other way.

In this chapter we begin with Noether's theorem, which shows that there exists a conserved quantity associated with every continuous symmetry and also how to find it. We then study the discrete symmetries which the Lagrangian may satisfy. Discrete symmetries are are single, isolated transformations with no infinitesimal form, and the three which are of great importance in field theory are parity (space inversion), charge conjugation (the interchange of particles and antiparticles), and time reversal (or, more correctly, the reversal of the direction of motion). After discussing each of these in turn, we discuss the famous PCT theorem, which states that the product of all three of these transformations must always be a symmetry of the system, even if individual members of this set are not symmetries. The non-trivial consequences of PCT invariance are among the most secure predictions of field theory. In Chapter 13 we continue, with a discussion of non-Abelian gauge symmetries, chiral symmetry, and spontaneous symmetry breaking, all ideas of paramount importance in modern physics.

8.1 NOETHER'S THEOREM

The foundation of all of our discussion of continuous symmetry will be Noether's theorem, which we now state and prove.

Theorem: For every continuous transformation of the field functions and coordinates which leaves the action unchanged, there is a definite combination of the field functions and their derivatives which is conserved (i.e., a constant in time).

8.1 NOETHER'S THEOREM 207

Fig. 8.1 Illustration of the active translation of a scalar function through a distance a in the jr-direction.

Fig. 8.1 Illustration of the active translation of a scalar function through a distance a in the jr-direction.

The infinitesimal transformations of the coordinates and fields will be written where A and fl are known functions of x and e* are infinitesimal parameters which describe the transformation. Note that the range of i is not specified. In particular, i need not range from 0 —> 3 and el may not be a four-vector.

Examples of Continuous Transformations before we proceed with the proof of Noether's theorem, we give some examples of transformations. (Review Sec. 2.7 and 5.8.)

Translations in space and time. The translation of a scalar function <j>(x) through a distance a is illustrated in Fig. 8.1. Here where in this case v [the i of Eq. (8.1) ] runs from 0 to 3 and av is a four-vector. If we translate both the function and the coordinates, everything is unchanged, so that x'» = x" + \»i(x)ei tfa(x') = ipa(x) + nai(x) S

In this case:

translations.

where ößi is the conserved current density

dip0

dx"

Of course this "current" density does not necessarily have anything to do with the usual electric current; the term used here refers to any general four-vector quantity which satisfies a local conservation law.

If the fields fall to zero at spatial infinity, we obtain the constant of the motion by integrating the four-divergence of O over an infinite slab bounded by any two times t\ and a^+üi integrates to zero

Integrating the first term from ti to t2 and noting that t\ and t2 could be any two times permit us to conclude that

This quantity is the conserved charge (i.e., the total time component of the conserved current), and the proof is now complete.

A nice feature of this proof is that it leads to the explicit construction of the conserved quantity, so that we know what is actually conserved as a consequence of the continuous symmetry. We now illustrate the consequences of this important theorem in a number of special cases.

8.2 TRANSLATIONS

Noether's theorem provides the ultimate justification for the definitions of the energy, momentum, and angular momentum operators, which we introduced in Chapters 1 and 2. To obtain these quantum field operators we take the classical c-number quantity (8.21) and substitute the quantum field operators for the classical fields. The order of the terms now matters, and in order to insure that the ground state has zero energy, momentum, and angular momentum, we normal order the terms as discussed in Chapter 1.

As an example, consider the implications of translational invariance for the Dirac theory. From Eq. (8.4), AM„ = and Vlau — 0, and the conserved current density, which is called the stress energy tensor, is dC dipo r\ f 9i)a d[d,^

The conserved "charges" are therefore

As a consequence, we obtain four conserved quantities associated with invariance under translations in any of the four space-time directions:

which is consistent with the usual definition of the Hamiltonian. (b) Space translations (f — ¿ = 1,2,3):

dC dip a

For example, the momentum operator in the Dirac theory is

I* = -J d3r l-.0 dip 1 dip . o ' 2 â? " 2 fl**7 *

k,s d and the orthogonality of the Dirac wave functions give

where we integrated by parts. Substituting the field expansions iP(r, t) = £ —L= lu(k, s)bktSe+ v(k, s)4,seifc'4

and using

pl = Efcî Ra-+ d-fc.-.difc,-} = £fci Rx« - d^dls} ■

8.3 TRANSFORMATIONS OF STATES AND OPERATORS

Hence, the infinitesimal change in any operator O under a symmetry transformation is given by the commutator of the generators of the symmetry group with the operator. This is a very general relation which we have used several times before.

If the symmetry in question is also a symmetry of the Lagrangian, then the transformations (8.30) will commute with the Hamiltonian, and the generator of the transformations will be a constant of the motion. In this case, the generator is the conserved "charge" associated with the symmetry, given in Eq. (8.21). In lieu of a proof of this statement, we will show that it is true for the translations.

The finite translations are constructed from the generators of translations, which are the momentum operators, in the following way:

Note that phase in the exponent is +i, instead of the —i used for the time translation operator; this is consistent with the covariant scalar product Ht — PV (and agrees with our definitions in Chapter 1; recall Prob. 1.2). For translations of a scalar operator O = O(x) = O'(x'), with x' = x + a, and we have

For infinitesimal a this reduces to

Equation (8.37) is an example of the general relation (8.34) and is consistent with Eq. (1.38). To see this, note that the time translation operator was U(At) = e-iHAt (for jj independent of time) and that therefore (1.38) could be written

Changing At -At and noting that U(-At) = £/t(Ai) permit us to rewrite (8.38) as

which agrees with (8.36).

We will return to the discussion of continuous symmetries in Chapter 13, where we discuss gauge invariance and chiral symmetry. Now we turn to a discussion of the three discrete transformations of great importance to the construction of interactions: space inversion, charge conjugation, and time inversion.

8.4 PARITY

We start with a discussion of space inversion, realized by the parity transformation. Assume that there exists a unitary operator V which transforms the spatial coordinates from r —> —r. Its representation on the Fock space of particle states will be denoted

Then the field operators transform according to Eq. (8.32),

Since parity is space inversion, the transformation law for the fields is

V4>(r,t)V* =<f>' = Ti%<t>(-r,t) Vip{r, — ip' = r)^S(P)ip{—r, t) — r]^*y°ip(—r, t) VA^r, t)pt = = A{P)"uAv{-r, t) ,

where the phases 77^ and 77'' are the intrinsic parities of the scalar and spinor fields and

0 0

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