M V 1STd gx

+ gn [mtfati - W'Mi + g*4npK} - g* - 2V'facfia + ig^r^tpa]

= iglH [-y6n4>a + ifa] 1> + - ig^r^] $ = 0. (13.124)

Now consider the effect of adding an explicit chiral symmetry breaking term of the form

to the linear Lagrangian. Then the axial vector current will no longer be conserved. The new term only affects the equation for 4>ay which becomes

U<t>„ + m2(j)a - 2V'<j>a + gnij>ip = -c ,

and this new equation, when used in the calculation of Eq. (13.124), adds an extra term to giving

The non-conservation is seen to be proportional to the pion field.

To determine c (in particular, to show that it is proportional to m2), consider the effect of this new term on the spontaneous symmetry breaking described in Sec. 13.6. Break the symmetry of the vacuum by choosing ft = 0 and ft ^ 0 as before. Then the new vacuum value of the sigma field is determined from the minimum of the sigma part of the meson Hamiltonian:

H<t>a = \m2<t>l + i\2<t>4a + cfo with a minimum at dH d<t>a rr^ft + A 2 ft + c = 0

•2-1012

Fig. 13.4 The Hamiltonian with a chiral symmetry breaking term. Now there is only one minimum [compare with Fig. 13.2],

The energy curve is now asymmetric, as shown in Fig. 13.4. Assuming c is small, we can solve for the minimum perturbatively by dividing (13.129) by cj>a and, in the small term, setting = v, its unperturbed value. We then obtain the new vacuum value of cj)a, which we will denote by v', from the equation m2 + \2<fl + - = 0 , v which has the solution

This solution is correct to first order in c. As Fig. 13.4 shows, c and v must have opposite signs.

The physical Lagrangian is now obtained by subtracting v' (instead of v) from <j>c

and the new Lagrangian, expressed in terms of s instead of becomes <—>

+ i [d^s + d^K] - \(rn2 + X2v'2) [s2 + <¡>1} A2

- — [s2 + eft}2 - A2v's [s2 + 4>l] - A2v'2s2 + constant. (13.132)

and gives the following specific form for the PCAC relation:

Note that, once again, the non-zero value of the pion mass is responsible for breaking chiral symmetry.

This concludes our discussion of symmetries. In the next chapter we discuss the path integral formulation of quantum mechanics and prepare the way for a discussion of the quantization of QCD.

PROBLEMS

13.1 Charged scalar theory.

(a) Find the current which is conserved as a consequence of global gauge invariance for a charged scalar field theory.

(b) If J^ is the current found in part (a), show that the interaction -J^A^ is not sufficient to insure that the total Lagrangian is invariant under local U(l) gauge transformations. An extra interaction term of the form

is needed. Show that with this extra term the Lagrangian is locally gauge invariant provided the constant A takes a particular value. Find A.

13.2 Consider the scattering of photons from a 7r+ meson, 7 + 7r+ —» 7 + n+ (this is Compton scattering with the pion replacing the electron).

(a) Using the Feynman rules for "scalar QED" given in Appendix B, draw all Feynman graphs which contribute to order e2. Label all momenta and write the correct A4-matrix corresponding to each diagram. Do not simplify your results at this stage.

(b) If M = M^e^'e", where e/ and e, are the four-polarization vectors of the final and initial photons, show that Mconserves current. In particular, prove that kjM„v = 0 = KM^ , where kj and fc, are the four-momentum vectors of the final and initial photons, respectively.

13.3 Redo Prob. 10.8(a) including the new diagram drawn in Fig. 13.3 (consult Appendix B for the Feynman rules for this case, if you need to). How do the results change? Discuss the significance of this calculation. What important physical principle does this problem illustrate?

13.4 Consider a classical field theory of massive neutral scalar mesons, <j>, and massless fermions, ip, described by the Lagrangian density

+\i>{x) [¿7^] ip(x) ~ gi>(x)ijj(x)<t>{x) , where A > 0.

(a) Find the Hamiltonian density for this theory.

(b) Consider the classical, near static case when d^<p(x) ~ 0 and xp(x) ~ 0. Find the value of the field <j> which minimizes the energy for m2 > 0 and for m2 < 0.

(c) Define a new scalar field s(x) = <p(x) — {</>}, where (4>) is the value at the minimum found in part (b), and rewrite the Lagrangian in terms of this field. What are the masses of the scalar and Fermi particles? Discuss and interpret your result.

Relativistic Quantum Mechanics and Field Theory

FRANZ GROSS Copyright© 2004 WILEY-VCH Verlag GmbH

CHAPTER 14

PATH INTEGRALS

In this chapter an alternative formulation of quantum mechanics and field theory, based on path integrals, is presented. The great advantage of this formulation is that it allows us to quantize a theory using only c-number fields, without the need to turn the fields into operators. Other advantages of this approach to quantum mechanics are:

• It provides the simplest, most direct way to obtain the Feynman rules for any field theory. In particular, we can obtain the Feynman rules for QCD (some of which were introduced in Chapter 13) using this method.

• It provides a method for obtaining exact, numerical solutions of strongly interacting field theories (where the perturbation expansion does not work). These methods, referred to as lattice gauge calculations, will not be discussed in this book.

• It provides a connection between field theory and statistical mechanics, which gives insight into the nature of both subjects.

• It provides a general theoretical framework of a systematic discussion of a number of advanced topics in field theory. Among these is the study of renormalization and the appearance of anomalies. Using path integrals one can prove general theorems about renormalization in a comparatively easy way, and anomalies, not discussed in this book, are most easily understood using these techniques.

Because of these many advantages, the path integral approach to quantum mechanics is an essential part of a modern study of field theory.*

In this chapter we introduce the idea of a path integral from a consideration of the propagator and then show how the ¿'-matrix can be expressed as a path integral. To show the power of this approach and to acquire needed experience,

'The use of path integrals for practical calculations was first proposed by Feynman [Fe 48]. For further reading, see Feynman and Hibbs (1965), Negele and Orland (1988), and the "new" books listed in the References.

we then obtain the Feynman rules for <p3 theory. In the following chapter we use this approach to obtain the Feynman rules for QED and QCD. Obtaining the rules for QED serves as another example of how the method works, but the quantization of QCD requires the power of path integrals. Several of the topics developed in subsequent chapters will depend on path integrals for their development.

14.1 THE WAVE FUNCTION AND THE PROPAGATOR

In this chapter we will work in the Heisenberg representation. In this representation the states are independent of time, while the operators depend on time. The operator which will be the center of attention is the generalized coordinate operator, denoted by Q = Q(t). In this section this operator represents the position of a particle, but in subsequent sections the discussion will be extended to fields, which are also generalized coordinates. Since this operator depends on time, its eigenfunctions, denoted by t), must also depend on time. Hence the coordinate space wave function for the state n is ipn(q,t) = (q,t\n)H , (14.1)

where the subscript H reminds us that the matrix element is in the Heisenberg representation. The corresponding Schrodinger states are equal to the Heisenberg states at some time t = t0 Irecall Eq. (1.31)], so that

where U is the familiar time translation operator equal to unity at t = t0. Demanding that the wave function be the same for either picture,

gives us the following relation between the eigenfunctions of the operator Q in the two pictures:

This relation, which will be used shortly, is identical to Eq. (14.2), but it looks different because the time dependence of the eigenfunctions of the coordinate operator is opposite to the usual rule that Heisenberg states are independent of time and Schrodinger states depend on time. From now on the subscript H will be implied but not written as we work in the Heisenberg representation unless explicitly stated to the contrary.

Using the completeness of the states, the wave function at a different position and later time can be written ipn(qf,tf) = (qj,tf\n) = J dqi(qf>tf\qi,ti)(qi,ti\n)

ei[pq-H(p,q)}

Note that the original operator H(P, Q) has been replaced by its equivalent c-number H(p,q). Because the path integral is an integral over all functions q(t) [and p(t)], it is referred to as a functional integral, and this whole approach is sometimes referred to as a functional method for handling quantum mechanics and field theory.

In deriving the path integral for the propagator, we used the Schrodinger equation when we took the infinitesimal time translation operator to be exp(—iHe). However, it is informative to prove directly that the wave function (14.5) satisfies the SchrOdinger equation. Substituting our final answer (14.18) into (14.5) and differentiating with respect to the last time, tn, gives

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