14 The Quantization of Wave Fields The theory of quantum mechanics presented thus far in this book has dealt with systems that, in the classical limit, consist of material particles. We wish now to extend the theory so that it can be applied to the magnetic field and thus provide a consistent ba.9is for the quantum theory of radiation .. ,The quantization of a wave field imparts to it some particle properties; in the case of the electromagnetic field, a theory of light quanta (photons) results. The field quantization technique can also be applied to a 1/1 fIeld, such as that described by the nonrelativistic Schrodinger equation (6.16) or by one of the relativistic equations (51.4) or (52.3). As ,ye shall see in the nonrelativistic case (Sec. 55), it then converts a one-particle theory into a many-particle theory, in a manner equivalent to the transition from Eq. (6.16) to (16.1) or (40.7). Because of this equivalence, it might seem that the quantization of 1/1 fields merely pro- vides another formal approach to the many-particle problem. However, the new formalism can deal as well with processes that involve the creation or destruction of material particles (radioactive beta decay, meson- nucleon interaction), 490 THE QUANTIZATION OF WAVE FIELDS ell This chapter is intended to serve a.'l an introduction to 'l.uum"", field theory.l We start in Sec. 54 ,vith a discussion of the classic:L1 lind quantum equations of motion for a wave field, without specifying the detailed nature of the field. The application to Eq. (6.16) is used as a first example in Sec. 55. Several other particle wave equations (including the relativistic Schrodinger and Dimc equations) have also been quantized but are not discussed here. The electromagnetic field is considered in the last two sections. 54DCLASSICAL AND QUANTUM FIELD EQUATIONS A geneml procedure for the quantization of the equations of motion of a classical system was obtained in Sec. 24. We start with the lagrangian function for the system and verify that it gives the correct classical equations. The momenta canonically conjugate to the coordinates of the system are found from the lagrangian, and a hamiltonian function is set up. The classical hamiltonian equations of motion are then con- verted into quantum equations by the substitution of commutator brackets for Poisson brackets; this gives the change of the dynamical;! variables with time in the Heisenberg picture. We now show how this procedure can be applied in its entirety to a wave field ",(r,t), which we assume for the present to be reaL 2 COORDINATES OF THE FIELD A wave field is specified by its amplitudes at all points of space and the dependence of these amplitudes on the time, in much the same W/l.y ns a system of particles is specified by the positional coordinates qi aud their dependence on the time. The field evidently hf.s an infinite n \lm ber of degrees of freedom and is!tnalogous to a system that (1ousisi,H of lUI infinite number of particles. It is natuml, t,hen, to lIKO tlw ",(r,t) at all points r as coordinates, in analogy wit.h the nates qi(t) of Sec. 24. It is not necessary, however, to pro(:()od in thiK wny. AK Itll alter- native, W() can expand'" in some comnlcte orthonormal Ket of fUllctions I For further discussion, see P. A. M. "The Principles of Quantum l\iechanics," 4t,1l ",I., chaps. X, XII (Oxford, New York, 1958); H. Goldstein, "Classical Meehan- ieR," "hnp. J1 (Addison-Wesley, Reading, Mass., J950) i J. D. Bjorkcn and S. D. Droll, "Hd",\'iviHl.ic: Quantum Fields" (McGraw-Hili, New York, 1965); E. Henley and W. Thin'illl(, "Elementary Quantum Ficld Theory" (McGraw-Hili, New York, 1962); S. H. H<:liwc(,cr, "An Introduction to Relativistic QUtmtum Field Theory" (Harper & Row, N.,w York, 1961); J. J. Sakurai, "Advanced Quantum Mcehanics" (Addison- W(,,,,,IllY, Mass., 1967). • W. lIei",mhcl'l( 1\.11(1 W. Pauli, Z. Physik 66, 1 (1929); 69, 168 (11)30). 1I $ 492 QUANTUM MECHANICS Uk: t/t(r,t) = Skak(l)uk(r) The expansion coefficients ak in (54.1) call be regarded as the field coordi- nates, and the field equations can be expressed in termR of either t/t or the ak. We shall use the wave amplitudes at all points as the field coordi- nates in this Rectioll. It will be convenient for some of the later work to make use of the coefficients ak. TIME DERIVATIVES It is important to have clearly in mind the meaning of time derivatives in classical and quantum field theories. In classical pl1rtici<" theory, both total and partial time derivatives were defined in cOlllleetioll wifb :1 function F(qi,P."t) of the coordinates, momenta, and time; theBe deriva- tives are related through Eq. (24.22). Similarly, both dcrivatjv('B were defined for a Heisenberg-picture operator and related to each ot,!lOr as in Eq. (24.10). In classical field theory, t/t(r) is the analog of q" and the only time derivative that can be defined is af/i)t; we refer to it a.N f in analogy with qi in the particle case. Thus, in the claHsieal hamil- tonian equations of motion of the field (54.19) beIO\'l], we illtcrpl'oI, ,f, and also 'Ii', as partial time derivatives. However a functional P(t/t,1f,t) can depend explicitly on the time as well as Oil the field, so that. it, i", important to distinguish between dF/dt and aF/at in (54.20). The same situation appears ill quantum field theory. No di:.;lill('.- tion can be made between dift/dt and aift/at, and both are referred 1.0 as ,f. On the other hand, a HeiRenberg-picture operator can depend on the time, and the distinction between the two time derivative;; must be made in Eq. (54.2:3). CLASSICAL LAGRANGIAN EQUATiON The lagrangian L(q;,rj;,t) used in Sec. 24 is a fUllction of the time and a functional of the possible paths qi(t) of the system. The actual paths are derived from the variational principle (24.17) : o(t, L dl 0 it. =0 By analogy, we expect the field lagrangian to be a functional of the lipid alllplitude f(r,!). It can usually be expressed as the integral over all of a lagrangian L: • f, JL(o/I,Vf,,f,t) dar (54.2) t Ill'! 1·('IIIi1.rlwd above, is at/t/at or dt/t/dt.' The appearance of Vt/t HK HII ItrIl.UIll()II(. of L is a consequence of the continuoUH dependence of t/t [I 493 THE QUANTIZATION OF WAVE FIELDS on r (continuously infinite number of of freedom); higher deriva- tives of ,y could also be present but, (10 tlo!, scem to ariHe in problems of physical interest. The variational Umt eorrcHponds to (24.17) is (54.3) f «,iL) dlll'l' 0 L dt = Il f' JL dt d'r to the restrictions where the infinitesimal variation o,y or ift (54.4) il,y(r,t 1) = o,y(r,i 2) = 0 If L has the form indicated ill it.s variation can be written aL aL ((lift) ill" (54.5) oL a,y o,y + L iJ{"iI,y/ il.t) f, iI.1' I ,It/: of xyz where the summation over ;/:, 1/, Z 11111""';' sum of three terms with y and z substituted for x. Now bt/: iI, II", dil'ft;rellce between the original and varied'" and hence is 1,1)(' tilll!' d'·J'lv:d.ive of the variation of ,yo This and the similar expressioll for .\{ .hI" ! (1,1'\ elm be written iI . a ilf) iI.!' (bift) Oif; = -(o,y)at f, (fI,1' Equation (54.3) then \lecollwH at ,I (o,y) dt dar = 0 (54.0) raL ,1.1' (llift) 1 La,y o,y + L clift {]. J X'IIZ The summation terms ill (I'd 1'0) t::ttl be integrated by parts with respect to the space COortiill:l,I,('H, 1.111' ,'\lrrae(' terms vanish, either because ,y falls off rapidly enough ILl. illiillik diNI.:I,llee, or because,y obeys periodic boundary conditions at Hw wnllH "fIt Inrge but finite box. The last term of (54.6) can be integrated hy I'll.d:; wit.h respect to the time, and the boundary terms vanish bee:J,\IHI' nr (£..1.,1). Equation (.')4.6) ean therefore be written 3 (54.7) (12 a (?!:;)} o,y di d r = 0 J{aL \' a r ilL \ ill. il,y )11 04' 1... ax L5«'ift/ il.r ) xyz Since (54.3) is valid for an arhitrnry VII,!'I;I.uon o,y at each point in space, Eq. (54.7) is equivalent to the dilTt·r"lll.i;d equation (54.8) al:. ii (ilL) 0 ao/; f:. ax o(o,ylax) ill at/: Equation (54.8) is the classical iie\d ,·/lII:I.LlIlI! derived from the lagrangian density L(4',V4',.j;,t). 494 QUANTUM MECHANICS FUNCTIONAL DERIVATIVE In order to pursue further ·the analogy with particle mechanics, it is deOlirable to rewrite Eq. (54.8) in terms of L rather than L. Since the aggregate of values of y; and J; at all points is analogous to the qi and Ii, of particle theory, we require derivatives of L with respect to y; and J; at particular points.