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. These are called functional derivatives and are denoted by i!LjiN and i!LjilJ;. Expressions for them can be obtained by dividing up all space into small eells and replacing volume integrals by summations over these cells. The average values of quantities such as y;, Vy;, and J; in the ith cell are denoted by subscripts i, and the volume of that cell by OT,. Then

L(Y;i, (Vy;)" .p" tJ OTt appronelwx Ii in the limit in which all the OT, approach zero. In ximilar ftu:lhioIl, the t integrand in Eq. (54.6) or (54.7) can be r

f - a LOY;i Or. + f of, OT, where the variation inL is now produeed by independent variations in the Y;i and the .pi. Suppose now that all the OY;i and ofi are zero except for a particular oY;j. It is natural to relate the functional derivative of L with respeet to Y; for a point in the jth cell to the ratio of oL to oY;j; we therefore define

ilL r oL aL "a [ aL ] (54.9) ily; ...oY;j OTj = ay; - L. aX iJ(oy;jox) xV' Similarly, the functional derivative of L with respeet to f is defined by setting all the OY;i and of, equal to zero except for a particular ofi: iJL ilL lim. oL (54.lO) iJf .Irj->O ofj Orj 'af

Here again the point r at which the functional derivative is evaluated is in the jth celL Substitution of 0';4.9) and O';4.lO) into (54.8) gives

o iJI, aL = 0 (54.11) at iJ.p iJy; whieh dosely resembles the lagrangian equations (24.18) for a system of partieltlH. I THE QUANTIZATION OF WAVE FIELDS 411

CLASSICAL HAMILT.ONIAN EQUATIONS The momentum canonically conjugate to 1/;j can be defined as in particle mechanics to be the of oL to the infinitesimal change o,h when all the other 0"" and all the OY;i are zero. We thus obtain

p. = Or 12) J o1/;j J iJ'" j

It follows from (54.11) and (154.12) that

Pj Or; (.14.13)

The analogy with Eq. (24.19) then gives for the hamiltonian

H = LPi"', L = L"'i Or, - L (54.14) i i iJ1/;.

We 'write H as the volume integral of a hamiltonian density Hand assume that the cells are small enough so that the difference between a volume integral and the corresponding cell summation can be ignored; we then have iJL ilL H = Hd3r H= L 71'==- - J iJ'" il", The approximate hamiltonian (54.14), with the relations (.14.12) and (.14.13), can be manipulated in precisely the same way as the hamiltonian for a system of particles. Instead of showing this explicitly, we now work with the true field hamiltonian H given in (54.15), which is a functional of 1/; and 71' from which", has been eliminated. The classical hamiltonian equations of motion will be derived without further recourse to the cell approximation. The variation of L produced by variations of 1/; and '" can be written, with the help of (54.11) and (.14.15),

3 i oL = J 01/; + 0"') d r J(ir01/; + 71'0",) dar = J[0(71'''') + iro1/; - "'07rJ dar = oH + aL + f(iro1/; - ",071') dar (54.16) r I '\ The variation of H produced the corresponding variations of 1/; and 71' :\ can be written

oH = (iJH 01/; + iJH 071') dar (54.17) J iJ1/; iJ7r

496 QUANTUM MECHANICS

It follow" from (,11.1'11('" discussion of functional derivatives that {,W i)// LilaH ilt/; ilt/; u;r; li(aNax) rut, (54.18) illl ./11 aH ,111 illf '-' ax XUt

('Ollllllll'i:;"11 or Eqs. 16) and (.54.17) for arbitrary variations at/; and r.1l 1.111'11 the field equations in hamiltonian form:

&// ilH if; 11- (.54.19) i-l1r at/;

The hamiltonian equation for the time rate of change of a functional '" of t/; alld 7f can now be found. We express F as the volume integral of II,,· functional densij,y F(t/; ,11", t) , which for simplicity is nNsullwd not to depend explicitly on the time or on the gradientR of t/; or 7f. The foregoing analysis can be used to show that

dF = + J+ aF + J(iJF aH _ ilF ilH) d3 at at/; iJ11" iJt/; r

= aF + {F,H} (54.20)

This equation also serves t.o define the Poisson bracket expression for two functionals of the field variables. The right side of Eq. (.54.20) is not changed if F also depends on Vt/; or Vrr Prob. 2), It is apparent from (54.20) that H is a constant of the motion if it does not depend explicitly on the time; in this case, H is the total energy of the field.

QUANTUM EQUATIONS FOR THE FIELD The analogy between particle coordinates and momenta qi, Pi and the cell averages t/;i, Pi suggests that we choose as quantum conditions for the lipid

[t/;"t/;,l = [Pi,Pj ] = 0 = ihOij (54.21)

1111.11.1114 that we have converted the wave field from a real numerical I'lIlIdioli 10 :I lH'rmitian operator in the Heisenberg picture. W(, IIOW that the cell volumcs are very small. Then Eqs. I) 1.11.11 Ion with the help of (54.12) and in terms of

•• THE QUANTIZATIQN OF WAVE FIELDS 417

1/1 and 7r: [1/1 (r,t),I/I(r' '"' ['IT(r,t) ,7r(r' =0 [I/I(r,t),7r(r' = t'lio(r,r')

where o(r,r') = 1/0T' ir r nnd r' are in the same cell and zero otherwise. The function o(r,r') haM the property that ff(r)o(r,r') d3r is equal to the average value of J fOl' the cell in which r' is situated. Thus, in the limit in which the cell volumes approach zero, Il(r,r') can bc replaced by the three-dimellsional Dirac 0 function 1l3 (r r'). The Quantum conditions. for tho canonical field variables then become

[I/I(r,t),I/I(r',t») = [7r(r,t),7r(r',t)] 0 (54.22) [1/1 (r,t) ,7r(r' = ihll 3(r

The equation of motion for any quantum dynamical variable F is obtained from Eq. 10) or by replacing the Poisson bracket in Eq. the commutator bracket divided by ih.

dF aF + 1 [F H) (54.23) dt at ' The commutator bracket can be evaluated with the help of (54.22) when explicit expressions for F and H in terms of 1/1 and 7r are Thus Eqs. and (54.23) completely describe the behavior of the quantized field that is specified by the hamiltonian H.

FIELDS WITH MORE THAN ONE COMPONENT Thus far in this section we have dealt with fields that can be described a single real amplitude. If the field has more than one component 1/11, 1/12, ... , the lagrangian density has the form L(I/Il, 4.1/11, ,Itt, 1/13, 4.1/12, ';'2, ... ,t). Then if each of the field components iH vlLried inlillpcnd. ently, the variational equation (54.3) leads to 1m eqllll,tloll of UIIl form (54.8) or (54.11) for ench of 1/11, 1/12, . . .. A mOnl(lllt,lIlll conjugatc to each 1/1, can be defilled lUI ill I';q. (M. The hamiltonian -density lU11l the form H = L7r,';', - L (54.24)

and the hamiltonian equations COlllliRt. of It pail' like (M.19) for each 8.

I Equation (54.23) is unchanged, and tho commutation relations "I are replaced [I/I.(r,t),1/1",(r',t)] = [7r.(r,t),7r•• (r',t)J = 0 (54.25) [I/I.(r,t),7r,.(r',t)) = ihll..,o3(r - r') 498 QUANTUM MECHANICS

COMPLEX FIELD Thus far we have dealt with fields that are real numerical functions in the classical case I1nd hermitian operators in the Heisenberg picture in the quantum case. A different situation that is of immediate interest for the nonrelativu.,t,j(l Hehrodinger equation is a single ifi field that is complex or nonhermhil1ll. ___c____c_____ .::·c". In t.he e1mll'lienJ case we can express ifi in terms of real fields "'1 and ifi2 as 1 '" e':"! +iifi2) ifi* 2- (ifi1 - iifi2) (54.26) Wo HImI\' Iir'HI, tim!, the lagrangian equations of the form (54.8) obtained hy ilI«iopmllJolII, variation of ifi and ifi* are equivalent to those obtained vnr'ial.ioll 01' ifil ILnd ifi2' It follows from (54.26) that ;/ ::ll - i ) & = 2-t + i ilifi &ifi1 &ifi2 &ifi1 &ifi2

'1'11111'1 LIIl! ifi, ifi* equations are obtained by adding and subtracting the ifi1l ifi. III I-limilaf fashion, the classical momenta canonically conjugate to ifi II.lld ifi"" arc seen to be

11' ::l- 1(11'1 - i1l'2) if = 2-'(11'1 + ill'Z) (54.27)

'1'111' Il0(lOlld momentum is written as if rather than 11'* in order to emphasize UIL\ f,l.d 01111. it. is defined as being canonically conjugate to ifi* and is not 1I1'III'Illlll.l'ily I,he complex conjugate of 11'. Indeed, as we shall see in the twx!. Hl'l'lcioll, if is identically zero for the nonrelativistic Schrodinger ('l1uul.ioll. Iinwever, whenever the lagrangian is real, 11'1 and 11'2 are inde- 11"111 h1ll I, or (llIdl other and if = 11'*. In this case 11'1"'1 + 11'2"'2 = + 11'*"'*, unci 1.111' IlItlnil1.onian is unchanged. '1'111' (',(IfT()HpOnding quantum case is obtained from the commutation (fd.::lii) with 8 = 1, 2. If 11'1 and 1l'2 are independent, then all \mlt',. of vHl'illhles except the following commute: lifi(r,I),IT(r',l)] [ifit(r,t),1ft(r',t)] = ihQ3(r r') (54.28)

51111QUANTIZATION OF THE NONRELATIVISTIC SCHRt)DINGER EQUATION

It tir'l-Il. mmrnple of the application of the field-quantization technique dllvplopOll ill Lhe preceding section, we consider here the quantization of I.Iw lIolIl'olnt.iviHLie Rchrodinger equation (6.16). The application implies !,Imt. Wil 11,1'0 Lroai.itlfJ: Eq. (6.16) as though it were a classical equation that dOH(ll'ihlll-l tlw llloti()1l of some kind of material fluid. As we shall see, the .. THE QUANTIZATION OF WAVE FIELDS 4.. resulting quantized field theory ie; equivalent to a many-particle Schl'il- dinger equation, (W.I) 01' (·10.7). For this reason, field quantization is often called second (tluwli;:lllion; this term implies that the transition from classical quantization.

CLASSICAL LAGRANGIAN AND HAMILTONIAN EQUATIONS The lagrangian dmlHiLy may he taken to be /t,2 L ?'hljl"'" , V.j;* . vljI V(r,t)ljI*ljI 2m (55,1)

As shown ai, til(, plld of the preceding section, ljI and ljI* can be varied separatdy 1,0 ubl.aill the lagrangian equations of motion. The equation of tho f(ll'll! ([,.1.1") Umt results from variation of ljI is

2 11. ill"," , , \i2lj1* V(r,t)ljI* 2m +

whiel! iH Ul(' wmplex conjugate of Eq. (6.16). Variation of ljI* gives Eq. (Ii.· .. ·

{.2I. \i2lj1 + V(r,t)ljI 'l'm (55.2)

'1'111, IIlOlllontum canonically conjugate to ljI is ilL ?r thljl* <"I", (55.3)

How(w(,1' "'. dOI'H not appear in the lagrangian density, so that i ie; identil,ally 1,"1'0. It therefore impossible to satisfy the of tho conuHutllliOIl rdatjoflS (54.28) (or the corresponding classical I'O[:-;SOII- bradwL n'ild,ioll), so that ljI*, i caIUlOt be regarded as a pail' of conj lI!l;lI.to They can easily be eliminated from the hamilLonian since?r" Ill'vt'r and Eq. (55.3) gives ljI* in terms of ?r.1 Tho Imllliitollinll density is itt i H L -V?r'VljI - VlI'l/t 2m II,

I Tlw "()tWIIlHi"" IJlltl " ""11 identified with >/;. is related to the appearance of only th" firH!. ..nl.·, 1.111'" d..... vlltive in the wave equation (55.2), since in this case'" can be expr""A"d III 1."IIIIiI of >I, nlld ;!,s space derivatives through the wave equation. If the wave "'IlIltl.lo" IH of H",,,,,,,I order in the time derivative, >/; and", are independent; then 'If ill ..1'111,1,,'<1 I.., J" I'Id.lll'f thau to >/;., and both !/I, 1<' and >/;., ii' are pairs of canonical variabl"H. '1'1". lIollrl'l",l.;v;HI,;e Hchrodinger equation and the Dirac equation are of the fOl'lIIer !,.VI"', witor.,,,,,, th" relativistic Schrodinger equation is of the latter type. SOD QUANTUM MECHANICS' The hamiltonian cqUlttjOllS of motion obtained from (54.19), with the I of (54.18). arc!

i £h if; - Vljt + '12ljt ft, 2m \ . i Viii, '12 Il' "'" fI, '11" 2m 11" 1&

Tlw or l.linHC (jquations is the same as (55.2), and the second equation, t,oll;ol.hm· wil.h (55.a), is the complex conjugate of (55.2). We have thus AhoWIl, 1'1'O1l1 the point of view of classical field theory, that the lagrangian (11!).1) and the canonical variables and hamiltonian derived from il. Itl'(l ill agreement with the wave equation (6.16) or (SS.2).

QUANTUM EQUATIONS

as the hamiltonian, (54.23) as the equation of motion, and linolt. of (54.28) as the quantum condition on the wave field. Since ljt is now a Heisenberg-picture operator rather than a numerical function, ljt* is replaced by ljtt, which is the hermitian adjoint of ljt rather than its complex conjugate. Further, as remarked above, the Heisenberg-picture , operators ljt, ljtt have no explicit dependence on the time, so that their " equations of motion are given by (54.23) or (24.10), with the first term \ on the right side omitted and #/dt on the left side identified with "'. The hamiltonian is conveniently written with replacement of 7r by ihljtt and becomes

H f Vljtt. vljt + Vljtt.p) d·r (55.5)

_ and (22.16) then shows that H is hermitian. Ul;t,Hlol""U hamiltonian given in (55.5) is the operator that represents the total energy of the field; it is not to be confused with the operator (23.2), which is the energy operator for a single particle that is described I by the wave equation (6.16). We have as given no explicit repre- j sentation for the new operators ljt and H and therefore cannot say on what they might operate. The choice of a particular representation is !lot necessary so far as the Heisenberg equations of motion are concerned hut is for the physical interpretation of the formalism that we .. lI;i ve irtter in this section. ThH commu.tation relations are =0 r/) THE QUANTIZATI.QN OF WAVE FIELDS lot

The omission of t from ['IH' tLl'l(llIIlOli t or the field varhthl{Jf; implies hoth fields in a com III II [,ttt,OI' 1I1'IWI(llI, 1'1'1'01' t,o Lhe same time, In accord- ance with the earlinr (liJolflltl'!l'!ioll, I,lio oqul\l,ioIl of motion for f is

rf,H)

'(1'f " V'f' dil/,' (5.5.7) = [f, J ' I If, J V'ft'f'

where primes indi

fV'(fft'f' fl'V/f) Itnl" .IT'(IPf" dar' JV'f' - r') (Pr' (55.8)

f eommutes with V, ",hill!! iii II, IIIlllwI·jmtl function. Evaluation of the first term Oil i.ho right JoIid •• of (MI.7) iH Hilllplified by performing a partial integration on f'(1fl' • Vf' dil" 1,0 ohLni II f f t 'V'2f' dJr'; the surface terms vanish bccaui-)() f Ilit.lw!' vlI,lIilllll'H /Lt, infinity or obeys periodic houndary conditions. W (l UtilI'! oill-Itill

[f,JV'ft' • '(1'f (til,,11 - 'f,N"V'2f' = - tinl" Jc'V'2f') /)3(r - r') d 3r' -V2f (55,9)

Substitutioll or (oIio.H) ILIlIi (MI.IJ) illl,!) (55.7) yields Eq. so that the eqmd,jullK ubl,ldunli 1'1'0111 (,I".NNionl nnd quantum field theories agree. A similu.r ealOlllntioll JoIhOWH 1,llId, UII' oquati'OIl ihJ;t = [ft,H) yields the hermitian adjoin!, of II)q. it mm also be seen directly that this equation is tho hormil,iltll Itlijoillt of the equation [f,H) so long as H is hermitian. If V is inUep()lIdollt, of t, /I ImH 110 explicit dependence on the and Eq. (54.2:J) HhoWH 1.11101, 1/ ill 11, of the motion. Thus the energy in the field iii UIIIlHI,UIlL AlIoLller interesting operator is

N = N t fd 8r

" (55.

The commutator of N wit,h i./Ill V pnrl. of II can be written as JfV'(ftfft'f' - ft'f'f'f) d,:lnPr'

• 502 QUANTUM MECHANICS

With the. help of (.55Jj) the parenthesis in the integrand is ";1",,,,t',,,' - ",t'",'",t", = ",t[",t'", + (P(r - r')]",' _ ",t'",'",t", ",t'",t",'", + ",t",'oJ(r - r') _ ",t'",'",t", + ",t",'o3(r r') _ ",t'",'",t", =0 since tho Il i'lilldion vanishes unless r r'. A similar but slightly more calculation shows that 1",1"" v'",t' . V'",'] [",tv'",' (v'",t')",] . V'/lJ(r - r') TJUI dOllhle integral of this over rand r' is zero. Thus Eq. (55.10) shows Lhal, N iH !1 constant of the motion. 1(, e:w also be shown that the commutator brackets in (55.6) are COIIH\'ItIlt,H of the motion, so that these equations are always valid if they ILI'O nt, n particular time. 1,

THE N REPRESENTATION We now specialize to a representation in which the operator N is diagonal. Since N is hermitian, its eigenvalues are real. A convenient and general way of specifying this representation is by me!1nS of an expansion like (54.1) in terms of some complete orthonormal set of functions Uk(r) , which we assume for definiteness to be discrete. We put ",(r,t) = 2: ak(t)uk(r) "'t(r,t) 2: akt(t)u:(r) (55.11) k k where the Uk are numerical functions of the space coordinates and the ak are Heisenberg-picture operators that depend on the time. Equations 11) can be solved for the ak: I, ak(t) Ju:(r)",(r,t) d 3r akt(t) JUk(r)",t(r,t) dar Thus, if we multiply the last of the commutation relations (55.6) by u:(r)ul(r') on both sides and integrate over rand r', we obtain

[ak(t),a/(t)] JJu;(r)ul(r') 83(r - r') d3rd3r' = Ilkl (55.12) of the orthonormality of the Uk. In similar fashion, it is apparent I.hnl. ltk and al commute and that akt and alt commute, for all k and 1. HuhHtiLution of (55.11) into the expression for N shows that t N='2, where Nk . (55.13) k

11. iM nu,.4ily thnt each Nk commutes vrith aJl the others, so that they call ho dilt/l:ollnlizmi Himultaneollsly. 101 THE QUANTIZATION' OF WAVE FIELDS

CREATION, DESTRlI.CTION, AND NUMBER OPERATORS T The commutatioll relaj,iollH for the operators ak and ak woro solved in Sec. 21) ill oOlllllidion the harmonic oscillator. There it was found that tlw 8olut.ion of (25.10), in the representation in which ata is diagonal, eO!lHiHj,s of the matrices (2.'5.12). It follows that the states of the qUltflLized field, in the representation in which each Nk is diagonal, are the kett! (55.14) inl,n2, ... nk, ... ) where each nk is an eigenvalue of Nk and must be a positive integer or zero. We also have the relations aklnl, nk, ) = nk!l n l, ... nk - 1, ... ) ) (n. + 1)*\n!, ... nk + 1, ... ) (55.15) a.tlnt, nk, Thus akt and ak are called creation and destruction operator8 for the state k of the field. The number operator Nk need not be a constant of the motion, although we have seen from Eq. (55.10) that N = :zl"h is a constant. The rate of change of Nk is given hy ihNk [akta.,II] where H is obtained from (55.5) and (55.11):

H a/al J Vui • VUl + VU;UI) dar

2 (55.16) = a/al JU;'" ( - :;. \7 + v) 11.1 dar

It is not difficult to show from (fiIU2) that a particular Nk is constant if and only if all the volume intogml,; in (55.16) arc zero for which either j or l is equal to k. These int,ogmlH are just the matrix elements of the one-particle hamiltonian (23.2), 1'40 I.h!tt the necessary and sufficient condi- tion that Nk be a constant of !;lw motion is that all such off-diagonal elements that involve the state Uk be zero.! The case in which the Uk are eigenfunctions of (23.2) with eigen- values Ek is of particular illterl!iiL The integrals in (55.16) are then

E10 jh and the field hamiltonian IW(:OU1CH .. (55.17) H LaktakEk LivkEk k k This particular N representation ill t,}10 one in which H is also diagonal;

1 ThiH for the quantized field is dOR"ly related to the corresponding result, containml in Eq. (35.5), for the one-partido prol)llbility amplitude. 504 QUANTUM MECHANICS the kef, In" . Ii/".,,) has j,he eigenvalue 'J:,nkEk for the toj.al energy OPOI':!'I.OI' fl. I t, it-! npl'llH'1I t. t.hat all the are constant ill thi" case.

CONNECTION WITH THE SCHROOINGER EQUATION

Thn <111/1.11 Li ,/,.,01 li.·ld UH,or,Y is closely related to the many-particle Sehrcid- iltfJ:4\t· cIll'litLioll in Sec, 40. If the Uk are eigenfunctions of the Ol!t'-I'II.I·Lled.. IlHluill,olliall (23.2), the field theory shows that Holill,iolll1 for which the number of particles n, in the kth state is 11

(iOtlHI.alll. I'mii Li VI' ilIi-eger or zero, and the energy is 'J:,nkEk • Each solution (,all 1,,- .h'H(\I'il.pd by ket ... nk, ., . these kets form a complete Ol'l.hollOI'lIIII,1 HnC, alld there is just one solution for each set of number" III, ()1I the other hand, a stationary many-particle wave function Iii", 1111' .p ill I';q. (-iO.1) can be written a.s a product of olle-particle wave fUlidiollH if there is no interaction hetween the '1'114' linolLI' combination of such products that is symmetric with of any of pa.rticle coordinates can be specified uniquely the number of particles in each state. Again, the number of in eaeh state is a positive integer or zero, and the energy is the Hum of alt the particle energies. We see then that the quantized field theory developed thus far in this section is equivalent to the Schrodinger equation for several non- interacting particles, provided that only the symmetric solutions are retained in the latter case. We are thm; led to It theory of that Einstein-Bose statistics, It can be shown that the two theories are completely equivalent even if interactions between narticles are taken into account. l It is natural to see if there is some way in which the quantized-field formalism can be modified to yield a theory of particles that obey Fermi- Dirac statistics. As discus;,;ed in Sec. 40, a system of such particles can be described by a many-particle wave function that is antisymmetric with to interchange of any pair of particle coordinates. The required linear combination of products of one-particle wave functions can be specified uniquely by stating the number of particles in each stat,e, pro- vided that each of these numbers is either 0 or 1. The desired modifica- bon of the must, limit the eigenvalues of each nnprfl.tor Nk to 0 and 1.

ANTICOMMUTATION RELATIONS '\' A review of the foregoing theory shows that the conclm;ion that the values of each Nk arc the positive arid zero stems from the com- 111lllation relations (.55.12) for the ak and akt. Equations (55.12) in turn

I H(',' W. Heisenberg, "The Physical Principles of the Quantum Theory," App., see. 11 (University of Chicago Press, Chicago, 1930). j THE QUANTIZATION OF WAVE FIELDS , " .. arise from the commutation relations (55.6) for"p and "pt. Thus we must modify Eqs. (55.6) if we are to obtain a theory of particles that obey exclusion principle. It is reasonable to require that this modification be made in such a way that the quantum equation of motion for"p is the wave equation (55.2) when the hamiltonian has the form (55.5). It was found by Jordan and Wigner1 that the desired modification consists in the replacement of the commutator brackets [A,B] == AB - BA in Eqs. (54.22) and (55.6) by anticommutator brackets

[A,BJ+ == AB,+ BA This means that Eqs. (55.6) are replaced by

["p(r),Hr')]+ "p(r)"p(r') + "p(r')"p(r) = 0 ["pt(r),,,pt(r')l+ = "pt(r)"pt(r') + "pt(r')"pt(r) = 0 (55.18) ["p(r),,,pt(r')l+ = "p(r)"pt(r') + "pt(r')"p(r) = a3 (r - r') It then follows directly from Eqs. (55.11) and (55.18) that [ak,ad+ = aka, + a,a", = 0 [akt,a,tl+ = a.l,ta,t + a,takt ... 0 (55.19) [ak,a/]+ = akazt + a/ak "" thl • We define Nk = a.ta", as before and notice first that each Nk com- mutes with all the others, so that they can be diagonalized simultane- ously. The eigenvalues of Nk can be obtained from the matrix equation Nk2 aktakaktak = akt(1 - aktak)ak = aktak = Nk (55.20) where use has been made of Eqs. (55.19). ,If Nk is in diagonal form and has the eigenvalues nr, ... ,it is apparent thM Nk2 is also in diag- onal form and has the eigenvalUes • . •• Thus the matrix equation (55.20) is equivalent to the algebraic equations

'2 I "2 II. nk nk nk nk for the eigenvalues. These are quadratic equations that have two roots: ,. oand 1. Thus the eigenvalues of each Nk are 0 and 1, and the particles obey the exclusion principle. The eigenvalues of N = T.Nk are the positive integers and zero, as before. The earlier expressions (55.16) and (55.17) for the hamiltonian are unchanged, and the energy eigenvalues are T.nkE k •

1 P. Jordan and E. Wigner"Z. Physik 47, 631 (1928) . ... . 508 QUANTUM MECHANICS

WO lilJ(1 LlI(' effects of operating with ak and ak! on a ket ... ,II", . .) (,Jmt has the eigenvalue 11k (= 0 or 1) for the operator Nt. 'I'lip d('.yir(·" f'('I:diolls would have the form (;j5.25) were it not that It "pril';; of HII<'II ('I!,llIlioIlS (with subscripts added) would not agree with the 1.\\0 .. I' 1':q:l. (f,!).1 W(' tlil'l'l,r..l'!' proeeed in the following way. \Ve order the states k but definite way: 1,2, ... ,k, . . .. Then has the form (55.2.5), except that l1 multiply- iiiI/, "hHI or IIlilltis sign is introduced, according as the kth state is preceded ill II,,· 11',:1111111'.1 order by an even or an odd number of occupied states. W.. 11111" !'I'plae!, the Einstein-Bose equations (55.15) bv the exclusion-

iI" III" ... nk., ...) ,1 nk,. .) (1"1111,, ... nk, ...) = Ok(l .. ,1 nk, ...) (55.26) k-l Ih (1)" Vk L nj i 1 As lm example, we calculate the effect of operating with akal and with Ifl"f. 011 ::lome ket, where we assume for definiteness that the order is such Chat l > k. If each operation is not to give a zero result, both nk and nl ill t.he original ket must equal unity. Operation with akal empties fir8t the tth and t.hen the kth state and introduces a fltctor OIOk. Operation wit,h alak empties the kth state first, so that. Ok is unchanged. But when the lth stat.e is emptied iu this case, there is one less particle in the states below the lth than there was in the previous case, since the kth state is now empty, whereas it was occupied before. Thus the sign of .01 is changed. We find in this way that

akad ... nk •.• nl ••.) = -alakl ... nk ..• rll .•.) in agreement with the first of Eqs. (5.5.19). In similar fashion, it can be shown that Eqs. (55.26) agree with the result of operating with the other two of Eqs. (.55.19) OD any ket. SiDce the aggregate of kets represents all possible states of the many-particle system, they constitute a eomplete sd, and Eqs. ([)f>.19) follow as operator equations from Eqs. (55.26).

561 IELECTROMAGNETIC FIELD IN VACUUM' W.· the methods developed in Sec. 54 to the quantization of II \I' field in vacuum. Since we are not coneerned with II", 1.... 1.1,,·,· di"""N>lion of the material in this section aild the next, see the references I',I,·d '" 1001"01... I. I'ag" 4!)], and also E. :Fermi, Rev. ilIod. Phys: 4, '137 (19:32); L. 1I0H,·"I',·ld .. 11111 / 118L 1l,.lI.l'i Poincare 1,25 (1981); W. Reitter, "The Quantum Theory oj' i!/l.diu.lioll," :\d .. d. (Ox[Ol'(t, New York, 1954).