Tus theory of radiation that was set up in Chapter X involved some approximations in its handling of the interaction of the radiation with matter. The object of the present chapter is to remove these approximations and get, as far as possible, an accurate theory of the electromagnetic field interacting with matter, subject to the limitation that the matter consists only of electrons and positrons. Too little is known about other forms of matter, protons, neutrons, etc., for one to attempt at the present time to get an accurate theory of their - interaction with the electromagnetic field. But there exists a precise theory of electrons and positrons, as given in the preceding chapter, which one can use for building up a precise theory of the interaction of the electromagnetic field with this form of matter. The theory must bring in the interaction of the electrons and positrons with one another, through their Coulomb forces, as well as their interaction with electromagnetic radiation, and it must, of course, conform to special relativity. For brevity in this chapter we shall take c = 1. We must first consider the electromagnetic field without interaction with matter. Now in § 63 we set up first a treatment of the field of radiation without interaction of matter. Dynamical variables were there introduced to describe the field, commutation relations were established for them, and a Hamiltonian was found which made them vary correctly with the time. No approximations were made in this piece of work. The resulting theory would therefore be a satisfactory, exact theory of radiation without interaction with matter, were it not for one feature in it, namely our taking the scalar potential to be zero. This feature spoils the relativistic form of the theory and makes it unsuitable as a starting-point from which to develop a precis¢ theory of the electromagnetic field in interaction with matter. We must therefore extend the treatment of §63 by leaving A, general and bringing it into the work along with the other potentials A,, A,, Aj. Thus we shall have the four 4, and they will satisfy, as the generalization of (62) of § 63, ; A, = 0, dA,,/Ox, = 0. (1), (2) For the present we shall ignore the second of these equations. § 74 THE ELECTROMAGNETIC FIELD 277 For the present we shall ignore the second of these equations and work only from the first. Equation (1) shows that each A, can be resolved into waves travelling with the velocity of light. Thus, corresponding to equation (63) of § 63, A,(e) = { (Ae, chr Ae, e~tkn) Bhp, (3) where k.x denotes the four-dimensional scalar product k.v = ky, x )—(k,x), k, being the 4-vector whose space components are the same as the components of the three-dimensional véctor k of § 63 and whose time component k, = |k|, and d?k denotes dk, dk, dks, as in § 63. The index c in the coefficients A¢, indicates that they are constant in time. We shall later introduce some other Fourier coefficients A,,,, not constant in time, which must be distinguished from the present ones. The Fourier component A‘, has a part Aj, coming from A,(x) and a part A‘, (r = 1,2, 3) which is a three-dimensional vector. The latter can be decomposed into two parts, a longitudinal part lying in the direction of k, the direction of motion of the waves, and a transverse part perpendicular to k. The longitudinal part is k,k,/k,?.Ag,. The transverse part is say. It satisfies k, f°, = 0. (5) It is known from the Maxwell theory of light that only the trans- verse part is effective for giving electromagnetic radiation. Chapter X dealt only with this transverse part, the A,, of § 63 being the same as the present 7%, and equation (65) of § 63 corresponding to the present equation (5). Nevertheless, the longitudinal part cannot be neglected in a complete theory of electrodynamics because of its connexion with the Coulomb forces, as will show up later. We can now decompose the three-dimensional vector A,(x) into two parts, a transverse part and a longitudinal part. The former is Af (a) = [ (tee +L, e-2) Pk and satisfies b.0,(x)/Ox, = 0. (6) The longitudinal part may be expressed as the gradient @V/éx, of a scalar V given by Vai J Ie| lig? (AS, et? — AS, eth) de, (7) Thus A, = A+6V /ax,. (8) 3595 57 T 278 QUANTUM ELECTRODYNAMICS § 74 The magnetic field is determined by the transverse part of A,, # = curlA = curl ¥&. It is convenient to count A,(x) as longitudinal, so that the complete potentials A,(z) are separated into a transverse part (x) and a longitudinal part A,, dV/dx,. This separation, of course, refers to a particular Lorentz frame of reference.and must not be used when one wants to keep one’s equations in a relativistic form. Each Fourier coefficient A¢, occurs in (3) combined with the time factor e%%. The product . Ao, Etkoty — Aix (9) say, forms a Hamiltonian dynamical variable in classical mechanics and a Heisenberg dynamical variable in quantum mechanics, like the the A,,, of § 63. ; The work of § 63 gives us the P.B. relations for the transverse part of A,x. To connect up with it, we pass over to discrete k-values in three-dimensional k-space and take, for example, a particular discrete k-value for which hk, = kh, = 0, kg = ky > 0. Then the polarization variable 1 can take on two values referring to the two directions 1 and 2 and equation (73) of § 63 gives, with the help of the commutation relations for the y’s and 4’s, equations (11) of § 60, [AA] = [Ax Art] = —18,/407ky. (10) The work of §63 gives us no information about Ag, and Ay. However, we can now obtain the P.B. relations for A,, and A, from the theory of relativity. Equations (10) have to be built up into a relativistic set and the only simple way of doing so is by adding to them the two further equations (Asx; Asie] = —[Aos Aon] = —t8,/47%ho, (11) so that the four equations (10) and (11), together with the conditions that Au and A,, commute for p ~v (as they must do since they refer to different degrees of freedom), combine to form the single tensor equation [ As Ay) = 1G yy 84/47 7%. (12) We get’in this way the P.B. relations for all the dynamical variables. Equation (12) ean be extended to [Aus Aye) = yy 5k Syxe/ 477g. (13) Let us now return to continuous k-values. To convert 6,,, to con- § 74 THE ELECTROMAGNETIC FIELD 279 tinuous k-values we note that, for a general function f(k) in three- dimensional k-space, SSK) Bey = S06’) = | f(s) 5k —-k) ah, (14) where 6(k--k’) is the three-dimensional 6 function B(K—K’) = 8(by—k,)8 (leq —k)8(ey—K5). In order that (14) may conform to the standard formula connecting sums and integrals, equation (52) of § 62, we must have Si Sig == 8(K—K’). (15) Thus (13) goes over to [Ayo Ave] == 19,,/477ky d(kK—k’). (16) This equation, together with the equations [Aye Aye] = [Aus Ay] = 0, (17) provide the P.B. relations in the theory with continuous k-values. It should be noted that these P.B. relations remain valid if we replace Au Ay by Aix, Aj. The same P.B. relations apply to the constant Fourier coefficients Ajy, Ajx- We must now obtain a Hamiltonian which makes each dynamical variable A,,, vary with the time ¢= a, in the Heisenberg picture according to the law (9) with Aé, constant. Calling this Hamiltonian Hp, we require [Aux Hp] = dA, /day = thy Ay (18) It is easily seen that this is satisfied by Hp = —4n? | kA, AY, dk. (19) We therefore take (19), with the possible addition of an arbitrary numerical term not involving any dynamical variables, as the Hamil- tonian for the electromagnetic field in the absence of matter. In § 63 we used our knowledge of the transverse part of the Hamil- tonian to obtain the P.B.s of the transverse variables. We have now applied the reverse procedure to the longitudinal variables, using our knowledge of their P.B.s, obtained by a relativistic argument, to find the part of the Hamiltonian, that refers to them so as to get agreement with (18). If we write out the Hamiltonian (19) it appears as Hyp = 4x” | ki Ar Ase + Aoy Aon t+ A ax Ase Aor Lox) ak. 280 QUANTUM ELECTRODYNAMICS § 74 The first three terms of the integrand here have a transverse part which is just equal to the transverse energy given by (71) of § 63. The last term of the integrand, which is the part of Hy referring to the scalar potential A), appears with a minus sign. This minus sign is demanded by relativitv and means that the dynamical system formed by the variables A,,, 4., is a harmonic oscillator of negative energy. Itis rather surprising that such an unphysical idea as negative energy should appear in the theory in this way. We shall see in § 77 that the negative energy associated with the degrees of freedom connected with A, is always compensated by the positive energy associated with the other longitudinal degrees of freedom, so that it never shows up in practice. 75. Relativistic form of the quantum conditions The theory of the preceding section has relativistic field equations, namely equations (1). To establish that the theory is fully relativistic we must show further that the P.B. relations are relativistic. This is not at all evident from the form (16) in which they are written in terms of Fourier components. We shall obtain a relativistic form for the P.B.s by working out [A,,(x), A,(z’)] with x and 2’ any two points in space-time. We must first, however, study a certain invariant singular function that exists in space-time. The function 8(x,, 2") is evidently Lorentz invariant. It vanishes everywhere except on the light-cone with the origin as vertex, i.e. the three-dimensional space x, 2" = 0. This light-cone consists of two distinct parts, a future part, for which x, > 0, and a past part, for which % <0. The function which equals 3(x,, 2") on the future part of the light-cone and —8(x, x") on the past part of the light-cone is also Lorentz invariant. This function, equal to 8(x, v)a/|z9|, plays an important role in the dynamical theory of fields, so we introduce a special notation for it. We define A(x) = 28(2, x)2x9/|2%o|. (20) This definition gives a meaning to the function A applied to any 4-vector. With the help of (9) of § 15, we can express 6(x,, 2”) in the form B(x, a) = $x|-48(ary— |x|) +8(2)+|XI)}, (21) |x| being the length of the three-dimensional part of «,, and then A(x) takes the form Ax) = |X|748(%— |X|) —8(%-+ |X])}. (22) §75 RELATIVISTIC FORM OF THE QUANTUM CONDITIONS 281 A(z) is defined to have the value zero at the origin, and evidently A(—2z) = —A(z). . Let us make a Fourier analysis of A(z). Using d*x to denote dz dx,dx,dx, and dz to denote dz,dx,dx,; we have, for any 4- vector ky» | A@e™ dtr =| |x|-B(ay— |X|) —8 (a+ |X|) }etleore—lool fa: =| |x| ~Letholx! _.¢—thalx!} 9 —a(kx) ax. By introducing polar coordinates |x|, 9, ¢ in the three-dimensional 2,2, 2%, space, with the direction of the three-dimensional part of k,, as pole, we get [ A@et dq = ify {eikalxl_— g~tkolxi}g—ilkiixicos 8 x sin @ dOddd|x| —_ 2a [ {eikaix|—g-tkuial d}x | e-tiklixieos 8) x|sin 8 dO i j foo} = 2Qni|k|- | fetkoixl_—e~tholxl} | x| {e~alkiixl_— gill} 0 wo = 2ni|k|- | {eilko—ikda__ gitke+ikda) dey —-a@ = 4n°4[k|-4{8(ty— |k |) —3(ko+ IK I)} = 4n77A(k). (23) Thus the Fourier analysis gives the same function again, with the coefficient 4721. Interchanging & and x in (23), we get A(z) == —i/4n?. | A(kje* dh. (24) Some of the important properties of A(x) can easily be deduced. from its Fourier resolution. In the first place equation (24) shows that A(a) can be resolved into waves all travelling with the velocity of light. To get an equation for this result we apply the operator [] to both sides of (24), thus A(x) = —i/4n®. | A(k)De** dtk = i/4n*. | k, keA(k)e** atk. fe Now k, k#A(k),= 0, and hence A(x) = 0. (25) This equation holds throughout space-time. We can give a meaning to (JA(x) at a point where A(x) is singular by taking the integral 282 QUANTUM ELECTRODYNAMICS § 75 of [JA(x) over a small four-dimensional space surrounding the point and transforming it to a three-dimensional surface integral by Gauss’s theorem. Equation (25) informs us that the three-dimensional surface integral always vanishes. The function A(x) vanishes all over the three-dimensional surface t= 0, Let us determine the value of GA(x)/ax, on this surface. It evidently vanishes everywhere except at the point 2, = 2, = 2, = 0, - where it has a singularity which can be evaluated as follows. Differ- entiating both sides of (24) with respect to 29, we get 6A (a) [Oxy = 1/4n?. | Fo A(hjet= dh = 1/4a?. [ dg |ke|-45(y— [he |) 8 (leg + Ik Je dt = 1/4e®. | {8(ibp— ||) +3(ko+ [ke|)}et dh. Putting x, = 0 on both sides here, we get [2A (x) /2xrg]oyao = A/4m®. | {8(kg— ||) +5(lby + |k|)}e-1 dh = 1/2n?, [ e-tte 4%, == 47 8(2r1)3(ar2)8(a'g) == 47 8(x). (26) Thus the ordinary 5 singularity, with the coefficient 47, appears at the point 2, = 2, = 2, = 0. Let us now evaluate [4,,(x), A,(x’)]. We have from (3), (16), and (17) [4, (x), A,(e}] = | i [Apne *+A,e-, Ay e24 A eth 2"| Bed h’ = ig,,[4n?. | i lig Me thngth’.2' _ gik.tre—ik' a §( kk’) dd 3h! = Agyy/4r®. f lg Mente) gikia—29} py, er) The k, here is defined. to be equal to |k| and is thus always positive. By putting —k for k in the second part of the integrand, one finds that (27) is equal to the four-dimensional integral 19 ,,[4n?. | [K|-1{3(ky— iK|)—8(ky+ [k[)}e-*-@-29 dk — 29 ,y[4n?. i A(k)e~t@-2) day, in which k, takes on all values, negative as well as positive. Evaluating this with the help of (24), we get finally [A, (2), A,(@’)] = 9,,A@—2"), (28) §75 RELATIVISTIC FORM OF THE QUANTUM CONDITIONS 283 a result which shows that the P.B. relations are invariant under Lorentz transformations. The formula (28) means that the potentials at two points i in space- time always commute unless the line joining the two points is a null line, ie. the track of a light-ray. The formula is consistent with the field equations (1A,,(x) = 0, because [] applied to the right-hand side gives zero, from (25). 76. The dynamical variables at one time As a basis for a theory with interaction we must use the dynamical variables at one time. The relationships between the dynamical variables at one time (i.e. their P.B.s) are not affected by the introduc- tion of interaction. On the other hand the relationships between the dynamical variables at different times (comprising the field equations as well as the P.B:s of variables at different times) are very much affected by the interaction. The dynamical variables at one time form a non-relativistic concept, but a very important concept in Hamil- tonian theory. For the case of the electromagnetic field the independent dynamical variables at one time are A, and 04,,/@x for all values of 24, ¥q, 3 for the given 2. The higher time derivatives 2°A,/éu,°,..., are not independent. Let us put 0A, B 29 ws By (29) Then we have Arxs B,x, with the suffix x denoting x, %, £3, a8 the dynamical variables at one time. The Fourier resolution of these variables is, from (3) and (9), Aux = [ (Apt Ay x)eO™ Pk Bux = | hol Aye—Ay, alot ah We may reverse the Fourier transformation and express A tA, lk and A, ,—A,-; in terms of A,, and B,, respectively. Thus 4, and A pk are determined by 4,,., B, x for all x (at a given 2). The. equa- tions connecting A,,, Aux with. A,x, By, do not involve the time explicitly. Thus the A,,,, Aux form an “alternative set of one-time dynamical variables, on the same footing as the A,x, Bux When we work with the variables A Bux we shall need to know (30) px 284 QUANTUM ELECTRODYNAMICS § 76 their P.B. relations. These may be obtained either from the Fourier expansions (30) together with (16) and (17) or from the general P.B. relation (28). The latter gives the required results more quickly. Putting xj = 2%) in (28), we get [A unr Aye] = 0. (31) pe Differentiating (28) with respect to x) and then putting 2 = 2%, we get, with the help of (26), Differentiating (28) with respect to both z, and 2 and then putting 2 = Ly, we get [Bux Bye] = 0, (33) since 6°A(x)/a5 = 0 for 2 = 0. Equations (31), (32), and (33) give all the P.B. relations between the A px Dy Variables, They show that, apart from numerical coefficients, the A ux can be looked upon as a set of dynamical coordinates and the B,,, a8 their conjugate momenta, there being a 5 function on the right-hand side of (32) instead of a two-suffix 6 symbol on account of the number of degrees of freedom being a continuous infinity. We can decompose A,, into a transverse and a longitudinal part, as shown by equations (8) and (6). We can do the same with B,, and get B=2,48 (34) Ox, with 0B, ox, = 0. (35) From (7) with —k substituted for k in the second term of the integrand, Vi=i i lig lig 2( Age t+ Ay, )e-*™ Bh. (36) The corresponding equation for U is, since U = aV /éa,, U = — | lig lig (A gy — Ay Je Bh, (37) The electric field is given by é,=—B, _ aA, ox, _ g HAD), (38) ox, Thus divé = 8B, V?Ag Ox, = —V%{A,+U). (39) It is evident that any longitudinal variable commutes with any § 76 THE SCHRODINGER DYNAMICAL VARIABLES 285 transverse variable. Some useful P.B. relations will now be worked out. We shall use the notation—for any field function f,, If in (32) we put» = 7, v = 6 and differentiate the equation with respect to x,, we get [Bucs Agy'] = 479,,5'(X—X') = —4n59(x—x’), or, from (39), [div &,, A] = 4765(x—x’). (41) Now (39) shows that div € is a function only of the longitudinal variables, so (41) gives [div &,, V.*] = 478° x—x’) = —478*(x—x’). Integrating with respect to x, we get [div €,,V%,] = —413(x—x’), (42) there being no constant of integration since the field functions &, and ° V, are made up of waves of non-zero wave length. From (42) and (39) WT. Vo] = 478(x—x’). Integrating with the help of formula (72) of § 38, we get (U,V) = —|x—x’|-7, (43) there being no constant of integration or other terms not vanishing at infinity on the right-hand side, because U, and Y, are made up of waves of non-zero wave length. We have from (38) and (43) [Ee Ve] = [Ug Vy] = — (0,2) |x! |. (44) 9°xX We shall now obtain the Hamiltonian in terms of the 4,, and B,, variables. We have from the second of equations (30) | Bux Br, Ba == fff bok Apa Aya) Abe Te yotene tO Bhd Be — 873 if kg ki A yr — Ay) (AM — AY ¢)8(K-+K’) Bhdsh! = —873 | kA ye—Ay~ (AY 4 —Aly) Bh.” Similarly, from the first of equations (30), | A, x’ At! Bax = fff by BA py MAb + Ae yee dled a = 83 | Keg" A get Ay (AM + AM) Bh 286 QUANTUM ELECTRODYNAMICS — § 76 Adding and dividing by —87, we get —(8m)-2 | (B, BeAr Aer) dix = —27? J leg(A ye AM + Aye AY 4) Bh. This is equal to A, given by (19), apart from an infinite numerical term. The formula (19) for 2, already involves an arbitrary numerical term, so we may take Hyp = —(87)-2 | (B,, Bu-+-A,7A*) dx (45) with an arbitrary numerical term, different from that of (19). The Hamiltonian (45) can, of course, be used to give the Heisenberg equations of motion, and the arbitrary numerical term in it does not have any effect. One can easily check, using (31), (32) and (33), that 0A,,/0% = [A,, Hp] = B,, (46) and. 6B,,/0x% = [B,, Hp] = VA,, agreeing with (29) and (1). It also gives the Schrédinger equation of motion ihd|P)|day = Hy|P> for a ket |P> representing a state in the Schrédinger picture. The arbitrary numerical term here has the effect of changing |P> by a phase factor, which is not of physical importance. We can decompose the expression (45) for H, into a transverse part Hp, and a longitudinal part H,,;. We have from (34) | BB, Bz = | (B,+ UY B,+U") Ba = | BB, Bx + | UrUr dx, since the cross terms vanish on account of [ 0, Ba = — [ UB? dx =0 from (35). Similarly we have from (8) | AfA,? Ba = { bhiof8 Ba + | preyre d3z, with the cross terms vanishing again. Thus (45) becomes Ap = ppt Hpy, with Hyp = (80)"} | (B, B+ Al) Be (47) and = Hy; = (8m)-2 | (UrUt- VrsVr*— B, By—A Ag’) Gx. (48) § 76 THE SCHRODINGER DYNAMICAL VARIABLES 287 it should be noted that the term (87) [ feeds da in Hyp can be transformed to = (81) [ of" Bx = — (80) | Alf — oh") Ba = (Sn) | cfslohs—ahe) Pa = (16m) | (oho — ahs) dx == (Sm)-2 | A? Bar, so this term is just the magnetic energy. Some further partial inte- grations give | Vrsyrs io = | prryss Bx, so (48) may be written Hy, = (8m) [ {(U—Ag¥(U + Ag+ (V"—By)(V"+ By)} Bx. (49) 77. The supplementary conditions We must now go back to the Maxwell equation (2), which we have ignored so far. We cannot take this equation over directly into the quantum theory without getting inconsistencies. The left-hand side of the equation does not commute with A,(x’), according to the quantum conditions (28), so this left-hand side cannot vanish. The way out of the difficulty was shown by Fermi.} It consists in adopting a less stringent equation, namely the equation (0A,,/2x,,)|P> = 9, (50) and assuming it to hold for any | P> corresponding to a state that can actually occur in nature. There is one equation (50) for each point in space-time and these equations must all hold for any ket corre- sponding to a state that can actually occur. We shall call a condition such as (50), which a ket has to satisfy to correspond to an actual state, a supplementary condition. The exis- tence of supplementary conditions in the theory does not mean any departure from or modification in the general principles of quantum mechanics. The principle of superposition of states and the whole of the general theory of states, dynamical variables, and observables, as given in Chapter II, apply also when there are supplementary + Fermi, Reviews of Modern Physics, 4 (1932), 125. 288 QUANTUM ELECTRODYNAMICS § 77 conditions, provided we impose a further requirement on a linear operator in order that it may represent an observable. We define a linear operator to be physical if it has the property that, when it. operates on any ket satisfying the supplementary conditions, it pro- duces another ket satisfying the supplementary conditions. In order that a linear operator may represent an observable it must evidently satisfy the requirement of being physical, in addition to the require- ments of § 10. We have already had an example of supplementary conditions in the theory of systems containing several similar particles. The con- dition that only symmetrical wave functions, or only antisymmetrical wave functions, represent states that can actually occur in nature, is precisely of the samé type as condition (50) and is what we are now calling a supplementary condition. In this theory the requirement that a linear operator shall be physical is that it shall be symmetrical between the similar particles. When we introduce supplementary conditions into our theory we must verify that they are consistent, i.e. not too restrictive to allow any ket at all to satisfy them. Ifwe have more than one supplementary condition, we can deduce further supplementary conditions from them by taking P.B.s of the operators in them; thus if we have U|P>=0, V\|P>=90, (51) we can deduce [U,V]|P>=9, [U,[U, VI]|P> = 0, (52) and so on. To verify that our supplementary conditions are consistent we have to look into all the further supplementary conditions obtain- able by this procedure to see that they can be satisfied, which we can usually do by showing that after a certain point the further supple- mentary conditions are all either identically satisfied or repetitions of the previous ones. We must also verify that the supplementary conditions are in agree- ment with the equations of motion. In the Heisenberg picture, for which the ket | P) in (51) is fixed, we shall have different supplementary conditions referring to different times and they must all be consistent, in the way discussed above. In the Schrodinger picture, for which the ket |P> varies with the time inaccordance with Schrédinger’s equation, we require that if |P) satisfies the supplementary conditions initially it satisfies them always. This means that d|P)/dé must satisfy the § 77 THE SUPPLEMENTARY CONDITIONS 289 supplementary conditions, or that H|P> must satisfy the supplemen- tary conditions, or that H must be physical. It is convenient when we have asupplementary condition U|P> =0 to write U0 (53) and to call (53) a weak equation, in distinction to an ordinary or strong equation. A weak equation gives another weak equation if it is multiplied by any factor on the left, but does not in general give a valid equation if it is multiplied by a factor on the right. Thus a weak equation must not be used in working out P.B.s. With this way of speaking, the requirement (52) that the supplementary conditions are consistent becomes the requirement that the P.B.s of the operators in the supplementary conditions shall vanish weakly. The condition for a dynamical variable é to be physical is that, for each supplementary condition 7|P> = 0, we have Ug|P> = 0, and hence [U,é]|P> = 0. Thus the condition is that the P.B. of the dynamical variable with each of the operators of the supplementary conditions shall vanish weakly. Let us now return to electrodynamics. We take equation (2) to be a weak equation, so it should be written 0A ,/Ox, = 0. (54), In the Heisenberg picture we have one of these equations for each point x. To check their consistency, we take two arbitrary points x and 2’ in space-time and form the P.B. [eu Se) gly) AND) Oat, a, On, OX, Evaluating it with the help of (28), we get 9 AA(a—z’) BY 026, Ox, from (25), so the requirements for consistency are satisfied strongly. As we have verified that the supplementary conditions are consistent at all times in the Heisenberg picture, we have verified that they are in agreement with the equations of motion. Since equation (54) is only a weak equation, any of its consequences 290 QUANTUM ELECTRODYNAMICS §77 in the ordinary Maxwell theory will be valid in the quantum theory only as weak equations. The equations divA! = 0, dF /ét = —curl € follow simply from the definitions of € and# in terms of the potentials, so they are valid strongly in the quantum theory. The other Maxwell equations for empty space, namely divé = 0, d&/at ~ curl#, (55) are weak equations in the quantum theory, because one needs the help of (54) as well as (1) in deriving them. The field quantities € and # are components of the antisymmetric tensor 0A”/dx,—dAr/dx,. The P.B. of the tensor with the operator of (54) at a general point x’ is "= 8Ar(x) | » PA(a—2’) F pO A(c—z’') , “da ’ OX, Ox Ox Ox, OX, az 0. Je dx, Ox, It follows that € and # are physical. The potentials A, are not physical. The supplementary conditions affecting the dynamical variables at a particular time are OA, 0 2 OA, 0. (56) ft ex, Oxy Ot, Higher differentiations with respect to x) do not give independent equations, but equations which are consequences of these and the strong equation (1). Thus in terms of the Schrédinger variables of § 76, the supplementary conditions are BytAy = (57) and (A jt+B,) = 0. (58) Equation (58) is the same as the first of equations (55) and may also be written, from (39), V%(Ag+U) = 0. Since this holds throughout three-dimensional space, it leads to A,+U = 0. (59) Noting that A,” = V"", we can now see from (49) that Hyp 0. (60) Thus there ts no longitudinal field energy for states that occur in nature. §77 THE SUPPLEMENTARY CONDITIONS 291 To set up a convenient representation, we introduce a standard ket [0,> satisfying the supplementary conditions (BtAs)0p> =0, — (Ap +-U)|0p> = 0, (61) and also satisfying Ay|\On> = 0. - (62) These conditions are consistent, because Ay, commutes with the operators in (61), and they are sufficient to fix |0,> completely, apart from a numerical factor, because the only independent dynamical variables that we have are Ay, By, U, A,’, %,, Z,, and of these A,+U, By+A,’, %, form a complete commuting set. With this standard ket we can express any ket as ‘P(A; Bo, Gx) |0n>. (63) Our representation is just the Fock representation so far as concerns the transverse dynamical variables 4,, 7,, so Y must be a power series in the variables %,, with different terms in the series corre- sponding to the presence of different numbers of photons. The number of variables occurring in ¥ is a continuous infinity, so ¥ is what mathematicians call a ‘functional’. If the ket (63) satisfies the supplementary conditions, ‘ must be independent of A, and By, and thus a function only of the o7,,. So physical states are represented by kets of the form F(4) On», (64) with a power series in the variables %,. The standard ket |0,> itself represents the physical state with no photons present, the perfect vacuum. Our Hamiltonian H, and its parts Hy,, Hpp have so far contained arbitrary numerical terms. It is convenient to choose these terms so that Hy,, Hpp are zero for the perfect vacuum. The result (60) shows that Hp; given by (48) or (49) has the numerical term in it correctly chosen to make Hp; have the value zero for the perfect vacuum, as well as for every other physical state. We must take Hpr to be App = 47° | he? ye Ay, Ph, (65) the transverse part of (19), in order that the numerical term in it may be correctly chosen to give no zero-point energy for the photons. (47) differs from (65) by an infinite numerical term, consisting of a half-quantum of energy for each photon state. 292 QUANTUM ELECTRODYNAMICS § 78 78. Electrons and positrons by themselves We now consider electrons and positrons in the absence of electro- magnetic field. The state of an electron is described, as in Chapter XI, by a wave function % with four components #, (a = 1, 2, 3, 4), satis- fying the wave equation th, im, ah ox, Ban be Xp, Ha (66) Tr To get a many-electron theory we shall apply the method of second’ quantization of § 65, which involves changing the one-electron wave function into a set of operators satisfying certain anticommutation relations. When we are dealing with at various places at a given time we may write it ¢,, with x denoting 2,, x, x5. Its components are then tu, We pass to the momentum representation with the wave function i$, by a three-dimensional Fourier resolution b, = h-3 | ceri dp, by = hei | e~iplty d8a, (67) #, has four components y¥,,, corresponding to the four components of #,. In this representation the energy operator is Po = Oy Dp Xm Ms in which the momentum operators p, are multiplying factors. We can separate y into a positive-energy part € and a negative- energy part ¢, p= ELL, € and ¢ each having four components like J. In the momentum representation they are given by I 7 Ppt Xm, M bp = 5 14 recency ho op _ f Se le (68) 1 2\ (p?-+m?)t since these equations lead to Bobp = (% Pp tom ME, = Hot Ppt Om M+ (D?-+-m*) hh, (p?+m*)ié,, and similarly Dob = —(p?-+m?)iE,, showing that &, and ©, are eigenfunctions of p, with the eigenvalues (p?-m?)? and —(p?-+-m?)! respectively. When one is working with the operators I Uy, Seed at Op Ppt Om M0 9 ’ a ED TONG Of? (p?-+-m?)t 2 § 78 ELECTRONS AND POSITRONS BY THEMSELVES 293 one should note that their squares are equal to themselves and their product in either order is zero. The second quantization makes the ’s into operators like the 4’s of § 65, satisfying anticommutation relations like (11’) of § 65. Using the notation for the anticommutator MN+NM = [M,N },, (69) we get _ + _ [Loses Pore + — 0, [Leas Pr le —_ 0, (70) [Laxs dow le = Sap 6(x—x’), the function 5(x— x’) appearing in the last equation owing to the x’s taking on continuous ranges of values. On transforming to the p- representation according to (67), we get [Laps Pop |e = 0, [Pap» Pop ls = 0, [Lap Pr ls = 5,,5(P— Pp’). With é and ¢ defined again by (68), the last of equations (71) gives (71) LL = 1 re an m MH 7, I 3 3 mm [fap Sop'l+ = 3(! | pint |, eo bugle + TTY fe 1 Xp PrP Xm, ’ and similarly [ap foyle = 51 ebR Sn) 8(p—p') (73) and [fap lop le = [Enp> lop le = 0. According to the interpretation of §65, the operators ,, are operators of annihilation of an electron of momentum p and the operators ¢,, are operators of creation of an electron of momentum p. To avoid the unphysical notion of negative-energy electrons, we must pass over to a new interpretation based on the positron theory of § 73. The annihilation of a negative-energy electron is to be understood as the creation of a hole in the sea of negative-energy electrons, or the creation of a positron. So the operators ¢,, become operators of creation of a positron. The positron has the momentum — p, because an amount p of momentum gets annihilated. Similarly the £,,, become operators of annihilation of a positron of momentum —p. The €,, and £,, are operators of annihilation and creation respectively of an ordinary, positive-energy electron of momentum p. 3595 57 Uv 294 QUANTUM ELECTRODYNAMICS § 78 It should be noted that, although €, has four components, only two of them are independent, because the four are connected by p ele, — 0, (p?-+-m?)! which involves two independent equations. The two independent components of €, correspond to the annihilation of an electron in each of the two independent states of spin. Similarly ¢, has only two independent components, because of the equations te Etat lt = 0, and they correspond to the creation of a positron in the two inde- pendent states of spin. The vacuum state, for which there are no electrons or positrons present, is represented by the ket |0p) satisfying We can use this ket as the standard ket of a representation. We then have any ket expressed as Y(Eap> Cap) 9p), in which the function, or rather functional, ‘¥ is a power series in the variables £,,, Cap. Hach term of ¥ is like (17') of § 65. It must not contain any ofits variables to a higher power than the first. It corre- sponds to the existence of certain (positive-energy) electrons and certain positrons, in states specified by the labels of the variables appearing in it. From (12’) of § 65, the total number of electrons is | Pap tap dp summed over a, We may write it in the notation of equation (12) of § 67 as { ¢i), Bp. Transforming it to the x-representation by (67), we get be [[f iomhe tort dhe Pad e'dp = | Phx de, showing that the density of the electrons is gi 4,. This result includes an infinite constant representing the density of the sea of negative- energy electrons. We get a quantity of more physical significance if we take the total charge Q, equal to the number of positive-energy electrons minus the number of holes or positrons, all multiplied by —e. Thus Q=—e| (e—-GG) a. (75) § 78 ELECTRONS AND POSITRONS BY THEMSELVES 295 We can evaluate this with the help of (68). Using the transpose of the second of these equations, namely Now for any matrix a whose cova sum is zero, the anticommutation relations (71) give Hh, cy pt, oth, = oan(Pap Pop + Pop bap) = qq O(P— P’) = 0, (76) a result which we may assume still holds for p’ = p. Then the expression for Q reduces to Q= —2{ Mohd, op Fp) ap. Transforming it to the x-representation as before, we get Q=—e[ Mid —ved,) Oe, showing that the charge density is The interpretation of the one-electron wave function in § 68 gives, besides the probability density ft, a probability current Plo, yp. With second quantization we shall have correspondingly a flow of electrons, given by the operator fia,y,. The sea of negative-energy electrons produces no resultant flow of electrons, from symmetry, and so the electric current is ™ —erp, Oy Pigs (78) The total energy of the electrons is, from formula (29) of § 60, which is valid also for fermions, Hp = | bh Do}, Pp = | Pt (ot, Dpto mb, Ep. (79) It becomes, when transformed to the x-representation, Hy = [ Pil —itiags,!-+ay my) Bx. (80) This total energy contains an infinite numerical term representing the energy of the sea of negative-energy electrons. We get a quantity of more physical significance if we take the energy 296 QUANTUM ELECTRODYNAMICS § 78 of all the electrons and positrons, reckoning the energy of the vacuum as zero. This quantity is Hp = : (pm)? a +042,) dp (81) oo pesnmlo + | (P+ m) AT bp toh Dy) Bp. (82) From (76), the first integral in (82) is the same as (79) and is just Hp. The second integral is an infinite constant and is minus the energy of all the negative-energy electrons of the vacuum distribution. We may take either Hp or Hp. as the Hamiltonian. The Heisenberg equation of motion for %,, is thus Oax/ Oty = [Pars Hy| = = [Paxs Hp], and if we work this out we just get back to the wave equation for +, namely (66). We must now look into the question of whether the theory is relativistic. It is built up from operators 4 which satisfy the field equations (66). These equations are the same as the wave equation for the one-electron wave function and are known to be invariant under Lorentz transformations, provided % transforms according to the law (20) of Chapter XI. Our present theory goes beyond the one-electron theory in that anticommutation relations are introduced for the %’s and #’s, and it becomes necessary to verify that these anticommutation relations are Lorentz invariant. We proceed by a method analogous to that of § 75. We take two general points x and 2’ in space-time and form the anticommutator Kay (a, 2") = pale) a") + byl’ fal). (83) We can evaluate it by working directly from the anticommutation relations (71) for the Fourier components of ys and J. A simpler way is to note certain properties that K,,(x,a’) must have, namely (i) it involves x, and vy only through their difference ty 25 § 78 ELECTRONS AND POSITRONS BY THEMSELVES 297 (ii) it satisfies the wave equation a a i — ih — “— (it a, the, én, Qn m) Kel x)= 0 (84) on account of %(x) satisfying (66); (iii) for vy) = a it has the value §,,5(x—x’), as follows from the third of equations (70). These properties are sufficient to fix K,,(x, x’) completely, since (iii) fixes it for 2) = 2%, (ii) shows how it depends on 2p, and (i) then shows how it depends on 2. The solution is easily seen to be where the >’ means a summation over the two values +(p?+m?)} for py with particular values for p,, D2, p3- Tt satisfies (ii) since the operator in (84) produces the factor (p)—a,~,—%,_,m) in the integrand of (85), which factor gives zero when multiplied on the left into the factor {}. It satisfies (iii) since, with x) = 2, the summation over py makes the second term in {} cancel out. The law of transformation for and ¢ given in § 68 has the effect of making the quantities £'(x')a,, (x) transform like the four com- ponents of a 4-vector and making #1(2’)o,, (x) invariant. Thus Leaf (20! )ow,, x(x) + SBE (ar! Jory, (a) (86) is invariant with l# any 4-vector and S any scalar. The invariance | of (86) must be sufficient to ensure the correct transformation law for and ys, since it enables one to deduce the invariance of the wave equation for %, by taking l+ = th d/éx,, S = —m. The invariance of (86) leads to the invariance of (Woy + Seem )avl Pal’ yoo(%) +p (e)Pq(@’)}. Thus (Ha, + Son)ap Kyal%, 2’) (87) should be invariant with K,,(x,x’) given by (85), and its invariance would be sufficient to ensure the invariance of the anticommutation relations. We get for (87) A | > 3 (Ma, + Sam )an( Pot % Prt %m Mog e~te-2).-piiy 1 Bp =h* | > H(y—l, Og tom) Pot Oy Ppt Mm ™)}aa ete-2)-pihiy > ap = hs [ > 20 Po—l, Prt Smee) Pps t Bp. (88) This is Lorentz invariant because the differential element pz! d°p is 298 QUANTUM ELECTRODYNAMICS § 78 Lorentz invariant. Thus the relativistic invariance of the theory is proved. 79. The interaction The complete Hamiltonian for electrons and positrons interacting with the electromagnetic field is H = Hp+Hp+Ho, (89) where Hy, is the Hamiltonian for the electromagnetic field alone, given by (19) or (45), Hp is the Hamiltonian for the electrons and positrons alone, given by (80) or (81), and Hp is the interaction energy, involving the dynamical variables of the electrons and positrons as well as those of the electromagnetic field. We take Ho = Ary, dx, . (90) with j, given by (77) and (78), as we shall see that this gives the correct equations of motion. Thus, with neglect of infinite numerical terms, H = [ (Bag —itr—ed"p) + Pray mp JeAPY— PP} de — —(87)-2 | (B, Bet A,tAv) dx. (91) Let us work out the Heisenberg equations of motion that follow from the Hamiltonian (91}. We have iT Bhg| Oh = thayg H— Hag. = Way (Hp + Ho) —(Hp-+ Ho Wax = / [Leases Pox’) +{a,( ~— tial” —eA ** by) + + Om mip —eA Oe! bet ax’ = {a.,( — fix, — eA’, Px) +m mip,.— eA Oe tae Thus {2u{ +e) oy, ml = 0. (92) This agrees with the one-electron wave equation (11) of Chapter XT. Since H is real, the equation of motion for ¢ will be the conjugate of the equation of motion for y and so will agree with (12) of Chapter XT. Thus the interaction (90) gives correctly the action of the field on the electrons and positrons. Further we have, making use of the P.B. relations in (46), = B, (93) § 79 THE INTERACTION 299 px? = VA nt f (Bue Aline Fo" = VAL +4: (94) (93) and (94) lead to A, = 40), (95) which agrees with the Maxwell theory and shows that the interaction (90) gives correctly the action of the electrons and positrons on the field. To complete the theory we must bring in the supplementary con- ditions (54). We must verify that they are in agreement with the equations of motion. The method used in § 77, which consisted in showing that the supplementary conditions at different times in the Heisenberg picture are consistent with one another, is no longer applicable, because the quantum conditions connecting dynamical variables at different times get altered by the interaction in a way that is too complicated to be worked out. So we shall obtain all the supplementary conditions affecting the dynamical variables at one instant of time and check whether they are consistent. We have again equations (56). A further differentiation with respect to %» gives ~ oo . dA, /ex, = 0. (96) Now the equation of motion for , namely (92), leads, as in § 68, to a(hta,, p)/ax, = 0, This is the same as j,/0x, = 9, (97) because the difference between —ezitys and j, is constant in time, even though it is infinite. From (95) we now see that (96) holds as a strong equation. Thus equations (56) are the only independent supplemen- tary conditions affecting the dynamical variables at one instant of time. The first of them gives (57), as before, and the second now gives, with the help of (95) for » = 0, (A,?+B,) +4rjy = 0. (98) This may be written (A> tU)’+4nj, 0 (99) or, from (39), div €—4nj, = 0, (100) and is just one of the Maxwell equations. 300 QUANTUM ELECTRODYNAMICS § 79 One can see without detailed calculation that, for any two points x and 2’ at the same time, [Jox>Jox'] = 9, since, from the form of (70), the P.B. must be a multiple of 5(x—x’) and cannot contain derivatives of 5(x— x’), while also it has to be anti- symmetrical between x and x’. Thus the extra terms 47j), in equa- tions (98) for various values of x, as compared with the corresponding equations (58), commute with one another as well as with all the other dynamical variables occurring in (58) and (57). It follows that these extra terms will not disturb the consistency of (58) and (57), and hence (98) and (57) are consistent. Our method of introducing interaction into the theory was not relativistic, since the interaction energy (90) involves the dynamical variables at an instant of time in some Lorentz frame. It therefore becomes questionable whether the theory with interaction is a rela- tivistic one. Our field equations, namely (92) and (95), are evidently relativistic and so are the supplementary conditions (54). It remains uncertain whether the quantum conditions are Lorentz invariant. We know the quantum conditions connecting all our dynamical variables A,,., Buss Wax bay, at a given time x. We cannot, as men- tioned above, work out the general quantum conditions connecting dynamical variables at any two points in space-time, because the interaction makes it too complicated. We shall therefore make an infinitesimal Lorentz transformation and work out the quantum con- ditions at a given time in the new frame of reference. If we can estab- lish that the quantum conditions are invariant under infinitesimal Lorentz transformations, their invariance under finite Lorentz trans- formations will follow. Let x* be the time coordinate in the new frame of reference. It is connected with the original coordinates by LE = Lyter,2,, (101) where ec is an infinitesimal number and », is a three-dimensional vector, ev, being the relative velocity of the two frames. We shall neglect terms of order «?. A field quantity « at the place x at the time x} in the new frame has the value K(X, @G) = K(X, %q)+ (uF —2) On, /x, = K(X, Xp) ev, a, [k,, A]. (102) § 79 THE INTERACTION 301 Its P.B. with another such field quantity A(x’, 2}) is [oe(x, 8), AC, 28] = Le, ap) Per, elke, HT, A(X, rp) +0, 2% Ey, HE] == [k(X, 2), A(X’, Xo) |+- ev, a[k,, (Ay, A]]+ tev, t,[ [ky H], Ay] == [k(X, #9), A(X’, Xp) ] + €%,(@,— 2, ) [kgs [Ayes AY + +ev,@ [ly Ay],H]. (103) If x and A are ¢ or } variables, we should be interested in their anti- commutator instead of their P.B. Using the notation (69) for the anticommutator, we have [x(x, #5), A(X’, #5) ] = [ee(X, LQ), A(x’, 7 ev, Xp [Kxs [Ags Ayl.+ Ev, 2A LK; #1), Agle == [(X, 29), A(X", Xp) |. + €Y,(2,— 2) [ks ges AT] + 60, OKs Ay das HI. (104) With « and A any two of the basic variables A,,, B,, $a; #,, the P.B. [x Ay] or anticommutator [x,,A,],, a8 the case may be, is a number, and so the last term in (103) or (104) vanishes. We are left with [e(X, xO), A(X’, xo) 1. = [«(X, 2X); A(x’, 2%) 144° +€v,(2,—2,)[ xs [Axes Hp + Her], + €(2,— 4) [x Aw Hola, (105) where [x,A], denotes the P.B. or the anticommutator, as the case may be. From the form (90) for Hg we see that [A,,, Hy] can involve only the dynamical variables A,,.-, fax’s $a, and cannot involve any deriva- tives of these variables. It follows that [k,, [A,, Hg]].., if it does not vanish, will be a multiple of 8(x— x’) and will not contain terms with derivatives of 5(x—x’). Hence the last term of (105) vanishes. We can conclude that [«(x, 2%),A(x’,v%)], has the same value as when there is no interaction, and is thus Lorentz invariant from our earlier work. A possible criticism of the above proof'should be noted. At several places we worked out expressions in powers of « and neglected ¢?. Such a procedure cannot be valid for calculating [k(x),A(z’)], with « and a’ two general points in space-time lying close together, so that €y— Ly is of order «, because the result of the calculation should be a function of the (x,—«',)’s having a singularity when the 4-vector a—z’ lies on the light-cone and such a function, of course, cannot be ep expanded as a power series in the (x,—a,)’s. xX 302 QUANTUM ELECTRODYNAMICS § 79 To validate the argument we should reformulate it so as to avoid the use of the 6 function. Instead of evaluating [x(x, 7), A(x’, x%)],., we should evaluate [ J ae te(x. ach) da, [by ACK, 28) Bx] , (106) + where a, and b, are two arbitrary continuous functions of 2,, Zp, X3. Then the quantities that we need to expand in powers of « all vary continuously with a continuous change in the direction of the time- axis, and the expansions are justifiable. The equations that we now get are those of the previous argument multiplied by a,b, d?xd32' and integrated. We are led to the same conclusion—that the P.B. or anticommutator has the same value as when there is no interaction. It will be seen that the reason why the interaction does not disturb the quantum conditions is because it is so simple, involving only the basic dynamical variables and not their derivatives. The P.B.s and anticommutators have the same values as with no interaction pro- vided they refer to variables at two points in space-time that are at the same time with respect to some observer. This means the two points must be outsidé each other’s light-cones and may approach coincidence only along a path lying outside the light-cone. 80. The physical variables A ket |P> that represents a physical state must satisfy the supple- mentary conditions (By+A,")|P) = 0, (div €—4ij,)|P> = 0. (107) A dynamical variable is physical if, when multiplied into any ket satisfying these conditions, it gives another ket satisfying these con- ditions. This requires that it shall commute with the quantities B,+A,, div €—4ij,. (108) Let us see what simple dynamical variables have this property. The transverse field variables 4, #, evidently commute with the quantities (108) and are physical. The variable 4, commutes with the first of the quantities (108) but not the second and is thus not physical. We have Mars Pr Dox = (tax Doe toy: Barbe? : = Sap 8(X— X’ bry: = Pax (X—X’). Thus [baxrSow] = 4e/Hi Wag 5(X—X'). (109) § 80 THE PHYSICAL VARIABLES 303 From (42) [eiePalt, div 6,.] = Arvie/fi.c%”2!"8(x—x’). Hence [e* V xl as div ey —_ Arrjoy:| = [e% Yall, div EO Wax — drrete lh [bar Jox’] Thus if we put fix = CM alhh (110) $*, commutes with both expressions (108) and is physical. Similarly ot, is physical. The variables 4, Z,, $*, $* are the only independent physical variables, apart from the quantities (108) themselves. We have Thus the charge density and current are physical. Also it is easily seen that € and # are physical, just as in the case when there are no electrons and positrons present. All those variables are physical that are unaffected by the arbitrariness that exists in the electro- magnetic potentials in the Maxwell theory. The operator ¢%,, represents the creation of a positron or the annihilation of an electron at the piace x. Let us see what is the physical significance of the operator Y*,. From (44) ihfeteralt, C,.] = celePal(x,—a1)/x—x’ |, and hence this Eng | = eb, (4,—2,)|K— x’ | F or Org! Pix = Parl Ere + (Lp —2,) |X! — x|-3}. (1 12) Take a state |P) for which €, at a certain point x’ certainly has the numerical value c,, so that €,4|P) = ¢,|P). Then from (112) Ore! bax|P) — {c, : e(2, ,) |X x|3}bax|P), so for the state Y%.|P), €, at the point x’ certainly has the value C+ e (a —a,.) |X’ — x |, This means that the operator i7,., besides creating a positron or annihi- lating an electron at the point x, increases the electric field at the point x’ by e(w,—2,)|x’—x[-°, which is just the classical Coulomb field at x’ of a positron with chatge e at the point x. Thus the operator $3, creates a positron at the point x together with tis Coulomb field, or else annihilates an electron at x together with its Coulomb field. - For electrons and positrons interacting with the electromagnetic 304 QUANTUM ELECTRODYNAMICS § 80 field it is the variables ¥*, {*, rather than the variables ¢, J, that correspond to the physical processes of creation and annihilation of electrons and positrons, since these processes must always be accom- panied by the appropriate Coulomb change in the electric field around the point where the particle is created or annihilated. It is easily seen that the variables %,, #%, satisfy the same anticommutation relations (70) as the unstarred variables. When we pass to the momentum representation the important quantities will be, not the unphysical variables 4, defined by (67), but the physical variables 4% defined by pe = h-i | eiemnys Bp, fk = ht [ entry dx. (113) We must now replace (68) by 1 Oy P+ a, 7) 5, 1 Op Deb Lay, eo * _ * & 5! ae ea ct p (p2-bm2)F Yip» and take & to represent the annihilation of an electron of momentum p, & the creation of an electron of momentum p, 5 the creation of a positron of momentum —p and ¢* the annihilation of a positron of momentum —p. The variables ut, J%, &, E, Cf, T* will all satisfy the same anticommutation relations as the corresponding unstarred variables. We can express the Hamiltonian entirely in terms of physical variables. We have pr = cM icf. Vp). Thus . Hp-+ Hg = | (pte —itiy eof" — Vr] tft, map-+-AY%jg} dx = i {hhc (— ithe? — ed Wf*) + p*ta, mp *+A%} Ba, The last term in the integrand here should be combined with Hp,. From (49) and (57) Fry, & —(8n)* [ (U~Ag)(U+Ag)” dx A [ (U—Ag) jp Bx with the help of (99). Thus Hp, + | A%y da = $ [ (T+ Ap) jg Ce. Integrating (99) with the help of formula (72) of § 38, we get AytU, & [ tox gay’, J |X—x’| § 80 THE PHYSICAL VARIABLES 305 and hence H, A%. Ba mz JoxJox’ gud dqy’. Thus we get H = H* with Ht = | (pita, (itp — esd) + Pq mp} Part 1 JoxJox’ i - OxJ/ 0x Bayan! Herts | | 2 x eee (114) We may use H* instead of H as our Hamiltonian. It leads to tne same Schrodinger equation for a physical ket, since if |P> is physical H*|PS = A\P). Also it leads to the same Heisenberg equations of motion for physical variables, since if € is a physical variable [6 H*] = [, A> Thus H* and H are equivalent Hamiltonians for the physical quanti- ties, and the others do not matter. H* involves only physical variables. The longitudinal field variables do not appear in it. Instead of them we have the last term of (114), which is just the Coulomb interaction energy of any charges that are present. The appearance of such a term in a relativistic theory is rather strange, as it is an energy associated with the instantaneous propagation of forces. It appears as a result of our having transformed the theory a long way from the Heisenberg form in which the relati- vistic invariance of the theory is manifest. We could set up a representation by taking as standard ket the product of the standard ket |0,> for the electromagnetic field alone, given by (61) and (62), with the standard ket |0p> for the electrons and positrons alone, given by (74). This representation would not be a convenient one, however, because its standard ket does not satisfy the second of the supplementary conditions (107 ). We get a more convenient representation if we take another stan- dard ket |Q) satisfying (By+4,")\Q> = 0, (div €—4njy)|@> = 0, (115) Dy\Q>=0, EIQ = 0, Lay @> = 0. (116) These conditions are consistent, because the operators on |Q> in them all commute or anticommute with each other, and there are enough of them to fix |Q) completely, apart from a numerical factor, 306 QUANTUM ELECTRODYNAMICS § 80 because there are as many of them as of the conditions for |0,)|0p). The conditions (115) show that |@Q> satisfies the supplementary con- ditions and so represents a physical state. The conditions (116) show that |Q> represents a state for which there are no photons, electrons, or positrons present. Any ket |P> that satisfies the supplementary conditions (107) and so represents a physical state can be expressed as some physical variable multiplied into |Q>. The only independent physical vari- ables that give non-vanishing results when applied to [Q> are rk? fap Cap. Hence |P> = Ya, ee ap) |@Q>- (117) Thus |.P> isrepresented by a wave functional ¥ involving the variables t,,, E*,, Ct. It is a power series in these variables, the various terms in it corresponding to the existence of various numbers of photons, electrons, and positrons, with the Coulomb fields around the electrons and positrons. Tn using the representation (117) together with the Hamiltonian H * we have a form of the theory in which we can ignore the conditions. (115), as,they have no effect on the kets (117). We must retain the conditions (116). The longitudinal variables then no longer appear in the theory. | 81. Interpretation The foregoing work establishes the basic equations of quantum electrodynamics. There are two forms of the theory, involving the Hamiltonians H and H* respectively. We must now: consider the interpretation and application of the theory. We shall take the H* form for definiteness. The argument would be essentially the same with the 7 form. ) The ket |Q> represents a state for which there are no photons, electrons, or positrons present. One would be inclined to suppose this state to be the perfect vacuum, but it cannot be, because it is not stationary. For it to be stationary we should need to have A*IQ> = CQ) with C a number. Now H* contains the terms —e | Ptta, fr Pe +5 { [ deste areata’, (118) x—-x"| which do not give numerical factors when applied to |Q>) and which therefore spoil the stationary character of IQ. § 81 DIFFICULTIES OF THE THEORY 307 Let us call the state @ represented by |Q> the no-particle state at a certain time. If we start with the no-particle state it does not remain the no-particle state. Particles get created where none previously existed, their energy coming from the interaction part of the Hamil- tonian. To study this spontaneous creation of particles, we take the ket |Q> as initial ket in the Schrodinger picture and treat the terms (118) as a perturbation giving rise to a probability of the state Q jumping into another state, in accordance with the theory of §44. The first of them, resolved into its Fourier components, contains a part —~e(aan [| 7 Eh Bossa Phd, (119) which causes transitions in which a photon is emitted and simul- taneously an electron-positron pair is created. After a short time the transition probability is proportional to the squared length of the ket formed by multiplying (119) into the initial ket |Q>, which is €?( a) an (Osea X . [Pf Ole ann Be Mc Le By Cl suenl@> Phd pdh'dp! = (&)arlasdea | fff Ql, Lex x [oop Sep lea psa SFpsenle|@> Phd? pdrh'd?p’. Using the values of the P.B. and anticommutators given by (4), (16), (72), (73), we get an integrand which depends on the k, k’ variables according to the law |k|-18(k~—-k’) for large values of k and k’. This gives an integral that diverges, so the transition probability is infinite. The second term of (118), resolved into its Fourier components, contains terms like & &%, Cf. CF. yp, which cause transitions in which two electron-positron pairs are created simultaneously. One can calculate the transition probability as before, and one finds again that it is infinite. From these calculations one can conclude that the state Q is not even approximately stationary. A theory which gives rise to infinite transition probabilities of course cannot be correct. We can infer that there is something wrong with quantum electrodynamics. This result need not surprise us, because quantum electrodynamics does not provide a complete description of nature. We know from experiment that there exist other kinds of particles, which can get created when large amounts of energy are available. All that we can expect from a theory of quantum 308 QUANTUM ELECTRODYNAMICS § 81 electrodynamics is that it shall be valid for processes in which there is not enough energy available for these other particles to be created to an appreciable extent, say for energies up to a few hundred MeV. Thus the high-energy part of the interaction energy (118) is quite unreliable, and it is this high-energy part that is responsible for the infinities. It appears that we must modify the high-energy part of the inter- action. At present there does not exist any detailed theory of the other particles and so it is not possible to say how it ought to be modified. The best we can do is to cut it out from the theory altogether, and so remove the infinities. The precise form of the cut-off and the energy where it is applied will be left unspecified. Of course, the cut-off spoils the relativistic invariance of the theory. This is a blemish which cannot be avoided in our present state of ignorance of high- energy processes. Even with a cut-off the no-particle state Q is not approximately stationary. It therefore differs very much from the vacuum state. The vacuum state must contain many particles, which may be pictured as in a state of transient existence with violent fluctua- tions. Let us introduce the ket |V> to represent the vacuum state. It is the eigenket of H* belonging to the lowest eigenvalue. Here and sub- sequently H* denotes the expression (114) modified by the cut-off. One might try to calculate |V> as a perturbation of the ket |Q), but such a method would be of doubtful validity, because the difference between |V> and |Q> is not small. No satisfactory way of calculating |V> is known. In any case the result would depend strongly on the cut-off, and since the cut-off is unspecified the result would not be a definite one. . It follows that we must develop the theory without knowing [V). This is not a great hardship, because we are not mainly interested in the vacuum state. We are mainly interested in states which differ from the vacuum through having a few particles present in addition to those associated with the vacuum fluctuations, and we want to know how these extra particles behave. For this purpose we focus our attention on an operator K representing the creation of the extra particles, so that the state we are interested in appears as KiV). We do not know how the ket |V> varies with the time in the Schré- dinger picture, since we do not know the lowest eigenvalue of H*. To § 81 DIFFICULTIES OF THE THEORY 309 avoid this difficulty we work in the Heisenberg picture in which |V> is constant. We then require K|V> to represent another state in the Heisenberg picture and thus to be another constant ket. This leads to dK |dt = 0. (120) Usually K will involve the time explicitly as well as Heisenberg dynamical variables, so (120) gives ihoK |ot-+-KH*—H*K = 0. (121) We now have each physical state determined by a solution K of (120) or (121). We obtained this result without knowing the vacuum ket |V>, and we can proceed to study K without knowing |V)>. The only further information about K that we would have if we did know |V> would be that two K’s, say K, and K,, would correspond to the same state if we had (K,—K,)|V> = 0. But we can get on without this further information and count all different K’s satisfying (121) as corresponding to different states. We are thus led to a drastic alteration of one of the basic ideas of quantum mechanics, namely to represent a state by a linear operator and not a ket vector. This alteration is brought about by the complexities of applying quantum mechanics to a field and by our ignorance of high-energy processes. A trivial solution of (120) or (121) is K = 1. This evidently corre- sponds to the vacuum state. A general solution may be put in the form of an explicit function of t and of the dynamical variables at time t. Let us use the symbol 7. to denote collectively the emission operators at time #. Thus 7 equals one of the variables 7,,, ip, lap at the time ¢ in the Heisenberg picture. The absorption operators are then 7,. A solution of (121) then “ppeses K = f (tempi). (122) We require some physical interpretation for the state represented by this K, as the usual physical interpretation of quantum mechanics, requiring a state to be represented by a ket, is no longer applicable. We shall need to make some new assumptions. Keeping to the Heisenberg picture, we introduce at each time t the ket |Q,> satisfying the conditions (116) with respect to the Heisenberg dynamical variables at time ¢. These conditions may now be written MQ = 0. The ket |Q,> corresponds to no particles existing at the time ¢ and it provides a reference ket for the discussion of general states at time t. 310 QUANTUM ELECTRODYNAMICS § 81 For any state fixed by a solution K of (121) we form K|Q> and assume that this ket determines what can be observed at the time t and is to be interpreted according to the standard rules. We obtain K in the form (122) and then arrange it so that in each term all the absorption operators 7, are to the right of all the emission operators 7. It is then said to be in the normal order. Any term in K containing an absorption operator then contributes nothing to K|Q,5. The surviving terms in K|Q,p will contain only emission operators, like (117). Each surviving term is associated with certain particles in particular states, and the square of the modulus of its coefficient (with the appropriate factors n! when there is more than one boson in the same state) is assumed to be, after normalization, the probability of these particles existing in these particular states at the time ¢. We now have a general method of physical interpretation which is rather similar to the usual one, but there are important differences. A term in &K with an absorption operator on the right will not con- tribute to K|Q,> and so will not contribute anything observable at time ¢. We may call it a latent term at the time?. Such a term cannot be discarded as non-existent, because it will contribute observable effects at other times. These latent terms are a new feature of the theory and are to be understood as an incompleteness in the descrip- tion of a state in terms merely of the particles which can be observed. to be present at a certain time. As a consequence of the occurrence of latent terms, if K|Q,> is normalized at one time, it will usually not be normalized at other times. We thus have to carry out a separate normalization for each time in order to derive the probabilities. 82. Applications There are two important applications of the foregoing theory in which effects are calculated that cannot be obtained from a more primitive theory. These applications are concerned with a single electron in a static electric or magnetic field. As a consequence of the interaction of the electron with electromagnetic waves, the energy levels are shifted somewhat from their values given by the elementary theory. The important cases are: (i) An electron in the Coulomb field of a proton. The theory here leads to a shift in the energy levels of the hydrogen atom, It is named the Lamb shift, after its discoverer. (ii) An electron in a uniform magnetic field. The extra energy is APPLIGATIONS 311 here interpreted as arising from an extra magnetic moment of the electron, called the anomalous magnetic moment. . To take a static field into account one mercly has to introduce potentials to describe it and add them on to the potentials in the Hamiltonian. The potentials of the static field are functions of %4, Xa, %, only, and are numbers for each x,, %, %, not dynamical variables, so their introduction does not increase the number of degrees of freedom. The calculations of the Lamb shift and anomalous magnetic moment are rather complicated. They are given in detail, working from the Hamiltonian H, in the author’s book Lectures on Quantum Field Theory (Academic Press, 1966). The results are in good agreement with experiment and provide a confirmation of the theory. These calculations were made in terms of the Heisenberg picture throughout. One may tackle quantum electrodynamics on the Schrédinger picture, looking for a solution of the Schrodinger equation by taking the no-particle ket, or a ket corresponding to just a few particles present, as the initial ket of a perturbation procedure and applying the standard perturbation technique. One finds that the later terms are large and depend strongly on the cut-off, or are infinite if there is no cut-off. The perturbation procedure is not logically valid under these conditions. Nevertheless people have developed this method a long way and have devised werking rules for discarding infinities (in a theory without cut-off) in a systematic manner, so that finite residual effects remain. The procedure is described in many books, e.g. Heitler’s Quantum. Theory of Radiation (Clarendon Press, 1954). The original calculations of the Lamb shift and anomalous magnetic moment were carried out on these lines, long before the corresponding calculations in the Heisenberg picture. The results are the same by both methods. I do not see how these calculations based on the Schrédinger picture, supplemented by some working rules, can be presented as a logical development of the standard principles of quantum mechanics. The Schrédinger picture is unsuited for dealing with quantum electro- dynamics, because the vacuum fluctuations play such a dominant role in it. These fluctuations present great mathematical difficulties, and also they are not of physical importance. They get bypassed when one uses the Heisenberg picture, and one is then able to concentrate on quantities that are of physical importance.