The application of quantum electrodynamics to multiple ...rspa.royalsocietypublishing.org/content/royprsa/187/1009/210.full.pdf · The application of quantum electrodynamics to multiple

210 C. Jayaratnam Eliezer

R e f e r e n c e s

Bhabha, H. J. 1942 Presidential Address, Agrp, Session, National Academy of Sciences (India).

Dirac, P. A. M. 1935 Quantum mechanics. Oxford University Press.Dirac, P. A. M. 1939 Ann. Inst. Poincare, 9, 13.Dirac, P. A. M. 1942 Proc. Roy. Soc. A, 180, 1-40.Heisenberg, W. & Pauli, W. 1929a Z. Phys. 56, 1—61.Heisenberg, W. & Pauli, W. 19296 Z. Phys. 59, 168-190.Pauli, W. 1943 Rev. Mod. Phys. 15, 175—207.

The application of quantum electrodynamics to multiple processes

B y C. J ay a r a tn a m E l ie z e r , University of Ceylon

{Communicated by P. A . M. Dirac, F.R.8.—Received 12 September 1945)

The higher approximation terms of the interaction of an electron and a radiation field, which were obtained by the author in the previous paper, are applied here to investigate the probability of multiple scattering processes. It is shown that the probability of certain of these processes could be large. In particular it is shown that the probability of a photon dividing into a number of photons in the presence of an electron is appreciable under certain circumstances. This result shows that quantum electrodynamics does not disallow, as was believed earlier, the existence of photon showers in which the quanta are emitted simultaneously.

I n tr o d u c tio n

1. The well-known difficulty of occurrence of infinite integrals in the higher approximation terms of the interaction of an electron and a radiation field has made it so far not possible to consider exactly various multiple scattering processes. Recently, the author (Eliezer 1946)* has investigated the interaction of an electron and a radiation field, on the basis of Dirac’s formulation of quantum electrodynamics (Dirac 1939, 1942), and shown that the higher approximations of this interaction are free from the usual infinite integrals. The three features of this theory which bring about the elimination of these divergent integrals are:

(i) the A-limiting process, which is used to express the equations of motion in Hamiltonian form, and which introduces a convergence factor of the form exp { — i(Jc, A)} in the commutation relations;

(ii) the use of negative-energy photons as well as positive-energy photons in the process of second quantization;

(iii) the use of only those solutions of the wave equation which correspond to outgoing waves of the electron.

* Referred to here as I.

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The usual difficulties in quantum electrodynamics being thus removed, it becomes now possible to investigate exactly various multiple processes involving any number of photons, whereas previously all exact calculations had to be restricted to pro­cesses in which at most two photons take part. These multiple processes are important for the following reasons. These are typical quantum effects, and their probability cannot.be estimated from the classical theory by using the correspondence principle. Also, the existence of showers in cosmic radiation makes it desirable to know if in photon showers the quanta are emitted simultaneously or one after another in a short range. By calculating the order of magnitude of the transition probabilities for these multiple processes we shall be able to see whether these probabilities as given by the present theory of quantum electrodynamics are large enough to explain the phenomenon of photon showers.

In this paper we consider scattering processes in which an electron and three photons are involved. The calculations can then be extended in a straightforward way to processes involving a larger number of photons. We follow here a method which was used by Dirac (1930) to calculate the scattering probability in the Compton effect. I t will be shown that the theory does not disallow the possibility of photon showers in which the quanta are emitted simultaneously in a single process. This conclusion is contrary to that of Heitler (1936), who has also considered this problem but who arrives a t the result that the theory does not explain the production of showers. We believe that this discrepancy is due to the fact that Heitler restricts all the frequency of all the emitted photons to be of the same order of magnitude. If, instead, we permit a t least one photon to have a low frequency while the frequencies of the rest can be of any order, then it will be seen that the probability of the multiple processes may become considerable. We shall show that the probability of a multiple process varies as &J"1, where k0 corresponds to an emitted photon of low frequency. In this way we see that quantum electrodynamics does support a theory of showers which supposes that all the light quanta are emitted simultaneously.

The application of quantum electrodynamics to multiple processes 211

T h e w ave eq uatio n a n d its solution

2. Following the notation employed in I, the wave equation for the motion of an electron in an electromagnetic field is

[p0- e A 0- a .( p - e A ) - m f i ] f t = 0, (1)

where A fl is the four-vector potential of the electromagnetic field. For the present problem A 0 may be taken to be zero. I t is convenient to take units in which c, the velocity of light, is unity, and h/27r, where h is Planck’s constant, is also unity, xjr is taken as a matrix with four rows and four columns, instead of the usual column matrix, since this representation is more convenient for averaging over the initial states, which averaging becomes necessary later.

1 4 - 2

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The equation (1) is solved by a perturbation method by expressing the wave function as a series in ascending powers of the charge e of the electron, thus

ir = ^ 0+ e^ri + e2 r2+ — (2)

I t is assumed that e is small enough for such an expansion to be valid. The successive terms i/r0, ... are connected by equations of the form

( P o - V ' P - m f l f t n = - (< * -A )^ » -i* (3)

The solution has been investigated in detail in I. I t has been shown that if we take only those solutions which correspond to outgoing waves of the electron then all the divergent parts, up to any order n in the approximations, become eliminated. We shall therefore assume that the solution is free from divergences, and consider only that part of the wave function which is appropriate for transition processes in which three photons take part.

In general, the wave function \Jrn will consist of a -sum bf terms of the type ftn,n> & n,n-2>fn,n-ir> • ••> where denotes tha t part of \Jrn which refers to mphotons. Hence, the wave function \]r that should be taken to deal with transition processes in which three photons take part is

f = «V * 3 + eV s , 8 + — + e2W+1f2 » + l, 8 + ••• • (4)

I t was shown in I tha t if we take only those solutions which correspond to out­going waves of the electron, then

xlrn ,m = 0 for all m =j= w. (5)

Hence, for the transition process under consideration, we take

f = eY3.s- (6)To obtain 3 we may take the vector potential A of the electromagnetic field

expressed in its Fourier components, thus,

A = ae**’*), (7)

where the scalar product notation

(a,b) = a ftp = a060- a . b

is employed, and kJ2n gives the frequency and the momentum and a the direc­tion of polarization of the photon, and k2 = J k 2 = 0.

Suppose that initially the electron is a t rest. The problem in which the electron is initially not at rest can be considered by an appropriate Lorentz transformation. The initial wave function is then

ifr0 =* ue~«, (8)

where uis a matrix of four rows and four columns, which satisfies the equation

(l-/? )w = 0. (9)

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The application of quantum electrodynamics to multiple processes 213

The wave equations tha t have to be solved are all of the form

(p0- a . p - rift) f ei(b> (iq)

where 6 = (60,b) is a four-vector. I t is easily seen tha t the solution of (10) is

^ — (i>o-'P2~ m2)-1 (i>o + a • P + mft) e<(6,— {(m — 60)2—b 2 — m2}-1 (m — — a . b + mfi) e^6* , (11)

provided (m — 60)2 — b 2 — m2 is not zero.

I t is convenient to employ the following notation:

Rb = m — b0 — a .b + m/3, y b = (m — b0)2—b2 — ra2. (12)

By solving the equations (3) successively we readily obtain= - y k l R k (a. a) u e Hk,x)-imx0j (13)

^ 2,2 = 7k+k'7klR k+Aa•a') R k(a •a) ueM+k'.ximx0, (14)1 8.3 = - y*+*'+fc'yfc+*'7fc fc+fc'+fc a • a") Rk+k’(<*.. a') I2fc(a . a) (15)

T h e t r a n sit io n pr o b a b il it y

3. In general ^ 3 3 is periodic in the time. Exceptions arise, however, when any of the terms of the denominator of (15) vanish, that’is, when any of y k, y k+k' and Yk+k'+k? vanish. When this happens the solution must be modified.

When y k is zero, k0 is zero; and this case is not of interest for our problem here. The vanishing of y k+k> is the condition of conservation of energy and momentum for a transition process in which the two photons Tc and k' are emitted or absorbed, according to the sign of k0 and k'0. This process is also not of interest to us here, since we are concerned with a process in which three photons take part. The vanishing of Yk+k’+k’ is the condition of conservation of energy and momentum for a process in which three photons k, k ', k" are emitted or absorbed. I t is easily seen that yk+k> and yk+k'+k' cannot both vanish simultaneously for non-zero k0, k'0 and

To determine the solution when any of the y’s vanish we see from the work in paper I that the solution of the equation (10) which is valid for a range in which yb is zero is

f = r r j ( „/ 1 " ~ inS{p'0-p ) ) Rbe ^ ~ imxo, (16)Po+Po lPo~ P o I

where p'0 = m — b0,= m2+ b 2, and the term in i7r8(p'0—p 0) is introduced to makethe solution correspond to outgoing waves of the electron.

Suppose the three photons concerned with in the problem are such that k0, k'0, K,yk'+kr, Yic’+k and Yk+k' are a^ non-zero, but yk+k’+k- is zero. In this case we shall

see that t/r3 3increases with the time, which shows that transition processes are taking place with the continual appearance of electrons of energy and momentum p , where

p'0 = m - ( k 0 + k'Q + ko), p ' = - (k + k '+ k " ) . (17)

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Since y &+&'+&» — p '^ —p '2 — m2 the vanishing of y k+k'+k~ is equivalent to

p'02- p ' 2- m 2 = 0, (18)

which may also be written as

m(iko+K + K) = - k ' . k #) + (&£ k) - k , k'}. (19)Probabilities with a physical significance are obtained by supposing tha t the

photons are such that (18) does not hold exactly and that a small correction q is made in p'0 for (18) to be satisfied. Then

tyo - - W - P ,2-™ 2)/2Po. (20)As in Dirac’s paper (1930, p. 370) we obtain as a solution which remains finite as Sp'0 tends to zero and which is appropriate to consider these transition processes,

1 3,3 = yk+k’7* 1 k+k'+kr(a •a ") Rk;+k’(a*a ) k(a -a) (1 — e~ltpoxo)/2p'0 8p0. (21)The term, which gives rise to the transition process under consideration, is obtained by summing (21) for all possible interchanges of k' and k". Let (1, T, 1") represent unit vectors along the direction of motion and (m, m', m") unit vectors along the direction of polarization of the three photons. Let (a, a', a") = (/cm, m', /c"m"). We also define the vectors (n, n', n") where n = 1 x m , with similar expressions for

214 C. Jayaratnam Eliezer

n' and n". The appropriate Wave function is then

\Jr = ezKK'k"Rk+k'+kr Uuei(p,’x\ l — e~iSp'ox»)j2p'0Sp'0, (22)

where U = tyk+vykl(« • m") Rk+k>(a. m ') . m), (23)

Z denoting summation of all terms obtained by interchanging k, k' and k".The density of electrons represented by (22) is the diagonal sum of the matrix

\Jr<j), where ^ is the Hermitian conjugate of ijf. Hence the density of electrons is

|e 6(# r W ac")2 F { \ - cos 8p'0x0)l8p'02, (24)

where r is the diagonal sum of

Rk+k’+k" UuuURk+k’+k*. (25)

For further computations it is convenient to express the matrix u in terms of a and /?. I f we take an initial distribution of electrons with no preferential direction of spin, then u must involve /? only, and hence we take, consistent with (9) and without loss of generality,

w = l+ /? . (26)

The initial density of electrons is given by the diagonal sum of fr0<p0 which is easily seen to be 8. The probability of transition is given by the ratio of the final and initial densities. We may express the probability in terms of the intensities and I" of the three photon beams by substituting k2 = (2n)21k^ 2 and similar expressions. We thus obtain for the probability

4e«n«(p'0k0k'0K)-2i r i ' T { \ - cos 8p'0x0)/8p'02, (27)

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To obtain a transition probability with a physical meaning we suppose one of the beam of photons, the ^"-photon say, to be not sharply monochromatic, but to have intensity per unit frequency range about the appropriate frequency for transi­tions. The total transition probability is thus

W )-2 i r r Kr j 1 1 <28>

To evaluate this integral we express the variation of in terms of the variations of Sp'o by using (20). Then the integral in (28) has the value

PoK f 1 - cos Sp'0 . . . 1 p'0 hi xQ2”7k+k'J W P° 2

where we have made yse of the condition (19).Substituting for = kl*/2n we get as the probability per unit time of a process

in which the "-photon is spontaneously emitted and the and -photons stimulated

e«n% UITI(4p'0klk'0*7k+k,). (29)

The above calculations will also apply to processes in which any of these photons may be absorbed, provided we change the sign of the frequency corresponding to each photon that is absorbed. Thus from (29) we may deduce the probability of annihilation of an electron and a positron by the emission of three photons, or of the splitting of one photon into two photons, or of the combination of two photons into one photon.

For further discussion we need to know the value of Its evaluation involves rather long calculation and is done in the Appendix. When we substitute the value of r and sum the resulting expression for all directions of polarization of the three photons, (29) takes the form

r r , to im2klk'0 *yk+k’L7l #+*' yfc+fc'Tfe+fc'J*

where the pCs are homogeneous expressions of the third degree in m, k0, k'0 and k^.If we substitute I = ky2ir, and similarly for I ', we obtain as the probability of a

triple spontaneous emission

r _ ^ _ + m4 m 'Yk+k'L rl+fc' Jk+ k’yk+ k 'A

per unit sohd angle of direction of emission of the ^-photon about 1, ^'-photon about I', and A: "-photon about 1", and also per unit frequency ranges of the k- and '-photons.

By expressing (31) as a probability per unit energy range for the final electron, and then integrating over all directions of emission, we obtain the total probability of the electron making a transition with emission of three photons. By assuming now that all the negative energy states are occupied except one, and that the un­occupied one corresponds to a positron, we obtain the probability of annihilation of an electron and a positron by the emission of three photons.

The application of quantum electrodynamics to multiple processes 215

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216

Suppose we now consider the probability of the double Compton scattering, tha t is, the probability of one photon, say the &-photon, splitting into two photons k' and k". We must then change the sign of k0 in all the above expressions. The prob­ability is then seen to be

^ w / j r _ i L . ___v i , , 212 »»ais y k+k. Lrl+n-

per unit solid angle of direction of emission of -photon about T and -photon about 1", and per unit frequency range of the -photon. The A’s are obtained by changing the sign of k0 in the corresponding s.

In estimating the order of magnitude of the probability there are two cases of interest to be considered separately.. One is when the frequencies of the emitted photons are of the same order of magnitude. This case has been considered by Heitler, who has shown that the probability of the double scattering is smaller than that of the single process by at least 1/137.

The other case of interest to be considered is when there is no restriction imposed on the smallness of the frequencies of the emitted photons. Suppose the emitted photons are such that the frequency of one of them can be as small as we please while the other frequencies can be of any order consistent with the condition of conservation of energy and momentum. I f k$ is small, then from the conservation of energy and momentum

‘Yw+K’-k — Tk'-k ~ 2K{m + &0(1 — 1.1*) — &o(l— i* • 1*)} = and therefore y^-k is also small. The expression (32) is then approximately

n W 0K*nil(2m *yl’- k), (33)where

K = 32?#{&jj + k'02 — &0 &o( 1 — 1. T2)} (1 — 1. T) — 1 6(&q — k0)x { 2 m 2( l + l . r a) - m ( ^ - * 0) ( l - l . r ) 2- f 2 ( ^ + ^ a) ( l ~ l . l / )+ I ;0^ ( l - l . r ) ( I + ! . r 2)}

+ 32 {(k'0 - k0)(1 - 1 . r ) - m( 1 +1. r 2)} {1. (k' - k)}2, (34)

where we have omitted certain terms which vanish in the limit k^ tends to zero.Since k^ is small, the expression (33) is rather large, since occurs as kl~x. The

order of k'0 is obtained from the condition that y k>-k is small. I f the incident photon is such that k0 is much smaller than the rest mass of the electron then k'0 is of the same order as k0. But if k0 is much larger than the rest mass of the electron, and if the angle between k and k ' is large then k'0 is of the same order as the rest mass, and if the angle is small then k'0 is of the same order as k0.

Hence the probability of a high-energy photon dividing into photons is large, if one of the emitted photons is of low frequency. The other emitted photon is then at least of moderately large frequency.

In the same way if we calculate the probability of a high-energy photon dividing into three or more photons, we see that this probability could be large provided one

C. Jayaratnam Eliezer

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at least of the emitted photons is of low frequency. The other emitted photons could be of moderately large frequencies. We see thus tha t the present quantum electro­dynamics does give large probability values for certain multiple processes. In particular, we see tha t the theory does not disallow, as has been believed earlier, the possibility of photon showers in which a large number of photons are emitted simul­taneously.

The author wishes to express his sincere thanks to Professor P. A. M. Dirac, who suggested this problem, for his guidance and supervision.

The application of quantum electrodynamics to multiple processes 217

A p p e n d ix

To evaluate T we need to know the value of From (23)

Uu = 1(«-m") Rk+k'(a.m ') RjXa.m) (1

= ^y*-M - 2mfc0)-1 (a ,m") Rk+Aa -m') {m(l - a . 1)} (a .m) (1

= (2w)-12 y j^ ^ (a . m") Rk+k.(a . m ') (1 + a . 1) ( a . m) (1 (35)

For further simplifications it is convenient if we express the matrices a and ft in terms of the Dirac matrices p and Pauli matrices a, where a = ft — p3, and wherethe p’s commute with the a ’s, and both sets satisfy the same commutation relations as the a ’s. Also, the following notation is helpful:

61 = 2(m '.m "), &2 = m '.n " + m " .n ', b = m 'x i i '+ m ’ xn ', (36)

with similar expressions for (b'x, b'2, b') and (6*, b\, b"), obtained by cyclic interchange. Then

(a . m ') (1 + a . 1) ( a . m) + ( a . m) (1 + a . T) ( a . m ') = + .b",

where we have made use of the formula (o . a) ( a . b) = a . b + . (a x b). Hence

Uu — (2m)~JlZ'yk+k'p1(a.in.")Rk+kf(bi + ip1b2 —p1a .h '')( l+ p gi).

Therefore UuuU = + PzY(37)

where

Pi = px{a. m )Rk'+kr{bx + ipxb2- p x<s.b) (1 + p3) (6j - .b )Rk'+kr{a . m )px,(38)

P2 = px{a . m") Rk+kr{b{ + ipxb'2- p xo .b") (1 + p3) - ipxb2- p xa .b ')Rk+k>(a. m

+px{a. m ') R k+k.(b'x + ip xb2 - p xa.b ') (1 +/>3) (bx - .b . m(39)

with similar expressions for P 2, P x and P 2.

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218

For our purpose in this paper it is sufficient to know the value of the Px’s. P2’s have the same order of magnitude as the Pj’s,

Pi = px(a. m ) P ^+ir{&!( 1 + pz) + (b\+ b 2) (1 - pz) + - 2bxpxa . b} . m)= A{\+p^) + B{l — pz) + Cpz +p xa.D, (40)

where

a = 2 (k'0* + k?) p - r . V) + *KK{b\v. r + + b 2 - bxb . ( r + r »

= 2(jfc' 2 + fcg2 + 2 ^ ( 1 - 2m ' . m "2)} (1 - 1 ' . V),

B = 4m26? - 4mk'0bx(bx—b. V) - 4m ^61(61 - b . 1") + 2 (&'2 + &o2) (1 - V 1")

+ 2k'0 K{b\+ (6i + b 2) ( r . v)- bxb . ( r + r )}

= 16m2(m '. m ")2- 8m(k'0 + K) (m '.m") ( m '. m" - n ' .n") + 2 + 1 - 1'. V)

- 4£;&o(1 - 1' • 1") {(m '. m") (n '. n") + ( m '. n") (m " . n')},

C = 46162{w(^o + K) - K k l(\ - r . 1")} = - 4 (m '. m") ( m '. n" + m " . n') y ^ +kr,

D *= 2(2(1'. m ) m - 1 ' } (2jfc'2( l - V. 1") - 2 m k’Qb + b 2 - 26xb .1")}

+ 2(2(1" . m ) m -1 " } ( 2 ^ 2( 1 - 1 '. V)- 2mkz b \+ k’Q kl{b\+ 6 f + b 2 - 26xb . 1')}

- 26x(2(b. m) m - b ) y v+k,

= + kz)(1 - 1 ' . 1") - 4m (m '. m")2} [k'0{2(l' m -1'} + K{2{\". m ) m -1"}]

- 4 (m '. m") (2(b. m ) m - b } y ft,+ r . (41)

The diagonal sum of

Rk+k'+k’{A (1 +P3) + R P ~~Pa) + C P 2 + P i ° • D} Rk+k'+kr

is the diagonal sum of

Rk+k'+k’Rk+k’+k"{A( 1 +^3) + P(1 ~ P z ) + Cp% + P i a • D}>that is, of

[2(m -fe0- f c '- ^ ) ( m ( l + /o3) - /o1o .(k + k '+ k ")} + (^0 + + ) 2 + (k + k '+ k '')2]

x {A(l + pz) + B(1 — pz) + Cp2+pxo .D},which is

8(m - k 0-k '0- k n0) {2mA - (k + k '+ k "). D} + 4{(&0 + k 'tf (k + k ' + k")2} (A + B)= 8(m — k0— k'0 — kQ)[2mA— (fc0 + &£ + fc3) (!d + P) — (k + k '+ k ") .D}, (42)

where we have made use of the condition y k+k>+k' = 0. Hence P is of the form

r = 4 ^ ~ 2p ,o^[Qi7k'+k' + Q zykh 'JkhA , (43)

where the Q’s are homogeneous expressions of the third degree in m, k0, k'0 and k'z,

Qx = 2 m A - (k0 + k '+ JQ (A - (k+ k '+ k"). D . (44)

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After substituting for A , B and D we obtain the total transition probability with either state of polarization for each of the three photons, by adding together the expressions obtained by replacing m by n and n by — m , and so on for the other two photons. The following formulae are helpful: I f 8 denotes the summation over all directions of polarization of the -photon, with similar meaning for S' and S", then

$ (a .m )(b .m ) = $ (a .n )(b .n ) = a .b — (l.a ) '(l.b ),

8{2l . m ) (b . n) = (1, a x b).

Hence it is seen tha t

y t = S S 'S nQ1 = S2m{k'0* + kno2 + k ’oro{ l - r . V 2) } { l - l 'A ,f)- 1 6(&0 + k'0 + k”0) (2m2( 1 + r . r 2) - m(k'0 + (1 - jfr

+ 2( t p + K*) (i - v . v) - KK{\-r . r ) ( i + r . r 2)}+ 32{(jfc' + ) ( l - r . n - m a + lM ^ K l.C k + k 'ljf l.C k + k '+ k ")}

- *Yv+A 1 + 1'-1") {1 • a # + n {l.(k+k'+k')}.By changing k0 to — k0, we see that

Mi = 32 m{k% + K2 - KKO- ~ 1 • l '2)} (1 — 1-1')- 16(^ - k0) {2m2(l + 1. 1'2) - m(k'0 - &0) (1 - 1. 1')2

+ 2( ^ + ^ 2) ( i - i . r ) + A : 0^ ( i - i - n ( i + i . r 2)}+ 32{(fc' - k0)(1 - 1. 1') - m(l + 1. 1'2)} {1. (k' - k)}2,

where certain terms which vanish with fcj have been omitted. This is the result quoted in the text.

The application of quantum electrodynamics to multiple processes 219

R e fe r e n c e s

Dirac, P. A. M. 1930 Proc, Camb. Phil. Soc. 26, 361. Dirac, P. A. M. 1939 Ann. Inst. Poincare, 9, 13.Dirac, P. A. M. 1942 Proc. Roy. Soc. A, 180, 1.Eliezer, C. J. 1946 Proc. Roy. Soc. A, 187, 197. Heitler, W. 1936 Quantum theory of radiation. Oxford.

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