(31)
where p · x21 = p · x2 − p · x1 = pµ x2µ − pµ x1µ, p = pµγµ, and d4p means (2π )−2dp1dp2dp3dp4, the integral over all p. That this is true can be seen immediately from (12), for the representation of the operator i∇ − m in energy (p4) and momentum (p1,2,3) space is p − m and the transform of δ(2, 1) is a constant. The reciprocal matrix (p − m)−1 can be interpreted as (p + m)(p2 − m2)−1 for p2 − m2 = (p − m)(p + m) is a pure number not involving γ matrices. Hence if one wishes one can writeK+(2, 1) = i (i ∇2 + m ) I+(2, 1),
where
(32)
is not a matrix operator but a function satisfying
(33)
where .
The integrals (31) and (32) are not yet completely defined for there are poles in the integrand when p2 − m2 = 0. We can define how these poles are to be evaluated by the rule that m is considered to have an infinitesimal negative imaginary part. That is m, is replaced by m − iδ and the limit taken as δ → 0 from above. This can be seen by imagining that we calculate K+ by integrating on p4 first. If we call then the integrals involve p4 essentially as which has poles at p4 = +E and p4 = −E. The replacement of m b m − iδ means that E has a small negative imaginary part; the first pole is below, the second above the real axis. Now if t2 − t1 > 0 the contour can be completed around the semicircle below the real axis thus giving a residue from the p4 = + E pole, or −(2E)−1 exp(−iE(t2 − t1)). If t2 − t1 < 0 the upper semicircle must be used, and p4 = − E at the pole, so that the function varies in each case as required by the other definition (17).
Other solutions of (12) result from other prescriptions. For example if p4 in the factor (p2 − m2)−1 is considered to have a positive imaginary part K+ becomes replaced by K0, the Dirac one-electron kernel, zero for t2 < t1. Explicitly the function is od(x, t = x21μ)
(34)
where for t2 > x2 and for t2 < x2, is the Hankel function and δ(s2) is the Dirac delta function of s2. It behaves asymptotically as exp(−ims), decaying exponentially in space-like directions.oe
By means of such transforms the matrix elements like (22), (23) are easily worked out. A free particle wave function for an electron of momentum p1 is u1 exp(−ip1 · x) where u1 is a constant spinor satisfying the Dirac equation p1u1 = mu1 so that The matrix element (22) for going from a state p1, u1 to a state of momentum p2, spinor u2, is −4π2i(ū2a(q)u1) where we have imagined A expanded in a Fourier integralA(1) = ∫ a(q) exp (−iq · x1) d4q,
and we select the component of momentum q = p2 − p1.
The second order term (23) is the matrix element between u1 and u2 of
(35)
since the electron of momentum p1 may pick up q from the potential a(q), propagate with momentum p1 + q (factor (p1 + q − m)−1) until it is scattered again by the potential, a(p2 − p1 − q), picking up the remaining momentum, p2 − p1 − q, to bring the total to p2. Since all values of q are possible, one integrates over q.
These same matrices apply directly to positron problems, for if the time component of, say, p1 is negative the state represents a positron of four-momentum −p1, and we are describing pair production if p2 is an electron, i.e., has positive time component, etc.
The probability of an event whose matrix element is (ū2Mu1) is proportional to the absolute square. This may also be written (ū1, where is M with the operators written in opposite order and explicit appearance of i changed to −i( is β times the complex conjugate transpose of βM). For many problems we are not concerned about the spin of the final state. Then we can sum the probability over the two u2 corresponding to the two spin directions. This is not a complete set because p2 has another eigenvalue, −m. To permit summing over all states we can insert the projection operator (2m)−1(p2 + m) and so obtain for the probability of transition from p1, u1, to p2 with arbitrary spin. If the incident state is unpolarized we can sum on its spins too, and obtain
(36)
for (twice) the probability that an electron of arbitrary spin with momentum p1 will make transition to p2. The expressions are all valid for positrons when p’s with negative energies are inserted, and the situation interpreted in accordance with the timing relations discussed above. (We have used functions normalized to (ūu) = 1 instead of the conventional (ūβu) = (u*u) = 1. On our scale (ūβu) = energy/m so the probabilities must be corrected by the appropriate factors.)
The author has many people to thank for fruitful conversations about this subject, particularly H. A. Bethe and F. J. Dyson.
APPENDIX
a. DEDUCTION FROM SECOND QUANTIZATION
In this section we shall show the equivalence of this theory with the hole theory of the positron.2 According to the theory of second quantization of the electron field in a given potential,of the state of this field at any time is represented by a wave function χ satisfyingi∂χ/∂t = Hx,
where H = ∫ ψ*(x)(α · (−i∇ − A) + A4 + mβ)ψ(x)d3x and ψ(x) is an operator annihilating an electron at position x, while ψ*(x) is the corresponding creation operator. We contemplate a situation in which at t = 0 we have present some electrons in states represented by ordinary spinor functions ƒ1 (x), ƒ2(x), . . . assumed orthogonal, and some positrons. These are described as holes in the negative energy sea, the electrons which would normally fill the holes having wave functions p1 (x), p2 (x), .... We ask, at time T what is the amplitude that we find electrons in states g1 (x) , g2(x) , . . .and holes at q1 (x), q2(x),
.... If the initial and final state vectors representing this situation are χi and χƒ respectively, we wish to calculate the matrix element
(37)
We assume that the potential A differs from zero only for times between 0 and T so that a vacuum can be defined at these times. If χ0 represents the vacuum state (that is, all negative energy states filled, all positive energies empty), the amplitude for having a vacuum at time T, if we had one at t = 0, is
(38)
writing S for exp(−i . Our problem is to evaluate R and show that it is a simple factor times Cν, and that the factor involves the functions in the way discussed in the previous sections.
To do this we first express χi in terms of χ0. The operator
(39)
creates an electron with wave function φ(x). Likewise Φ = ∫ φ*(x) × ψ(x)d3x annihilates one with wave function φ(x). Hence state χi is χi = while the final state is where Fi, Gi, Pi, Qi are operators defined like Φ, in (39), but with ƒi, gi, pi, qi replacing φ; for the initial state would result from the vacuum if we created the electrons in ƒ1, ƒ2 . . ., and annihilated those in p1, p2,.... Hence we must find
(40)
To simplify this we shall have to use commutation relations between a Φ* operator and S. To this end consider and expand this quantity in terms of ψ*(x), giving ∫ ψ* (x)φ(x, t)d3x, (which defines φ(x, t)). Now multiply this equation by and find
(41)
where we have defined . As is well known ψ(x, t) satisfies the Dirac equation, (differentiate ψ(x, t) with respect to t and use commutation relations of H and ψ)
(42)
Consequently φ (x, t) must also satisfy the Dirac equation (differentiate (41) with respect to t, use (42) and integrate by parts).
That is, if φ (x, T) is that solution of the Dirac equation at time T which is φ(x) at t = 0, and if we define Φ* = ∫ ψ*(x)φ(x)d3x and Φ′* = ∫ ψ*(x)φ(x, T)d3x then Φ′* = SΦ* S−1, or
(43)
The principle on which the proof will be based can now be illustrated by a simple example. Suppose we have just one electron initially and finally and ask for
(44)
We might try putting F* through the operator S using (43), SF* = F′* S , where ƒ′ in F′* = ∫ ψ*(x) ƒ′(x)d3x is the wave function at T arising from ƒ(x) at 0. Then
(45)
where the second expression has been obtained by use of the definition (38) of Cν and the general commutation relationGF* + F*G = ∫g*(x) ƒ(x) d3x,
which is a consequen
ce of the properties of ψ(x) (the others are FG = −GF and F*G* = −G*F*). Now in the last term in (45) is the complex conjugate of . Thus if ƒ′ contained only positive energy components, would vanish and we would have reduced r to a factor times Cv. But ƒ′, as worked out here, does contain negative energy components created in the potential A and the method must be slightly modified.
Before putting F* through the operator we shall add to it another operator F′′* arising from a function ƒ′′(x) containing only negative energy components and so chosen that the resulting ƒ′ has only positive ones. That is we want
(46)
where the “pos” and “neg” serve as reminders of the sign of the energy components contained in the operators. This we can now use in the form
(47)
In our one electron problem this substitution replaces r by two terms
The first of these reduces to
as above, for is now zero, while the second is zero since the creation operator gives zero when acting on the vacuum state as all negative energies are full. This is the central idea of the demonstration.
The problem presented by (46) is this: Given a function ƒpos(x) at time 0, to find the amount, , of negative energy component which must be added in order that the solution of Dirac’s equation at time T will have only positive energy components, . This is a boundary value problem for which the kernel is designed. We know the positive energy components initially, ƒpos, and the negative ones finally (zero). The positive ones finally are therefore (using (19))
(48)
where t2 = T, t1 = 0. Similarly, the negative ones initially are
(49)
where t2 approaches zero from above, and t1 = 0. The ƒpos(x2) is subtracted to keep in only those waves which return from the potential and not those arriving directly at t2 from the K+(2, 1) part of , as t2 → 0. We could also have written
(50)
Therefore the one-electron problem, r = , gives by (48)
as expected in accordance with the reasoning of the previous sections (i.e., (20) with replacing K+).
The proof is readily extended to the more general expression R, (40), which can be analyzed by induction. First one replaces by a relation such as (47) obtaining two terms
In the first term the order of and G1 is then interchanged, producing an additional term times an expression with one less electron in initial and final state. Next it is exchanged with G2 producing an addition − times a similar term, etc. Finally on reaching the with which it anticom-mutes it can be simply moved over to juxtaposition with where it gives zero. The second term is similarly handled by moving through anti commuting , etc., until it reaches P1. Then it is exchanged with P1 to produce an additional simpler term with a factor or from (49), with t2 = t1 = 0 (the extra ƒ1(x2) in (49) gives zero as it is orthogonal to p1 (x2)). This describes in the expected manner the annihilation of the pair, electron ƒ1, positron p1. The is moved in this way successively through the P’s until it gives zero when acting on χ0. Thus R is reduced, with the expected factors (and with alternating signs as required by the exclusion principle), to simpler terms containing two less operators which may in turn be further reduced by using in a similar manner, etc. After all the F* are used the Q*’s can be reduced in a similar manner. They are moved through the S in the opposite direction in such a manner as to produce a purely negative energy operator at time 0, using relations analogous to (46) to (49). After all this is done we are left simply with the expected factor times Cν (assuming the net charge is the same in initial and final state.)
In this way we have written the solution to the general problem of the motion of electrons in given potentials. The factor Cν is obtained by normalization. However for photon fields it is desirable to have an explicit form for Cν in terms of the potentials. This is given by (30) and (29) and it is readily demonstrated that this also is correct according to second quantization.
b. ANALlYSIS OF THE VACUUM PROBLEM
We shall calculate Cv from second quantization by induction considering a series of problems each containing a potential distribution more nearly like the one we wish. Suppose we know Cv for a problem like the one we want and having the same potentials for time t between some t0 and T, but having potential zero for times from 0 to t0. Call this Cv(t0), the corresponding Hamiltonian Ht0 and the sum of contributions for all single loops, L(t0). Then for t0 = T we have zero potential at all times, no pairs can be produced, L(T) = 0 and Cv(T) = 1. For t0 = 0 we have the complete problem, so that Cv(0) is what is defined as Cv in (38). Generally we have,
since Ht0 is identical to the constant vacuum Hamiltonian HT for t < t0 and χ0 is an eigenfunction of HT with an eigenvalue (energy of vacuum) which we can take as zero.
The value of Cv(t0 − Δt0 ) arises from the Hamiltonian Ht0 − Δt0 which differs from Ht0 just by having an extra potential during the short interval Δt0. Hence, to first order in Δt0, we have
we therefore obtain for the derivative of Cv the expression
(51)
which will be reduced to a simple factor times Cv(t0) by methods analogous to those used in reducing R. The operator ψ can be imagined to be split into two pieces ψpos and ψneg operating on positive and negative energy states respectively. The ψpos on χ0 gives zero so we are left with two terms in the current density, and . The latter is just the expectation value of βA taken over all negative energy states (minus which gives zero acting on χ0). This is the effect of the vacuum expectation current of the electrons in the sea which we should have subtracted from our original Hamiltonian in the customary way.
The remaining term , or its equivalent can be considered as ψ*(x)fpos(x) where fpos(x) is written for the positive energy component of the operator βAψ(x). Now this operator, ψ*(x)fpos(x), or more precisely just the ψ*(x) part of it, can be pushed through the in a manner exactly analogous to (47) when f is a function. (An alternative derivation results from the consideration that the operator ψ(x, t) which satisfies the Dirac equation also satisfies the linear integral equations which are equivalent to it.) That is, (51) can be written by (48), (50),
where in the first term t2 = T, and in the second t2 → t0 = t1. The (A) in K + (A) refers to that part of the potential A after t0. The first term vanishes for it involves (from the ) only positive energy components of ψ* , which give zero operating into χ0 . In the second term only negative components of ψ*(x2) appear. If, then ψ*(x2) is interchanged in order with ψ(x1) it will give zero operating on χ0, and only the term,
(52)
will remain, from the usual commutation relation of ψ * and ψ.
The factor of Cv(t0) in (52) times − Δt0 is, according to (29) (reference 10), just L(t0 − Δt0) − L(t0) since this difference arises from the extra potential ΔA = A during the short time interval Δt0. Hence −dCv(t0 )/d t0 = +(dL(t0)/dt0)Cv(t0) so that integration from t0 = T to t0 = 0 establishes (30).
Starting from the theory of the electromagnetic field in second quantization, a deduction of the equations for quantum electrodynamics which appear in the succeeding paper may be worked out using very similar principles. The Pauli-Weisskopf theory of the Klein-Gordon equation can apparently be analyzed in essentially the same way as that used here for Dirac electrons.
THE RADIATION THEORIES OF TOMONAGA, SCHWINGER, AND FEYNMAN
BY
FREEMAN DYSON
A unified development of the subject of quantum electrodynamics is outlined, embodying the main features both of the Tomonaga-Schwinger and of the Feynman radiation theory. The theory is carried to a point further than that reached by these authors, in the discussion of higher order radiative reactions and vacuum polarization phenomena. However, the theory of these higher order processes is a program rather than a definitive theory, since no general proof of the convergence of these effects is attempted.
The chief results obtained are (a) a demonstration of the equivalence of the Feynman and Schwinger theories, and (b) a considerable simplification of the proc
edure involved in applying the Schwinger theory to particular problems, the simplification being the greater the more complicated the problem.
I. INTRODUCTION
AS a result of the recent and independent discoveries of Tomonaga,og Schwinger,oh and Feynman,oi the subject of quantum electrodynamics has made two very notable advances. On Reprinted with permission from the American Physical Society: Dyson, Physical Review, Volume 75, p. 486, 1949. ©1949 by the American Physical Society.
The Dreams That Stuff is Made of Page 73