The Dreams That Stuff is Made of

by Stephen Hawking

  FIG. 2 The Dirac equation permits another solution K+(2, 1) if one considers that waves scattered by the potential can proceed backwards in time as in Fig. 2 (a). This is interpreted in the second order processes (b), (c), by noting that there is now the possibility (c) of virtual pair production at 4, the positron going to 3 to be annihilated. This can be pictured as similar to ordinary scattering (b) except that the electron is scattered backwards in time from 3 to 4. The waves scattered from 3 to 2’ in (a) represent the possibility of a positron arriving at 3 from 2’ and annihilating the electron from 1. This view is proved equivalent to hole theory: electrons traveling backwards in time are recognized as positrons.

  This alternative is already included in (14) as contributions for which t4 < t3, and its study will lead us to an interpretation of K+(4, 3) for t4 < t3. The factor K+(2, 4) describes the electron (after the pair production at 4) proceeding from 4 to 2. Likewise K+(3, 1) represents the electron proceeding from 1 to 3. K+ (4, 3) must therefore represent the propagation of the positron or hole from 4 to 3. That it does so is clear. The fact it in hole theory the hole proceeds in the manner of and electron of negative energy is reflected in the fact that K+ (4, 3) for t4 < t3 is (minus) the sum of only negative energy components. In hole theory the real energy of these intermediate states is, of course, positive. This is true here too, since in the phases exp(−iEn (t4 − t3)) defining K+(4, 3) in (17), En is negative but so is t4 − t3. That is, the contributions vary with t3 as exp(−i|En |(t3 − t4)) as they would if the energy of the intermediate state were |En|. The fact that the entire sum is taken as negative in computing K+(4, 3) is reflected in the fact that in hole theory the amplitude has its sign reversed in accordance with the Pauli principle and the fact that the electron arriving at 2 has been exchanged with one in the sea.ny To this, and to higher orders, all processes involving virtual pairs are correctly described in this way.

  The expressions such as (14) can still be described as a passage of the electron from 1 to 3 (K+(3, 1)), scattering at 3 by A(3), proceeding to 4 (K+ (4, 3)), scattering again, A(4), arriving finally at 2. The catterings may, however, be toward both future and past times, an electron propagating backwards in time being recognized as a positron.

  This therefore suggests that negative energy components created by scattering in a potential be considered as waves propagating from the scattering point toward the past, and that such waves represent the propagation of a positron annihilating the electron in the potential.nz

  With this interpretation real pair production is also described correctly (see Fig. 3). For example in (13) if t1 < t3 < t2 the equation gives the amplitude that if at time t1 one electron is present at 1, then at time t2 just one electron will be present (having been scattered at 3) and it will be at 2. On the other hand if t2 is less than t3, for example, if t2 = t1 < t3, the same expression gives the amplitude that a pair, electron at 1, positron at 2 will annihilate at 3, and subsequently no particles will be present. Likewise if t2 and t1 exceed t3 we have (minus) the amplitude for finding a single pair, electron at 2, positron at 1 created by A(3) from a vacuum. If t1 > t3 > t2, (13) describes the scattering of a positron. All these amplitudes are relative to the amplitude that a vacuum will remain a vacuum, which is taken as unity. (This will be discussed more fully later.)

  FIG. 3 Several different processes can be described by the same formula depending on the time relations of the variables t2 , t1 . Thus Pv|K(A)+ (2, 1)| is the probability that: (a) An electron at 1 will be scattered at 2 (and no other pairs form in vacuum). (b) Electron at 1 and positron at 2 annihilate leaving nothing. (c) A single pair at 1 and 2 is created from vacuum. (d) A positron at 2 is scattered to 1. (K(A)+ (2, 1) is the sum of the effects of scattering in the potential to all orders. Pv is a normalizing constant.)

  The analogue of (2) can be easily worked out.oa It is,

  (18)

  where d 3 V1 is the volume element of the closed 3-dimensional surface of a region of space time containing point 2, and N(1) is Nµ(1)γµ where Nµ(1) is the inward drawn unit normal to the surface at the point 1. That is, the wave function ψ (2) (in this case for a free particle) is determined at any point inside a four-dimensional region if its values on the surface of that region are specified.

  To interpret this, consider the case that the 3-surface consists essentially of all space at some time say t = 0 previous to t2, and of all space at the time T > t2. The cylinder connecting these to complete the closure of the surface may be very distant from x2 so that it gives no appreciable contribution (as K +(2, 1) decreases exponentially in space-like directions). Hence, if γ4 = β , since the inward drawn normals N will be β and −β ,

  (19)

  where t1 = 0, t’1 = T . Only positive energy (electron) components in ψ (1) contribute to the first integral and only negative energy (positron) components of ψ (1’) to the second. That is, the amplitude for finding a charge at 2 is determined both by the amplitude for finding an electron previous to the measurement and by the amplitude for finding a positron after the measurement. This might be interpreted as meaning that even in a problem involving but one charge the amplitude for finding the charge at 2 is not determined when the only thing known in the amplitude for finding an electron (or a positron) at an earlier time. There may have been no electron present initially but a pair was created in the measurement (or also by other external fields). The amplitude for this contingency is specified by the amplitude for finding a positron in the future.

  We can also obtain expressions for transition amplitudes, like (5). For example if at t = 0 we have an electron present in a state with (positive energy) wave function f (x), what is the amplitude for finding it at t = T with the (positive energy) wave function g(x)? The amplitude for finding the electron anywhere after t = 0 is given by (19) with ψ (1) replaced by f (x), the second integral vanishing. Hence, the transition element to find it in state g (x) is, in analogy to (5), just (t2 = T , t1 = 0)

  (20)

  since g* = ḡβ.

  If a potential acts somewhere in the interval between 0 and T, K+ is replaced by K +(A). Thus the first order effect on the transition amplitude is, from (13),

  (21)

  Expressions such as this can be simplified and the 3-surface integrals, which are inconvenient for relativistic calculations, can be removed as follows. Instead of defining a state by the wave function f(x), which it has at a given time t1 = 0, we define the state by the function f (1) of four variables x1 , t1 which is a solution of the free particle equation for all t1 and is f (x1) for t1 = 0. The final state is likewise defined by a function g (2) over-all space-time. Then our surface integrals can be performed since ∫ K+(3, 1)βf(x1)d3x1 = f(3) and ∫ḡ(x2)βd3x2K+(2, 3) = ḡ(3). There results

  (22)

  the integral now being over-all space-time. The transition amplitude to second order (from (14)) is

  (23)

  for the particle arriving at 1 with amplitude f(1) is scattered (A(1)), progresses to 2, (K+(2, 1)), and is scattered again (A(2)), and we then ask for the amplitude that it is in state g(2). If g(2) is a negative energy state we are solving a problem of annihilation of electron in f(1), positron in g(2), etc.

  We have been emphasizing scattering problems, but obviously the motion in a fixed potential V, say in a hydrogen atom, can also be dealt with. If it is first viewed as a scattering problem we can ask for the amplitude, φk (1), that an electron with original free wave function was scattered k times in the potential V either forward or backward in time to arrive at 1. Then the amplitude, after one more scattering is

  (24)

  An equation for the total amplitude

  for arriving at 1 either directly or after any number of scatterings is obtained by summing (24) over all k from 0 to ∞;

  (25)

  Viewed as a steady state problem we may wish, for example, to find that initial condition φ0 (or better just the ψ ) which leads to a periodic motion of ψ. This is most practically done, of course, by solv
ing the Dirac equation,

  (26)

  deduced from (25) by operating on both sides by i∇2 − m, thereby eliminating the φ0, and using (12). This illustrates the relation between the points of view.

  For many problems the total potential A + V may be split conveniently into a fixed one, V, and another, A, considered as a perturbation. If is defined as in (16) with V for A, expressions such as (23) are valid and useful with K+ replaced by and the functions f(1), g(2) replaced by solutions for all space and time of the Dirac Eq. (26) in the potential V (rather than free particle wave functions).

  4. PROBLEMS INVOLVING SEVERAL CHARGES

  We wish next to consider the case that there are two (or more) distinct charges (in addition to pairs they may produce in virtual states). In a succeeding paper we discuss the interaction between such charges. Here we assume that they do not interact. In this case each particle behaves independently of the other. We can expect that if we have two particles a and b, the amplitude that particle a goes from x1 at t1, to x3 at t3 while b goes from x2 at t2 to x4 at t4 is the productK (3, 4; 1, 2) = K+a (3, 1) K+b (4, 2) .

  The symbols a, b simply indicate that the matrices appearing in the K+ apply to the Dirac four component spinors corresponding to particle a or b respectively (the wave function now having 16 indices). In a potential K+a and K +b become and where is defined and calculated as for a single particle. They commute. Hereafter the a, b can be omitted; the space time variable appearing in the kernels suffice to define on what they operate.

  The particles are identical however and satisfy the exclusion principle. The principle requires only that one calculate K(3, 4; 1, 2) − K(4, 3; 1, 2) to get the net amplitude for arrival of charges at 3, 4. (It is normalized assuming that when an integral is performed over points 3 and 4, for example, since the electrons represented are identical, one divides by 2.) This expression is correct for positrons also (Fig. 4). For example the amplitude that an electron and a positron found initially at x1 and x4 (say t1 = t4) are later found at x3 and x2 (with t2 = t3 > t1) is given by the same expression

  (27)

  The first term represents the amplitude that the electron proceeds from 1 to 3 and the positron from 4 to 2 (Fig. 4(c)), while the second term represents the interfering amplitude that the pair at 1, 4 annihilate and what is found at 3, 2 is a pair newly created in the potential. The generalization to several particles is clear. There is an additional factor for each particle, and anti-symmetric combinations are always taken.

  No account need be taken of the exclusion principle in intermediate states. As an example consider again expression (14) for t2 > t1 and suppose t4 < t3 so that the situation represented (Fig. 2(c)) is that a pair is made at 4 with the electron proceeding to 2, and the positron to 3 where it annihilates the electron arriving from 1. It may be objected that if it happens that the electron created at 4 is in the same state as the one coming from 1, then the process cannot occur because of the exclusion principle and we should not have included it in our term (14). We shall see, however, that considering the exclusion principle also requires another change which reinstates the quantity.

  FIG. 4 Some problems involving two distinct charges (in addition to virtual pairs they may produce): is the probability that: (a) Electrons at 1 and 2 are scattered to 3, 4 (and no pairs are formed). (b) Starting with an electron at 1 a single pair is formed, positron at 2, electrons at 3, 4. (c) A pair at 1, 4 is found at 3, 2, etc. The exclusion principle requires that the amplitudes for processes involving exchange of two electrons be subtracted.

  For we are computing amplitudes relative to the amplitude that a vacuum at t1 will still be a vacuum at t2. We are interested in the alteration in this amplitude due to the presence of an electron at 1. Now one process that can be visualized as occurring in the vacuum is the creation of a pair at 4 followed by a re-annihilation of the same pair at 3 (a process which we shall call a closed loop path). But if a real electron is present in a certain state 1, those pairs for which the electron was created in state 1 in the vacuum must now be excluded. We must therefore subtract from our relative amplitude the term corresponding to this process. But this just reinstates the quantity which it was argued should not have been included in (14), the necessary minus sign coming automatically from the definition of K+. It is obviously simpler to disregard the exclusion principle completely in the intermediate states.

  All the amplitudes are relative and their squares give the relative probabilities of the various phenomena. Absolute probabilities result if one multiplies each of the probabilities by Pv, the true probability that if one has no particles present initially there will be none finally. This quantity Pv can be calculated by normalizing the relative probabilities such that the sum of the probabilities of all mutually exclusive alternatives is unity. (For example if one starts with a vacuum one can calculate the relative probability that there remains a vacuum (unity), or one pair is created, or two pairs, etc. The sum is .) Put in this form the theory is complete and there are no divergence problems. Real processes are completely independent of what goes on in the vacuum.

  When we come, in the succeeding paper, to deal with interactions between charges, however, the situation is not so simple. There is the possibility that virtual electrons in the vacuum may interact electromagnetically with the real electrons. For that reason processes occuring in the vacuum are analyzed in the next section, in which an independent method of obtaining Pv is discussed.

  5. VACUUM PROBLEMS

  An alternative way of obtaining absolute amplitudes is to multiply all amplitudes by Cv, the vacuum to vacuum amplitude, that is, the absolute amplitude that there be no particles both initially and finally. We can assume Cv = 1 if no potential is present during the interval, and otherwise we compute it as follows. It differs from unity because, for example, a pair could be created which eventually annihilates itself again. Such a path would appear as a closed loop on a space-time diagram. The sum of the amplitudes resulting from all such single closed loops we call L. To a first approximation L is

  (28)

  For a pair could be created say at 1, the electron and positron could both go on to 2 and there annihilate. The spur, Sp, is taken since one has to sum over all possible spins for the pair. The factor arises from the fact that the same loop could be considered as starting at either potential, and the minus sign results since the interactors are each −iA. The next order term would beob

  etc. The sum of all such terms gives L.oc

  In addition to these single loops we have the possibility that two independent pairs may be created and each pair may annihilate itself again. That is, there may be formed in the vacuum two closed loops, and the contribution in amplitude from this alternative is just the product of the contribution from each of the loops considered singly. The total contribution from all such pairs of loops (it is still consistent to disregard the exclusion principle for these virtual states) is L2/2 for in L2 we count every pair of loops twice. The total vacuum-vacuum amplitude is then

  (30)

  the successive terms representing the amplitude from zero, one, two, etc., loops. The fact that the contribution to Cv of single loops is −L is a consequence of the Pauli principle. For example, consider a situation in which two pairs of particles are created. Then these pairs later destroy themselves so that we have two loops. The electrons could, at a given time, be interchanged forming a kind of figure eight which is a single loop. The fact that the interchange must change the sign of the contribution requires that the terms in Cv appear with alternate signs. (The exclusion principle is also responsible in a similar way for the fact that the amplitude for a pair creation is −K+ rather than + K+.) Symmetrical statistics would lead toCv = 1 + L + L 2/2 = exp (+ L ) .

  The quantity L has an infinite imaginary part (from L(1), higher orders are finite). We will discuss this in connection with vacuum polarization in the succeeding paper. This has no effect on the normalization constant for the probability that a vacuum remain vacuum
is given byPv = |Cv|2 = exp (−2 · real part of L),

  from (30). This value agrees with the one calculated directly by renormalizing probabilities. The real part of L appears to be positive as a consequence of the Dirac equation and properties of K+ so that Pv is less than one. Bose statistics gives Cv = exp(+L) and consequently a value of Pv greater than unity which appears meaningless if the quantities are interpreted as we have done here. Our choice of K+ apparently requires the exclusion principle.

  Charges obeying the Klein-Gordon equation can be equally well treated by the methods which are discussed here for the Dirac electrons. How this is done is discussed in more detail in the succeeding paper. The real part of L comes out negative for this equation so that in this case Bose statistics appear to be required for consistency.3

  6. ENERGY-MOMENTUM REPRESENTATION

  The practical evaluation of the matrix elements in some problems is often simplified by working with momentum and energy variables rather than space and time. This is because the function K+(2, 1) is fairly complicated but we shall find that its Fourier transform is very simple, namely (i/4π 2( p − m)−1 that is