The Dreams That Stuff is Made of

by Stephen Hawking

  4. GENERALIZED PROBABILITY AMPLITUDE

  We must now find the physical meaning of the functional Ψ[C]. As regards this we can follow a method similar to that of Bloch in the case of ordinary many-time theory. Besides the fact that in our case there appear infinitely many time variables, one point differs from Bloch’s case: in (16) the unitary operator u is commutable with the coordinates q1, q2, . . . , qN, while our U is not commutable with the field quantities v1 (xyz) and v2(xyz). Noting this difference and treating the continuum infinity as the limit of a denumerable infinity by some artifice, for instance, by the procedure of Heisenberg and Pauli,(8) Bloch’s consideration can be applied also here almost without any alteration. We shall give here only the results.

  Let us suppose that the fields are in the state represented by a vector Ψ[C]. We suppose that we make measurements of a function f(v1, v2, λ1, λ2) at every point on a surface C1 in the space-time world. Let P1 denote the variable point on C1, then, if f(P1) at any two “values” of P1 commute with each other, the measurements of f at each of these two points do not interfere with each other. Our first conclusion says that in this case the expectation value of f(P1) is given by

  (32)

  where f(P1) means f(V1(P1), . . ., according to our convention on page 8, and the symbol ((A, B)) with double parentheses is the scalar product of two vectors A and B. It is impossible in cases of continuously many degrees of freedom to represent this scalar product by an integral of the product of two functions. For this purpose we must replace the continuum infinity by an at least denumerable infinity.

  More generally, we suppose a functional F[f(P1)] of the independent variable function f(P1), regarding f(P1) as a function of P1. Then the expectation value of this F is given by

  (33)

  A physically interesting F is the projective operator M[, ; , V2(P1)] belonging to the “eigen-value” , v2(P1) of V1(P1), V2(P1). Then its expectation value

  (34)

  gives the probability that the field 1 and the field 2 have respectively the functional form and v2(P1) on the surface C1. As C1 is assumed to be space-like, the measurement of the functional M is possible (the measurements of V1 (P1) and V2(P1) at all points on C1 mean just the measurement of M).

  Thus far we have made no mention of the representation of Ψ[C]. We use now the special representation in which V1 (P1) at all points on C1 are simultaneously diagonal. It is always possible to make all V1 (P1) and V2(P1) diagonal when the surface C1 is space-like. In this representation Ψ[C1] is represented by a functional Ψ[, ; C1] of the eigenvalues and of V1(P1) and V2(P1). The projection operator M has in this representation such diagonal form that (34) is simplified as follows

  (35)

  In this sense we can call Ψ[, ; C1] the “generalized probability amplitude.”

  5. GENERALIZED TRANSFORMATION FUNCTIONAL

  We have stated above that between Ψ[C1] and Ψ[C2] the relation (31) holds, where C1 and C2 are two space-like surfaces in the space-time world. We see thus that the transformation operator

  (36)

  plays an important role. It is evident that this operator also has a space-time meaning.

  Just as the special representative of the ψ-vector, the probability amplitude, has a distinct physical meaning, so there is a special representation in which the representative of the transformation operator T[C2; C1] has a distinct physical meaning.

  We now introduce the mixed representative of T[C2; C1] whose rows refer to the representation in which V1(P1) and V2(P1) at all points on C1 become diagonal and whose column refer to the representation in which V1(P2) and V2(P2) at all points on C2 become diagonal. We denote this representation by

  (37mo)

  or simpler:

  (38)mo

  If we note here the relation (35), we see that we can give the matrix elements of this representation the following meaning: One measures the field quantities V1 and V2 at all points on C2 when the fields are prepared in such a way that they have certainly the values and at all points on C1. Then

  (39)

  gives the probability that one obtains the result and in this measurement. In this proposition we have assumed that C2 lies afterward against C1.

  From this physical interpretation we may regard the matrix element (37), or (38), considered as a functional of , and , , as the generalization of the ordinary transformation function .

  As a special case it may happen that C2 lies apart from C1 only in a portion S2 and a portion S1 of C2 and C1 respectively, the other parts of C1 and C2 overlapping with each other.

  In this case the matrix elements of T[C2; C1] depend only on the values of the fields on the portions S1 and S2 of the surfaces C1 and C2. In this case we need for calculating T[C2; C1] to take the product in (36) only in the closed domain surrounded by S1 and S2, thus

  (40)

  The matrix elements of the mixed representation of this T is a functional of , and , (p2), where p1 denotes the moving point of the portion S1, and p2 the moving point on the portion S2. This matrix is independent on the field quantities on the other portions of the surfaces C1 and C2.

  The matrix element of T [S2;S1] regarded as a functional of , and , has the properties of g.t.f. (generalized transformation functional) of Dirac. But in defining our g.t.f. we had to restrict the surfaces S1 and S2 to be space-like, while Dirac has required his g.t.f. to be defined also referring to the time-like surfaces. As mentioned above, however, such a generalization as required by Dirac is superfluous so far as concerns the relativity theory.

  It is to be noted that for the physical interpretation of , |, ] it is not necessary to assume C 2 to lie afterward against C1. Also when the inverse is the case, we can as well give the physical meaning for W of (39): One measures the field quantities V1 and V2 at all points on C2 when the fields are prepared in such a way that they would have certainly the values and at all points on C1 if the fields were left alone until C1 without being measured before on C2. Then W gives the probability that one finds the results and in this measurement on C2.

  6. CONCLUDING REMARKS

  We have thus shown that the quantum theory of wave fields can be really brought into a form which reveals directly the invariance of the theory against Lorentz transformations. The reason why the ordinary formalism of the quantum field theory is so unsatisfactory is that it has been built up in a way much too analogous to the ordinary non-relativistic mechanics. In this ordinary formalism of the quantum theory of fields the theory is divided into two distinct sections: the section giving the kinematical relations between various quantities at the same instant of time, and the section determining the causal relations between quantities at different instants of time. Thus the commutation relations (1) belong to the first section and the Schrödinger equation (2) to the second.

  As stated before, this way of separating the theory into two sections is very unrelativistic, since here the concept “same instant of time” plays a distinct role.

  Also in our formalism the theory is divided into two sections, but now the separation is introduced in another place. One section gives the laws of behaviour of the fields when they are left alone, and the other gives the laws determining the deviation from this behaviour due to interactions. This way of separating the theory can be carried out relativistically.

  Although in this way the theory can be brought into more satisfactory form, no new contents are added thereby. So, the well-known divergence difficulties of the theory are inherited also by our theory. Indeed, our fundamental equations (29) admit only catastrophic solutions, as can be seen directly in the fact that the unavoidable infinity due to non-vanishing zero-point amplitudes of the fields inheres in the operator H12(P). Thus, a more profound modification of the theory is required in order to remove this fundamental difficulty.

  It is expected that such a modification of the theory could possibly be introduced by some revision of the concept of interaction, because we meet no such difficulty when we deal with the non-interact
ing fields. This revision would then have the result that in the separation of the theory into two sections, one for free fields and one for interactions, some uncertainty would be introduced. This seems to be implied by the very fact that, when we formulate the quantum field theory in a relativistically satisfactory manner, this way of separation has revealed itself as the fundamental element of the theory.

  PHYSICS DEPARTMENT,

  TOKYO BUNRIKA UNIVERSITY.

  REFERENCES

  1 H. Yukawa, Kagaku, 12, 251, 282 and 322, 1942.

  2 P. A. M. Dirac, Phys. Z. USSR., 3, 64, 1933.

  3 W. Pauli, Solvay Berichte, 1939.

  4 P. A. M. Dirac, Proc. Roy. Soc. London, 136, 453, 1932.

  5 F. Bloch, Phys. Z. USSR., 5, 301, 1943.

  6 E. Stueckelberg, Helv. Phys. Acta, 11, 225, § 5, 1938.

  7 W. Heisenberg, Z. Phys., 110, 251, 1938.

  8 W. Heisenberg and W. Pauli, Z. Phys., 56, 1, 1929.

  SPACE-TIME APPROACH TO QUANTUM ELECTRODYNAMICS

  BY

  RICHARD FEYNMAN

  In this paper two things are done. (1) It is shown that a considerable simplification can be attained in writing down matrix elements for complex processes in electrodynamics. Further, a physical point of view is available which permits them to be written down directly for any specific problem. Being simply a restatement of conventional electrodynamics, however, the matrix elements diverge for complex processes. (2) Electrodynamics is modified by altering the interaction of electrons at short distances. All matrix elements are now finite, with the exception of those relating to problems of vacuum polarization. The latter are evaluated in a manner suggested by Pauli and Bethe, which gives finite results for these matrices also. The only effects sensitive to the modification are changes in mass and charge of the electrons. Such changes could not be directly observed. Phenomena directly observable, are insensitive to the details of the modification used (except at extreme energies). For such phenomena, a limit can be taken as the range of the modification goes to zero. The results then agree with those of Schwinger. A complete, unambiguous, and presumably consistent, method is therefore available for the calculation of all processes involving electrons and photons.

  The simplification in writing the expressions results from an emphasis on the over-all space-time view resulting from a study of the solution of the equations of electrodynamics. The relation of this to the more conventional Hamiltonian point of view is discussed. It would be very difficult Co make the modification which is proposed if one insisted on having the equations in Hamiltonian form.

  Reprinted with permission from the American Physical Society: Feynman,

  Physical Review, Volume 76, p. 769, 1949. ©1949, by the American Physical Society.

  The methods apply as well to charges obeying the Klein-Gordon equation, and to the various meson theories of nuclear forces. Illustrative examples are given. Although a modification like that used in electrodynamics can make all matrices finite for all of the meson theories, for some of the theories it is no longer true that all directly observable phenomena are insensitive to the details of the modification used.

  The actual evaluation of integrals appearing in the matrix elements may be facilitated, in the simpler cases, by methods described in the appendix.

  This paper should be considered as a direct continuation of a preceding onemp (I) in which the motion of electrons, neglecting interaction, was analyzed, by dealing directly with the solution of the Hamiltonian differential equations. Here the same technique is applied to include interactions and in that way to express in simple terms the solution of problems in quantum electrodynamics.

  For most practical calculations in quantum electrodynamics the solution is ordinarily expressed in terms of a matrix element. The matrix is worked out as an expansion in powers of e2/ħc, the successive terms corresponding to the inclusion of an increasing number of virtual quanta. It appears that a considerable simplification can be achieved in writing down these matrix elements for complex processes. Furthermore, each term in the expansion can be written down and understood directly from a physical point of view, similar to the space-time view in I. It is the purpose of this paper to describe how this may be done. We shall also discuss methods of handling the divergent integrals which appear in these matrix elements.

  The simplification in the formulae results mainly from the fact that previous methods unnecessarily separated into individual terms processes that were closely related physically. For example, in the exchange of a quantum between two electrons there were two terms depending on which electron emitted and which absorbed the quantum. Yet, in the virtual states considered, timing relations are not significant. Only the order of operators in the matrix must be maintained. We have seen (I), that in addition, processes in which virtual pairs are produced can be combined with others in which only positive energy electrons are involved. Further, the effects of longitudinal and transverse waves can be combined together. The separations previously made were on an unrelativistic basis (reflected in the circumstance that apparently momentum 7but not energy is conserved in intermediate states). When the terms are combined and simplified, the relativistic invariance of the result is self-evident.

  We begin by discussing the solution in space and time of the Schrödinger equation for particles interacting instantaneously. The results are immediately generalizable to delayed interactions of relativistic electrons and we represent in that way the laws of quantum electrodynamics. We can then see how the matrix element for any process can be written down directly. In particular, the self-energy expression is written down.

  So far, nothing has been done other than a restatement of conventional electrodynamics in other terms. Therefore, the self-energy diverges. A modificationmq in interaction between charges is next made, and it is shown that the self-energy is made convergent and corresponds to a correction to the electron mass. After the mass correction is made, other real processes are finite and-insensitive to the “width” of the cut-off in the interaction.mr

  Unfortunately, the modification proposed is not completely satisfactory theoretically (it leads to some difficulties of conservation of energy). It does, however, seem consistent and satisfactory to define the matrix element for all real processes as the limit of that computed here as the cut-off width goes to zero. A similar technique suggested by Pauli and by Bethe can be applied to problems of vacuum polarization (resulting in a renormalization of charge) but again a strict physical basis for the rules of convergence is not known.

  After mass and charge renormalization, the limit of zero cutoff width can be taken for all real processes. The results are then equivalent to those of Schwingerms who does not make explicit use of the convergence factors. The method of Schwinger is to identify the terms corresponding to corrections in mass and charge and, previous to their evaluation, to remove them from the expressions for real processes. This has the advantage of showing that the results can be strictly independent of particular cut-off methods.

  On the other hand, many of the properties of the integrals are analyzed using formal properties of invariant propagation functions. But one of the properties is that the integrals are infinite and it is not clear to what extent this invalidates the demonstrations. A practical advantage of the present method is that ambiguities can be more easily resolved; simply by direct calculation of the otherwise divergent integrals. Nevertheless, it is not at all clear that the convergence factors do not upset the physical consistency of the theory. Although in the limit the two methods agree, neither method appears to be thoroughly satisfactory theoretically. Nevertheless, it does appear that we now have available a complete and definite method for the calculation of physical processes to any order in quantum electrodynamics.

  Since we can write down the solution to any physical problem, we have a complete theory which could stand by itself. It will be theoretically incomplete, however, in two respects. First, although each term of increasing order in e2 /ħ c can be wr
itten down it would be desirable to see some way of expressing things in finite form to all orders in e2/ħc at once. Second, although it will be physically evident that the results obtained are equivalent to those obtained by conventional electrodynamics the mathematical proof of this is not included. Both of these limitations will be removed in a subsequent paper (see also Dyson5).

  Briefly the genesis of this theory was this. The conventional electrodynamics was expressed in the Lagrangian form of quantum mechanics described in the Reviews of Modern Physics.mt The motion of the field oscillators could be integrated out (as described in Section 13 of that paper), the result being an expression of the delayed interaction of the particles. Next the modification of the delta-function interaction could be made directly from the analogy to the classical case.mu This was still not complete because the Lagrangian method had been worked out in detail only for particles obeying the nonrelativistic Schrödinger equation. It was then modified in accordance with the requirements of the Dirac equation and the phenomenon of pair creation. This was made easier by the reinterpretation of the theory of holes (I). Finally for practical calculations the expressions were developed in a power series in e2/ħc. It was apparent that each term in the series had a simple physical interpretation. Since the result was easier to understand than the derivation, it was thought best to publish the results first in this paper. Considerable time has been spent to make these first two papers as complete and as physically plausible as possible without relying on the Lagrangian method, because it is not generally familiar. It is realized that such a description cannot carry the conviction of truth which would accompany the derivation. On the other hand, in the interest of keeping simple things simple the derivation will appear in a separate paper.