In Feynman’s theory the graph corresponding to a particular matrix element is regarded, not merely as an aid to calculation, but as a picture of the physical process which gives rise to that matrix element. For example, an electron line joining x1 to x2 represents the possible creation of an electron at x1 and its annihilation at x2, together with the possible creation of a position at x2 and its annihilation at x1. This interpretation of a graph is obviously consistent with the methods, and in Feynman’s hands has been used as the basis for the derivation of most of the results, of the present paper. For reasons of space, these ideas of Feynman will not be discussed in further detail here.
To the product Pn correspond a finite number of graphs, one of which may be denoted by G; all possible G can be enumerated without difficulty for moderate values of n. To each G corresponds a contribution C(G) to the matrix element of (31) which is being evaluated.
It may happen that the graph G is disconnected, so that it can be divided into subgraphs, each of which is connected, with no line joining a point of one subgraph to a point of another. In such a case it is clear from (47) that C(G) is the product of factors derived from each subgraph separately. The subgraph G1 containing the point x0 is called the “essential part” of G, the remainder G2 the “inessential part.” There are now two cases to be considered, according to whether the points xk and xrk lie in G2 or in G1 (they must clearly both lie in the same subgraph). In the first case, the factor C(G2) of C(G) can be seen by a comparison of (31) and (32) to be a contribution to the matrix element of the operator S(∞) for the transition A → B. Now letting G vary over all possible graphs with the same G1 and different G2, the sum of the contributions of all such G is a constant C(G1) multiplied by the total matrix element of S(∞) for the transition A → B. But for one-particle states the operator S(∞) is by (21) equivalent to the identity operator and gives, accordingly, a zero matrix element for the transition A → B. Consequently, the disconnected G for which xk and xrk lie in G2 give zero contribution to the matrix element of (31), and can be omitted from further consideration. When xk and xrk lie in G1, again the C(G) may be summed over all G consisting of the given G1 and all possible G2; but this time the connected graph G1 itself is to be included in the sum. The sum of all the C(G) in this case turns out to be just C(G1) multiplied by the expectation value in the vacuum of the operator S(∞). But the vacuum state, being a steady state, satisfies (21), and so the expectation value in question is equal to unity. Therefore the sum of the C(G) reduces to the single term C(G1), and again the disconnected graphs may be omitted from consideration.
The elimination of disconnected graphs is, from a physical point of view, somewhat trivial, since these graphs arise merely from the fact that meaningful physical processes proceed simultaneously with totally irrelevant fluctuations of fields in the vacuum. However, similar arguments will now be used to eliminate a much more important class of graphs, namely, those involving self-energy effects. A “self-energy part” of a graph G is defined as follows; it is a set of one or more vertices not including x0, together with the lines joining them, which is connected with the remainder of G (or with the edge of the diagram) only by two electron lines or by one or two photon lines. For definiteness it may be supposed that G has a self-energy part F, which is connected with its surroundings only by one electron line entering F at x1, and another leaving F at x2; the case of photon lines can be treated in an entirely analogous way. The points x1 and x2 may or may not be identical. From G a “reduced graph” G0 can be obtained by omitting F completely and joining the incoming line at x1 with the outgoing line at x2 to form a single electron line in G0, the newly formed line being denoted by λ. Given G0 and λ, there is conversely a well determined set Γ of graphs G which are associated with G0 and λ in this way; G0 itself-is considered also to belong to Γ. It will now be shown that the sum C(Γ) of the contributions C(G) to the matrix element of (31) from all the graphs G of Γ reduces to a single term C’(G0).
Suppose, for example, that the line λ in G0 leads from a point x3 to the edge of the diagram. Then C(G0) is an integral containing in the integrand the matrix element of
(52)
for creation of an electron into the state B. Let the momentum-energy 4-vector of the created electron be p; the matrix element of (52) is of the form
(53)
with aα independent of x3. Now consider the sum C(Γ). It follows from an analysis of (31) that C(Γ) is obtained from C(G0) by replacing the operator (52) by
(54)
(This is, of course, a consequence of the special character of the graphs of Γ.) It is required to calculate the matrix element of (54) for a transition from the vacuum state O to the state B, i.e., for the emission of an electron into state B. This matrix element will be denoted by Zα; C(Γ) involves Żα in the same way that C(G0) involves (53). Now Zα can be evaluated as a sum of terms of the same general character as (47); it will be of the form
where the important fact is that Ki is a function only of the coordinate differences between yi and x3. By (53), this implies that
(55)
with R independent of x3. From considerations of relativistic invariance, R must be of the form
where p2 is the square of the invariant length of the 4-vector p. But since the matrix element (53) is a solution of the Dirac equation,
and so (55) reduces toZα = R1Yα (x3) ,
with R1 an absolute constant. Therefore the sum C(Γ) is in this case just C’(G0), where C’(G0) is obtained from C(G0) by the replacement
(56)
In the case when the line λ leads into the graph G0 from the edge of the diagram to the point x1, it is clear that C(Γ) will be similarly obtained from C(G0) by the replacement
(57)
There remains the case in which λ leads from one vertex x3 to another x4 of G0. In this case C(G0) contains in its integrand the function
(58)
which is the vacuum expectation value of the operator
(59)
according to (43). Now in analogy with (54), C(Γ) is obtained from C(G0) by replacing (59) by
(60)
and the vacuum expectation value of this operator will be denoted by
(61)
By the methods of Section VI, (61) can be expanded as a series of terms of the same character as (47); this expansion will not be discussed in detail here, but it is easy to see that it leads to an expression of the form (61), with S’F(x) a certain universal function of the 4-vector x. It will not be possible to reduce (61) to a numerical multiple of (58), as Zα was in the previous case reduced to a multiple of Yα. Instead, there may be expected to be a series expansion of the form
(62)
where □2 is the Dalembertian operator and the a, b are numerical coefficients. In this case C(Γ) will be equal to the C’(G0) obtained from C(G0), by the replacement
(63)
Applying the same methods to a graph G with a self-energy part connected to its surroundings by two photon lines, the sum C(Γ) will be obtained as a single contribution C’(G0) from the reduced graph G0, C’(G0) being formed from C(G0) by the replacement
(64)
The function D’F is defined by the condition that
(65)
is the vacuum expectation value of the operator
(66)
and may be expanded in a series
(67)
Finally, it is not difficult to see that for graphs G with self-energy parts connected to their surroundings by a single photon line, the sum C(Γ) will be identically zero, and so such graphs may be omitted from consideration entirely.
As a result of the foregoing arguments, the contributions C(G) of graphs with self-energy parts can always be replaced by modified contributions C’(G0) from a reduced graph G0. A given G may be reducible in more than one way to give various G0, but if the process of reduction is repeated a finite number of times a G0 will be obtained which is “totally reduced,” contains no self-energy part, and is uni
quely determined by G. The contribution C’(G0) of a totally reduced graph to the matrix element of (31) is now to be calculated as a sum of integrals of expressions like (47), but with a replacement (56), (57), (63), or (64) made corresponding to every line in G0. This having been done, the matrix element of (31) is correctly calculated by taking into consideration each totally reduced graph once and once only.
The elimination of graphs with self-energy parts is a most important simplification of the theory. For according to (22), HI contains the subtracted part H S , which will give rise to many additional terms in the expansion of (31). But if any such term is taken, say, containing the factor HS (xi) in the integrand, every graph corresponding to that term will contain the point xi joined to the rest of the graph only by two electron lines, and this point by itself constitutes a self-energy part of the graph. Therefore, all terms involving HS are to be omitted from (31) in the calculation of matrix elements. The intuitive argument for omitting these terms is that they were only introduced in order to cancel out higher order self-energy terms arising from Hi , which are also to be omitted; the analysis of the foregoing paragraphs is a more precise form of this argument. In physical language, the argument can be stated still more simply; since δm is an unobservable quantity, it cannot appear in the final description of observable phenomena.
VIII. VACUUM POLARIZATION AND CHARGE RENORMALIZATION
The question now arises: What is the physical meaning of the new functions D’F and S’F, and of the constant R1? In general terms, the answer is clear. The physical processes represented by the self-energy parts of graphs have been pushed out of the calculations, but these processes do not consist entirely of unobservable interactions of single particles with their self-fields, and so cannot entirely be written off as “self-energy processes.” In addition, these processes include the phenomenon of vacuum polarization, i.e., the modification of the field surrounding a charged particle by the charges which the particle induces in the vacuum. Therefore, the appearance of D’F , S’F , and R1 in the calculations may be regarded as an explicit representation of the vacuum polarization phenomena which were implicitly contained in the processes now ignored.
In the present theory there are two kinds of vacuum polarization, one induced by the external field and the other by the quantized electron and photon fields themselves; these will be called “external” and “internal,” respectively. It is only the internal polarization which is represented yet in explicit fashion by the substitutions (56), (57), (63), (64); the external will be included later.
To form a concrete picture of the function D’F, it may be observed that the function DF(y − z) represents in classical electrodynamics the retarded potential of a point charge at y acting upon a point charge at z, together with the retarded potential of the charge at z acting on the charge at y. Therefore, DF may be spoken of loosely as “the electromagnetic interaction between two point charges.” In this semiclassical picture, D’F is then the electromagnetic interaction between two point charges, including the effects of the charge-distribution which each charge induces in the vacuum.
The complete phenomenon of vacuum polarization, as hitherto understood, is included in the above picture of the function D’F. There is nothing left for S’F to represent. Thus, one of the important conclusions of the present theory is that there is a second phenomenon occurring in nature, included in the term vacuum polarization as used in this paper, but additional to vacuum polarization in the usual sense of the word. The nature of the second phenomenon can best be explained by an example.
The scattering of one electron by another may be represented as caused by a potential energy (the Møller interaction) acting between them. If one electron is at y and the other at z, then, as explained above, the effect of vacuum polarization of the usual kind is to replace a factor DF in this potential energy by D’F. Now consider an analogous, but unorthodox, representation of the Compton effect, or the scattering of an electron by a photon. If the electron is at y and the photon at z, the scattering may be again represented by a potential energy, containing now the operator SF(y − z) as a factor; the potential is an exchange potential, because after the interaction the electron must be considered to be at z and the photon at y, but this does not detract from its usefulness. By analogy with the 4-vector charge-current density jµ which interacts with the potential DF, a spinor Compton-effect density uα may be defined by the equationuα (x) = Aµ(x) (γµ)αβψβ (x),
and an adjoint spinor by
These spinors are not directly observable quantities, but the Compton effect can be adequately described as an exchange potential, of magnitude proportional to SF (y − z), acting between the Compton-effect density at any point y and the adjoint density at z. The second vacuum polarization phenomenon is described by a change in the form of this potential from SF to S‘F. Therefore, the phenomenon may be pictured in physical terms as the inducing, by a given element of Compton-effect density at a given point, of additional Compton-effect density in the vacuum around it.
In both sorts of internal vacuum polarization, the functions DF and SF, in addition to being altered in shape, become multiplied by numerical (and actually divergent) factors R3 and R2; also the matrix elements of (31) become multiplied by numerical factors such as R1 . However, it is believed (this has been verified only for second-order terms) that all n’th-order matrix elements of (31) will involve these factors only in the form of a multiplier
this statement includes the contributions from the higher terms of the series (62) and (67). Here e is defined as the constant occurring in the fundamental interaction (16) by virtue of (37). Now the only possible experimental determination of e is by means of measurements of the effects described by various matrix elements of (31), and so the directly measured quantity is not e but . Therefore, in practice the letter e is used to denote this measured quantity, and the multipliers R no longer appear explicitly in the matrix elements of (31); the change in the meaning of the letter e is called “charge renormalization,” and is essential if e is to be identified with the observed electronic charge. As a result of the renormalization, the divergent coefficients R1, R2, and R3 in (56), (57), (62), and (67) are to be replaced by unity, and the higher coefficients a, b, and c by expressions involving only the renormalized charge e.
The external vacuum polarization induced by the potential is, physically speaking, only a special case of the first sort of internal polarization; it can be treated in a precisely similar manner. Graphs describing external polarization effects are those with an “external polarization part,” namely, a part including the point x0 and connected with the rest of the graph by only a single photon line. Such a graph is to be “reduced” by omitting the polarization part entirely and renaming with the label x0 the point at the further end of the single photon line. A discussion similar to those of Section VII leads to the conclusion that only reduced graphs need be considered in the calculation of the matrix element of (31), and that the effect of external polarization is explicitly represented if in the contributions from these graphs a replacement
(68)
is made. After a renormalization of the unit of potential, similar to the renormalization of charge, the modified potential takes the form
(69)
where the coefficients are the same as in (67).
It is necessary, in order to determine the functions D‘F , S’F, and , to go back to formulas (60) and (66). The determination of the vacuum expectation values of the operators (60) and (66) is a problem of the same kind as the original problem of the calculation of matrix elements of (31), and the various terms in the operators (60) and (66) must again be split up, represented by graphs, and analyzed in detail. However, since D‘F and S’F are universal functions, this further analysis has only to be carried out once to be applicable to all problems.
It is one of the major triumphs of the Schwinger theory that it enables an unambiguous interpretation to be given to the phenomenon of vacuum polari
zation (at least of the first kind), and to the vacuum expectation value of an operator such as (66). In making this interpretation, profound theoretical problems arise, particularly concerned with the gauge invariance of the theory, about which nothing will be said here. For Schwinger’s solution of these problems, the reader must refer to his forthcoming papers. Schwinger’s argument can be transferred without essential change into the framework of the present paper.
Having overcome the difficulties of principle, Schwinger proceeded to evaluate the function D‘F explicitly as far as terms of order α = (e2/4πħc) (heaviside units). In particular, he found for the coefficient c1 in (67) and (69) the value to this order.oo It is hoped to publish in a sequel to the present paper a similar evaluation of the function S’F; the analysis involved is too complicated to be summarized here.
The Dreams That Stuff is Made of Page 76