The Dreams That Stuff is Made of

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by Stephen Hawking


  (24)

  and (S(σ0))−1 by a corresponding expression analogous to (14). Substitution of these series into (20) gives at once

  (25)

  The repeated commutators in this formula are characteristic of the Schwinger theory, and their evaluation gives rise to long and rather difficult analysis. Using the first three terms of the series, Schwinger was able to calculate the second-order radiative corrections to the equations of motion of an electron in an external field, and obtained satisfactory agreement with experimental results. In this paper the development of the Schwinger theory will be carried no further; in principle the radiative corrections to the equations of motion of electrons could be calculated to any desired order of approximation from formula (25).

  In the Feynman theory the basic principle is to preserve symmetry between past and future. Therefore, the matrix elements of the operator HT are evaluated in a “mixed representation;” the matrix elements are calculated between an initial state specified by its state vector Ω1 and a final state specified by its state vector Ωʹ2. The matrix element of HT between two such states in the Schwinger representation is

  (26)

  and therefore the operator which replaces HT in the mixed representation is

  (27)

  Going back to the original product definition of S(σ) analogous to (10), it is clear that S(∞) × (S(σ))−1 is simply the operator obtained from S(σ) by interchanging past and future. Thus,

  (28)

  The physical meaning of a mixed representation of this type is not at all recondite. In fact, a mixed representation is normally used to describe such a process as bremsstrahlung of an electron in the field of a nucleus when the Born approximation is not valid; the process of bremsstrahlung is a radiative transition of the electron from a state described by aCoulomb wave function, with a plane ingoing and a spherical outgoing wave, to a state described by a Coulomb wave function with a spherical ingoing and a plane outgoing wave. The initial and final states here belong to different orthogonal systems of wave functions, and so the transition matrix elements are calculated in a mixed representation. In the Feynman theory the situation is analogous; only the roles of the radiation interaction and the external (or Coulomb) field are interchanged; the radiation interaction is used instead of the Coulomb field to modify the state vectors (wave functions) of the initial and final states, and the external field instead of the radiation interaction causes transitions between these state vectors.

  In the Feynman theory there is an additional simplification. For if matrix elements are being calculated between two states, either of which is steady (and this includes all cases so far considered), the mixed representation reduces to an ordinary representation. This occurs, for example, in treating a one-particle problem such as the radiative correction to the equations of motion of an electron in an external field; the operator HF(x0), although in general it is not even Hermitian, can in this case be considered as an effective external potential energy acting on the particle, in the ordinary sense of the words.

  This section will be concluded with the derivation of the fundamental formula (31) of the Feynman theory, which is the analog of formula (25) of the Schwinger theory. IfF1(x1),...,Fn (xn)

  are any operators defined, respectively, at the points x1, . . . , xn of space-time, then

  (29)

  will denote the product of these operators, taken in the order, reading from right to left, in which the surfaces σ(x1), ..., σ(xn) occur in time. In most applications of this notation Fi (xi ) will commute with Fj(xj) so long as xi and xj are outside each other’s light cones; when this is the case, it is easy to see that (29) is a function of the points x1, . . ., xn only and is independent of the surfaces σ(xi ). Consider now the integral

  Since the integrand is a symmetrical function of the points x1, . . ., xn, the value of the integral is just n! times the integral obtained by restricting the integration to sets of points x1, . . ., xn for which σ(xi) occurs after σ(xi+1) for each i. The restricted integral can then be further divided into (n+1) parts, the j’th part being the integral over those sets of points with the property that σ(x0) lies between σ (xj−1) and σ(xj) (with obvious modifications for j = 1 and j = n + 1). Therefore,

  (30)

  Now if the series (24) and (28) are substituted into (27), sums of integrals appear which are precisely of the form (30). Hence finally

  (31)

  By this formula the notation HF (x0 ) is justified, for this operator now appears as a function of the point x0 alone and not of the surface σ. The further development of the Feynman theory is mainly concerned with the calculation of matrix elements of (31) between various initial and final states.

  As a special case of (31) obtained by replacing He by the unit matrix in (27),

  (32)

  VI. CALCULATION OF MATRIX ELEMENTS

  In this section the application of the foregoing theory to a general class of problems will be explained. The ultimate aim is to obtain a set of rules by which the matrix element of the operator (31) between two given states may be written down in a form suitable for numerical evaluation, immediately and automatically. The fact that such a set of rules exists is the basis of the Feynman radiation theory; the derivation in this section of the same rules from what is fundamentally the Tomonaga-Schwinger theory constitutes the proof of equivalence of the two theories.

  To avoid excessive complication, the type of matrix element considered will be restricted in two ways. First, it will be assumed that the external potential energy is

  (33)

  that is to say, the interaction energy of the electron-positron field with electromagnetic potentials which are given numerical functions of space and time. Second, matrix elements will be considered only for transitions from a state A, in which just one electron and no positron or photon is present, to another state B of the same character. These restrictions are not essential to the theory, and are introduced only for convenience, in order to illustrate clearly the principles involved.

  The electron-positron field operator may be written

  (34)

  where the φuα (x) are spinor wave functions of free electrons and positrons, and the au are annihilation operators of electrons and creation operators of positrons. Similarly, the adjoint operator

  (35)

  where āu are annihilation operators of positrons and creation operators of electrons. The electromagnetic field operator is

  (36)

  where bv and are photon annihilation and creation operators, respectively. The charge-current 4-vector of the electron field is

  (37)

  strictly speaking, this expression ought to be antisymmetrized to the formon

  (38)

  but it will be seen later that this is not necessary in the present theory.

  Consider the product P occurring in the n’th integral of (31); let it be denoted by Pn. From (16), (22), (33), and (37) it is seen that Pn is a sum of products of (n + 1) operators ψα , (n + 1) operators , and not more than n operators Aµ, multiplied by various numerical factors. By Qn may be denoted a typical product of factors ψα, , and Aµ, not summed over the indices such as α and µ, so that Pn is a sum of terms such as Qn. Then Qn will be of the form (indices omitted)

  (39)

  where i0, i1, . . . , in is some permutation of the integers 0, 1, . . . , n, and j1, . . . , jm are some, but not necessarily all, of the integers 1,...,n in some order. Since none of the operators and ψ commute with each other, it is especially important to preserve the order of these factors. Each factor of Qn is a sum of creation and annihilation operators by virtue of (34), (35), and (36), and so Qn itself is a sum of products of creation and annihilation operators.

  Now consider under what conditions a product of creation and annihilation operators can give a non-zero matrix element for the transition A → B. Clearly, one of the annihilation operators must annihilate the electron in state A, one of the creation operators must create the ele
ctron in state B, and the remaining operators must be divisible into pairs, the members of each pair respectively creating and annihilating the same particle. Creation and annihilation operators referring to different particles always commute or anticommute (the former if at least one is a photon operator, the latter if both are electron-positron operators). Therefore, if the two single operators and the various pairs of operators in the product all refer to different particles, the order of factors in the product can be altered so as to bring together the two single operators and the two members of each pair, without changing the value of the product except for a change of sign if the permutation made in the order of the electron and positron operators is odd. In the case when some of the single operators and pairs of operators refer to the same particle, it is not hard to verify that the same change in order of factors can be made, provided it is remembered that the division of the operators into pairs is no longer unique, and the change of order is to be made for each possible division into pairs and the results added together.

  It follows from the above considerations that the matrix element of Qn for the transition A → B is a sum of contributions, each contribution arising from a specific way of dividing the factors of Qn into two single factors and pairs. A typical contribution of this kind will be denoted by M. The two factors of a pair must involve a creation and an annihilation operator for the same particle, and so must be either one and one ψ or two A; the two single factors must be one and one ψ . The term M is thus specified by fixing an integer k, and a permutation r0, r1, . . . , rn of the integers 0, 1, . . . , n, and a division (s 1, t1), (s2, t2 ) , . . . , (sh , th ) of the integers j1, . . . , jm into pairs; clearly m = 2h has to be an even number; the term M is obtained by choosing for single factors ψ (xk) and ψ (xrk), and for associated pairs of factors (ψ (xi), ψ (xri )) for i = 0, 1, . . . , k − 1, k + 1, . . . , n and (A(xsi), A(xti)) for i = 1, . . . , h . In evaluating the term M, the order of factors in Qn is first to be permuted so as to bring together the two single factors and the two members of each pair, but without altering the order of factors within each pair; the result of this process is easily seen to be

  (40)

  a factor ε being inserted which takes the value ±1 according to whether the permutation of ψ and ψ factors between (39) and (40) is even or odd. Then in (40) each product of two associated factors (but not the two single factors) is to be independently replaced by the sum of its matrix elements for processes involving the successive creation and annihilation of the same particle.

  Given a bilinear operator such as Aµ(x)Aν (y), the sum of its matrix elements for processes involving the successive creation and annihilation of the same particle is just what is usually called the “vacuum expectation value” of the operator, and has been calculated by Schwinger. This quantity is, in fact (note that Heaviside units are being used)

  where D(1) and D are Schwinger’s invariant D functions. The definitions of these functions will not be given here, because it turns out that the vacuum expectation value of P(Aµ(x), Aν(y)) takes an even simpler form. Namely,

  (41)

  where DF is the type of D function introduced by Feynman. DF (x) is an even function of x, with the integral expansion

  (42)

  where x2 denotes the square of the invariant length of the 4-vector x. In a similar way it follows from Schwinger’s results that

  (43)

  where

  (44)

  κ0 is the reciprocal Compton wave-length of the electron, η(x, y) is −1 or +1 according as σ (x) is earlier or later than σ (y) in time, and ΔF is a function with the integral expansion

  (45)

  Substituting from (41) and (44) into (40), the matrix element M takes the form (still omitting the indices of the factors , ψ , and A of Qn)

  (46)

  The single factors and ψ(xrk) are conveniently left in the form of operators, since the matrix elements of these operators for effecting the transition A → B depend on the wave functions of the electron in the states A and B. Moreover, the order of the factors and ψ(xrk) is immaterial since they anticommute with each other; hence it is permissible to write

  Therefore (46) may be rewritten

  (47)

  with

  (48)

  Now the product in (48) is (−1)p, where p is the number of occasions in the expression (40) on which the ψ of a P bracket occurs to the left of the . Referring back to the definition of ε after Eq. (40), it follows that ε’ takes the value +1 or −1 according to whether the permutation of and ψ factors between (39) and the expression

  (49)

  is even or odd. But (39) can be derived by an even permutation from the expression

  (50)

  and the permutation of factors between (49) and (50) is even or odd according to whether the permutation r0 , . . . , rn of the integers 0, . . . , n is even or odd. Hence, finally, ε’ in (47) is +1 or −1 according to whether the permutation of r0, . . . , rn is even or odd. It is important that ∈’ depends only on the type of matrix element M considered, and not on the points x0, . . . , xn; therefore, it can be taken outside the integrals in (31).

  One result of the foregoing analysis is to justify the use of (37), instead of the more correct (38), for the charge-current operator occurring in He and Hi . For it has been shown that in each matrix element such as M the factors and ψ in (38) can be freely permuted, so that (38) can be replaced by (37), except in the case when the two factors form an associated pair. In the exceptional case, M contains as a factor the vacuum expectation value of the operator jµ(xi) at some point xi; this expectation value is zero according to the correct formula (38), though it would be infinite according to (37); thus the matrix elements in the exceptional case are always zero. The conclusion is that only those matrix elements are to be calculated for which the integer ri differs from i for every i ≠ k, and in these elements the use of formula (37) is correct.

  To write down the matrix elements of (31) for the transition A → B , it is only necessary to take all the products Qn , replace each by the sum of the corresponding matrix elements M given by (47), reassemble the terms into the form of the Pn from which they were derived, and finally substitute back into the series (31). The problem of calculating the matrix elements of (31) is thus in principle solved. However, in the following section it will be shown how this solution-inprinciple can be reduced to a much simpler and more practical procedure.

  VII. GRAPHICAL REPRESENTATION OF MATRIX ELEMENTS

  Let an integer n and a product Pn occurring in (31) be temporarily fixed. The points x0 , x1 , . . . , xn , may be represented by (n + 1) points drawn on a piece of paper. A type of matrix element M as described in the last section will then be represented graphically as follows. For each associated pair of factors with i ≠ k , draw a line with a direction marked in it from the point xi to the point xri . For the single factors , ψ(xrk), draw directed lines leading out from xk to the edge of the diagram, and in from the edge of the diagram to xrk. For each pair of factors (A(xsi), A(xti)), draw an undirected line joining the points xsi and xti . The complete set of points and lines will be called the “graph” of M; clearly there is a one-to-one correspondence between types of matrix element and graphs, and the exclusion of matrix elements with ri − i for i ≠ k corresponds to the exclusion of graphs with lines joining a point to itself. The directed lines in a graph will be called “electron lines,” the undirected lines “photon lines.”

  Through each point of a graph pass two electron lines, and therefore the electron lines together form one open polygon containing the vertices xk and xrk, and possibly a number of closed polygons as well. The closed polygons will be called “closed loops,” and their number denoted by l. Now the permutation r0, . . . , rn of the integers 0, . . . , n is clearly composed of (l + 1) separate cyclic permutations. A cyclic permutation is even or odd according to whether the number of elements in it is odd or even. Hence the parity of the permutation r0, ... , rn is the pa
rity of the number of even-number cycles contained in it. But the parity of the number of odd-number cycles in it is obviously the same as the parity of the total number (n + 1) of elements. The total number of cycles being (l + 1), the parity of the number of even-number, cycles is (l − n). Since it was seen earlier that the ε’ of Eq. (47) is determined just by the parity of the permutation r0, ..., rn, the above argument yields the simple formula

  (51)

  This formula is one result of the present theory which can be much more easily obtained by intuitive considerations of the sort used by Feynman.

 

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