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Approximations to the QED Fermion Green's Function in a Constant External Field

An exact representation of the causal QED fermion Green's function, in an arbritrary external electromagnetic field, derived by Fried, Gabellini and McKellar, and which naturally allows for non-perturbative approximations, is here used to calculate non-perturbative approximations to the Green's function in the simple case of a constant external field. Schwinger's famous exact result is obtained as the limit as the order of the approximation approaches infinity.


I. INTRODUCTION
An exact representation for the causal QED fermion Green's function, G c (x, y |A), in an arbitrary external field, was derived by Fried et al. [1] in such a way as to obtain an exact representation which naturally allows for non-perturbative approximations. The rather intimidating exact representation is where Ω(s ′ ) is the solution to the "map" and ζ(s ′ ) is given by ′ represent the product and sum respectively over all odd natural numbers. It is easy to see that we may approximate the Green's function non-perturbatively by retaining a finite number of integrations, in particular, for the nth approximation G (n) c (x, y |A), where n = 0, 1, 2, . . ., we retain 2n + 3 integrations. This means that for n = 0, the odd number N takes the values N = 1, 3, . . . , 2n−1, while for the zeroth approximation all N-dependence is neglected. The exact result is of course recovered in the limit n → ∞, that is, G c (x, y |A) = lim n→∞ G (n) c (x, y |A). It is well known that there is an exact expression, first obtained by Schwinger [2], for the fermion Green's function in the simple case of a constant, but otherwise arbitrary, external field, which (in an arbitrary gauge) reads where the holonomy factor Φ(x, y |A) is and carries the complete gauge-dependence of the Green's function. It is convenient to employ matrix notation, in which we regard the field strength tensor as a constant, antisymmetric 4 × 4 matrix F . It is obvious that Schwinger's result must somehow be contained as a special case of the exact representation (1.1)-(1.3). Furthermore, the Fradkin representation [3], [4], from which the above representation was derived, almost trivially yields Schwinger's result. This leads us to expect that the latter may be extracted, analytically, from the above representation, and here we demonstrate that this is indeed the case.
We proceed by evaluating that nth non-perturbative approximation to the Green's function in a constant field as given by (1.1)-(1.3) Schwinger's result is then recovered in the limit as the order of this approximation approaches infinity. That we may carry out this programme relies on two things: for the case of constant F the ordered exponential of (1.1) becomes an ordinary exponential, and the 2n + 3 integrations we must perform are all Gaussian. Indeed a large part of the evaluation of the nth approximation is an extension of our matrix notation to account for this latter fact. 2n + 3 Gaussian integrals may be expressed as one Gaussian integral over a 2n + 3-dimensional space. The integration is then, with the appropriate notation, essentially trivial. In Section II we make the change of notation, perform the integration, and express the nth approximation in terms of the two resulting, order 2n + 3, determinants. In Section III we evaluate the determinants and take the limit n → ∞ to recover Schwinger's result.
We emphasise that the result when stopped at a finite n (except n = 0) is essentially non-perturbative, in that it is not a polynomial in g. The result of (1.1)-(1.3) is thus, as n is increased, a systematically improving, but non-perturbative, method for calculating the Green's function in an external field.

II. THE INTEGRATION
We must first choose a gauge to work in. It shall be convenient to work exclusively in the Schwinger-Fock (SF ) gauge the initial motivation for which is that we may forget about the holonomy factor, which reduces to 1. An immediate conseqence is that ∂ ∂zµ A SF µ (z) = 0, due to the antisymmetry of F ; the factor in (1.1) containing this term in the exponent also reduces to 1. With the above simplifications, the nth approximation to the fermion Green's function in a constant field, in the SF gauge, as given by (1.1), is thus where Ω (n) (s ′ ) is the solution to (1.2), (1.3), but with the sum in (1.3) terminating at N = 2n − 1, and in the spirit of using matrix notation we have used dp instead of d 4 p and so on. In the SF gauge, (1.2), (1.3) yield a simple integral equation for Ω (n) (s ′ ): 3) The equivalent differential equation plus boundary condition may be solved with only elementary integrals. It is the combination p − Ω (n) (s ′ ) which appears in (2.2), we find where Λ N = N π gF s . Our choice of gauge allows us to make the change of variable z − y → z, after which p − Ω (n) (s ′ ) is independent of y, so that the only x and y dependence in the exponent of (2.2) appears in the term ip · (x − y).
If we imagine substituting (2.4) into (2.2), we recognise that all of the terms in the exponent of the integrand, save the term ip · (x − y), are able to be expressed in the form where the X i are the 2n + 3 4-vector variables we must integrate over, the A (n) ij are some matrix functions of F , and the i 2 is a convenient normalisation factor. This suggests that we extend our matrix notation, and write this part of the exponent of (2.2) Let us define the column vector X such that The elements of the row vector C (n)T (s ′ ) are read straight from (2.4). The transpose of the row vector is , and omitted the last two N = 2n − 1 elements for brevity. We also write e iP ·z+i s (2.9) Our change of notation complete, (2.2) and (2.4) become All that we have done is to re-express, in the usual way, the product of 2n + 3 Gaussian integrals as one (2n + 3)-dimensional Gaussian integral. After a change of variable X → X − A (n)−1 B (n) , the integration is trivial, we obtain where Det A (n) is the determinant of A (n) . The matrix A (n) is a 4(2n + 3) × 4(2n + 3) matrix which is naturally partitioned into the (2n + 3) × (2n + 3) 4 × 4 matrices A ij , as (matrix) functions of F only, commute, and are thus referred to as the "elements" of A (n) . Since A (n) is symmetric, we have the relations A ij , the result will be a 4 × 4 matrix, also some function of F , which we denote det A (n) . It is easy to convince oneself that the determinant of A (n) , which may be partitioned in this way, is the determinant of the matrix we have called det A (n) , that is, Det A (n) = det(det A (n) ).
Recalling that B (n) i = (x − y)δ i1 , and with the notation discussed above, we may write 1 is the matrix obtained by replacing the first row of A (n) by C (n)T (s). Similarly, where detÃ (n) is the (1, 1) cofactor of A (n) , that is,Ã (n) is the matrix obtained by deleting the first row and the first column of A (n) . In fact it will be shown in the next section that det A (n) symmetric with respect to the interchange of space-time indices, for all n, and denoted by a superscript S. The nth approximation to the Green's function, in terms of the two determinants det A (n) and det A (n) 1 , is then det A (n) provides a non-perturbative approximation to − 1 2 (gF coth gF s + gF ), the symmetric part of the former a non-perturbative approximation to the symmetric part of the latter. In the next section we calculate exact expressions for these non-perturbative approximations, and show that we can obtain (2.16) and (2.17), and thus Schwinger's result, in the limit n → ∞.

III. THE DETERMINANTS
An obvious but important fact is that we only need to find the determinants of the matrices A (n) and A (n) 1 , not the matrices themselves. This means that we can simplify C (n) (s ′ ), its transpose, and D (n) , with any row and column operations which do not alter the determinant, before using (2.17) to find A (n) . We now find the reduced form of A (n) , from this it will be easy to obtain the reduced form of A (n) 1 and ofÃ (n) . We perform the following sets of row operations on the column vector C (n) (s ′ ): use the first element to remove those terms proportional to e −gF s ′ from all other elements, noting that the second element requires the row operation row 2 → row 2 + 1 2 gF row 1 ; then ; the last two sets for all rows N = 1, 3, . . . , 2n − 1.
The row operations are performed on C (n) (s ′ ) and D (n) . To keep things symmetric we perform the transposed operations on C (n)T (s ′ ) and D (n) . The integrals required by (2.17) are elementary and we easily obtain the reduced form of the matrix A (n) : gF (1 + e gF s ) and we have further used the (2, 3) and (3, 2) elements of the reduced A (n) to eliminate those elements to the right of the former and below the latter. After trivially expanding the determinant along the second row and down the second column we are left with the determinant of a (2n + 1) × (2n + 1) bordered matrix. The first set of row and column operations and the orthogonality of the sine and cosine functions have ensured the matrix is block diagonal, the other sets of operations that every second element of the border is zero. Expanding along the first row and down the first column we obtain the following expression for the nth non-perturbative approximation to (2.16): where and we have written the approximation in terms of λ = gF s, (Λ N = N π λ ). That (3.3) is an approximation to (2.16) can be seen with the help of the relations [5] The nature of the approximation is now evident. The function sinh λ λ is rewritten as , the foremost factor is expressed exactly as the infinite product of the α N s, and the expression in the square brackets as the infinite sum of the β N s. The nth approximation is then defined by including the first n terms in the product and in the sum. Note that det A (n) is an even function of λ (of F ), and hence symmetric, for all n. This is desirable since the exact (2.16) is symmetric. The zeroth approximation, in which (2.16) is approximated by det A (0) = 2s, is the only perturbative result, of order (gF s) 0 . Every approximation order greater than zero contains all (natural number) powers of gF s.
The matrix (3.2) shall be our starting point for finding det A (n) 1 . Before replacing the first row of (3.2) with the column-reduced form of C (n)T (s) (the transpose of (3.1) with s ′ = s), we must reinstate the second row via row 2 → row 2 − 1 2 gF row 1 . This procedure is valid since the second element of C (n) (s ′ ) in (2.8) is proportional to the first element, and we could have used the second element to eliminate the e −gF s ′ terms instead of the first element. After thus reinstating the second row and replacing the first row with the column-reduced C (n)T (s), we obtain the reduced form of the (neither symmetric nor antisymmetric) matrix A (n) 1 : We may now expand the determinant to obtain the following nth approximation to (2.17): (3.9) Using the second relation of (3.5), we find (3.10) The function 2e λ is thus rewritten exactly as cosh λ the first factor again expressed as the infinite product of the α N s, the expression in the square brackets as the infinite sum of the γ N s, and approximated by taking the first n terms in the product and in the sum. Note that the zeroth approximation det A The nth non-perturbative approximation to − 1 2 (gF coth gF s + gF ) is the ratio of (3.8) and (3.3) (note that the product over the α N s cancel) is and is the object that appears in the nth approximation to the Green's function (2.15). It is not obvious from the reduced form of the matrix A (n) 1 that the symmetric part of the determinant of the same is the determinant of the matrixÃ (n) . That is no problem, we find detÃ (n) in much the same manner as we found det A (n) 1 : take the reduced form of A (n) 1 , reinstate the second column with the operation col 2 → col 2 + 1 2 gF col 1 , and discard the first row and column to obtain the symmetric, reduced form ofÃ (n) : 2 (e gF s + 1) 0 0 0 · · · 0 0 · · ·  Upon comparison of (3.12) and (3.8), noting that α N , β N , and γ N are all even functions of λ (F ), and using 1 2 (1 + cosh λ) = cosh 2 λ 2 , our earlier statement that detÃ (n) is the symmetric part of det A (n) 1 is apparent. Finally then, (3.8), the symmetric part thereof, (3.12), and (3.3) substituted into (2.15) give the nth approximation to the fermion Green's function in a constant external field, in the Schwinger-Fock gauge, as defined through the exact representation (1.1)-(1.3). The approximation easily leads to Schwinger's exact result (1.4) in the limit n → ∞, and we repeat and emphasise that only the zeroth approximation corresponds to a perturbative result.