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Analytic Formulae for the Feynman Propagator in Coordinate Space Howard E

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Analytic Formulae for the Feynman Propagator in Coordinate Space Howard E Analytic formulae for the Feynman propagator in coordinate space Howard E. Haber Santa Cruz Institute for Particle Physics University of California, Santa Cruz, CA 95064, USA April 17, 2020 Abstract In these notes, we provide an explicit calculation of the Feynman propagator of scalar field theory in coordinate space in four spacetime dimensions. Two different methods of the computation are provided. The methods employed demonstrate that the integral representation of the Feynman propagator cannot be interpreted as an ordinary function but rather as a generalized function (or more precisely a tempered distribution). We begin with the integral representation of the free-field Feynman propagator of scalar field theory in four spacetime dimensions, d4p 1 ∆ (x)= e−ipx F (2π)4 p2 m2 + iǫ Z − ∞ 1 · 1 = d3p dp e−ip0x0+i~p ~x . (1) 4 0 2 2 2 (2π) −∞ p ~p m + iǫ Z Z 0 − − where p and x are four vectors, px p x and ǫ is a real positive infinitesimal quantity. Consider the integral ≡ · ∞ e−ip0x0 = lim dp . (2) + 0 2 2 2 I ǫ→0 −∞ p ~p m + iǫ Z 0 − − First, we consider the case of x > 0. In the limit of ǫ 0, the integrand has poles at p = p , 0 → 0 ± where p ~p 2 + m2 . ± ≡ ± q To evaluate , we shall close the contour in the lower half of the complex p -plane since e−ip0x0 I 0 is exponentially small along the semicircle at infinity when x0 > 0. Only the pole p+ lies inside the closed contour. Hence by the residue theorem of complex analysis, πi exp ix ~p 2 + m2 = − 0 , for x > 0, (3) 2 0 I − ~p p+ m2 where the minus sign is due to the factp that the closed contour is clockwise. Second, we consider the case of x0 < 0. In this case, we shall evaluate by closing the −ip0x0 I contour in the upper half of the complex p0-plane so that e is exponentially small along 1 the semicircle at infinity when x0 < 0. Only the pole p− lies inside the closed contour. Hence by the residue theorem of complex analysis, πi exp ix ~p 2 + m2 = 0 , for x < 0. (4) 2 0 I − ~p p+ m2 In this case, the closed contour is counterclockwisep and the minus sign arises due to the fact that p p is negative. − − + Without loss of generality, we may choose the z-axis to lie along the vector ~x, in which case ei~p·~x = eipr cos θ, where p ~p and r ~x . Hence, it follows that ≡| | ≡| | 2π 4π sin(pr) d cos θdφeipr cos θ = eipr e−ipr = . (5) ipr − pr Z Collecting all of our results above, if follows that ∞ 2 2 1 p sin(pr) exp i x0 p + m i∆F (x0; ~x)= − | | dp . (6) 2 2 2 4π r 0 p + m Z p Note that the integral above is not convergent due top the oscillatory behavior of the integrand as p . This is not surprising since ∆ (x) is not an ordinary function. In fact, ∆ (x) is a →∞ F F tempered distribution, which is an example of a generalized function. Thus, one must regard the integral representation given in eq. (6) in the same way as the integral representation of a delta function given in eq. (A.1). To perform the integral exhibited in eq. (6), we consult I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products [1], henceforth denoted as G&R. We note the following formula 3.914 no. 9 on p. 495 of G&R which states that1 ∞ 2 2 1 p exp( z p + m ) m 2 2 − sin(pr) dp = K1 m√r + z , for Re m> 0, Re z > 0. 2 2 2 2 r 0 p + m √r + z Z p (7) p Since z = i x0 in eq. (6), one cannot immediately employ eq. (7) to evaluate the integral of interest.| However,| one can define a generalized function that is represented by eq. (6) by replacing x0 x0 iǫ, where ǫ is a positive infinitesimal quantity [which is unrelated to the ǫ that| appears|→| in| eqs.− (1) and (2)]. Hence we set z = i( x iǫ) in eq. (7), which satisfies | 0| − the condition that Re z > 0. Indeed, this ensures the necessary damping of the integrand as p in order to guarantee that eq. (7) is convergent. Thus, we shall assign the following result→ ∞ to the otherwise divergent integral, ∞ 2 2 1 y exp( i x0 p + m ) m 2 2 − | | sin(py) dp = lim K1 a r x + iǫ . (8) 2 2 + 2 2 0 r 0 p + m ǫ→0 r x iǫ − Z p 0 + − q We can identify pr2 x2 = xµx = x2. Hence, itp follows that − 0 − µ − ∞ 2 2 1 y exp( i x0 p + m ) m 2 − | | sin(pr) dp = K1 m√ x + iǫ , (9) 2 2 2 r 0 p + m √ x + iǫ − Z p − 1The constraints on thep parameters m and z were omitted by G&R, but they can be found in the corre- sponding reference cited by G&R. See formula (36) on p. 75 of Ref. [2]. 2 where the ǫ 0+ limit is henceforth implicitly assumed. Consequently, eq. (6) yields, → 2 m K1 m√ x + iǫ i∆F (x)= − . (10) 4π2 √ x2 + iǫ − Note that in the case of x2 > 0, one must be careful in interpreting both the square root and the Bessel function K1 of an imaginary argument. Here, I shall employ eq. (5.7.6) of Ref. [3], K (z)= 1 iπe−iπν/2H(2)(ze−iπ/2) , for 1 π< arg z < π. (11) ν − 2 ν − 2 One can apply eq. (11) to eq. (10) with z m√ x2 + iǫ in the cases of x2 > 0 and x2 < 0, ≡ − respectively, since in either case the condition on arg z is satisfied. We now compute, √ x2 + iǫ = √x2 eiπ + iǫ = (x2 iǫ)eiπ = eiπ/2√x2 iǫ . (12) − − − In the case of x2 < 0, note that x2 iǫ liesp just below the branch cut that runs along the − negative real axis, which implies that arg √x2 iǫ 1 π. In contrast, x2 + iǫ lies just − ≃ − 2 − above the branch cut in the case of x2 > 0, which implies that arg √ x2 + iǫ 1 π. Thus, in − ≃ 2 both cases, it follows that the last step of eq. (12) is valid. Hence, independently of the sign of x2, 2 1 −iπ/2 (2) 2 (2) 2 K1 m√ x + iǫ iπe H (m√x iǫ) H (m√x iǫ) − = − 2 1 − = 1 iπ 1 − . (13) √ x2 + iǫ eiπ/2√x2 iǫ 2 √x2 iǫ − − − It then follows that an equivalent form of eq. (10) is given by, (2) 2 im H1 (m√x iǫ) i∆F (x)= − . (14) 8π √x2 iǫ − Admittedly, eq. (10) is more convenient in the case of x2 < 0, whereas eq. (14) is more convenient in the case of x2 > 0. Hence, one can replace eqs. (10) and (14) with the more convenient expression,2 m K m√ x2 + iǫ iπH(2) m√x2 iǫ 1 − 2 1 − 2 i∆F (x)= 2 Θ( x )+ Θ(x ) , (15) 4π " √ x2 + iǫ − 2√x2 iǫ # − − where we have employed the step function, Θ(z) =1for z > 0 and Θ(z)=0for z < 0, subject to the condition that Θ(z)+Θ( z) = 1.3 Indeed, the form of ∆ (x) is consistent with our − F previous assertion that ∆F (x) is a generalized function. An alternative derivation of eq. (15) is provided in Appendix A. 2A different technique for evaluating the integrals of this section is presented in Ref. [4]. In this work, the authors also demonstrate that both eqs. (10) and (14) are separately valid, independently of the sign of x2. 3One need not specify the values of Θ(0+) and Θ(0−). Indeed, when Θ(z) is regarded as a generalized function, the specification of the value of Θ(z) at the origin has no significance (e.g., see p. 63 of Ref. [5]). 3 It is instructive to examine the leading singularities of the Feynman propagator near the (2) light cone, This is most easily done by employing the expansions for H1 (z)) and for K1(z) which are either given or can be deduced from the results on pp. 927–928 of G&R, 1 z z 1 iπ 1 iπH(2)(z)= + ln + γ + + (z3) , (16) 2 1 −z 2 2 − 2 2 O 1 z z 1 K (z)= + ln + γ + (z3) . (17) 1 z 2 2 − 2 O (2) Since H1 (z) and K1(z) possess branch cuts along the negative real axis, eqs. (16) and (17) are valid for arg z < π. | | To obtain the leading singularities of i∆F (x), one can either insert the expansion given in eq. (17) into eq. (10) or the expansion given in eq. (16) into eq. (14) to obtain, 1 m2 + ln 1 m√ x2 + iǫ + γ 1 , as x2 0−, 4π2( x2 + iǫ) 8π2 2 − − 2 → i∆ (x) − (18) F 2 h i ∼ 1 m 1 1 1 2 + + ln m√x2 iǫ + γ + iπ , as x 0 . −4π2(x2 iǫ) 8π2 2 − − 2 2 → − h i In light of eq. (12), we see that the two limiting cases above are analytic continuations of each other. Indeed, it is a simple matter to check that if x2 > 0 then 1 iπ + lim ln √x2 iǫ = 1 iπ + lim ln(x2 iǫ) = 1 iπ + ln x2 iπΘ( x2) 2 ǫ→0 − 2 ǫ→0 − 2 | | − − = ln x2 + 1 iπ 1 Θ( x2) = ln x2 + 1 iπ Θ(x2) , (19) | | 2 − − | | 2 where we have employed the identity,p Θ(x2)+Θ( x2) = 1.
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