Method for calculating the imaginary part of the Hadamard elementary function G 1 in static, spherically symmetric spacetimes

Transcription

1 Method or calculating the imaginary part o the Hadamard elementary unction G in static, spherically symmetric spacetimes Rhett Herman* Department o Chemistry and Physics, Radord University, Radord, Virginia 2442 Received 8 March 998; published 22 September 998 Whenever real particle production occurs in quantum ield theory, the imaginary part o the Hadamard elementary unction G () is non-vanishing. A method is presented whereby the imaginary part o G () may be calculated or a charged scalar ield in a static spherically symmetric spacetime with arbitrary curvature coupling and a classical electromagnetic ield A. The calculations are perormed in Euclidean space where the Hadamard elementary unction and the Euclidean Green unction are related by 2 G () G E. This method uses a4 th order WKB approximation or the Euclideanized mode unctions or the quantum ield. The mode sums and integrals that appear in the vacuum expectation values may be evaluated analytically by taking the large mass limit o the quantum ield. This results in an asymptotic expansion or G () in inverse powers o the mass m o the quantum ield. Renormalization is achieved by subtracting o the terms in the expansion proportional to non-negative powers o m, leaving a inite remainder known as the DeWitt-Schwinger approximation. The DeWitt-Schwinger approximation or G () presented here has terms proportional to both m and m 2. The term proportional to m 2 will be shown to be identical to the expression obtained rom the m 2 term in the generalized DeWitt-Schwinger point-splitting expansion or G (). The new inormation obtained with this method is the DeWitt-Schwinger approximation or the imaginary part o G (), which is proportional to m in the DeWitt-Schwinger approximation or G () derived in this paper. S PACS numbers: v PHYSICAL REVIEW D, VOLUME 58, I. INTRODUCTION Despite the absence o a ull quantum theory o gravity, there are well known methods to calculate useul quantities involving quantum ields in curved space. One o the most successul o these methods is the semiclassical approach, wherein the gravitational and electromagnetic ields are treated classically while the various matter ields that act as sources are treated quantum mechanically. In this approach, the evolution o the spacetime rom a set o given initial conditions is described by the combined Einstein-Maxwell ield equations and G 8T, F ;4 j, where T and j are the vacuum expectation values VEVs o the stress-energy tensor and the current due to a charged quantized scalar ield. This paper ollows the sign conventions o Misner, Thorne, and Wheeler and uses natural units (Gc) throughout. In the semiclassical regime, the calculation o quantities such as 2 and T may begin with calculation o the Feynman Green unction G F (x,x) deined by 2 ig F x,x0txx0, where T... is the time ordering operator, (x) is the quantum ield evaluated at the stationary spacetime point x, and *Electronic address: rherman@runet.edu 2 3 (x) is the quantum ield evaluated at the nearby point x. One o the major diiculties in using the semiclassical method in quantum ield calculations is the act that ininities appear in the VEVs in Eqs., 2 in both lat 3 5 and curved 2,6 spacetimes. Various methods have been used to isolate those ininities and remove them rom the physical theory 2, yet arguably the most powerul o these methods is based on the classic work o Schwinger 7. Schwinger calculated the Feynman Green unction or a quantized ermion ield by introducing a ictitious, nonquantum mechanical Hilbert space. This (4)-dimensional Hilbert space was constructed rom the our spacetime dimensions in addition to a ith dimension which was identiied as the proper time parameter s within this ictitious space. Working within this proper time space, Schwinger was able to both isolate the divergences in the quantum ield integrals or the ermion current j (x) lim x x ie tr G F (x,x) produced by an external electromagnetic ield in lat spacetime, and to use those divergences to renormalize the charge e o the quantum ield. Schwinger s calculations o the quantum action unctional W were perormed by transorming the integrals involving the our spacetime dimensions into the momentum representation. The integrals were evaluated using perturbation expansions in powers o ea and ef, where A and F are the gauge ield vector and the electromagnetic ield tensor, respectively. Were these integrals to be evaluated without urther modiication, Schwinger noted 8 that conservation o energy and momentum would dictate that no pair creation would occur, or j 0. Schwinger s solution to this situation was to add an ininitesimal imaginary part to the action integral, resulting in W acquiring a positive imaginary part. Schwinger associated this imaginary part o W with pair production by stating that /98/588/084028/$ The American Physical Society

2 RHETT HERMAN PHYSICAL REVIEW D e iw 2 2 Im[W] e 4 represented the probability that no pair creation would occur. Ater this identiication was made, Schwinger s calculation o the pair creation rate yielded a series expansion or the probability o pair creation per unit our volume by a constant external electric ield, e2 4 3 E 2 n n 2 e nm2 /ee, where m and e are the mass and charge o the ermion ield, respectively, and E is the magnitude o the electric ield. The expansion o Eq. 5 yields a series involving inverse powers o m 2. Schwinger s calculation or the pair creation rate assumed that spacetime was lat. Building on the work o Schwinger, DeWitt 9,6 extended the proper time method to include curved spacetime results. Using what is now called the DeWitt-Schwinger proper time method, DeWitt calculated an asymptotic expansion or the Feynman Green unction or a real scalar ield in an arbitrary curved spacetime. This asymptotic expansion was an expansion in inverse powers o m 2, where m is the mass o the quantum ield. This expansion o G F (x,x) yielded an expression with both real and imaginary parts. The real and imaginary parts o G F (x,x) could then be identiied with other Green unctions by using the identity G F x,xḡx,x 2 ig x,x, where Ḡ(x,x) is one-hal the sum o the advanced and retarded Green unctions, and the Hadamard elementary unction G () (x,x) 2 is deined by G x,x0x,x0. Thus, the DeWitt-Schwinger asymptotic series expansion or G () (x,x) has the general orm G x,xb 2 x,xm 2 B 0 x,xm 0 B 2 x,xm 2, where the B n are coeicients constructed rom curvature tensors. In the semiclassical method, quantities such as 2 and T may be calculated rom G () (x,x) by taking the appropriate derivatives with respect to either the stationary spacetime point x or the nearby point x 6,9,2; e.g., and 2 lim x x G x,x, 9 T x lim x x Re 2 2 g G g G 4 g g G In Eq. 0, the vertical bar indicates gauge covariant dierentiation, and g is the bivector o parallel transport 9, which is used to transport the inormation rom the nearby point x to the stationary point x. The ininities appearing in the VEV in Eq. 9 lead to the well known divergences in the unrenormalized expression or T (x) in Eq De- Witt s asymptotic expansion o G () (x,x) in powers o m 2 isolated the ininities, or the counterterms, o both G () (x,x) and T, with the ininities being ound as coeicients o the m 2 and m 0 terms in the series o Eq Once these ininities have been subtracted rom the unrenormalized VEVs in Eqs. 9, 0, then the inite remainder carries the inormation about the physics o the spacetime. While these subtractions may be perormed in principle, the remaining expressions are oten so complicated that they can not be evaluated analytically 2. Investigators have used the semiclassical method to calculate renormalized expressions or both the vacuum polarization 2 ren 2 7, and the stress energy tensor T ren 8 2 in speciic spacetimes. In each o these cases, the point-splitting counterterms were used to renormalize the divergent VEVs. These calculations were perormed in spacetimes whose high degrees o symmetry allowed the subtractions o the counterterms to be evaluated analytically. See Re. 2 or a more thorough discussion o the diiculties encountered in these subtractions. The DeWitt-Schwinger point-splitting expansion or G () may also do more than simply isolating the divergent counterterms necessary to renormalize the VEVs encountered in semiclassical quantum ield theory. I the terms proportional to nonnegative powers o m are discarded, the remaining terms in Eq. 8 constitute the DeWitt-Schwinger approximation or G () 2 23 or ields whose masses are large when compared to the magnitude o the coeicients B n. Retaining the irst one or two o these terms has been shown to provide a close approximation to the exact results in several cases 4,9,7,2. In the case o charged scalar ields, G () involves the complex quantity G x,x0x,*x0, where (x) is the charged scalar ield. Even though G () is generally complex, since all o the coeicients B n are real, Eq. 8 only contains part o the inormation about G (). DeWitt has pointed out 6 that the point-splitting asymptotic expansion o G () in inverse powers o m 2 is incapable o yielding the imaginary part o G (). Yet whenever real particle production occurs G () will have a non-zero imaginary part. This is readily seen by examining the expression or the current due to a charged scalar ield,

3 METHOD FOR CALCULATING THE IMAGINARY PART... PHYSICAL REVIEW D j 4 F ; ie 2 D,*D,**, 2 where e is the charge o the ield, D ( iea ) is the gauge covariant derivative, A is the background gauge ield, and the asterisk* denotes the complex conjugate. Using Eq., and making the transition rom classical to quantum ields 24, gives the vacuum expectation value o the current due to the coupling between the charged scalar ield and the background gauge ield, j x lim x x ie 4 G g G G g G *. 3 Since the coeicients B n in Eq. 8 are real, using the DeWitt-Schwinger point-splitting expansion involving only inverse powers o m 2 to construct the current j rom Eq. 3 can only yield a value o zero. Yet the current is not identically zero, as evidenced by Schwinger s calculation in lat space, thus illustrating the limitation the DeWitt- Schwinger point-splitting expansion suers by yielding only the imaginary part o G (). In this paper, a method is presented to obtain an analytic expression or the imaginary part o G () in a static spherically symmetric spacetime with a gauge ield A. The imaginary part arises in a straightorward manner due to the presence o the gauge ield A and does not require the addition o any extra terms during the course o the calculation. The () renormalized expression G ren would ordinarily be constructed by subtracting the DeWitt-Schwinger point-splitting counterterms rom the unrenormalized expression or G () : G ren G unren G DS. 4 The mode sums and integrals in Eq. 4 cannot be evaluated analytically at present due to the complexity o the terms in the ininite sums and integrals. A method recently described by Anderson, Hiscock, and Samuel AHS 2 yields a way to construct an approximate expression or G () ren. All subtractions are perormed in Euclidean space where the relationships between various Green unctions are given by G x,xig F x,xg E x,x. 5 Using the Euclideanized metric, the mode unctions in the VEV o Eq. may be rewritten in a WKB orm 7,2. The renormalized expression G E,ren is constructed by moving into Euclidean space and perorming the subtraction G E,ren G E,WKB G E,WKB div, 6 where G E,WKB is the expression obtained in Euclidean space upon substituting the Euclidean WKB mode unctions into Eq., and G E,WKB div contains the ultraviolet divergences ound in G E,WKB. The mode sums and integrals in Eq. 6 still cannot be evaluated analytically in their present orm. In order to construct an expression or G E,ren, the ourth order WKB approximations or the exact mode unctions are substituted into the quantum ield expressions. Then, the mode sums and integrals in the subtraction o Eq. 6 may be evaluated analytically by expanding the integrands and summands in powers o the mass m o the quantum ield. This mass is assumed to be large when compared to the inverse o the radius o curvature o the spacetime. By working within this large mass limit, a DeWitt-Schwinger asymptotic expansion results or G E,ren similar to Eq. 8; G E,ren G E,WKB G E,WKB div large m B 2 x,xm 2 B x,xm B 0 x,xm 0 B x,xm B 2 x,xm Equation 9 diers rom Eq. 8 in that the presence o the gauge ield in this method is directly responsible or the presence o the new terms involving odd powers o m. Renormalization is achieved by discarding terms proportional to non-negative powers o m, leaving a inite DeWitt- Schwinger approximation or G E,ren ; G E,ren B x,xm B 2 x,xm In Euclidean space, all o these terms are real. When the inal expression is rotated back to the Lorentzian sector, the real and imaginary parts o G () are obtained. The terms proportional to m are all imaginary and are proportional to both odd powers o the gauge ield A and its derivatives along with odd powers o the charge e o the quantum ield. The terms proportional to m 2 are all real and are exactly those obtained rom the m 2 term o the DeWitt-Schwinger pointsplitting expansion 0,7,22. In Sec. II, an expression is derived or the unrenormalized Euclidean Green unction G E in a static spherically symmetric spacetime. In this paper, the ield is assumed to be at zero temperature, although this method would extend to the analysis o ields at nonzero temperature 7. Section III renormalizes the expression or G E derived in Sec. II. The renormalization subtractions are evaluated analytically by evaluating the mode sums and integrals in the unrenormalized expression in the limit that the mass o the quantum ield is large when compared to the inverse o the radius o curvature o the spacetime. The resulting DeWitt-Schwinger approximation will be shown to contain both the real and imaginary parts o G (), with the main goal o this paper being obtaining an expression or the imaginary part. Section IV contains the results o the calculation o the asymptotic expansion o G (). The terms proportional to m 2 will be shown to give the same results as the m 2 term in the generalized DeWitt-Schwinger point-splitting expansion when the general expression is evaluated in a static spherically symmetric spacetime 0,7. The terms proportional to m are all real in Euclidean space but are all imaginary when rotated back to Lorentzian space. All o the

4 RHETT HERMAN PHYSICAL REVIEW D m terms are proportional to odd powers o the charge e o the complex scalar ield, as well as odd powers o the gauge ield A and derivatives, and thus these terms all vanish when either the quantum ield is uncharged or the spacetime is uncharged. This connection to previous work will serve as a check on the validity o the method presented here. II. UNRENORMALIZED GREEN FUNCTIONS FOR THE CHARGED SCALAR FIELD IN EUCLIDEAN SPACE The goal o this paper is to derive an expression or the imaginary part o G () so that curved space phenomena involving real particle production due to the presence o an electromagnetic ield may be investigated. Thus it is useul to begin with the electromagnetic vector potential or a static spherically symmetric electric ield like that o the Reissner- Nordström spacetime: A A t r,0,0,0. 2 The calculations will be simpliied by moving to Euclidean space 7. The Euclidean metric or a static spherically symmetric spacetime is ds 2 rd 2 hrdr 2 r 2 d 2, 22 where it, (r) and h(r) are the same unctions as in the Lorentzian metric or a static spherically symmetric spacetime ds 2 rdt 2 hrdr 2 r 2 d 2, 23 and t is the coordinate whose derivative (/t) is the timelike Killing vector in the Lorentzian sector 7. The Lorentzian and Euclidean expressions or the invariant A A are given by A A g tt A L t A L t ra L t A L t ria L t ia L t g A E A E ra E A E, 24 where the subscripts L and E reer to the Lorentzian and Euclidean sectors, respectively. Eq. 24 indicates that A t E ia L. 25 This shows the relation between the gauge invariant derivative operators in both sectors is g D L D L g tt t iea t t iea t r t iea t t iea t ri t ieia t i t ieia t r iea iea D iea 27 or the Euclidean sector. An explicit orm or the Euclidean Green unction or ields at zero temperature is needed. The case o zero temperature is chosen here or simplicity, and the derivation here ollows those o Anderson 7 and AHS 2. The action o the wave operator on the Euclidean Green unction is given by g D E D E m 2 RG E x,x g /2 4 x,x,r,r,, r 2 h /2. 28 The delta unction (,) may be expanded using the Legendre polynomials with the result, 2lP 4 l cos, l0 29 where cos()cos()cos()sin()sin()cos(). With the scalar ield chosen to be at zero temperature, the delta unction or the split in the direction is given by, de i Thus, the Euclidean Green unction or a ield at zero temperature is G E x,x d cos l0 2lP l coss l r,r, 3 where S l (r,r) is an unknown unction o the coordinate r. Applying the wave operator o Eq. 26 to G E gives a dierential equation or S l (r,r), h d 2 S dr 2 2 rh d 2 h dr dh 2h 2 ea 2 r,r r 2 h /2, dr ds dr ll r 2 m 2 RS 32 g D D g D E D E 26 where m is the mass o the ield, and A is the only non-zero component o the Euclideanized vector potential. S l (r,r) is given by indicating S l r,rc l p l rq l r,

5 METHOD FOR CALCULATING THE IMAGINARY PART... PHYSICAL REVIEW D where C l is a normalization actor, and p l (r) and q l (r) are the solutions o the homogeneous orm o Eq. 32. In Res. 7 and 2, S l (r,r) is written in the orm S l r,rc l p l r q l r. S l (r,r) satisies the Wronskian condition C l dq l p l dr q dp l l dr r 2 34 h /2, 35 obtained by integrating Eq. 32 once with respect to r rom r to r, with 0 in the end. The unctions p l and q l may be put into a WKB orm using 7 p l r 2r 2 Wr e q l r 2r 2 Wr e r Wrh/ /2 dr, r Wrh/ /2 dr In Eq. 36, p l (r) is well behaved at r0 and the event horizons, but is divergent at r. The unction q l (r) is divergent at r0 and the event horizons, but is well behaved at r. See Anderson 7 or a more complete discussion o the properties o these unctions. Equations show that C l. Using Eqs. 36 and 37, Eq. 32 assumes the amiliar WKB orm where W 2 2 rv rv 2 r d 2 2 W hw dr d 2 h dr dh h dr 2 dw 2W dr 3 2 h 2 dw W dr, 2 rea 2 m 2 l 22 r 2, V r d 2rh dr dh 2rh 2 dr 4r 2, V 2 rr d 2 h dr 2 2 h d 2 dr 2 d dh 2 h 2 dr dr 2 d rh dr 2 dh rh 2 dr 2 r 2 h 2 r 2, and where the product l(l) has been actored into two pieces according to ll l The simple orm o the vector potential in Eq. 2 causes the WKB equation to assume a new but straightorward orm. Previous work had WKB equations just like Eqs with A 0 see the reerences in Re. 2 or a more detailed list o work by others. The new eature in the WKB dierential equations is the presence o the electromagnetic vector potential. Equation 38 indicates how the complex quantities necessary or determining the charged scalar ield current j arise in the present method. The WKB unction W(r), while real in the Euclidean sector, is now complex in Lorentzian space due to the electromagnetic ield o Eq. 25. Equation 38 is not in general exactly solvable. Fortunately, it may be solved approximately by iteration. For example, the zeroth order solution or W is W. Substituting W into the right hand side o Eq. 38 and solving or W yields the second order WKB solution W (2) : W 2 2 V V V d 4 2 h 2 dr d 2 h dr dh h dr 2 d 2 2 dr 3 2 d 2 h 3 dr. 43 The ourth order solution, W (4), is obtained by making the substitution W W (2) in the right hand side o Eq. 38 and solving or W. Just as in point-splitting, this iteration procedure could continue indeinitely. However, in the present case o obtaining the (m ) and (m 2 ) terms in the DeWitt-Schwinger expansion or G (), the iterations need only be continued until the ourth order WKB solution, W (4), is reached. As pointed out by AHS 2, a second order WKB expansion contains the inormation o the (m 2 ) and (m 0 ) terms in the DeWitt-Schwinger point-splitting expansion, while a ourth order WKB expansion contains the inormation o the (m 2 ) terms. Thus, the ininite counterterms in the theory may be reproduced by perorming the present calculation using only W (2), while the DeWitt- Schwinger approximations o the (m ) and (m 2 ) terms require the use o W (4). For purposes o simplicity and to make connections to previous work 2, the choice is made to split the points along the direction such that (x,x) (,r,,;,r,,). Making the deinition 2 A lim r r l0 2 l 2 p l rq l r l0 l/2 r 2 W, 44 with the limit r r being taken in the last expression, allows the unrenormalized Euclidean Green unction to be written as

6 RHETT HERMAN PHYSICAL REVIEW D G E,unren x,;x, 2 unren d 4 2 cosa, 0 45 G E,unren x,;x, d cos l/2 r 2 W r /2. l0 48 or the separation along. The next section discusses the identiication and removal o the two types o divergences that appear in this unrenormalized expression or G E (x,;x,). III. RENORMALIZATION OF THE 4 th ORDER WKB APPROXIMATION IN THE LARGE MASS LIMIT There are two types o divergences which appear in Eq. 45. The irst o these is a supericial divergence which appears when evaluating the sum over l with an upper limit o l, while the second is an ultraviolet divergence due to perorming the integrations over with an upper limit o. This section presents the details o how the subtractions o these divergent terms rom the unrenormalized Green unction may be perormed and how the resulting expression may be evaluated analytically. As discussed more ully elsewhere 2,7,5,8, the supericial divergence due to the sums over l cannot be real since G E,unren must remain inite with the points separated as they are in Eq. 45. The sum and integral present in Eq. 45 will be evaluated later by assuming the mass m o the quantum ield is large enough to expand the summand o A in inverse powers o m. At present, it is useul to consider how the alse divergence over l is isolated assuming the sum and integral o Eq. 45 were to be evaluated without such a large mass expansion. The supericial divergence that appears in A may be identiied by evaluating the sum l0 2 l p 2 l q l l0 l/2 r 2 W 46 in the limit l. The only part o W that contributes as l is the zeroth order term o W, or; terms in W proportional to negative powers o do not diverge as l. Substituting W into (l/2)/r 2 W and expanding in inverse powers o l, and then truncating the expansion at (l 0 ) gives l0 lim l 2l/2 2r 2 W l0 r /2. 47 Thus, the counterterm /(r /2 ) must be subtracted rom the summand o Eq. 44 in order to remove the supericial divergence over l. This now indicates the unrenormalized expression or G E (x,;x,) is given by This technique will serve to remove any supericial divergences over l that may appear. In Eq. 48, only the term /(r /2 ) is subtracted rom the single summand (l /2)/r 2 W. Later, when the WKB approximation or W is substituted into Eq. 45 and the summands and integrands are expanded in the large mass limit, many terms involving actors o l will arise. Only some o those terms will contain a divergence in l, and those that do must have the appropriate counterterms in l subtracted rom them to remove this supericial divergence. Equation 45 also contains the ultraviolet divergences that are known to arise in semiclassical quantum ield theory. These divergences occur as and must be subtracted rom the unrenormalized expression. As in the case o the divergence over l, these subtractions must be perormed term by term ater the large mass expansions o A have been perormed. Thus, the generalized orm or the renormalized expression or G E is given by G E,ren d cos l0 l/2 r 2 W WKB expansion terms l terms terms, 49 where the subscript WKB expansion terms indicates all o the terms that result rom a 4 th order WKB expansion o (l /2)/r 2 W, and the terms identiied as divergent in l and are subtracted rom each o these terms o the expansion. The calculation o all the terms in Eq. 49 is a straightorward but tedious process. The general procedure ollows Appendixes D, F, and G in the recent work o AHS 2 and will be described here. The presence o the gauge ield increases the diiculty o the calculations substantially, but the method o AHS can be modiied to account or the gauge ield. The calculation o G E,ren starts with the substitution o the 4 th order WKB approximation or W into the deinition o A and the subsequent identiication o the various l counterterms. To keep track o the order in the WKB expansions o quantities it is useul to introduce the dimensionless parameter, letting in the end. In Eq. 38, is ( 0 ), V (r) is( ), and the rest o the terms are ( 2 ). The quantity (l/2)/(r 2 W) is expanded in powers o the WKB parameter, truncating the expansions at ( 4 ), and setting. This will result in over 600 terms whose general orm is given by

7 METHOD FOR CALCULATING THE IMAGINARY PART... PHYSICAL REVIEW D L hjk l0 h 2 j 2l/2 k l terms, 50 where (r) is deined in Eq. 39, and various actors o (r),v (r),ea (r),..., that are present in ront o each o the L hjk have been omitted. This deinition o L hjk is similar to that deined as L jk in Re. 2. In the present work, the presence o the gauge ield has led to the need to keep track o additional actors o which now enter into the deinition o the L hjk. The notation here is chosen so that it most closely matches the notation o AHS. Determining the l subtraction terms proceeds as beore by expanding the summands in each L hjk in inverse powers o l and truncating the expansion at (l 0 ). For example, the expression or L 00 requires one subtraction term; 2l/2 2r /2. 5 L 00 l0 Determining the subtraction terms requires using the Plana sum ormula 2,25 lk gl 2 gk k gd dt i 0 e 2t gkitgkit. 52 The irst two terms o Eq. 52 may be computed analytically, while the third term is not in general able to be computed analytically. Since the ultraviolet, or, behavior o these sums is dominant, the third term o Eq. 52 may be expanded in inverse powers o. The expansions are truncated at ( ) since terms o ( 3 ) are not ultraviolet divergent 2. Then, the integrals that arise rom the expansion in inverse powers o may be computed analytically. Applying this procedure to L 00, or example, gives L 00 l0 2l/2 2r /2 2eA 2 m 2 /4r 2 /2 2r2 ea 2 m 2 2 /2 4 t dt 4r 0 e 2t 2r2 2er2 A 2 m2 r The only L hjk or which there will be subtraction terms are those or which j,k have the values 0,, 0,3,,3,,5, 2,5, and 2,7; only h0 occurs or these combinations o j,k. Comparison o L 00 in this example with L 0 o AHS, along with Eq. 25, show how the gauge ield will bring complex quantities into the expressions or the L hjk and eventually into G (). Once the terms divergent in both l and have been determined or each o the L hjk, the expression or G E,ren is now given by G E,ren d cos FrL hjk L hjk,, 54 where each o the F(r) above represent the unctions such as (r),v (r),ea (r),..., that appear in the calculation but are not aected by the summations and integrations, and the curly brackets... indicate the large number o terms arising due to the WKB expansions. In Eq. 54, the counterterms or each L hjk are indicated symbolically by L hjk,. Since the terms which diverge as are subtracted in Eq. 54, then the unction cos() will be dominated by the ininitesimal separation (). This indicates that cos() is valid. For the purposes o completeness, it should be noted that uture work involving both j and T will involve more actors o than those that arise within the L hjk. Thus, it is useul to make the deinition S ihjk d 4 2 i L hjk L hjk,, 0 55 where the indices ihjk have been chosen so the present work is more easily compared to that o AHS. In the calculation o the Green unction in this paper, i0 throughout; in AHS, i0 or the Green unction calculation while i0 or 2 when calculating T. The sums and integrals in Eq. 55 still cannot be evaluated analytically in general in their present orm. They may be put into a orm that can be evaluated analytically by taking the large mass limit o the quantum ield. Each o the S ihjk may be computed in the large mass limit by irst expanding each h 2(l/2) 2 j / k term in the L hjk using the Plana sum ormula o Eq. 52. The irst two terms o the Plana ormula may be computed exactly. Again, the third term may not be computed analytically, but once expanded in the large mass limit each piece o the expansion may be computed analytically. The large mass limit o this

8 RHETT HERMAN PHYSICAL REVIEW D third term is obtained by expanding the integrand o the third term in inverse powers o the large quantity (ea ) 2 m 2 (l/2) 2 /r 2. Each integral may then be evaluated analytically. Ater the integrations, the entire expression or each S ihjk is expanded in inverse powers o m. In Appendix G o Re. 2, AHS give an example o this type o calculation when they calculate the large mass expansion o S 00. Here, the large mass expansion o S 000 will be shown or comparison. The l limit or L 00 is ound in Eq. 5, while the subtraction term is ound in Eq. 53. This gives the expression to be evaluated in the large mass limit as S 000 d l0 2l/2 ea 2 m 2 l/2 2 /r 2 /2 2r /2 2r2 2 2 m2 r 2er2 A. 56 Using the Plana sum ormula and expanding in inverse powers o the large quantity (ea ) 2 m 2 (l/2) 2 /r 2 yields S 000 d ea 2 m 2 l/2 2 /r 2 /2 2r /2 dl 0 ea 2 m 2 l/2 2 /r 2 /2 2r /2 i 0 2r2 dt 2it/2 2it/2 e /2 2t ea 2 m 2 it/2 2 /r 2 /2 ea 2 m 2 it/2 2 /r 2 2 m2 r 2 2er2 A, 57 S 000 d r2 ea 2 m 2 2 /2 4r 2 05r 4 ea 2 m 2 /4r 2 5/2 2r2 ea 2 m 2 /4r 2 /2 30r 2 ea 2 m 2 /4r 2 3/2 2 m2 r 2 2er2 A. 58 Perorming the integrals over and expanding in the large mass limit yields S 000 m2 r 2 8 ln m mr2 ea 2 2 /2 96 ln 2 m2 2 r2 ea m 3 7eA r 2 /2 ea /2 r2 ea 5/ m 4 2 r 4 m 5. m ea 48 2 /2 r2 ea 3 3/ m 2 2 r 2 59 In Eq. 59, the inrared cuto parameter has been introduced due to the lower limit 0 on the integrals over 2,2. Comparison o Eq. 59 with Eq. G2 o Re. 2 shows that the presence o the gauge ield has brought odd powers o m into the asymptotic expansions or the S ihjk. Fortunately, when the large mass expansions o all o the S ihjk are computed, many are ound to be o such a high order in inverse powers o m that they do not contribute to the inal expression or G E,ren since the DeWitt-Schwinger expansions are truncated at (m 2 ). Note that the gauge ield has also contributed terms o (m ) and (m 0 ) to S 000, as it does or some o the other S ihjk. In this work, only terms which contain negative powers o m will be considered in the inal DeWitt-Schwinger approximation. Finally, all o the S ihjk are substituted into the general expression or G E,ren G E,ren FrS ihjk, 60 where the curly brackets... indicate the great number o terms to be added together, and like powers o m are collected. IV. RESULTS When all the substitutions are made and terms o like powers o m are collected, the result is that the coeicients

9 METHOD FOR CALCULATING THE IMAGINARY PART... PHYSICAL REVIEW D o the negative powers o m constitute the DeWitt- Schwinger approximation to G E,ren or a complex scalar ield in a static spherically symmetric spacetime. A complex scalar ield was chosen in order to make a connection to the earlier work o DeWitt 9,6 and Christensen 0 involving a real scalar ield since the results o the present work must reduce to their results when the charge o the ield vanishes. In their work, the generalized DeWitt-Schwinger approximation or the Hadamard elementary unction or a scalar ield is 2 G 2 a m 2, 6 where a 2 is given by 9,24 a 2 80 R R 80 R R 6 5 R ; 2 6 2R 2 e2 2 F F. 62 In the present work, the m 2 term in the DeWitt-Schwinger approximation or G E,ren 2 G () 2 is given in Euclidean space by G E,m 2 m r 4 240r 4 h 2 2 R 2 32 e2 A 2 96h 96r 3 h 2 96r 3 h r 2 2 h r 2 2 h r 3 h h 2 3h 60r 3 h 3 h 96r 3 h 2 h 96r 2 h 3 h 576r 2 h 2 2 h 80r 2 h h h 3 3h2 h2 640r 2 h 4 640rh 4 52 h h 4 7h3 240rh 5 7h3 960h 5 288r 2 h 2 288r 2 h 7 320r 2 h h h 2 960rh 3 5h h 3 9h2 920h h 2 h 240r 2 h 3 h 20rh 3 2 h h 3 3hh 3 hh 480rh 4 920h 4 h 240h 3 R rh 92h h 2 R R 92h 96h 20rh 2 h 92 2 h 2 60h 3 h 240rh 3 h 960h 3 480h 2, 63 where the primes indicate dierentiation with respect to r. This is exactly the same result that is obtained by writing Eqs in terms o the metric unctions o Eq. 22. Since the gauge ield only appears here in the orm (ea ) 2, rotating back to Lorentzian space to obtain G () simply results in the appearance o a negative sign or this term. This demonstrates the correspondence between the present method in a static spherically symmetric spacetime and the method o the generalized DeWitt-Schwinger expansion; they yield the same inormation or the real part o G () in the limit that the mass o the quantum ield is large when compared to the inverse o the radius o curvature o the spacetime. With the inormation in Eq. 63 previously known and accepted, it is ortunate that the present method not only duplicates that inormation but also yields more inormation. The DeWitt-Schwinger point-splitting expansion is ully covariant and yields complete inormation about each o the coeicients in the expansion, indicating that all o the terms in a large mass DeWitt-Schwinger expansion proportional to inverse, even powers o m have been determined by the generalized expansion. These terms are constructed out o real curvature and electromagnetic ield tensors and thus are incapable o yielding the imaginary part o G (). Yet the deinition o G () or a complex scalar ield, G x,x0x,*x0, 64 shows that it is, in general, a complex quantity. Any method or calculating G () that does not yield both real and imaginary pieces is incomplete in its determination o G (). DeWitt emphasized this limitation existed or the point-splitting procedure by explicitly stating that the expansion in inverse powers o m 2 was incapable o yielding the imaginary part o G () 6. Yet this imaginary part is non-vanishing even in lat space with a constant gauge ield. Classical ield theory calculations o the current due to a complex scalar ield under these conditions is not identically zero 26. Schwinger s work 7 incorporated these lat space, constant ield conditions speciically with the result being the non-vanishing Schwinger pair creation rate o Eq. 5. The only way in which the imaginary part o G () may be obtained in a DeWitt-Schwinger expansion is by obtaining the terms proportional to inverse, odd powers o m. It is possible to anticipate the orm o the terms in the expansion or G (). Moving to units where c but G, then G () and curvature tensors such as R are second order quantities with units o (length) 2, while m and A have units o (length). Using such power counting, Davies et al. 27 constructed the stress energy tensor o a conormally invariant scalar ield in a conormally invariant spacetime. In the present case o the expansion or G () the same procedure may be used. The term proportional to m must be proportional to (length) 3 in order or G () to remain second order. Since Eq. 64 holds in both lat and curved

10 RHETT HERMAN PHYSICAL REVIEW D spacetimes, the present method should yield a non-vanishing imaginary part when (r)h(r) and when the gauge ield A constant. Under these conditions, the m term o G E should be proportional to (ea ) 3 and (ea )r 2. This power counting may be expanded by allowing the gauge ield to be a unction o r, with the result that additional terms such as (ea)r should appear in the expansion. The m term in the DeWitt-Schwinger approximation or G E,ren in the present work is given in Euclidean space by G E,m m 2 ea /2 ea 24r 2 /2 era 8 /2 ea 24r 2 /2 h ea 24r /2 h ea 6r 3/2 h ea 48 3/2 h 3eA /2 h ea h 24r /2 h 2 ea h 96 /2 h 2 ea h 64 3/2 h 2 ea 48 /2 h ea 32 h 3/2. 65 All o these terms are proportional to odd powers o A and its derivatives and, due to Eq. 25, are thus all imaginary when rotated back to Lorentzian space. All o these terms are also proportional to odd powers o the charge e o the quantum ield. Upon imposing charge conjugation symmetry on 2 2 G () 28, these terms will not contribute to the calculation o the vacuum polarization in the spacetime. It is remarkable that the two methods yield exactly the same result or the m 2 term in their respective DeWitt- Schwinger expansions. As pointed out by Gibbons 29, while the DeWitt-Schwinger point-splitting expansion is widely used in renormalization theory, there is still much that is unknown about the type o series expansion that it really is. The nonzero results o Eq. 65 highlight the major dierence between the present method and the generalized DeWitt-Schwinger point-splitting expansion. A major limitation o the DeWitt-Schwinger point-splitting expansion is that it is an asymptotic expansion limited to inverse powers o m 2. This dictates that it is restricted to the calculation o only the real part o G () which, as stated above, is generally complex. Yet the DeWitt-Schwinger expansion has the distinct advantage o being a completely general expansion with its pieces constructed rom curvature and electromagnetic tensors. Thus, once the point-splitting calculations have been perormed and renormalization achieved by discarding terms proportional to non-negative powers o m, the remainder is a completely general expression which may then be used to study any spacetime or which a metric exists. The main limitation o the present method is that it is restricted to spacetimes or which the basis unctions o the quantum ield can be put into the WKB orm like that o Eq. 38. The wave equation must thereore be separable. Since this has been achieved or the Kerr metric 30, the present work could conceivably be extended to Kerr spacetimes. The major advantage o the present method over the DeWitt-Schwinger point-splitting expansion is that it can yield the imaginary part o G (). The calculation o the imaginary part o G () is important due to the deinition o the current j in Eq. 3, which is non-vanishing even in lat space with a constant gauge ield. The actor o i in ront o the expression, along with the subtraction o the complex conjugate o the derivatives o G (), show that only the imaginary part o G () will contribute to the current. The actor o e in ront o the expression or j requires the presence o the odd powers o e in the imaginary part o G () i the current is to be constructed rom G (). The inal expression or the current must be proportional to even powers o e i the current is not to vanish under charge conjugation symmetry 28. Work is in progress to calculate the DeWitt- Schwinger approximation or the current in two ways; irst, directly rom Eqs. 65 and 3, and second, starting rom Eq. 3 and proceeding through the steps outlined in this paper to calculate the DeWitt-Schwinger approximations or quantities such as G () that are required to construct j. Much o the previous research in quantum ield theory in curved spacetimes dealt with the production o virtual particles due to vacuum polarization. Whenever real particle production due to an electromagnetic ield was to be studied, investigators most oten used the pair creation rate derived by Schwinger or lat spacetime 3. While Schwinger s work assumed a constant electric ield E, the authors in Re. 3 replaced that ield with the spherically symmetric radial ield o the Reissner-Nordström spacetime. With the curved space results in Eq. 65, it should now be possible to determine how the eects o curved space will cause the rate o charged particle production to dier rom that o the lat space Schwinger rate. ACKNOWLEDGMENTS The author would like to thank William A. Hiscock and Paul R. Anderson or helpul discussions concerning this work. This work was supported by a Radord University Seed Grant. C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation Freeman, San Francisco, N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space Cambridge University Press, Cambridge, England, S. S. Schweber, An Introduction to Relativistic Quantum Field Theory Row, Peterson, and Company, New York, 96. See Chap. 6 or a discussion o the degree o divergences in QED. 4 J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics McGraw-Hill, New York, J. D. Bjorken and S. D. Drell, Relativistic Quantum Fields McGraw-Hill, New York,

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