THE ENERGY-MOMENTUM TENSOR, THE TRACE IDENTITY AND THE CASIMIR EFFECT

Transcription

1 THE ENERGY-MOMENTUM TENSOR, THE TRACE IDENTITY AND THE CASIMIR EFFECT S.G.Kamath * Department of Mathematis, Indian Institute of Tehnology Madras, Chennai ,India Abstrat: ρ The trae identity assoiated with the sale transformation x x = e x on the Lagrangian density for the noninterating eletromagneti field in the ovariant gauge is shown to be violated on a single plate on whih the Dirihlet boundary ondition ( A t, x, x, x = a = 0is imposed. It is however respeted in free spae, i.e. in the absene of the plate; these results reinfore our assertions in an earlier paper where the same exerise was arried out using the Lagrangian density for the free, massive, real salar field in + dimensions. * amath@iitm.a.in PACS:.0.-j

2 .Introdution ρ The trae identity [] assoiated with the sale transformation x x = e x on a salar field φ(x, for example, is onventionally expressed in terms of the energy- momentum tensor Θ ˆ ν,and for the massless free real salar field Lagrangian density in + dimensions, namely, L = φ φ the trae identity is given by (4 (4 ( ν gνγ ( y; x, z + id δ ( y x + δ ( y z G( x, z = 0 ( ν with the anonial sale dimension d = ; G ( x, z and Γ ( yxz ;, being the onneted T * -ordered produts ( φ φ and T * ( ˆ ν y φ x φ z * 0 T ( x ( z 0 0 Θ ( ( ( 0 respetively. Θ ˆ ν is defined [] in terms of the anonial energy-momentum tensor Θ ˆ ν = φ ν φ g ν L by ˆ ν ˆ =Θ + g φ ( 6 Θ ν ( ν ν with the anonial dilatation urrent D ˆ = Θ + ( x φ As emphasized by Coleman and Jaiw[], one an now rewor ( to get a modified version of the dilatation urrent given by D = x Θ ˆ (4 as the seond term defining the field virial V is a total divergene. For the free eletromagneti field on the other hand, it is well nown [] that the orresponding Θ ˆ ν assoiated with the Lagrangian density

3 L= F F, F β β = A A 4 (5 is the Belinfante tensor ˆ ν Θ B,the latter being defined [] in terms of the anonial energymomentum tensor Θ ˆ ν by ν ν ˆ ˆ ν Θ =Θ + (6 B X with ν ˆ ν ν Θ =Π A g L (6a = Π Σ Π Σ Π Σ A (6b and X ( β β β ν ν β ν νβ the ounterpart of (4 now being D ˆ β B = x Θ (7 In eqs.(6 Π =F and Σ = δ δ δ δ. ν ν ν β β β This paper extends the derivation leading to eqs.(6 and (7 and assoiated with the Coulomb gauge Lagrangian (5 to the Lagrangian density for the free eletromagneti field in the ovariant gauge given by L= F F ( A (8 4 While this is a routine exerise using the methods of Coleman and Jaiw[] it has been inluded here - with the relevant steps gone through in some detail - merely for the sae of ompleteness; more importantly, for the issues whih this paper addresses and whih motivate this paper, the relevant energy-momentum tensor is not the Belinfante version ˆ ν Θ B but the energy - momentum tensor Θ ˆ ν defined by []

4 where ˆ ˆ Θ =Θ B + X ν ν ν β (9a X ν = g σ ν g σ βν g ν σ β + g ν σ ( g g ν g g βν σ (9b with ν g ν A A A σ = A ν, σ = g σ (9 As is obvious, σ of Θˆ ν ν is symmetri in and ν and its nonzero value underlines the preferene over ˆ ν Θ B for the Lagrangian in the ovariant gauge given by (8. While relegating these details to the text below, let s motivate this paper with the following question: Given that the trae identity (see eq.(0 below assoiated with the sale transformation ρ x x = e x on the Lagrangian (8 is maintained, is the same trae identity also maintained on a boundary? To elaborate, we shall first use standard methods in the sequel to he that the trae identity given by ( ˆ ν ν Θ + δ ( * * 0 ( ( ( 0 ( 0 ( ( 0 g T za sa t i z s T A sa t ν ( + iδ z t T A s A t =0 * ( 0 ( ( 0 (0 is preserved in free spae. Subsequently, we he if eq.(0 also holds on a single plate at x = - a on whih the Dirihlet boundary ondition A ( t x x x a,,, = = 0is introdued; note that x denotes the spatial z-omponent of the four vetor x. For the sae of larity we shall label these twin hes below as being arried out without and with a boundary respetively. While this exerise is inspired by the Casimir effet [4,5] in whih the fore 4

5 between a pair of plates is alulated indeed, this explains the use of the term in the title of the paper the objetive here is learly different: Namely to determine how a minimal hange from an unonstrained onfiguration represented by the absene of a boundary to a onstrained one haraterized by the presene of a single plate at Dirihlet boundary ondition holds A ( t x x x a x = a and on whih the,,, = = 0 affets, if at all, the validity of the trae identity given by eq.(0 above. Needless to say so, within the above framewor the Casimir onfiguration of a pair of plates shall be understood as a not - so minimal hange. Sine the objetive of this paper has been spelt out above, the alulation of the Casimir energy on the single plate will not engage our attention here, more so as the literature [4,5] on this aspet as well as on the experimental support [6,7] for the Casimir effet is vast, with little or no mention to the best of our nowledge, of the issues raised here. Let s reiterate at this stage that it is the suseptibility, if any, of the trae identity given by eq.(0 to a minimal hange as explained above from the unonstrained onfiguration that is on test in this paper. We must also emphasize here that the departure from the use of a pair of parallel onduting plates separated by a distane that was used in the original Casimir alulation [8] to a single plate in this paper has been done here deliberately. The impetus for this exerise omes from an analogous report by the author[9] reently on the validity of the trae identity assoiated with the Lagrangian density for a noninterating massive real salar field viz. L= φ φ m φ in + dimensions when one moves from an unonstrained onfiguration, wherein the identity has been found to hold, to a onstrained onfiguration in whih the trae identity is shown to be violated - where a boundary in the form of a single plate at x = - a is introdued and on whih the Dirihlet ( boundary ondition φ tx,, x = a = 0 is imposed. The present wor is thus an extension to the Lagrangian given by (8 of the wor done in Ref.9. The plan of this paper is as follows: In Se. we shall present very briefly following the methods of Ref. a rationale for the hoie of the energy-momentum tensor given by eq.(9a so as to obtain a modified dilatation urrent in a form analogous to eq.(4 for the 5

6 Lagrangian for the massless free salar field and follow it up with an expliit verifiation of the trae identity given by (0 for the unonstrained onfiguration; Se. is the main part of this paper wherein we use a modified generating funtional following Bordag, Robashi and Wiezore[0] whih expliitly inorporates the Dirihlet boundary ( ondition A t, x, x, x = a = 0to derive the onneted Green s funtions relevant for the verifiation of the trae identity. The paper onludes with a short Appendix that inludes a derivation of some of the relevant equations disussed in Se..A he of eq.(0 without a boundary The anonial dilatation urrent assoiated with the Lagrangian density (8 is given by D x ˆ ν ν = Θ +Π A (a ν ν with Π =F g ( A.This extra seond term in the definition of Π relative to just and Π =F for the Coulomb gauge Lagrangian (5 now implies that with eqs.(6,(6a one an rewor eq.(a above as X ( β β β = Π Σ Π Σ Π Σ A (b ν ν β ν νβ ( D x ˆ ν ν β β = Θ +Π A + g X x X ( ν B ν β Note that the r.h.s. of eq.(b is now twie that of the r.h.s. of (6b; this differene will turn out to be ruial below. From the seond and third terms in ( one obtains ν β ν 5 ( Π Aν + gx = F Aν + A A ν ν 5 = ( A A Aν + ( A A = σ ( 6

7 when = - 5; the last equality defines the field virial V = σ,with σ given by (9 for the ovariant gauge Lagrangian density given by (8.We an now use eqs.(9a and (9b above to define the dilatation urrent in terms of Θ ˆ ν as D x ˆ = Θ as in eq.(a. of Ref. ;and then derive the trae identity given in (0 following the steps given in Appendix B of Ref..By ontrast, retaining the same form as (6b yields for the l.h.s. of ( ν 4 A g X β ν ν Π ν + = ( Π +Π A ν + ( A A = ( AA 4 ( AA ν + = ( AA ν ( Unlie the r.h.s. of eq.(, the r.h.s. of the last step of ( annot be written as a total derivative; while this underlines the advantage here in the hoie of (b over (6b it also emphasizes the utility of the F ν A ν term in obtaining the desired form of the field virial in (.Let s also point out here that we do not laim originality in deriving eq.( above for the assoiated value of, as we are onfident that the steps that have been wored through above must have also been obtained in the unpublished literature. For the sae of ompleteness let s now hint at a short derivation of the trae identity given by eq.(0. It is onvenient to start from the following Ward identities, firstly for the anonial energy momentum tensor ˆ ν Θ : ( ˆ ν β ν δ ( ( 0 T Θ ( z A ( s A ( t 0 + i zs 0 T A ( s A ( t 0 + ( z * (4 * β ν ( ( (4 * iδ z t T A s A t 0 ( ( 0 =0 and seondly for the anonial dilatation urrent D : ν ν ( δ ( ( δ + (4 * ν iδ ( z t T ( A s δa t 0 T D ( z A ( s A ( t 0 + i zs 0 T A ( s A ( t 0 ( z * (4 * 0 ( ( 0 = 0 7

8 ( with δ A ( x = + x A ( x.with the derivation of the dilatation urrent as D x ˆ = Θ explained in detail in the earlier part of this setion it is now a simple matter to repeat verbatim the steps of Appendix B of Ref. and obtain the trae identity given by eq.(0 in this paper. As a matter of aution it is useful in this exerise to bear in mind the remars on p.589 and 590 of Ref.. We shall now he the trae identity in the absene of a boundary and begin with the familiar expression for the generating funtional for the onneted Green s funtions that is appropriate for the disussion below, viz. ( ˆ ν exp iz J, K, M = N DA expi L + J A + K A + M ν ν Θ x n i ( ˆ n ν = N DA M Θ expi ( L+ J A + K A 0 n! ν (4 x where J,K ν and M ν are external soures for A (x, A ν and Θ ˆ ν, and N a normalizing onstant whih ensures that the l.h.s. is unity when the external soures are zero. On expressing Θ ˆ ν in terms of funtional derivatives given further below - ating on the exponential one writes (4 as ν exp iz J, K, M = 0 n i ( ˆ M Θ n! n N DA expi ( L + J A K ν + Aν x = n i ( ˆ n M Θ ( ( ( ( ( 0 n! exp i ( ρ β ( J x + ikρ x D x y Jβ y + i y K β x y (5 In eqs.(4 and (5 the symbol x denotes 4 d x,with ( e id ( x y = i 6 g + i xy β (6 iε 8

9 being the propagator assoiated with the Lagrangian (8 when = - 5 ; also in (6 is shorthand for 4 d ( π 4.Thus δ Z δj ( x δj ( y * β i i T ( A x A y i β soures= 0 = 0 ( ( 0 =D ( x y (7 It is also easy to he that 6 = 5 ( (4 g D β ( x y β δδ x y (8 the derivative operators being taen with respet to x.we now give below Θ ˆ in terms of funtional derivative operators whih will at on the exponential on the r.h.s. of eq.(5 ( ( Θ ˆ ( z = i C ( z + S ( z + T ( z (9 with: C δ δ βρ δ δ g g g βρ β δk δk 5 δk δk = δ δ δ δ δ + g + g 4 δk δk δkρ δk ρ 0 δk ρ ρ S δ δ τν δ ρ δ δ τν δ ρ gρ g gρ g ρ ρ τν Σ + ρ ρ τν Σβ δk δk 5 δk δk δk 5 δk δ = δ δ τν δ ρ δ J gβρ g + βρ ρβ τν Σ δk δk 5 δk 9

10 T δ δ δ δ gν g g g β νβ ν β ν δj δj δj δj δj δj δj δj = δ δ + gνβ g ν δj δj δj δj Note that the partial derivative and funtional derivative operators above are taen with respet to z δ and should be understood as ( z and, for instane, respetively. In (9 δ J ( z C β, S β and T β orrespond respetively to the funtional derivative representation of the anonial energy-momentum tensor Θ ˆ, the divergene X of the ovariant version of the tensor X β given in (b and, lastly, to a ovariant version of the last term in (9a but after neessary simplifiation. To verify the trae identity given in (0 it is enough to δ Z alulatei δm ( z δj ( s δjν ( t Soures fator of = 0 from (4 as that will yield, besides a multipliative i, the T* - produt in the first term in eq.(0. On ontration with g simplifiation one immediately obtains the answer as β and ( ν δ ( ν iδ zs T A s A t i zt T A s A t 0 * * ( 0 ( ( 0 ( 0 ( ( thus verifying the trae identity in the unonstrained onfiguration. We have gone through the alulation above in a rather perfuntory fashion as there are no surprises to be expeted here, but will exerise aution in the sueeding setion as there will be qualitative hanges in eqs.(7 and (8..A he of eq.(0 with a boundary A simple way to inorporate the Dirihlet boundary ondition A ( t x x x a,,, = = 0 in to the generating funtional given by (4 is to first rewrite it using the method of Bordag, Robashi and Wiezore[0]as ν = δ ( ( = ( + ν ˆ + ν + Θ exp iz J, K, M N DA A t, x, x, x a expi L J A K A M x (0 0

11 whih an be rewored as ν ν ( ˆ ( = Θ + δ + ( ν exp iz J, K, M C DA Db expi L J A K A M x a b A x with C a normalizing fator and b (x an auxiliary field that exists on the plate at x = - a only; thus b (x is a funtion of the three variables t, x and x only. One now rewrites ( as n i ˆ n ( ( ν δ exp iz J, K, M = ( M Θ N DA Db expi L + J A + K A + x + a b A 0 n! x ν ν ( with L denoting the same Lagrangian as given by (8 but with = - 5,and Θ ˆ ν now expressed in terms of funtional derivatives as in (9.It is now easy to obtain from ( n ν i exp,, ( ˆ n i iz J K M = M Θ N Db exp M ( x D ( x y M β ( y 0 n! x y ( i y M ( y J ( y K ( y y a e b ( y with = + + δ ( + β β β i x M ( x J ( x K ( x x a e b ( x and = + + δ ( + (4a (4b In eq.(, D ( x y has the same form as in (6. Note that the derivative operators in (4a and (4b are with respet to the variables y and x respetively. For the sae of larity we rewrite the exponent in ( below as: Q ( x D ( x y Q ( y + Q ( x D ( x y b ( y + b ( x D ( x y Q ( y β β β xy + + xy x y + b ( x D ( x y b ( y + + x y β (5

12 β with Q ( x J ( x + K ( x ; also β is shorthand for d 4 xδ ( x + a d 4 yδ ( y a + + x y 4 4 is a label for d x d y δ ( y + a,so that xy + + and represents the familiar double xy integral without the Dira δ - funtion, with y denoting the spatial z- omponent of the four vetor y. With the definition x b x Q z D z u D u x ρ ( ( + ( ( ρ( + zu (6 (5 beomes ρ Q( x D ( x y D ( xz Dβ ρ ( zu D ( u y Q ( y + ( x D ( x y ( y β x y z u x y (7 Eq.( below defines the inverse of D ( x y and (7 leads to n ν i exp,, ( ˆ n i iz J K M = M Θ exp Q ( x D ( x y Qβ ( y 0 n! (8 x y with D ( x y denoting the expression in parentheses in (7.The seond term in (7is independent of the external soures and the resulting path integral due to the shift in (6 will be absorbed in to the normalization onstant N. It is not diffiult to he that in the presene of the boundary adopted in this paper, we obtain from (8 δ Z δj ( x δj ( y * β i i T ( A x A y i β soures= 0 = 0 ( ( 0 =D ( xy (9 Eq.(9 thus defines the propagator for our alulation below; note that the onneted Green s funtion has been given an additional subsript to distinguish it from that given in eq.(7.further, it is the ounterpart of eq.(.7 in Bordag et al. [0] or to use a more reent referene, eq.(6 in Bordag and Lindig []. Using (8 one obtains with

13 D ( x y = D ( x y D ν ( xz D ( z u D ρβ νρ ( u y (0 + + z u 6 = ( uy (4 g D ( x y β x y ν ρβ δδ δδ( x z Dν ρ( z u D ( z u β = δδ ( δ ( (4 ρβ x y x + a D ( xu D ( u y β (4 = ( x y + u ρ β ( δ δ δ δ ( x y δ ( x a + (a when y = - a; or, to rewrite (a 6 5 (4 g D ( x y β β ( = δδ ( xy δ δ ( x y δ ( x y (b when y = - a. While the first term in (b arises from the r.h.s.of eq (8, the latter being assoiated with D ( x y,the seond term despite its resemblane to the first is valid only when y = - a,i.e. on the single plate introdued in this paper. Indeed, it is the latter term that will play a deisive role in invalidating the trae identity as given by (0. The inverse of D ( x y, namely, D ( x y satisfies the relation ρ = ( ( D ( u y D β ρ ( y z δρ δ ( u z + y when u = - a and z = - a ;and the form of D ρ ( x y is wored out in the Appendix to this paper. Shelving these details it is enough to antiipate here that given the following form for D ( x y viz., i ( xy i ( x y qq D ( x y = i e η ρ β ( with: i. the momentum integration being -dimensional so that d denotes ( π

14 ν ν ii. η being a diagonal matrix given by η = (,,,5 iii. = 0 q (,,, for = 0,, and, and, iv. 0,, β ρ = 0,, β =, 0 ( + x y one an show that i ( x y + i x y D ( x y = i e b β + p p p β (4 with: i. b being a diagonal matrix given by b = (,, for,β =0, and ii. 0 p = 5 0,5 0,5 0, 0 and, iii. = 50 0 with 0 defined as above. Note that eqs.( and (4 define D ( x y and D ( x y on the three dimensional subspae x ( = 0,, only. Also eqs. ( and (4 are the respetive ounterparts of eqs. (.0 and (.4 of Ref. 0. We shall now tae up the verifiation of the trae identity in the presene of the boundary remembering see eq.(6 that D ( x y = D β ( x y. The steps below use this symmetry property as well as eqs.(0and (b ; they do not use eqs.( and (4 for D ( x y and D ( x y respetively. In other words, the main onlusions of this paper do not depend on eqs.( and (4, with the latter as well as the ontent of the Appendix to this paper only serving the limited purpose of woring out a possible form for D ( x y, at least partly inspired by the wor of Bordag et al.[0] and Bordag and Lindig[] and as a omplement to our wor in Ref.9. 4

15 δ Z As in the previous setion it is enough to alulatei δm ( z δj ( s δjν ( t Soures as that will yield, besides a multipliative fator of = 0 from (8, the T* - produt in the first term in eq.(0.to mae the disussion reader-friendly, we shall now present below the results got from eah of the three terms in (9, but after ontration with g β,remembering that the partial derivative operators are with respet to the variable z : i Term : ρ ρ ρ i D ( zs D ρν ( zt D ( zs D ρ ν ( zt D ( zs D ρν ( zt 5 Term : 6 β β β D ( zs D βν ( z t + D β ( zs ( D ν ( zt D ν ( zt 5 8 β β 8 β β i + D ( zs D βν ( zt D β ν ( z t + D ν ( zt D β ( zs β D ( zs D ν ( z t ( D ( z s D ( z s Adding the two terms above results in ( ( ( ( ( ( β D zs D βν z t + D ν zt D zs D zs i 8 β β 8 β β D ( zs D βν ( zt β D ν ( z t + D ν ( zt D β ( zs β D ( zs 5 5 Finally, we have for { } Term : i ( D ( z s D ρ ( z t β ρ ν ( D ( zs D βν( zt The addition of Term to the sum preeding it now yields the final answer for the first term in the trae identity given by (0, viz. 5

16 6 β β 6 β β i D ( zs D βν ( zt D β ν ( z t + D ν ( zt D β ( zs β D ( zs 5 5 (5 Note that the first entry in the parentheses in (5 beomes 6 β β D βν ( zt D β 6 β β ν ( zt = D β ν( zt D βν( zt 5 5 β ρ 6 ρ = βg D ρν( zt 5 (4 ( ( ν δν δ ( z t δ ( z t = δ δ ( z t (6a with the seond entry similarly given by 6 β β Dβ ( zs β D ( zs 5 = δ δ ( z s (4 ( ( δ δ ( z s δ ( z s (6b With eqs.(6a and (6b, (5 now beomes the sum of (4 (4 R id ( t s δ ( z t id ( s t δ ( z s ν ν = id (4 (4 ( t s δ ( z t δ ( z s ν ( + (7 and ( ν ( ( δ z t δ z t Q id ( t s = id ( t s + id ( s t ( ν ( ( δ zs δ z s ν ( δ ( ( ( ( ( ( z t δ z t δ z s δ z s + (8 Thus the trae identity given by (0 now beomes ν ( g 0 T ( z A ( s A ( t 0 * ˆ Θ = i( Q+ R 6

17 i.e., ν ( g 0 T ( z A ( s A ( t 0 * ˆ * Θ + T ( A β x A y 0 ( ( ( (4 (4 0 iδ ( z t + iδ ( z s = i Q (9 * with iq = T ( A s A ν t i δ ( ( z t δ ( z t i δ ( ( z s δ ( z s 0 ( ( 0 ( + (9a Clearly, the r.h.s. of (9 is not zero on the plate at z = - a and the trae identity given by (0 is thus invalid on the single plate we have introdued as a boundary in this paper. Note that (9a is formally similar to and with the same sign as the seond term on the l.h.s. of the same equation but originates from the seond term on the r.h.s. of eq.(0.we shall now view the above result in another ontext below, namely, as a support to our wor in Ref.9 Disussion: In our earlier report [9] assoiated with the Lagrangian density of a real massive noninterating salar field in + dimensions, viz. L φ φ m φ = (40 we had dealt with the trae identity ( ˆ Θ + ( * ( * 0 ( ( ( 0 ( 0 ( ( 0 g T z φ s φ u iδ z s T φ s φ u ( * * + iδ ( z u 0 T ( φ( s φ( u 0 = m 0 T ( φ ( z φ( s φ( u 0 (4 To mae our point, let s reall here from Ref.9 that while eq.(4 held in the unonstrained onfiguration, it did not on a boundary on whih the Dirihlet boundary ondition φ ( tx,, x = a = 0 was imposed. In other words, in the latter ase (4 was modified to 7

18 ( ˆ Θ + ( * ( * 0 ( ( ( 0 ( 0 ( ( 0 g T z φ s φ u iδ z s T φ s φ u ( * * + iδ ( z u 0 T ( φ( s φ( u 0 = m 0 T ( φ ( z φ( s φ( u 0 + ik (4 where { } ik = 0 T ( s ( u 0 i z u z u i z s z s ( φ φ δ ( δ ( δ ( δ ( * ( ( (4 with z being the spatial y-omponent of the -vetor z, with the same understanding for u and s. Eq.(4 is thus the ounterpart of the r.h.s. of (9,but for the Lagrangian in + dimensions given by (40. Apart from a differene of the anonial sale dimension whih is respetively for the salar field and appears in (4 and (4, and for the vetor field as seen in (9 and (9a, both eqs.(9a and (4 are similar despite the fat that the two equations refer in details to two different theories. Put differently, the observed violation in Ref.9 of the naïve trae identity given by (4 above on a plate on whih the Dirihlet boundary ondition has been imposed is now sustained by the failure of the ounterpart of eq.(4 - namely eq.(0 on a plate on whih a similar boundary ondition is imposed in this paper. Both these naïve identities are otherwise, needless to say so, maintained in free spae. For the sae of ompleteness, we give below the form of the anonial dilatation urrent for the Lagrangian given by (40, viz. D ˆ = Θ + (44 4 x φ while the ounterpart of (9b is ν ν ν β X = ( g g g g φ (44a 4 8

19 Eqs.(44 and (44a together lead as shown in Ref.9 to a modified version of the dilatation urrent namely, D given by (44a. = x Θ ˆ with Θ ˆ ν ν ν defined in terms of ˆ ν Θ by eq. (9a but with X ν as To onlude, the twin hes on the validity of the trae identity given by (0 in Ses. and of this paper for the Lagrangian of the noninterating eletromagneti field given by (8 in the ovariant gauge, reinfore our onlusions obtained in the earlier paper[9] that the ρ trae identity assoiated with the sale transformation x x = e x for the Lagrangian (40 beomes anomalous in the sense that it is not maintained when one moves from an unonstrained onfiguration where the identity is respeted to a onstrained one, the latter being haraterised by a boundary on whih a Dirihlet boundary ondition is imposed; a notable feature of this anomalous term being the appearane of the anonial sale dimension as a numerial oeffiient. As another example of the latter observation, let us reonsider the Lagrangian given by (40 in + dimensions in whih ase the anonial sale dimension of the salar field beomes zero, and the trae identity given by (4 now reads as ( ˆ ( * * 0 ( ( ( 0 0 ( ( ( 0 g T Θ z φ s φ u = m T φ z φ s φ u (45 with the anonial dilatation urrent now given by D ˆ = x Θ ; the absene of the field virial in this ase, in the sense that it is zero, now implies that: One, ˆ ν ˆ ν Θ =Θ with Θ ˆ being the anonial energy-momentum tensor, and seondly, while the trae identity given by (44 is respeted in the unonstrained onfiguration, it is also maintained in the ase when one introdues a plate on whih the Dirihlet ondition ( tx, a ν φ = = 0 holds. We refer the interested reader to Ref.9 for a somewhat more detailed disussion on these issues and onlude this paper below with an Appendix wherein a fairly detailed evaluation of D ( x y is presented. 9

20 Appendix We begin with the eq.(6 of the text, namely, ( e id ( x y = i 6 g + i xy β (A iε and perform a Wi rotation to obtain for = 00 i x y 0 r 00 D x y i e re dr g e r dr ( ( 6( 0 0 (A We shall now integrate over in (A and get after doing the r integral the result D ( x y =i e g + + x y 0 ( ( ( 00 i xy x y 00 A similar effort on the other omponents of D ( x y leads to ( 0 j i ( xy x y ( 0 j D ( x y = i e + x y (A for j.liewise for ij D ( x y for i,j we get ij i ( xy x y ij D ( x y = i e g + ( + x y i j (A4 It is not diffiult now to obtain a general expression for D ( x y after undoing the Wi rotation and this is given by i ( xy i ( x y qq D ( x y = i e η ρ β (A5 0

21 with : the momentum integration being -dimensional so that ν a diagonal matrix given by η = (,,,5, 0 q (,,, denotes ( π d ν, η being = for = 0,, and, and, 0,, β ρ = 0,, β = where 0 ( + x y. We now use the following form for D ( x y ip ( x y i p ( x y r r + β D( x y =i e p d + b (A6 p p so as to reflet the symmetry properties of D ( x y exhibited by eq.(6 and mentioned earlier in the text of the paper. The d β, b and r will be fixed by using the definition βρ ρ ( D ( x z D ( z y = δ δ ( x y (A7 + z when x = - a and y = - a ; z + being a symbol for d 4 zδ ( z + a and z being the spatial - z omponent of the 4-vetor z with a similar understanding for x and y,it being additionally understood that d β is a diagonal matrix. Repeated use of (A7 for =,ρ = 0, and enable us to arrive at 0 r0 = 5 0, r = 50 and r = 0.A similar exerise with = 0 and but with ρ = 0,, and yields d 0 0 =,d = - = d, r = 5 0 and b =. 50 0

22 Referenes. S.Coleman and R. Jaiw, Ann.Phys.(N.Y. 67, 55(97.. See remars following eq.(a.6 in Ref.. See eq.(a.8 in Ref. 4. H. B.G. Casimir, Pro. Kon. Ned. Aad. Wetensh., 5,79(948;for additional referenes see K. A. Milton, The Casimir effet :Physial Manifestations of Zero- Point Energy, World Sientifi(New Jersey, G. Plunien, B. Muller and W. Greiner, Phys. Rep.4,87( M.J. Spaarnay, Physia 4,75( See also, M. Bordag, U. Mohideen and V. M. Mostepaneno, Phys.Rep.5,(00 for additional referenes. 8. H. B.G. Casimir,Ref.4 9. S.G. Kamath, submitted for publiation. 0. M. Bordag, D. Robashi and E. Wiezore, Ann. Phys.(N.Y.65,9(985.. M. Bordag and J. Lindig, Phys. Rev. D 58,04500(998.