Vacuum energy density and pressure of a massive scalar field
aa r X i v : . [ qu a n t - ph ] A p r version of 2 April 2015 Vacuum energy density and pressure of a massivescalar field
Fernando Daniel Mera and S A Fulling Department of Physics, Northeastern University, Boston, MA 02115 USA Departments of Mathematics and Physics, Texas A&M University, College Station,TX 77843-3368 USA
Abstract.
With a view toward application of the Pauli–Villars regularization methodto the Casimir energy of boundaries, we calculate the expectation values of thecomponents of the stress tensor of a confined massive field in 1 + 1 space-timedimensions. Previous papers by Hays and Fulling are bridged and generalized. TheGreen function for the time-independent Schr¨odinger equation is constructed from theGreen function for the whole line by the method of images; equivalently, the one-dimensional system is solved exactly in terms of closed classical paths and periodicorbits. Terms in the energy density and in the eigenvalue density attributable to thetwo boundaries individually and those attributable to the confinement of the fieldto a finite interval are distinguished so that their physical origins are clear. Then thepressure is found similarly from the cylinder kernel, the Green function associated mostdirectly with an exponential frequency cutoff of the Fourier mode expansion. Finally,we discuss how the theory could be rendered finite by the Pauli–Villars method.PACS numbers: 03.70.+k, 11.10.GhAMS classification scheme numbers: 81T55 acuum energy of a massive scalar field
1. Introduction
The Casimir energy [1, 2, 3] of a massive scalar field in two space-time dimensions,despite the seeming simplicity of the model, has not been completely studied. The 1979paper of Hays [4] calculated the energy and the force but did not look at the local energydensity, a subject of much interest today. The more recent paper of Fulling [5] treatedthe energy density for a massless scalar field from a viewpoint of spectral theory andasymptotics, but did not consider the massive field. Neither paper calculated pressuredirectly. The present article generalizes the works [4] and [5] and uses methods fromeach.The primary reason for studying massive fields in this context is to be able toconduct a Pauli–Villars regularization [6, 7, 8, 9, 10] (see Appendix A). It has becomeclear [11] that the traditional ultraviolet cutoff produces unphysical results, dependenton the direction of “point-splitting”, for the counterterms in energy density and pressurenear perfectly reflecting boundaries; this development casts some doubt on the claim thatsuch approaches to divergences are more “physical” than the analytic ones (dimensionalor zeta). The Pauli–Villars method (which occupies a place somewhere between theanalytic and the cutoff methods) preserves Lorentz invariance, and hence one hopes thatit will avoid this problem. A serious implementation of this strategy requires calculationsin four space-time dimensions, which are deferred to future work, but here we give ita test drive. The previous applications of the method that are most pertinent to ourproblem are those to gravitational backgrounds, and we review the relevant literaturein Appendix A.In section 2 the local energy densities E Weyl ( t ), E per ( t ), and E bdry ( x, t ), relatedrespectively to zero-length, periodic, and closed reflected classical paths, are expressed interms of Macdonald functions. (Here t is a temporary regularization parameter.) Theseare expanded in various limits in section 3. As expected, the m → t → m → ∞ limits provide neededinput into the Pauli–Villars construction. Section 4 deals with the (regularized) totalenergy and its nontrivial relation to the nonconstant density term, E bdry ( x, t ). Section 5deals with the eigenvalue density and counting function. Section 6 and section 8 use thecylinder-kernel method pioneered by Hays [12, 4] to find the various contributions to theexpectation value of the pressure; section 7 presents the dependence on the parameter ξ that labels different possible gravitational couplings. Finally, section 9 applies thePauli–Villars procedure.The key results of the paper are the formulas (12), (16), and (17) for energy density;(44), (47), and (49) for pressure; and (51) for the conformal correction to the energydensity (the correction to the pressure being zero). acuum energy of a massive scalar field
2. Vacuum energy density from closed and periodic orbits
We consider a finite interval with either a Dirichlet or a Neumann boundary conditionat each end, following the notation of [5], which allows the two boundary conditions tobe treated simultaneously. Thus H = − d dx + m acts in L (0 , L ) on the domain definedby u (1 − l ) (0) = 0 , u (1 − r ) ( L ) = 0 , l, r ∈ { , } . (1)The superscript is the number of derivatives in the boundary condition. Thus l = 0means that the left endpoint is Neumann, etc. In nonrelativistic terms we are solving aSchr¨odinger equation with potential V = m . The Green function can be constructedfrom G ∞ , the Green function on the whole real line, by the method of images: G ( ω , x, y ) = G ∞ ( y ) + ( − l G ∞ ( − y ) + ( − r G ∞ (2 L − y ) + ( − l + r G ∞ (2 L + y )+ ( − l + r G ∞ ( − L + y ) + ( − l + r G ∞ ( − L − y )+ ( − l +2 r G ∞ (4 L − y ) + ( − l +2 r G ∞ (4 L + y ) + · · · = ∞ X n =0 ( − n ( l + r ) G ∞ ( − nL + y ) + ∞ X n =0 ( − l + n ( l + r ) G ∞ ( − nL − y )+ ∞ X n =1 ( − − l + n ( l + r ) G ∞ (2 nL − y ) + ∞ X n =1 ( − n ( l + r ) G ∞ (2 nL + y ) . (2)(Here and occasionally elsewhere we suppress some function arguments to avoid clutter.)The only difference from [5] is that in G ∞ the energy parameter λ must be replaced by λ − m . Thus, many formulas in [5] remain valid if we replace ω ( ≡ √ λ ) by κ ≡ √ ω − m (hence ω dω = κ dκ ) , (3)and the basic Green function is G ∞ ( ω , x, y ) = i κ e iκ | x − y | . (4)It is easy to check from first principles that this new G ∞ , and hence G , satisfy the rightequation, − ∂ G∂x − κ G = ( H x − ω ) G ( ω , x, y ) = δ ( x − y ) . (5)The spectral densities in terms of λ for this problem are the same as in [5] except forthe shift of the argument variable λ by − m . This is exactly to be expected, because weknow that adding a constant to the potential in the Schr¨odinger equation merely addsthat constant to all the energies. Note that only values of ω ≥ m need to be considered,because we know that H has no spectrum below m . This even comes out automaticallyin the formalism, because if κ is imaginary, then the imaginary part of G ∞ is zero anddoesn’t contribute to the density of states. When we go to the variable ω the situationis slighly more complicated: κ is not just ω minus a constant, and that is where someinteresting new behavior arises. acuum energy of a massive scalar field ∞ X j δ ( λ j − λ ) = 1 π Z L dx [Im G ( λ + iǫ, x, y )] x = y . (6)It is more convenient to work with the density with respect to ω = √ λ , which carriesan additional factor 2 ω . Then ρ ( ω ) dω = 2 ωπ dω Z L dx Im G ( ω , x, x ) = 2 κπ dκ Z L dx Im G ( ω , x, x ) . (7)We have by definition ρ ( ω ) = Z L dx σ ( ω, x ) , σ ( ω, x ) = 2 ωπ Im G ( ω , x, x ) , (8)and hence πκω σ ( ω, x ) = 2 κ Im G ( ω , x, x )= ∞ X n =0 ( − n ( l + r ) cos(2 κnL ) + ∞ X n =0 ( − l + n ( l + r ) cos(2 κ ( nL + x ))+ ∞ X n =1 ( − − l + n ( l + r ) cos(2 κ ( nL − x )) + ∞ X n =1 ( − n ( l + r ) cos(2 κnL )= 1 + 2 ∞ X n =1 ( − n ( l + r ) cos(2 κnL ) + ∞ X n = −∞ ( − l + n ( l + r ) cos(2 κ ( x + nL )) ≡ πκω ( σ Weyl + σ per + σ bdry ) . (9)The paths connecting x to an image charge in (2) can be folded back into theoriginal interval as paths connecting x to y after some number of reflections from theendpoints. In (9) these paths connect x to itself. The first term, coming from a path ofzero length, provides the bulk spectral density of Weyl’s famous theorem. Paths withan even number of reflections are periodic and provide a spatially homogeneous Casimirenergy. Terms with an odd number of reflections “bounce” off one of the boundariesand yield energy distributions somewhat concentrated there.The stress tensor of a scalar field contains a free parameter, ξ , reflecting anambiguity in its coupling to the gravitational field. The relevant formulas are reviewedin section 6 and appendix C. Until section 6 we confine attention to the value ξ = , forwhich the energy expressions are maximally simple. In particular, the contribution ofthe space derivatives to the energy density is identical to that of the time derivatives,so we can write (following [5]) h T ( t, x ) i ≡ E ( t, x ) = − ddt Z ∞ σ ( ω, x ) e − ωt dω ≡ E Weyl ( t ) + E per ( t ) + E bdry ( t, x ) . (10)Here t is an ultraviolet cutoff parameter, which can be related by a Wick rotation to adifference of physical time coordinates. acuum energy of a massive scalar field Z ∞ e − t √ m + κ cos(2 nLκ ) dκ = mt p t + (2 nL ) K ( m p t + (2 nL ) ) , (11)where K is a Macdonald function (see Appendix B). In particular, if n = 0 (the Weylterm), (11) reduces to K ( mt ). Thus, doing the change of variables (3), we get E Weyl ( t ) = − ddt Z ∞ σ Weyl ( ω ) e − ωt dω = − π ddt Z ∞ √ κ + m κ · κ √ κ + m e − t √ κ + m dκ = − π ddt mK ( mt ) = m π (cid:18) mt K ( mt ) + K ( mt ) (cid:19) (12)(see (B.3)). Similarly, the periodic term is E per ( t ) = − π ddt ∞ X n =1 ( − n ( l + r ) Z ∞ σ per ( ω ) e − ωt dω = − π ddt ∞ X n =1 ( − n ( l + r ) mt p (2 nL ) + t K ( m p (2 nL ) + t ) . (13)Finally, the boundary term is E bdry ( t, x ) = − ( − l π ddt ∞ X n = −∞ ( − n ( l + r ) Z ∞ ωκ cos(2 κ ( x + nL )) e − ωt dω = − ( − l π ddt ∞ X n = −∞ ( − n ( l + r ) mt p (2( x + nL )) + t K ( m p (2( x + nL )) + t ) . (14)
3. Asymptotic behaviors t To put the energy expressions (12)–(14) into the usual form for renormalizationcalculations, we need to expand them in power (Laurent) series in t . Using (B.1) onegets E Weyl ( t ) = − π ddt h t + m t (cid:16) mt (cid:17) + m t γ −
1) + O (cid:0) ( mt ) (cid:1)i = 12 π h t − m (cid:16) mt (cid:17) − m γ + 1) i + O ( t ) . (15)When the derivatives in (13) and (14) are calculated, only one term survives in the limit t →
0. Furthermore, the resulting limits are finite (no negative powers or logarithm of t ): E per (0) = − π ∞ X n =1 ( − n ( l + r ) m nL K (2 nLm ) , (16) acuum energy of a massive scalar field E bdry (0 , x ) = − ( − l π ∞ X n = −∞ ( − n ( l + r ) m | x + nL | K (2 | x + nL | m ) . (17)In the case (14), this argument assumes x = 0 and x = L , and the limit is not uniformin distance from the boundary. Therefore, we shall need to revisit this case whenconsidering the total energy in section 4. The same expansion (15) shows that when m = 0, E Weyl ( t ) = 12 πt ( m = 0) , (18)as expected [5]. (The only interesting fact is that (15) includes less trivial terms when m > E per (0) and E bdry (0 , x ) iscomplicated by the conflict between the m → n → ∞ limits in the individualMacdonald functions. However, (16) when l + r is even is a special case of [14, (2.10)],a complicated formula from which only one term survives when m = 0: E per (0) = − π L ( m = 0) , (19)the well known one-dimensional Casimir energy. In exactly the same way, [14, (2.12)]gives E per (0) = + π/ L when l + r is odd (one Dirichlet and one Neumann end). The behavior when m → ∞ is critical for the Pauli–Villars analysis. Using (B.2) onesees that all the limits are zero: From (12) we havelim m →∞ E Weyl ( t ) = lim m →∞ m π r π mt e − mt = 0 (20)when t >
0. Similarly, the terms of (16) and (17) (or even (13) and (14)) for fixed m vanish sufficiently rapidly with n to make the series converge, and for fixed n decreasemonotonically to 0 as m → ∞ ; therefore, by standard arguments [15, pp. 3 and 8] thesum of the series approaches 0 as m → ∞ . The only exceptions are the terms in (17)with n = 0, x = 0 and with n = − x = L , where the energy density is singular, aspreviously noted.
4. Total energy
The energy equals the integral of the energy density over x from 0 to L , at least formally.Departing somewhat from the notation of [5], we denote a total energy by the letter E . E Weyl and E per are constant in x , so their energies are obtained by multiplying by L andthere is nothing more to be said. acuum energy of a massive scalar field l + r is even, the boundary formula (14) yields E bdry ( t ) = − ( − l π ddt ∞ X n = −∞ Z L mt p x + nL ) + t K ( m p x + nL ) + t ) dx. (21)Making a change of variables x ′ = x + nL in each term, we have E bdry ( t ) = − ( − l π ddt ∞ X n = −∞ Z L ( n +1) Ln mt √ x ′ + t K ( m √ x ′ + t ) dx ′ = − ( − l π ddt Z ∞−∞ mt √ x ′ + t K ( m √ x ′ + t ) dx ′ . (22)After another change of variables, u = 4 x ′ + t , we get with (B.4) E bdry ( t ) = − ( − l π ddt ( mt ) Z ∞ t K ( mu ) √ u − t du = − ( − l ddt e − mt = ( − l me − mt . (23)One can now take t to 0, obtaining in the Dirichlet case ( l = 1) E bdry (0) = − m , (24)in agreement with Hays [4] and with the conclusion in [5] that the net boundary energyvanishes in the massless case. It definitely does not agree (for any m ) with an attempt tointegrate E bdry (0 , x ) over the interval (that is, to take the cutoff away before integrating),which encounters divergences at the endpoints. For later use note also thatlim m →∞ E bdry ( t ) = 0 if (and only if) t >
0. (25)When l + r is odd, E bdry ( t ) vanishes for an elementary reason: The middle memberof (22) acquires a factor ( − n , and hence term n cancels term − ( n + 1).
5. Counting eigenvalues
For completeness of the comparison with the massless theory in [5], we look here atthe global eigenvalue density, ρ ( ω ), and its integral, the counting function N ( ω ). Insection 2 we started from the local spectral density, σ ( x, ω ), and integrated in frequencyspace to get the energy density, E ( t, x ) ; then in section 4 we integrated over x to get atotal energy. Here we shall perform the integrations in the opposite order. Looking atthe spectral and eigenvalue densities is interesting because (unlike most problems) theimage method determines them exactly, and the eigenvalues are known, so that one candirectly compare the eigenvalue densities. Knowing the eigenvalues, one can then sumover the frequencies, in one’s favorite regularization scheme, to get the total energy inthe traditional Casimir manner, but we shall not do that explicitly.The local spectral density (and hence all the other quantities) is divided into threequalitatively different parts in the defining formula (9). The eigenvalue density is thus ρ ( ω ) = Z L σ ( ω, x ) dx = ρ Weyl ( ω ) + ρ per ( ω ) + ρ bdry ( ω ) , (26) acuum energy of a massive scalar field ρ Weyl ( ω ) = Z L σ Weyl dx = Z L ωπκ dx = Lωπκ , (27)and similarly ρ per ( ω ) = 2 Lωπκ ∞ X n =1 ( − n ( l + r ) cos(2 κnL ) , (28) ρ bdry ( ω ) = ( − l π ∞ X n = −∞ ( − n ( l + r ) ωκ [sin(2 κL ( n + 1)) − sin(2 κLn )] , (29)where κ = √ ω − m . The eigenvalue counting function equals zero for ω < m and R ω ρ (˜ ω ) d ˜ ω for ω > m . Therefore, it is (for ω > m ) N Weyl ( ω ) = Lπ Z κ ω ˜ κ · ˜ κω d ˜ κ = Lκπ = L √ ω − m π , (30) N per ( ω ) = 2 Lπ ∞ X n =1 ( − n ( l + r ) Z κ √ ˜ κ + m ˜ κ ˜ κ √ ˜ κ + m cos(2 nL ˜ κ ) d ˜ κ = 1 π ∞ X n =1 ( − n ( l + r ) n sin(2 nL κ ) , (31)and similarly N bdry ( ω ) = ( − l π ∞ X n = −∞ ( − n ( l + r ) Z ˜ κ κ [sin(2˜ κL ( n + 1)) − sin(2˜ κnL )] d ˜ κ. (32)The Fourier series in (31) can be evaluated as in [5] to a sawtooth function, givenin (35)–(36) below. It is easy to see (as at the end of section 4) that N bdry = 0 if l + r is odd (that is, equals 1). When l + r is even, we manipulate (32) to the form N bdry ( ω ) = ( − l π lim n →∞ Z κ sin(2 κLn ) κ dκ and hence N bdry ( ω ) = ( − l π lim n →∞ Z nLκ sin ˜ z ˜ z d ˜ z = ( − l π Z ∞ sin zz dz = ( − l . (33)Adding the three counting functions gives N ( ω ) = Lπ √ ω − m + N per ( ω ) + ( − l δ l + r, (for ω > m ) , (34)where N per ( ω ) = 12 − Lκπ if l + r is even and 0 < κ < πL , (35) N per ( ω ) = − Lκπ if l + r is odd and − π L < κ < π L ; (36)both functions are extended periodically to all other intervals on the positive axis oflength πL in the variable κ . acuum energy of a massive scalar field N ( ω ) is indeed the number of eigenvalues less than or equal to ω . The true counting function must be 0 for ω < m and constant and integer-valuedon the interval between two eigenvalues. Comparing (34) with (35)–(36), we see that N is indeed constant except at the places where N per has a discontinuity. At each suchpoint, N per jumps from − to + , indicating the addition of one new eigenvalue. In theodd case, these points occur at κ = (cid:18) n − (cid:19) πL , ω = (cid:18) n − (cid:19) (cid:16) πL (cid:17) + m ( n = 1 , , . . . ) , (37)and immediately to the right of such a point, N ( ω ) evaluates to Lκπ + 12 = (cid:18) n − (cid:19) + 12 = n. That is, the jumps occur at the correct eigenvalues of the mixed Dirichlet–Neumannproblem, and N counts them correctly. In the even case, the jumps are at numbers ofthe form κ = nπL , ω = (cid:16) nπL (cid:17) + m , (38)and the limit from the right is N ( ω ) = Lκπ + 12 + ( − l ( n if l = 1, n + 1 if l = 0.That is, we get the correct eigenvalues for the Dirichlet and Neumann problems,including the extra eigenvalue at n = 0, ω = m , in the Neumann case; N per and N bdry conspire beautifully to make things come out right at the bottom of the spectrum.
6. Pressure
Because of the need to deal with spatial derivatives, the spectral density σ ( ω, x ) is notconvenient for calculating the expectation value of the pressure, p ≡ h T i . Therefore,we revert to the formalism of the cylinder kernel, T ( t, r , r ′ ) = − ∞ X n =1 ω n φ n ( r ) φ n ( r ′ ) ∗ e − ω n t (39)in terms of the eigenfrequencies and eigenfunctions of the cavity. The cylinder kernelfor the massive field in infinite space is [4, (2.2), (3.1)–(3.2)] T ∞ ( t, x, y ) = − π Z ∞−∞ dω e − iωt √ ω + m e −√ ω + m | x − y | = − π K ( m p t + ( x − y ) ) . (40)The kernel for the Casimir slab is then formed by the same construction as in (2), whichagain generates Weyl, periodic, and boundary terms. Here and henceforth we confineattention to the pure Dirichlet case ( l = r = 0). acuum energy of a massive scalar field T are E ( t, x ) = − ∂ T∂t + β (cid:16) ∂ T∂x + ∂ T∂y + 2 ∂ T∂x ∂y (cid:17) , (41) p ( t, x ) = 18 (cid:16) ∂ T∂x + ∂ T∂y − ∂ T∂x ∂y (cid:17) , (42)where it is understood, formally, that y is set equal to x and t to 0 at the end; β = ξ − is the curvature (or conformal) coupling constant, hitherto assumed to be 0. In arrivingat (41)–(42) several routine intermediate steps have been omitted: Passing from theexpectation of the product of two fields to T requires inserting a compensating factor ;field products need to be symmetrized; physical time derivatives need to be converted to t derivatives, and in that process φ can be converted to − φφ , so that, in particular,the β term in p turns out to vanish identically.The pressure function for the Weyl term, according to (42) and (40) and thediscussion at the end of Appendix B, is given by p Weyl ( t, x, y ) = m π ( t + x − y )( t − x + y ) K ( m p t + ( x − y ) )( t + ( x − y ) ) / − m ( x − y ) K ( m p t + ( x − y ) ) t + ( x − y ) ! . (43)When y = x it simplifies to p Weyl ( t ) = m πt K ( mt ) . (44)The periodic terms are calculated similarly: p per ( t, x, y ) = − π d dx ∞ X n =1 K ( m p ( x − y + 2 nL ) + t ) . (45)After setting y = x and suppressing the redundant argument, we get p per ( t, x ) = − mπ ∞ X n =1 L mn K (cid:0) m √ L n + t (cid:1) L n + t − ( t − Ln )(2 Ln + t ) K (cid:0) m √ L n + t (cid:1) (4 L n + t ) / ! (46)(which actually is independent of x ). In fact, here we can immediately set t = 0, becausethere is no divergence in that limit: p per (0 , x ) = − mπ ∞ X n =1 (cid:18) K (2 mLn )2 Ln + mK (2 mLn ) (cid:19) = m π ∞ X n =1 K ′ (2 mLn ) (47) acuum energy of a massive scalar field L of the total periodic energy, which is L times quantity (16): − ddL E per (0) = ddL (cid:16) π ∞ X n =1 m K (2 mLn )2 mn (cid:17) = m π ∞ X n =1 K ′ (2 mLn ) = p per (0 ,
0) or p per (0 , L ) . (48)The argument at the end of Appendix B shows that the boundary terms in thepressure vanish: p bdry ( t, x, y ) = 0 . (49)This result is analogous to the vanishing of p on p. 5 of [11]; it reflects the fact thatmoving the boundary does not change the boundary energy.
7. Conformal correction to the energy
We digress to discuss the “ β terms” in (41). The same argument from Appendix Bshows that the periodic and Weyl β terms add to 0, the sign change on the third termbeing compensated by the replacement of x + y by x − y , whereas the boundary β termsare nonzero, in close analogy with (45)–(46):∆ E β bdry ( t, x ) = − βmπ ∞ X n = −∞ m (2( Ln + x )) K ( m p t + (2( Ln + x )) )(2( Ln + x )) + t + (2( Ln + x ) − t )(2( Ln + x ) + t ) K ( m p t + (2( Ln + x )) )((2( Ln + x )) + t ) / ! . (50)Combining (50) with (17), we get E β bdry (0 , x ) = E bdry (0 , x ) + ∆ E β bdry (0 , x )= − π (cid:18)
12 + 2 β (cid:19) ∞ X n = −∞ m | x + nL | K (2 m | x + nL | ) − βm π ∞ X n = −∞ K (2 m | x + nL | ) . (51)If β = − ( ξ = 0), which counts as both conformal and minimal coupling in space-timedimension 2, then the first term in (51) vanishes. The surviving term is less singularat the boundary, and it vanishes when m = 0, as expected for a conformally coupledmassless field at a flat (here, 0-dimensional) boundary. acuum energy of a massive scalar field
8. Asymptotics of the pressure t or small m From (44) and (B.1) we have p Weyl ( t ) ∼ π (cid:20) t + m (cid:18) mt (cid:19) + m γ − (cid:21) , (52)and thus p Weyl ( t ) = 12 πt ( m = 0) . (53)These formulas are to be compared with (15) and (18). In fact, we have p Weyl ( t ) − E Weyl ( t ) ∼ m π (cid:20) ln (cid:18) mt (cid:19) + γ (cid:21) , (54) p Weyl ( t ) + E Weyl ( t ) ∼ π (cid:20) t − m (cid:21) . (55)From (54) we see that the zero-point stress tensor of the massless theory, with the t cutoff, is traceless ( T µµ = − E + p = 0), as befits a conformally invariant theory. Onthe other hand, (55) shows that this stress tensor does not satisfy the “principle ofvirtual work” (energy-pressure balance) [16, 11], p = − dEdL (which is − E in this case).A cutoff procedure that respects Lorentz invariance [17] must yield a zero-point stressproportional to the metric tensor (“dark energy”), replacing (55) by 0 but destroyingthe tracelessness (unless it makes p and E identically 0).For the periodic term we have already taken the cutoff away at (47), so the onlyremaining task is to check the massless limit in analogy with section 3.2. In the middlemember of (47), the first term is the same as (16) (in the Dirichlet case), and the secondterm can be shown to vanish as m → p = E in the masslesslimit. But in this case we also have the right pressure balance: p per (0) = + E per (0) = − π L = − ddL ( LE per ) ( m = 0) . (56)We have already seen that the boundary pressure vanishes identically (as does the L derivative of the boundary energy) and that the conformally coupled boundary energydensity, (51), vanishes when m = 0, as does the “renormalized” boundary energy, (24). As m → ∞ , the periodic pressure (47) approaches 0 because each Macdonald functionvanishes exponentially rapidly. The same is true of the Weyl pressure (44) so long as t = 0. The boundary pressure is identically zero. acuum energy of a massive scalar field
9. Applying the Pauli–Villars method
In sections 3.1 and 8.1 we have shown that the stress tensor’s expectation value hasdivergences of orders t − and ln t . In dimension 2 these arise only from the zero-point(Weyl) energy, apart from a caveat about a nonuniform limit at the endpoints in theboundary energy, to which we shall return. The structure is most clearly shown in(54)–(55).In these sections, the display of formulas with coincident spatial coordinates andsmall, imaginary time separation is purely for calculational and expository convenience;in principle, the coordinates are arbitrary. It may appear that we have done a kind of“point-splitting” regularization at an intermediate step; this perception is wrong. Thespirit of Pauli–Villars regularization is to do the subtractions “at the theory level”. Inpractice, this means that the subtractions involve Green functions as a whole , regardedas distributions (or, in other words, they involve the operators that the Green functionsrepresent). Thus the potential divergences are removed before the issue of evaluatingthe Green functions at coincident arguments ever arises.Following Appendix A, consider the effect of superposing the stress tensors forseveral (or many) values of m : E = Z E [ m ] f ( m ) dm, p = Z p [ m ] f ( m ) dm, (57)where the function or distribution f is independent of t . If (A.7) and (A.8) are satisfied,the terms in (55) are obliterated; thus the Weyl part of the vacuum stress satisfies p Weyl = − E Weyl (58)(a nontrivial result, in view of [17] and [11]). We have already observed (section 8.1)that the periodic and boundary parts of the stress are also nonanomalous, thoughthe relations expressing this health are different from (58) because the respective totalenergies have different dependences on L .If, in addition, Z f ( m ) m ln m dm = 0 , (59)then (54) is also obliterated. If one requires merely that this logarithmic integral befinite, as in (A.10), then the stress tensor is finite for all t but its Weyl part may be anontrivial multiple of the metric tensor, a two-dimensional version of the cosmologicalconstant.The model as it stands is unlikely to be physically realistic, because it contains theeffects of unphysical fields with negative energies. Therefore, one studies the effect oftaking the unphysical auxiliary masses very large, in hopes that their effects will becomeunobservable. We verified in sections 3.3 and 8.2 that the periodic and boundary termsvanish in this limit; only the vacuum stresses of the original physical field will survive.If one can guarantee that the integral on the left of (59) remains finite in the limit,then the final theory has no divergences but does have a “cosmological” term with an acuum energy of a massive scalar field m limit before the t limit would be inconsistent withour treatment of the Weyl term. So, we are stuck with (24), a boundary energy linearin m . Recall that it arose because of the nonuniform limiting behaviors of the boundarystress at the endpoints of the interval; in some sense it is concentrated on the endpoints and has become independent of the stress in the interior, which we have succeededin regularizing. Obliterating it appears to require yet another constraint on the massdistribution f .In conclusion, we have shown that the Pauli–Villars construction is mathematicallyfeasible and eliminates the only pressure anomaly that arises in dimension 2, thedirection dependence of Christensen [17]. Physically, whether this maneuver is any moreconvincing than the “analytic” methods (zeta functions and dimensional regularization)is open to debate. Further philosophical discussion probably should wait until animplementation in four-dimensional space-time, where the anomaly of Estrada et al. [11]will be encountered in the ultraviolet-cutoff theory. Acknowledgements
This research was supported by National Science Foundation Grant PHY-0968269, andby the renewed hospitality of the Mathematics Department of Texas A&M Universitytoward F.D.M. while some of the work was done. We thank Klaus Kirsten and KimMilton for critical remarks on a preliminary presentation of the results.
Mathematica was useful at various stages of the work.
Appendix A. Varieties of Pauli–Villars regularization
Pauli–Villars regularization introduces auxiliary fields whose divergences have onbalance opposite sign from those of the original, physical fields, so that the totalpredictions of the theory are finite. Usually the auxiliary masses are taken very large,so that the new fields have no noticeable effects on the finite predictions.The original paper of Pauli and Villars [6] (which remarks, “This method hasalready a long history,”) deals with quantum electrodynamics in Minkowski space-time.Later the method was applied in cosmological contexts [7, 8, 9] and in quantum gravity[10]. There are major differences in philosophy and procedure among these works.Pauli and Villars distinguish between “realistic” and “formalistic” regularizations.In realistic theories the auxiliary masses are assumed to belong to real (physical) fields,whose vacuum energies for some reason do not all have the same sign; these masses arekept finite. In formalistic theories the auxiliary fields are fictitious, and their massesare sent to infinity at the end of the calculations. The realistic approach replaces the acuum energy of a massive scalar field Λ →∞ π Z Λ0 p p + m p dp (A.1)and postulates a mass distribution function f ( m ) (possibly a finite sum of deltafunctions) such that Z f ( m ) dm = Z f ( m ) m dm = Z f ( m ) m dm = 0 , (A.2)so that Z dm f ( m ) Z Λ0 p dp p p + m (A.3)has an asymptotic expansion containing no positive powers of Λ. It may contain termsproportional to Z f ( m ) m p ln(Λ m ) dm ( p = 0 , , , (A.4)but these are actually independent of Λ by virtue of (A.2). Thus, given a fixed f for which the integrals in (A.4) with Λ = 1 converge, the ultraviolet divergences havebeen eliminated. Because it is not required that the integrals (A.4) equal 0, arbitraryrenormalization constants appear.The intention of Zel’dovich was that f ( m ) represent a spectrum of real particles,with negative values of f arising from fermions. This theory, therefore, is of the“realistic” type; it is a forerunner of supersymmetry. Zel’dovich’s main motivationwas to produce a nonzero, but finite, cosmological constant from the integral Z f ( m ) m ln m dm. (A.5)Note, however, that there is a possibility of creating a “formalistic” theory bymoving the support of the negative part (at least) of f off to infinity at the end of theargument, provided that any integrals like (A.5) that arise still converge in this limit.It is not immediately obvious that this can be done, and especially whether the finitelimiting values of the renormalization constants can be different from 0. It is rathereasy to see that the minimal finite sum consistent with (A.2) will not work: Take f ( m ) = δ ( m ) − δ ( m − m ) + δ ( m − m ) − δ ( m − m ) , (A.6) acuum energy of a massive scalar field ∞ . Consider for simplicity a two-dimensional space-time, so that the only constraints from (A.2) and (A.4) that must besatisfied are Z f ( m ) dm = 0 , (A.7) Z f ( m ) m dm = 0 , (A.8) Z f ( m ) ln m dm < ∞ , (A.9) Z f ( m ) m ln m dm < ∞ . (A.10)(In (A.9) the term with m = 0 should be omitted. The precise meaning of (A.9) and(A.10) is that the sums remain bounded as the m j go to infinity.) It is clear that tosatisfy (A.7) the total number of masses must be even, and then to satisfy (A.8) also,the number must be at least 4. Let us first consider the case m = m ; then by (A.6),(A.8) becomes m = 2 m . (A.11)Then (A.10) states that m ln m − m ln m = 2 m ln √ m → ∞ , which is false. Now try m = νm , with ν > F ( ν ) ≡ (1 + ν ) ln(1 + ν ) − ν ln ν = 0 . (A.13)But F (1) = 2 ln 2 > F ′ ( ν ) = 2 ν ln(1 + ν ) − ν ln ν > , (A.14)so splitting the masses can only make the problem worse.Bernard and Duncan [9] take a formalistic approach from the beginning. Theyconsider a two-dimensional cosmological space-time. Unlike [7, 8], who impose (A.3) aposteriori, they start with a Lagrangian and explicitly construct a Fock space. Theirnegative-energy fields (corresponding to masses m and m in the foregoing) are notordinary fermion fields, but anticommuting scalar fields producing states with negativemetric. In the infinite-mass limit this sector of the state space decouples, leaving aunitary dynamics in a Hilbert space. This construction apparently requires m = m ,so the mass spectrum in [9] is the same as (A.6), except that they allow the physicalfield to have a mass, µ ; then m = 2 M − µ and (in effect) m = m = M . Thus(A.7) and (A.8) are satisfied. It turns out that (A.9) is unnecessary because of thetriviality of two-dimensional gravity. But (A.10) is not satisfied in the limit; instead,Bernard and Duncan explicitly introduce a cosmological counterterm to cancel thisdivergence. They remark that the analogous construction in dimension 4 would require acuum energy of a massive scalar field M . The regulated (finite- M )expressions are free of the direction dependence [17] and resulting pressure anomalies[11] characteristic of point-splitting calculations of the stress tensor.Anselmi [10] takes the further step of cancelling the large- M divergences by addingstill more regulator fields. He requires that the logarithmic sums (A.4) vanish. He insertsthe regulator fields into a path integral in a nonstandard way, which permits (in effect)spectra like (A.6) with coefficients not necessarily equal to ±
1. This additional freedomallows the conditions to be satisfied by solving a linear system for those coefficients,instead of the nonlinear system for the masses; this significantly simplifies the study ofthe existence question. The result is that, with enough regulator fields, a formalisticformulation without counterterms is achieved. (Nevertheless, because of the need tomodify (A.4) for p = 0 when the physical field is massless, the logarithmic divergencesinevitably result in two arbitrary renormalized coupling constants in the final equationof motion of the gravitational field. In the present paper this complication does notconcern us.) Appendix B. Calculus with Macdonald functions
In [13] or any similar reference one finds the approximations K ( z ) = 1 z + z z (cid:18) γ − (cid:19) z O ( z ln z ) (B.1)for small z and K ν ( z ) ∼ r π z e − z (B.2)for large z .Derivatives can be eliminated by [13, (8.486.12)] K ′ ( z ) = − K ( z ) , K ′ ( z ) = − z K ( z ) − K ( z ) . (B.3)The integral Z ∞ t K ( mu ) √ u − t du = πe − mt mt (B.4)does not appear in [13] but is known to Mathematica and can be deduced from [13,(6.596.3)].In section 6 and section 7 we repeatedly encounter second derivatives of K ( m p ( x ± y + 2 nL ) + t ) . (B.5) acuum energy of a massive scalar field − . For the β terms in (41) the role of the sign is precisely the reverse. Appendix C. The stress tensor in dimension 2
The general form of the scalar stress tensor, defined by variation of the gravitationalLagrangian with respect to the metric, is given (in the sign convention where g <
0) in[19, 17, 20]. After specializing to flat space (and glossing over operator symmetrizations),it is T µν = (1 − ξ ) φ µ φ ν + (2 ξ − ) g µν φ σ φ σ − ξφφ µν + 2 ξg µν φφ σσ − m g µν φ = [ φ µ φ ν − φφ µν + g µν φφ σσ − m g µν φ ] + 2 β [ − φ µ φ ν + g µν φ σ φ σ − φφ µν + g µν φφ σσ ] , (C.1)where ξ ≡ β + is the curvature coupling constant (and indices on φ denote derivatives).Using the equation of motion, φ σσ = m φ , to rewrite the first term (but not the second),one arrives at T µν = [ φ µ φ ν − φφ µν ] + 2 β [ − φ µ φ ν + g µν φ σ φ σ − φφ µν + g µν φφ σσ ] . (C.2)The advantages of this form are (a) the mass (more generally, a scalar potential [21])does not appear at all, (b) the first term of T µµ contains only µ derivatives, and (c) the β term is manifestly a total derivative. Specializing henceforth to dimension 1 + 1, wehave T = [ φ − φφ ] − β [ φ + φφ ] , (C.3) T = [ φ − φφ ] − β [ φ + φφ ] . (C.4) References [1] Milton K A 2001
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