Local and Global Casimir Energies: Divergences, Renormalization, and the Coupling to Gravity
From the beginning of the subject, calculations of quantum vacuum energies or Casimir energies have been plagued with two types of divergences: The total energy, which may be thought of as some sort of regularization of the zero-point energy, , seems manifestly divergent. And local energy densities, obtained from the vacuum expectation value of the energy-momentum tensor, , typically diverge near boundaries. These two types of divergences have little to do with each other. The energy of interaction between distinct rigid bodies of whatever type is finite, corresponding to observable forces and torques between the bodies, which can be unambiguously calculated. The divergent local energy densities near surfaces do not change when the relative position of the rigid bodies is altered. The self-energy of a body is less well-defined, and suffers divergences which may or may not be removable. Some examples where a unique total self-stress may be evaluated include the perfectly conducting spherical shell first considered by Boyer, a perfectly conducting cylindrical shell, and dilute dielectric balls and cylinders. In these cases the finite part is unique, yet there are divergent contributions which may be subsumed in some sort of renormalization of physical parameters. The finiteness of self-energies is separate from the issue of the physical observability of the effect. The divergences that occur in the local energy-momentum tensor near surfaces are distinct from the divergences in the total energy, which are often associated with energy located exactly on the surfaces. However, the local energy-momentum tensor couples to gravity, so what is the significance of infinite quantities here? For the classic situation of parallel plates there are indications that the divergences in the local energy density are consistent with divergences in Einstein’s equations; correspondingly, it has been shown that divergences in the total Casimir energy serve to precisely renormalize the masses of the plates, in accordance with the equivalence principle. This should be a general property, but has not yet been established, for example, for the Boyer sphere. It is known that such local divergences can have no effect on macroscopic causality.
For more than 60 years it has been appreciated that quantum fluctuations can give rise to macroscopic forces between bodies casimir (). These can be thought of as the sum, in general nonlinear, of the van der Waals forces between the constituents of the bodies, which, in the 1930s had been shown by London london () to arise from dipole-dipole interactions in the nonretarded regime, and in 1947 to arise from the same interactions in the retarded regime, giving rise to so-called Casimir-Polder forces casimirandpolder (). Bohr casimir50 () apparently provided the incentive to Casimir to rederive the macroscopic force between a molecule and a surface, and then derive the force between two conducting surfaces, directly in terms of zero-point fluctuations of the electromagnetic fields in which the bodies are immersed. But these two points of view—action at a distance and local action—are essentially equivalent, and one implies the other, not withstanding some objections to the latter Jaffe:2003ji ().
The quantum-vacuum-fluctuation force between two parallel surfaces—be they conductors or dielectrics lifshitz (); dzyaloshinskii0 (); dzyaloshinskii () —was the first situation considered, and still the only one accessible experimentally. (For a current review of the experimental situation, see Bordag:2009zz (); Klimchitskaya:2009cw ()) Actually, most experiments measure the force between a spherical surface and a plane, but the surfaces are so close together that the force may be obtained from the parallel plate case by a geometrical transformation, the so-called proximity force approximation (PFA) derpt (); derpt2 (); blocki (). However, it is not possible to find an extension to the PFA beyond the first approximation of the separation distance being smaller than all other scales in the problem. In the last few years, advances in technique have allowed quasi-analytical and numerical calculations to be carried out between bodies of essentially any shape, at least at medium to large separation, so the limitations of the PFA may be largely transcended. (For the current status of these developments, see the contributions to this volume by Emig, Jaffe, and Rahi, and by Johnson; for earlier references, see, for example Milton:2008st ().) These advances have shifted calculational attention away from what used to be the central challenge in Casimir theory, how to define and calculate Casimir energies and self-stresses of single bodies.
There are, of course, sound reasons for this. Forces between distinct bodies are necessarily physically finite, and can, and have, been observed by experiment. Self-energies or self-stresses typically involve divergent quantities which are difficult to remove, and have obscure physical meaning. For example, the self-stress on a perfectly conducting spherical shell of negligible thickness was calculated by Boyer in 1968 Boyer:1968uf (), who found a repulsive self-stress that has subsequently been confirmed by a variety of techniques. Yet it remains unclear what physical significance this energy has. If the sphere is bisected and the two halves pulled apart, there will be an attraction (due to the closest parts of the hemispheres) not a repulsion. The same remarks, although exacerbated, apply to the self-stress on a rectangular box lukosz (); lukosz1 (); lukosz2 (); ambjorn (). The situation in that case is worse because (1) the sharp corners give rise to additional divergences not present in the case of a smooth boundary (it has been proven that the self-energy of a smooth closed infinitesimally thin conducting surface is finite balian (); Bernasconi ()), and (2) the exterior contributions cannot be computed because the vector Helmholtz equation cannot be separated. But calculational challenges aside, the physical significance of self-energy remains elusive.
The exception to this objection is provided by gravity. Gravity couples to the local energy-momentum or stress tensor, and, in the leading quantum approximation, it is the vacuum expectation value of the stress tensor that provides the source term in Einstein’s equations. Self energies should therefore in principle be observable. This is largely uncharted territory, except in the instance of the classic situation of parallel plates. There, after a bit of initial confusion, it has now been established that the divergent self-energies of each plate in a two-plate apparatus, as well as the mutual Casimir energy due to both plates, gravitates according to the equivalence principle, so that indeed it is consistent to absorb the divergent self-energies of each plate into the gravitational and inertial mass of each Fulling:2007xa (); Milton:2007ar (). This should be a universal feature.
In this paper, for pedagogical reasons, we will concentrate attention on the Casimir effect due to massless scalar field fluctuations, where the potentials are described by -function potentials, so-called semitransparent boundaries. In the limit as the coupling to these potentials becomes infinitely strong, this imposes Dirichlet boundary conditions. At least in some cases, Neumann boundary conditions can be achieved by the strong coupling limit of the derivative of -function potentials. So we can, for planes, spheres, and circular cylinders, recover in this way the results for electromagnetic field fluctuations imposed by perfectly conducting boundaries. Since the mutual interaction between distinct semitransparent bodies have been described in detail elsewhere Milton:2007gy (); Milton:2007wz (); Wagner:2008qq (), we will, as implied above, concentrate on the self-interaction issues.
A summary of what is known for spheres and circular cylinders is given in Table 1.
|EM||Boyer:1968uf () DeRaad:1981hb ()|
|D||Bender:1994zr ()gosrom ()|
|Brevik:1998zs ()Cavero-Pelaez:2004xp ()|
|Klich:1999df ()nesterenko ()|
|Kitson:2005kk ()kitsonromeo ()|
|Milton:2002vm ()CaveroPelaez:2006rt ()|
2 Casimir Effect Between Parallel Plates: A -Potential Derivation
In this section, we will rederive the classic Casimir result for the force between parallel conducting plates casimir (). Since the usual Green’s function derivation may be found in monographs miltonbook (), and was for example reviewed in connection with current controversies over finiteness of Casimir energies Milton:2002vm (), we will here present a different approach, based on -function potentials, which in the limit of strong coupling reduce to the appropriate Dirichlet or Robin boundary conditions of a perfectly conducting surface, as appropriate to TE and TM modes, respectively. Such potentials were first considered by the Leipzig group hennig (); bkv (), but more recently have been the focus of the program of the MIT group graham (); graham2 (); Graham:2002fw (); Graham:2003ib (). The discussion here is based on a paper by the author Milton:2004vy (). (See also Milton:2004ya ().) (A multiple scattering approach to this problem has also been given in Milton:2007wz ().)
We consider a massive scalar field (mass ) interacting with two -function potentials, one at and one at , which has an interaction Lagrange density
where the positive coupling constants and have dimensions of mass. In the limit as both couplings become infinite, these potentials enforce Dirichlet boundary conditions at the two points:
The Casimir energy for this situation may be computed in terms of the Green’s function ,
which has a time Fourier transform,
Actually, this is a somewhat symbolic expression, for the Feynman Green’s function (3) implies that the frequency contour of integration here must pass below the singularities in on the negative real axis, and above those on the positive real axis kantowski (); Brevik:2000hk (). Because we have translational invariance in the two directions parallel to the plates, we have a Fourier transform in those directions as well:
The reduced Green’s function in (5) in turn satisfies
This equation is easily solved, with the result
for both fields inside, , while if both field points are outside, ,
Here, the denominator is
Note that in the strong coupling limit we recover the familiar results, for example, inside
Here , denote the greater, lesser, of . Evidently, this Green’s function vanishes at and at .
Let us henceforward consider , since otherwise there are no long-range forces. (There is no nonrelativistic Casimir effect.) We can now calculate the force on one of the -function plates by calculating the discontinuity of the stress tensor, obtained from the Green’s function (3) by
Writing a reduced stress tensor by
we find inside, just to the left of the plate at ,
From this we must subtract the stress just to the right of the plate at , obtained from (7b), which turns out to be in the massless limit
which just cancels the 1 in braces in (12b). Thus the pressure on the plate at due to the quantum fluctuations in the scalar field is given by the simple, finite expression
while for large it approaches half of Casimir’s result casimir () for perfectly conducting parallel plates,
We can also compute the energy density. Integrating the energy density over all space should give rise to the total energy. Indeed, the above result may be easily derived from the following expression for the total energy,
where the first term is regarded as an irrelevant constant ( is constant so the can be scaled out), and the second term coincides with the massless limit of the energy first found by Bordag et al. hennig (), and given in Graham:2003ib (); Weigel:2003tp (). When differentiated with respect to , (17), with fixed, yields the pressure (14). (We will see below that the divergent constant describe the self-energies of the two plates.)
If, however, we integrate the interior and exterior energy density directly, one gets a different result. The origin of this discrepancy with the naive energy is the existence of a surface contribution to the energy. To see this, we must include the potential in the stress tensor,
and then, using the equation of motion, it is immediate to see that the energy density is
so, because the first two terms here yield the last form in (16), we conclude that there is an additional contribution to the energy,
where the derivative is taken at the boundaries (here , ) in the sense of the outward normal from the region in question. When this surface term is taken into account the extra terms incorporated in (17) are supplied. The integrated formula (16) automatically builds in this surface contribution, as the implicit surface term in the integration by parts. That is,
(These terms are slightly unfamiliar because they do not arise in cases of Neumann or Dirichlet boundary conditions.) See Fulling Fulling:2003zx () for further discussion. That the surface energy of an interface arises from the volume energy of a smoothed interface is demonstrated in Milton:2004vy (), and elaborated in Sect. 2.2.
In the limit of strong coupling, we obtain
which is exactly one-half the energy found by Casimir for perfectly conducting plates casimir (). Evidently, in this case, the TE modes (calculated here) and the TM modes (calculated in the following subsection) give equal contributions.
2.1 TM Modes
To verify this last claim, we solve a similar problem with boundary conditions that the derivative of is continuous at and ,
but the function itself is discontinuous,
and similarly at . (Here the coupling has dimensions of length.) These boundary conditions reduce, in the limit of strong coupling, to Neumann boundary conditions on the planes, appropriate to electromagnetic TM modes:
It is completely straightforward to work out the reduced Green’s function in this case. When both points are between the planes, ,
while if both points are outside the planes, ,
where the denominator is
It is easy to check that in the strong-coupling limit, the appropriate Neumann boundary condition (23c) is recovered. For example, in the interior region, ,
Now we can compute the pressure on the plane by computing the component of the stress tensor, which is given by (12a), so we find
and the flux of momentum deposited in the plane is
and then by integrating over frequency and transverse momentum we obtain the pressure:
In the limit of weak coupling, this behaves as follows:
which is to be compared with (15a). In strong coupling, on the other hand, it has precisely the same limit as the TE contribution, (15b), which confirms the expectation given at the end of the previous subsection. Graphs of the two functions are given in Fig. 2.1.
For calibration purposes we give the Casimir pressure in practical units between ideal perfectly conducting parallel plates at zero temperature:
2.2 Self-energy of Boundary Layer
Here we show that the divergent self-energy of a single plate, half the divergent term in (17), can be interpreted as the energy associated with the boundary layer. We do this in a simple context by considering a scalar field interacting with the background
where the background field expands the meaning of the function,
with the property that . The reduced Green’s function satisfies
This may be easily solved in the region of the slab, ,
Here , and
Let us proceed here with more generality, and consider the stress tensor with an arbitrary conformal term ccj (),
in dimensions, being the number of transverse dimensions, and is an arbitrary parameter, sometimes called the conformal parameter. Applying the corresponding differential operator to the Green’s function (35), introducing polar coordinates in the plane, with , , and
we get the following form for the energy density within the slab.
We can also calculate the energy density on the other side of the boundary, from the Green’s function for ,
and the corresponding energy density is given by
which vanishes if the conformal value of is used. An identical contribution comes from the region .
Integrating over all space gives the vacuum energy of the slab
Note that the conformal term does not contribute to the total energy. If we now take the limit and so that , we immediately obtain the self-energy of a single -function plate:
which for precisely coincides with one-half the constant term in (17).
There is no surface term in the total Casimir energy as long as the slab is of finite width, because we may easily check that is continuous at the boundaries . However, if we only consider the energy internal to the slab we encounter not only the integrated energy density but a surface term from the integration by parts—see (21). It is the complement of this boundary term that gives rise to , (43), in this way of proceeding. That is, as ,
with the normal defining the surface energies pointing into the slab. This means that in this limit, the slab and surface energies coincide.
Further insight is provided by examining the local energy density. In this we follow the work of Graham and Olum Graham:2002yr (); Olum:2002ra (). From (39) we can calculate the behavior of the energy density as the boundary is approached from the inside:
For for example, this agrees with the result found in Graham:2002yr () for :
Note that, as we expect, this surface divergence vanishes for the conformal stress tensor ccj (), where . (There will be subleading divergences if .) The divergent term in the local energy density from the outside, (41), as , is just the negative of that found in (46). This is why, when the total energy is computed by integrating the energy density, it is finite for , and independent of . The divergence encountered for may be handled by renormalization of the interaction potential Graham:2002yr ().
Note, further, that for a thin slab, close to the exterior but such that the slab still appears thin, , the sum of the exterior and interior energy density divergences combine to give the energy density outside a -function potential:
for small . Although this limit might be criticized as illegitimate, this result is correct for a -function potential, and we will see that this divergence structure occurs also in spherical and cylindrical geometries, so that it is a universal surface divergence without physical significance, barring gravity.
For further discussion on surface divergences, see Sect. 3.
3 Surface and Volume Divergences
It is well known as we have just seen that in general the Casimir energy
density diverges in the neighborhood of a surface. For flat surfaces
and conformal theories (such as the conformal scalar theory considered
above Milton:2002vm (), or electromagnetism) those divergences are not
This result was given more recent rederivations in Actor:1996zj (); Milton:2002vm (). Dowker and Kennedy dowkerandkennedy () and Deutsch and Candelas deutsch () considered the local stress tensor between planes inclined at an angle , with the result, in cylindrical coordinates ,
where for a conformal scalar, with Dirichlet boundary conditions,
and for electromagnetism, with perfect conductor boundary conditions,
Although for perfectly conducting flat surfaces, the energy density is finite, for electromagnetism the individual electric and magnetic fields have divergent RMS values,
a distance above a conducting surface. However, if the surface is a dielectric, characterized by a plasma dispersion relation, these divergences are softened
The null energy condition ()
is satisfied, so that gravity still focuses light.
Graham grahamqfext (); Graham:2005cq () examined the general relativistic energy conditions required by causality. In the neighborhood of a smooth domain wall, given by a hyperbolic tangent, the energy density is always negative at large enough distances. Thus the weak energy condition is violated, as is the null energy condition (56). However, when (56) is integrated over a complete geodesic, positivity is satisfied. It is not clear if this last condition, the Averaged Null Energy Condition, is always obeyed in flat space. Certainly it is violated in curved space, but the effects always seem small, so that exotic effects such as time travel are prohibited.
However, as Deutsch and Candelas deutsch () showed many years ago, in the neighborhood of a curved surface for conformally invariant theories, diverges as , where is the distance from the surface, with a coefficient proportional to the sum of the principal curvatures of the surface. In particular they obtain the result, in the vicinity of the surface,
and obtain explicit expressions for the coefficient tensors and in terms of the extrinsic curvature of the boundary.
For example, for the case of a sphere, the leading surface divergence has the form, for conformal fields, for ,
in spherical polar coordinates, where the constant is for a scalar field satisfying Dirichlet boundary conditions, or for the electromagnetic field satisfying perfect conductor boundary conditions. Note that (58) is properly traceless. The cubic divergence in the energy density near the surface translates into the quadratic divergence in the energy found for a conducting ball miltonballs (). The corresponding quadratic divergence in the stress corresponds to the absence of the cubic divergence in .
This is all completely sensible. However, in their paper Deutsch and Candelas deutsch () expressed a certain skepticism about the validity of the result of mildersch () for the spherical shell case (described in part in Sect. 4.2) where the divergences cancel. That skepticism was reinforced in a later paper by Candelas candelas (), who criticized the authors of mildersch () for omitting function terms, and constants in the energy. These objections seem utterly without merit. In a later critical paper by the same author candelas2 (), it was asserted that errors were made, rather than a conscious removal of unphysical divergences.
where and are the principal curvatures of the surface.
The question is
to what extent are they observable. After all, as has been shown
in miltonbook (); Milton:2002vm () and in Sect. 2.2,
we can drastically change the local structure
of the vacuum expectation value of the energy-momentum tensor
in the neighborhood of flat plates by merely
exploiting the ambiguity in the definition of that tensor, yet each
yields the same finite, observable (and observed!) energy of interaction
between the plates. For curved boundaries, much the same is true.
A priori, we do not know which energy-momentum tensor to employ,
and the local vacuum-fluctuation energy density is to a large extent
meaningless. It is the global energy, or the force between distinct bodies,
that has an unambiguous value. It is the belief of the author that divergences
in the energy which go like a power of the cutoff are probably unobservable,
being subsumed in the properties of matter. Moreover, the
coefficients of the divergent terms depend on the regularization scheme.
Logarithmic divergences, of course, are of another class bkv ().
Dramatic cancellations of these curvature terms can occur. It might
be thought that the reason a finite result was found for the Casimir
energy of a perfectly conducting spherical
shell Boyer:1968uf (); balian (); mildersch () is that the term involving the squared difference of curvatures
in (59) is zero only in that case. However,
it has been shown that at least for
the case of electromagnetism the corresponding term
is not present (or has a vanishing coefficient) for an arbitrary smooth
cavity Bernasconi (), and so the Casimir energy for a perfectly conducting
ellipsoid of revolution, for example, is finite.
4 Casimir Forces on Spheres via -Function Potentials
This section is an adaptation and an extension of calculations presented in Milton:2004vy (); Milton:2004ya (). This investigation was carried out in response to the program of the MIT group graham (); graham2 (); Graham:2002fw (); Graham:2003ib (); Weigel:2003tp (). They first rediscovered irremovable divergences in the Casimir energy for a circle in dimensions first discovered by Sen sen (); sen2 (), but then found divergences in the case of a spherical surface, thereby casting doubt on the validity of the Boyer calculation Boyer:1968uf (). Some of their results, as we shall see, are spurious, and the rest are well known bkv (). However, their work has been valuable in sparking new investigations of the problems of surface energies and divergences.
We now carry out the calculation we presented in Sect. 2 in three spatial dimensions, with a radially symmetric background
which would correspond to a Dirichlet shell in the limit . The scaling of the coupling, which here has dimensions of length, is demanded by the requirement that the spatial integral of the potential be independent of . The time-Fourier transformed Green’s function satisfies the equation ()
We write in terms of a reduced Green’s function
We solve this in terms of modified Bessel functions, , , where , which satisfy the Wronskian condition
The solution to (63) is obtained by requiring continuity of at each singularity, at and , and the appropriate discontinuity of the derivative. Inside the sphere we then find ()
Here we have introduced the modified Riccati-Bessel functions,
Note that (65) reduces to the expected Dirichlet result, vanishing as , in the limit of strong coupling:
When both points are outside the sphere, , we obtain a similar result:
which similarly reduces to the expected result as .
Now we want to get the radial-radial component of the stress tensor to extract the pressure on the sphere, which is obtained by applying the operator
to the Green’s function, where in the last term we have averaged over the surface of the sphere. Alternatively, we could notice that Cavero-Pelaez:15kq ()
where is the angle between the two directions. In this way we find, from the discontinuity of across the surface, the net stress
The same result can be deduced by computing the total energy (16). The free Green’s function, the first term in (65) or (68), evidently makes no significant contribution to the energy, for it gives a term independent of the radius of the sphere, , so we omit it. The remaining radial integrals are simply
Then using the Wronskian (64), we find that the Casimir energy is
If we differentiate with respect to we immediately recover the force (71). This expression, upon integration by parts, coincides with that given by Barton barton03 (), and was first analyzed in detail by Scandurra Scandurra:1998xa (). This result has also been rederived using the multiple-scattering formalism Milton:2007wz (). For strong coupling, it reduces to the well-known expression for the Casimir energy of a massless scalar field inside and outside a sphere upon which Dirichlet boundary conditions are imposed, that is, that the field must vanish at :
because multiplying the argument of the logarithm by a power of is without effect, corresponding to a contact term. Details of the evaluation of (74) are given in Milton:2002vm (), and will be considered in Sect. 4.2 below. (See also benmil (); lesed1 (); lesed2 ().)
The opposite limit is of interest here. The expansion of the logarithm is immediate for small . The first term, of order , is evidently divergent, but irrelevant, since that may be removed by renormalization of the tadpole graph. In contradistinction to the claim of graham2 (); Graham:2002fw (); Graham:2003ib (); Weigel:2003tp (), the order term is finite, as established in Milton:2002vm (). That term is
The sum on can be carried out using a trick due to Klich klich (): The sum rule