#### Abstract

The low temperature expansion of the free energy in a Casimir effect setup is considered in detail. The starting point is the Lifshitz formula in Matsubara representation and the basic method is its reformulation using the Abel-Plana formula making full use of the analytic properties. This provides a unified description of specific models. We rederive the known results and, in a number of cases, we are able to go beyond. We also discuss the cases with dissipation. It is an aim of the paper to give a coherent exposition of the asymptotic expansions for . The paper includes the derivations and should provide a self-contained representation.

#### 1. Introduction

The Lifshitz formula is the basic tool for the calculation of van der Waals and Casimir forces between two material half spaces. It emerged in 1956 for the description of the electromagnetic dispersion forces. Together with Casimir’s approach of zero-point or vacuum fluctuations these are two sides of one coin. In the language of quantum field theory these are one-loop corrections to a classical background which may be given by boundary conditions or by classical fields as well.

For the configuration of two parallel interfaces with a gap of widths between them (see Figure 1), the Lifshitz formula provides the separation dependent part of the free energy of the electromagnetic field at temperature in terms of the reflection coefficients of the two interfaces. In (1), the prime on the sum indicates that the -contribution must be taken with a factor . Initially this formula was written for dielectric half spaces with permittivity behind the interfaces with the well known reflection coefficients which must be inserted for according to the polarization. In (1), the integration is over the wave numbers in the directions in parallel with the interfaces; the summation is over the Matsubara frequencies and the notations are used. Here and in (1) the permittivity is allowed to be frequency dependent, , and must be taken at imaginary frequency, .

The Lifshitz formula (1) is basic for the calculation of the force acting between the interfaces (actually the pressure since (1) is the free energy per unit surface), At present, this force can be compared with results from force measurements on a high level of precision; see [1] or the recent review [2]. Also, the Lifshitz formula was generalized to arbitrary, nonflat geometry of the interfaces in terms of the scattering approach; see, for example, Chapter 10 in [1].

Along with the great success, it must be mentioned that a decade ago a problem with the Lifshitz formula appeared when including dissipation. For instance, when inserting the permittivity of the Drude model, with a temperature dependent dissipation parameter , decreasing for sufficiently fast (see (172)), the free energy (5) violates the third law of thermodynamics (Nernst’s heat theorem) [3]. This means that the separation dependent part of the entropy, has a nonvanishing limit for . The physical interpretation of this problem is still under discussion. Another closely related problem appears when decreasing the dissipation parameter at fixed temperature. The naive expectation would be to reobtain in the limit the free energy of the plasma model. While the permittivity (6) turns into that of the plasma model, the free energy has an additional contribution for ; that is, it is not perturbative in . This is counterintuitive from the physical point of view since a small dissipation should have a small effect on the dispersion force. Also, there is growing evidence for a disagreement between the predictions following from the Lifshitz formula with dissipation and measurements [2, 4–7].

At finite temperature, the vacuum fluctuations appear together with the thermal fluctuations of the electromagnetic field. For the latter, one defines a characteristic temperature, which, at room temperature, corresponds to a separation . The free energy depends typically on a dimensionless combination, It is, for most measurements, a small parameter making the low temperature regime relevant.

At low temperature, the Matsubara sum in (5) becomes inefficient since high give significant contributions. Instead, one uses the Abel-Plana formula. For a sequence integer of numbers, the representation holds, where is the analytic continuation to the complex -plane with for integer. Thereby it is assumed that the initial sum converges and that does not have poles or branch points for . In case the derivatives exist, the right hand side of (10) has an expansion, which, if applied to (5), gives the expansion for . Regrettably, in a number of cases, especially those related to the Drude model, these derivatives do not exist and a more elaborate treatment is needed. A detailed discussion is given in the beginning of Section 3. It should be mentioned that (11) was used in the early calculations of the Casimir force, for instance in [8], to perform the limit of vanishing regularization parameter.

Using the Abel-Plana formula (10), the free energy (5) can be rewritten in the form where resulting from the first integral in the right hand side of (10), is the vacuum energy, that is, the zero temperature contribution, and the second integral with is the thermal contribution. We introduced the notation which will be used throughout this paper. In (13) the integration is over imaginary frequencies whereas in (14) it is over real frequencies and the integrand involves the Boltzmann factor .

Using representation (10) or (11), the low temperature expansion was calculated in nearly all cases of interest at least in the leading order. In the present paper we give a detailed representation of this expansion in all cases of interest based on (14). We make full use of the analytic properties of the function , (16), and are able to improve a number of known expansions. Using this method we also reconsider the derivation of the contributions violating the third law of thermodynamics as well as the nonperturbative contribution appearing for vanishing relaxation parameter.

It is the aim of the present paper to review the low temperature expansion to the free energy for the basic models and represent them unified in the framework of (14). Thereby we consider the* asymptotic* expansion of the free energy as given by the Lifshitz formula, (1), for . This means that we do not consider any corrections which are exponentially small and we also do not discuss the applicability of the expansion to the one or the other situation. On the other side, we try to cover all relevant models and try to give a self-contained representation which includes all derivations. It should enable the reader to follow all calculations nearly without consulting other sources. Therefore, the experienced reader may find some places too much going into detail for which the author asks for indulgences. A part of the calculations is made machinized, using a standard tool. We tried to explain all steps in such detail that the reader should be able to repeat the calculations easily.

As for the models considered, these cover the most frequently used for the Casimir effect between real material bodies. We do not discuss their ranges of applicability. In this sense, the asymptotic expansions for low considered in this paper must be understood primarily as properties of these models. Also, we do not consider all models. The model describing a metal by impedance boundary conditions and the model describing graphene by the Dirac model are not considered here.

In the next section we collect the basic formulas for the free energy and, in a number of subsections, the specific models we are going to consider. These are, after shortly recapitulating of the ideal conductor case, a dielectric with fixed permittivity as the simplest example for a medium. Next is the plasma model as the simplest model describing a metal beyond the ideal conductor. Then we add dissipation by considering a Drude model permittivity. The next subsection is devoted to an insulator, followed by a subsection for dielectric with dc conductivity as another example for dissipation. Finally we consider the hydrodynamic model for graphene. In Section 3 we derive the low temperature expansion and the specific representation we are using. In Section 4 we derive the low frequency expansions for the specific models. In Section 5 we collect the low temperature expansions for all models. Conclusions are given in the last section. Some calculations are presented in the appendices.

Following the theoretical approach of this paper we use units with . Throughout the paper denotes the Riemann zeta function and is Euler’s constant.

#### 2. Basic Formulas and Models

In this paper, the basic formula is the Lifshitz formula, mentioned already in the introduction. We consider two plane parallel interfaces perpendicular to the -axis, with an empty gap of widths between them. On the interfaces, boundary or matching conditions are assumed to be given or the space behind the interfaces is assumed to be filled with a homogeneous medium or to be empty in case of graphene. Also we assume homogeneity and isotropy in any plane parallel to the interfaces. We denote the Lifshitz formula in the form with We allow for different reflection coefficients and on the two interfaces. One has to insert according to the model considered and to perform the summation over the polarizations in case of the electromagnetic field. In general, (17) is valid for any field when inserting the corresponding reflection coefficients in (18). This formula represents the free energy per unit area of an interface. The summation is over the Matsubara frequencies (3) and for frequency dependent reflection coefficients one has to use their analytic continuation to .

The Lifshitz formula was originally derived from the fluctuations of the electromagnetic field in the gap introducing a random field into the Maxwell equations. At once it was mentioned that this field is associated with the “zero point vibrations” [9]. Indeed, writing the vacuum energy (13) in the form it is seen that this is just the half sum of the excitation of the electromagnetic field in the sense introduced by Casimir in [8] after Wick rotation. After that, the step from (19) to (17) follows simply by applying the Matsubara formalism.

In (18), as compared with (16), we dropped a factor , This is equivalent to putting in all formulas. In fact, this is no restriction since this exponential is the only place where the separation enters the Lifshitz formula. Formally, this can be achieved by the substitution . The dependence on can be restored at any time by dimensional consideration. Since the free energy is a density per unit area, its dependence is restored by and all dimensional quantities entering must be made dimensionless by the factor . For instance, one has to substitute the temperature by and any frequency, like the plasma frequency, or dissipation parameter, or , by In fact, there are no other parameter to be restored in this paper. At this place we also mention how to restore the fundamental constants. For the free energy, in place of (21), one has to substitute (this is an energy divided by an area) and with to be measured in Kelvin and in .

In the configuration of two parallel interfaces, considered in this paper, the polarizations of the electromagnetic field always separate and are commonly chosen as transverse electric (TE) and transverse magnetic (TM) modes. The corresponding scalar amplitudes, , satisfy the Maxwell equations and, on the interfaces, boundary or matching conditions. These amplitudes can be chosen as linear combinations of plane waves, with a dependence on the coordinate in perpendicular to the interfaces, Here, is the frequency, are the wave vectors, respectively, momenta (since we have ), in directions parallel to the interfaces, and , respectively, , are the wave vectors in perpendicular direction outside, respectively, inside, the gap. The dispersion relations, follow from inserting (26) into the Maxwell equations. Here, the permittivity is allowed to depend on the frequency and we use the notation . For nontransparent boundary conditions like Dirichlet or ideal conductor conditions, the waves outside the gap do not contribute to the free energy (17). For transparent boundary conditions like in the hydrodynamic model one has to put outside the gap and the corresponding momenta are equal, .

After Wick rotation, one has always an imaginary wave vector inside the gap. The wave vector outside the gap may remain real or it may become imaginary, The dispersion relations turn into We will use notations (30)–(32) throughout the paper.

In the Lifshitz formula, different choices of the integration variables are possible. In (17), these are and . In this case, one needs to express all other in terms of these, Below, the integration over in (18) will be changed for , In that case one has to express with the range .

As mentioned above, the properties of the interacting bodies enter the Lifshitz formula only through the reflection coefficients. For the interface at , that is, for the right one if looking on Figure 1, the corresponding mode function is where and are the reflection and transmission coefficients. These are to be determined from the boundary or matching conditions in . The reflection coefficient , determined this way, must be inserted for in (18). The reflection coefficient follows, accordingly, from the scattering from the right on the interface in with the choice for the mode function. We mention that the reflection coefficients and are independent of one another. The Lifshitz formula stays correct for any combinations. In case of nonphysical choices, like one from the TE and the other from the TM polarization, there would be, however, no physical realization for.

As defined by (36) and (37) together with (26) mode, the reflection coefficients are functions of real , whereas and may be real or imaginary. For real , the function describes scattering states and for imaginary both, and , these are surface modes (for more details see [10]).

In the remaining part of this section we specify the models which we will consider.

##### 2.1. Ideal Conductor

For ideal conducting surfaces, the boundary conditions are Dirichlet for the TE polarization and Neumann for the TM polarization. The reflection coefficients are resulting in equal contributions from the two polarizations to the free energy. In this case, the mode functions terminate on the interfaces. Equivalently, one may put in (36) and (37).

##### 2.2. Fixed Permittivity

For two dielectric half spaces with fixed permittivity , the reflection coefficients are in terms of the real wave numbers related to (28) with the frequency . In terms of the imaginary wave numbers we note and the relation (32) applies, including In this form, the reflection coefficients enter the Lifshitz formula (17) through formula (34).

The reflection coefficients (38) of ideal conducting interfaces follow from the above with (39) or (40) in the limit . However, this does not imply that the free energy behaves the same way. This can be seen already in representation (17). Consider , that is, the lowest contribution to the Matsubara sum. From (3) we have and with (41) we note in this case and the reflection coefficients become Hence, in the limit , the -contribution of the TE polarization does not deliver any contribution to the free energy while all other contributions deliver the corresponding ideal conductor contributions. This behavior was first observed in [11] and motivated “Schwinger’s prescription” to take the limit before putting . Also, the question on whether this single mode can influence the result much can be answered quite easily. In the high temperature limit, the -contribution delivers the leading order contribution and with the vanishing TE contribution half of the result is missing. The physics behind this behavior is transparent. A fixed permittivity implies a dielectric material keeping its properties at all, including highest frequencies, which, of course, does not happen in physics.

##### 2.3. The Plasma Model

The plasma model appears if one considers the whole space, or a half space behind an interface, being filled with a charged fluid (e.g., electrons) coupled to the electromagnetic field while the half space before the interface is empty. Eliminating the dynamical variables of the fluid from the equations of motion (or, in a functional integral approach, integrating them out), one comes to the same reflections coefficients (39) or (40) as above where one has to insert the frequency dependent permittivity of the plasma model,
The causality of this permittivity was shown in [12]. Here is the so-called* plasma frequency* and from (35) or (41) we note
The spectrum, that is, the mode content, of this model is well known. A recent discussion in the context of vacuum energy was given in [10] and here we mention only that it is the same as for fixed permittivity with, in addition, a surface mode in the TM polarization. This is a mode with real frequency , propagating on the interface and decaying exponentially on the vacuum side of the interface.

The plasma model describes some basic properties of the electrons in a metal. Typical values of the plasma frequency are approximately equal 8-9 eV. Its inverse is the skin depths The reflection coefficients for ideal conducting boundary conditions can be obtained from (43) in the limit . Also the free energy of ideal conductors is recovered in this limit due to the sufficiently fast decrease of the permittivity for large frequencies.

##### 2.4. The Drude Model

This model is an extension of the plasma model allowing for dissipation. Physical reasons may be Ohmic losses or scattering of the electrons on the lattice or on impurities. These are accounted for by a phenomenological dissipation parameter , entering the permittivity, of the model. In the formal limit one recovers the permittivity , (43), of the plasma model. The corresponding free energy does not follow in this limit; see (176).

At the moment it is not clear whether the use of the Drude permittivity in the Lifshitz formula gives correct results or not [1, 2]. Since we do not enter this discussion in the present paper, we take this model as is and make only a few comments.

The permittivity (46) is complex. Being inserted into the Maxwell equations (28), a nonvanishing imaginary part of the frequency results. For , which one needs to assume, this describes dissipation of energy. This is in accordance with the intention of the model describing losses, finally resulting in heat. The model is in accordance with causality and the permittivity; , (46), obeys the Kramers-Kronig relation. This model, taken alone, has no unitarity and a mode expansion of the free energy in terms of real frequencies is not possible [13]. In line with this, it must be mentioned that after Wick rotation, which is possible for , the permittivity is real delivering with (17) a real free energy. In this way one obtains an easy-to-use formula. Derivations of this procedure were given in [14] and recently discussed, for example, in [15].

##### 2.5. Insulator Described by Oscillator Model

The response of insulators to the electromagnetic excitations is, beyond a fixed permittivity, frequently described by the permittivity of an -oscillator model, where are the oscillator frequencies, their strengths, and their damping parameters. Here one excludes the case since that would rather be described by a plasma or Drude model. In the low frequency limit, , one comes to a constant permittivity, as considered in Section 2.2.

It must be mentioned that (48) includes also dissipation processes like the Drude model. However, the free energy calculated from (17) is real and for all nonvanishing oscillator frequencies, , this model is not known to have problems [1].

##### 2.6. The Case of dc Conductivity

At nonzero temperature, dielectrics possess, as a rule, some conductivity due to dissipation processes like in the Drude model. These are, in the simplest case, accounted for by an additional contribution to the permittivity , (48), where is the static conductivity resulting in a dc current. Being inserted into the Lifshitz formula, (17), this model results in a real free energy. This conductivity is typically a function of temperature, , vanishing at . This model has problems similar to that in the Drude model mentioned in Section 2.4 [1]. Below we consider for completeness also the case of a fixed , although it may be physically less interesting.

##### 2.7. Hydrodynamic Model for Graphene

In the hydrodynamic model one assumes a charged fluid, like in the plasma model, but confined to a plane, that is, being two-dimensional. Again, eliminating the dynamical variables of the fluid, the Maxwell equations appear and the field strengths obey matching conditions on the pane. These result in reflection coefficients In the mode expansions (36) and (37) one has to put since from both sides of an interface we have empty space. Accordingly, from the Maxwell equations (28), only the second applies.

The matching condition for the TE mode is equivalent to a scalar field with a repulsive delta function potential on the interface obeying the wave equation For the TM mode, the corresponding scalar problem can be formulated in terms of a -potential.

The mode content of this model is quite similar to that of the plasma model considered in Section 2.3. For instance, there is, for each interface, a surface mode. In the limit of infinite plasma frequency, , the reflection coefficients turn into that of an ideal conductor, (38), and the free energy of this model turns into that of ideal conductors.

This model was first considered in [16]. In [17] and subsequent papers it was used to describe the -electrons of graphene and . It provides a quite good description of their properties in interaction with electromagnetic fields at large frequencies. For small frequencies the Dirac model [18–20] provides a better description.

#### 3. Low Temperature Expansion for the Free Energy

We take the Lifshitz formula in Matsubara representation, (17), as starting point for the low temperature expansion. The convergence of the sum in (17) and of the integration over in (18) comes from the exponential factor (we restored, for a moment, the dependence on the gap’s width ) making especially the sum over fast convergent. This picture changes with decreasing temperature since enters through the Matsubara frequency , (3). Obviously, for becoming large, numbers must be accounted for. Thus, for decreasing , the convergence slows down and (17) becomes, in the limit, unusable.

A way out can be found if an analytic continuation of to noninteger, in general complex, can be found. This gives the possibility of defining , (18), as a function in the complex -plane and, using the Cauchy theorem, of representing the Matsubara sum in (17) as an integral,
Here the path encircles the nonnegative integers, , and crosses the real axis in with . The next step is a deformation of the integration path towards the imaginary axis. For , that is, on the upper half of the path, one substitutes
with and the exponential in the denominator becomes large for . For , that is, on the lower half of the path, one substitutes
Since, in this case, the exponential does not grow for large , one needs to rewrite it,
In the contribution from the first term on the right hand side it is meaningful to change the integration variable according to and to write down this contribution separately. The integration can go along the real axis since there are no poles in this contribution. The second term can be joined with the contribution from the upper half of the path. In both cases runs from zero till infinity. One comes to the representation
This is the well-known* Abel-Plana formula*.

In moving the integration path towards the imaginary axis and performing the limit , from the origin, that is, from , a contribution appeared which just cancels the first term in the right hand side of (54). We mention that in (58) there is no pole for due to the compensation in the parentheses. Further we mention that in (58) it is assumed that the function is continuous in . If this is not the case, one cannot move the path completely to the imaginary axis. However, such situation does not appear in the examples considered in this paper.

A further assumption in deriving (58) concerns the function . It is assumed that it does not have poles or branch points in the half plane . Otherwise, from moving the path there would be additional contributions. This property is always guaranteed if the modes of the electromagnetic field are subject to an elliptic scattering problem. It holds also for the model with dissipation where the corresponding poles are all located in the half plane . For vanishing dissipation parameter, these poles move towards the imaginary axis from the left and the path must pass them on the right side, for instance, by adding an infinitesimal amount, which is necessary for all models anyway.

In application to the free energy (17), (58) defines a split, into the vacuum energy (19), resulting from the first term in the right hand side of (58), and, from the second term, the temperature dependent part, involving the Boltzmann factor and The integration variable has the meaning of a frequency like that entering (28) provided a mode expansion makes sense. As already mentioned, this is not the case for models with dissipation (see the remark at the end of this section). However, independently on the interpretation, representation (60) with (62) and the property (63) are valid for these too.

At this place, an important remark on the direction of the contour rotations is in order. The rotation (56) is the inverse of the usual Wick rotation (29), whereas (55) is the inverse of an Anti-Wick rotation. Since it is customary to write the Abel-Plana formula just with the order of terms as in the parentheses in (58) with first, in application to the free energy (63), the term corresponding to the inverse Anti-Wick rotation goes first. Of course, this does not change anything except for notations.

Below we will find it convenient, in a number of occasions, especially after some variable substitutions, to use the reflection property, this function with reflection coefficients following from a scattering problem has to represent the difference in (63) in the form where . denotes the complex conjugate of what is in front to be inserted. In doing so one has only to pay attention to signs in some places, especially in the permittivity, which changes under Wick rotation , but which enters after Anti-Wick rotation, This sign shows up in models with dissipation only.

In some simple cases it is possible to use the Abel-Plana formula, formally not entering the complex plane. Assume the function has a Taylor series expansion, one gets for , (63), an expansion directly in terms of real quantities. This can be inserted into (62). Interchanging the orders of integration and summation, the integration can be carried out. One obtains which is known as Euler-Maclaurin summation formula, with the Riemann zeta function in even integers, in terms of the Bernoulli numbers . Obviously, this is an expansion for . For most systems, however, the function does not have a Taylor expansion. Typically, in the examples considered in this paper, exists, but not the derivatives. Nevertheless, even in the case there is no Taylor expansion, the derivatives, as far as they exist, give with (68) the lowest contributions to the asymptotic expansion for .

We add a remark on the convergence of the vacuum and the free energies. In general, the vacuum energy and with it also the free energy have ultraviolet divergences resulting from slow convergence for large frequencies or momenta. In the situation of a Casimir effect setup, considered here, these divergences do not depend on the width of the gap and the Casimir force is always finite. The split (60) is, of course, valid beyond the Casimir effect setup. In that case the divergences in the free and in the vacuum energies are the same and the thermal part , (62), does not have any divergences. Its convergence follows, obviously, from the Boltzmann factor, whereas before the contour rotation, that is, in (54), this factor is bounded. It is just the contour rotation, which, without changing the integral, redistributes for small the main contribution to the integral towards small . As a result, for , the integral over is fast convergent and the contributions from determine the asymptotic expansion for low . This is opposite to the situation in the initial Matsubara representation where large imaginary frequencies were needed for.

At this place we mention the Poisson resummation formula which is yet another way to redistribute the convergence. In order to use that formula either one has an explicit representation, typically a Gaussian exponential, or one needs to make an analytic continuation from integer to, at least, real ones. One obtains, in place of the Matsubara sum, another sum, which is fast converging for small . We do not use this approach in the present paper.

Due to the convergence properties just discussed, (62) allows, in a simple way, for the low temperature expansion of . For this, it is sufficient to assume the function , (63), has an asymptotic expansion for , Here we allowed for a logarithmic contribution in the first order and for half-integer orders since these will appear below in the Drude model (Section 4.4) and for the insulator (Section 4.5). If the low frequency expansion (70) is found, the asymptotic expansion, for , of the free energy can be easily written down by inserting (70) into (62) and using for the integration over . For logarithmic contributions one may take the derivative of this formula with respect to . In this way one comes to By using the explicit values (69) and multiplying out the square bracket, this formula can be rewritten as It should be mentioned that expansion (72) can be derived for any model if does not depend on . It starts always at least from For this reason, it cannot come in contradiction with thermodynamics. However, in case of the Drude model with a dissipation parameter vanishing for , the expansion (72) is incomplete as discussed in detail in Section 4.4.2. It should be mentioned that expansion (68) is a special case of (72) since it has only odd powers of and no logarithmic contributions.

With representation (72), respectively, (73) at hand, the “remaining task” is the calculation of the coefficients . For this we return to (18), assuming the analytic continuation to the complex -plane is done. Following from (28) and (32), the variables , , and are related to It is possible to change the integration in (74) from to , which runs from till infinity parallel to the real axis. In Figure 2 this integration path is denoted by . In deriving the expansion for the various models in the next section, it turns out to be convenient to change the integration path for the sum of two, one running from to and the other from along the real axis till infinity. These two are shown in Figure 2 as and . Of course, the integral does not change. In representation (74) with integration over , this corresponds to a subdivision of the integration region into two regions, where, at once, wave numbers are shown which are real in the given region, as it follows from (28) and (32). As a convention, we will mark all quantities calculated in these regions correspondingly by subscripts (a) or (b).

According to (76), the integral in (74) splits into two, which will be treated separately. With (63), this induces a corresponding split and the relations In region (a) we have with shown in (76). The reflection coefficients must be expressed in terms of and . For instance, for the wave number we note which may be both imaginary or real in dependence on the model. Changing for the integration variable , (80) can be written in the form where one needs to express everything in terms of and ; for instance and . In this formula, the are the reflection coefficients for scattering states.

In region (b) we have In the second line we changed the integration variable for using (76). This integration corresponds to the path in Figure 2. The reflection coefficients entering the second line must be expressed in terms of and . For instance, from (32) we note In (82), the are the reflection coefficients analytically continued into region . We remind that we keep the relations and in all calculation.

#### 4. The Low Frequency Expansion for Specific Models

In this section we obtain the low frequency expansions (70) of the function for various models. This section comprises the main technical part of the paper. Some calculations are banned to the appendices. As mentioned in the Introduction, we use the simplest form of notations; we especially drop the factor everywhere as announced.

##### 4.1. Ideal Conductor

This is the simplest model and is well known. We consider it for completeness. At once we illustrate the technique used, especially the division of the integration in into two regions. Also it allows for an easy checking of the overall factors.

The reflection coefficients for ideal conductors are given by (38) and their product, entering , is for both polarizations. The contribution from region (a) can be written in the form where we changed the integration variable for using (76). The logarithm can be written in the form Now, as long as , the sine does not change sign and the and the logarithm in the right side stays real. Now we calculate according to (79) the imaginary part, The remaining integration is trivial, In region (b) we note This expression is completely real. Hence it does not contribute, , and we get from (78) and (88) We mention that this formula not only provides the asymptotic expansion for but also is exact for .

From (90), in the context of (70), we have nonvanishing coefficients Inserted into (73), these deliver the expansion for ideally conducting interfaces, Here we included a factor of 2 to account for the two polarizations of the electromagnetic field. The dots represent exponentially decreasing contributions that we do not care about in the present paper.

It should be mentioned that the method used in this paper is the only one out of quite a number of equivalent ones applicable for this simple model. More details can be found, for example, in [1, chap. 7.4].

##### 4.2. Fixed Permittivity

Dielectrics with fixed permittivity represent the simplest model for an insulator. While the ideal conductor considered in the preceding subsection is in the aim of Casimir’s original idea, an insulator is rather in the spirit of Lifshitz’s approach. Here both are treated within the same formalism. Also, the model with a fixed may serve as a good approximation for more complicated permittivities at low frequency.

The reflection coefficients are given by (39), or by (40), which will be used in regions 1 and 2, accordingly. As already mentioned, the limit does not turn the free energy into that of ideal conducting interfaces. Hence, a situation with one interface ideal conducting, the other with finite permittivity behind, is different from a situation with both interfaces having finite permittivities and behind and cannot be obtained by any limiting process. For this reason we consider 4 cases as shown in Table 1 and denote the case as an index in parentheses, .

As discussed in Section 3 we will perform the calculations separately in regions (a) and (b). We add this information to the index such that denotes the contribution from region to case . We use his notation also for the functions and in the relations For a given case, the contributions from both regions must be added, In the final result for the electromagnetic field, cases 1 and 3 or cases 2 and 4 must be added.

###### 4.2.1. Region (a)

Here we use the reflection coefficients as given by (39). In (80) we change, for convenience, the integration variable for . In this case, in (39), one has to use and we get where for and one has to insert according to Table 1.

In this expression, a direct expansion of the integrand in powers of with subsequent integration over delivers the expansion of and, by means of (94), that of . In the latter only odd powers of remain. Defining expansion coefficients in parallel with (70), these can be calculated easily by machine. The nonvanishing coefficients are shown up to the order , which corresponds to an order in the expansion (73),

###### 4.2.2. Region (b)

In this region we use (83) and insert the reflection coefficients as given by (40) with , which follows from (35). Next we observe that in the part of the integration with delivers a real contribution and does not contribute in (94). Thus we restrict the integration region, In this expression is imaginary, with . Now we make the substitution and get with and for the reflection coefficients, in dependence on the polarization, one has here to insert For the expansion of the function , (112), for , we need to know the expansion of the function for .

It is not possible to expand the integrand in (101) since a subsequent integration would not converge for since for the reflection coefficients (102) the relations hold, which result in singular terms in an expansion of the logarithm in (101). We proceed by factorizing the logarithm in (101) and representing as a sum, of two functions, The functions , defined in (93), can be reobtained from these as follows: where we used , which holds for the coefficients (102).

For the function , the expansion can be obtained easily by expanding the integrand with subsequent integration. The singularities resulting from the expansion of the logarithm appear for and are compensated by factors from the expansion of the exponential. We denote this expansion in the form The coefficients up to order 4 are shown in Table 2.

For the function , which carries the above mentioned singularities when expanding the integrand, we define an auxiliary function In this function, the integration can be carried out explicitly with subsequent expansion in powers of . We use this function to represent in the form with In this function, the integrand can be expanded up to the order of with convergent subsequent integration over . In this way we obtain the expansion All these operations can be carried out by machine. The coefficients up to order 4 are shown in Table 2.

Now, using the expansions (107) and (111), we can calculate from (106) the expansion where For the four cases defined in Table 1, these expansions read

In this way, the calculation of the contributions from region 2 to the low temperature expansion is finished.

###### 4.2.3. The Low Frequency Expansion

Here we collect the contributions from the two regions calculated above. According to (95) one has to add them. It is seen that from region (a), (97), only odd powers of come, whereas from region (b), (114), even powers come in too. Together, the expansions coefficients for the four cases in Table 1 are For the electromagnetic case we have to add the polarizations. In case of two dielectric half spaces we have to add cases 2 and 4 from (115), For one interface ideal conducting in front of a dielectric half space we have to add cases 1 and 3, Inserted into (73) these coefficients give the expansion of the free energy up to ; see Section 5. As already mentioned, the limit does not turn these coefficients into that of the ideal conductor. Also, the two cases (117) and (116) are independent from one another.

##### 4.3. The Plasma Model

The plasma model is described by the reflection coefficients (39) or, equivalently, (40) and by the permittivity (43), Since in the limit of infinite plasma frequency, , the free energy turns into that of the ideal conducting interfaces, there is no need here to introduce the cases used in the preceding subsection. Instead, we use on each interface its own plasma frequency, denoted by and . The case of one ideal conducting interface can be restored afterwards by sending one of these frequencies to infinity.

For the calculations, we divide the contributions according to regions (a) and (b), defined in Section 3, (76).

###### 4.3.1. Region (a)

In this region we have a real and we use representation (82), where we have from (28) with the permittivity (43) an imaginary with a real . The reflection coefficients are given by (39), where one has to insert for the corresponding interface, which we indicate by an additional index , with . For sufficiently small , , these reflection coefficients are pure phase factors; that is, their modula are equal to unity. Thus we can rewrite them in the form with This allows to cast (119) in the form where one has to insert Like in the case of ideal conducting interfaces, (86), this simple structure allows rewriting the logarithm as As before, the remaining log does not contribute since its argument does not change sign for sufficiently small and we get with (63) In these expressions, the integrations can be carried out explicitly and we come to where result from the integrations in (126).

The expansion of these functions, can be used in (127) to obtain the expansions where we used the notation for TE and TM not to write down nearly the same formula twice. From this representation it is seen that the ideal conductor result (90) can be reobtained in the limit of both plasma frequencies becoming large.

###### 4.3.2. Region (b)

In this region we have a real and, from (41) with (118) inserted, a real . The reflection coefficients (40) are, explicitly written, In the case, the two interfaces have different plasma frequencies; we have to insert and into the corresponding reflection coefficients (40) and in These coefficients are real. Thus the function (83), seems to be real too. If this would be the case, it would not contribute to However, it happens that the argument of the logarithm in (134) changes sign not only for a finite as in the case of an ideal conductor, but also for arbitrarily small .

Physically this can be understood from the spectrum. We consider real with imaginary both momenta, inside and outside the gap. The plasma model is known to have, for the TM polarization, such excitations, the surface modes (or surface plasmons). Further, the argument of the logarithm in (133) is proportional to the transmission coefficient of the system of two interfaces; thus it has zeros at the wave numbers corresponding to these excitations at a given frequency . Moreover, the argument of the logarithm has poles corresponding to the surface plasmons on the interfaces taken individually, where their reflection coefficients become infinite.

Since there are no surface plasmons for the TE polarization, we can be sure that the corresponding logarithm in (133) stays real and that this polarization does not contribute to (134). For the TM polarization we rewrite (133) using (40), Here, the arguments of the logarithms have no poles but only zeros. We denote the wave number for the single plasmons by . These follow from the poles of ; that is, these are solutions of Using (118) and (132), these equations can be solved for resulting in We denote the wave numbers of the plasmons in the system of two interfaces by and . These notations account for the properties of the corresponding wave functions to have a symmetry in case of equal plasma frequencies. These wave numbers are solutions of the equation or, equivalently, of equating the argument of the third logarithm in (135) to zero.

It is known (see, e.g., Figure 2 in [10]) that is not real for small . Therefore it does not contribute in region (b). The solution of (138) exists for arbitrarily small and it has an expansion whose coefficients can be easily calculated by inserting (139) into (138) and expanding in powers of . One obtains In case of equal frequencies, this expression simplifies The analytic continuation in