The Casimir effect for fields with arbitrary spin
The Casimir force between two perfectly reflecting parallel plates is considered. In a recent paper we presented generalised physical boundary conditions describing perfectly reflecting parallel plates. These boundary conditions are applicable to a field possessing any spin, and include the well-known spin- and spin- boundary conditions as special cases. Here we use these general boundary conditions to show that the allowed values of energy-momentum turn out to be the same for any massless fermionic field and the same for any massless bosonic field. As a result one expects to obtain only two possible Casimir forces, one associated with fermions and the other with bosons. We explicitly verify that this is the case for the fields up to spin-. A significant implication of our work is that periodic boundary conditions cannot be applied to a fermionic field confined between two parallel plates.
In 1948 Casimir and Polder published a long, technically complex paper about the influence of retardation on the Van der Waals force Casimir and Polder (1948). The stated goal of their work was to account for discrepancies between experiments and theory concerning colloidal suspensions of large particles Verwey and Overbeek (2001). However, the work took on a whole new significance when, after discussing the results with Bohr, Casimir was inspired to try and re-explain his and Polder’s results using the relatively new idea that the quantized electromagnetic field undergoes vacuum fluctuations. A short time later, Casimir published his now-famous paper Casimir (1948) on the force of attraction between two infinite perfectly conducting parallel plates, whose presence modifies the quantized electromagnetic vacuum field. This force came to be known as the Casimir force. The calculation in Casimir (1948) reproduces the results of the much more involved calculation in Casimir and Polder (1948), but is remarkable in its simplicity and elegance, while also providing one of the very few macroscopic manifestations of quantum field theory. The Casimir force is extremely weak so was initially nothing more than a theoretical curiosity, but as experimental methods improved the effect became measurable, and this led to rapidly increasing attention from the 1970s onwards. Since then there has been a profusion of extensions of Casimir’s original work, which look into imperfectly conducting plates and different physical geometries Landau et al. (1960); Boyer (1968); Rahi et al. (2009); Schwinger et al. (1978); Philbin (2011). There have also been a number of experimental confirmations of the effect Lamoreaux (1997); Mohideen and Roy (1998); Decca et al. (2007). The existence of the Casimir effect is often cited in standard quantum field theory textbooks as the primary justification for the reality of vacuum fluctuations, though such interpretations carry some controversy Jaffe (2005).
A fluctuating vacuum is a general feature of quantum fields, of which the free Maxwell field considered in Casimir and Polder (1948); Casimir (1948); Verwey and Overbeek (2001); Landau et al. (1960); Boyer (1968); Rahi et al. (2009); Schwinger et al. (1978); Philbin (2011); Lamoreaux (1997); Mohideen and Roy (1998); Decca et al. (2007); Jaffe (2005) is but one example. Fermionic fields such as that describing the electron, also undergo vacuum fluctuations, consequently one expects to find Casimir effects associated with such fields whenever they are confined in some way. Such effects were first investigated in the context of nuclear physics, within the so-called “MIT bag model” of the nucleon Chodos et al. (1974). In the bag-model one envisages the nucleon as a collection of fermionic fields describing confined quarks. These quarks are subject to a boundary condition at the surface of the ‘bag’ that represents the nucleon’s surface. Just as in the electromagnetic case, the bag boundary condition modifies the vacuum fluctuations of the field, which results in the appearance of a Casimir force Milton (1983, 1983); Oxman et al. (2005); Milonni (1994); Fosco and Losada (2008). This force, although very weak at a macroscopic scale, can be significant on the small length scales encountered in nuclear physics. It therefore has important consequences for the physics of the bag-model nucleon Thomas and Weise (2010).
The Maxwell and Dirac fields are both spinor fields, though the former is not usually described as such. It is possible to write Maxwell’s equations in a form identical to the Dirac-Weyl equation that describes massless spin- fermions Bialynicki-Birula (1996). This naturally leads one to the question as to whether it is possible to use a spinor formalism to describe the Casimir effect for the Dirac (spin-) and the Maxwell (spin-) fields in a unified way. We have shown Stokes and Bennett (2014) that such a unification can be accomplished using the two-spinor calculus formalism introduced by Van der Waerden van der Waerden (1928). Moreover, this unification naturally lends itself to a generalization, which is applicable to confined higher-spin fields. These fields include the spin- field associated with the so-called graviton, which appears in linearized quantum gravity, and its supersymmetric partner the spin- gravitino.
In this paper we will present specific results for the Casimir force associated with the fields up to spin-. We organize our work by noting that calculations of Casimir forces broadly follow the following three steps:
The statement of one or more boundary conditions governing how the considered field behaves at material surfaces. These can be mathematically convenient (examples include Dirichlet Graham et al. (2004), Neumann Alves et al. (2003), Robin Romeo and Saharian (2002) and periodic Wen-Biao et al. (2007); Langfeld et al. (1995) BCs) or physically-motivated (those imposed by electromagnetism Landau et al. (1960) or by the bag model Chodos et al. (1974) for example).
The determination of a set of field solutions that obey the boundary conditions specified in step 1. In the simplest cases this can be achieved by direct solution of the equations of motion. However, this step is usually non-trivial, and has resulted in the development of numerous techniques including the so-called macroscopic QED Dung et al. (1998), worldline numerics Gies and Klingmüller (2006), the “proximity-force approximation” Derjaguin (1934); Gies and Klingmüller (2006), certain scattering theory based methods Jaekel and Reynaud (1991), and many more.
The substitution of the field solutions found in step 2 into an expression for the vacuum energy of the relevant field. Upon suitable regularisation and the dropping of any boundary-independent terms, one is left with the Casimir force for some combination of: a field (Maxwell, Dirac, etc), a boundary condition, and a physical geometry.
We will begin in section II by reviewing the generalized, physically-motivated boundary conditions presented in Stokes and Bennett (2014). We then find explicit field solutions for the parallel-plate geometry, and hence accomplish steps and given above. In section III we will carry out the final step above by computing specific values for the Casimir force associated with the massless fields up to spin-.
Ii Generalised physical boundary conditions
In this section we review our generalisation of the boundary conditions (BCs) employed in the calculation of the Casimir effect associated with the spin- and spin- fields. To do this we use the two-spinor calculus formalism presented in appendix A. Further details of the two-spinor calculus formalism can be found, for example, in Penrose and Rindler (1987); Barut (1964); Dreiner et al. (2010) and the references therein.
ii.1 Unified physical boundary conditions for massless spin- and spin- fields
We begin by considering the simplest spin field—the spin- massless Dirac-Weyl field. We adopt the two-spinor calculus formalism laid out in appendix A. The spin- massless field is described by a pair of square-root Klein-Gordon equations, which in the massless case are decoupled;
where and are spin- massless quantum fields describing right and left-helicities respectively. The usual Dirac bispinor can be constructed through a direct sum of fields proportional to and .
We wish to calculate the Casimir effect associated with the massless spin- field due to the presence of two perfectly reflecting parallel plates orthogonal to the -axis. We assume that one plate is located at and that the other is located at as shown in Fig. 1. The physical constraint we impose on the fields is that there be no particle-current normal to the surfaces at Milonni (1994);
where are components of the outward-pointing unit normals to the surfaces at and respectively, and are the components of the spin- particle-current vector. In terms of the two-spinors in Eq. (1), Eq. (2) reads [c.f. Eq. (A.3)]
This condition will hold if we impose the BCs Milonni (1994)
where the (normalised) Pauli matrices are defined in Eq. (A.3). The sign in Eq. (4) corresponds to the case , and the sign to the case . To prove that Eq. (4) implies Eq. (3), one first multiplies Eq. (4) through by . The left-hand-side then equals while the right-hand-side equals , where is the symplectic form on the spinor-space [c.f. appendix A]. This completes the proof. We note that the BCs in Eq. (4) are nothing but the usually employed BCs in the calculation of the spin- Casimir effect Chodos et al. (1974); Milonni (1994).
We now Fourier-expand the fields and in plane-wave superpositions as follows
Here , and the label (for right) corresponds to the helicity while the label corresponds to the helicity . The function and its charge-conjugate are (suitably normalised) momentum-space single-particle right and left-helicity wavefunctions respectively. The and are annihilation and creation operators for particles with momentum and helicity , and the and are the corresponding anti-particle operators. Altogether these operators satisfy the fermionic anti-commutation relations
If this solution is to be non-trivial, i.e., such that is not identically zero, then it cannot satisfy the BCs in Eq. (4). In other words the possibility of a completely free solution is negated by the presence of the plates, which evidently must modify the free solution in some way. We make the ansatz Chodos et al. (1974); Milonni (1994)
where denotes the projection of the three-vector onto the - plane. Making this ansatz constitutes the physically reasonable assumption that the effect of the plates is to flip the sign of in Eq. (8). The modified solution (9) satisfies the BC in Eq. (4) at identically. It will also satisfy the BC corresponding to provided that
When used in conjunction with Eqs. (7) and (11), the above restricted values of yield the usual Casimir force associated with the spin- field between two perfectly reflecting parallel plates (c.f. section III.1.1). In section II.2 it will be shown that with a straightforward generalisation of the BCs in Eq. (4), the above values of turn out to be the same for any fermionic field.
The massless spin- field is the familiar field of Maxwell electrodynamics. In the two-spinor calculus formalism the Maxwell field is described by a pair of symmetric spin-tensors and corresponding to the right and left-helicity states of the photon respectively. The equations of motion analogous to those in (1) are the square-root Klein-Gordon equations
The relation of the spin-tensors above to the more conventional electromagnetic three-vectors is most easily achieved through the complex Riemann-Silberstein vector , where and are the electric and magnetic fields respectively. The vectors and can be viewed as complex three-vectors corresponding to the right and left-helicity states of the photon Bialynicki-Birula and Bialynicka-Birula (2013). If one replaces the imaginary unit in these definitions with the volume form on Minkowski space-time and views the fields and as bivectors and over , then one obtains the familiar electromagnetic field tensor and its reverse . Here the juxtaposition denotes the Clifford (geometric) product of and Doran and Lasenby (2003). One can also describe the electromagnetic field in terms of the dual tensor where denotes the Hodge-dual. Clearly each of the objects , and (or ) constitutes a different organisation of the six real degrees of freedom (including gauge degrees of freedom) that describe the physical electromagnetic field.
Relating the Maxwell spin-tensor to the Riemann-Silberstein vector we have Bialynicki-Birula and Bialynicka-Birula (2013)
Although either of the above relations suffices in order to relate the two-spinor treatment of electrodynamics to the more widely-known approaches, we choose to make use of the former relation (13). Using Eq. (13) it is straightforward to show that the equations in (12) are equivalent to the free Maxwell equation
and its complex-conjugate.
In determining the appropriate BCs to impose for the calculation of the Casimir effect in the spin- case, one must contend with the fact that there exists no local particle-current vector for massless fields with spin greater than Penrose (1965). Thus, for the Maxwell field in particular, an alternative physical current must be chosen in order to obtain a condition analogous to Eq. (2). As is well-known, one of the few local observables associated with photons is their energy-density Sipe (1995); Bialynicki-Birula and Bialynicka-Birula (2013). A natural choice for in the spin- case is therefore the energy-current
where as before and are normal to the surfaces at and respectively. The analogy between the spin- and spin- cases becomes obvious when Eqs. (II.1.2) and (17) are written in terms of the Maxwell spin-tensor . To this end we note that using Eq. (13) the energy-momentum tensor can be written [c.f. Eq. A.3]
Within the two-spinor calculus formalism there is a clear analogy between the spin- energy-momentum spin-tensor , and the spin- particle-current . As such, substituting the expression for given by Eq. (18) with , into Eq. (2) yields
which is the spin- version of Eq. (3). This condition will necessarily hold if we impose the BCs
Using Eq. (13) it is easy to show that these conditions are equivalent to Eq. (17). The BCs in Eq. (II.1.2) written in terms of , are therefore completely equivalent to the usual BCs [in Eq. (17)] employed in the calculation of the electromagnetic Casimir effect Casimir (1948).
There still remains the proof that the BCs in Eq. (II.1.2) imply . This proof is not as straightforward as the proof given in the spin- case. There we were able to make use of the anti-symmetry of the symplectic form on the spinor space , but such a strategy will obviously fail if the field under consideration is bosonic, i.e., is described by an evenly ranked spin-tensor. However, a more brute force proof involving the use of Eq. (21) is available in the spin- case. In section II.2 we specify generalised BCs associated with an arbitrary spin- field [Eq. (II.2)], and prove in appendix B that they imply the vanishing of the normal component of the relevant local current [Eq. (28)]. In this context the Maxwell field simply corresponds to the special case .
Before moving on to determine the allowed values of for the spin- field, we note that a significant difference between Eq. (II.1.2) and Eq. (4), is that Eq. (II.1.2) involves an even number of factors of rather than an odd number. Thus, the boundary condition is the same for both the and cases, rather than differing by a minus sign as is the case in Eq. (4) for the spin- field. We will see quite generally in section II.2, that this is the crucial difference between the BCs for fermionic (half odd-integer spin) and bosonic (integer-spin) fields.
Now, using Eq. (II.1.2) the determination of the allowed values of in the spin- case exactly mirrors the procedure used above for the spin- field. We employ the usual expression for the quantised energy of the Maxwell field
where and are bosonic annihilation and creation operators satisfying the commutation relation
where is a momentum-space left-helicity single-particle spin-tensor, analogous to for the spin- field. As in the spin- case the modified solution satisfies the BC in Eq. (II.1.2) at identically. It will also satisfy the BC at provided
which is the case if and only if
These values of are usually found by more conventional means involving the electric and magnetic fields and the BCs in Eq. (17). Along with Eq. (22) they can be used to obtain the well-known electromagnetic Casimir force first found in Casimir (1948) (c.f. section III.1.2). In the following section (II.2) we generalise the BCs in Eq. (II.1.2) to show that the above values of turn out to be the allowed values for any bosonic field.
ii.2 Generalised physical boundary conditions for massless fields with higher spin
Given the analogy between the spin- and spin- fields described above, the extension of the physical BCs in Eqs. (4) and (II.1.2) to higher-spin fields now naturally presents itself. Initially for higher spin fields the identification of a local physical current to use in conjunction with Eq. (2) may seem problematic, but we have effectively already tackled this problem in adapting the spin- BCs to the spin- case. The BCs for any massless higher spins should evidently involve the physical fields directly involved in the description of the right and left-helicity states. These fields belong to the (carrier spaces of the) “outer” (rather than the “inner”) irreducible representations of the symplectic group . This terminology is clarified by Fig. 2. The physical fields belonging to the outer representations are those that appear explicitly in the square-root (massless) Klein-Gordon equations, which describe the physical dynamics of the free system. For a spin- field these equations read
Specifying the appropriate BCs in terms of the outer field clearly avoids any discussion regarding the use of unphysical inner field potentials, which are often used in the description of higher spins.
Generalising the local currents encountered in the spin- and spin- cases, we begin by defining for a spin- field the local current
which are the same generalised BCs first given in Stokes and Bennett (2014). We prove in appendix B that these BCs imply that satisfies the physical constraint given in Eq. (2), i.e., . For fermionic fields is odd, which implies that the BCs in Eq. (II.2) contain an odd number of factors of . This means that when the BC differs by a minus sign compared with when . In contrast, for bosonic fields is even, so the BC is the same for both the and cases.
To determine the allowed values of due to the BCs in Eq. (II.2), we use in analogy to the spin- and spin- cases, the following modified single-particle positive-energy solution to the first of the equations in (27)
where is a momentum-space single-particle positive-energy spin-tensor. In complete analogy to the spin- and spin- cases, substituting the solution above into Eq. (II.2) implies
This gives the allowed values of and as
In Fig. 3 we give a schematic representation of field solutions corresponding to the first few allowed values of specified above.
According to the generalised BCs in Eq. (II.2) the two sets of energy-momentum values in (II.2) are the only two possible, and they correspond to fermionic and bosonic fields respectively. A significant implication of these general results is that periodic BCs cannot be applied to a physically confined fermionic field between two parallel plates, because for fermionic fields the BCs are necessarily different at the two surfaces as is lucidly illustrated in Fig. 3. This means for example, that the approach adopted in Wen-Biao et al. (2007) where periodic BCs were imposed on the spin- field, is unphysical.
Iii The Casimir effect for arbitrary spin fields
Having obtained according to the BCs in Eq. (II.2), the allowed values of energy-momentum for a field with arbitrary spin, we now look to calculate the resulting Casimir force.
iii.1 The Casimir effect for spin- and spin- fields
Calculating the Casimir forces associated with the spin- and spin- fields is a textbook exercise, so we review it here only very briefly, with an eye towards extending the calculation to the case of higher spin fields.
The energy associated with the massless spin- field is given in Eq. (7). Our strategy in calculating the Casimir force is to initially restrict the field to a fictitious cavity upon which we impose periodic BCs. We will take the continuum limit in the and -directions after having employed the allowed values of given in Eq. (11). The vacuum energy inside the quantisation cavity is given by
The integral above is divergent and requires regularisation. We adopt a conventional regularisation, which yields the finite vacuum energy
In the last line above a term linear in , which gives a constant contribution to has been ignored. The final result for the Casimir force associated with the massless spin- field between two perfectly reflecting parallel plates is therefore
which is well-known Chodos et al. (1974).
For the massless spin- field in a box with periodic BCs, the vacuum energy according to Eq. (22) is
It must be understood in the above expression that when any of the components vanish, there is only one independent polarisation . Substituting into Eq. (37) the allowed values of given in Eq. (26), and as in the spin- case taking the continuum limit with respect to the and -directions, one obtains Milonni (1994)
where following Milonni (1994) we use a prime on the summation to indicate that for the value a factor of must be inserted. The vacuum energy is as expected divergent, and we adopt the same regularisation as in the spin- case. This yields the finite vacuum energy
where as in the spin- case a term linear in has been ignored. Thus, the final result for the Casimir force associated with the massless spin- field between two perfectly reflecting parallel plates is
which is well-known Casimir (1948).
The calculations above pertaining to the spin- and spin- fields reveal that the only ingredients necessary in order to obtain a value for the Casimir force, are an expression for the vacuum energy of the field under consideration, and a set of energy-momentum values allowed by the BCs.
iii.2 The Casimir effect for spin- and spin- fields
To calculate the Casmir force associated with a higher spin field, one must first obtain an expression for the associated vacuum energy. This is much more problematic for higher spin fields than it is for the fields with spin . Higher spin fields possess additional gauge freedom, and the local energy-momentum tensors associated with such fields are generally gauge-dependent. Fortunately the total energy may still be gauge-invariant, and this is the working assumption we will make here.
The most common description of massless spin- particles is through the Rarita-Schwinger field Rarita and Schwinger (1941), which consists of four four-component Dirac bispinors where the index labels the four distinct bispinors. The Rarita-Schwinger field belongs to the reducible representation , which is build from the two outer spin- irreducible representations and the single inner spin- irreducible representation of Weinberg (1995) (c.f. Fig. 2). The representation has (complex) dimension sixteen, whereas the two outer irreducible representations and , that directly correspond to the right and left-helicity states of the physical spin- field, have dimension four (c.f. Fig. 2). This gives a total of eight degrees of freedom, but not all of these are dynamically independent. Gauge-fixing reduces the total number of physical degrees of freedom to four, which is precisely the number required to describe the right and left-helicity states of a massless spin- particle and its anti-particle. The Rarita-Schwinger field contains some twelve redundant degrees of freedom, and accordingly the associated equations of motion are invariant under the gauge transformation
Although largely redundant the Rarita-Schwinger field is advantageous, because it can be manipulated in much the same way as the familiar Dirac field for spin- particles. It is for this reason that we use it to obtain an expression for the energy of the massless spin- field.
The Rarita-Schwinger Lagrangian can be written
where , the are totally anti-symmetric with , the set consists of the usual Dirac matrices with , and denotes the relativistic adjoint of . Under the gauge transformation in Eq. (41) the Lagrangian varies by a total divergence.
The twelve redundant degrees of freedom are eliminated using constraints. In particular, the generalised Coulomb gauge is obtained by imposing the constraints Das and Freedman (1976)
where denotes the four-column with all zero entries. Using the identity
it is a straightforward exercise to verify that for the Lagrangian in Eq. (42) along with the constraints in (43), the Euler-Lagrange equations yield the Dirac equation and its adjoint . This in turn ensures that (and its adjoint) satisfy the correct relativistic wave equation
The constraints in (43) reduce the number of (complex) physical degrees of freedom to the four required in order to describe the right and left-helicity states of a massless spin- particle and its anti-particle. As an ansatz for in Eq. (45), we therefore make the following Fourier expansion Das and Freedman (1976)
where the and are annihilation and creation operators for particles and anti-particles respectively, which as in the spin- case satisfy the anti-commutation relations in Eq. (6). It is of course possible to invert Eq. (46) and its hermitian-conjugate, and this means it is possible to define the and in terms of . The single-particle positive and negative-energy wavefunctions and are defined by
where and satisfy the momentum-space counterparts of the constraints in (43). The positive and negative-energy wavefunctions are not linearly-independent and are related by . The orthonormality requirements for and can be deduced from the required properties of and under a Lorentz boost, and read Das and Freedman (1976)
Since the spin- field is fermionic, according to Eq. (II.2) the allowed values of for the calculation of the Casimir force, are the same as in the spin- case. Moreover, because the vacuum energies in Eqs (7) and (III.2.1) are identical, the calculation of the Casimir force from Eq. (III.2.1) is also the same as in the spin- case, with the final result given by Eq. (36).
The spin- field is most commonly described using a symmetric traceless tensor field , which corresponds to the inner irreducible representation of the symplectic group [c.f. appendix A.3 and Fig. 2]. This field can also be viewed as the first-order gravitational correction to the components of the Minkowski inner-product. If we expand the general “metric” tensor of curved spacetime as