A generalized bag-like boundary condition for fields with arbitrary spin

Boundary conditions for the Maxwell and Dirac fields at material surfaces are widely-used and physically well-motivated, but do not appear to have been generalised to deal with higher spin fields. As a result there is no clear prescription as to which boundary conditions should be selected in order to obtain physically-relevant results pertaining to confined higher spin fields. This lack of understanding is significant given that boundary-dependent phenomena are ubiquitous across physics, a prominent example being the Casimir effect. Here, we use the two-spinor calculus formalism to present a unified treatment of boundary conditions routinely employed in the treatment of spin-1/2 and spin-1 fields. We then use this unification to obtain a boundary condition that can be applied to massless fields of any spin, including the spin-2 graviton, and its supersymmetric partner the spin-3/2 gravitino.

The coupling of a quantised field to matter causes the spectrum of its vacuum fluctuations to change. The range of resulting phenomena includes what are variously known as Casimir forces, energies and pressures. Often, the matter involved is idealised through the assumption that its only effect is to place boundary conditions on the quantised field -in fact, so much of the literature pertaining to the Casimir effect is based on such idealizations, that at its core Casimir physics is the study of boundary conditions. The simple case of two perfectly reflecting, infinite, parallel plates interacting with the Maxwell field was investigated by Casimir in [1]. Casimir's seminal paper has since resulted in a wide range of extensions, generalizations and experimental confirmations over the last half-century or so [2][3][4][5][6][7]. This has lead, for example, to new constraints on hypothetical Yukawa corrections to Newtonian gravity [8]. Casimir's relatively simple and intuitive calculation has provided an enormously fruitful link between real-world experiments and the abstract discipline of quantum field theory. In fact, boundary-dependent effects are often cited in standard quantum field theory textbooks as the primary justification for the reality of vacuum fluctuations. Such interpretations however, are not without controversy [9].
Boundary-dependent vacuum forces are not specific to electromagnetism, and are in fact a general feature of quantised fields. A striking example of nonelectromagnetic Casimir effects can be found in nuclear physics. Early attempts to model the nucleon without considering boundary conditions at its surface ran into a variety of problems [10]. Many of these were solved by the introduction of the 'bag model' [11], which describes a nucleon as a collection of free massless quarks [12] confined to a region of space (the 'bag'), with a postulated boundary condition that governs their behaviour at the surface. This model, subject to sensible choices of a small number of free parameters, correctly predicts much of the physics of the nucleon [10]. The boundary-dependent vacuum contribution to the energy (the Casimir energy) has important consequences for the stability of the bag [13][14][15]. This further emphasises the importance of using physically-motivated boundary conditions. A further example of the need to impose physical boundary conditions on fermionic fields is provided by graphene and carbon nanotubes, both of which are the subject of intense contemporary interest. These structures support a two-dimensional gas of massless fermions [16] and the resultant fermionic Casimir force has been found to cause non-trivial edge effects [17]. Given that Casimir effects associated with the Maxwell (spin-1) field and the Dirac (spin-1 /2) field are of experimental and theoretical interest, one is naturally led to the question as to whether the Casimir effect for these fields can be calculated in a unified way. Specifically, since Casimir physics is essentially the study of boundary conditions, can we construct boundary conditions that includes those used for the spin-1 /2 and spin-1 fields as special cases? Furthermore, can we generalise this unified boundary condition to one that applies to higherspin fields? This would represent a significant advance as compared to existing calculations involving boundary conditions for higher-spin fields. For example, in [18] ar-bitrary boundary conditions (periodic) are applied to the spin-3 /2 field -no physical justification is attempted.
To answer these questions we will begin by outlining the boundary conditions assumed within the bag model, i.e., those usually employed in the treatment of massless spin-1 /2 particles. In this model, one envisages a fermionic field confined to some region of space that is surrounded by an impenetrable barrier. Thus, a physically reasonable constraint to impose (which can also be motivated by an appropriate choice of Lagrangian [10,19]) is that there be no particle current across the surface; where n µ is a spacelike unit four-vector normal to the surface defining the bag, and we have employed the summation convention for repeated upper and lower indices. Rather than using the usual notationψ to denote the Dirac adjoint ψ † γ 0 of ψ, we have used φ ≡ ψ † γ 0 in order to avoid confusion later on. The constraint (1) This can be shown by multiplying Eq. (2) by φ from the left, and the Dirac adjoint of Eq. (2) by ψ from the right. Adding these two quantities, one finds 2in µ φγ µ ψ = 2in µ j µ = 0. This shows that the boundary condition (2) implies n µ j µ = 0, which is the constraint imposed in the bag model. What about higher spins? It is well-known that the description of fields with arbitrary spin can be constructed using elementary two-spinors via the so-called two-spinor calculus formalism [20][21][22]. This means that, for example, the Maxwell field can be described on the same footing as the massless Dirac field. As a result, we should be able to find a spin-1 analog of the constraint (1). Initially this might seem hopeless, because no local particlecurrent exists for fields with spin greater than 1 /2 [23]. However, we shall see that there is a natural adaptation of the Dirac-field boundary condition to the Maxwell field, which moreover coincides with the boundary condition usually employed in the calculation of the electromagnetic Casimir force. This allows us to generalise the boundary condition (2) to arbitrary spin.
The two-spinor calculus allows one to build irreducible representations of the homogeneous Lorentz group using two-dimensional complex symplectic vector spaces S andS, where a bar is used to denote the complex conjugate space. The space S is the pair (V, ω), where V is a two-dimensional complex vector space and ω is a complex symplectic (non-degenerate) form. Choosing a basis {f a } ⊂ V we can write arbitrary elements (spinors) of S andS as where we use bars rather than the more commonly used dots to distinguish between a spinor index and a conjugate-spinor index. Furthermore we rely entirely on the different indices in order to distinguish between the components of ψ andψ. With these index conventions matrix operations become particularly simple. If a matrix v has elements v ab then we have the following representations where ⊺ and † denote matrix transposition and hermitian conjugation respectively. The symplectic form ω is used to raise and lower spinor indices. We adopt the convention that ω ab = −ω ba can only be used to lower an index when the repeated index is in the first slot. Similarly ω ab only raises the index when the repeated index is in the second slot. The same rules apply for barred indices, so altogether We note that these identities imply the following identity for the contraction of a rank-n spin tensor with its dual φ a1a2...an φ a1a2...an = (−1) n φ a1a2...an φ a1a2...an .
The above ingredients allow one to write a spacetime tensor of rank (i, j) in terms of Hermitian matrices as with σ i the ith Pauli matrix. We have now laid out the formalism that allows us to use a unified language to describe fields of arbitrary spin. This will eventually enable us to determine a unified physical boundary condition applicable to any massless spinor field. We begin this process by rewriting the right-helicity component of the Dirac current in (1) as In terms of the two-spinor calculus formalism, the boundary condition (2) for the right-helicity component becomes We can demonstrate that Eq. (10) implies n µ j µ = 0 by multiplying both sides by ψā and using the identity (6), which gives This shows that Eq. (10) is indeed the two-spinor calculus version of the bag boundary condition (2) for a righthelicity spinor. A similar calculation holds for the lefthelicity spinor. The next-lowest spin field after the Dirac field (s = 1 /2) is of course the Maxwell field (s = 1). Just as in our discussion of the Dirac field, we will begin by casting the usual statements of the boundary conditions (in this case given by restrictions on the electric and magnetic fields E and B) in the language of two-spinor calculus. The electromagnetic boundary condition for a perfect conductor requires that n × E and n · B vanish at the surface. This in turn implies that n · S also vanishes, where S = E × B is the Poynting vector. Using the Riemann-Silberstein (RS) vector F ≡ E + iB, the electromagnetic boundary conditions can be written We can assume without loss of generality that n µ = (0,ẑ) so that the RS vector obeying the boundary conditions (12) is Following [24], we now introduce the spin tensor φ ab such that in terms of which (13) can be written Using Eq. (7) we can write a symmetric tensor T µν as T µν = σ μ aa σ νb b Tā abb (16) where Tā abb is a symmetric spin-tensor. If we define Tā abb = φābφ ab , then T µν in Eq. (16) is the familiar electromagnetic energy-momentum tensor, with components In terms of T µν the constraint n · S becomes which for n µ = (0,ẑ) can be written Comparing this with (1), we see that T µ0 plays the role of the Dirac current j µ for the Maxwell field. The physical constraint, analogous to (1), that we impose on the Maxwell field is therefore which will necessarily hold if We can easily demonstrate that the boundary condition (21) implies the constraint (20) by again taking n µ = (0,ẑ), so that the boundary condition becomes Using the explicit form of the Pauli matrices, Eq. (22) immediately yields Eqs. (15), which themselves followed from having written the the boundary conditions (10) and (21) in terms of the RS vector. The fact that the above procedure is exactly analogous to that for the Dirac field is remarkable and unexpected. As already mentioned, no local particle-current exists for massless fields with spin greater than 1 /2. However, one of the few local observables associated with photons is their energy current, which is precisely the quantity that naturally appears in the spin-1 constraint (20).
The generalization of the boundary condition to arbitrary spinor fields is now clear. For spinm /2 we write for is the local current for the spinor field concernedthe Dirac field has J µ (1) = j µ , the Maxwell field has J µ (2) = T µ0 and so on. In terms of the spin-tensor φ, the current J is defined by J a1ā1a2ā2...amām ≡ φ a1a2...am φā 1ā2...ām .
The boundary condition (23) ensures that which we can prove by using the rules (5). These allow (23) to be written (27) Substituting this into n µ J µ and using the explicit forms of the Pauli matrices along with the matrix representation ω = iσ 2 , we find Using σ 1 = (σ 1 ) ⊺ and relabelling the indices a i ↔ a ′ i for i = 2, ..., m, the last line of Eq. (28) is equal to which is the negative of the first line in Eq. (28). This proves that the boundary condition (23) implies n µ J µ (m) = 0 for an arbitrary spinm /2 field, which is the main result of our work.
As we have already noted, the identification of a physical current J for higher spin fields seems at first problematic, due to the non-existence of a local particle current for spin > 1 /2. We have in fact already tackled this problem by adapting the spin-1 /2 boundary conditions to the spin-1 case. This enables us to inductively determine the appropriate J for higher spin fields.
Particularly noteworthy is identification of J for the spin-2 field that describes linearised quantum gravity. This field is most commonly described using a symmetric traceless tensor field h µν that results from the firstorder expansion g µν (u) = g µν + uh µν + ... of the general metric tensor of curved spacetime. In this first-order approximation, Einstein's vacuum equations in terms of h µν are equivalent to the correct relativistic wave equation for a massless spin-2 particle (the so-called graviton). The right and left-helicities of the graviton are described by symmetric spin-tensors ψ abcd and ψābcd respectively. These can be used to define the Bel-Robinson tensor, which is a strong candidate for the gravitational version of a symmetric energy-momentum tensor [22]. While it is well-known that the gravitational field does not possess a unique local energy-momentum tensor, the Bel-Robinson tensor T µνρσ possesses many of the properties usually associated with such objects, namely, total-symmetry, tracelessness and certain positivity properties [22]. It is also the natural spin-2 analog of the symmetric energymomentum tensor T µν of electrodynamics. The generalised boundary condition in Eq. (23) therefore implies the vanishing of the local current T µ000 . Analogously to the currents encountered in the spin-1 /2 and spin-1 cases, T µ000 could be viewed as a natural quantity in terms of which the physical boundary condition should be specified for the spin-2 field.
A short example of the impact our boundary condition has on the physics of higher-spin fields is found by considering its consequences for a fermionic field confined between two plates. Since the normals at each plate point in opposite directions, for odd m the factor n µ1 n µ2 ...n µm appearing in the boundary condition (23) differs by a minus sign from one plate to the other, meaning that the the fields are, in general, different at each plate. This in turn means that the usual assumption of periodic boundary conditions is unphysical when applied to any massless fermionic field. Results related to this were previously seen in the rather more abstract context of certain periodic topological spaces in which the vacuum polarisation [25] and various Casimir energies [26] were calculated. In particular, the authors of [25] found that attempting to impose periodic boundary conditions on fermionic fields leads to problems with causality. We have provided a physical explanation for the origin of this effect.
Finally, we observe that the results reported in [18] for the Casimir force associated with the spin-3 /2 field cannot correspond to any physical effect due to the periodic boundary conditions employed therein.
In this Letter we have reported the first unified treatment of physical boundary conditions for fields with arbitrary spin. This was achieved by writing well-known boundary conditions for the Maxwell and massless Dirac fields in a unified language, and then carrying out a natural generalisation. The very existence of a unified boundary condition for the Maxwell and Dirac fields is a remarkable result on its own, because of the fundamental differences between the conserved currents for the two fields. However, we have shown that such a boundary condition does exist, and the unification proceeds in such a way that it can be naturally extended to find physical boundary conditions for higher-spin fields. The derived generalised boundary condition opens up a whole landscape of study for physically confined higher-spin fields -here we have demonstrated just one of their consequences, which is that periodic boundary conditions cannot be applied to any physically confined fermionic field.