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

Abstract

Boundary conditions (BCs) for the Maxwell and Dirac fields at material surfaces are widely-used and physically well-motivated, but do not appear to have been generalized to deal with higher spin fields. As a result there is no clear prescription as to which BCs 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 BCs routinely employed in the treatment of spin-1 2 and spin-1 fields. We then use this unification to obtain a BC 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 quantized 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. The simple case of two perfectly reflecting, infinite, parallel plates, that impose boundary conditions (BCs) on 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 led, 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 quantized fields. Here we provide a unified and physically well-motivated treatment of the effects that perfectly reflecting material boundaries have on any quantum field.A striking example of non-electromagnetic Casimir effects can be found in nuclear physics, wherein early attempts to model the nucleon without considering BCs at its surface ran into a variety of problems [10]. Many of these problems were solved by the introduction of the 'bag model' [11], which describes a nucleon as a collection of free massless quarks 2 confined to a region of space (the 'bag'), with a postulated BC that governs their behavior at the surface (see figure 1). 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 [12][13][14]. This further emphasizes the importance of using physically-motivated BCs. Another example of...

Casimir energy for spinor fields with δ-shell potentials

“…Before we proceed with the analysis of the spectral zeta function we would like to provide a physical interpretation of the matching conditions (22) and (23) (see [19,20]). A δ-shell potential is defined as impenetrable if the spinors confined either in the interior or the exterior of the shell at t = 0 remain confined in their respective regions for all t ∈ R + .…”

Localization for Dirac fermions

“…Then, the extension of the 1 dimensional treatment presented here to the more realistic 3 dimensional problem simply follows from adding the angular part of the problem (including the spin) and adapting the treatment of the spatial part given by us to the radial component. Our treatment also seems suitable to further generalizations dealing with higher spin fields, where a unified treatment of bag like BC were recently developed, [29], and applied to the study of the Casimir effect, [30]. Now, we construct the orthonormal stationary modes in our cavity, i.e., we solve the eigenproblem for the Dirac hamiltonian (4) in the domain:…”

“…) are given in appendix B, expressions (B5), (B6). This solves the Cauchy problem posed by (28), (29).…”