Schwinger's 1975 source-theory derivation obtained the Casimir force between conducting plates and its temperature dependence without any reference to vacuum zero-point energy, but under acknowledged idealizations: a scalar (spin-zero) photon model with a polarization-doubling factor of two, the neglect of transient effects through a macroscopic-time limit, ideal dispersionless conductors, and a Planck factor inserted into the propagation function by substitution rather than by derivation. Later source-theory work substantially discharged the vector-electromagnetic and material-response strands of this gap (notably 'Casimir effect in dielectrics', 1978), but two strands were never completed within the formalism: a first-principles source-theoretic construction of finite-temperature propagation functions DD and their inverses D−1D^{-1} that eliminates the heuristic thermal substitution, and a source-theoretic analysis of transient and time-dependent boundary processes (the dynamical Casimir regime). The problem is this surviving core: complete the source-theoretic, zero-point-energy-free formulation of the Casimir effect by constructing the thermal propagation functions from source-theoretic principles and extending the energy-shift relation beyond the long-time limit to transients and time-dependent boundaries, recovering the established thermal and dynamical Casimir results as limits.

Background

The Casimir effect is the attraction between two neutral, perfectly conducting parallel plates separated by a distance aa, predicted by H. B. G. Casimir in 1948; the electromagnetic Casimir pressure is −π2ℏc/(240a4)-\pi^2\hbar c/(240a^4). In the standard derivation one computes the change in the sum of zero-point energies 12ℏω\frac{1}{2}\hbar\omega of the field's normal modes caused by the boundary conditions and subtracts the divergent free-space value. A neighboring route to the same physics is the theory of dispersion forces between dielectric half-spaces (the Lifshitz formula and its general theory due to Dzyaloshinskii, Lifshitz and Pitaevskii), in which the force is expressed through frequency-dependent dielectric response functions rather than vacuum mode sums.

Julian Schwinger's source theory is an operator-free, phenomenological reformulation of quantum field theory organized around external sources and propagation functions rather than quantized oscillators and vacuum fluctuations. The basic objects are the propagation function (Green's function) DD relating a source JJ to the field AA it induces, the inverse kernel D−1D^{-1} defined by J=D−1AJ=D^{-1}A, and the vacuum persistence functional WW, the amplitude that the vacuum remains the vacuum in the presence of sources. An infinitesimal change of the physical parameters changes WW by δW0=12i∫dx dx′ D(x,x′) δD−1(x′,x)\delta W_0=\frac{1}{2}i\int dx\,dx'\,D(x,x')\,\delta D^{-1}(x',x), and for a macroscopic duration T\mathcal{T} one identifies δW0=−δε T\delta W_0=-\delta\varepsilon\,\mathcal{T}, which defines an observable energy shift δε\delta\varepsilon; differentiating with respect to a geometric parameter gives the force. This identification is a long-time statement and discards transients.

In 'Casimir effect in source theory' (Lett. Math. Phys. 1, 43 (1975)), Schwinger used this framework to derive the plate-plate force without any reference to zero-point energies. Expanding the scalar Green's function between plates at z=0z=0 and z=az=a in the modes 2/asin⁡(nπz/a)\sqrt{2/a}\sin(n\pi z/a), regulating the short-time singularity with a time-splitting parameter τ→0\tau\to0, and canceling the outer-region divergence against cutoff plates at z=±Lz=\pm L with L→∞L\to\infty, he obtained the scalar pressure −π2/480a4-\pi^2/480a^4 (in units with ℏ=c=1\hbar=c=1) and multiplied by two for the two transverse electromagnetic polarizations to recover −π2/240a4-\pi^2/240a^4. Finite temperature entered by substituting into DD a Planck occupation factor, e−iEτ→e−iEτ+2eβE−1cos⁡(Eτ)e^{-iE\tau}\to e^{-iE\tau}+\frac{2}{e^{\beta E}-1}\cos(E\tau) with β=1/kBT\beta=1/k_BT, leaving D−1D^{-1} unchanged; this reproduces the known classical high-temperature limit. The derivation rests on acknowledged idealizations: a spin-zero (scalar) photon with polarization doubling; omission of transient effects through the macroscopic-T\mathcal{T} limit; ideal dispersionless conductors; and the thermal factor inserted by substitution.

Much of this program was completed in the following two decades. The follow-up paper 'Casimir effect in dielectrics' (Ann. Phys. 115, 1 (1978), with L. L. DeRaad and K. A. Milton) rederived the Lifshitz formula from the source-theoretic action principle using the electromagnetic Green's dyadic (the matrix-valued Green's function carrying polarization indices), recovering the perfect-conductor Casimir force, the known thermal limits, and the van der Waals limits; Jaffe's 2005 review credits this paper with the electromagnetic generalization of the 1975 scalar derivation. Later works ('Casimir effect in source theory II' and 'III', 1992; the 1993-94 papers documented by Milton) refined the derivation and treated dielectric slabs with explicit electric/magnetic mode decompositions, although some geometries were still handled with scalar models (DeRaad's 1985 review computes the force between conducting spheres 'assuming spin zero photons'), and an independent vector-electromagnetic treatment of conducting bodies by multiple scattering was given at the same time by Balian and Duplantier (1978).

Two of the 1975 paper's gaps, however, were never closed within source theory. First, the identification δW0=−δεT\delta W_0=-\delta\varepsilon\mathcal{T} is a long-time statement, so the source-theoretic treatment of the transient regime — the insertion, modification, or removal of boundaries, and time-dependent boundaries generally — does not exist; this is the regime of the dynamical Casimir effect, in which moving or switching boundaries produce real quanta (observed in a superconducting circuit in 2011), which is otherwise well developed through time-dependent quantum field theory. Second, the finite-temperature propagation function was never constructed from source-theoretic first principles: the Planck factor is inserted by hand, tracing back to an earlier source-theory formula, and D−1D^{-1} is left unchanged without derivation; Schwinger's mid-1980s source-theoretic work on thermal Green's functions was never completed. Meanwhile, the same physics has been thoroughly developed outside source theory — equilibrium thermal Casimir forces through the Matsubara formalism (finite-temperature quantum field theory built on sums over discrete imaginary frequencies; including the resolved controversies over the transverse-electric zero mode, the static mode that dominates the classical high-temperature limit, and real-material response, reviewed by Milton in 2004), and dynamical Casimir phenomena through time-dependent quantum field theory. What survives is therefore a completion task within the source-theoretic formalism itself.

Problem Statement

Within Schwinger's source theory — the formulation based on sources, propagation functions DD and their inverse kernels D−1D^{-1}, and the vacuum persistence functional WW, with no reference to quantum oscillators or vacuum zero-point energy — complete the treatment of the Casimir effect left unfinished by the 1975 parallel-plate derivation, restricted to the two strands that later source-theory work did not discharge:

(1) Finite temperature from first principles. Construct the finite-temperature propagation function DD and its inverse D−1D^{-1} from source-theoretic principles, deriving (rather than substituting) the Planck-factor structure in DD and exhibiting a D−1D^{-1} consistent with it, and use them to rederive the equilibrium thermal Casimir force between ideal conducting plates, recovering as limits the zero-temperature pressure −π2ℏc/(240a4)-\pi^2\hbar c/(240a^4) and the classical high-temperature pressure −ζ(3)kBT/(4πa3)-\zeta(3)k_BT/(4\pi a^3).

(2) Transients and time-dependent boundaries. Extend the energy-shift relation δW0=−δεT\delta W_0=-\delta\varepsilon\mathcal{T} beyond the long-time (macroscopic-T\mathcal{T}) limit so that transient and time-dependent processes — the insertion, modification, or removal of boundary conditions, and time-dependent boundaries generally — are described within source theory, including the production of real quanta by time-dependent boundaries, with recovery of the static result at long times, consistency with energy conservation, and correct adiabatic and sudden limits.

These are two facets of one objective: establishing that thermal and dynamical Casimir phenomenology, like the static force, is derivable within source theory alone, without invoking vacuum zero-point energies. The completion must derive the thermal structure in (1) from source-theoretic principles rather than by substitution and extend the energy-shift relation in (2) beyond the long-time identification, without zero-point-energy subtraction. The intrinsic vector-electromagnetic and material-response strands of the original 1975 gap are excluded because they were substantially discharged within the source-theory program itself ('Casimir effect in dielectrics', 1978, and successors); the question concerns the surviving core.

References

  1. H. B. G. Casimir, 'On the attraction between two perfectly conducting plates', Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen 51, 793-795 (1948), no DOI registered; open digitized full text at https://dwc.knaw.nl/DL/publications/PU00018547.pdf, INSPIRE record at https://inspirehep.net/literature/24990
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