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Macroscopic Quantum Electrodynamics

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Book cover Surprises in Theoretical Casimir Physics

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Abstract

In his quantum theory of light, Dirac proposed a quantisation of the electromagnetic field, as described by Maxwell’s equations. We briefly propound a simple version of quantum electrodynamics that is suitable for describing some of the effects of quantised electromagnetic fields in media, and determine an expression for the Casimir stress between two interacting objects embedded in a fluid.

Photon, photon, shining bright! Diffracting through the lab. at night. Can even God, with all his might, measure thy position right?

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Notes

  1. 1.

    Monochromatic modes are stationary modes that conserve energy. We therefore expect the Hamiltonian to be the sum of the Hamiltonians of the individual modes.

  2. 2.

    See Sect. 1.1.2.

  3. 3.

    The basic approach adopted in this section is developed in detail in [8–10]. However, we will not use the Lorentz force expression, which does not recover the standard results for forces in media, but an analogue of the Minkowski stress tensor. Pitaevskii’s comments here are relevant [13, 14].

  4. 4.

    See Appendix B and [21].

  5. 5.

    The accepted use of the ‘Minkowski-like’ stress tensor for computing Casimir forces in the Lifshitz theory [22] was challenged in [23], resulting in some debate [13, 24, 25]. A new argument for the disputed result can be found in [3], where the Casimir stress was derived in the context of the canonical theory of macroscopic quantum electrodynamics [4].

  6. 6.

    Lifshitz theory, in this case, refers to the more general results obtained in [12].

  7. 7.

    This result is not valid when there is absorption. Note that the expression for the variation of the free energy in Lifshitz theory takes a similar form [12]:

    $$ \delta F=\delta F_{0}-\frac{T}{4\pi }\sum _{n=0}^{\infty }\int D_{ii}(\mathbf {r},\mathbf {r},\xi _{n})\delta \epsilon (\mathbf {r},i\xi _{n})\text {d}^{3}\mathbf {r}. $$
  8. 8.

    We note that the first contribution \(\mathbf {f}_{P}\) contains a pressure term which is present in the absence of electromagnetic fluctuations, and a contribution due to the deformation of the medium which vanishes in the limit of an incompressible medium.

  9. 9.

    We retain only the real parts. To take proper account of absorption in media requires a more sophisicated formulation of macro-QED [4]. However, the stress tensor is the same [3].

  10. 10.

    For a discussion of Wick rotation to the imaginary axis, see Appendix A.

  11. 11.

    The full details of this approach can be found in [1].

  12. 12.

    In the limit as \(x'\rightarrow x\), this amounts to subtracting the self-interacting terms. Only waves which have explored the environment contribute to the Casimir force.

  13. 13.

    The Lifshitz result was not originally cast in terms of reflection coefficients, until Kats observed that it was natural to do so [26]. Indeed, expressed in this form [1] it remains valid even when the optical response of the mirrors cannot be described by a local dielectric response function [27].

  14. 14.

    Metamaterials incorporate arrays of micro-engineered circuitry, and can be engineered to produce a strong magnetic response at certain frequencies.

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  1. Department of Physics of Complex Systems, The Weizmann Institute of Science, Rehovot, Israel

    William M. R. Simpson

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Simpson, W.M.R. (2015). Macroscopic Quantum Electrodynamics. In: Surprises in Theoretical Casimir Physics. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-09315-4_2

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