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Scale limited fields and the Casimir effect

Šimon Vedl, Daniel J. George, and Gavin K. Brennen
Phys. Rev. D 109, 016018 – Published 12 January 2024
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  • INTRODUCTION
  • CONTINUOUS WAVELET TRANSFORM
  • THE CASIMIR EFFECT
  • DETECTING A SCALE CUTOFF
  • DISCUSSION AND CONCLUSION
  • ACKNOWLEDGMENTS
    • References

    Abstract

    We revisit the calculation of the Casimir effect from the perspective of scale limited resolutions of quantum fields. We use the continuous wavelet transform to introduce a scale degree of freedom and then restrict it to simulate either an observational or fundamental limitation of resolution. The Casimir force is derived in this setting for a free complex massless scalar field between two infinite plates with both Dirichlet and periodic boundary conditions. The dependence of the force on the choice of wavelet and size of scale cutoff is extensively discussed for several examples of wavelets.

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    • Received 12 October 2023
    • Accepted 12 December 2023

    DOI:https://doi.org/10.1103/PhysRevD.109.016018

    © 2024 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Casimir effectResummation methods
    1. Techniques
    Functional analytical methodsMathematical physics methods
    Particles & Fields

    Authors & Affiliations

    Šimon Vedl1,2,3,*, Daniel J. George1,2,3, and Gavin K. Brennen1,3

    • 1Department of Physics and Astronomy, Macquarie University, Sydney, New South Wales 2109, Australia
    • 2Sydney Quantum Academy, Sydney, New South Wales 2000, Australia
    • 3ARC Centre of Excellence in Engineered Quantum Systems, Macquarie University, Sydney, New South Wales 2109, Australia

    • *Corresponding author: simon.vedl@hdr.mq.edu.au

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    Issue

    Vol. 109, Iss. 1 — 1 January 2024

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    Images

    • Figure 1
      Figure 1

      Cross section of the first Hermitian wavelet (45) in both position (a) and momentum (b) representation.

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    • Figure 2
      Figure 2

      Dependence of the Casimir force FC on the separation of the plates s. The scale cutoff is set A=1. The solid line corresponds to the usual result −π215s4 and the other lines represent the Casimir force perceived by the Hermitian wavelets for different choices of n.

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    • Figure 3
      Figure 3

      Cross section of the wavelet associated with exponential cutoff (50) in both position (a) and momentum (b) representation.

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    • Figure 4
      Figure 4

      Dependence of Casimir force on the separation of the two plates when the wavelet (50) is considered. The scale cutoff is set A=1. The solid line corresponds to the exact expression (55), the dashed-dotted line on the left corresponds to (34) where the sum runs between −3 and 3. Finally, the dashed line on the right is the expression (52) obtained from the Euler-Maclaurin formula.

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    • Figure 5
      Figure 5

      Dependence of the Casimir force on the separation of the two plates with the scale cutoff set to A=1. The solid line corresponds to the usual result −π215s4, the dashed line is the correction (48) when the first Hermitian wavelet is considered, and the dotted dashed line is the correction (52) obtained from the wavelet (50).

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    • Figure 6
      Figure 6

      Cross section of the wavelet that is defined as a bump function (57) in the momentum representation (b). The cross section in position representation (a) is computed numerically directly from the inverse Fourier integral.

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    • Figure 7
      Figure 7

      The cross section of the wavelet (61) associated with the nonanalytic cutoff function 1−e−1/k10 both in position (a) and momentum (b) representation. The momentum representation is zero only for k=0, in the neighborhood of zero the value of the function is so small that the plotting software rounds it down to zero.

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    • Figure 8
      Figure 8

      Comparison of the corrections to Casimir force obtained from the wavelet wbump (61) associated with cutoff function f˜(k)=1−e−1/k10 (dashed) and the wavelet wMeijer (57) associated with the smoothed cutoff function (solid), both exhibiting oscillatory behavior. The scale cutoff is set to A=1.

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