STM-D-0853Paper2017Published and peer-reviewed

Is zero-point energy physical? A toy model for Casimir-like effect

Hrvoje Nikolić

Abstract and summary · read the original at the source

In one page

Hrvoje Nikolić, of the Rudjer Bošković Institute in Zagreb, asks what a Casimir force is actually made of, and answers with a model small enough to hold in your head: three moving parts instead of an infinite field. One stands for the electromagnetic field, one for the polarisation of the plates — the charges that shift when a field pushes on them — and one for the gap between the plates. Quantise the first two, let the third move, and a Casimir-like attraction falls straight out. Nikolić then computes that force twice, once by differentiating the vacuum energy and once by taking the average of the force operator itself, and the two answers agree exactly. Along the way he settles three points of bookkeeping that get muddled in the wider literature: the Casimir vacuum is a lowest-energy state only while the gap is held fixed, it is not a state without photons, and the fluctuations doing the work are unambiguously physical — set them to zero and the force disappears with them.

Why it matters hereChapter 2 needs the strongest version of the vacuum claim, which is the one no critic can take away, and this paper draws the line precisely: the fluctuations are real and they push, while the ledger that books them as a ground-state energy is an effective one that only holds while the plates are pinned. Read it next to Robert Jaffe’s 2005 note at /library/stm-dc5bf2b401, which reaches the same place from quantum electrodynamics, and read both against the site’s precision rules — the zero in zero-point energy means zero thermal temperature and nothing else, and the negative energy between the plates is a naming convention for a level below the ambient vacuum, not a substance.

What it claims

01The Casimir vacuum is a ground state only for an effective Hamiltonian that treats the plates as given and their separation as fixed; it is not the ground state of the Hamiltonian in which the separation is free to move, and it is not the ground state of a fundamental Hamiltonian that also allows states with no plates at all.Sections 2.2 and 2.3
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02To describe a force the separation must be a dynamical variable, and the same separation cannot be fixed and unfixed at once: hold it fixed and the Casimir energy reads as an effective ground-state energy but there is no force, let it move and there is a force but the energy is no longer a ground-state energy.Section 2.3
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03In a toy model with only three degrees of freedom — two coupled oscillators standing for the electromagnetic and polarisation fields, plus a slow heavy coordinate standing for the plate separation — the force can be obtained two independent ways, by differentiating the interacting vacuum energy and by taking the expectation value of the force operator in that same vacuum, and the two results coincide exactly with no divergence anywhere in the calculation.Sections 3 and 4; the force in Equations 49 and 53
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04The quantum fluctuations that generate the force are physical: normal-ordering the squared-displacement term away would give a force of zero, which is the wrong answer, and in the free uncorrelated vacuum the two oscillators are uncorrelated and the force vanishes — so it is the correlation between field and polarisation, not a bare zero-point sum, that does the work.Section 4.3, the remarks following Equation 53, with Equations 54 and 55
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05The Casimir vacuum is not a state without photons: it is related to the photon vacuum by a non-trivial Bogoliubov transformation, so it holds zero quanta of the mixed field-plus-matter excitations that a real material actually supports — polaritons — while holding an uncertain number of photon and polarisation pairs.Sections 5.2 and 5.3; Section 6, the paragraphs on polaritons; Conclusions, third point
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06Whether ground-state energy matters for gravitation is set aside here rather than answered: the paper states plainly in its opening pages that it does not study the relevance of ground-state energy to gravitational physics, and points the reader elsewhere for it.Section 1, closing disclaimer
What to watch

Read it · abstract

Abstract

Zero-point energy is generally known to be unphysical. Casimir effect, however, is often presented as a counterexample, giving rise to a conceptual confusion. To resolve the confusion we study foundational aspects of Casimir effect at a qualitative level, but also at a quantitative level within a simple toy model with only 3 degrees of freedom. In particular, we point out that Casimir vacuum is not a state without photons, and not a ground state for a Hamiltonian that can describe Casimir force. Instead, Casimir vacuum can be related to the photon vacuum by a non-trivial Bogoliubov transformation, and it is a ground state only for an effective Hamiltonian describing Casimir plates at a fixed distance. At the fundamental microscopic level, Casimir force is best viewed as a manifestation of van der Waals forces.

Keywords zero-point energy; vacuum; Casimir effect; van der Waals force.

(Abstract only — see the rights note above. The full 21-page text is free to read at arXiv:1702.03291, where the arXiv comment line records the version as the one accepted for publication in Annals of Physics. Its closest companion in this library is Robert Jaffe’s 2005 paper at /library/stm-dc5bf2b401, which computes Casimir forces from quantum electrodynamics alone.)

The way in

https://arxiv.org/abs/1702.03291LICENCE CHECKED. The manuscript is arXiv:1702.03291, version 2 posted 16 May 2017, and the arXiv record carries the arXiv.org perpetual non-exclusive distribution licence version 1.0 — read on the arXiv abstract page on 2026-09-08 — which is not a Creative Commons licence and grants no redistribution. No Creative Commons statement appears anywhere in the text. So this sheet carries the summary, the claims and the author’s own abstract, and sends the reader to the source. The arXiv comment line on that same record reads ‘21 pages, version accepted for publication in Annals of Physics’. The claims below are read against the complete 21-page manuscript and the locators use the paper’s own section and equation numbering; equations are described in words because the page is MDX. Hrvoje Nikolić writes from the Theoretical Physics Division of the Rudjer Bošković Institute, Zagreb, Croatia; the work was supported by the Ministry of Science of the Republic of Croatia and by H2020 Twinning project 692194, RBI-T-WINNING. REGISTRY NOTE: the record reached the library with the byline spelled H. Nikolic and with chapters ch02 and ch06. The diacritics on the author’s own title page are restored here, and chapter 6 is dropped, because the paper is about how a Casimir force arises and says nothing about drawing energy from the vacuum — it explicitly sets the gravitational and cosmological question aside as well.

How to cite it

Hrvoje Nikolić (2017) Is zero-point energy physical? A toy model for Casimir-like effect. arXiv:1702.03291

Where it sits in the curriculum

What the vacuum is