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In the usual example of the Casimir effect we have two metal plates seperated by some small distance and a quantum field that lives both outside and inside of these plates. The boundary condition for the field inside constraints the allowed modes and thus alters the energy expectation value. The energy both inside and outside is of course infinite but by means of regularization we can assign a negative energy density to the inside, where this quantity makes sense in comparison to the vacuum without the plates.

I wonder now what happens if we don't allow for a quantum field to live outside? So considering an alternate spacetime where one spatial coordinate has a finite (small) range. By imposing vanishing boundary conditions we obtain a similar problem as above. Will there be Casimir force in that scenario? And how could a negative energy make sense, when there is no "unperturbed vacuum" in that case to compare to?

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    $\begingroup$ Can you clarify what you mean by 'one spatial coordinate has a finite (small) range'? Do you mean periodic boundary conditions, like compactifying on a cylinder? If so, I don't see how that's the same as not allowing quantum fields to live on the outside. $\endgroup$
    – user34722
    Commented Feb 14, 2022 at 3:16
  • $\begingroup$ I was thinking more of an extension of a strip to (3+1)D. In different CFT books I found the vacuum energy of a 2D strip to be negative and I wonder what that means. $\endgroup$
    – korni1990
    Commented Feb 14, 2022 at 14:50

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My best guess (please correct me if I'm wrong, I'm just learning):

If the absolute value (not relative to in-between the plates) of the quantum field energy outside the two plates can be assumed to be zero, then a net outward force would be exerted on the plates via the Casimir effect.

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  • $\begingroup$ Since the quantum field energy within the plates (>>0) would be greater than outside the plates (0). $\endgroup$
    – user250486
    Commented Feb 14, 2022 at 12:22
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    $\begingroup$ what I understood by a diagonal reading of this link (formula,(2,24)with mixed conditions) books.google.be/… $\endgroup$
    – The Tiler
    Commented Feb 14, 2022 at 13:02

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