"Heuristics" for Casimir: how close to reality do they have to be?

[nomadreid]

According to the dictionary, a "heuristic" is a tool to allow someone to figure out on her own the full explanation.
If I say that explanation A is a heuristic for a complicated concept B, shouldn't B be at least a close, even if incorrect, explanation?

Specifically, I have read (sorry about no source, but finding this on the net is easier than one even wants it to be) "explanations" of the Casimir effect which are then excused as being false, but OK as "heuristics".

Even more specifically, the correct explanation, I read, is that
--it is a van der Waals effect due to the boundary conditions, while
the "heuristics" say that
---the energy vacuum expectation inside the plates is lower than the vacuum energy expectation outside,
---usually going further and saying this is because there is a lower cutoff of the wavelengths that can fit in there,
---while some even try with a cardinality argument with an uncountable number of waves outside and a countable number of waves inside.

I don't see the connection between the correct explanation and the incorrect ones, so I do not see how the incorrect one can be termed heuristics (especially when it uses the cardinality argument).

Has the word "heuristic" come to mean "replacing the correct argument with an incorrect argument to make the reader at least think that she understands it", or is there a connection that I do not see?

Thanks for any guidance.

 

[hutchphd]

---while some even try with a cardinality argument with an uncountable number of waves outside and a countable number of waves inside.

I don't know this argument but maybe.

--it is a van der Waals effect due to the boundary conditions, while
the "heuristics" say that
---the energy vacuum expectation inside the plates is lower than the vacuum energy expectation outside,
---usually going further and saying this is because there is a lower cutoff of the wavelengths that can fit in there,

These arguments are all based upon boundary conditions.
Begs the question: How close do you want them to be?

 

[nomadreid]

Thanks, hutchphd

I don't know this argument but maybe.

There are actually two cardinality arguments, if I understand them correctly.
The first one appears to be the weakest, so I start with it:

If the distance between the plates is d, then a standing wave (identified with virtual particles) along the axis of a line perpendicular to the plates could have wavelengths 2d, d, 2d/3, d/2, ...2d/n... that is, a countable number of possibilities, whereas in open space they can have any real-number wavelength, that is an uncountable number; and so before any cutoff, a higher energy density; after the cutoff the density remains higher outside, hence the pressure. This argument can be found, for example, after 2:20 of (address split up in order not to insert the video itself into this text box)
https:// (no space) www. (no space) youtube.com/watch?v=nDxW9ZF5wGs

The other cardinality argument seems to start the same with the countable wavelengths, but this in free space, whereas some of these leak out, leaving a finite number of wavelengths, hence the pressure. This seems to be the argument in Equations 11 & 12 in
https://towardsdatascience.com/the-energy-of-the-vacuum-5f70ed72f820

These arguments are all based upon boundary conditions.

That they all refer to boundary conditions is clear to me, but I am having problems seeing that they are the same boundary conditions, as the discussion in
https://physics.stackexchange.com/questions/11544/van-der-waals-and-casimir-forces
writes
"The charges in the plate will have to redistribute and polarize the plate to generate a corresponding field."

where the "corresponding" is expressed in this way in the above cited
https://towardsdatascience.com/the-energy-of-the-vacuum-5f70ed72f820

"The Casimir effect is a consequence of the change Δε in the vacuum expectation ε of the electromagnetic field due to the presence of the perfectly conducting plates. The change is what generates the Casimir force between the plates (since the vacuum expectation itself is not observable)."

Why the electric field should correspond to a lower vacuum expectation value is not something which is immediately obvious, but I suppose that just means that I have a few hours of working through the corresponding equations...

Begs the question: How close do you want them to be?

I withdraw my use of the word "false" in the original post, so I admit that this is a sort of heuristic. But it would be nice if the heuristic explanation included the role of the electric field instead of the bald statement about lower expectation value.

 

[Demystifier]

Sec. 2 of my https://arxiv.org/abs/1702.03291 contains a lot of heuristic arguments on Casimir effect.

 

[nomadreid]

Super! I began to read it, realized that this was just what the doctor ordered, and downloaded it for reading carefully a bit later. Many thanks, Demystifier! (The excellence of the presentation did not surprise me, as I have read some of your other stuff -- always superbly explained.)