The corrected derivation of the Casimir effect (the physical reality)
The physical setup consists of two idealised conducting plates at a distance \(d\). What is sought is not the absolute vacuum energy, but the energy difference between the vacuum with plates and the corresponding reference without plates. In the interior, the longitudinal modes are discretised by the boundary conditions; outside, or in the reference geometry, the corresponding continuous modes occur.
The hyperreal vacuum energy and the plasma frequency
In the discretised nonstandard space, the eigenfrequencies do not run into an indefinite infinity, but up to an exactly specified midfinite maximum frequency \(\omega\).
Real plates, however, are not perfect mirrors up to arbitrarily high frequencies. Above their characteristic material frequency, or plasma frequency, they become increasingly transparent. The physically appropriate target function therefore contains a damping factor \(g(x)\), which is approximately \(1\) for small frequencies and vanishes at the upper midfinite boundary:
\[ g(0)=1,\qquad g(\omega)=0. \]For a smooth cut-off function, it is additionally assumed that the relevant derivatives also vanish at the upper boundary. Schematically, the longitudinal summation kernel is then
\[ E_{\mathrm{inside}} \propto {\LARGE{\textbf{+}}}_{n=1}^{\omega} n^3\;g(n). \]In hyperreal space, this sum is not a mysterious negative quantity, but an exactly defined hyperfinite sum. The finite Casimir contribution arises only through the structured separation of the common background terms.
Structural deflation by Euler-Maclaurin
For the analysis, the first Euler-Maclaurin formula is used in its hyperreal extension. It reads:
\( {\LARGE{\textbf{+}}}_{\check{q}=0}^{\check{r}}f(\check{q})= \uparrow_0^{\check{r}}{f\left(x\right){\downarrow}x} +\check{f}(\check{r}) +\check{f}(0)\)\( +{\LARGE{\textbf{+}}}_{\check{m}=1}^{\check{n}-1}{\widetilde{m!}}B_m \left({}^{\acute{m}}f(\check{r})-{}^{\acute{m}}f(0)\right) +\mathcal{O} \left( {\widetilde{n!}}B_n \left({}^{\acute{n}}f(\check{r})-{}^{\acute{n}}f(0)\right) \right). \)
Substituting
\[ f(x)=x^3\;g(x) \]separates the relevant contributions into the integral term, the upper boundary, and the lower boundary.
Integral:
The integral term
\[ {\uparrow}_{0}^{\omega} x^3\;g(x){\downarrow}x \]yields the dominant hyperreal background contribution. It belongs to the common vacuum core and is not, by itself, the measurable Casimir energy.
Upper boundary at \(\omega\):
Since real plates become transparent at sufficiently high frequencies, the cut-off function is chosen so that, at the upper boundary,
\[ f(\omega)=0,\qquad {}^1 f(\omega)=0,\qquad {}^2 f(\omega)=0,\qquad {}^3 f(\omega)=0 \]holds. Hence the relevant Euler-Maclaurin boundary terms at the upper boundary vanish.
Lower boundary at \(0\):
At the lower boundary, \(g(0)=1\) implies
\[ f(0)=0,\qquad {}^1 f(0)=0,\qquad {}^2 f(0)=0. \]The third derivative, however, remains:
\[ {}^3 f(0)=6. \]The finite regular contribution therefore arises from the fourth-order Bernoulli term:
\[ R_{\mathrm{constant}} = \widetilde{4!}\,B_4 \left( {}^3 f(\omega)-{}^3 f(0) \right). \]With
\[ B_4=-\widetilde{30}, \qquad {}^3 f(\omega)=0, \qquad {}^3 f(0)=6 \]it follows that
\[ R_{\mathrm{constant}} = \widetilde{4!}\cdot \left(-\widetilde{30}\right) \cdot (0-6). \]Thus the longitudinal sum deflates into a dominant hyperreal background contribution and a positive, singularity-free remainder:
\[ R_{\mathrm{constant}}=\widetilde{120}. \]The decisive point is this: the hyperfinite sum itself is not equal to \(\widetilde{120}\). Rather, \(\widetilde{120}\) is the regular remainder that remains after the common background structure has been separated off.
The exterior-space integral as reference
Outside the plates, or in the reference geometry, the mode spectrum is continuous. The energy density there is described by the corresponding nonstandard integral.
Since the same vacuum and the same physical high-frequency cut-off are used, the same dominant hyperreal pole is generated as in the interior. This contribution describes the common background energy and is not directly measurable.
Schematically, therefore,
\[ E_{\mathrm{inside}} = P_{\mathrm{inside}}(\omega,d)+R_{\mathrm{constant}}, \]\[ E_{\mathrm{outside}} = P_{\mathrm{outside}}(\omega,d). \]Under consistent regularisation, the common divergent contributions agree:
\[ P_{\mathrm{inside}}(\omega,d) = P_{\mathrm{outside}}(\omega,d). \]Transverse integration and the physical sign
The longitudinal constant
\[ R_{\mathrm{constant}}=\widetilde{120} \]is positive. This, however, does not yet determine the sign of the physical Casimir energy.
In the real three-dimensional system, the continuous transverse momenta \(k_\perp\) parallel to the plates must also be integrated. Only this phase-space integration supplies the physical prefactor of the mode difference.
Schematically, the transverse integration generates a negative geometrical factor in front of the regular longitudinal remainder. In the usual normalisation this gives
\[ \frac{\Delta E}{A} = -\frac{\hbar c\pi^2}{720d^3}. \]The negative sign therefore does not arise from the longitudinal sum being negative. It arises from the complete three-dimensional mode calculation, in particular from the transverse phase-space integration and the subsequent subtraction of the reference geometry.
Exact algebraic cancellation of the background contributions
The physically measurable Casimir energy is the difference between the interior space and the reference space:
\[ \Delta E = E_{\mathrm{inside}}-E_{\mathrm{outside}}. \]Inserting the deflated expressions gives, schematically,
\[ \Delta E \propto \left( P_{\mathrm{inside}}+R_{\mathrm{constant}} \right) – P_{\mathrm{outside}}. \]Since both calculations use the same vacuum, the same cut-off, and the same regularisation, the common hyperreal poles cancel exactly:
\[ P_{\mathrm{inside}}-P_{\mathrm{outside}}=0. \]Only the finite regular remainder remains, multiplied by the physical geometrical prefactor:
\[ \Delta E \propto -\widetilde{120}. \]In full physical normalisation, the result per unit area is
\[ \frac{\Delta E}{A} = -\frac{\hbar c\pi^2}{720d^3}. \]The associated force per unit area is
\[ \frac{F}{A} = -\frac{\partial}{\partial d} \left( \frac{\Delta E}{A} \right) = -\frac{\hbar c\pi^2}{240d^4}. \]The negative sign means: the plates attract one another.
Conclusion of the derivation
The Casimir effect does not arise because
\[ {\LARGE{\textbf{+}}}_{n=1}^{\omega} n^3 = -\widetilde{120} \]were true. This statement is false in nonstandard space.
The hyperfinite vacuum sum remains positive and exactly determined. The decisive point is the structured deflation:
\[ \text{hyperreal background} + \text{regular remainder}. \]For the longitudinal cubic summation kernel, the lower boundary of the Euler-Maclaurin formula, together with the physical high-frequency cut-off, yields the positive regular remainder
\[ R_{\mathrm{constant}} = \widetilde{120}. \]The physically negative sign of the Casimir energy then does not arise from a negative sum, but from the complete three-dimensional mode difference:
\[ \frac{\Delta E}{A} = -\frac{\hbar c\pi^2}{720d^3}. \]Real materials additionally enforce this clean view, because they become transparent at very high frequencies. Nonstandard analysis thereby makes visible what is often hidden in the usual regularisation: the enormous hyperreal background contributions are not arbitrarily discarded, but are exactly balanced against one another under the same physical regularisation. What remains measurable is only the finite, distance-dependent remainder of the altered mode structure.
© 2026 by Boris Haase