A nonstandard derivation of the electron self-energy (the physical interpretation)
In quantum electrodynamics (QED), the interaction of an electron with its own virtual photons contributes to its self-energy. In perturbative QED this contribution contains divergent terms. Established physics handles these terms by renormalisation: the bare parameters of the theory are related to the finite, experimentally measured quantities by a regularisation and renormalisation prescription.
In the present nonstandard formulation, the aim is not to introduce an undefined infinity. Instead, the divergent contribution is represented as an exactly specified hyperreal pole, which is then separated from the finite regular remainder.
The hyperreal momentum sum and the physical cut-off
The electron is not treated here as living in an unrestricted continuum, but in a discretised nonstandard space whose momenta extend up to an exactly specified midfinite maximum momentum \(\omega\). Such a scale may be interpreted as a physical or effective cut-off, for example near the Planck scale, although this interpretation is an additional modelling assumption.
The interaction of the electron with vacuum fluctuations is described by a sum over virtual momenta \(p\). To express that the model is only trusted up to its effective resolution scale, the summand is multiplied by a smooth cut-off function or damping factor \(g(p)\). For low momenta this factor is approximately \(1\); at the upper midfinite boundary it vanishes, together with the relevant derivatives:
\[ g(0)=1,\qquad g(\omega)=0. \]The dominant mass-renormalisation kernel behaves schematically like \(\tilde{p}\) at large momentum. Thus the hyperfinite model sum has the form
\[ E_{\mathrm{self}} \propto {\LARGE{\textbf{+}}}_{p=1}^{\omega} \tilde{p}g(p). \]In the standard continuum limit this expression has logarithmic growth. In the hyperreal setting, however, it is an exactly defined hyperfinite quantity: very large, but not indeterminate.
Structural deflation by Euler-Maclaurin
To isolate the finite regular part, the target function
\[ f(x)=\tilde{x}g(x) \]is deflated by the Euler-Maclaurin formula. Since the kernel has a pole at \(x=0\), the sum is taken from \(1\) rather than from \(0\). This gives exactly
\( {\LARGE{\textbf{+}}}_{\check{q}=1}^{\check{r}}f(\check{q})= \uparrow_1^{\check{r}}{f\left(x\right){\downarrow}x} +\check{f}(\check{r}) +\check{f}(1)\)\( +{\LARGE{\textbf{+}}}_{\check{m}=1}^{\check{n}-1}{\widetilde{m!}}B_m \left({}^{\acute{m}}f(\check{r})-{}^{\acute{m}}f(1)\right) +\mathcal{O} \left( {\widetilde{n!}}B_n \left({}^{\acute{n}}f(\check{r})-{}^{\acute{n}}f(1)\right) \right). \)
Here \(B_m\) are the Bernoulli numbers, and \(\mathcal{R}\) denotes the corresponding remainder term.
Integral: the hyperreal pole
The integral term
\[ P_{\mathrm{self}} = {\uparrow}_{1}^{\omega} \tilde{x}g(x) {\downarrow}x \]contains the dominant hyperreal background contribution. In the simplest undamped approximation it is proportional to \({}_e\omega\), and is therefore the nonstandard representative of the usual logarithmic divergence. By itself, however, this pole is not a directly measurable mass.
Boundary values at \(\omega\)
Because the cut-off function suppresses the spectrum at the upper boundary, the relevant Euler-Maclaurin boundary terms vanish there:
\[ f(\omega)=0,\qquad {}^1 f(\omega)=0,\qquad {}^2 f(\omega)=0,\qquad \dots \]Boundary values at \(1\): the regular remainder
The lower boundary at \(x=1\) avoids the singularity at \(x=0\) and supplies the finite boundary contributions. These terms depend on the chosen cut-off function and on the chosen renormalisation condition. They form a well-defined, finite, singularity-free remainder:
\[ R_{\mathrm{mass}} = \check{f}(1) – {\LARGE{\textbf{+}}}_{\check{m}\ge 1} {\widetilde{m!}}B_m{}^{\acute{m}}f(1) + \mathcal{R}_{\mathrm{finite}}. \]Thus the self-energy is separated algebraically into
\[ E_{\mathrm{self}} = P_{\mathrm{self}} + R_{\mathrm{mass}}. \]The first term is the hyperreal pole; the second term is the finite regular contribution after the common divergent structure has been separated off.
The bare mass as a counterterm
In standard QED, the bare mass \(m_0\) is not an independently observable mass. It is a parameter in the regularised theory. Its task is to absorb the divergent part of the self-energy so that the measured mass remains finite.
In the present nonstandard interpretation, this bare mass is represented by a hyperreal counterterm. It is not an arbitrary mystical infinity, but a precisely specified pole contribution fixed by the same cut-off structure:
\[ m_0 = P_{\mathrm{bare}} + R_{\mathrm{bare}}. \]The renormalisation condition requires that the pole in the bare mass cancel the pole in the self-energy:
\[ P_{\mathrm{bare}} = – P_{\mathrm{self}}. \]This equation is not an additional physical force law. It is the algebraic statement that the regularised theory is calibrated to the observed electron mass.
Exact algebraic cancellation and the physical mass
The physical electron mass is obtained from the bare mass plus the self-energy correction:
\[ m_{\mathrm{phys}} = m_0+E_{\mathrm{self}}. \]Substituting the deflated quantities gives
\[ m_{\mathrm{phys}} = \left( P_{\mathrm{bare}}+R_{\mathrm{bare}} \right) + \left( P_{\mathrm{self}}+R_{\mathrm{mass}} \right). \]Using the cancellation condition
\[ P_{\mathrm{bare}} = – P_{\mathrm{self}}, \]the hyperreal poles cancel exactly:
\[ m_{\mathrm{phys}} = \left( -P_{\mathrm{self}} \right) + P_{\mathrm{self}} + R_{\mathrm{bare}} + R_{\mathrm{mass}}. \]Therefore
\[ m_{\mathrm{phys}} = R_{\mathrm{bare}} + R_{\mathrm{mass}}. \]The measurable electron mass is thus not the hyperreal pole itself, but the finite remainder fixed by the renormalisation condition.
Interpretation
This nonstandard formulation replaces the informal language of divergent infinities by an exact hyperreal juxtaposition:
\[ \text{bare mass} + \text{self-energy} = \text{pole} + \text{counter-pole} + \text{finite remainder}. \]The pole and counter-pole cancel algebraically:
\[ P_{\mathrm{bare}}+P_{\mathrm{self}}=0. \]What remains is finite and measurable:
\[ m_{\mathrm{phys}} = R_{\mathrm{bare}} + R_{\mathrm{mass}}. \]The finite remainder is not obtained from the self-energy sum alone. It also depends on the renormalisation condition, that is, on how the theoretical parameters are matched to the experimentally observed electron mass.
Conclusion of the derivation
The electron self-energy need not be read as an undefined infinity. In the nonstandard formulation it becomes an exactly specified hyperreal pole plus a finite regular remainder.
The essential structural decomposition is
\[ E_{\mathrm{self}} = P_{\mathrm{self}} + R_{\mathrm{mass}}. \]The bare mass supplies the corresponding counter-pole,
\[ P_{\mathrm{bare}} = – P_{\mathrm{self}}, \]so that the physical mass remains finite:
\[ m_{\mathrm{phys}} = R_{\mathrm{bare}} + R_{\mathrm{mass}}. \]Thus the renormalisation of QED can be interpreted as an exact algebraic cancellation of hyperreal pole terms under a common cut-off. The measurable electron mass is not a divergent quantity, but the finite remainder left after this cancellation and after the theory has been matched to experiment.
© 2026 by Boris Haase