A Geometric Conjecture on Compressing QED Loops into a Single Functional Integral

J. Rogers, SE Ohio

Abstract

We propose a conjecture that the perturbative expansion of quantum electrodynamics (QED), traditionally expressed as an infinite series of Feynman loop diagrams, can be reformulated as a single geometric functional integral over spacetime configurations. By interpreting the running of the coupling constant, loop integrals, and momentum-space variables as reflections of real spacetime length contractions, time dilations, and field geometries, we hypothesize that all loop contributions can be simultaneously integrated. This approach suggests a unified geometric representation of QED, offering potential conceptual clarity, computational simplification, and a bridge between relativistic geometry and quantum field theory.

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1. Introduction

The perturbative expansion of QED has provided remarkably precise predictions, yet its standard interpretation relies on abstract notions of virtual particles and loop integrals over momentum space. While the mathematical formalism is well established, the physical ontology of these integrals remains opaque.

Recent work in Planck-normalized coordinates suggests that energy, momentum, mass, and force are all projections of a single underlying spacetime process, where $E\sim p\sim 1\mathrm{/}L\sim m\sim F\sim$f in natural units. In this view, the running of the fine-structure constant can be interpreted geometrically as the response of the electromagnetic field to relativistic length contraction and time dilation, rather than as virtual particle fluctuations.

This paper conjectures that the full set of loop integrals in QED may be expressible as a single functional integral over geometric configurations, rather than as a perturbative series evaluated loop by loop.

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2. Background

2.1 Perturbative QED and Loop Integrals

In conventional QED, the Dyson series expansion leads to terms with increasing numbers of loop integrals:

$$S=\mathcal{T}\exp \text{\hspace{0.17em}⁣}[i\text{\hspace{0.17em}⁣}\int d^{4}x{\mathcal{L}}_{\mathrm{i}\mathrm{n}\mathrm{t}}(x)]=1+i\int d^{4}x{\mathcal{L}}_{\mathrm{i}\mathrm{n}\mathrm{t}}+\frac{(i{)}^2}{2}\int d^4{x}_1{d}^4{x}_2{\mathcal{L}}_{\mathrm{i}\mathrm{n}\mathrm{t}}(x_1){\mathcal{L}}_{\mathrm{i}\mathrm{n}\mathrm{t}}(x_2)+\dots$$

Each loop is typically interpreted as a sum over virtual particle momenta. UV divergences arise in the short-distance limit, requiring renormalization.

2.2 Geometric Interpretation

In natural units, $p \sim 1/L$, so momentum-space integrals are equivalently integrations over spacetime configurations:

$$\int d^{4}p⟷\int d^4(1\mathrm{/}L)$$

Thus, each loop integral can be interpreted as an integration over:

• All possible length contractions ($\gamma$ factors)

• All local time dilation scales

• All orientations and configurations of the flattened charge distribution

Renormalization then imposes a physically motivated cutoff at small length scales rather than an abstract mathematical trick.

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3. Conjecture

Conjecture: There exists a reformulation of QED in which the entire perturbative expansion of loop diagrams is equivalent to a single functional integral over geometric spacetime configurations, which simultaneously accounts for all length, time, and field deformations that the loops individually encode.

Formally, if $Z$ is the generating functional of QED:

$$Z=\int \mathcal{D}\psi \mathcal{D}{A}_{\mu }{e}^{i\int d^{4}x\mathcal{L}[\psi ,A_{\mu }]}$$

then there exists a change of variables $(\psi, A_\mu) \to \mathcal{G}$, where $\mathcal{G}$ encodes local geometric variables (length contraction, time scaling, field shape), such that

$$Z=\int \mathcal{D}\mathcal{G}{e}^{i{S}_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}}[\mathcal{G}]}$$

with $S_{\rm geom}$ capturing the same physics as the sum over all loop diagrams.

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4. Implications

• Conceptual clarity: The virtual particle interpretation becomes unnecessary; all integrals are real geometric contributions.

• Computational potential: Instead of summing loops sequentially, the entire functional integral might be approximated or evaluated directly in geometric coordinates.

• Unification: Bridges relativistic geometry with quantum field scaling, giving a physical interpretation to the renormalization group and β-functions.

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5. Challenges and Open Questions

1. Measure definition: How to define $\mathcal{D}\mathcal{G}$ rigorously to capture all degrees of freedom without redundancy.

2. UV cutoff: What geometric principle replaces ad-hoc regularization, and how does the Compton or Planck scale emerge naturally?

3. Non-Abelian generalization: Can this approach extend to QCD or the electroweak theory?

4. Practical evaluation: How can one construct numerical or analytical methods to approximate $Z$ in geometric variables?

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6. Conclusion

We propose that QED loop expansions are not inherently “quantum” but encode geometric integrations over spacetime contractions and dilations. If true, this reformulation could compress all perturbative loops into a single functional integral in geometric variables, providing a new conceptual and computational framework for quantum field theory. We invite the community to explore this conjecture, define the measure rigorously, and test whether this geometric perspective can reproduce known QED predictions and perhaps suggest new avenues for unification.

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References:

1. Peskin, M. E., & Schroeder, D. V. An Introduction to Quantum Field Theory. 1995.

2. Weinberg, S. The Quantum Theory of Fields, Vol. I. 1995.

3. The other recent papers on this blog.