Finite Formulation of QFT

• Finite formulation of QFT is a framework ensuring all observables remain manifestly finite by using spectral algebraic methods and novel regularization techniques.
• It employs nonlocal entire-function regularization and the θ-operation to damp high-momentum contributions and eliminate ultraviolet divergences.
• These approaches enable consistent treatment of interacting theories and curved spacetimes, addressing the hierarchy problem without fine-tuning.

A finite formulation of quantum field theory (QFT) seeks to construct the theory so that all observables and intermediate quantities—particularly loop integrals in Feynman diagrams and correlation functions—are manifestly finite at every stage. This program addresses longstanding difficulties with ultraviolet (UV) divergences, the necessity of renormalization, and the influence of high-energy physics on low-energy observables ("hierarchy problem"). A variety of approaches exist, leveraging different mathematical structures, regularization schemes, and reformulations of the underlying fields and spacetime, each tailored to maintain locality, unitarity, gauge invariance, and compatibility with observed low-energy physics.

1. Algebraic, Spectral, and Functional Approaches to Finiteness

Spectral and algebraic approaches replace the traditional expansion of quantum fields in terms of plane-wave momentum eigenstates with expansions over complete orthogonal polynomials in energy or spectral variables. The energy variable, often expressed as $z = E^2 - M^2$, becomes the key parameter, and the central object is a set of orthogonal polynomials $p_n(z)$ generated via a symmetric three-term recursion relation:

$$z\, p_n(z) = a_n\, p_n(z) + B_{n-1}\, p_{n-1}(z) + B_n\, p_{n+1}(z),\quad n=0,1,2,\dots$$

These polynomials are orthogonal with respect to a weight function $\rho(z)$ as

$$\int_\Omega \rho(z) p_n(z) p_m(z) dz + \sum_{z_i} \omega(z_i) p_n(z_i) p_m(z_i) = \delta_{n,m}$$

(as in (Alhaidari, 2022, Alhaidari, 7 Jul 2024)).

The expansion of fields

$$Y(t, \mathbf{r}) = \int dE\, e^{-iEt}\, y(E, \mathbf{r})\, a(E) + \sum_{n=0}^{N} \phi_n(\mathbf{r})\, a_n$$

captures both the continuum (“free” part) and discrete (“internal structure”) components. All physical information about particle propagation, structure, and interactions is encoded in the spectral polynomials and their weight functions.

The closed-loop (Feynman diagram) integrals, when recast in this spectral framework, become sums of products of spectral propagators of the form $p(z) p_n(z) p_m(z)$, and, by orthogonality, these are always bounded. In typical calculations one finds, for example,

$$S_{n,m,...}(a,b,...) = \int p(w) p(w \oplus a) p(w \oplus b) ... dw$$

where the “$\oplus$” operation encodes spectral parameter conservation at vertices. Numerical evidence demonstrates that such integrals remain finite as the degrees $n$ increase (Alhaidari, 2022, Alhaidari, 7 Jul 2024), ensuring manifest UV finiteness.

The same strategy underpins finite field-theory loop calculations in both 1+1 and 3+1 spacetime dimensions, and extends to interacting theories via the modification of the recurrence relations by interactions.

2. Nonlocal Entire-Function Regularization and Holomorphic Extension

A class of finite QFTs employs nonlocal regularization via insertion of entire functions of the d'Alembert operator $\Box$ in the kinetic or interaction terms,

$F(\Box) = \exp\left(\frac{\Box}{M_*^2}\right),$

where $M_*$ is a physical UV scale. These entire functions are holomorphic everywhere in the complex plane and of order $\gamma > 1/2$, ensuring that they damp high-momentum contributions exponentially without introducing additional poles or ghosts (Moffat et al., 14 Jul 2025, Green et al., 2020).

The nonlocal regulator is inserted into the gravitational and gauge actions:

$$S_\text{grav}^{(\text{reg})} = -\frac{1}{16\pi G_N} \int_C d^4z\, \sqrt{-\det g_{\mu\nu}(z)}\, g^{\mu\nu}(z)\, F(\Box/M_*^2) R_{\mu\nu}(z)$$

for gravity (see also gauge and Dirac sectors). BRST invariance and holomorphic gauge symmetry are preserved because the regulator commutes with diffeomorphisms and gauge transformations.

At the quantum level, each vertex in a Feynman diagram is exponentially suppressed at high momentum, rendering every loop integral finite to all orders. The one-loop effective action is computed as:

$$\Gamma^{(1)} = \frac{i}{2} \sum_r (-)^{F_r} \text{Tr} \ln[F(\Box/M_*^2) \Delta_r^{(0)}]$$

By construction, there are no new counterterms or pathological complex poles. As $|p^2| \ll M_*^2$, $F(-p^2/M_*^2) \approx 1$, and infrared physics—i.e., the predictions of General Relativity and Standard Model gauge theory—are precisely recovered (Moffat et al., 14 Jul 2025).

3. Local Covariance, Operator Expansion, and States in Curved Spacetime

In strongly curved backgrounds, standard QFT constructs depending on Poincaré symmetry (unique vacuum, global energy) are unavailable. The finite formulation is achieved via:

• Algebraic formulation: The field algebra $\mathcal{A}$ is constructed locally from the spacetime metric, and states are positive linear maps $\omega: \mathcal{A} \rightarrow \mathbb{C}$. The GNS construction then yields a Hilbert space.
• Microlocal spectrum condition: Replaces the global spectrum condition by constraining the singularity structure of $n$-point functions using wavefront sets to mimic local positive-frequency behavior.
• Operator product expansion (OPE): Serves as a substitute for vacuum expectation values, encoding local information via covariant distributions such as locally defined Hadamard functions (0907.0416).

This ensures that quantization and renormalization remain finite by design, and the framework is robust even in the absence of a vacuum, as captured in the projective limit approach (Lanéry, 2016).

4. Finite Formulation via θ-Operation and Callan–Symanzik Equations

A pragmatic approach to finiteness is the systematic differentiation of $n$-point functions with respect to bare parameters (e.g., $m_0^2$), dubbed the “θ-operation”:

$$\Gamma_\theta^{(n)} \equiv -i \frac{d}{dm_0^2} \Gamma^{(n)}$$

This derivative lowers the superficial degree of divergence, and further iterations ($kθ$) render all relevant Green's functions finite (Mooij et al., 1 Aug 2024, Ageeva et al., 23 Sep 2024).

These θ-differentiated correlators satisfy “finite Callan–Symanzik equations,” e.g.:

$$2i m^2(1+\gamma)\Gamma_\theta^{(n)} = [2m^2\frac{\partial}{\partial m^2} + \beta \frac{\partial}{\partial \lambda} + n\gamma]\Gamma^{(n)},$$

with analogous equations for higher θ-derivatives. Because the divergence is eliminated at each stage, renormalization counterterms are rendered unnecessary.

The method generalizes to nonrenormalizable theories, e.g., $\lambda\phi^4 + g\phi^6/M^2$, producing uniformly finite effective potentials and correlators. The scalar mass parameter $m^2$ receives no $M^2$-dependent corrections (Ageeva et al., 23 Sep 2024), obviating the conventional hierarchy problem and fine-tuning associated with heavy scales.

5. Consequences for Hierarchies, Naturalness, and Physical Observables

The finite formulations above eliminate the possibility of large quantum corrections to low-energy parameters from higher mass scales. For instance, when analyzing scalar field theories with multiple vacuum expectation values (vevs) $\langle \phi \rangle = v$, $\langle \Phi \rangle = V$ with $v\ll V$, no large quantum corrections to $m^2$ proportional to $M^2$ arise. The hierarchy between $v$ and $V$ is maintained without fine-tuning (Mooij et al., 1 Aug 2024, Ageeva et al., 23 Sep 2024).

Decoupling of heavy degrees of freedom is automatic, and all physical parameters—masses, couplings, vacuum expectation values—are determined by finite boundary conditions imposed on the physically measurable (renormalized) quantities.

6. Operator, Projective, and Discrete Approaches

Alternative formulations guarantee finiteness by other means:

• Operator form with complexification: Introducing both self-adjoint (real) and anti self-adjoint (imaginary) coordinate and momentum operators, arranged to achieve cancellation of infinities in the Hamiltonian through identification, e.g., $k_\text{im} = i k_\text{re}$ (Prvanovic, 2022).
• Projective limits: Building the state space as a projective limit of finite-dimensional Hilbert spaces; the resulting theory is inherently finite and does not require a vacuum (Lanéry, 2016).
• Process algebra and causal set approaches: Quantum fields as combinatorially-generated, discrete objects (informons) embedded in a causal tapestry, resulting in manifest finiteness and a unified description of particles and fields (Sulis, 2015).
• Finite element discretizations: Hierarchical finite element meshes discretize spacetime, and integrating out centroid field variables yields renormalized actions that converge to the continuum Laplacian; the renormalization flow is driven entirely by the integration procedure (Kar et al., 2013).

7. Phenomenology, Testability, and Open Challenges

These finite frameworks are constructed to reproduce the correct low-energy limit (Standard Model and General Relativity) in the appropriate domain. Quantities such as Hawking spectra, Lamb shift measurements, and gravitational wave phase shifts have calculable, finite corrections linked to the regularization scale or spectral structure (Moffat et al., 14 Jul 2025). UV-completeness does not introduce acausal artifacts; regulators are chosen to preserve locality, Lorentz invariance, unitarity, and gauge symmetry (including BRST invariance).

Matching observed quantities (such as vacuum energy) becomes a question of physical parameter input rather than regularization scheme; for example, the vacuum energy density after quantum coordinate averaging is finite yet necessitates the presence of additional bosonic modes to align with cosmological data (Cahill, 12 Jun 2024).

In summary, modern finite formulations of quantum field theory employ spectral algebraic methods, nonlocal entire-function regularization, local-covariant operator algebra, and projectively-constructed state spaces to eliminate ultraviolet divergences, ensure the mathematical rigor of all constructions, and avoid the need for renormalization counterterms, thereby delivering a controlled framework applicable to both renormalizable and nonrenormalizable interactions, curved spacetime, and quantum gravity.