Nonlocal Scalar Field Theory

Updated 4 November 2025

Nonlocal scalar field theory is a framework where scalar fields interact through nonlocal operators and kernels, influencing UV regularization and altering causality.
The approach employs analytic functions of the d’Alembertian and integral kernels to modify propagators, ensuring ghost-free dynamics and convergence in loop integrals.
Applications extend to quantum gravity, cosmology, and condensed matter, offering insights into soliton behavior, operator expansions, and thermal effects.

A nonlocal scalar field theory is a quantum or classical field theory for scalar fields in which the dynamics or interactions involve nonlocal operators, kernels, or structures—meaning that the field at a spacetime point can depend directly on the values of the field at other, possibly distant, spacetime points. Nonlocality can be introduced via analytic functions of differential operators, integral kernels, or action principles that couple field arguments at different locations. This framework is motivated by a range of contexts including string field theory, causal set theory, attempts to resolve ultraviolet pathologies in quantum field theory and gravity, and condensed-matter phenomena. Nonlocality in scalar theories yields profound consequences for ultraviolet regularity, operator mixing on the lattice, the structure of conservation laws, and the phenomenology of physical observables.

1. Algebraic and Integral Constructions of Nonlocality

Nonlocality arises in scalar field theory primarily via two technical routes. The most common class involves generalizing the kinetic, mass, or interaction terms by analytic functions of the d’Alembertian operator ( $\Box$ ), leading to models where the Lagrangian includes terms like $F(\Box)\phi$ for an entire or polynomial function $F$ (Briscese et al., 2015, Frolov, 2020, Erbin et al., 2021, Boos et al., 2021). The Fourier transform of the kinetic term is altered non-trivially, modifying the propagator and spectrum. Alternatively, nonlocality may be realized using delocalization kernels, where the field in the interaction term is replaced by a convolution: $\tilde{\phi}(x) = \int d^dy\, F(x - y)\phi(y)$ and the action involves $V(\tilde{\phi}(x))$ , with $F$ determining the range and type of nonlocal interactions (Tomboulis, 2015, Kegeles et al., 2015).

Fractional Laplacians and their extension to quantum field theory introduce another nonlocal structure where the kinetic term involves $(-\Delta)^{\alpha/2}\phi$ ; here, nonlocality is intrinsic and is connected to scale-invariant spectral representations and unparticle physics (Frassino et al., 2019).

Causal set theory leads to radical nonlocality: due to the lack of locality in the causal set, the d'Alembertian is manifestly nonlocal, expressed via summations over the causal past and depending on the entire causal structure (Belenchia et al., 2014, Sorkin, 2011, Carone et al., 2023).

2. Nonlocality in the Kinetic and Interaction Terms: Ghosts, UV Regularization, and Propagators

Nonlocal scalar field theories admit broad generalizations of the kinetic operator: $\mathcal{L} = -\frac{1}{2}\phi F(\Box/\Lambda^2)\phi - V(\phi)$ with $F(z)$ entire and $\Lambda$ the nonlocal scale (Briscese et al., 2015). Choosing $F$ so that the propagator $G(k^2)=1/(F(-k^2/\Lambda^2)(k^2-m^2))$ has only a single pole and no ghostlike or tachyonic singularities ensures unitarity at tree level (Frolov, 2020). A sufficient requirement is $F(z)=e^{g(z)}$ with $g(z)$ entire and $g(0)=0$ . Typical kernels in integral form must be entire in momentum space, and of rapid decay, for perturbative unitarity and UV finiteness (Tomboulis, 2015).

Nonlocality provides natural UV regularization: the propagators are exponentially suppressed at large momenta, rendering loop integrals convergent and, in specific cases, making the theory super-renormalizable or finite at the quantum level (Modesto, 2021). In the "asymptotic nonlocality" construction, a hierarchy of Lee-Wick-like higher-derivative theories yields, in the limit of infinitely many partners with masses scaled appropriately, a nonlocal propagator $e^{-\ell^2p^2}/p^2$ whose nonlocality scale regulates mass corrections independently of the Lee-Wick mass thresholds (Boos et al., 2021, Boos et al., 2021).

In interacting theories where the delocalized field enters the potential, the distinctions between "quasi-local" (compact-support) and "strictly nonlocal" (rapid decay but non-compact support) kernels control the spread and severity of acausal and UV effects (Tomboulis, 2015).

3. Initial Value Problem, Causality, and Quantum Structure

The introduction of nonlocal operators and kernels fundamentally alters the nature of the initial value problem (IVP) and causality. In general, the IVP for partial integro-differential equations in nonlocal theories admits existence of solutions but fails to ensure uniqueness: in both strictly and quasi-local kernel cases, acausal behavior arises, with the field at a point depending not only on past but also future data within a finite "delay region" determined by the kernel support. This acausality is strictly confined for quasi-local kernels, but for strictly nonlocal kernels it extends arbitrarily far though with vanishing weight (Tomboulis, 2015).

In the context of analytic (infinite derivative) nonlocalities, time-nonlocality may threaten causality and well-posedness. However, if the nonlocal operator is constructed to be an entire function (no additional poles), the IVP is well-behaved: field redefinitions can be used to map purely time-dependent configurations onto local theories with deformed potentials, yielding a canonical Hamiltonian and restoring the IVP, as in string-inspired models (Erbin et al., 2021). In more general situations with spacetime nonlocality, only a "local in time" but spatially nonlocal structure can be achieved; causality is guaranteed at the wavefront by appropriate analysis of the dispersion relations.

In causal set theory, the d'Alembertian is nonlocal and retarded; the field equations for a causal set element involve all elements in its causal past. The decoherence functional for quantizing the theory is built in a histories-based formalism, and the emergence of nonlocal equations is unavoidable (Sorkin, 2011, Belenchia et al., 2014).

4. Phenomenological Manifestations: Spectra, Solitons, and Thermodynamics

Nonlocality induces testable deviations in quantum and classical field properties.

Bound states and Tunneling: The spectrum of particles in an external potential is markedly altered. In WKB analysis, nonlocality leads to compressed or non-equidistant spectra for bound states due to the non-polynomial Hamiltonian, and enhances tunneling rates through potential barriers, increasing barrier transparency at high energies (Frolov, 2020).
Domain Walls and Kinks: The existence and properties of soliton solutions, such as kinks in $\phi^4$ -like models, are quantitatively altered in nonlocal scalar field theory. Perturbative corrections to the kink profile and energy density are computable, yielding nontrivial constraints on the form factor for the expansion to converge. Generic nonlocal potentials reconstructed to support given solitons involve infinite series in the field (Andrade et al., 31 Jul 2024).
Vacuum and Thermal Properties: One-loop corrections to the effective potential in nonlocal scalar models are typically extremely small, suppressed by the scale of nonlocality, and are unobservable in cosmological data even at high energies (e.g., inflation) for realistic scales ( $\Lambda \sim M_{\rm Pl}$ ) (Briscese et al., 2015). At finite temperature, nonlocality affects the partition function only at the level of interactions (not in the free theory, provided no new poles are present), leading to phenomena such as breakdown of thermal duality and emergence of Hagedorn-like behavior at high temperature (Dijkstra, 2019).
Unruh and Observer Effects: In nonlocal field theory, observer-dependent phenomena such as the Unruh-Fulling effect are inert to nonlocality in the Lorentz-invariant case, but modified by overall form factors (multiplicative suppression or enhancement) in Lorentz-noninvariant cases. The Unruh temperature itself remains universal when using Unruh quantization (Das et al., 2022).
Nonlocality from Geometry: In curved momentum space ("relative locality"), scalar field theory is fundamentally nonlocal in spacetime, with the nonlocality dictated by a nontrivial, noncommutative star product arising from the geometry of momentum space. The kinetic term can be made local by a choice of base point, but interactions are always nonlocal (Freidel et al., 2013).

5. Symmetries, Conservation Laws, and Operator Expansions

Nonlocal scalar field theories modify basic structural aspects of field theory:

Conservation Laws: Standard Noether's theorem does not yield exactly conserved currents in nonlocal field theories. Instead, symmetries lead to generalized conservation laws, where the divergence of the Noether current is balanced by nonlocal correction terms depending on the kernel or analytic structure of the action (Kegeles et al., 2015, Scomparin, 2022). In certain cases with internal symmetries or for specific nonlocal structures, the correction term vanishes (e.g., $U(1)$ phase invariance in a symmetric kernel).
Operator Product Expansions (OPE): Nonlocal scalar field theory enables the development of "smeared" operator product expansions (sOPE). Nonlocal operators can be systematically decomposed into locally smeared operators (defined, e.g., via gradient flow), with perturbatively calculated Wilson coefficients depending on the smearing scale. This decomposition eliminates power-divergent mixing on the lattice and provides a practical link between lattice results and continuum nonlocal operators, crucial for QCD and hadronic physics (Monahan et al., 2015).

6. Existence and Solutions of Nonlocal Nonlinear Equations

Nonlocal scalar equations of elliptic and parabolic type—including those with competing nonlocal terms—admit both radial and nonradial solutions under broadly applicable growth conditions on the nonlinearities. The analysis, often via variational methods with mountain pass geometry, is more delicate than in the local case due to failure of compactness, requiring nonlocal Brezis-Lieb lemmas, truncation, and symmetry-based arguments. Thresholds for the existence of solutions become functions of the relative strength of the competing nonlocalities (d'Avenia et al., 2020).

On causal sets, the presence of "defects" or subregions with altered propagation rules leads to characteristic modifications to Green's functions: different types of endpoints (ordinary points, defects) yield multivalued propagators for the same invariant interval, with coarse-grained effects interpretable as defect-induced mass and wavefunction renormalization (Carone et al., 2023).

7. Applications and Directions

Nonlocal scalar field theories have impactful roles in:

UV-complete Models: As building blocks for UV-complete models in particle physics and gravity, particularly in frameworks such as infinite-derivative gravity and asymptotically nonlocal gauge theories, where quadratic divergences are softened by emergent nonlocal scales independent of new physical resonances (Boos et al., 2021, Boos et al., 2021).
Quantum Gravity/Condensed Matter: In group field theory and hydrodynamic approximations of Bose-Einstein condensates, nonlocality arises structurally in the action, demanding generalized symmetry and conservation law frameworks (Kegeles et al., 2015).
Cosmology and String Theory: String field theory naturally motivates nonlocal scalar field theories, especially in cosmological applications (rolling tachyons, cosmological acceleration), for which initial value problems and Hamiltonian formulations have been clarified (Erbin et al., 2021, Vernov, 2010).

Representative Table: Kinetic Terms and Propagator Structure

Model Type	Kinetic Term	Propagator Structure
Local	$\Box - m^2$	$1/(k^2 - m^2)$
Infinite-derivative	$\Box f(\Box)$ , $f$ entire	$1/(f(-k^2) k^2 - m^2)$ , entireness avoids extra poles
Kernel (integral)	$F(x-y)$ via convolution	$\hat{F}(k)^2/(k^2 - m^2)$
Causal set (retarded)	Retarded/interval sums	Branch cut yields continuum of masses
Curved momentum space	Geometry-induced, nonlinear	Nonlocal star product in position space

Nonlocal scalar field theory thus encompasses a rich and varied set of models, mathematical tools, and physical consequences. The interplay between analytic nonlocal operators, integral kernels, nonlocal conservation laws, and the structure of fundamental interactions makes this subject a central arena for contemporary field-theoretical research across quantum gravity, cosmology, and high-energy physics.