Nonlocal Energy-Momentum Tensor

• Nonlocal energy-momentum tensors are defined by incorporating integral kernels, nonlocal differential operators, and convolution structures that extend conventional local descriptions.
• They enable gauge invariance and consistent formulations of energy and momentum in theories like modified gravity, dispersive electromagnetism, and noncommutative field settings.
• Their applications include resolving angular momentum decomposition in quantum gauge theories and addressing anomalies via nonlocal corrections in effective quantum actions.

A nonlocal energy-momentum tensor is a generalization of the standard energy-momentum tensor (EMT) wherein the local functional dependence (typically algebraic or differential in the spacetime point) is replaced or supplemented by integral or convolution structures, or nonlocal differential operators. Nonlocal terms in the EMT can arise from fundamental nonlocalities (e.g., in noncommutative geometry or modified gravity), from higher-derivative Lagrangians, from gauge-invariant constructions that involve Wilson lines or phase factors, or from the analytic structure of quantum effective actions. The role and construction of nonlocal EMTs is critically important in the paper of quantum gauge theories, nonlocal gravity, dispersive electromagnetism, and the analysis of angular momentum–spin decompositions.

1. Conceptual Definition and Motivation

A nonlocal energy-momentum tensor is any EMT whose definition—either via the canonical Noether procedure or by functional metric variation—depends on fields (or their derivatives) at multiple spacetime points, or involves nonlocal differential or integral operators. Formally, this can arise via structures such as:

• Convolution kernels—e.g.,

$$T^{\mu\nu}(x) = \int d^4y \, K^{\mu\nu}(x,y) \, \mathcal{O}[A(y)]\,,$$

• Nonlocal differential operators—e.g., powers of the d’Alembertian or its inverse in modified gravity,
• Wilson line integrals in gauge-invariant high-energy QCD operator definitions,
• Frequency-dependent response kernels in dispersive media.

A primary motivation for considering nonlocal EMTs is to enforce physical symmetries (such as gauge invariance in gauge theories or diffeomorphism invariance in gravity) that are otherwise not compatible with strict locality, or to handle the fundamentally nonlocal structure of the underlying spacetime or medium.

2. Nonlocal EMTs in Gauge Theories and High-Energy QCD

The canonical EMT derived from Noether’s theorem is generally not gauge invariant because the underlying fields transform nontrivially under local symmetry groups. To recover gauge invariance, it is necessary to give up strict locality and introduce nonlocal constructs such as Wilson lines along appropriate directions (e.g., the light front):

• The gauge-invariant canonical energy-momentum tensor takes the schematic form

$$\mathcal{T}^{\mu\nu}_{\text{GI}}(x) = \bar{\psi}(x) \gamma^\mu \left[ P \exp \left( ig\int_x^{x+\infty} A_\rho(z) dz^\rho \right) \right] D^\nu\psi(x) + \text{h.c.}$$

Such nonlocality encodes the dynamical soft-gluon contributions required by factorization in QCD (Lorcé, 2015, Lorcé, 2016).

Parametrizations of nonlocal EMTs for nucleon matrix elements require a full basis of Lorentz structures, including dependence on auxiliary light-like vectors specifying Wilson line directions (n), yielding 32 complex-valued scalar functions (for a spin-1/2 target) as in

$$\langle p',S' | T^{\mu\nu}_a(0) | p,S \rangle = \bar{u}(p',S') \, \Gamma^{\mu\nu}_a(P,\Delta,N) \, u(p,S)$$

with detailed constraints from nonlocality and conservation laws (Lorcé, 2015).

Crucially, the nonlocal parameterizations impose LF constraints, which relate scalar functions by enforcing properties such as the vanishing of certain matrix element projections, and lead to well-known sum rules (e.g., Burkardt sum rule).

3. Nonlocality in Gravitational Energy-Momentum Tensors

In classical general relativity, the localization of gravitational energy is obstructed by the equivalence principle and the lack of a generally covariant local density. However, in higher-derivative and nonlocal theories of gravity, the Lagrangian can be a function of the metric and its derivatives to arbitrary (even infinite) order, or may contain explicitly nonlocal operators:

$$L_g = \sqrt{-g} \left[ R + a_0 R^2 + \sum_{k=1}^p a_k R \Box^k R \right]$$

or in the limit $p\to\infty$,

$L_g = \sqrt{-g} R f(\Box^{-1} R)$

with $\Box^{-1}$ defined via a retarded Green’s function (Capozziello et al., 2023, Capozziello et al., 2017, Capozziello et al., 2022).

Applying Noether’s theorem to such actions yields an energy-momentum complex (pseudo-tensor) of the form:

$$\tau^{\eta}_\alpha = \frac{1}{2\sqrt{-g}} \sum_{m=0}^{n-1} (-1)^m \partial_{i_1\ldots i_m} \left[ \frac{\partial L}{\partial g_{\mu\nu, \eta i_1 \ldots i_m}} g_{\mu\nu,\alpha} \right] - \delta^\eta_\alpha L$$

which is an affine object, not a true tensor, and is only locally conserved under affine coordinate transformations (Capozziello et al., 2017).

Nonlocal operators such as $\Box^{-1}$ are handled using integral definitions:

$$\Box^{-1} p(x) = \int d^4x' \sqrt{-g(x')} G(x,x') p(x')$$

which, combined with the metric dependence, gives rise to nonlocal energy-momentum pseudo-tensors. This energy-momentum complex inherits local conservation properties from generalized contracted Bianchi identities and underpins the flux formulae for gravitational wave emission—including possible non-Einsteinian polarizations (Capozziello et al., 2022, Capozziello et al., 2023).

4. Nonlocal EMTs in Dispersive and Non-Commutative Field Theories

Dispersive electrodynamics and noncommutative field theories furnish further examples where EMTs are fundamentally nonlocal:

• In dispersive media, the electromagnetic Lagrangian involves convolution with response kernels,

$$H^{ab}(x) = \int d^4y \tilde{M}^{abcd}(x-y) F_{cd}(y)$$

which leads to a nonlocal Lagrangian and corresponding EMT derived via an extension of Noether’s theorem (Heredia et al., 2020). These tensors may be symmetrized using a generalized Belinfante–Rosenfeld procedure, but inherit additional convolution or integral dependence on the field history.

• In noncommutative (Moyal) space, the EMT must be defined with the star-product,

$$(f * g)(x) = \exp\left[\frac{i}{2} \theta^{\mu\nu}\partial_\mu^x \partial_\nu^y\right] f(x) g(y) \big|_{y=x}$$

and is in general neither locally conserved nor fully gauge invariant. Gauge invariance can be restored via nonlocal “smeared” constructions involving Wilson lines, but these in turn spoil strict local conservation—a precise manifestation of emergent nonlocality (Balasin et al., 2015).

This mutual incompatibility between local conservation and gauge invariance is a robust signature of nonlocality induced by the deformation of spacetime structure (e.g., via a nonvanishing $\theta^{\mu\nu}$).

5. Relations to Angular Momentum Decomposition and Spin-Orbital Separation

Nonlocality in the EMT is crucial to the proper separation of spin and orbital angular momentum in gauge field theories and optics:

• The canonical (nonsymmetric) EMT separates orbital and intrinsic spin contributions, while the symmetric Poynting-type or Belinfante tensor “mixes” them.
• Experimental evidence from photon diffraction patterns demonstrates that only orbital angular momentum generates net momentum flow distortions, supporting a nonsymmetric (and potentially nonlocal) canonical EMT for spin-polarized photons (Chen et al., 2012).
• In high-energy QCD, only by employing nonlocal, gauge-invariant canonical EMT constructions can the separate spin and OAM contributions to the nucleon spin decomposition be unambiguously defined and related to observables (Lorcé, 2015, Lorcé, 2016).

The appearance of Wilson lines (nonlocal phase factors) is essential for restoring gauge invariance in these decompositions, and the corresponding correlation functions are sensitive to path choices, linking nonlocality directly to measurable matrix elements and distributions (GPDs, TMDs).

6. Nonlocal Corrections from Effective Quantum Actions

Quantum corrections in massless field theory, notably when computed via background field methods, can yield genuine nonlocal modifications to the effective action and hence to the energy-momentum tensor:

• In $ϕ^3$ theory in six dimensions, a one-loop computation leads to a nonlocal correction of form

$$\tilde{T}_{\mu\nu}(x) = -\frac{\lambda^2}{(4\pi)^3} \ln \frac{\Box}{\mu^2} \left( \text{local combinations of } \phi, \partial_\mu \phi \right)$$

which modifies either the virial current or the traceless part, depending on the coupling to gravity (Wu, 2015). In effective actions, such terms appear as logarithms of the d’Alembertian, yielding spatially extended (nonlocal) contributions to the EMT and, if coupled to gravity, to the trace anomaly.

A plausible implication is that infrared physics and anomaly phenomena in QFT may often be controlled by nonlocal structures in composite operators such as the EMT.

7. Physical Implications and Applications

Nonlocal energy-momentum tensors are essential for:

• Providing gauge-invariant and physically meaningful definitions of momentum, energy flow, and angular momentum in quantum chromodynamics and quantum electrodynamics, especially when relating partonic structure to experimental observables (Lorcé, 2015, Lorcé, 2016),
• Handling energy and momentum conservation and radiation fluxes in higher-derivative or nonlocal theories of gravity, where the standard local pseudo-tensor constructions fail or need to be generalized (Capozziello et al., 2022, Capozziello et al., 2023),
• Formulating electromagnetism in complex materials, dispersive or absorptive media, where the response functions encode long-range memory and retardation effects (Heredia et al., 2020),
• Interpreting anomalies, spin decomposition, and mass renormalization phenomena via the nonlocal behavior of composite operators.

Table: Summary of Major Contexts for Nonlocal EMTs

Physical Context	Source of Nonlocality	Key Manifestation
Gauge-invariant EMT in QCD, QED	Wilson lines (LF, path dependence)	Parametric dependence on n, constraints from nonlocality
Noncommutative field theory (Moyal space)	Star-product, noncommuting coords	Obstructed local conservation and/or gauge invariance
Dispersive electromagnetism	Integral response kernel	Convolutional form of the EMT
Higher-derivative / nonlocal gravity	$\Box^{-1}$, analytic operators	Pseudotensor includes integral or Green's function terms
Quantum loop corrections in massless fields	Logarithmic nonlocalities	Nonlocal corrections to trace, virial current, effective action

The interplay between locality, symmetry, and physical observability makes nonlocal energy-momentum tensors central to both formal and experimental fields ranging from high-energy QCD to gravitational-wave astrophysics to condensed matter and photonics. Their paper directly connects the mathematical structure of field theory to the architecture of observable conservation laws.