field effective theory wave theories

The gap

Supporting evidence — 4 representative gaps

• Theories on classical electromagnetic metamaterials (2026) · doi

  Metamaterial can be viewed as composite materials. Subsequently, the effective permittivity εeff can be determined. This framework includes theories such as: temporal and spatial factor, homogenization theory, Maxwell–Garnett theory, Lord-Rayleigh approximation, and Bruggeman mixing equations. enhancement electric-field 4.1.1.1 Theoretical formulation For metamaterials composed of two different materials, the dielectric constants of the two materials are ε1 and ε2, and their proportions are f2 = 1 − f, respectively. We can derive the equivalent dielectric constant (derivations are provided in Supplementary Material 8): f1 = f and εeff = f1ε1A + f2ε2 f1A + f2 (4.1.1) Here, A is Electric Field Enhancement Factor, which is defined as the ratio of the electric field strength of the medium. When A = 1, the original effective medium theory is obtained: εeff = f1ε1 + f2ε2 f1 + f2 (4.1.2) , the equivalent dielectric constant described by When A = 3ε2 ε1+2ε2 the Maxwell–Garnett model is derived as [125, 126]. εeff = ε2 + 3 f (ε1 − ε2) ε1 + 2ε2 − f (ε1 − ε2) (4.1.3) By adding higher-order terms of f, we obtain the following expression of the Lord–Rayleigh model [127]: εeff = ε2 + 3 f (ε1 − ε2) ε1 + 2ε2 − f (ε1 − ε2) + α2 f2 +  α3 f3 + ⋅ ⋅ ⋅ (4.1.4) Here, αn(n = 2, 3, …) denotes the higher-order coefficients associated with higher-order mixing terms. If the two materials have the same proportion, according to self-consistent Bruggeman theory [128], the Bruggeman Mixing Equation is adopted [128]: f1 ε1 − εeff ε1 + 2εeff + f2 ε2 − εeff ε2 + 2εeff = 0 (4.1.5) 4.1.1.2 Assumptions and limitations This section treats the composite material formed by two pure components and determines the effective permittivity based on the filling fraction. The starting point and objective of this theory are complete; however, each equation in the derivation introduces its own assumptions, which restrict its applicability. Equations 4.1.1–4.1.4 are the general forms suitable for arbitrary cases. However, the former contains an undetermined parameter A, whereas the latter involves αn. Both methods require experimental determination or additional assumptions. From another viewpoint, the latter can be regarded as a higher-order expansion of the former with respect to f. Assuming that the interaction between the two materials is negligible, A = 1, yielding Equation 4.1.2. Such an assumption holds only when the wavelength of the electromagnetic wave and the characteristic scale differ significantly. when the wavelength is much larger than the structural scale, scale related effects can be ignored, and when the characteristic scale is large, the composite can be treated as consisting of two independent constituents. Assumi

  Keywords: materials theory higher order scale composite effective bruggeman mixing electric field dielectric equation assumptions permittivity
• Gravitational radiations from periodic orbits around a black hole in the effective field theory extension of general relativity (2026) · doi

  While the references document periodic orbits in numerous modified gravity theories (Horndeski theory, Einstein-Æther theory, Horava-Lifshitz gravity, quantum-corrected spacetimes), a systematic comparison of gravitational radiation signatures across these distinct EFT frameworks has not been performed. Direct numerical comparisons of gravitational wave spectra from periodic orbits in different effective field theory extensions would clarify which modifications are detectable by space-based observatories.

  Keywords: gravitational radiation Horndeski theory Einstein-Æther Horava-Lifshitz quantum-corrected effective field theory detection
• Primes and Particles (2024) · doi

  The relationship between effective field theory approaches and the proposed framework of primes and particles needs further clarification and development.

  Keywords: relationship effective field theory approaches proposed framework primes particles needs further clarification development
• Wellness Effects of Ultra-Weak Photon Emission Convergence Evidenced by Quantitative EEG Analysis: Innovation in Noninvasive Neuromodulation (2026) · doi

  Direct biophoton field measurements during sessions are required to validate proposed mechanisms.

  Keywords: direct biophoton field measurements sessions required validate proposed mechanisms