Engineered Wightman Power Spectra

• Engineered Wightman power spectra are tailored two-point correlators whose features are modulated via system parameters, boundary conditions, and topological defects to probe quantum field dynamics.
• They are engineered using techniques such as topological modifications, spectral shaping, and quantum state control, resulting in distinctive analytic structures like exponential decay and sharp cutoffs.
• These spectra influence operator growth and observables by affecting Krylov complexity and have practical applications in precision sensing, quantum control, and theoretical diagnostics.

Engineered Wightman power spectra are precisely constructed two-point correlation functions (in frequency or momentum space) whose analytic, algebraic, or geometric features are tailored—by adjusting the physical system, boundary conditions, topology, or quantum state—to explore or control specific aspects of quantum field theory (QFT), statistical mechanics, condensed matter, or quantum information science. These engineered spectra serve as diagnostic tools, probes of dynamics and vacuum structure, and fundamental observables in both theoretical research and experimental design.

1. Formal Construction and Characterization

The Wightman function $G(x, x') = \langle \Phi(x) \Phi(x') \rangle$ encodes the two-point correlations of quantum fields and, upon Fourier transformation, yields a power spectrum $f^{(W)}(\omega)$ that describes the distribution of fluctuation power across frequencies or momenta. Engineered Wightman spectra are achieved by modulating the physical system's input parameters or boundary conditions to produce desired spectral properties.

A canonical example is provided by scalar quantum fields in the presence of global or topological modifications such as cosmic strings. The metric is altered to encode angular deficits—for de Sitter and anti-de Sitter space, this leads to the introduction of an integer parameter $q$ representing the conical singularity, and a Wightman function expressed as a sum over $k$-images:

$G(x, x') = \sum_{k=0}^{q-1} G_k(x, x')$

Each $G_k$ term has explicit dependence on the geometric/topological features of the background, e.g., for de Sitter space:

$$G_k(x, x') = \frac{\Gamma(3/2 + \nu) \Gamma(3/2 - \nu)}{8\alpha^2\pi^2} \frac{P^{(-1)}_{\nu-1/2}(u_k)}{\sqrt{1-u_k^2}}$$

with $u_k$ encoding the proper distance and image indexing (Mello, 2012).

In quantum dynamical systems, the engineered power spectrum $f^{(W)}(\omega)$ is experimentally realized or theoretically constructed with exponential decay or sharp cutoffs, often via step functions or filters, thus controlling the support and analytic structure (e.g., $\Theta(\mu-\omega)$ for chemical potential $\mu$, or exponential factors for thermal envelopes) (He et al., 18 Sep 2025).

2. Engineering Techniques: Topology, Spectral Modulation, and Quantum State Preparation

Engineered power spectra can be generated by several mechanisms:

• Topological Engineering: The insertion of defects (e.g., cosmic strings) leads to extra image sums in the Wightman function, with the string-induced contributions being finite and highly spatially dependent. This alters the near-string behavior and introduces anisotropy in the power spectrum (Mello, 2012).
• Spectral Shaping: In quantum dynamical and sensing systems, the spectrum is intentionally modulated by system-bath couplings, input noise spectra, or measurement settings. The resulting engineered spectrum can optimize sensitivity or probe noise structure. Quantum Fisher information analysis determines the quantum limit for estimation accuracy in such settings (Ng et al., 2016).
• Quantum State Control: The choice of initial states (thermally populated, squeezed, or non-equilibrium density matrices) directly affects the cumulant structure and thus the Wightman correlators and power spectrum; generating asymmetry or non-Gaussian features is possible by engineering the state $\rho$ (Vinayak, 2022).
• Polymer Quantization and Discreteness: Non-standard quantization procedures such as polymer quantization introduce fundamental scales, leading to bounded Wightman functions and effective minimum length scales—this engineering regularizes short-distance divergences and encodes putative spacetime discreteness (1705.01431).

3. Influence on Krylov Complexity and Operator Growth

The structure of engineered Wightman spectra imposes strong constraints on the evolution of Krylov complexity, a measure of operator growth in quantum systems. Key findings include:

• Single-sided Exponential Decay: For spectra decaying exponentially and cut off on one side (as with positive chemical potentials in fermionic systems), the Lanczos recursion coefficients $a_n$ and $b_n$ transition from near-flat behavior to asymptotic linear decay, with $b_n$ showing a two-stage slope—a unique signature of a non-symmetric, engineered spectrum. This results in quadratic growth of Krylov complexity:

  $K(t) = 1 + (\kappa t)^2$

  where $\kappa$ is set by the spectral decay parameter. Such behavior matches the SL(2,ℝ) algebraic case where $\gamma^2=4\alpha^2$ (He et al., 18 Sep 2025).

• Double-sided Exponential Decay: For symmetric, double-exponential spectra, the Lanczos coefficients yield exponential growth of complexity consistent with the maximal chaos bound:

  $$K(t) \propto \sinh^2(\pi t/\beta) \sim e^{2\pi t/\beta}$$

• Thermalization and Quasinormal Modes: In holographic (AdS/CFT) settings, the engineered relaxation of Wightman correlators is dictated by the lowest quasinormal mode of the dual black hole; the engineered spectrum thereby determines the exponential relaxation rate of both correlators and effective occupation numbers, independent of the microscopic details of the quench or system (Keranen et al., 2015).

These structures can be summarized in the table below:

Engineering Type	Spectral Structure	Complexity Growth
Single-sided exponential	$$f^{(W)}(\omega) \sim e^{\kappa \omega}\Theta(\mu-\omega)$$	Quadratic ($K(t) \propto t^2$)
Double-sided exponential	$f^{(W)}(\omega) \sim e^{-\beta|\omega|}$	Exponential ($K(t)\propto e^{2\pi t/\beta}$)
Topological (cosmic string)	Image-sum; spatial anisotropy	Local enhancements; alters $⟨\Phi^2⟩_{ren}$

4. Connections to Geometry, Topology, and Singularity Structure

Engineered Wightman spectra are deeply sensitive to the underlying geometry and topology:

• Geometric Focusing and Conjugate Planes: In spacetimes exhibiting impulsive gravitational waves, the Wightman function and its singularities are controlled by the van Vleck determinant and the world function $\sigma(z,z')$. Focusing at conjugate planes leads to transitions in the singularity structure, with engineered power spectra exhibiting twofold or fourfold changes in divergence structure when crossing these critical surfaces—this is a direct imprint of the engineered metric on quantum correlations (Cho, 2023).
• Polymer-induced Discreteness: In polymer quantization, the engineered suppression of short-distance correlation amplitudes produces a finite effective zero-point length, reflected in the boundedness of the Wightman function; this regularizes the ultraviolet, introduces nonperturbative corrections, and suppresses detector signatures such as the Unruh effect (1705.01431).

5. Experimental and Theoretical Applications

Applications of engineered Wightman power spectra span sensing, quantum control, and fundamental field theory:

• Precision Sensing and Quantum Fisher Information: In optomechanics and gravitational wave detection, engineering the noise spectrum maximizes parameter estimation precision; spectral photon counting can outperform standard homodyne detection by approaching the quantum limit, especially in low SNR scenarios (Ng et al., 2016).
• Quantum Linear Systems Identification: For quantum linear systems driven by stationary inputs (vacuum, squeezed), the frequency-domain power spectrum encodes all identifiable system parameters. System realization algorithms allow one to reconstruct a physical QLS realization unique up to symplectic equivalence, providing a concrete route to engineering desired output correlations (Levitt et al., 2016).
• Conformal and Topological Field Theory: In conformal field theory, all three-point Wightman functions and their power spectra can be constructed in momentum space using the R-product and Appell $F_4$ functions, with analytic continuation and boundary conditions (symmetry, OPE coefficients) serving as the main engineering tools (Gillioz, 2021). Topologically nontrivial settings (e.g. cosmic strings in (A)dS) allow deliberate introduction of anisotropies and finite corrections to the spectrum (Mello, 2012).

6. Diagrammatic and Algebraic Advances

Diagrammatic expansions—originating in generalizations of Wick’s theorem for non-vacuum or interacting systems—enable efficient computation and characterization of engineered Wightman spectra:

• Cumulant-based Diagrams: For general density matrices (beyond Gaussian states), Wightman correlators are decomposed into sums of cumulants representing connected multi-point contractions. Diagrammatic formalism for harmonic and anharmonic oscillators allows systematic expansion in powers of interactions and incorporates state engineering directly via cumulant coefficients (Vinayak, 2022).
• Algebraic Structure and CCR Algebras: In bosonic Wightman quantum field theories, algebraic engineering is achieved by fixing the two-point functions, which via the symplectic structure and the Weyl CCRs uniquely define the spectrum of operators in the associated $C^*$-algebra; thus, the engineered power spectrum is embedded in the spectral properties of the unique CCR algebra (Raab, 2020).

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Engineered Wightman power spectra provide both a theoretical framework and practical methodology for modulating and diagnosing physical, algebraic, and geometric features of quantum fields. Advances in both mathematical construction (image sums, spectral engineering, analytic continuation) and experimental realization (measured spectra, quantum-limited estimation, noise tailoring) drive the field forward, unifying insights across quantum field theory, quantum information, and condensed matter physics.