Moment-to-Moment Theory

• Moment-to-Moment Theory is a framework describing evolving moment sequences using mathematical, probabilistic, and experiential insights.
• It employs convex geometry, moment mapping, and optimization to quantify atomic representations and system dynamics.
• The theory offers practical applications in quantum dynamics, cognitive science, and electronic structure, illuminating emergent temporal behavior.

Moment-to-Moment Theory encompasses a set of interconnected frameworks and results that describe phenomena evolving as sequences of moments—ranging from probability theory and convex geometry of moment cones, to quantum dynamics, neuroscience, and the phenomenology of time. In mathematics and physics, it is often formalized in the analysis and synthesis of moment sequences, the geometry of their associated representation spaces, and the stochastic or structural dynamics that govern their evolution and stability. In cognitive science and philosophy, it undergirds mechanistic accounts for temporality, experience, and decision-making as processes organized "moment by moment." What unites these diverse threads is the explicit focus on the analysis, propagation, and interplay of moments—whether statistical, algebraic, or experiential—across different levels of abstraction and physical implementation.

1. Foundations: The Structure of Moment Cones and Moment Sequences

The core mathematical formalism for moment-to-moment theory is the moment cone $\mathcal{S}_A$, defined for a finite set of measurable functions $A = \{a_1, ..., a_m\}$ on a space $X$ as

$$\mathcal{S}_A = \left\{ s \in \mathbb{R}^m ~:~ s = \int_X s_A(x)\, d\mu(x) \text{ for some positive measure } \mu \right\},$$

where $s_A(x) = (a_1(x), ..., a_m(x))^T$. This convex cone encodes all possible moment sequences arising from integrating $A$ against positive measures $\mu$. A central result is Richter's theorem, stating that every $s$ in $\mathcal{S}_A$ admits a finitely atomic representing measure with at most $m$ atoms (if $m = \dim \operatorname{span} A$) (Dio et al., 2018).

For unbounded domains, distributions on moment spaces (e.g., $\mathcal{M}_n([0, \infty))$, $\mathcal{M}_n(\mathbb{R})$) are constructed as limiting objects of compact-support moment densities. Canonical moments and $z$-variables (with specific bounds) allow explicit characterization, leading to the identification of independent Gamma-distributed recursion variables in the unbounded limit (Dette et al., 2010). The structure and complexity of moment cones are further refined via analysis of facial structure (faces, exposed faces), which dictate the uniqueness and minimality of atomic representations.

2. Carathéodory Number, Differential Structure, and Optimization

The Carathéodory number $C_A$ quantifies the minimal number of atoms needed in the representing measure for any $s \in \mathcal{S}_A$. Precisely, $C_A$ is the smallest $k$ such that every $s$ can be written as a convex combination of at most $k$ points on the moment curve. In univariate cases, $C_A = \lceil (d+1)/2 \rceil$ for polynomial degrees $d$. For multivariate and higher degree truncated problems, tight bounds for $C_A$ follow from the geometry and differential properties of $\mathcal{S}_A$ (Dio et al., 2018).

The moment map $$S_{k,A}: \mathbb{R}_+^k \times X^k \to \mathbb{R}^m$$ defined by

$S_{k,A}(C, X) = \sum_{i=1}^k c_i s_A(x_i)$

admits a derivative whose full rank condition determines a lower bound on the possible atomic length (minimal $k$) and regularity of a representing measure. Regular moment sequences are those admitting a measure such that the Jacobian $DS_{k,A}$ attains maximal rank; others are considered singular. Analysis of the moment map—and tools such as the flat extension property—provide decisive criteria for the atomicity and structure of solutions to truncated moment problems, including in tensor decomposition and semidefinite representability (Curto et al., 2017, Huang et al., 28 Apr 2024).

The maximal mass problem, formulated as

$$\rho_s(x) = \sup \{ \mu(A_x) ~|~ \mu \text{ representing measure for } s\}$$

and its dual infimum over positive polynomials provides conic optimization strategies for determining "most representative" atoms in a measure, with deep connections to cubature, tensor decompositions, and sum-of-squares decompositions.

3. Probabilistic and Asymptotic Views: Random Moment Sequences and Universality

The probabilistic aspect of moment-to-moment theory is exemplified by central limit theorems for random moment vectors. For random measures on $[0, \infty)$ and $\mathbb{R}$, standardized moment vectors converge in distribution to Gaussians centered at the moments of universal distributions (Marchenko–Pastur and Wigner’s semicircle law, respectively). Explicitly,

$$\sqrt{n} C^{-1} (\mathbf{m}_k^{(n)} - \mathbf{m}_k(\eta)) \xrightarrow{d} \mathcal{N}_k(0, I_k)$$

for $[0, \infty)$, and a similar theorem for measures on $\mathbb{R}$ (Dette et al., 2010). These limiting behaviors establish a universality principle: regardless of initial measure specifics, random moment sequences converge in distribution to the canonical moments of these fundamental laws.

The role of random matrix theory is pivotal: Jacobi, Laguerre, and Hermite ensembles provide random spectral measures whose canonical moments and recursion variables align with the derived distributions on moment spaces. The combinatorial approach via the moment method and its orthogonal polynomial–modified extension allows precise characterization of global and local eigenvalue statistics—culminating in universality results such as convergence to the Airy point processes at spectral edges (Sodin, 2014).

4. Moment Functionals: Intrinsic Characterizations and Positivstellensatz

A significant aspect is the intrinsic characterization of moment functionals $L: A \to \mathbb{R}$ in real algebras. Sufficient and necessary conditions for a linear functional $L$ to admit a representing positive measure (possibly on compact sets determined by families of bounds $C_a$) are formulated in terms of "moment growth" boundedness,

$$C_a = \sup_{n \in \mathbb{N}} |L(a^{2n})|^{1/2n} < \infty \quad \forall a \in A,$$

and (sometimes) positivity on suitable cones or semirings (Atanasiu, 13 Oct 2024). Archimedean Positivstellensatz statements for cones of the form

$C = \{ (C_a - a)(C_a + a) : a \in A \} \cup \dots$

allow geometric and algebraic characterization of feasible moment functionals, generalizing both positivity and support conditions.

These tools enable new proofs and generalizations of foundational theorems (Berg-Maserick, Schüdgen) as well as extension to functionals on semigroups with involution, with implications for harmonic analysis and representations on more general algebraic objects.

5. Algorithmic and Applied Directions: Control, Recovery, and Computation

Moment-to-moment theory directly informs methods for recovery and control in high-dimensional and structured systems:

• Moment-SOS relaxations provide convex (SDP) hierarchies for moment/tensor recovery under semialgebraic constraints. By optimization over truncated moment sequences with positivity enforced via localizing matrices and sum-of-squares certificates, these relaxations yield atomic measures or low-rank tensor decompositions—often guaranteed by flat extension conditions (Huang et al., 28 Apr 2024). Efficient implementation and cost-effective performance across a range of problems is supported by rigorous convergence theorems.
• Moment-based ensemble control translates the state of a continuum of dynamical systems to a moment representation, enabling reduced-order (but exact) control design in moment space. The equivalence of ensemble and moment controllability, the Banach manifold structure, and Lyapunov-based feedback constructions are grounded in the interplay between classical moment problems (Hausdorff and extensions) and geometric control theory (Narayanan et al., 2020).
• Quantum dynamics reformulated via propagation of moments (e.g., $\langle x^a y^b z^c \rangle$) instead of the full wavefunction allows for classical molecular dynamics–like numerical schemes. Theoretical advances include reduction to cumulant or Edgeworth expansions and the incorporation of neural network models to approximate higher derivatives, leading to computational efficiency for quantum simulations (Boyer et al., 9 Jan 2024).

6. Extensions to Cognitive Science and Philosophy: Moment-to-Moment Dynamics in Mind and Experience

The moment-to-moment framework generalizes to cognitive and experiential domains:

• In psychology, hidden Markov models with regime switching formalize cognitive processing as transitions between discrete latent states, accounting for both short-term fluctuations and long-term trends in behavior in trial-based experiments. The probabilistic likelihood structures provide rigorous model selection and time series analysis (Gunawan et al., 2019). In human–automation trust, the temporal progression of trust is quantified as a sequence of momentary updates, modulated by outcome bias and contrast effects, with explicit computation of trust increments post-interaction informing adaptive automation (Yang et al., 2021).
• In the philosophy of time and consciousness, moment-to-moment theory (especially as formalized in recent neuroscience) posits that the perceived flow or succession of time arises from the structure of causal distinctions in neural substrates. Integrated Information Theory (IIT) models the present as a directed grid of causal distinctions, producing the phenomenology of elapsed, flowing time without requiring an external “moving now” (Comolatti et al., 30 Nov 2024). This is contrasted with tenseless accounts of time, where moment-to-moment updating is emergent from the brain’s integration of events within a static sequence of physical occurrences (Romero, 2012).
• In process algebras for quantum mechanics, the moment, or "duron," is regarded as an extended region where past and future are entangled; dual dynamical equations in a bi-algebraic setting yield both Schrödinger and thermodynamic arrows of time, linking algebraic structure, irreversibility, and the origin of temporal phenomena (Hiley, 2013). Other proposals see events and their information-rich “views” as the primitive ontology—governing the emergence of quantum potentials and dynamical diversity (Smolin, 2017).

7. Connection to Many-Body Electronic Structure and Spectroscopy

In modern electronic structure theory, moment-to-moment approaches, particularly moment-functional based spectral density–functional theory (MFbSDFT), exploit low-order spectral moments (e.g., first four) to reconstruct the spectral function and describe both ground- and excited-state properties efficiently. The spectral moments, given as functionals of local density and its correlations, encapsulate information beyond standard density-functional approximations—correcting for deficiencies in predicted bandwidth, exchange splitting, and the existence of correlation-induced satellites in strongly correlated materials like Ni, Pd, and SrVO$_3$ (Freimuth et al., 2022, Freimuth et al., 2022). Advanced algorithms based on block-matrix diagonalization scale linearly with the number of moments used and support systematic refinement and universality across materials classes.

Summary Table: Central Concepts and Mathematical Structures

Concept	Mathematical Object / Statement	Key Feature or Result
Moment cone $\mathcal{S}_A$	${ s : s= \int s_A(x) d\mu }$	Convex, may admit atomic measure, facial
Carathéodory number $C_A$	Minimal $\# \{x_i\}$: $s=\sum c_i s_A(x_i), c_i>0$	Bounds minimal atomic cardinality
Central limit law for moments	$$\sqrt{n}C^{-1}(\mathbf{m}_k^{(n)}-\mathbf{m}_k(\eta)) \to N$$	Universal fluctuation (MP or semicircle)
Flat extension property	$M_{k+1}$ extends $M_k$ flatly: rank($M_{k+1}$)=rank($M_k$)	Implies finite atomic representation
Moment-SOS relaxation	Minimize $R(y)$, subject to $M_k[y]\geq 0$, localizing constraints	Tractable SDP for moment/tensor recovery
Trust increment ΔTrust	$$\Delta\text{Trust}(i) = \text{Trust}(i)-\text{Trust}(i-1)$$	Time-series quantification of trust
Phenomenal moment (IIT)	Causal distinction, directed inclusion/connection/fusion	Structurally grounds temporal experience

A plausible implication is that moment-to-moment theory supplies a unifying mathematical and conceptual language to analyze evolving structure—across probability, operator algebras, control, quantum theory, psychological modeling, and temporal phenomenology—by tracking and relating the fine-grained behavior and synthesis of their moments or associated functional distinctions. The architecture, stability, and propagation of these moments encode the complexity, universality, and emergent phenomena observed at every scale.