Transdimensional Mutation & Paraxial Projection

• Transdimensional mutation and paraxial projection operations are methodologies that systematically enable transitions between variable-dimensional models while preserving essential dynamics.
• These methods are applied across fields such as optics, quantum field theory, and cosmology to manage effective dimensionality and resolve projection-induced biases.
• Their use enhances analytical and computational tractability by reducing complex behavior to simpler models, as seen in spin transport phenomena and inverse problems.

Transdimensional mutation and paraxial projection operations constitute a class of methodologies and formalisms that systematically address changes of system dimension—whether spatial, parametric, or algebraic—together with projection techniques that map system behaviors within or across these variable-dimensional spaces. Such concepts have emerged in diverse domains including optics, quantum field theory, geometry, inverse problems, and computational design, linking the mechanics of multidimensional transition (“mutation”) to structural or analytical projections (“paraxial operations”).

1. Dirac-like Paraxial Equations and Spin Transport

In weakly inhomogeneous optical media, Maxwell’s equations can be recast in a Dirac-like formalism by introducing the Riemann–Silberstein vector and suitable “matrix operators” obeying Dirac algebra (Mehrafarin et al., 2010). Under the paraxial approximation (dominant propagation along one axis and weak transverse dynamics), the Maxwell equations reduce to an eigenvalue structure with an effective Hamiltonian:

$$H = -n_0 \beta - i M_L \cdot p_\perp + \text{(small terms)}$$

where $n_0$ is the background refractive index, $\beta$ a diagonal matrix, and $M_L$ encodes the algebraic structure. The Foldy–Wouthuysen (FW) transformation is then applied to decouple positive and negative “energy” states (circular polarizations), resulting in a diagonalized Hamiltonian in the adiabatic limit. Berry connection terms, arising from the momentum dependence of the transformation matrix, encode a geometric phase—leading to observable phenomena:

• Spin Hall Effect: Semiclassical rays split, depending on polarization, with the physical position operator covariant under an emergent Berry gauge field.
• Rytov Rotation Law: The Berry phase induces polarization plane rotation proportional to the integrated gauge potential along the momentum trajectory.

The coupled spin and spatial degrees of freedom form a “transdimensional mutation” in the paraxial projection: formerly independent spatial and polarization (spin) dimensions are intermixed by FW transformation and Berry curvature, leading to nontrivial “blended” beam dynamics.

2. Generalized Measures and Cutoffs in Transdimensional Frameworks

In cosmology, the problem of defining probability measures over landscapes with variable effective dimension is prominent. When modeling a transdimensional multiverse, regions (“bubbles”) can differ in the number of large (noncompact) dimensions, labelled by $D$ (Schwartz-Perlov et al., 2010). Generalizing the standard scale factor cutoff measure ($a = \mathcal{V}^{1/3}$ for $(3+1)$d), straightforward extension ($a = \mathcal{V}^{1/D}$) induces exponential biases against slow-roll inflation in lower $D$ bubbles, conflicting with observed density parameters ($\Omega$).

A “volume factor” cutoff is instead introduced:

• Volume Time: $T = \ln \mathcal{V}$
• Cutoff Surface: Level sets of $T$ project all bubbles onto a unified expansion history, circumventing the projection-induced bias.
• Projection Operation: The transition from a scale-factor to volume-factored cutoff is interpreted as a paraxial-like projection: global dynamics are mapped onto a coordinate system that obviates dimension-dependent exponential weighting.

This operation regulates infinities in a heterogeneous, transdimensionally-mutating spacetime, yielding observational distributions commensurate with data (e.g., $\Omega \approx 1$).

3. Operators and Algebraic Projections in Matrix Genetics

The matrix genetics approach employs projection operators as a central tool for representing and manipulating the genetic code within high-symmetry matrices generated by repeated Kronecker products of a fundamental $2 \times 2$ alphabet matrix (Petoukhov, 2010). Black-and-white mosaics denote strong and weak code degeneracies, recoded as binary matrices. After scaling, these mosaics yield genoprojectors—idempotent matrices ($P^2 = P$) that underpin the code’s internal algebraic structure, symmetry, and direct sum decompositions.

Algebraic “transdimensional mutations” correspond to permutations or reordering of genetic triplet positions, effectively moving the genotype’s representation between subspaces of the full vector space (modifying “dimension” in an algebraic sense), but preserving projection properties and direct sum decomposability. These ideas extend beyond genomics to system modeling in logic, signal processing, and even musical composition—the paraxial projection operation here being a general discrete transform that projects a higher-dimensional construct onto independent, biologically meaningful components.

4. Paraxial Projection in Quantum, Optical, and Geometric Contexts

Paraxial projection refers to the mapping or reduction of a higher-dimensional (often full-wave or nonparaxial) system to a paraxial regime—typically by factoring out rapid variation along a privileged axis and retaining only slow (envelope) dynamics. Variants of this procedure appear:

• Optics and Quantum Optics: The paraxial approximation projects electromagnetic or Dirac-like wave equations (or, row vectors in ABCD matrix formalism (Ornigotti et al., 2012)) onto a reduced functional space, simplifying evolution while retaining essential physics. Generalizations elevate the ABCD matrix to an infinite-dimensional operator equivalent to the translation operator ($e^{L\,d/dx}$), constituting a “mutation” from a finite- to functionally infinite-dimensional analytic structure.
• Spin Optics: Incorporating photon spin into paraxial wave dynamics leads to a projected effective “exotic” phase space with a twisted symplectic structure and curvature-induced shifts in ray position and momentum (including momentum shifts at curved interfaces), which are higher-dimensional effects imprinted on the projected plane (Duval et al., 2012).
• Energy–Momentum in Wave Propagation: Conservation laws derived from paraxial-approximated Lagrangian densities reveal that “projection” from full to paraxial models introduces corrections and necessitates quality measures (e.g., $Q = \|\mathbf{D}\|/\|\mathbf{S}\|$) to quantify paraxiality—a practical diagnostic indicating dimensional mutation from the original dynamical structure (Mahillo-Isla et al., 2018).

5. Transdimensional Algorithms and Discrete–Continuous Inversion

Transdimensional mutation underpins reversible-jump Markov Chain Monte Carlo (rjMCMC) methods in inverse problems, where the model parameter count varies dynamically (Somogyvári et al., 2019). Mutations are performed via “birth” and “death” moves, with conversion to fixed-dimensional representations (scalar, vector, rasterized projections) enabling convergence diagnostics. Paraxial projection operations, in this stochastic context, refer to the geometric or image-based projection of high-dimensional geophysical models onto planes or grids for ensemble statistical analysis.

In automated lens design (Teh et al., 28 Sep 2025), a related paradigm emerges: discrete topology changes (e.g., altering the number and type of elements) are implemented as transdimensional mutations. To ensure these discrete changes yield “warm start” candidates, a paraxial projection operation is used—mutated designs are projected onto the manifold of designs with equivalent first-order (paraxial) ray matrices by solving constrained optimization problems:

$$\min_{\theta^*} \|\theta - \theta^*\|^2 \quad \text{subject to} \quad (T(\theta) - T(\theta^*))\begin{bmatrix}0 \ 1\end{bmatrix} = 0$$

The combined optimization (gradient-based for continuous and MCMC-based for discrete mutations) enables effective exploration of an expanded design space.

6. Field-Theoretic and Geometric Transdimensional Defects

Recent advances in conformal field theory introduce the notion of transdimensional defects: defects in CFTs whose dimension $p$ is continuously deformed via a parameter $\delta$ ($p = 2 + \delta$) (Sabbata et al., 26 Nov 2024). This continuous interpolation enables analytic continuation between defects of various integral dimensions, thus uncovering new conformal fixed points and allowing the paper of operator spectra and RG flows as a function of defect dimensionality. Key aspects include:

• Renormalization of defect couplings with explicit $\delta$-dependence of counterterms and beta functions.
• Diagrammatic summation and recurrence relations for defect–defect correlators with all-order resummation in $\delta$.
• The “mutation” in defect dimension is analogous to the $\varepsilon$-expansion in spacetime but is implemented at the level of submanifold embedding.

“Paraxial projection” in this context refers to the projection of bulk field correlators (or propagators) onto the non-integer-dimensional defect, executed via integrals over the defect’s fractional-dimensional subspace—closely following paraxial (nearly parallel) projection concepts in wave physics.

7. Unified Themes and Applications

Across these disciplines, the interplay between transdimensional mutation and paraxial projection operations systematically enables:

• Control over Effective Dimensionality: By allowing mutation or projection, systems can interpolate between different effective dimensions, as required by context (e.g., energy scale in quantum gravity (Mazumdar et al., 2015), spatial or algebraic structure in matrix genetics).
• Enhanced Analytic and Computational Tractability: Projections (e.g., paraxial, rasterized, or functional) reduce model complexity without discarding essential physics or information, permitting clearer understanding of critical phenomena (e.g., spin Hall effect, energy conservation, or conformal operator scaling).
• Novel Optimization and Inference: By integrating discrete (topological or algorithmic) mutation with projective regularization (maintaining essential equivalence under projection), computational frameworks can escape local optima and survey broader solution spaces (as in lens design and transdimensional inversion).
• Cross-Disciplinary Transfer: The formalisms developed in quantum field theory, optics, computational geometry, and stochastic sampling are observed to possess deep structural parallels, with paraxial projection and mutation providing a common analytic thread.

In sum, transdimensional mutation and paraxial projection operations represent a powerful unifying paradigm for analyzing, transforming, and optimizing multidimensional systems whose relevant “dimensions” may be physical, abstract, or algorithmically encoded.