Process-Theoretic Quantum Mechanics

• Process-theoretic quantum mechanics is a framework that reconstructs quantum phenomena from explicit process models and algebraic compositions.
• It utilizes discrete informons, categorical diagrams, and stochastic square-root processes to recover emergent quantum states and dynamics.
• The approach provides novel insights into measurement, entanglement, and indefinite causal order, linking quantum theory with noncommutative geometry.

Process-theoretic quantum mechanics is a paradigm in which quantum phenomena and formalism are reconstructed from explicit models of processes, their compositions, and their generative algebraic structures. Eschewing a priori wave-function realism in favor of operational or ontological notions of process, these approaches recast states, measurements, dynamics, entanglement, and even spacetime itself as emergent, often discrete, from a processual substrate. This framework underpins several distinct but interlocking threads: algebraic process frameworks, categorical quantum mechanics, stochastic "square-root" processes, groupoid and Clifford algebraic structures, process matrices for indefinite causal order, and tomographic representations of processes. The result is a family of highly technical, mathematically robust models with the capacity to reproduce traditional quantum theory as an emergent or asymptotic limit, while often providing new insights into measurement, nonlocality, classicality, and background independence.

1. Algebraic and Discrete Process Frameworks

Early constructions, such as the Sulis process algebra and its extensions (Sulis, 2014, Sulis, 2015, Sulis, 2013), define the foundational entities of quantum mechanics as discrete, generative objects called informons. A process is a rule-based, non-deterministic generator of informons, each of which carries a local embedding in space-time, a “wavelet” contribution to the global state, a strength (amplitude), and a causal ancestry.

The algebra of processes admits a rich structure, including:

• Sequential and concurrent composition: exclusive vs. free sums/products.
• Interaction and concatenation: non-commutative composition allows encoding of measurement and coupling.
• Configuration-space process covering maps: for multipartite and entangled systems, multi-informon configurations directly yield standard configuration-space wavefunctions in the continuum limit.

States and observables arise at the emergent level via interpolation theory—process-level compositions, after many rounds, yield the familiar Hilbert-space wavefunction and observable algebra, with the Born rule reproduced as an asymptotic quotient. Probabilities at the process level are non-Kolmogorov due to interference among informon strengths.

Emergence of all quantum mechanical phenomena—including superposition, interference, entanglement, and state collapse—can be modeled combinatorially via these algebraic rules, with macroscopic classicality resulting from superselection enforced by the complexity of large-scale process interactions (Sulis, 2014, Sulis, 2015, Sulis, 2013).

2. Categorical and Diagrammatic Process Theory

The categorical quantum mechanics (CQM) program constructs quantum theory as a process theory: a symmetric monoidal category in which “systems” are objects and “processes” are morphisms (Coecke et al., 2015, Coecke et al., 2016, Szawiel, 2017, Gogioso, 2019). Key features include:

• String-diagram calculus: captures sequential composition and parallel (monoidal) product, enabling reasoning about interactions, broadcasting, and no-signalling.
• Doubling and discarding: to model quantum (CP) maps, every system is doubled and a discard operation is adjoined, recovering complete positivity and the Born rule without Hilbert-space vectors.
• Spiders (†-SCFA structures): operationally model classicality and data-processing; quantum systems arise by “doubling” classical wires and constraining composition via “spider fusion.”
• Purification principle: every mixed process arises as the marginal of a pure higher-dimensional process (Stinespring dilation), with uniqueness up to unitaries (purification equivalence).
• Categorical composition: fully captures measurements, decoherence, entanglement, protocols, and yields diagrammatic proofs of results such as no-broadcasting and the existence of Stinespring dilation (Coecke et al., 2015, Coecke et al., 2016).

Through categorical reconstructions, quantum theory is recovered functorially from categories of physical processes—states, processes, dynamics, and symmetries emerge from the data of the symmetric monoidal functor GNS, with classical limit and deformation quantization accessible through categorical extensions (Szawiel, 2017, Gogioso, 2019).

3. Square-root Stochastic Process and Emergence of the Schrödinger Equation

Frasca’s “square-root” construction demonstrates that quantum mechanics is not a direct stochastic process (e.g., Brownian motion), but rather the square root of such: for a real Wiener increment $dW(t)$, the quantum increment is given by $dX(t) = [dW(t)]^{1/2}$, with the “square-root process” involving a complex Bernoulli-valued phase variable (Frasca, 2012).

This expansion yields, after regularization, a process where each step includes both a random modulus and a local phase “flip” by the complex Bernoulli variable. Over time, the accumulation of these complex phases yields unitary quantum propagators, and the Kolmogorov forward equation for the amplitude becomes precisely the Schrödinger equation for an appropriate choice of drift and diffusion. Introducing a potential extends the construction to the full Schrödinger equation with interactions.

The process-theoretic interpretation posits that quantum amplitudes are the complex “square-root” of classical probability distributions on paths. Interference, phase, and measurement outcomes become rooted in the statistical history of complex phases aggregated over process histories, providing a rigorous stochastic foundation for quantum dynamics (Frasca, 2012).

4. Process Matrices and Indefinite Causal Structure

The process matrix formalism is a process-theoretic extension of quantum mechanics that enables indefinite causal order between spacetime regions (“laboratories”), with process matrices $W$ encoding the operational statistics of all local CP maps and generalizing the Born rule to multiple parties (Parker et al., 2021). Unlike standard quantum circuits (which impose fixed causal relations), process matrices can encode situations where no definite order exists between events.

Advances include the development of background-independent process formulations: requiring permutation invariance (discrete diffeomorphism invariance) across event labels leads to non-trivial indefinite causal structures even when all laboratory reference frames are “gauge-fixed” out. This removes operational distinctions between space-time points unless a material “reference frame” is encoded into the system, mirroring approaches to background independence in quantum gravity.

A key feature is the failure of the superselection rule paradigm: in the S_n-invariant setting, charge sectors (labeled by irreducible representations of the permutation group) may contain no valid process matrices at all, necessitating new conceptual tools for interpreting quantum symmetry and locality under process-theoretic background independence (Parker et al., 2021).

5. Groupoid, Clifford Algebra, and Noncommutative Geometry of Process

Hiley’s approach builds quantum theory upon a groupoid algebra of elementary processes, which are then equipped with suitable bilinear forms giving rise to orthogonal and symplectic Clifford algebras (Hiley, 2012, Hiley, 2018). These algebras not only encode the basic structure of Schrödinger, Pauli, and Dirac quantum mechanics—including spinors and relativistic dynamics—but also unify quantum phase space via symplectic Clifford (Weyl) algebras.

• Points as idempotents: manfiold points and phase-space points are algebraic idempotents; dynamical variables correspond to algebraic generators.
• Dynamics from connections: Dirac connections on Clifford bundles generate quantum dynamical equations, reproducing both the Liouville (continuity) equation and the quantum Hamilton–Jacobi equation (including the quantum potential).
• Noncommutative “implicate order”: empirically observable (or “explicate”) manifolds—such as configuration or momentum space—arise as projections (“shadow manifolds”) of the full noncommutative process algebra, providing a rigorous mathematical structure for Bohm's implicate–explicate order.

This framework unifies standard operator quantum mechanics with Bohmian formulations and accommodates multiple classical–quantum limits as different explicate projections (Hiley, 2012, Hiley, 2018).

6. Tomographic and Probability-Representation Process Formalisms

Quantum processes can be reformulated entirely in the language of probability theory by expressing states and processes in tomographic (symplectic) representations (Przhiyalkovskiy, 2021). Here, the Kraus (operator-sum) representation is translated into convolution kernels acting on measurable tomograms:

• Process kernels: each completely positive map is associated with a kernel that acts by convolution on the input tomogram to yield the output tomogram.
• Partial kernels and decompositions: process kernels decompose into partial kernels reflecting the underlying operator sum, allowing explicit modeling of projective measurements, Gaussian blurring, and error processes.
• Experimental link and star-product structure: process action is directly accessible via measured probability distributions (tomograms); operator noncommutativity is recovered as a star-product structure on symbols.

This operationalizes quantum information processing and control entirely at the level of classical probability distributions while retaining the process-theoretic formalism of quantum channels (Przhiyalkovskiy, 2021).

7. Synthesis and Conceptual Implications

Process-theoretic quantum mechanics provides mathematically rigorous, ontologically flexible frameworks capable of reconstructing all central features of conventional quantum theory—including mixed and pure dynamics, entanglement, measurement, nonlocality, and spacetime structure—as emergent or asymptotic from more fundamental processual, algebraic, or categorical data (Sulis, 2014, Coecke et al., 2015, Gogioso, 2019).

Crucially, this approach demystifies nonlocality, contextuality, and the measurement problem by rooting their origin in the algebraic and combinatorial properties of processes, rather than in observer-centric dynamics or intrinsic randomness. Non-Kolmogorovian probability, the square-root relation to diffusion, and the unification of symmetry and quantum logic highlight why process-theoretic frameworks are increasingly central in discussions of quantum foundations, quantum information, and prospectively, quantum gravity (Frasca, 2012, Parker et al., 2021, Hiley, 2012, Gogioso, 2019).