Theory of Physical Probability

• Theory of Physical Probability is a framework that defines probability through physical laws, symmetries, and measurement processes.
• It integrates approaches from classical symmetry, dynamical descriptive statistics, and quantum mechanics to form rigorous, context-dependent probability measures.
• Recent developments extend the theory using information geometry, microstate counting, resource-based computation, and even signed probabilities in complex systems.

The theory of physical probability seeks to ground the use of probability in physical science by connecting probabilistic concepts directly to the laws, symmetries, structures, and measurement processes of physical systems, rather than adopting probability as a purely abstract or subjective notion. Diverse and sometimes competing frameworks have been developed within mathematics, physics, and the philosophy of science to articulate this connection. Below, representative theories and their foundational arguments are described, highlighting core axioms, methodologies, and the implications for statistical mechanics, classical and quantum physics, and mathematical probability theory.

1. Symmetry and the Classical Foundation of Physical Probability

A canonical approach is to define physical probability in terms of physical symmetry and the principle of indifference. For well-understood physical systems (e.g., tossing a fair coin, rolling a symmetric die), outcomes are judged equally likely by virtue of the system's symmetry group. When the finite outcome set is $O = \{O_1, ..., O_k\}$ and symmetry implies indistinguishability between singletons $\{O_i\}$ and $\{O_j\}$, the probability of an event $E \subseteq O$ is given by

$P(E) = \frac{|O(E)|}{k}.$

This assignment recovers the classical axioms:

• $0 \leq P(A) \leq 1$,
• $P(\emptyset) = 0$, $P(\Omega)=1$,
• Additivity for disjoint events: $P(A \cup B) = P(A) + P(B)$, for $A \cap B = \emptyset$,
• Conditional probability: $P(B|A) = \frac{P(A \cap B)}{P(A)}$ when $P(A)>0$.

For continuous cases, symmetry under translation or rotation (as in the spinning wheel) leads to the Lebesgue measure, with $P(E) = |A(E)|$ for measurable sets $A(E) \subset (0,1)$, where $|A(E)|$ denotes Lebesgue length. This construction, as codified in Bowater's analysis, underpins the objective and additive character of physical probability within its domain of application, but fails to address cases lacking such symmetry, including most inferential statistical tasks and hypothesis testing (Bowater, 2022).

2. Probability as Emergent From Dynamical Descriptive Statistics

Johnson proposes a foundation rooted in the descriptive statistics of deterministic high-dimensional systems, inspired by Hilbert's Sixth Problem and Khintchine's conjectures. Here, the long-run statistics of observables evolving under deterministic (but possibly non-ergodic) dynamics are used to define the probability of events via their frequencies or correlation functions in the thermodynamic limit. The autocorrelation functions

$$R_n^{(m)}(t_1, ..., t_m) = \lim_{T\to\infty} \frac{1}{2T}\int_{-T}^{T} x_n(t + t_1)...x_n(t + t_m) dt$$

are shown to coincide for "almost all" trajectories as $n \rightarrow \infty$ (i.e., for almost all initial conditions), even in non-ergodic linear systems. The set of limiting correlation functions $\{R^{(m)}\}$ uniquely determines a probability measure on an abstract probability space (e.g., $[0,1]$ with Lebesgue measure): $P(A) = \mu(\{x \in M_0 : f_0(x) \in A\}),$ where $f_0$ is the limit observable with moments given by the $R^{(m)}$. Events correspond to measurement outcomes, and their probabilities describe the measure of initial conditions producing those outcomes in long-time evolution.

This approach avoids logical circularity inherent in frequentist definitions and grounds the probability measure in the physical properties of large deterministic systems, providing a physically meaningful axiomatization of probability that is both mathematically rigorous and tied to empirical measurement (Johnson, 2014).

3. Quantum and Information-Geometric Theories

3.1 Quantum Probability via Logic and Geometry

The quantum framework generalizes the event structure to the orthomodular lattice of projection operators $\mathcal{P}(\mathcal{H})$ on Hilbert space, with fundamental axioms:

• $s(0) = 0$, $s(I) = 1$,
• $s(P^\perp) = 1 - s(P)$,
• Countable additivity: for mutually orthogonal $\{P_i\}$, $s(\bigvee P_i) = \sum s(P_i)$.

Gleason's theorem ensures that any such measure arises uniquely from a density operator $\rho$ as $s(P) = \operatorname{Tr}(\rho P)$ for systems with $\dim \mathcal{H} \ge 3$. This measure-theoretic structure is intimately connected to the geometry of $\mathcal{H}$: transition probabilities are given by squared inner products of rays, and the full probability calculus follows from the projective (convex) geometry of state space (Holik, 2013).

3.2 Fisher-Information Geometry and Extremal Principles

Hung advances a theory wherein every physical system carries an "intrinsic" probability distribution whose information geometry—encoded via the Fisher metric—reflects the system's physical spacetime metric,

$$g_{\mu\nu}^{\rm phys}(x) \longleftrightarrow g_{\mu\nu}(x) = \int p(x) \partial_\mu \ln p(x) \partial_\nu \ln p(x) dx,$$

and the actual $p(x)$ is determined by extremizing total Fisher information under normalization and physical constraints: $$\mathcal{I}[p] = \int p(x) g^{\mu\nu}(x)\partial_\mu \ln p(x)\partial_\nu \ln p(x) dx.$$ The resulting variational equations generically yield wave-type PDEs for $p(x)$. This construction unifies quantum equations (e.g., Klein–Gordon) with information-theoretic principles and offers a geometric underpinning for the emergence of physical probability densities (Hung, 2014).

4. Microstate Counting, Locality, and the Born Rule in Quantum Theory

Saunders synthesizes a physically realistic notion of probability for no-collapse (Everettian) quantum mechanics via four axioms:

• Boolean event structure,
• Instantaneous state dependence,
• Kolmogorov additivity,
• Locality (no action at a distance).

He demonstrates that, for any equi-amplitude orthogonal expansion $|\Psi\rangle = \sum_{i=1}^n |\phi_i\rangle$, equiprobability and local invariance imply that each microstate $|\phi_i\rangle$ carries probability $1/n$. Probabilities for measurement projectors $P$ are then obtained as the limit

$$\mu_\Psi(P) = \lim_{n\to\infty} \frac{m}{n} = |\langle a|\Psi\rangle|^2$$

where $m$ is the number of $|\phi_i\rangle$ in the support of $P$. This microstate-counting approach delivers the Born rule within a fully local, collapse-free ontology (Saunders, 11 May 2025, Saunders, 29 Nov 2025). Bell-inequality violation, in this view, is due not to nonlocality but to the lack of outcome independence in entangled states.

5. Probability As Physical Resource or Complexity Measure

Hagar & Sergioli propose that physical probability is an objective measure of the "distance"—quantified via computational complexity or required physical resources—between physical states. Given initial and target states $S_\text{init}, S_\text{target}$ and a resource bound (e.g., energy, time), objective chance is defined as the fraction of allowable dynamical evolutions (or computations) that achieve the specified transition within the resource budget: $$P_{n, \text{Pw}}(A) = \frac{|A|}{|S_{n, \text{Pw}}|},$$ where $S_{n, \text{Pw}}$ is the set of all allowed processes. This view eliminates "ignorance" from statistical physics: non-trivial chance arises solely due to finite resources and the (objective) computational limitations imposed by the laws of physics (Hagar et al., 2011).

6. Signed Measures, Extended Probability, and Negative Probabilities

In certain physical contexts—especially quantum field theory with gauge redundancy, or in turbulence theory describing transitions from deterministic to stochastic dynamics—the assignment of probability requires signed (possibly negative) measures. Noldus formulates a generalized framework where "raw" probabilities $\tilde{P}(E)$ may lie outside $[0,1]$, with observable (renormalized) probabilities defined by

$$P(E) = \frac{|\widetilde{P}(E)|}{\sum_F |\widetilde{P}(F)|},$$

assuming the denominator converges. The physical import of negative probabilities is tied to coarse-graining and detector response—"holes" or "debts" in detector micro-states can manifest as apparent negative assignment, subsequently averaged out (Noldus, 2015). Similarly, in turbulence, genesis of a stochastic process (the transition from deterministic to random behavior due to resonance) is represented by a signed probability density function with dynamic fractal structure (Yang et al., 8 Oct 2025).

7. Decoherence, Natural Probability, and Observer-Dependent Cutoffs

Parker introduces "natural probability" as a framework that grounds the emergence of classical probabilistic behavior in quantum decoherence and the selective visibility of projection statements. In this approach:

• A projection $P$ localized in spacetime is "classical" if it admits many independent records in spacelike-separated regions.
• The norm-square $\|P\psi\|^2$ defines Born weight, but actual recordability (and thus meaningful probability assignment) is limited by decoherence-induced errors.
• There exists a cutoff: events with probability below the noise threshold cannot be observed or recorded; all probability identities (additivity, Bayes' rule) hold only up to these decoherence-limited errors.

Thus, "natural probability" is observer-centric and context-dependent: only sufficiently robust, macro-recorded outcomes are available to agents, and low-probability (rare) events are physically suppressed at the phenomenological level (Parker, 6 Dec 2024).

8. Probability in Thermodynamics and Quantum Field Theory

Zhang et al. extend the theory of physical probability by equating probability distributions with spectral density functions of (potentially unknown) operators. In this view:

• The spectrum is randomly sampled from a distribution $p(\lambda)$.
• Thermodynamic and quantum field quantities (partition function, vacuum energy, etc.) become integrals over $p(\lambda)$ (the "probability thermodynamics" and "probability quantum field" program).
• Spectral zeta-function, generalized heat kernels, and related spectral functions are all determined by $p(\lambda)$.
• Negative and unbounded spectra require analytical continuation and renormalization.

This correspondence augments classical probability with tools from spectral theory, enabling analysis of phenomena where conventional moments or generating functions fail to exist (Zhang et al., 2019).

9. Superdeterminism and the Causal Foundations of Probability

Vervoort advocates that genuine physical probability is underpinned by deterministic hidden-variable dynamics. The principle of superdeterminism posits that all current probabilistic dependencies—independence, central limit law, Bell-type correlations—are manifestations of deep causal constraints inherited from the universe's initial state. Probability is thus not an ontological primitive but reflects deterministic processes unfolding beyond current epistemic access, with independence and randomness always traceable to, and contingent upon, common causation (Vervoort, 2018).

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Summary Table: Select Physical Probability Theories

Approach	Defining Feature	Core Reference
Symmetry/Indifference	Equally likely outcomes by physical symmetry	(Bowater, 2022)
Descriptive Statistics	Probability from thermodynamic limit of time series	(Johnson, 2014)
Quantum Logical/Geometric	Measures on Hilbert lattice, geometry of $\mathcal{H}$	(Holik, 2013)
Fisher Information	Intrinsic extremal-information distributions	(Hung, 2014)
Microstate Counting (Everett)	Frequentist microstate ensemble in no-collapse QM	(Saunders, 11 May 2025, Saunders, 29 Nov 2025)
Complexity/Resource-Based	Objective chance via computation/resource distance	(Hagar et al., 2011)
Signed/Extended Probability	Signed and negative probabilities, renormalization	(Noldus, 2015, Yang et al., 8 Oct 2025)
Decoherence/Natural Prob.	Observer recordability and decoherence errors	(Parker, 6 Dec 2024)
Spectral/Probability Thermodynamics	Probability via operator spectra and spectral functions	(Zhang et al., 2019)
Superdeterminism	All observed probabilities deterministic in origin	(Vervoort, 2018)

Each of these approaches provides a rigorous and explicit connection between probabilistic measures and physical structures, symmetries, or constraints. The choice of framework is sensitive to physical context: equilibrium versus non-equilibrium statistical mechanics, classical versus quantum, measurement theory, and cosmological or field-theoretic regimes. The theory of physical probability is thus a pluralist and evolving field, with ongoing debate on its interpretation, scope, and physical implications.