Prequantum Classical Statistical Field Theory

• PCSFT is a framework where quantum states are represented as covariance operators of classical random fields, offering an alternative view of quantum phenomena.
• It predicts measurable deviations from Born's rule using higher-order corrections, enabling refined detection thresholds and experimental probes.
• The theory unifies quantum channels and correlations through classical Gaussian processes, providing new insights for quantum information processing.

Prequantum Classical Statistical Field Theory (PCSFT) is a theoretical framework in which quantum systems are represented as statistical ensembles of classical random fields, rather than as fundamental quantum objects. PCSFT models quantum phenomena—including the correlations observed in entangled systems—as consequences of classical signal theory applied to a special class of stochastic fields. The approach treats the formalism of quantum mechanics (QM) as an effective, approximate theory that emerges from deeper, physically meaningful averages taken over fluctuations in prequantum fields. This perspective asserts that the probabilistic predictions of quantum mechanics, including Born's rule, can be viewed as limits or approximations, with PCSFT providing quantitatively precise corrections that, in principle, are testable.

1. Core Theoretical Framework

PCSFT postulates that every quantum system corresponds to a classical random field, typically modeled as a Gaussian random process in an appropriate complex Hilbert space (e.g., $L^2(\mathbb{R}^3; \mathbb{C})$ or its finite-dimensional analogs). In this scheme:

• Quantum states are represented by the covariance operators of these random fields. For a pure state $|\psi\rangle$, the associated density matrix is $\rho = |\psi\rangle\langle\psi|$, and in the PCSFT formalism, one has a covariance operator $D$ satisfying $\rho = D/ \mathrm{Tr}\,D$.
• Quantum observables are associated with quadratic forms of the field: given a self-adjoint operator $A$, the corresponding classical observable is $f_A(\phi) = (A\phi, \phi)$.
• Statistical predictions in PCSFT reproduce those of quantum mechanics: the ensemble average $$\langle f_A \rangle_\mathrm{CL} = \mathbb{E}\,f_A(\phi)$$ satisfies $$\langle f_A \rangle_\mathrm{CL} = \mathrm{Tr}\, D\,A = \mathrm{Tr}\,\rho\,A$$, after normalization.

In the open system setting, PCSFT interprets the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation as the evolution of covariance operators under positive trace-preserving semigroups, with normalization corrections leading to nonlinear dynamical generalizations (Khrennikov, 2013).

2. Quantum Probabilities and the Role of Born's Rule

Born's rule, which states that the probability of finding a system in a region $I$ is $p(x \in I) = \int_I |\psi(x)|^2 dx$, is viewed in PCSFT as an approximation. The detection of a quantum outcome in PCSFT involves integrating not only quadratic but also higher-order contributions of the prequantum field:

$T_{2,4}(\psi) = |\psi(x)|^2 + \alpha |\psi(x)|^4,$

with $\alpha > 0$ a parameter proportional to the prequantum field’s fluctuation dispersion. The generalized probability is:

$$p(x \in I) \approx \int_I |\psi(x)|^2 dx + \alpha\left[\int_I |\psi(x)|^4 dx - \left(\int_I |\psi(x)|^2 dx\right)^2\right].$$

This higher-order correction implies that Born's rule is the leading term in an expansion; deviations are generally small but can, in principle, be amplified under certain state preparations (e.g., Gaussian wave packets with small dispersion) (Khrennikov, 2010). PCSFT thus predicts subtle but potentially measurable violations of Born's rule.

3. Detection, Thresholds, and Discreteness

PCSFT explains the apparent discreteness of quantum detections (e.g., “clicks” in photodetectors) as the result of thresholding in the measurement apparatus. Classical random signals are registered only when their energy—possibly amplified with random gain in the detector—exceeds a set threshold. The statistics of detection events are mathematically modeled via first passage (hitting) times for a Brownian (Wiener) process in Hilbert space, leading to detection probabilities that reduce to the Born rule under high-threshold, low-pulse energy asymptotics (Khrennikov, 2012, Khrennikov, 2012, Khrennikov, 2014). However, the exact detection statistics (including random gain and threshold effects) lead to corrections that go beyond the standard quantum formalism, again with experimental implications.

4. Quantum Correlations, Entanglement, and Nonlocality

PCSFT reconstructs quantum correlations—including those of entangled systems—not by invoking nonlocal hidden variables but by relating them to the structure of the covariance matrix (block structure) of the composite classical random field. For a bipartite system with components $S_1$ and $S_2$:

• The random field $\phi = (\phi_1, \phi_2)$ has a covariance operator $D$ with off-diagonal blocks reflecting classical correlations.
• For entangled quantum states, the off-diagonal block $D_{12}$ encodes nontrivial cross-correlations; for factorizable states (unentangled), the off-diagonal blocks persist, but the quadratic forms representing observables become uncorrelated in the quantum sense (Khrennikov, 2011).

The omnipresence of a common background field ensures that all prequantum fields are correlated—PCSFT thus interprets quantum nonlocality as a manifestation of underlying classical (but generalized) correlations, demystifying “spooky action at a distance.”

5. Representation of Quantum Channels and Dynamics

Quantum channels (including unitary evolution, open system dynamics, and measurements) are represented in PCSFT as classical linear (or affine) transformations of Gaussian random fields. For a channel represented by operator $V$, the covariance transforms as $D_\mathrm{out} = V D_\mathrm{in} V^\ast$. This framework directly maps quantum error correction, decoherence, and measurement (von Neumann-Lüders projections) to classical analogues in signal theory (Khrennikov et al., 2010). The universality of this mapping suggests possible applications of classical signal processing techniques to quantum information problems.

6. Practical and Experimental Implications

• Testing Born's Rule: PCSFT predicts small deviations from the Born rule via higher-order detection nonlinearities. Carefully prepared quantum states (asymmetric toy models or Gaussian packets with minimal dispersion) may serve as sensitive probes (Khrennikov, 2010).
• Threshold Detection: The detailed predictions depend on both the detection threshold and the random amplification (gain) characteristics of the detector, providing a testable distinction from standard QM in regimes where these effects are not negligible (Khrennikov, 2012).
• Photon Detection and Coherence: Numerical simulations confirm that PCSFT reproduces both single-event statistics and two-point (second order) coherence functions in line with experiment, refuting semiclassical models that cannot account for photon antibunching (Khrennikov, 2014).
• Quantum Information Processing: The equivalence of quantum and classical Gaussian channels may enable new perspectives on quantum error correction and communication, borrowing methods from classical communication theory (Khrennikov et al., 2010).

7. Philosophical and Methodological Foundations

PCSFT implements the “Bild-conception” (picture/model conception) of scientific theory structuring (Khrennikov, 2018, Khrennikov, 2023), distinguishing between the “observational model” (QM) and the deeper “causal theoretical model” (PCSFT). The quantum formalism is interpreted as an effective summary (OM) of statistical correlations in the underlying causal theory (CTM)—here, given by normalized covariances of classical fields. This framework resists simplistic identification of classical “hidden variables” with quantum observables or their spectra (contrary to the premises of Bell-type no-go theorems), accommodating the many-to-one mapping from subquantum configurations to quantum states.

PCSFT’s embrace of classical field theory as ontic (real) and quantum mechanics as epistemic (observational) provides new context for foundational debates, reframing quantum randomness, nonlocality, and measurement as emergent and contextual phenomena, rather than inexplicable features of nature.

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In summary, PCSFT is a mathematically precise, physically motivated framework proposing that quantum mechanics arises as an effective theory for averages over classical random fields. It reproduces QM’s statistical predictions—including entanglement—while predicting small deviations from the standard quantum formalism in appropriately designed experiments. PCSFT thereby opens a route to classical explanations of quantum phenomena, challenges to Born’s rule, and potentially unifies quantum and classical information theories under a common signal-theoretic umbrella.