Interaction–Collapse Law in Quantum Systems

• The Interaction–Collapse Law is a framework that asserts entangling interactions via conservative potentials induce stochastic, non-unitary wave function collapse while preserving conservation laws.
• It is mathematically formulated using a stochastic drift equation that integrates non-unitary evolution with exact conservation of momentum, energy, and angular momentum in every event.
• The law has broad applications in quantum measurement, Bose–Einstein condensates, thermodynamics, and particle physics, offering a unified perspective on quantum dynamics.

The Interaction–Collapse Law posits that entangling interactions between quantum subsystems are the fundamental source of wave function collapse, driving stochastic, non-unitary evolution of the total system state without violating conservation laws at the level of individual events. The law asserts that whenever two or more quantum systems interact through a conservative potential, they become entangled, and it is these entangling interactions—not external observation—that induce physical collapse. This perspective yields a unified, dynamical, and conservation-respecting understanding of quantum measurement and collapse, with broad implications ranging from the foundations of quantum theory to condensed matter, thermodynamics, and field theory.

1. Fundamental Principle and Mathematical Formulation

The central assertion of the Interaction–Collapse Law is as follows: whenever quantum subsystems interact via a conservative potential $V_{\rm int}$, they become entangled, and the entangling interaction itself induces a real, non-unitary collapse of the wave function. Collapse is not a projection onto a subsystem eigenstate, but rather a nonlocal selection of a correlated branch of the full entangled state determined by the interaction potential. Conserved quantities—momentum, orbital angular momentum, and energy—are exchanged only through these interactions, so the total value of these quantities is preserved across all subsystems in any collapse event (Gillis, 2018).

The stochastic collapse equation is given by: $$d|\psi(t)\rangle = \Bigl[ -\frac{i}{\hbar} H \, dt + \sum_{j<k} (L_{jk} - \langle L_{jk} \rangle) dW_{jk}(t) - \frac{1}{2} \sum_{j<k} (L_{jk} - \langle L_{jk} \rangle)^2 dt \Bigr] |\psi(t)\rangle$$ where $H$ is the Hamiltonian,

$$H = \sum_j \left[-\frac{\hbar^2}{2m_j} \nabla_j^2\right] + \sum_{j<k} V_{jk}(|\mathbf{r}_j - \mathbf{r}_k|)$$

and

$L_{jk} = V'_{jk} - \langle V'_{jk} \rangle$

with $$V'_{jk} = \sqrt{\frac{1}{(m_j + m_k) c^2}} V_{jk}(\mathbf{r}_j, \mathbf{r}_k)$$ and $\langle \cdot \rangle$ denotes the quantum expectation value. The $dW_{jk}(t)$ are independent Wiener processes for each interacting pair.

Collapse dynamics thereby arise from the very same potentials responsible for entanglement and exchange of conserved quantities, avoiding the introduction of new operators or constants (Gillis, 2018, Gillis, 2021, Gillis, 2022).

2. Conservation Laws in Individual Collapse Events

Strict conservation of total momentum, orbital angular momentum, and energy is enforced in each realization of the collapse process—not just on average. This is a direct consequence of the commutation relations: $[P, L_{jk}] = 0$ for total momentum $P$, and similar for the total angular momentum operator, since $V_{jk}$ depends only on relative coordinates. For energy, conservation is exact within the regime of validity of the nonrelativistic Hamiltonian; leading-order corrections due to collapse are of order $(V/(mc^2))^2$, which are negligible in standard regimes and commensurate with other well-known subleading corrections (relativistic mass increase, radiative losses) (Gillis, 2018, Gillis, 2021).

The exact enforcement can be schematically summarized:

Conserved Quantity	Mechanism in Collapse Law	Conservation Status
Total Momentum	Collapse operators commute with P	Exact in each event
Orbital Angular Momentum	Collapse operators commute with L	Exact in each event
Total Energy	Stochastic drift suppressed; small errors	Exact to O[(V/(mc^2))^2]

Apparent violations in subsystems are always accompanied by precisely compensating changes in other entangled systems, so that the global conservation law is maintained in every branch selected by the collapse process (Gillis, 2018).

3. Pre-existing Entanglement and the Role of Generic Correlations

It is a central aspect of the law that all quantum interactions, even those considered part of state "preparation," create generic pre-existing entanglement with environments or measurement devices, as proven by Gemmer & Mahler and Durt. When a measurement induces collapse into one branch, all pre-existing correlations collapse as well, ensuring that apparent "losses" or "gains" of conserved quantities in the measured system are exactly compensated by correlated changes in the broader environment or apparatus. This "global accounting" is required for strict conservation in single runs (Gillis, 2018).

This principle provides a universal explanation for the conservation of observable quantities in projective measurements, regardless of the apparent local discontinuity.

4. Comparison With Alternative Collapse and Measurement Models

Traditional spontaneous collapse models typically introduce collapse operators tied to particular observables (e.g., position or energy) and include new parameters or fundamental constants. In contrast, the Interaction–Collapse Law derives collapse operators directly from pairwise interaction potentials already present in $H$, and the strength of collapse is fixed by these interactions and system masses. No new constants are introduced (Gillis, 2018, Gillis, 2022).

Furthermore, energy and momentum conservation are not enforced only after averaging over many collapse events but hold in every stochastic realization—an essential distinction.

In operational models, such as stroboscopic collapse in a soft-impact oscillator, alternative postulates for the criterion and localization of collapse all result in energy-conserving dynamics when appropriate care is taken, with ensemble maps that can be written as generalized Lindblad equations (Acharya et al., 2021).

5. Physical Applications and Case Studies

The law applies to diverse quantum and condensed-matter scenarios, such as:

• Interacting Polymer Chains: The collapse transition in self-avoiding walks and trails, where the collapse exponent and scaling of the free energy can be derived from the form and strength of the pairwise monomer attraction (Carmona et al., 2013, Bedini et al., 2013).
• Bose–Einstein Condensates: The stability, critical number, and scaling laws for collapse in condensates with singular (e.g., $1/r^b$) interactions are precisely dictated by the interaction law; regime distinctions (critical, supercritical, subcritical) follow from the interplay of kinetic and interaction energies (Lushnikov, 2010, Morris et al., 2024).
• Quantum Measurement and Decoherence: In models involving particles coupled via point interactions to detectors, collapse emerges dynamically as the exponential excitation of detector degrees of freedom, with the measured particle's distribution converging to a delta function at the interaction point—offering a formal model of position measurement (Guarneri, 2011).
• Thermodynamics and the Second Law: In thermodynamic cycles, the amount of work extractable via measurement-induced collapse is strictly bounded by the entropy generated in the collapse process, cementing a fundamental link between quantum randomness and the second law of thermodynamics (Hormoz, 2012).
• Particle Physics and Symmetry Testing: Collapses induced by strong interaction (e.g., in meson-antimeson pairs) enforce orthogonality constraints dictated by statistics, enabling the isolation and identification of direct CP violation exclusively through the collapse selection (Stodolsky, 2014).

6. Relativistic Generalization and Structural Significance

Recent work extends the stochastic collapse law to relativistic quantum theory by adapting the collapse operator to a stochastic process over space-like hypersurfaces. The nonlocal, probabilistic nature of the collapse mechanism ensures compatibility with both relativistic causality and strict conservation, while local commutativity (the vanishing of operator commutators for spacelike separation) emerges as a necessary dynamical outcome, not a separate axiom. This structure supports the elevation of local commutativity to a third postulate of relativity (Gillis, 2024).

Collapse terms, constructed from Lorentz-scalar combinations of potentials and rates, transform appropriately under Lorentz transformations, and their effect is to randomly but consistently amplify one branch while suppressing others within the constraints set by physical interactions.

7. Experimental Implications and Universality

The law provides concrete predictions for experimentally accessible quantities. For engineered systems such as quantum ratchets and box-trapped Bose gases, scaling relations for energy, spatial extension, density, and atom loss follow directly from the interaction–collapse dynamics and have been validated against numerical simulations and measurements (Acharya et al., 2021, Morris et al., 2024). Experimental differences arising from the choice of collapse criterion, the presence or absence of multi-particle conservation, or the introduction of structure-specific forces (e.g., in molecular braids) can be directly calculated and compared to observation (Lee, 2014).

In summary, the Interaction–Collapse Law delivers a unified, dynamical, and rigorously conservation-respecting account of quantum collapse, drawing structural and predictive power directly from the nature and range of physical interactions themselves. It dispenses with the need for observer-centric or ad hoc collapse mechanisms, offering instead a stochastic, norm-preserving evolution rooted in standard quantum couplings and consistent with both fundamental symmetries and empirical reality across scales (Gillis, 2018, Gillis, 2021, Gillis, 2022, Gillis, 2024).