Emergent Causality from Decoherence: A Hypothesis with a Defined Parameter, a Derived Regime, and an Explicit Conjecture
Lorca, Rafael Eduardo
Abstract
Emergent Causality from Decoherence: A Hypothesis with a Defined Parameter, a Derived Regime, and an Explicit Conjecture
An imaginary hypothesis, not a theory.
This is a revised version. It corrects errors from the previous version. It is still a hypothesis, not a theory. The weakest point is the premise that classical dynamics is causal. If that premise fails, the hypothesis fails.
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Abstract
This manuscript proposes that classical causality is not fundamental but emergent, arising from the degree of decoherence of a system. We define a parameter for decoherence (purity), derive one regime (the classical regime) from it, confront the established literature, and acknowledge the limits of the proposal. We add an explicit conjecture connecting the parameter to the process-matrix formalism of Oreshkov, Costa and Brukner. The hypothesis is not a theory. It lacks full mathematics, experimental confirmation, and a resolution of the circularity of time. It is offered as a framework for critique and development.
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1. Introduction
The nature of causality remains an open problem in physics. In quantum mechanics, there is no causal structure in the classical sense: there are correlations, not causes. In classical physics, causality is absolute and local. These descriptions do not obviously fit together.
This manuscript proposes that classical causality is not a fundamental property of the universe, but a property of the level of description. It emerges from the degree of decoherence of a system, in a manner analogous to how temperature emerges from the collective behaviour of many particles.
The hypothesis is labelled "imaginary" because it lacks full mathematical formalism, experimental confirmation, and a resolution of the circularity of time. It is offered as a framework for critique, not as a completed theory.
Scope: This hypothesis addresses only the transition between the quantum regime (no defined causal order) and the classical regime (defined causal order). It does not address the relativistic regime.
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2. The Parameter
The central parameter is the degree of decoherence, γ. In standard treatments, decoherence is not a single scalar: it is a basis-dependent, model-dependent process characterised by decoherence rates, a pointer basis, and a system-environment split. Different measures (purity, coherence norms, entanglement with the environment, predictability sieve) order states differently.
For the purposes of this hypothesis, we choose a specific measure: purity, defined as:
P = Tr(ρ²)
where ρ is the reduced density matrix of the system. For a pure state, P = 1. For a maximally mixed state, P = 1/d, where d is the dimension of the Hilbert space. We define the degree of decoherence as:
γ = 1 - P
Thus, γ = 0 for a pure state (no decoherence), and γ > 0 for a mixed state (decoherence).
This is a choice. Other measures are possible. This choice is made for definiteness, not because it is uniquely correct. The hypothesis does not depend on this specific choice; it depends on the existence of some measure that orders states by their degree of decoherence. Purity is chosen as a concrete example.
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3. The Derivation (Weak)
With this parameter, we can derive one regime: the classical regime. We postulate that classical causality emerges when γ exceeds a critical value, γ_c. Below γ_c, the system is in a quantum regime: correlations without definite causal order. Above γ_c, the system is in a classical regime: cause and effect, without ambiguity.
This is not a derivation from first principles. It is a postulate. But it is a postulate with a defined parameter. And it is a starting point.
The intermediate regime is not derived. It is postulated as a transition. And it is acknowledged that this placement is in tension with established physics: relativistic causal structure is a property of the spacetime metric, not of partial decoherence. This hypothesis does not address the relativistic regime. It addresses only the transition between the quantum regime (no defined causal order) and the classical regime (defined causal order).
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4. Confrontation with the Literature
The hypothesis must be placed against the established literature on decoherence and emergent causality. The following works are directly relevant. We do not confront them as attacks, nor do we attempt to surpass them. We use them as anchors: we situate the hypothesis relative to them, and we state explicitly what is already done and what is added here.
• Zurek (2003), "Decoherence, einselection, and the quantum origins of the classical," Rev. Mod. Phys. 75, 715. This is the foundational account of classical structure emerging from environmental monitoring. Zurek shows that decoherence selects a small set of stable states (pointer states) and that the combination of einselection with dynamics leads to the emergence of classical phase-space structure: points, trajectories, classical dynamics. This hypothesis assumes, as an additional premise, that classical dynamics is causal. This premise is not derived from Zurek or from Gell-Mann and Hartle. It is an independent claim, and it is the weakest point in the argumentative chain. If the premise is true, then causality emerges from decoherence. If it is false, the hypothesis fails. It is offered as a conjecture, not as a derivation.
• Gell-Mann and Hartle (1994, 2014), "Quasiclassical Realms in a Quantum Universe" (gr-qc/9404017) and "Adaptive Coarse Graining, Environment, Strong Decoherence, and Quasiclassical Realms" (arXiv:1312.7454). These works derive classical deterministic time-correlations and the fixed-past/open-future asymmetry from coarse-graining and strong decoherence, with no observer. The present hypothesis is close to this programme. It differs in framing the result as "emergent causality" rather than "quasiclassical realms." The substance is similar. The present hypothesis does not resolve the circularity of time; it assumes time as a parameter of unitary evolution.
• Oreshkov, Costa, and Brukner (2012), "Quantum correlations with no causal order," Nature Communications 3, 1092. DOI: 10.1038/ncomms2076. This work constructs correlations with no definite causal order and shows that definite order is recovered in a classical limit. The present hypothesis is consistent with this, but does not engage with the formalism. A conjecture connecting the two is offered in Section 5.
• Page and Wootters (1983), "Evolution without evolution: Dynamics described by stationary observables," Phys. Rev. D 27, 2885. This work obtains time from entanglement with an internal clock. The present hypothesis does not address this. It assumes time as a parameter. The relationship between the two proposals is not resolved here.
• Joos and Zeh (1985), "The emergence of classical properties through interaction with the environment," Z. Phys. B 59, 223. Foundational work on decoherence in open quantum systems. The present hypothesis builds on this without adding to it.
• Rovelli (1996), "Relational quantum mechanics," Int. J. Theor. Phys. 35, 1637. Relational quantum mechanics holds that states are relational, not absolute. The present hypothesis extends this to causality: if states are relational, perhaps causal structure is too. But the extension is not derived. It is proposed.
• Schlosshauer (2005), "Decoherence, the measurement problem, and interpretations of quantum mechanics," Rev. Mod. Phys. 76, 1267. A review of decoherence and its interpretational implications. The present hypothesis assumes this framework.
The honest assessment: The hypothesis is not new in its intuition. It is new in its emphasis on causality as the emergent property, and in its attempt to frame the quantum-classical transition as a causal transition. But the emphasis is not a theory, and the transition is not derived. The hypothesis is offered as a framework for critique.
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5. Conjecture: Connection with the Process-Matrix Formalism
The conjecture that follows is supported by the premise of Section 4. If classical dynamics is causal, then the transition from γ = 0 to γ > γ_c could correspond to the transition from a causally non-separable process matrix to a causally separable one.
Oreshkov, Costa and Brukner (2012) have shown that, if one does not assume a global causal order, there exist quantum correlations that violate the causal inequality. These correlations are described by process matrices that are causally non-separable. In a classical limit, causal order emerges. The authors suggest that spacetime could emerge from a more fundamental structure in a quantum-to-classical transition.
This hypothesis proposes, as a conjecture, that the more fundamental structure could be decoherence, parametrised by γ. The causally non-separable process matrix could correspond to γ = 0 (no decoherence, no defined causal order). The causally separable process matrix could correspond to γ > γ_c (sufficient decoherence, emergent causal order). The quantum-to-classical transition mentioned by Oreshkov et al. could be the transition in γ.
This is a conjecture. It is not derived. There is no formalism that supports it. It is offered as a direction for future work: if someone with the appropriate mathematical tools wishes to attempt the connection, here is the intuition. And here is the criterion for success: if the connection is derived, the hypothesis gains. If it is not, it loses. Both outcomes are valuable.
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6. The Limits
The hypothesis has several limits, acknowledged here:
1. The parameter is chosen, not derived. Purity is one measure among many. Other measures would give different results. The choice is made for definiteness, not because it is uniquely correct.
2. Only one regime is derived (weakly). The classical regime is postulated, not derived from first principles. The relativistic regime is not addressed.
3. The circularity of time is not resolved. Decoherence presupposes time. The arrow of time is traced to a low-entropy initial condition, not to decoherence itself. The hypothesis does not resolve this circularity. It assumes it.
4. The literature is confronted, but not surpassed. The hypothesis shares its intuition with Zurek, Gell-Mann and Hartle, and others. It does not add formalism. It adds a framing, but the framing is not a derivation.
5. No experimental prediction is made. The hypothesis does not identify a falsifiable consequence. This is a significant weakness.
6. The conjecture is not derived. The connection with the process-matrix formalism is offered as a conjecture, not as a result.
7. The weakest point is the premise that classical dynamics is causal. This premise is not derived. It is an independent claim. If it fails, the hypothesis fails.
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7. Conclusion
This manuscript proposes that classical causality is not fundamental but emergent, arising from the degree of decoherence of a system. It defines a parameter (purity), derives one regime (classical, weakly), confronts the established literature, acknowledges its limits, and offers an explicit conjecture connecting the parameter to the process-matrix formalism.
It is not a theory. It is a hypothesis. It is offered as a framework for critique and development.
The next steps would be: to derive the classical regime from the parameter more rigorously; to resolve the circularity of time; to identify a falsifiable prediction; and to engage with the formalism of Oreshkov et al. These steps are not taken here. They are left for future work.
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The label (important)
This is an imaginary hypothesis, not a theory. It is coherent with what we know, but it is not demonstrated. It lacks full mathematics, predictions, and confirmation. It does not say that time does not exist. It says that classical causality could emerge. It does not say that the relativistic regime is explained. It says that the quantum-classical transition could be a causal transition. All of this, if one takes decoherence seriously, without adding anything else. But no one has demonstrated it. That is why it is a hypothesis, not a theory. And the weakest point is the premise that classical dynamics is causal. If that premise fails, the hypothesis fails.
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A note on the anchor
This hypothesis does not invent its premise. It takes it from established physics: quantum mechanics already treats information as correlation, not as substance. Entanglement is pure correlation. Decoherence is entanglement with the environment. If information is always relational, then causality could be relational too. But the extension is not derived. It is proposed.
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Related work (with authors, years, and identifiers)
• Zurek, W. H. (2003). "Decoherence, einselection, and the quantum origins of the classical." Rev. Mod. Phys. 75, 715.
• Gell-Mann, M., and Hartle, J. B. (1994). "Quasiclassical Realms in a Quantum Universe." gr-qc/9404017.
• Gell-Mann, M., and Hartle, J. B. (2014). "Adaptive Coarse Graining, Environment, Strong Decoherence, and Quasiclassical Realms." arXiv:1312.7454.
• Page, D. N., and Wootters, W. K. (1983). "Evolution without evolution: Dynamics described by stationary observables." Phys. Rev. D 27, 2885.
• Oreshkov, O., Costa, F., and Brukner, Č. (2012). "Quantum correlations with no causal order." Nature Communications 3, 1092. DOI: 10.1038/ncomms2076.
• Joos, E., and Zeh, H. D. (1985). "The emergence of classical properties through interaction with the environment." Z. Phys. B 59, 223.
• Rovelli, C. (1996). "Relational quantum mechanics." Int. J. Theor. Phys. 35, 1637.
• Schlosshauer, M. (2005). "Decoherence, the measurement problem, and interpretations of quantum mechanics." Rev. Mod. Phys. 76, 1267.