The Informational Substrate Hypothesis (ISH): Bridging Quantum and Classical Realities

AI can generate plausible-sounding theories and explanations and these essays are posted here with that in firmly in mind.

These ideas were elicited from ChatGPT and Bard, as an effort to learn about physics (as a hobby), using a collaborative approach.

Introduction:

The historical tug-of-war between the deterministic world of classical mechanics and the probabilistic realm of quantum mechanics has been a foundational challenge in physics. While our series has systematically unveiled the intricacies of this dichotomy, we now introduce the Informational Substrate Hypothesis (ISH). This hypothesis attempts to redefine the conversation, venturing into an underlying fabric that possibly mediates between the known classical and quantum domains.

The Need for an Intermediary Layer:

In our exploration of the quantum-classical divide, we recognized that while both realms are mathematically sound and experimentally validated, their fundamental incompatibilities hint at a deeper layer. An intermediary layer that can potentially harmonize determinism and probability, offering insights into how and why the universe chooses between them.

The Core of the ISH:

Classical Layer: A deterministic realm where events are predictable and outcomes certain. The celestial ballet of planets and the deterministic unfolding of classical events reside here.

Quantum Layer (QUM Layer): A dance of probabilities. Here, particles exist in superpositions and states are defined by likelihoods rather than certainties.

Informational Substrate: More than just an intermediary layer, the ISH suggests this is the foundation. It hypothesizes that information, as a conserved quantity, drives the universe’s decisions, resolving the interplay between the Classical and QUM layers.

The Dynamics of Resolution:

The ISH isn’t merely a passive layer. It’s an active participant in the universe’s evolution. It proposes that the universe, in its essence, is constantly “deciding” based on the informational substrate. This decision-making, balancing between quantum possibilities and classical outcomes, defines the universe’s state.

Implications for Space-Time:

One of the boldest propositions of the ISH is the idea that space-time, rather than being fundamental, is emergent. It emerges from the interactions and decisions made within the informational substrate. This gives a fresh perspective on phenomena like quantum entanglement, which defy classical space-time constraints.

Potential Predictions and Challenges:

No theory is complete without predictions and testable outcomes. If the ISH is on the right track, we might anticipate:

1. Observational effects hinting at the transition from quantum to classical states.
2. New particles or phenomena that serve as mediators between quantum and classical realms.

Yet, the ISH remains in its infancy. Rigorous mathematical formulations, consistent with both quantum mechanics and general relativity, are still needed. And empirical validation is paramount.

Conclusion:

The Informational Substrate Hypothesis, while nascent, introduces a fresh narrative in the quantum-classical dialogue. By proposing an underlying realm of information, it raises captivating questions about the nature of reality. As we advance in this series, the ISH will serve as a stepping stone, guiding our exploration into the uncharted territories of the cosmos.

P. Delaney, August 2023

Disclaimer:
This essay presents a speculative and theoretical framework regarding The Informational Substrate Hypothesis (ISH), and Quantum Uncertainty Mathematics (QUM). The ideas and concepts discussed herein are exploratory in nature and are intended to provoke thought and discussion. They have not been validated by formal mathematical or scientific research. Readers are encouraged to approach the content with an open mind and to engage in constructive dialogue about its potential implications and applications. Feedback, critiques, and collaborative insights are warmly welcomed.

Appendix A: Elaborating the Informational Framework

The Informational Substrate Hypothesis (ISH) posits a deeper layer to our reality, governed by principles of information conservation and processing. This appendix delves into the intricacies of this informational framework, providing a blueprint for how such a structure might underlie and mediate between the classical and quantum realms.

1. Nature of Information:

In the context of ISH, information isn’t merely an abstract concept; it’s the essence of the substrate. It’s both quantifiable and qualitative:

Quantifiable: Measurable in units that might be akin to (but not necessarily the same as) classical bits or quantum qubits.

Qualitative: Representing the intrinsic characteristics and states of systems, be they classical or quantum.

2. Information Conservation Principle:

Just as energy is conserved in classical mechanics and quantum systems, the ISH posits that information is conserved within the substrate. This principle suggests that:

- Information cannot be created or destroyed but can transition between states.

- The total information content of the universe remains constant, despite the dynamic interplay between the Classical and QUM layers.

3. Information Processing and Decision-Making:

The substrate doesn’t just store information; it processes it. The universe’s evolution, in light of the ISH, is a continual decision-making process guided by:

Probabilistic Computations: Drawing from the quantum layer’s inherent probabilities.

Deterministic Outcomes: Influenced by the established states in the classical realm.

The Role of Observers:

While traditional quantum mechanics places significant emphasis on the role of observers in determining quantum states, the ISH suggests observers are also informational entities within the substrate. Thus, observing is an informational transaction, potentially influencing the decision-making process but not external to the informational dynamics.

5. Implications for Quantum Entanglement:

Quantum entanglement, with its “spooky action at a distance,” becomes more intuitive within the ISH. Entangled particles might share informational links within the substrate, allowing for instantaneous correlation regardless of spatial separation.

6. The Informational Architecture:

The substrate is likely not homogenous. It may possess:

Informational Pathways: Channels or routes that facilitate the flow and processing of information.

Nodes of Intensity: Regions of concentrated information processing, potentially correlating with phenomena in both quantum and classical layers.

The informational framework of the ISH paints a picture of a universe deeply rooted in the principles of information. It offers a fresh perspective, suggesting that beneath the dance of particles and waves, equations and uncertainties, there lies a cohesive informational tapestry guiding the cosmos’s evolution.

Appendix B: Mathematical Formulations of the Informational Framework

To make sense of the Informational Substrate Hypothesis (ISH) and its intricacies, it’s vital to explore its potential mathematical manifestations. This appendix provides a speculative framework that might underpin the principles of the ISH.

1. Quantifying Information:
For an informational substrate, the first task is to represent information in a quantifiable manner. If \( I \) represents the information content:
\[ I = \sum_{i} p(i) \log \left( \frac{1}{p(i)} \right) \]
Where:
- \( p(i) \) is the probability distribution of a particular state or outcome \( i \).
- The summation spans all possible states or outcomes.

2. Conservation of Information:
Building on the principle of information conservation:
\[ \Delta I = I_{final} — I_{initial} = 0 \]
This suggests that the net change in information over any transaction or transition is zero, emphasizing the conservation principle.

3. Probabilistic Processing:
The dynamics of information processing in the substrate can be represented using a hypothetical operator \( \hat{P} \):
\[ \hat{P} \Psi = \lambda \Psi \]
Where:
- \( \hat{P} \) is the probabilistic operator acting on a system.
- \( \Psi \) is an informational state vector.
- \( \lambda \) represents the eigenvalues, detailing the outcomes or resolutions of the probabilistic processing.

4. Deterministic Pathways:
The deterministic outcomes and their influence can be formulated using a deterministic operator \( \hat{D} \):
\[ \hat{D} \Phi = \omega \Phi \]
Where:
- \( \hat{D} \) is the deterministic operator.
- \( \Phi \) is a state vector in the classical layer.
- \( \omega \) are the eigenvalues representing deterministic resolutions.

5. Observer Dynamics:
Incorporating the role of observers as informational entities:
\[ \hat{O} \Gamma = \alpha \Gamma \]
Where:
- \( \hat{O} \) is the observer operator.
- \( \Gamma \) represents the observer’s informational state.
- \( \alpha \) signifies the eigenvalues that correspond to the outcomes of observational interactions.

6. Entanglement Information Metrics:
To understand entanglement in this framework:
\[ E(\Psi_1, \Psi_2) = -\log_2 \left( \max_i |\langle \Psi_i | \Psi_1 \otimes \Psi_2 \rangle |² \right) \]
Where:
- \( E(\Psi_1, \Psi_2) \) represents the entanglement measure between states \( \Psi_1 \) and \( \Psi_2 \).
- \( \langle \Psi_i | \Psi_1 \otimes \Psi_2 \rangle \) denotes the overlap between the combined state and a particular basis state \( \Psi_i \).

The mathematical formulations outlined here offer a glimpse into the possible mathematical structure of the ISH. While these equations are speculative and need rigorous vetting, they serve as a starting point for envisioning how the principles of the ISH could be represented in a formalized manner.

Appendix C: In-depth Exploration of Operators and Metrics within ISH

Introduction:

One of the defining features of the Informational Substrate Hypothesis (ISH) is its reliance on operators and metrics to delineate and analyze the interplay between deterministic and probabilistic realms. This appendix aims to provide a deeper understanding of these mathematical tools and their implications within the ISH framework.

1. The Probabilistic Operator \( \hat{P} \):

Function:
The \( \hat{P} \) operator captures the essence of the quantum uncertainty that permeates the ISH. It acts on the informational state vectors, representing potential outcomes and their respective likelihoods.

Characteristics:
- Non-commutative nature: \( \hat{P} \) may not necessarily commute with other operators, signifying a deep-rooted quantum behavior.
- Hermitian: Ensuring the eigenvalues \( \lambda \) are real, reflecting measurable outcomes.

2. The Deterministic Operator \( \hat{D} \):

Function:
In contrast to \( \hat{P} \), \( \hat{D} \) represents the deterministic, classical dynamics of the system. It operates on states where outcomes are definite and certain.

Characteristics:
- Commutative with most other operators in its realm, symbolizing the predictable nature of classical physics.
- Positivity: All eigenvalues \( \omega \) are positive, indicating the constructive and additive nature of deterministic processes.

3. The Observer Operator \( \hat{O} \):

Function:
The act of observation, particularly in quantum mechanics, has profound implications. The \( \hat{O} \) operator provides a mechanism to represent and analyze this process within the ISH.

Characteristics:
- Entangled States: \( \hat{O} \) may generate entangled states when interacting with other state vectors, capturing the observer’s influence on the observed.
- Complex Eigenvalues: Reflecting the inherently intricate nature of observation in quantum systems.

4. Metrics of Information:

The process of quantifying and analyzing information is pivotal to the ISH. Some proposed metrics include:

Entropy (S): It represents the amount of uncertainty or disorder in a system.
\[ S = — k_B \sum_i p(i) \log p(i) \]
Where \( k_B \) is the Boltzmann constant.

Mutual Information (I): This metric quantifies the information shared between two systems.
\[ I(X;Y) = \sum_{x,y} p(x,y) \log \left( \frac{p(x,y)}{p(x)p(y)} \right) \]
Where \( p(x,y) \) is the joint probability distribution of X and Y.

Quantum Discord (D): A measure of the quantumness of correlations in a system, hinting at how much quantum mechanics deviates from classical intuitions.

5. Linking Operators and Metrics:

To understand the complex dance of information in the ISH framework, one must consider how operators and metrics intertwine:

- The action of \( \hat{P} \) on a state could increase its entropy, reflecting the inherent uncertainty.
- Observational processes governed by \( \hat{O} \) might generate mutual information between the observer and the observed.
- The deterministic realm, steered by \( \hat{D} \), tends to minimize quantum discord, pushing systems towards classicality.

Conclusion:

Understanding the detailed characteristics and interplay of the operators and metrics within the ISH is crucial for a comprehensive grasp of the hypothesis. These tools not only provide insights into the foundational dynamics of the informational substrate but also help bridge the gap between quantum mechanics and classical physics.

Appendix D:

The Observer within the Informational Landscape:

1. Entangled Dynamics: In the ISH, the observer is not an isolated entity outside the system but a part of the very substrate being observed. When an observation is made, the informational state of the observer becomes entangled with the observed. This entanglement represents a sharing of information, where the state of one is inevitably tied to the state of the other.
2. Transactional Nature of Observation: Observation isn’t a one-way process. In the ISH perspective, it’s a transaction where both the observer and the observed exchange information. This means when we “observe” a quantum state, we’re not just passively receiving information; we’re also imparting information onto the state, which can influence its evolution.
3. Decision-making as Informational Processing: When we think of decision-making, especially in the context of quantum systems like Schrödinger’s cat, it’s often seen as a binary outcome based on observation. Within ISH, decision-making becomes a dynamic computational process within the informational substrate. The observer, in making a decision or observation, is essentially running a computation within the broader informational matrix.
4. Eliminating the “Outside” Observer Problem: One of the perennial challenges of quantum mechanics is the role of consciousness in observations. Some interpretations suggest that consciousness causes the collapse of the wave function. In ISH, since everything, including consciousness, is an informational process within the substrate, the distinction between conscious and unconscious observation becomes moot. Everything is a part of the same informational tapestry.

Implications for Reality and Perception:

Understanding observers as informational entities reshapes our perception of reality. If everything, including our very consciousness, is a manifestation of informational processes, then our experiences, decisions, and perceptions are all transactions within this substrate. It hints at a deeper interconnectedness of all things, where separation is more a matter of perception than a fundamental reality.

The ISH, in integrating the observer into the very fabric of its framework, offers a more holistic view of the universe. It acknowledges the complexities of observation and decision-making, not as anomalies or externalities but as integral components of the informational dynamics that govern our reality.

Appendix E: Sources

Foundational Texts on Quantum Mechanics:
Feynman, R. P., Leighton, R. B., & Sands, M. (1965). The Feynman Lectures on Physics, Vol. 3: Quantum Mechanics. Addison-Wesley.
Griffiths, D. J. (2004). Introduction to Quantum Mechanics. Pearson Prentice Hall.
Classical Mechanics:
Goldstein, H., Poole, C., & Safko, J. (2001). Classical Mechanics. Addison Wesley.
Information Theory:
Shannon, C. E. (1948). A Mathematical Theory of Communication. Bell System Technical Journal, 27(3), 379–423.
Cover, T. M., & Thomas, J. A. (2006). Elements of Information Theory. John Wiley & Sons.
Quantum Information Theory:
Nielsen, M. A., & Chuang, I. L. (2010). Quantum Computation and Quantum Information. Cambridge University Press.
Quantum Entanglement:
Horodecki, R., Horodecki, P., Horodecki, M., & Horodecki, K. (2009). Quantum entanglement. Reviews of Modern Physics, 81(2), 865.
Observer in Quantum Mechanics:
Wigner, E. P. (1961). Remarks on the mind-body question. In Symmetries and Reflections: Scientific Essays. MIT Press.
Space-Time and Quantum Mechanics:
Rovelli, C. (2004). Quantum Gravity. Cambridge University Press.
Quantum Uncertainty:
Heisenberg, W. (1927). Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. Zeitschrift für Physik, 43(3–4), 172–198.
Quantum-Classical Transition:
Zurek, W. H. (2003). Decoherence, einselection, and the quantum origins of the classical. Reviews of Modern Physics, 75(3), 715.
Speculative Theories in Physics:
Penrose, R. (1989). The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press.