Beyond Particles and Waves: A Reflection on "Everything is a Lagrangian Submanifold"
Zero Kelvin Moralist — January 2025
Introduction
This document explores the idea that “everything is a Lagrangian submanifold,” a concept introduced by Alan Weinstein and expanded upon by Curt Jaimungal in his article "Everything is a Lagrangian Submanifold..." (source). Building on these foundations, discussions leading to this work proposed a compelling metaphor: a shifted wire grid model. This metaphor captures the nuanced structure and dynamics of symplectic phase space and its Lagrangian submanifolds, offering a geometric and visual framework for understanding physical systems.
The Shifted Wire Grid Model
The shifted wire grid model was conceived as a way to visualize the layered and interconnected nature of phase space. In this model:
- Grid Nodes: Represent states within phase space, which combine position (q) and momentum (p) coordinates.
- Connections: Depict the symplectic relationships between states, governed by the symplectic form (\(\omega\)).
- Layers: Symbolize Lagrangian submanifolds, which are subspaces where the symplectic form vanishes (\(\omega|_L = 0\)). These layers describe the constraints or conserved quantities of physical systems.
The metaphor of “slightly shifted” grids highlights the dynamic and multi-dimensional interplay of states, where constraints and transformations shape the system’s evolution. The overlapping grids correspond to the interconnection of different Lagrangian submanifolds, revealing how symplectic geometry unites seemingly distinct states or subsystems.
Applications to Quantum Mechanics and Entanglement
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Entangled States in Lagrangian Submanifolds:
- Entangled states are seen as correlated configurations that exist within or between overlapping Lagrangian submanifolds.
- The geometric overlap between these submanifolds could provide a foundation for the non-local correlations observed in entangled systems.
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Non-Classical Proximity:
- In the shifted wire grid model, proximity is not spatial in the classical sense but exists within the phase space or an abstract Hilbert space.
- This aligns with the idea that entangled particles, despite being spatially separated, remain “close” within a shared symplectic structure.
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Superposition and Overlap:
- The model also captures the notion of quantum superposition, where states are not discrete points but overlapping configurations. These overlaps could represent the probabilistic nature of quantum states, constrained by the symplectic geometry.
Dynamic and Infinite Nature of Phase Space
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Infinite Possibilities:
- Phase space allows for a near-infinite number of configurations, constrained only by symplectic geometry and quantization conditions.
- The symplectic form (\(\omega\)) governs how these states transform and evolve, ensuring the conservation of areas within phase space.
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Continuous Shifts and Interactions:
- The idea of “slightly shifted grids” reflects the continuous transformations and interactions between Lagrangian submanifolds. These shifts correspond to the dynamic processes that define physical systems, such as quantum fluctuations or classical dynamics.
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Nuances and Directions:
- Small variations or “nuances” in position and momentum can lead to entirely new trajectories and configurations. This highlights the sensitivity of systems to initial conditions and the richness of their potential evolutions.
The Shifted Wire Grid Model as a Visualization Tool
- It provides a conceptual framework to visualize the layered structure of phase space.
- It highlights the dynamic relationships and constraints that govern the evolution of states.
- It captures the transition from classical mechanics to quantum mechanics, illustrating how quantization restricts the infinite possibilities of classical systems to discrete states.
For more about these concepts, visit: Everything is a Lagrangian Submanifold
How These Formulas Reflect the Article’s Ideas
- Phase Space and Symplectic Geometry: As defined in the article, phase space is a space of \( (q, p) \) pairs representing position and momentum. Symplectic geometry equips this space with a structure (\(\omega\)) that preserves dynamics.
- Lagrangian Submanifolds: These are subspaces where the symplectic form vanishes (\(\omega|_L = 0\)), encoding the constraints and dynamics of systems.
- Quantization: The transition from classical to quantum mechanics is viewed as selecting specific Lagrangian submanifolds corresponding to quantized states.
- Shift from “Things” to Relationships: Inspired by the article, this work emphasizes the relationships between states (e.g., energy and time, position and momentum) as captured by symplectic geometry.
Symplectic Phase Space Formulas with Explanations
1. Symplectic Form
\(\omega = \sum_{i=1}^n dp_i \wedge dq_i\)
Explanation: As described in the article, the symplectic form \(\omega\) provides the geometric foundation for phase space, defining how areas and dynamics are preserved.
2. Poisson Bracket
\(\{f, g\} = \sum_{i=1}^n \left( \frac{\partial f}{\partial q_i} \frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i} \frac{\partial g}{\partial q_i} \right)\)
Explanation: The Poisson bracket expresses how two quantities interact dynamically in phase space. The article highlights how this reflects conserved quantities and the evolution of systems.
3. Lagrangian Submanifold
\(L \subset (M, \omega) \quad \text{such that} \quad \omega|_L = 0\)
Explanation: A key concept from the article, Lagrangian submanifolds encode constraints and represent physically meaningful sets of states, serving as the “building blocks of nature.”
4. Hamiltonian Vector Field
\(\forall X \in \Gamma(TM), \quad \exists f \in C^\infty(M) \quad \text{s.t.} \quad \iota_X \omega = df\)
Explanation: Hamiltonian vector fields describe the flow of systems in phase space, connecting symplectic geometry to dynamics.
5. Quantization Condition
\(\frac{1}{2\pi} \int_S \omega \in \mathbb{Z}\)
Explanation: Quantization bridges classical and quantum mechanics, selecting discrete states corresponding to specific Lagrangian submanifolds.
6. Shifted Wire Grid Model
\(G(q, p, \omega, L) = \text{\emph{shifted wire grid model}}\)
Explanation: Inspired by the relationships emphasized in the article, this conceptual model visualizes symplectic phase space as a dynamic grid of quantized states and trajectories.
Acknowledgments
This document builds on Curt Jaimungal’s article "Everything is a Lagrangian Submanifold..." (source) and integrates insights from Alan Weinstein’s foundational work on symplectic geometry and Lagrangian submanifolds.
This document was created with the assistance of Calix, utilizing the READ Framework to enhance clarity and interdisciplinary integration. Learn more at: READ Framework.
References
- Curt Jaimungal, Everything is a Lagrangian Submanifold, January 2025. https://curtjaimungal.substack.com/p/everything-is-a-lagrangian-submanifold
