NO OTHERNESS

Abstract  

The Trouton-Noble Paradox has long puzzled physicists. When a charged object such as a capacitor is placed in an electric field, classical theories suggest it should experience a continuous torque because of its electric dipole moment. But if the object is in an isolated system, where would the energy for this motion come from?

Introduction

In this paper we address the Trouton-Noble Paradox with both classical electromagnetism and quantum mechanics. We depict the role of decoherence, which shows how quantum systems transition to classical behavior, and of information theory, especially in showing how changes in entropy can explain energy redistribution in quantum systems

Theoretical Framework 

— Torque and Force: According to classical physics, a charged dipole in an electric field experiences a torque (\(\tau\)) given by:

\[\tau = p \times E\]

where \(p\) is the electric dipole moment and \(E\) is the electric field. 

The expectation is that this torque leads to continuous angular motion, which raises questions about energy input and conservation.

— Energy Conservation Dilemma: The work done (\(W\)) on the dipole due to the torque is expressed as:

\[W = \int \tau \, d\theta\]

This raises concerns about how energy balance is maintained in an isolated system, as classical physics does not adequately address the energy sources involved.

Applying Quantum Mechanics and Decoherence 

— Decoherence Perspective  

Decoherence explains how quantum systems transition to classical behavior through interactions with their environment. The mathematical description of decoherence is captured by the master equation for the density matrix (\(\rho(t)\)):

\[\frac{d\rho(t)}{dt} = -\frac{i}{\hbar} [H, \rho(t)] + \mathcal{D}[\rho(t)]\]

where \(\mathcal{D}[\rho(t)]\) accounts for environmental interactions. 

In the context of the Trouton-Noble Paradox, the charged object interacts with both the electric field and its surrounding environment, leading to classical-like behavior as quantum coherence is lost.

— Information Theory Framework  

In information theory, the entropy (\(S\)) of a quantum system can be quantified using von Neumann entropy:

\[S(\rho) = -\text{Tr}(\rho \log \rho)\]

The reduction in entropy following a measurement reflects the information gained about the system’s state. This perspective can clarify how energy is redistributed rather than lost, as the work done on the dipole can be analyzed through the lens of information gain and contextuality.

Reconciling the Paradox 

— Nonlocality and Contextuality  

Quantum Nonlocality: The correlations observed in entangled particles suggest that these phenomena arise from pre-existing conditions rather than nonlocal interactions. In the case of the Trouton-Noble Paradox, the torque experienced by the charged object can be understood as a manifestation of quantum mechanical principles where entangled states yield correlations that extend beyond classical boundaries.

Contextual Outcomes: Measurement outcomes depend on the context in which they occur. The torque experienced by the charged object is influenced by the specific conditions of the measurement setup, suggesting that the classical view of torque misses critical components attributed to the overall quantum state.

— Energy Redistribution  

The interaction of the charged object with its environment entails energy being redistributed rather than simply consumed. This shows that the change in the system’s state does not violate conservation laws but that energy transitions involve complex interdependencies.

Conclusion 

The Trouton-Noble Paradox can be understood by applying principles of classical electromagnetism with quantum mechanics, decoherence, and information theory. By recognizing that torque, work done, and energy changes are fundamentally influenced by environmental interactions and decoherence processes, we can see how charged systems behave in an electromagnetic field. This demonstrates the limitations of classical explanations and reaffirms the necessity of quantum mechanical principles.

References

– Trouton, F. T., & Noble, H. (1901). The Torque on a Charged Body in an Electric Field. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 200, 1-20. 
– Feynman, R. P., Leighton, R. B., & Sands, M. (2011). The Feynman Lectures on Physics, Vol. II: Mainly Electromagnetism and Matter. Basic Books. 
– Zurek, W. H. (2003). Decoherence, Einselection, and the Quantum Origins of the Classical. Reviews of Modern Physics, 75(3), 715-775. 
= Nielsen, M. A., & Chuang, I. L. (2010). Quantum Computation and Quantum Information. Cambridge University Press