How is the Obidi Spectral Action Different from Alain Connes Spectral Action?

The Obidi Spectral Action and Alain Connes' Spectral Action differ in their approach to treating entropy and the dynamics of gravity. The Obidi Spectral Action is part of the Theory of Entropicity (ToE) and is designed to provide more physics and predictive structure than the original action by Ginestra Bianconi. It differs from Bianconi's action by not treating the entropy field as a fundamental field but as a variable that generates dynamics. The Obidi Spectral Action is framed as an action principle, generating equations of motion, which is a significant shift in theoretical physics.

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On the other hand, Alain Connes' Spectral Action is a paradigm of geometry that starts from quantum mechanics and uses the spectral action as an action functional for the theory. It is designed to encode the dynamics of gravity and has been shown to have remarkable accuracy in approximating the spectral action. Connes' Spectral Action is also related to the spectral triple, which is a unitary Hilbert space representation of "something" that allows for the manipulation of coordinates and the measurement of distances.

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In summary, the Obidi Spectral Action is a theoretical framework within the Theory of Entropicity that seeks to treat entropy as a field variable, generating dynamics, while Alain Connes' Spectral Action is a paradigm of geometry that uses the spectral action as an action functional for the theory of gravity. Both frameworks aim to provide a new understanding of the dynamics of gravity and the role of entropy in physics.