The principle of dimensional capacity can be characterized two ways. (1) A given amount of energy divides equally among available dimensions for homologous differently dimensioned systems. (2) If there is a fixed amount of energy per dimension, energy for homologous systems varies with the number of dimensions. I don't know enough about QM (though I wish I did) to investigate whether the principle of dimensional capacity might apply to QM. Does it?
More on the principle of dimensional capacity is at:
https://www.researchgate.net/project/The-Principle-of-Dimensional-Capacity
While there's no such notion as the principle of dimensional capacity, there does exist a notion of similarity, which describes how quantities behave upon scaling of space and time-and such properties do, of course, depend on the number of dimensions.
As does the notion of ``extensive'' quantities-that scale with the ``volume'', thereby defining corresponding densities.
So, indeed, in classical, as well as in quantum, mechanics energy is an ``extensive'' quantity and a corresponding density can be defined.
That's nothing new, though. It suffices to consult any textbook on classical mechanics and/or thermodynamics.
Dr. Nicolis
Thank you for your helpful and instructive reply.
Probably true, no principle of dimensional capacity is part of current physics. Reasons, however, to suspect it should be are discussed in various articles on RG, including recentlyPreprint Size, scaling, and invariant ratios
and related articles.As you indicate, similarity connects to scaling, and scaling connects to dimension. Then if scaling is induced by dimension, analyze size changes from a dimensional point of view rather than a scaling point of view. It is not so much theory or math, as a change in point of view.
New ideas or perspectives can fail and if not capable of rescue, must be discarded. Does that apply to the principle of dimensional capacity? Might you take a look?
Thanks again for your insights.
You can read about the equipartition theorem in [1]. Briefly, in quantum mechanics the equipartition theorem applies only at high temperature [2]. Here what amounts to 'high temperature' depends on the characteristic energy scale of the system under investigation.
[1] https://en.wikipedia.org/wiki/Equipartition_theorem
[2] https://en.wikipedia.org/wiki/Equipartition_theorem#Failure_due_to_quantum_effects
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