Author Commentary Below: A few notations from this notebook to indicate topics covered. Author’s page numbering is ignored, any page numbers referenced below are from the scanned PDF documents
Local Translations and Thermodynamics, The Thermodynamic-Mechanic Duality
(p.01) Substantial evidence has mounted that gravitation and inertia arise as emergent concepts from underlying degrees of freedom whose dynamics is unspecified. Also substantial evidence has accumulated that gravitation is a gauge theory of local translations. We must understand the relationship between these two approaches, as they are both extremely convincing.
(p. 3) If gravitation is an entropic form and not a fundamental one, fluctuations can cause behavior that is at odds with the behavior expected on traditional grounds. Typically two masses in proximity will attract each other, however due to random statistical fluctuations we should be able to calculate the probability that they will actually repel each other.
(p. 7) Is there a connexion, possibly through noncommutative geometry, between gravity as a gauge theory and gravity as an emergent phenomena?
(p. 9) If we begin with an approach to gauge gravity that is modeled by the work of the Chinese and with the added algebroid structure we have … a triple. Two algebroids and a Hilbert space carrying representation information. I think Wienstein has written considerably about the representation theory of algebroids …
(p. 10) Notes on geometric quantization.
(p. 12-14) Thermodynamic-Mechanic Duality Perhaps we can be bold enough to interpret Verlinde’s calculation as the consequence of a grand duality: Thermodynamic-Mechanic (TM) duality.
According to this proposed duality, (which may, I now realize, have been anticipated in the book Chance in Physics by David Bohm) every physical system has both an elementary, or mechanic description and a thermodynamic description. In the mechanic description, whose prototype is Newton’s mechanics, the description tends to be geometric in nature with few degrees of freedom postulated. In the thermodynamic description, many degrees of freedom are postulated and the description is completed by statistical mathematical arguments and by thermodynamic relations.
If this idea is true, it raises the status of thermodynamical laws to the status of basic mechanical laws. Some ideas have always been floating around about this from figures like Ilya Prigogine and others, but they have never taken hold in the particle physics or Relativity communities. This is quite understandable, as the reductionist goal of these communities is to find THE basic degrees of freedom and the laws by which they evolve and make predictions that can be compared with experiment. If there are always two widely differing descriptions this goal is phrased in too narrow a way.
There are already examples of this sort of behavior within high energy physics that fall under the rubric: duality.
Another duality, although it is not referred to in that way, is the Lagrangian – Hamiltonian duality. The well known dual fomalisms for mechanics related by the Legendre transformation. This dual description is remarkable and leads to dual descriptions in Q.M. and QFT. Lagrangian_____________spacetime approach to Q.M. Hamiltonian____________canonical approach to Q.M.
In the dualities the question of which description is correct does not arise because the two formalisms are mathematically equivalent and therefore at the level of “final prediction’ yield the same result. Sometimes progress is faster when one or another approach is adopted, but too often it has come to pass that the unpopular approach leads to just as many insights and is superior in many ways to the initially more promising one. The main insight is, however, that physical events may proceed according to various pictures according to various theories so that there does not seem to be a unique picture of the phenomena, while there is a unique final numerical result.
Another duality involves solitons which are localized but extended field configurations that have finite total energy and are typically non-singular; Coleman has called them ‘classical lumps’. When these lumps are quantized they become traditional elements of quantum field theory: quanta or monopoles. [I can add detail here to sharpen things up.] So in this case an extended field configuration, which is an aggregate system: it has interacting parts and in some sense can be thought of as a thermodynamic system, is precisely dual to a quantum. The quantum is the quintessential non-divisible elementary entity. So here we have an excellent example of TM duality.
(p. 32) The picture I am developing of ‘generalized gauge theory’ will either require the apparatus of atlases, charts, etc., or it will not. This determination will depend upon the relationship of the local translation group to the diffeomorphism group. The local structure of the diffeomorphism group actually determines the need for charts–not the global structure. In order to determine the mathematical needs of the theory we must examine the Frefeild (1968) proof and attempt to write the F-diffeomorphism in terms of local translations.