I have listed some requirements for a complete theory of quantum gravity. Anything anyone would like to add, subtract, or illuminate would be welcome. 1. A gauge group large enough to contain all known interactions, and all interactions predicted by the theory. 2. An action that generates and describes the motions of all particles and fields included in the theory. 3. The theory must recover quantum mechanics at small distances and slow speeds. 4. The theory must recover general relativity at high speeds and macroscopic distances. 5. The theory must recover newtonian mechanics at low speeds and large distances. This is already implied by 3 and 4. 6. The Lagrangian must spontaneously break the large gauge group into the subgroups that exist in current partial theories. 7. The theory must predict all known particles. 8. The theory must not predict particles that cannot exist, or are not known to exist at the energies available to current accelerators. 9. The theory must be background independent. 10. The theory must be able to solve symbolically what can now only be solved perturbatively.
Minor points: 2)HUP will defeat your action. 3)HUP will nullify any observation made at high or constant speeds. 4Quantum Mechanics will nullify any General Relativity effects when brought into local fields(see 2). 5)See 4 ,3 + 2. 6)See 5. 7)See 2. 8)Observations would have to be made with your Eye's shut, whilst cranking up the accelerator peddle! and with at least one finger placed deep into your Ear! 9)Remove the Background. 10 A + B = B ~ A.
2. "motions" should probably read "behaviors" in order to bypass ranyart's uncertainty principle objection. 3. Delete slow speeds from (3). Small distances correlate to high momenta and quantum particles travel not much below c. 4. Instead of high speeds and long distances, put "where h can be ignored in comparison to the size of actions encountered". 5. The criterion is "where h can be ignored and c is much larger than any speed." 7. add "and fields". 8. Change to "The theory must be completely consistent with experiment." 10. Change to "The theory must be non-perturbative. It will reproduce existing perturbative results under the appropriate restrictions."
I think it deserves some mention that a quantum theory of gravity doesn't necessarily have to have anything to do with a "theory of everything."
Perhaps also a solution to the age old measurement problem in QM, in a neat, logically satisfying way. I'd also like to be able to derive wave functions for various fundamental particles. Not just gross attributes (such and such a thing must be antisymetric), but actual written down formulas that will describe the potential. Finally, an equation that describes the quantum evolution of the big bang, and 'explains' inflation, SUSY symmetry breaking, galaxy formation. Btw, I realize this is abuse of terminology, TOE's as typically defined really arent intended to solve *all* these problems
Well, since we're dreaming, I suppose the theory would be rigorous as well as nonperturbative and would settle the question of whether particles really exist or are just a construction to handle the Haag theorem problems with field theory.
Re: Re: Requirements for a theory of Quantum Gravity, or Theory of Everything But due to General Relativity, one expects it to be a "theory of all the geometrical things".
Excellent points everyone. Here is a reformulation. The Theory of Everything must have: 1. A gauge group large enough to contain all known interactions, and all interactions predicted by the theory. 2. An action that generates and describes the behaviors of all particles and fields included in the theory. 3. The theory must recover quantum mechanics at small distances. 4. The theory must recover general relativity where h can be ignored in comparison to the size of actions encountered. 5. The theory must recover newtonian mechanics where h can be ignored and c is much larger than any speed. This is already implied by 3 and 4. 6. The Lagrangian must spontaneously break the large gauge group into the subgroups that exist in current partial theories. 7. The theory must predict all known particles and fields. 8. The theory must not predict particles that cannot exist, or are not known to exist at the energies available to current accelerators, and the theory must be completely consistent with experiment. 9. The theory must be background independent. 10. The theory must be non-perturbative. It will reproduce existing perturbative results under the appropriate restrictions, and where they have been successful. 11. The theory must not only make general predictions, but it must be possible to formulate the specific wavefunctions and potentials of all particles, fields, and situations from within the theory. 12. The theory must be consistent with what is known of the large-scale structure of the universe, and it must make accurate predictions about known aspects of the universe that current theories cannot explain. 13. The theory must be able to solve, and to explain, the question as the whether space is at bottom continuous or discrete, whether it is isomorphic to some improper subset of R or some improper subset of Z.
What I would like to do here is to begin collecting the canonical equations of all the subtheories, and eventually put everthing together into a flow chart that shows how the subtheories flow from the fundamental theory, which of course is not known. This flow chart will necessarily be a patchwork as it reflects the current state of knowledge. Many of the theories will overlap parts of one another, such as Kaluza-Klein and electroweak. Some parts will contradict one another in places, such as string theory and the standard model. The idea is to collect staements such as: " Kaluza-Klein posits a five-dimensional spacetime containg one compact space dimension. The action principle is a, the gauge group b. Kaluza-Klein decomposes into general relativity and electromagnetism. In general relativity the action principle is c, the gauge group d. Electromagnetism is described by the canonical equations e. In inertial frames general relativity becomes special relativity, where the action principle is f, the gauge group g. At slow speeds special relativity becomes newtonian physics, where the action principle is h, the gauge group i..." and so on. The idea would be to delineate each piece as succinctly as possible.
I have just had ocassion to read Smolin Scientific American article, and it should be noted that he refers to Quantum Gravity in the sense of TOE, ie a theory unifyng Quantum [Field] Theory and General Relativity. Thus very in the sense of this thread :-)