Restate my assumptions
Posted Aug23-09 at 05:55 PM by Finbar
1) classical theory is described by an action
2) By minmising the action we find the equations of motion
3) With boundry conditions we can find unique solutions to these equations.(be they analytical or numerical)
4) All classical theories in nature are geometrical theories, hence the solutions 3) are in general geometries
5)Quantisation can be understood through the renormalisation group as going from a single classical action to a set of actions parameterized by some energy scale k.
6) All actions in set 5) have the same set of symmetries
7) Each of the actions in 5) obeys 1)-4)
8)The real physical world is made up of many process each of which has a different energy
scale k these process cannot be seperated.
conclusion: although we can have a complete set of actions parameterized by 0<k<infinity that describe theories on all energy scales these actions are of little use in understanding natural process that have more than one energy scale. Nonetheless these classical theories have geometrical meaning and under the flow of k these meanings are invariant. Thus it seems that some kind of scale dependent geometrical setting can shed light on what QFT really is. Further more by understanding the geometrical nature that underlies QFT we may learn interesting facts relating to the renormalisatibily of a given theory.
What I conjecture is the exsitence of some general mathematical structure that is in some sense a scale dependent geometry. By understanding this structure theorems should exist that tell us whether given a set of symmeteries/gauge principles whether a theory is renormalisable/unitary.
2) By minmising the action we find the equations of motion
3) With boundry conditions we can find unique solutions to these equations.(be they analytical or numerical)
4) All classical theories in nature are geometrical theories, hence the solutions 3) are in general geometries
5)Quantisation can be understood through the renormalisation group as going from a single classical action to a set of actions parameterized by some energy scale k.
6) All actions in set 5) have the same set of symmetries
7) Each of the actions in 5) obeys 1)-4)
8)The real physical world is made up of many process each of which has a different energy
scale k these process cannot be seperated.
conclusion: although we can have a complete set of actions parameterized by 0<k<infinity that describe theories on all energy scales these actions are of little use in understanding natural process that have more than one energy scale. Nonetheless these classical theories have geometrical meaning and under the flow of k these meanings are invariant. Thus it seems that some kind of scale dependent geometrical setting can shed light on what QFT really is. Further more by understanding the geometrical nature that underlies QFT we may learn interesting facts relating to the renormalisatibily of a given theory.
What I conjecture is the exsitence of some general mathematical structure that is in some sense a scale dependent geometry. By understanding this structure theorems should exist that tell us whether given a set of symmeteries/gauge principles whether a theory is renormalisable/unitary.
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Comments
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It is true that the least action principle can be formulated for many dynamical systems. However there are two things to keep in mind:
1) The dynamics was discovered (proposed) first, and the least action principle came next. In Physics it is more important to guess correctly the equations, not the Lagrangian. Not every Lagrangian leads to meaningful equations. For example, in the Classical Electrodynamics there is a Lagrangian for the electron and the electromagnetic filed, but the equations do not have physical solutions: there are runaway solutions. The Noether theorems work fine if there are physical solutions. So the equations are "truncated" to be physical.
2) The upper limit of the integral in the least action principle is never used since it is not physical set of the problem. One obtains good equations but physically wrong "boundary" conditions. So, not all parts of the least action principle are used in practice.Posted Dec2-09 at 02:54 PM by Bob_for_short 
