Use Noether theorem, not moment map
Peter Woit has posting entitled "Use moment map, not Noether's theorem" in Not Even Wrong. Peter Woit argues that moment map which is the meth of dealing with conservation laws associated with continuous symmetries is superior to the Noether's theorem. His claim is wrong.
Noether's theorem is more general than moment map since it requires only Lagrangian formalism. Moment map requires the existence of also Hamiltonian formalism. In Lagrangian formalism the invariance of the action under infinitesimal symmetries leads to the expressions for conserved quantities using standard recipes of variational calculus easy to understand.
The use of moment map requires that the system allows transition from Lagrangian to Hamiltonian formalism which requires the introduction of phase space whose points are labelled by generalized positions and momenta (field values and canonical momentum densities in field theory). In Lagrangian formalism one as generalized positions and velocities (field values and their time derivatives in field theory). Moment map assigns to continuous symmetry a Hamiltonian. The continuous flow generated by this Hamiltonian is identifiable as the time development of the system. The value of the Hamiltonian is constant along the orbits of the flow and defines a conserved quantity if the dynamics is time-independent.
What Peter Woit does not notice is that this transition is not always possible.
- Already in gauge theories gauge symmetries produce difficulties since some canonical momentum densities vanish identically and one must perform gauge fixing. Same happens in general relativity: now one must introduce preferred space-time coordinates. The lesson number one is that Hamiltonian formalism is essentially Newtonian notion raising time in preferred position. In relativistic theories this is not natural.
- Also the non-linearity of the basic action principle can cause difficulties and this tends to be the case for general coordinate invariant action principles which have geometric content and have surfaces as the extrema of the action. In the case of string models with action identified as surface area of string world sheet the situation can be still handled but TGD represents a school example about failure of Hamltonian formalism. The time derivatives of imbedding space coordinates as functions of canonical momentum densities for Kähler action are many-valued functions and there is no hope of solving them.
For vacuum extremals this difficulty becomes especially acute. All space-time surfaces having CP2 projection, which is Lagrangian manifold having vanishing induced Kähler form. In the generic situation Lagrangian manifolds of CP2 are 2-dimensional so that vacuum degeneracy is gigantic. For them canonical momentum densities vanish identically whereas time derivatives of imbedding space coordinates do not. The variation of vacuum extremal keeping it as vacuum extremal changes time derivatives but keeps canonical momentum vanishing.
- 3-surface would represent the point of the configuration space and the possibility of topology change for 3- surface destroys even the idea about Hamiltonian dynamics which should keep the topology unchanged along the orbit of this point (4-surface).
The extreme non-lineary and vacuum degeneracy have also other implications, which deserve to be mentioned.
- Minkowski space would be the vacuum extremal around which perturbation would be carried out if one would take perturbative quantization of general relativity as a model. The action density however vanishes up to fourth order around Minkowski space in the derivatives of CP2 coordinates so that linearization makes no sense and perturbative approach fails completely since there are no kinetic terms and one cannot define propagators.
- Path integral approach fails- this irrespective of whether it is based on Lagrangian formalism or the hopeless attempt to construct Hamiltonian formalism. This eventually led to the realization that the notion of "world of classical worlds" as infinite-dimensional geometry consisting of pairs of 3-surfaces at the opposite ends of causal diamond (in zero energy ontology) provides the only imaginable manner to consctruct quantum TGD. It means generalization of Einstein's geometrization of physics program from classical physics to entire quantum physics.
- The vacuum degeneracy is very much analogous to gauge degeneracy since the symplectic transformations of CP2 generate new vacua and act like U(1) gauge transformations on Kähler gauge potential. These symmetries are not however gauge symmetries and this makes the situation hopeless if one wants to stay in Newtonian framework. Vacuum degeneracy and the associated non-determinism also leads to various notions like 4-D spin glass degeneracy and hierarchy of effective Planck constants allowing an interpretation as hierarchy of dark matter phases playing a key role in the understanding of living matter as macroscopic quantum system.
- Critical reader could of course ask critical questions. Why to stick to K\"ahler action? Why not use say 4-volume?
The answer is that 4-volume is wrong choice for WCW geometry: it would allow only rather small space-time surfaces (also in temporal direction) and would force to introduce dimensional fundamental coupling which does not give much hopes for divergence free theory. K\"ahler action does not suffer from these problems and has many other merits. For instance, Kähler-Dirac action - supersymmetric fermionic counterpart of K\"ahler action - leads to the emergence of string world sheets at which induced spinor modes are localized to guarantee well-definedness of em charge and which carry vanishing weak gauge potentials. This guarantees also absence of strong parity breaking effects above weak scale (which however is proportional to effective Planck constant so that strong parity breaking in living matter is obtained).