About symmetries once again

I have been analyzing the basic visions of TGD trying to identify weak points. Symmetries are central in TGD. The basic motivation for TGD was the loss of Poincare invariance in GRT and in the following I will analyze the claim of Phil Gibbs that one obtains energy conservation in GRT. Quite recently, I learned that the generalized holomorphies of space-time surfaces define non-trivial conserved charges. Also a generalization of Super-Kac-Moody charges associated with certain embedding space isometries emerges. This suggests a very close connection with string models and provides a possibility to provide answers to the longstanding questions relating to the identification of the isometry group of the “world of classical worlds” (WCW).

About the proposal of Phil Gibbs

The following considerations are inspired by discussions with Marko Manninen and related to whether in general relativity it could be possible to define conserved quantities associated with at least some general coordinate transformations as proposed by Phil Gibbs (see this). This is certainly in conflict with the general idea that the choice of coordinates cannot have any physical effect and personally I am skeptic. I however decided to analyze the proposal in detail and found that it relates to a possible generalization of the notion of Newtonian gravitational flux, which gives the gravitational mass of the system.

1. Gibbs’ proposal for Noether charges associated with general coordinate transformations says nothing detailed about charges and the straightforward application of the basic formula gives vanishing charges since these currents turn out to be proportional to T-G which vanishes by Einstein’s equations. However, the action includes a term containing second derivatives of the metric. Could this give an anomalous contribution to the Noether charge?
2. In electrodynamics and gauge theories, charges are obtained in connection with gauge transformations that become constant at a distance. The gauge charge density is a total divergence and gauge charge can be expressed as an electric flux across a very large sphere. On the other hand, in Newton’s theory, the gravitational flux far enough from the system gives its mass. Could mass correspond to a time translation as a symmetry? Could the transformation of the charge into total divergence generalize to other general coordinate transformations?
3. Einstein action (curvature scalar) contains terms proportional to the second order partial derivatives of the metric: these terms come from the part of the curvature scalar linear in Riemann connection, which serves as analogs of non-abelian gauge potentials. However, this does not give third derivatives to the equations of motion. The reason is that the second derivatives occur linearly. If the square of the curvature tensor would define the action as an analog of Yang-Mills action, the situation would be different. This term is analogous to a dissipative and might relate to the general features of GRT dynamics (blackholes as asymptotic states).

Is the divergence term taken into account automatically in the straightforward Noetherian guess for the conserved currents? Or could the charges associated with the general coordinate transformations emerge as analogs of electric charge as a flux integral over a very large sphere. This would certainly contradict the fact that general coordinate transformations do nothing to the system, so that they cannot relate physically non-equivalent configurations.

J^μ(j)= L^αβμ Dg_αβ + L^αβμν ∂_νDg_αβ

=L^αβμDg_αβ +∂_νL^αβμ Dg_αβ – ∂_ν[L^αβμDg_αβ],

where one has

L^αβμ= ∂ L/∂(∂_μ g_αβ),
L^αβμν= ∂ L/∂(∂_μνg_αβ),
Dg_αβ=j^ρ∂_ρ g_αβ .

The third term at the second line is a total divergence and this contribution, call it Q₃(j) to the expression for the charge as a 3-D integral of the μ=t component of the current can be transformed to a surface integral.

Q₃(j)= -∫_S²[∂ L/∂(∂_tr ∂_ρ g_αβ)] ∂_ρ g_αβj^ρ]dS .

Q₃(j)= ∫_S² C(t,rρ) j^ρdS .

where C(t,rρ) denotes Christoffel symbol. For Schwarzschild metric this gives Newtonian gravitational flux for time translation j^ρ=δ^ρ,t.

The conserved charges associated with holomorphies

Generalized holomorphy not only solves explicitly the equations of motion but, as found quite recently, also gives corresponding conserved Noether currents and charges.

1. Generalized holomorphy algebra generalizes the Super-Virasoro algebra and the Super-Kac-Moody algebra related to the conformal invariance of the string model. The corresponding Noether charges  are conserved. Modified Dirac action allows to construct the supercharges having interpretation as WCW gamma matrices. This suggests an answer to a longstanding question related to the isometries of the “world of the classical worlds” (WCW).
2. Either the generalized holomorphies or the symplectic symmetries of H=M⁴× CP₂ or both together define WCW isometries and corresponding super algebra. It would seem that symplectic symmetries induced from H are not necessarily needed and might actually correspond to symplectic symmetries of WCW. This would give a close similarity with the string model, except that one has half-algebra for which conformal weights are proportional to non-negative integers and gauge conditions only apply to an isomorphic subalgebra. These are labeled by positive integers and one obtains a hierarchy.
3. By their light-likeness, the light cone boundary and orbits of partonic 2-surfaces allow an infinite-dimensional isometry group. This is possible only in dimension four. Its transformations are generalized conformal transformations of 2-sphere (partonic 2-surface) depending on light-like radial coordinate such that the radial scaling compensates for the usual conformal scaling of the metric. The WCW isometries would thus correspond to the isometries of the parton orbit and of the boundary of the light cone! These two representations could provide alternative representations for the charges if the strong form of holography holds true and would realize a strong form of holography. Perhaps these realizations deserve to be called inertial and gravitational charges.

Can these transformations leave the action invariant? For the light-cone boundary, this looks obvious if the light-cone is sliced by a surface parallel to the light-cone boundary. Note however that the tip of this surface might produce problems. A slicing defined by the Hamilton-Jacobi structure would be naturally associated with partonic orbits.

For a summary of earlier postings see Latest progress in TGD.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.