John Baez about Noether's theorem in algebraic approach

John Baez gives very nice summary (see this of the triple of states, observables, and generators of symmetries from purely algebraic point of view. Jordan Banach algebra with commutative product A*B= (AB+BA)/2 would play the role of observables. The operators correspond to symmetry generators commuting with observables and the unitary evolutions generated by them in this algebra are trivial. One could say that this defines analogy of Noether’s theorem usually deduced for the symmetries of action principle.

To me the weakness of the algebraic approach is that it says very little about the dynamics- it woul be just unitary evolution generation by some generator of symmetry. Second problem is mentioned at the end of the posting is how classical relates to quantal. And there is nothing about quantum measurement problem so that basically an attempt to reproduce wave mechanics using operators is in question.

My own view – zero energy ontology (ZEO) – goes much beyond quantum mechanics of simple systems.

1. The basic problem of quantum measurement theory is the starting point. The notion of quantum state is modified. In wave mechanics and quantum field theories it is based on initial value problem in configuration space (space of positions for particle). Initial state is wave function – a superposition of possible initial values in configuration space. Time evolution is formulated in terms of unitary evolution defined by exponential of Hamiltonian and reduces to Schroedinger equation.

What happens in quantum measurements is not consistent with this time evolution. This is the problem.

1. In TGD one replaces initial value problem with a boundary value problem with boundary corresponding to values at times t1 and t2 (this is a simplification).
2. One defines states as superpositions of deterministic classical time evolutions – preferred extremals – analogous to Bohr orbits having the property that boundary value problem is equivalent to initial value problem. Once on knows configuration at t1, one knows it at t2.
3. Quantum states are superpositions of these preferred extremals from t1 to t2 and quantum jump replaces this kind of superposition with a new one. I call this approach zero energy ontology (ZEO).
4. The basic problem of quantum measurement theory disappears since there are two causalities: that of quantum jump and classical causality, and there is no need to break the deterministic time evolution analogous to that given by Schroedinger equation in quantum jump. There are two times: experienced time as sequence of quantum jumps and geometric time. One ends up with a theory of consciousness without moment of consciousness identifiable as state function reduction. Also a ZEO theory of self-organization emerges. This is of course only the basic idea. For instance, one must understand how correlation between experienced time and geometric time emerges.
5. One the many implications of ZEO is new view about quantum tunnelling: it must have classical time evolution as quantum correlate. This leads to a new view about tunnelling in nuclear reactions relying essentially on the change of the arrow of time in ordinary state function reduction. Just today I received link telling about strange phenomenon occurring in what is believe to be ordinary electron tunnelling. The electron getting through the barrier radiates energy which increases with the height of the barrier. I discusse the ZEO based explanation in a related posting New support for TGD view about quantum tunnelling .

How to realize this picture and how unique is it? Here one must leave the realm of wave mechanics.

1. Loosely speaking, in TGD framework point-like particle is replaced with 3-D surface in M⁴× CP₂ and its orbit as preferred extremal of action principle, whatever it might be, defines space-time region. A generalization of string model is in question. Also a generalization of general relativity solving its problem due to the loss of Poincare symmetries is in question.
2. This leads to a generalization of Einstein’s geometrization program: replace configuration space with the “world of classical worlds” (WCW) and give it Kaehler geometry to realize geometrization of quantum theory. Points of WCW are 3-surfaces or equivalently 4-surfaces: this reduces holography and reduces it to general coordinate invariance.

WCW spinor fields would represent physical states as “wave functions”. Configuration space gamma matrices would be superpositions of fermionic oscillator operators so tha also fermions are geometrized. The mere existence of WCW Kaehler geometry requires maximal isometries and this fixes TGD highly uniquely. Freed realize the uniqueness for loop spaces. How to realize the crucial preferred extremal property making initial value problem equivalent with boundary value problem?

1. Here the maximal isometry group of WCW enter the game. The symmetry algebra is replaced with an analog of infinite-D symplectic algebra acting as isometries of WCW induced from symplectic transformations at delta M⁴₊xCP₂ labelled by integer valued conformal weights assignable to the radial light-like coordinate of light-cone bounary δM⁴₊ defining second boundary of causal diamond cd identified as the intersection of future and past directed light-cones.
2. The crucial point is that this algebra – call it A – has fractal hierarchy of sub-algebras A_n with conformal weights coming as multiples of n=1,2,… which very probably corresponds to a hierarchy of hyper-finite factors of type II₁ forming inclusion hierarchies labelled by sequences …n₁ divides n₂ divides….
3. Infinite-D sub-algebra A_n appears would have vanishing classical Noether charges in the class of preferred extremals associated with A_n. Also [A_n,A] would have the same property.This is like posing the condition that analogous sub-algebra of say Kac-Moody algebra annilates physical states. The space-time surfaces in question would be minimal surfaces satisfying the additional condition that they extremize also what I call Kaehler action, and being analogous to Maxwell action.

See the article Some comments related to Zero Energy Ontology (ZEO).

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

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