Monday, June 29, 2020

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 An 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 An appears would have vanishing classical Noether charges in the class of preferred extremals associated with An. Also [An,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.

    Articles and other material related to TGD.


Crow- said...

Matti, I was reading about this guy who wrote a paper on pedal coordinates in 2017 and came up with some new theorem and relates it to the "Dark kepler problem".

"one of the main advantages of pedal coordinates is that the operation of making pedal curve(which would in general require solving a differential equation in Cartesian coordinates) can by done by simple algebraic manipulation.

I've updated a paper im writing on the topic at

pedal curves are related to circle inversion. I think it should be possible to show an algebraic relationship between the real and imaginary parts of tanh(ln(1+Z(t)^2))

Matti Pitkänen said...

Unfortunately the topic is so complex that it would take a lot of time and I do not have it. n any case an alebraic relationship beween real and imaginary parts of complex function does not seem possible. Cauchy-Riemann equations allow only non-local relatioship between real and imaginary parts in f= u+iv. To see that algebraic or more geneal local relationship between u and v is impossible substitute u=f(v) to C-R equations. You get two equations: partial_xu= partial_uf partial_y u and partial_yu= -partial_ufpartial_xu. Together these gives (partial_f)^2=-1, which is contradicton since f is real.

Crow- said...

I see what you are saying, the relationship I am suggesting would be nonlocal but it would still be algebraic . The reason is that, lemniscates are the pedal curves of hyperbolas, and hyperbolas are the pedal curves of lemniscates. My hunch is that the curves I uncovered have this same relationship, I just need to work out the formulas for the pedal coordinate transforms and see how they work out. Maybe my terminology is not correct in calling it algebraic? I would need to evaluate the 1st derivatives of the function to make the coordinate transform.

Crow- said...
This comment has been removed by the author.
Crow- said...


I mistyped, it must be the negative pedal curve, since the lemniscate is the pedal curve of the hyperbola and the hyperbola is the negative pedal curve of the lemniscate

Crow- said...

its a lost cause i know, but its pretty :)