Could space-time or the space of space-time surfaces be a Lagrangian manifold in some sense?

Gary Ehlenberg sent a link to a tweet to X (see this) by Curt Jainmungal. The tweet has title "Everything is a Lagrangian submanifold". The title expresses the idea of Alan Weinstein (see this), which states that space-time is a Lagrangian submanifold (see this) of some symplectic manifold. Note that the phase space of classical mechanics represents a basic example of symplectic manifold.

Lagrangian manifolds emerge naturally in canonical quantization. They reduce one half of the degrees of freedom of the phase space. This realizes the Uncertainty Principle geometrically. Also holography= holomorphy principle realizes Uncertainty Principle by reducing the degrees of freedom by one half.

What about the situation in TGD (see this, this and this). Does the proposal of Alan Weinstein have some analog in the TGD framework?

Consider first the formulation of Quantum TGD.

1. The original approach of TGD relied on the notion of Kähler action (see this). The reason was that it had exceptional properties. The Lagrangian manifolds L of CP₂ gave rise to vacuum extremals for Kähler action: any 4-surface of M⁴×L ⊂ H= M⁴×CP₂ with M⁴ is a vacuum extremal for this action. At these space-time surfaces, the induced Kähler form vanishes as also Kähler action as a non-linear analog of Maxwell action.

   The small variations of the Kähler action vanish in order higher than two so that the action would not have a kinetic term and the ordinary perturbation theory in QFT sense (based on path integral) would completely fail. The addition of a volume term to the action cures the situation and in the twistorialization of TGD it emerges naturally and does not bring in the analog of cosmological constant as a fundamental constant but as a dynamically generated parameter. Therefore scale invariance would not be broken at the level of action.

2. This was however not the only problem. The usual perturbation theory would be plagued by an infinite hierarchy of infinities much worse than those of ordinary QFTs: they would be due to the extreme non-linearity of any general coordinate invariant action density as function of H coordinates and their partial derivatives.

These problems eventually led to the notion of the "world of classical worlds" (WCW) as an arena of dynamics identified as the space of 4-surfaces obeying what I call now holography and realized in some sense (see this, this, this and this). It took decades to understand in what sense the holography is realized.

1. The 4-D general coordinate invariance would be realized in terms of holography. The definition of WCW geometry assigns to a given 3-surface a unique or almost unique space-time surface at which general coordinate transformations can act. The space-time surfaces are therefore analogs of Bohr orbits so that the path integral disappears or reduces to a sum in the case that the classical dynamics is not completely deterministic. The counterparts of the usual QFT divergences disappear completely and Kähler geometry of WCW takes care of the remaining diverges.

   It should be noticed in passing, that year or two ago, I discussed space-times surfaces, which are Lagrangian manifolds of H with M⁴ endowed with a generalization of the Kähler metric. This generalization was motivated by twistorialization.

2. Eventually emerged the realization of holography in terms of generalized holomorphy based on the idea that space-time surfaces are generalized complex surfaces of H having a generalized holomorphic structure based on 3 complex coordinates and one hyper complex coordinate associated which I call Hamilton-Jacobi structure.

   These 4-surfaces are universal extremals of any general coordinate invariant action constructible in terms of the induced geometry since the field equations reduce to a contraction of two complex tensors of different type having no common index pairs. Space-time surfaces are minimal surfaces and analogs of solutions of both massless field equations and of massless particles extended from point-like particles to 3-surfaces. Field particle duality is realized geometrically.

   It is now clear that the generalized 4-D complex submanifolds of H are the correct choice to realize holography (see this).

3. The universality realized as action independence, in turn leads to the view that the number theoretic view of TGD in principle could make possible purely number theoretic formulation of TGD (see this). There would be a duality between geometric and number theoretic views (see this), which is analogous to Langlands duality. The number theoretic view is extremely predictive: for instance, it allows to deduce the spectrum for the exponential of action defining vacuum functional for Bohr orbits does not depend on the action principle.

   The universality means enormous computational simplification as also does the possibility to construct space-time surfaces as roots for a pair of (f₁,f₂) of generalized analytic functions of generalized complex coordinates of H. The field equations, which are usually partial differential equations, reduce to algebraic equations. The function pairs form a hierarchy with an increasing complexity starting with polynomials and continuing with analytic functions: both have coefficients in some extension of rationals and even more general coefficients can be considered.

So, could Lagrangian manifolds appear in TGD in some sense?

1. The proposal that the WCW as the space of 4-surfaces obeying holography in some sense has symplectomorphisms of H as isometries, has been a basic idea from the beginning. If holography= holomorphy principle is realized, both generalized conformal transformations and generalized symplectic transformations of H would act as isometries of WCW (see this). This infinite-dimensional group of isometries must be maximal possible to guarantee the existence of Riemann connection: this was already observed for loop spaces by Freed. In the case of loop spaces the isometries would be generated by a Kac-Moody algebra.
2. Holography, realized as Bohr orbit property of the space-time surfaces, suggests that one could regard WCW as an analog of a Lagrangian manifold of a larger symplectic manifold WCW_ext consisting of 4-surfaces of H appearing as extremals of some action principle. The Bohr orbit property defined by the holomorphy would not hold true anymore.

   If WCW can be regarded as a Lagrangian manifold of WCW_ext, then the group of Sp(WCW) of symplectic transformations of WCW_ext would indeed act in WCW. The group Sp(H) of symplectic transformations of H, a much smaller group, could define symplectic isometries of WCW_ext acting in WCW just as color rotations give rise to isometries of CP₂.

See the article Could space-time or the space of space-time surfaces be a Lagrangian manifold in some sense? or the chapter with the same title.

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.