TGD and Quantum Hydrodynamics

This work is devoted to the question of what quantum hydrodynamics could mean in the TGD framework. In the standard picture quantum hydrodynamics (see this) is obtained from the hydrodynamic interpretation of the Schrödinger equation. Bohm theory involves this interpretation.

1. Quantum hydrodynamics appears in TGD as an exact classical correlate of quantum theory. Modified Dirac equation forces as a consistency condition classical field equations for X⁴. Actually, a TGD variant of the supersymmetry, which is very different from the standard SUSY, is in question.
2. TGD itself has the structure of hydrodynamics. Field equations for a single space-time sheet are conservation laws. Minimal surfaces as counterparts of massless fields emerge as solutions satisfying simultaneously analogs of Maxwell equations. Beltrami flow for classical Kähler field defines an integrable flow. There is no dissipation classically and this can be interpreted as a correlate for a quantum coherent phase.
3. Induced Kähler form J is the fundamental field variable. Classical em and Z⁰ fields have it as a part. For S³⊂ CP₂ em and Z⁰ fields are proportional to J: which suggests large parity breaking effects. Hydrodynamic flow would naturally correspond to a generalized Beltrami flow and flow lines would integrate to a hydrodynamic flow.
4. The condition that Kähler magnetic field defines an integrable flow demands that one can define a coordinate along the flow line. This would suggest non-dissipating generalized Beltrami flows as a solution to the field equations and justifies the expectation that Einstein's equations are obtained at QFT limit.
5. If one assumes that a given conserved current defines an integrable flow, the current is a gradient. The strongest condition is that this is true for all conserved currents. The non-triviality of the first homotopy group could allow gradient flows at the fundamental level. The situation changes at the QFT limit.
6. Beltrami conditions make sense also for fermionic conserved currents as purely algebraic linear conditions stating that fermionic current is a gradient of some function bilear in oscillator operators. Whether they are actually implied by the classical Beltrami conditions, is an interesting question.
7. The requirement that modified Dirac operator at the level of space-time surface is in a well-defined sense a projection of the Dirac operator of H implies that for preferred extremals the isometry currents are proportional to projections if the corresponding Killing vectors with proportionality factor constant along the projections of their flow lines. This implies as generalization of the energy conservation along flow lines of hydrodynamical flow (ρ v²/2+p=constant).

   This also leads to a braiding type representations for isometry flows of H in theirs of their projections to the space-time surface and it seems that quantum groups emerge from these representations. Physical intuition suggests that only the Cartan algebra corresponding to commuting observables allows this representation so that the selection of quantization axes would select also space-time surface as a higher level state function reduction.

   One also ends up to a generalization of Equivalence Principle stating that the charges assignable to "inertial" or "objective" representations of H isometries in WCW affecting space-time surfaces as analogs of particles are identical with the charges of "gravitational" or subjective representations which act inside space-time surfaces. This has also implications for M⁸-H duality.

8. Minimal surfaces as analogs of solutions of massless field equations and their additional property of being extremals of Kähler action gives a very concrete connection with Maxwell's theory öcite{btart/minimal}.

In the sequel some key challenges of hydrodynamics are considered from TGD point of view.

1. The generation of turbulence is one of the main problems of classical hydrodynamics and TGD inspired quantum hydrodynamics suggests a solution to this problem. Not only "classical" is replaced with "quantum" but also quantum theory is generalized.

   The key notion is magnetic body (MB): MB carries dark matter as h_eff=nh₀ phases and controls the flow at the level of ordinary matter. Magnetic flux tubes would be associated with the vortices. The proposal inspired by super-fluidity is that velocity field is proportional to Kähler gauge potential and that the cores of vortices corresponds to monopole flux tubes whereas their exteriors would correspond to Lagrangian flux tubes with a vanishing Kähler field so that velocity field is gradient. Vorticity field would correspond to the Z⁰ magnetic field so that a very close analogy with superconductivity emerges.

   The model is applied to several situations. The generation of turbulence and its decay in a flow near boundaries is discussed. ZEO suggests that the generation of turbulence could correspond to temporary time reversal associated with a macroscopic "big" (ordinary) state function reduction (BSFR).

   Also the connection with magnetohydrodynamics (MHD) is considered. The reconnection of the field lines is replaced with the reconnection of flux tubes. The fact that monopole flux tubes require no current to generate the magnetic field provides a new insight to the problem of how magnetic fields in astrophysical scales are generated.

   The topological picture based on flux tubes can be applied to the collisions of circular vortices. Also the violations of the circulation theorem of Kelvin is discussed.

2. Second section is devoted to hydrodynamic quantum analogs studied by Bush et al. These intriguing phenomena, in particular Couder walker bounces along a Faraday wave that it generates. Also surfing mode is possible. The energy feed comes from shaking the water pool and plays a role of metabolic energy feed leading to self-organization. This phenomenon allows in the TGD framework a modelling based on quantum gravitational hydrodynamics. MB serves as a "boss" and therefore takes the role of the pilot wave proposed by Bush. The key prediction that the Faraday wave length analogous to Compton wavelength equals to the gravitational Compton length Λ_gr= GM/v₀ is correct.
3. Also the electromagnetic and Z⁰ analogs of ℏ_gr make sense and it asked whether in these scales the gravitational, Z⁰ and electromagnetic Compton lengths are identical at gravitational flux tubes and that particles are at flux tubes with length of order this wavelength. The twistor lift predicts that also M⁴ has Kähler structure and M⁴ Kähler form could give contribution to electromagnetic and Z⁰ fields. Kähler currents for M⁴ and CP₂ parts are separately conserved and this leads to ask whether Magnus forces resembling Lorentz force could reflect the presence of classical Z⁰ force or M⁴ contribution to the Kähler force.
4. One section is devoted to the attempt to understand the origin of viscosity and interpret critical Reynolds numbers in the TGD framework. In TGD quantum gravitation involves quantum coherence in astrophysical scales so that it is not totally surprising that the critical Reynolds numbers associated with the turbulence in pipe flow and flow past a plate relate directly to the gravitational Compton lengths of Earth and Sun: In the case of Sun ℏ_gr involves two values of the velocity parameter β₀ appearing in the Nottale formula. Also a model for the ordinary viscosity and its increase with a decreasing temperature is discussed.
5. Also nuclear and hadron physics suggests applications for QHD. The basic vision about what happens in high energy nuclear and hadron collisions is that two BSFRs ("big" state function reductions changing the arrow of time) take place. The first BSFR creates the intermediate state with h_eff>h: the entire system formed by colliding systems need not be in this state. In nuclear physics this state corresponds to a dark nucleus which decays in the next BSFR to ordinary nuclei. The basic notions are the notion of dark matter at MB and ZEO, in particular the change of the arrow of time in BSFR.

   See the article TGD and Quantum Hydrodynamics or the chapter with the same title.

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

   Articles and other material related to TGD.