Tuesday, September 28, 2021

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. 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 heff=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 Λgr= GM/v0  is correct.
  3. The last 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  ℏgr involves two values  of the velocity parameter β0 appearing in the Nottale formula.

    Also the electromagnetic and Z0 analogs of ℏgr make sense and  tt is proposed that 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.  Also a model for the ordinary viscosity and its increase with a decreasing temperature is discussed.

    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. 


Crow- said...

first link is broken.

Matti Pitkänen said...

Thank you for noticing. It should work now.