Chern-Simons action in hydrodynamical interpretation and Maxwell hydrodynamics as toy model for TGD

Today Kea told about Terence Taos's posting 2006 ICM: Etienne Ghys, “Knots and dynamics”. Posting tells about really amazing mathematical results related to knots.

1. Chern-Simons as helicity invariant

Tao mentions helicity as an invariant of fluid flow. Chern-Simons action defined by the induced Kähler gauge potential for lightlike 3-surfaces has interpretation as helicity when Kähler gauge potential is identified as fluid velocity. This flow can be continued to the interior of space-time sheet. Also the dual of the induced Kähler form defines a flow at the light-like partonic surfaces but not in the interior of space-time sheet. The lines of this flow can be interpreted as magnetic field lines. This flow is incompressible and represents a conserved charge (Kähler magnetic flux). The question is which of these flows should define number theoretical braids. Perhaps both of them can appear in the definition of S-matrix and correspond to different kinds of partonic matter (electric/magnetic charges, quarks/leptons?,...). Second kind of matter could not flow in the interior of space-time sheet. Or could interpretation in terms of electric magnetic duality make sense?

Helicity is not gauge invariant and this is as it must be in TGD framework since CP₂ symplectic transformations induce U(1) gauge transformation which deforms space-time surface an modifies induced metric as well as classical electroweak fields defined by induced spinor connection. Gauge degeneracy is transformed to spin glass degeneracy.

2. Maxwell hydrodynamics

In TGD Maxwell's equations are replaced with field equations which express conservation laws and are thus hydrodynamical in character. With this background the idea that the analogy between gauge theory and hydrodynamics might be applied also in the reverse direction is natural. Hence one might ask what kind of relativistic hydrodynamics results if assumes that the action principle is Maxwell action for the four-velocity u^α with the constraint term saying that light velocity is maximal signal velocity.

1. For massive particles the length of four-velocity equals to 1: u^α u_α=1. In massless case one has u^α u_α=0. This condition means the addition of constraint term

   λ(u^α u_α-ε)

   to the Maxwell action. ε=1/0 holds for massive/massless flow. In the following the notation of electrodynamics is used to make easier the comparison with electrodynamics.

2. The constraint term destroys gauge invariance by allowing to express A⁰ in terms of Aⁱ but in general the constraint is not equivalent to a choice of gauge in electrodynamics since the solutions to the field equations with constraint term are not solutions of field equations without it. One obtains field equations for an effectively massive em field with Lagrange multiplier λ having interpretation as photon mass depending on space-time point:

   j^α= ∂_βF^αβ= λA^α,

   A^α==u^α,

   F^αβ= ∂^βA^α-∂^αA^β.

3. In electrodynamic context the natural interpretation would be in terms of spontaneous massivation of photon and seems to occur for both values of ε. The analog of em current given by λA^α is in general non-vanishing and conserved. This conservation law is quite strong additional constraint on the hydrodynamics. What is interesting is that breaking of gauge invariance does not lead to a loss of charge conservation.

4. One can solve λ by contracting the equations with A_α to obtain λ= j^αA_α for ε=1. For ε=0 one obtains j^αA_α=0 stating that the field does not dissipate energy: λ can be however non-vanishing unless field equations imply j^α=0. One can say that for ε=0 spontaneous massivation can occur. For ε=1 massivation is present from beginning and dissipation rate determines photon mass: a natural interpretation would be in terms of thermal massivation of photon. Non-tachyonicity fixes the sign of the dissipation term so that the thermodynamical arrow of time is fixed by causality.

5. For ε=0 massless plane wave solutions are possible and one has ∂_α∂_βA^β=λA_α. λ=0 is obtained in Lorentz gauge which is consistent with the condition ε=0. Also superpositions of plane waves with same polarization and direction of propagation are solutions of field equations: these solutions represent dispersionless precisely targeted pulses. For superpositions of plane waves λ with 4-momenta, which are not all parallel λ is non-vanishing so that non-linear self interactions due to the constraint can be said to induce massivation. In asymptotic states for which gauge symmetry is not broken one expects a decomposition of solutions to regions of space-time carrying this kind of pulses, which brings in mind final states of particle reactions containing free photons with fixed polarizations.

6. Gradient flows satisfying the conditions A_α =∂_α Φ and A^α A_α=ε give rise to identically vanishing hydrodynamical gauge fields and λ=0 holds true. These solutions are vacua since energy momentum tensor vanishes identically. There is huge number of this kind of solutions and spin glass degeneracy suggests itself. Small deformations of these vacuum flows are expected to give rise to non-vacuum flows.

7. The counterparts of charged solutions are of special interest. For ε=0 the solution (u⁰,u^r)= (Q/r)(1,1) is a solution of field equations outside origin and corresponds to electric field of a point charge Q. In fact, for ε=0 any ansatz (u⁰,u^r)= f(r)(1,1) satisfies field equations for a suitable choice of λ(r) since the ratio of equations associate with j⁰ and j^r gives an equation which is trivially satisfied. For ε=1 the ansatz (u⁰,u^r)= (cosh(u),sinh(u)) expressing solution in terms of hyperbolic angle linearizes the field equation obtained by dividing the equations for j⁰ and j^r to eliminate λ. The resulting equation is

   ∂_r²u+ 2∂_ru/r=0

   for ordinary Coulomb potential and one obtains (u⁰,u^r)= (cosh(u₀+k/r), sinh(u₀+k/r)). The charge of the solution at the limit r→ ∞ approaches to the value Q=sinh(u₀)k and diverges at the limit r→ 0. The charge increases exponentially as a function of 1/r near origin rather than logarithmically as in QED and the interpretation in terms of thermal screening suggests itself. Hyperbolic ansatz might simplify considerably the field equations also in the general case.

3. Similarities with TGD

There are strong similarities with TGD which suggests that the proposed model might provide a toy model for the dynamics defined by Kähler action.

1. Also in TGD field equations are essentially hydrodynamical equations stating the conservation of various isometry charges. Gauge invariance is broken for the induced Kähler field although Kähler charge is conserved. There is huge vacuum degeneracy corresponding to vanishing of induced Kähler field and the interpretation is in terms of spin glass degeneracy.

2. Also in TGD dissipation rate vanishes for the known solutions of field equations and a possible interpretation is as space-time correlates for asympotic non-dissipating self organization patterns.

3. In TGD framework massless extremals represent the analogs for superpositions of plane waves with fixed polarization and propagation direction and representing targeted and dispersionless propagation of signal. Gauge currents are light-like and non-vanishing for these solutions. The decomposition of space-time surface to space-time sheets representing particles is much more general counterpart for the asymptotic solutions of Maxwell hydrodynamics with vanishing λ.

4. In TGD framework one can indeed consider the possibility that four-velocity assignable to a macroscopic quantum phase is proportional to Kähler potential. In this kind of situation one could speak of quantal Maxwell hydrodynamics. In this case however ε could be function of position.

If TGD is taken seriously, these similarities force to ask whether Maxwell hydrodynamics might be interpreted as a nonlinear variant of real electrodynamics. One must however notice that in TGD em field is proportional to the induced Kähler form only in special cases and is in general non-vanishing also for vacuum extremals.

For the construction of extremals of Kähler action see the chapter Basic Extremals of Kähler action of "Classical Physics in Many-Sheeted Space-time".