Constant torque as a manner to force phase transition increasing the value of Planck constant

The challenge is to identify physical mechanisms forcing the increase of effective Planck constant h_eff (whether to call it effective or not is to some extent matter of taste). The work with certain potential applications of TGD led to a discovery of a new mechanism possibly achieving this. The method would be simple: apply constant torque to a rotating system. I will leave it for the reader to rediscover how this can be achieved. It turns out that the considerations lead to considerable insights about how large h_eff phases are generated in living matter.

Could constant torque force the increase of h_eff?

Consider a rigid body allowed to rotated around some axes so that its state is characterized by a rotation angle φ. Assumed that a constant torque τ is applied to the system.

1. The classical equations of motion are

   I d²φ/dt²= τ .

   This is true in an idealization as point particle characterized by its moment of inertia around the axis of rotation. Equations of motion are obtained from the variational principle

   S= ∫ Ldt , L= I(dφ/dt)²/2- V(φ) , V(φ)= τφ .

   Here φ denotes the rotational angle. The mathematical problem is that the potential function V(φ) is either many-valued or dis-continuous at φ= 2π.

2. Quantum mechanically the system corresponds to a Scrödinger equation

   - hbar²/2I× ∂²Ψ/∂φ² +τ φ Ψ = -i∂Ψ/∂ t .

   In stationary situation one has

   - hbar²/2I× ∂²Ψ/∂φ² +τ φ Ψ = EΨ .

3. Wave function is expected to be continuous at φ=2π. The discontinuity of potential at φ= φ₀ poses further strong conditions on the solutions: Ψ should vanish in a region containing the point φ₀. Note that the value of φ₀ can be chosen freely.

   The intuitive picture is that the solutions correspond to strongly localized wave packets in accelerating motion. The wavepacket can for some time vanish in the region containing point φ₀. What happens when this condition does not hold anymore?

   • Dissipation is present in the system and therefore also state function reductions. Could state function reduction occur when the wave packet contains the point, where V(φ) is dis-continuous?
   • Or are the solutions well-defined only in a space-time region with finite temporal extent T? In zero energy ontology (ZEO) this option is automatically realized since space-time sheets are restricted inside causal diamonds (CDs). Wave functions need to be well-defined only inside CD involved and would vanish at φ₀. Therefore the mathematical problems related to the representation of accelerating wave packets in non-compact degrees of freedom could serve as a motivation for both CDs and ZEO.

   There is however still a problem. The wave packet cannot be in accelerating motion even for single full turn. More turns are wanted. Should one give up the assumption that wave function is continuous at φ=φ₀+ 2π and should one allow wave functions to be multivalued and satisfy the continuity condition Ψ(φ₀)=Ψ(φ₀+n2π), where n is some sufficiently large integer. This would mean the replacement of the configuration space (now circle) with its n-fold covering.

The introduction of the n-fold covering leads naturally to the hierarchy of Planck constants.

1. A natural question is whether constant torque τ could affect the system so that φ=0 ja φ=2π do not represent physically equivalent configurations anymore. Could it however happen that φ=0 ja φ= n2π for some value of n are still equivalent? One would have the analogy of many-sheeted Riemann surface.
2. In TGD framework 3-surfaces can indeed be analogous to n-sheeted Riemann surfaces. In other words, a rotation of 2π does not produce the original surface but one needs n2π rotation to achieve this. In fact, h_eff/h=n corresponds to this situation geometrically! Space-time itself becomes n-sheeted covering of itself: this property must be distinguished from many-sheetedness. Could constant torque provide a manner to force a situation making space-time n-sheeted and thus to create phases with large value of h_eff?
3. Schrödinger amplitude representing accelerated wave packet as a wavefunction in the n-fold covering would be n-valued in the ordinary Minkowski coordinates and would satisfy the boundary condition

   Ψ(φ)= Ψ(φ+ n2π) .

   Since V(φ) is not rotationally invariant this condition is too strong for stationary solutions.

4. This condition would mean Fourier analysis using the exponentials exp(imφ/n) with time dependent coefficients c_m(t) whose time evolution is dicrated by Schröndinger equation. For ordinary Planck constant this would mean fractional values of angular momentum

   L_z= m/n hbar .

   If one has h_eff=nhbar, the spectrum of L_z is not affected. It would seem that constant torque forces the generation of a phase with large value of h_eff! From the estimate for how many turns the system rotates one can estimate the value of h_eff.

What about stationary solutions?

Giving up stationary seems the only option on basis of classical intuition. One can however ask whether also stationary solutions could make sense mathematically and could make possible completely new quantum phenomena.

1. In the stationary situation the boundary condition must be weakened to

   Ψ(φ₀)= Ψ(φ₀+ n2π) .

   Here the choice of φ₀ characterizes the solution. This condition quantizes the energy. Normally only the value n=1 is possible.

2. The many-valuedness/discontinuity of V(φ) does not produce problems if the condition

   Ψ(φ₀,t)=Ψ(φ₀+ n2π,t) =0 , & 0<t<T .

   is satisfied. Schrödinger equation would be continuous at φ=φ₀+n2π. The values of φ₀ would correspond to a continuous state basis.

3. One would have two boundary conditions expected to fix the solution completely for given values of n and φ₀. The solutions corresponding to different values of φ₀ are not related by a rotation since V(φ) is not invariant under rotations. One obtains infinite number of continous solution families labelled by n and they correspond to different phases if h_eff is different from them.

The connection with WKB approximation and Airy functions

Stationary Schrödinger equation with constant force appears in WKB approximation and follows from a linearization of the potential function at non-stationary point. A good example is Schröndinger equation for a particle in the gravitational field of Earth. The solutions of this equation are Airy functions which appear also in the electrodynamical model for rainbow.

1. The standard form for the Schrödnger equation in stationary case is obtained using the following change of variables

   u+e= kφ , k³=2τ I/hbar² , e=2IE/hbar²k² .

   One obtains Airy equation

   d²Ψ/du²- uΨ =0 .

   The eigenvalue of energy does not appear explicitly in the equation. Boundary conditions transform to

   Ψ(u₀+ n2π k )= Ψ(u₀) =0 .

2. In non-stationary case the change of variables is

   u= kφ , k³=2τ I/hbar² , v=(hbar²k²/2I)× t

   One obtains

   d²Ψ/du²- uΨ =i∂_v Ψ .

   Boundary conditions are

   Ψ(u+ kn2π,v )= Ψ(u,v) , 0 ≤ v≤ hbar²k²/2I× T .

An interesting question is what h_eff=n× h means? Should one replace h with h_eff=nh as the condition that the spectrum of angular momentum remains unchanged requires. One would have k ∝ n^-2/3 ja e∝ n^4/3. One would obtain boundary conditions non-linear with respect to n.

Connection with living matter

The constant torque - or more generally non-oscillatory generalized force in some compact degrees of freedom - requires of a continual energy feed to the system. Continual energy feed serves as a basic condition for self-organization and for the evolution of states studied in non-equilibrium thermodynamics. Biology represents a fundamental example of this kind of situation. The energy feeded to the system represents metabolic energy and ADP-ATP process loads this energy to ATP molecules. Also now constant torque is involved: the ATP synthase molecule contains the analog of generator having a rotating shaft. Since metabolism and the generation of large h_eff phases are very closely related in TGD Universe, the natural proposal is that the rotating shaft forces the generation of large h_eff phases.

For details and background see the chapter Macroscopic quantum coherence and quantum metabolism as different sides of the same coin: part II" of "Biosystems as Conscious Holograms".

For detals and background see the chapter Macroscopic quantum coherence and quantum metabolism as different sides of the same coin: Part II for details.