Sunday, November 17, 2013

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 heff (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 heff phases are generated in living matter.

Could constant torque force the increase of heff?

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, heff/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 heff?

  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 cm(t) whose time evolution is dicrated by Schröndinger equation. For ordinary Planck constant this would mean fractional values of angular momentum

    Lz= m/n hbar .

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

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 heff 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 heff=n× h means? Should one replace h with heff=nh as the condition that the spectrum of angular momentum remains unchanged requires. One would have k ∝ n-2/3 ja e∝ n4/3. 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 heff phases are very closely related in TGD Universe, the natural proposal is that the rotating shaft forces the generation of large heff 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".

Addition: The old homepage address has ceased to work again. As I have told, I learned too late that the web hotel owner is a criminal. It is quite possible that he receives "encouragement" from some finnish academic people who have done during these 35 years all they can to silence me. It turned out impossible to get any contact with this fellow to get the right to forward the visitors from the old address to the new one (which by the way differs from the old one only by replacement of ".com" with ".fi"). The situation should change in January. I am sorry for inconvenience. Thinking in a novel way in Finland is really dangerous activity!


At 8:52 PM, Blogger Hamed said...

Dear Matti,

I am not sure that understand your purpose. Does that means if we provide the system that a rigid body is rotating around some axes and we wait, the space time sheet of the rigid body(in really quantum state of it) splits to n space time sheets, each one with a different Planck constant? Or does that means phase transition of Planck constant? Hence when an object is rotating, there is an evolution with respect to Planck constant?(it seems magic!)
or maybe this is not really phase transition rather just the hierarchies of allowed phase transitions.

also as I understand in TGD this evolution appears in the rotating magnetic systems that leads to long ranged weak magnetic fields with large Planck constant.

At 9:48 PM, Anonymous Matti Pitkanen said...

Dear Hamed,

there is a slight misunderstanding.

The point is that there is constant torque acting on the system! System is open and there is energy and angular momentum feed to it! In absence of torque the standard description would work. System could be in eigenstate of angular momentum and totally delocalised in angular variable. Nothing exotic.

In the case of constant torque the first observation is that classically the system corresponds mathematically to an accelerating wave packet moving along circle. For a narrow wave packet classical picture is excellent approximation.

The serious mathematical problem is that the potential describing the situation is V(phi)= tau*phi and many-valued as function of phi or discontinuous at 2*pi. A solution to the problem is sharply localised wave packet vanishing at region containing the discontinuity of V. It can propagate at most one turn. This is however not consistent with the physical picture. We want many turns!

The solution is that the configuration space - now circle - is replaced with its n-fold covering so that the system can be n turns in accelerating motion. In TGD ZEO and CDs forces just this replacement. n, the maximal number of turns, corresponds the value of dark Planck constant which indeed corresponds to n-fold coverings for space-time surface for which M^4 is covered n times.

The magic is that all the new elements of TGD follow automatically from the construction of quantum description of open systems, which represents a lacking chapter in the text books of quantum theory!

We are indeed considering systems which are not open: there is feed of energy and angular momentum and living matter is fundamental example of this kind of systems as all self-organising systems discussed in non-equilibrium thermodynamics.

And amazingly: ATP synthase, the basic molecule of metabolism, contains a generator with a rotating shaft!

At 10:04 PM, Anonymous Matti Pitkanen said...

Dear Hamed,

still a little comment. In ZEO ontology one sees the time evolution of 3-D patter at space-time level as a 4-D pattern and quantum jumps recreate the quantum superpositions of these patterns.

Therefore one need not say that time evolution for a rotating system *in accelerating motion* would increase h_eff steadily . Rather, the period of accelerated rotation lasting for some finite time and corresponds to space-time sheet with minimal value of h_eff dictated by the number of turns.

The change of heff would mean addition of new sheets to the existing n-fold covering and also this is of course possible. It would correspond to a transition increasing Planck constant. A possible selection rule is that every sheet suffers same phase transition becoming n_1 sheeted. The final number of sheets would n_f= n*n_1: n would divide n_f. Prime values of n would represent "irreducible" Planck constants just as Hilbert spaces with prime dimension are primes for Hilbert spaces under tensor product operation.


At 2:00 PM, Blogger Ulla said...

What happen with Plancks constant in a quantum tunnelling (is the tunnel enlarged or compressed?), seen in the light that it vanish in classic physics. How is that happening, btw.?

One answer I got: The conventional mechanism is just uncertainty... basically the uncertainty relation holds for energy-time in the same way it does for position-momentum, so the particle can basically 'borrow' energy because its energy is uncertain. Since we require Et <= hbar, the Planck limit defines a constraint on how much energy the particle can borrow for how long. It can get away with this trick because the particle can't be observed while it's actually tunneling...
If we're willing to go beyond conventional ideas, though, there's actually a much simpler mechanism... First, consider that special relativity doesn't actually specifically forbid things from traveling faster than light... what it actually says is that IF anything travels faster than light, the result will be a violation of both energy conservation and classical causality... The reasoning is quite straightforward... if a particle, such as a tachyon, travels faster than light, in terms of the way simultaneity (the 'now') is defined in relativity, it will arrive at its destination before it leaves its source - i.e. it travels 'backwards in time' and since the 'effect' thus precedes the 'cause', causality is violated... if the particle carries energy, as it would have to, while its traveling there'll be more energy in the system than there should be, so energy conservation is violated...
Now consider, that a tunneling particle, say an electron, also violates energy conservation, it 'borrows' energy that isn't there... the idea that it also violates causality is somewhat more abstract... but relates to the simple question, how does it 'know' it can tunnel through the barrier? Putting it all together is actually quite direct... we just need an elementary 'time paradox'... Having tunneled through the barrier, the tunneling particle emits a tachyon, which arrives to be 'absorbed' by the particle (as in, its prior self) just as it's about to tunnel, giving it both the energy and the 'foreknowledge' it actually needs to tunnel through the barrier...

At 7:31 PM, Anonymous Matti Pitkanen said...

To Ulla:

The hierarchy of Planck constants leaves all existing quantum theory remains intact as far predictions are considered. This applies also to tunnelling. The new quantum physics is forced by the situation like the one considered here and is especially relevant to open systems, in particular living matter. All new physics elements in TGD solve some problem of existing physics: this applies also to the hierarchy of Planck constants.

The simplest picture about tunnelling is based on Schrodinger equation in standard QM. Everything is well-defined and simple mathematically. The wave nature of the particle implies realises uncertainty principle and justifies the argument that you represent. Lubos would say that interpretational problems relate to taking classical picture too far and add something nasty about anti quantum zealots;-).

Lubos cannot however silence me or whoever it is I am listening;-). What quantum classical correspondence in TGD sense could give to the understanding of tunnelling? There should exist the analog of classical orbit through the wall. A purely classical process but system defined more generally in terms of space-time sheets. What could this mean?

Could ZEO provide insights? Could particle emit negative energy photon (say) describable as space-time sheet received by another system and get the needed energy to overcome the barrier as a recoil (note that here I would assume energy conservation unlike the uncertainty principle inspired argument based on wave nature). The exchanged virtual photon is tachyon but in TGD framework can be said to consist of fundamental fermions which are on mass shell and massless but possibly with negative energy. This would resemble the Feynman diagrammatic description but in TGD framework the altered arrow of geometric time could bring in something new, maybe only at microscopic level.

About "how does it "know" how to tunnel through the barrier?". "Knowing" happens in subjective time and requires quantum jumps. In Schrodinger equation itself there is nothing about consciousness, it is law obeyed with respect to geometric time and wave nature alone explains tunnelling. Only the measurement telling at what side of the barrier the particle is, gives rise to conscious experience, perhaps that of "knowing".

At 8:00 PM, Blogger Hamed said...

Dear Matti,

Thanks, it is more clear now. I write my understanding: my body that is not dark hasn’t any fold covering but there is hierarchy of (dark) magnetic bodies associated with my body. They correspond to hierarchy of Planck constants. Suppose a magnetic body with hbar =n*hbar_0. This magnetic body is in really n-fold covering of M4*Cp2. That is n=n_a*n_b space time sheets common in M^2 and S^2.

The spinors of the 3-surfaces of WCW at given CD are induced from CP2 spinor connection. Corresponding to this CD at the moduli measurement resolution, there are 2 kind of coset algebra: G/H and N/N , the first one acts on the arguments of the spinor fields that are 3-surfaces and another one acts on the spinor fields themselves. at the measurement resolution, one must approximate 3-surfaces as points and G/H means the algebra corresponding to diffeomorphisms of one dimensional paths. Now spinor fields are averaged on the points of 3-surfaces. and give us one averaged spinor. There is any incorrect?

How anti-commutation relations for fermionic oscillator operators correspond to anti-commutation relations for the gamma matrices of the configuration space. in really how second quantization appear in quantum TGD by transition from classical TGD ?

At 3:15 AM, Anonymous Matti Pitkanen said...

It seems that your view about WCW spinor fields is essentially correct.

WCW gamma matrices satisfy anticommutation relations just like ordinary gamma matrices. Kahler structure of WCW makes the gamma matrices analogous to sermonic oscillator operators. These properties are obtained if gammas are linear combinations of fermionic oscillator operators for the second quantized spinor fields at space-time surfaces. This gives a deep geometric meaning for the fermionic anti commutation relations.

My belief is that fermionic oscillator operators relate to Boolean cognition: oscillator operator basis corresponds to infinite-D Boolean algebra.

One has induced spinor fields and modified Dirac operator defining their dynamics. One performs second quantisation for these spinor fields. What this exactly means is far from trivial! I have pondered this a lot and proposed formulas but I am not sure whether I have the final answer.

Finite measurement resolution suggests strongly that the number of fermionic oscillator operators is actually finite (finite Boolean resolution) corresponding to a finite number of braid strands effectively replacing orbit of the partonic 2-surface and carrying fermion number.

On the other hand, stringy picture suggests that stringy degrees of freedom give infinite number of indices just as in string models. Very probably a cut also in stringy modes emerges from finite measurement resolution.


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