Calculating the Quantum Kepler Length of a Particle

[arivero]

Originally posted by Orion1

Impose that at Planck scales, gravitational exchanges occur via a 'vibrating space-time string' called a 'harmonic graviton'

Hmm could you expand (ie, with more words than formulae...) on the physics attached to the math of this toy-graviton?

 

[Orion1]

Harmonic Resonance...



Binary rotational velocity:
$$d \omega_1 = d \omega_2 = \omega_t$$

The equal rotational frequencies for any given binary gravitational system can be explained if a harmonic resonator has been established between the binary mass centers.

Given a classical college lab experiment in which a mass is suspended by a 'guitar string' of given length L, and linear density $$\mu$$, the resulting resonant frequency is:

$$f_n = \frac{n}{2L} \sqrt{ \frac{F_g}{\mu}} = \frac{n}{2L} \sqrt{ \frac{mg}{\mu}}$$
g - gravitational acceleration

Two possible methods of inducing energy upon the string is to either 'guitar pick' the string, or displace the mass via a measurable vertical distance and release the mass under its own gravitational acceleration.

The resulting sound frequency is the ground resonance frequency for this particular harmonic resonator. The resulting harmonic amplitude decay of the string results in a pendulum motion of the suspended mass.

The pendulum motion of the suspended mass translates as an eccentric orbital on an astrophysical scale.

Given a classical microgravity experiment is to tension a 'guitar string' between two differential suspended masses sufficient to maintain the string properties.

Two possible methods of inducing energy upon the string is to either 'guitar pick' the string, or displace one of the masses via a measurable displacement distance away from the opposite mass. The resulting ground resonant frequency is:

$$f_n = \frac{n}{2L} \sqrt{ \frac{F_g}{\mu}}$$

F_g - string tension (implied gravitational force)

The resulting harmonic amplitude decay of the string results in the two differential masses 'orbiting' each other around their center of mass with the same rotational velocity:

$$d \omega_1 = d \omega_2 = \omega_t$$

The result is that a resonator has been established between the differential masses. The 'guitar string' is the 'Energy Carrier' between the differential masses in this system.

String Resonator Energy:
$$E_\omega = \mu v \omega \Psi_m^2 = \mu \overline{\lambda} \omega^2 \Psi_m^2 = \hbar \omega$$

$$\Psi_m$$ - maximum amplitude
$$\hbar$$ - harmonic string constant
$$\omega$$ - string angular frequency

$$v = \omega \overline{\lambda} = \sqrt{ \frac{F_g}{\mu}}$$

$$\Psi_m^2 = \frac{ \hbar}{ \mu v} = \frac{\hbar}{ \mu \omega \overline{\lambda}} = \frac{ \hbar}{ \sqrt{ \mu F_g}}$$

$$\hbar = \mu v \Psi_m^2 = \mu \omega \overline{\lambda} \Psi_m^2 = \sqrt{ \mu F_g} \Psi_m^2$$

$$\hbar = \sqrt{ \mu F_g} \Psi_m^2$$

Displayed here, $$\hbar$$ is not strictly implied as 'Planck's Constant', but as the 'harmonic string constant' for this particular harmonic resonator.

The 'harmonic string constant' displayed here is described as the fundamental property between a string's linear density, tension and maximum displacement.

I suppose the real challenge here is to prove that the 'harmonic string constant' is in fact constant for a particular string harmonic resonator, however, not necessarily a universal constant for all harmonic resonators.

 

[Orion1]

Quantum Mass...



Keplerian-Planck Mass Binary System

Conditions:
$$\frac{dA_p}{dt_p} = \frac{}{2} \sqrt{ \frac{ \hbar G}{c}} = K_k$$

Quantum Planck Mass:
$$M_t = 2nm_p$$
n - quantum integer

$$r_1 = \frac{(2 K_k M_t)^2}{Gm_2^3} = \frac{(4K_k)^2}{nGm_p}$$

$$m_1 = m_2 = nm_p$$
$$r_1 = r_2 = \frac{4r_p}{n}$$

$$L_t = m_1 r_1 \sqrt[3]{G M_t \omega_t} = 4m_p r_p \sqrt[3]{n2 G m_p \omega_t}$$

$$L_t = 2n \hbar$$

Truth Table:
$$n, L_t, M_t, r_1$$
$$.5, 1 \hbar, 1m_p, 8r_p$$ - forbidden binary mass state - special case
$$1, 2 \hbar, 2m_p, 4r_p$$
$$2, 4 \hbar, 4m_p, 2r_p$$
$$3, 6 \hbar, 6m_p, 1.334r_p$$
$$4, 8 \hbar, 8m_p, 1r_p$$

Limits:
$$n = 1 \to 4$$
$$L_t = 2 \hbar \to 8 \hbar$$
$$M_t = 2m_p \to 8m_p$$
$$r_1 = 4r_p \to r_p$$
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All other quantum Planck mass states forbidden.
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[arivero]

Sorry to revive this thread, but I think that a bibliographic reference to Barut is in order. Barut & Bracken proposed, Phys Rev D v 23 n 10 p 2454 proposed to see Zitterbewegung as if reflecting some internal structure of the electron. Some very nice off-academia physicists in Geneva have followed Barut's thread and even implemented quarks.

Well, the point is that Barut's internal structure uses Compton length to represent the electron as a pair of orbiting points, one carrying the charge, another the mass. Our Kepler length argument applies there fully, and the orbiting rule of this system should be related to Planck Area per Planck time.


PS: Remember that part of our discussion was already uploaded to the ArXiV, gr-qc/0404086

 

[arivero]

Well, the point is that Barut's internal structure uses Compton length to represent the electron as a pair of orbiting points, one carrying the charge, another the mass.

Hmm in general, as Dirac equation for the free electron only carries Planck's $$\hbar$$ through the Compton length, we can foresee that our Kepler Length argument let's one to derive free Dirac equation from quantum gravity. A different issue is to get the classical and not relativistic limits.

Could we define $$\hbar$$ as two times the angular moment of the Dirac particle?

 

[arivero]

see saw

An amusing consequence of the Quantum Haiku is the well knwon neutrino mass bound from see saw. The point is, any orbiting (read, interacting) particle with a mass less than plank mass should violate the Haiku rule if it were bound only by gravity. So it needs to enjoy also a new force with a coupling K at least greater than $$m/m_P$$, ie of order unity if the particle were at Planck mass, and not necessarily so big if the particle has smaller mass. Th strong force and electromagnetism do the work well, but for neutrinos we only have electroweak force. A neutrino will interact only with a force coupling $$K \approx ({m_\nu\over m_{EW}})^2$$

Thus combining the neutrino interaction force with the Haiku bound we have
$$m_\nu m_P \approx m_{EW}^2$$

As you know, nowadays this bound is though to be saturated for GUT mass instead of Planck mass.