Calculating the Quantum Kepler Length of a Particle

[arivero]

I am still a bit worried about
L^{WRONG}_{m_1>>m_2}\approx 2 n_1 \hbar { m_1\over m_P} ({m_1\over m_2})^{\frac32}
There is not a way to put another wrong to do a right?

Perhaps some argument exchanging particles or so. Hmm.

 

[Orion1]

Harmonic Gravitons...



Conditions:
\frac{dA}{dt} = \frac{}{2} \sqrt{ \frac{\hbar G}{c}} = K_k = 2.422*10^{-27} m^2s^{-1}

dt_1 = dt_2
d \omega_1 = d \omega_2 = \omega_t
r_1m_1 = r_2m_2

M_t = \frac{R_t^3 \omega_t^2}{G} = (m_1 + m_2)
R_t = \sqrt[3]{ \frac{GM_t}{ \omega_t^2}} = (r_1 + r_2)
\omega_t = \sqrt{ \frac{GM_t}{R_t^3}}

L_t = (m_1 \omega_1 r_1^2 + m_2 \omega_2 r_2^2)
d \omega_1 = d \omega_2 = \omega_t
L_t = \omega_t (m_1r_1^2 + m_2r_2^2)
r_1m_1 = r_2m_2
L_t = \omega_t m_{1,2} r_{1,2} (r_1 + r_2)
L_t = \omega_t m_{1,2} r_{1,2} R_t
L_t = \omega_t m_{1,2} r_{1,2} \sqrt[3]{ \frac{GM_t}{ \omega_t^2}}
L_t = m_{1,2} r_{1,2} \sqrt[3]{GM_t \omega_t}

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

harmonic graviton wave velocity:
c = \sqrt{ \frac{F_g}{\mu}}
F_g - Gravitational Force
\mu - Energy per distance

E_g = \frac{Gm_1m_2}{R_t}
\mu = \frac{E_g}{R_tc^2} = \frac{Gm_1m_2}{(R_tc)^2}

harmonic graviton normal mode frequency:
f_n = \frac{c}{ \lambda_n} = n \frac{c}{2R_t}
n - harmonic graviton vibration mode

normal modes:
n = \frac{2f_nR_t}{c} = \frac{2R_t}{\lambda_n}

R_t = \sqrt[3]{ \frac{GM_t}{ \omega_t^2}}

f_n = n \frac{c}{2} \sqrt[3]{ \frac{ \omega_t^2}{GM_t}}

Limit M_t = 2m_p
Limit r₁ = r_p

harmonic graviton Planck Frequency/Wavelength:
f_p = n \frac{c}{2} \sqrt[3]{ \frac{c^2}{2Gm_pr_p^2}}
f_p = 7.361*10^{42} Hz
\lambda_p = \frac{c}{f_p} = 4.072*10^{-35} m

harmonic graviton Planck quantum number:
E_g = n_g \hbar f_p
n_g = \frac{E_g}{ \hbar f_p}
n_g = 1.26

Expected/Predicted:
n_g = 1/1.26

P = 1.26^-1 = 79.365% accuracy
T_l = +,- 20% tolerance

harmonic graviton Classical Planck Frequency/Wavelength:
f_p = \frac{Gm_p^2}{2 \hbar r_p}
f_p = 9.274*10^{42} Hz
\lambda_p = \frac{c}{f_p} = 3.232*10^{-35} m
n_g = 1

Normalization:
f_n = n n_g \frac{c}{2} \sqrt[3]{ \frac{ \omega_t^2}{GM_t}}
n_g = 1.26

 

[marcus]

Originally posted by marcus
Alejandro has found online the historically first description of Planck units in a reproduction of a Spring 1899 document of the
"Royal Prussian Academy of Sciences"

http://www.bbaw.de/bibliothek/digital/struktur/10-sitz/1899-1/jpg-0600/00000494.htm

I think we should try to understand how Planck could have thought of the Planck units in 1899
when the Radiation Law (which has the constant h or hbar in it)
only occurred to him in late 1900. The Radiation Law paper was
published only in December of 1900.

So in March 1899, when he presented this paper to the Academy, he did not have "Planck's constant" as we understand it.
But he could still smell something universal in some radiative thermodynamics quantities he was finding and he got rather excited
as one can hear in the unusual tone of voice on page 600 of the
document which Alejandro has given us.

I have been trying to imagine what the old Prussians of the
Koenigliche Preussiche Akademie would have been thinking
when the 40-year not-so-famous-yet Planck started telling them
about some universal units intrinsic to nature.

(even after the Radiation Law discovery and acceptance of the h or hbar constant, the scientific community continued to disregard the Planck units. they were largely forgotten by everybody but Planck himself for 50 years or more)

 

[marcus]

...the historically first description of Planck units in a reproduction of a Spring 1899 document of the
"Royal Prussian Academy of Sciences"
(Königlich Preußischen Akademie der Wissenschaften)

http://www.bbaw.de/bibliothek/digital/struktur/10-sitz/1899-1/jpg-0600/00000494.htm
...

Here are a couple of exerpts from Planck's March 1899 paper, from around page 480 of the document Alejandro gave the link to:

----------------------

...Diese Grössen behalten ihre natürliche Bedeutung, so lange bei, als die Gesetze der Gravitation, der Lichtfortpflanzung I am Vacuum und die beiden Hauptsätze der Wärmtheorie in Gültigkeit bleiben,sie müssen also, von den verschiedensten Intelligenzen nach den verschiedensten Methoden gemessen, sich immer wieder als die nämlichen ergeben...

...ihre Bedeutung für alle Zeiten und für alle, auch ausserirdische und ausser menschliche Culturen nothwendig behalten und welche daher als "natürliche Maasseinheiten" bezeichnet werden können...

-----a rough translation into English---

...These quantities retain their natural meaning as long as the Law of Gravitation, the Speed of Light in Vacuum and the Laws of Thermodynamics remain valid, they must therefore appear as such, again and again, measured by the most various intelligences and by the most various methods...

...they therefore necessarily retain their meaning for all times and for all, even extraterrestrial and non-human, civilizations---and hence may be referred to as "natural units"...


------------------

"natürliche Maasseinheiten"
I believe Maass means "measure" in this context
and "Einheiten" are "unities" or "units".
So Maasseinheiten means "measurement-units" or "units of measure"

 

[Orion1]

Modern Translation...



Modern Equasion Translation:

I translated the equations into modern SI units, however for nostalgia, I normalized and kept the old standard units in brackets.

K_b = Boltzmann's Thermal Constant.

a = \frac{ \hbar}{K_b} = 7.638*10^{-12} K*s (Kelvin*Seconds)
b = \hbar (m^2*kg*s^-1)
c = c (m*s^-1)
f = G (m^3*kg^-1*s^-1)

r_p = \sqrt{ \frac{bf}{c^3}} = \sqrt{ \frac{ \hbar G}{c^3}}
m_p = \sqrt{ \frac{bc}{f}} = \sqrt{ \frac{\hbar c}{G}}
t_p = \sqrt{ \frac{bf}{c^5}} = \sqrt{ \frac{ \hbar G}{c^5}}
T_k = \frac{a}{t_p} = a \sqrt{ \frac{c^5}{bf}} = \frac{ \hbar}{K_b} \sqrt{ \frac{c^5}{ \hbar G}}

Interesting to note that Planck used Planck Time to calculate Planck Temperature, as a modern Physicist would use Planck Mass to calculate Planck Temperature, which would have been the next step:
T_k = \frac{ \hbar}{K_b} \sqrt{ \frac{c^5}{ \hbar G}} = \frac{}{K_b} \sqrt{ \frac{ \hbar c^5}{ G}}

Planck units:
\hbar = aK_b (j*s) (m^2*kg*s^-1)

However note that due to the time base in Planck units, explains this approach.

Reference:
http://www.bbaw.de/bibliothek/digital/struktur/10-sitz/1899-1/jpg-0600/00000494.htm

 

[marcus]

Orion this looks right
exept for the factor of 2pi.
What you say about Planck's a,b,c,f
in this paper is all clear and makes
good sense exept for his b turning out
not to be hbar but being h = 2pi hbar.

that had me confused earlier when I hadnt
noticed it

still don't understand how he can have come up
with b
when it wasnt even the year 1900 yet and he
had not figured out the Radiation Law in which
that constant (I thought) first made its appearance

thanks for the clear exposition of the formulas
in Planck's paper!

 

[arivero]

Originally posted by marcus
"the mountains of Algebraic Geometry are very beautiful at
this season. While there is still snow on the heights of Mount Galois
the three dimensions of space are beginning to appear on the lower slopes. Wish you were here"

By the way, the quotes are just stilistic, or were you actually quoting someone?

Lets inspect a bit more the lower slopes. As I put a QM argument in the other thread, and we have followed that area quantization plus classical gravity implies at least some bit of quantum mechanics, let's try now from this point of view.

We had that for a force G {m m' \over r^q}, the Area/time rule implies
2n=G^{\frac12-\frac1q} m^\frac12 r^{\frac32-\frac{q}2} <br /> c^{\frac3q-1} \hbar^{-\frac1q}

G, h and c are constants, but mc a momentum, and the other variable object is the distance r. So we obtain
p \approx {1 \over r^{3-q}}
And directly from r we have a natural interval t=r/c. Joining both equations we have a natural force
f=Dp/Dt\approx {1\over r^{4-q}}
If this force is to be identifyed with our initial force \approx {1 /r^q}, we get
4-q=q
And the only consistent force (excluding q=0, where the equation at h becomes undefined) is q=2, inverse square law.

The argument works for any area quantization, ie the G in Planck length does not need to coincide with the G in Newton force. Still, it is weaker than the one directly from quantum mechanics (at new thread) because here the momentum is not defined for the mediating particle, but for the kepler centre. Thus a bit of flaw.

[Edited]If we aim for a force from m', we can do a distintion between inertial mass and gravitational mass. Then the two forces can not be of the same exponent anymore; if d=2 (gravity as input) the new force is a confining F=K mm', with K=c^3/h. The magnitude is not unphysical (order of QCD string tension) but the meaning is, er, lacking. Surely one should concentrate in mediating particles.

 

[Orion1]

Planck Genesis...


This formula is the first mention of the universal constant (b) via reverse tracing the constant (b) in this thesis.

Pg. 465 - Paragraph 2 - Translation:
[/color]
"The entropy S of a resonator with rotational oscillation frequency (v) and the energy (U) we definition the following measurement:

S = -\frac{U}{av}log_e \frac{U}{bv} (41)

"Whereby (a) and (b) designate two universal positive constants, whose total value in the absolute C.G.S. system in the following section (\delta 25)(pg. 478-479) on thermodynamic pathway is determined; (e), which is the source of the natural logarithms, only from out formulation purpose massively superthermal mass turbulent gases course joins."

Formula: (41) - C.G.S. to SI conversion:
S = -\frac{U}{av}log_e \frac{U}{bv} = -\frac{UK_b}{ hf}ln \frac{U}{hf} = -\frac{UK_b}{ \hbar \omega}ln \frac{U}{\hbar \omega}

S = -\frac{UK_b}{ \hbar \omega}ln \frac{U}{\hbar \omega}

E_p = K_bT_k = \hbar \omega

a = \frac{T_k}{\omega} = \frac{\hbar}{K_b}

E_i = E_p

dS = - \frac{dU}{T_k} ln \frac{U_f}{E_i}

\hbar = \frac{E_p}{\omega}

S - resonator entropy
U - potential energy
E_p - Planck Thermodynamic Energy
f, \omega - rotational oscillation frequency
e - natural base e

Planck Mass Entropy:
T_k = \frac{}{K_b} \sqrt{ \frac{ \hbar c^5}{ G}}
E_i = E_p = \sqrt{ \frac{ \hbar c^5}{ G}}

dS_p = - \frac{dU}{T_k} ln \frac{U_f}{E_i} = -dUK_b \sqrt{ \frac{ G}{ \hbar c^5}} ln \left(U_f \sqrt{ \frac{G}{ \hbar c^5}} \right)

dS_p = -dUK_b \sqrt{ \frac{ G}{ \hbar c^5}} ln \left(U_f \sqrt{ \frac{G}{ \hbar c^5}} \right)

Universe Entropy:
dU = E_p
U_f = K_bT_u = K_b(2.725 K) = 3.762*10^{-23} J
S_u = K_b ln \left(K_bT_u \sqrt{ \frac{G}{ \hbar c^5}} \right)

S_u = 1.008*10^{-21} J*K^{-1}

Reference:
http://www.bbaw.de/bibliothek/digital/struktur/10-sitz/1899-1/jpg-0800/00000479.htm
[/color]

 

[marcus]

A.R.,
the quotes around the imagined text
of that postcard from the future
were stylistic, in fact, as you suggested

Orion, you seem to have found the
place in Planck's paper where he
first encounters his h constant, which
at that point in time he was calling "b".
It is very obscure to me where this "b"
constant (which Planck recognized as universal
in some sense) came from, or if it would
even make sense in the light of presentday
understanding. Maybe he stumbled on
Planck's Constant a year early by "mistake"?

 

[Orion1]

Universe Entropy...



'Expanding' the Planck Mass Entropy equation for volume expansion results in the following equasion:

Planck Mass Entropy equation for spherical universe containing an ideal gas:
S_u = K_b \left( ln \left(K_bT_u \sqrt{ \frac{G}{ \hbar c^5}} \right) + ln \left( \frac{r_f}{r_i} \right)^3 \right)

r_i = r_p = \sqrt{ \frac{ \hbar G}{c^3}}
r_f = \frac{c}{H_o}

H_o - Hubble Constant

\Delta S_{u1} = K_b \left( ln \left(K_bT_u \sqrt{ \frac{G}{ \hbar c^5}} \right) + ln \left( \frac{c}{H_o} \sqrt{ \frac{c^3}{\hbar G} \right)^3 \right)

\Delta S_{u1} = 4.810*10^{-21} J*K^{-1}

T_i = T_p - Planck Temperature
T_f = 3000 K CBR photo-transparency temperature
r_i = r_p = \sqrt{ \frac{ \hbar G}{c^3}}
r_f = 7*10^5 Ly CBR photo-transparency range

\Delta S_{u2} = K_b \left( ln \left(K_bT_f \sqrt{ \frac{G}{ \hbar c^5}} \right) + ln \left(r_f \sqrt{ \frac{c^3}{\hbar G} \right)^3 \right)

\Delta S_{u2} = 4.487*10^{-21} J*K^{-1}

T_i = 3000 K CBR photo-transparency temperature
T_f = 2.725 K CBR temperature
r_i = 7*10^5 Ly CBR photo-transparency range
r_f = \frac{c}{H_o} = 1.761*10^{10} Ly Universe range

Classical entropy equation for spherical universe containing an ideal gas:
\Delta S_{u3} = K_b \left( ln \frac{T_f}{T_i} + ln \left( \frac{r_f}{r_i} \right)^3 \right) = K_b \left( ln \frac{T_f}{T_i} + ln \left( \frac{c}{H_o r_i} \right)^3 \right)

\Delta S_{u3} = 3.230*10^{-22} J*K^{-1}

\Delta S_{u1} = \Delta S_{u2} + \Delta S_{u3}

[/color]

"We cannot order men to see the truth or prohibit them from indulging in error." - Max Planck, Philosophy of Physics, 1936

 

[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 verticle 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
[/color]

All other quantum Planck mass states forbidden.
[/color]

 

[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.