Exploring Dimensionless Numbers and Natural Ratios in the Standard Model

[jimjohnson]

I have been reading about dimensionless numbers. My question is: are there any natural dimensionless numbers?
All seem to be either equations, like the fine structure constant, or ratios, like β - mass/mass.
22 of the 26 standard model inputs are mass related and only become dimensionless when divided by the Planck mass.

 

[SteamKing]

Last time I checked, pi was dimensionless. So was e.

 

[jimjohnson]

Yes, I forgot about thise two. How about more?

 

[JayJohn85]

Well from wikipedia;

Certain fundamental physical constants, such as the speed of light in a vacuum, the universal gravitational constant, Planck's constant and Boltzmann's constant can be normalized to 1 if appropriate units for time, length, mass, charge, and temperature are chosen. The resulting system of units is known as the natural units. However, not all physical constants can be normalized in this fashion. For example, the values of the following constants are independent of the system of units and must be determined experimentally:

http://en.wikipedia.org/wiki/Dimensionless_quantity

 

[BruceW]

I'm not totally sure what jimjohnson is looking for. Is it purely mathematical dimensionless numbers that you are looking for? The golden ratio is another one.

 

[HallsofIvy]

I think the question itself is backwards. Strictly speaking all numbers are "dimensionless". It is only when we are measuring something that we give them "units" or "dimensions".

 

[Myslius]

Some of purely mathematical constants:

Physical constants:
https://www.physicsforums.com/showthread.php?t=721243

 

[mikeph]

think of any number
do not put a unit after it

...you have a dimensionless number

 

[jimjohnson]

Appreciate the responses but I should have been more specific on my question. Basically, I have read Wikipedia and a dozen articles on Dimensionless Numbers (DN). My objective is to find a hidden relationship or symmetry in DN. This is a task not requiring mathematical sophistication, something an amateur can pursue. I defined criteria for what would define a DN satisfying this objective:
1. Are fundamental constants (c, ħ, G, H) included?
2. Are they based on physical constants (electron charge, elementary particle masses, the four forces)?
3. Is there a physical significance to the inputs or ratios?
4. Are inputs or ratios supported by equations?
5. Are the ratios exact and not approximations?
6. Do the ratios link the Standard Model of Particle Physics including Quantum Mechanics (via the Planck constant) and General Relativity (via the Hubble constant)?
7. Do the ratios apply in multiple contexts (mass, force, radius, density)?
Ratios by Dirac's and others do not meet all the criteria, especially 5,6,and7. Also, as I initially said 22 of the 26 inputs to the Standard model are not dimensionless. Other attempts to find relationships among elementary masses have not been successful. I did find one article using the four fundamental constants (c, ħ, G, H) which meets the criteria but it is based on basically two ratios ( Vixra.org/abs/1308.0143 ).
I was confused on "natural DN" which the above posts clarified, they are pure math numbers that do not meet the criteria.
Anyway, please comment on my approach.

 

[mikeph]

You can divide any two quantities with the same dimension to get a dimensionless number.

Jupiter is 0.000009 times the mass of the sun. That's a dimensionless number, but what does it have to do with the relative permeability of liquid oxygen, for example?

 

[dauto]

You gave us a series of questions, bu not a single criterion. Specific desirable answers to those questions would form a set of criteria. What are the answers you're looking for?

 

[jimjohnson]

You can divide any two quantities with the same dimension to get a dimensionless number.
Jupiter is 0.000009 times the mass of the sun. That's a dimensionless number, but what does it have to do with the relative permeability of liquid oxygen, for example?

Yes, your number is a good example of a meaningless number. The one from the article referenced satisfies the criteria (a yes answer),except maybe the seventh:
MH/ mPL = RU/ lPL = (MH/mH)1/2 = (c5/2 ħ GH2)1/2 = N = 6.04x1060
The dimensionless ratios are derived from equations based on both the Planck constant and the Hubble constant.
Again, the goal is to find a hidden relationship, an inherent feature of nature represented by a number.

 

[dauto]

Yes, your number is a good example of a meaningless number. The one from the article referenced satisfies the criteria (a yes answer),except maybe the seventh:
MH/ mPL = RU/ lPL = (MH/mH)1/2 = (c5/2 ħ GH2)1/2 = N = 6.04x1060
The dimensionless ratios are derived from equations based on both the Planck constant and the Hubble constant.
Again, the goal is to find a hidden relationship, an inherent feature of nature represented by a number.

You're never going to get anything useful out of this, trust me.

 

[Myslius]

1. Are fundamental constants (c, ħ, G, H) included?
Those constants are the least fundamental constants of nature. It's just the ratio between units, nothing more. Such constants are only good for unit conversion, mass to energy etc. In Plank's units c, G, h and some other constants are equal to 1, math becomes simpler this way, you don't have to worry about units.
H0 isn't even a constant, it's a constant in space, but not a constant in time.
5. Basically the only exact constant is c, due to redefinition of the meter.
6. You can't link cosmology, because of high uncertainties, some goes for high energy physics constants.

I think what you're trying to do isn't possible to achieve due to high uncertainties. You can't distinguish patterns between constants from random noise. Try it yourself. We did.

 

[jimjohnson]

I do not think your points necessarily invalidate the equation I quoted.
"1. Are fundamental constants (c, ħ, G, H) included?
Those constants are the least fundamental constants of nature. It's just the ratio between units, nothing more. Such constants are only good for unit conversion, mass to energy etc. In Plank's units c, G, h and some other constants are equal to 1, math becomes simpler this way, you don't have to worry about units."
In each equation, the units cancel to form dimensionless numbers.
"2.H0 isn't even a constant, it's a constant in space, but not a constant in time."
Agree, but the relationship holds for a different H.

"3. You can't link cosmology, because of high uncertainties"
The link is via H and ħ in the equations.

 

[jimjohnson]

You're never going to get anything useful out of this, trust me.

Response follows:
The ratios use Planck values and Hubble values.
Planck length* = lPL = (2ħG/c3)1/2 = 2.28 x 10-33 cm
Planck mass* =mPL = (ħc/2G)1/2 = 1.53 x 10-5 gm
Planck density = ρPL = mPL /(4/3 π lPL 3) = 3 c5/16 π ħG2 = 3.1 x 1093 gm/cm
Planck time = tPl = (ħG/c5)1/2 = lPL/c =7.6 x 10-44 sec
* Planck length/mass calculated from setting Compton wavelength, λ, equal to the Schwarzschild radius: λ = ħ/mc = rs = 2Gm/c2.

Mass, time/age, critical density, and radius of the universe and are calculated from the Hubble constant (H = 2.18 x 10-18/sec ,converting from H = 67.15 km/sec/Mpc):
Mass of Hubble sphere = MH = c3/2GH = 9.25 x 1055gm - mass of the gravitationally connected universe
Hubble time = TH = 1/H = 4.6 x 1017 sec - age of universe
Hubble distance = RH = c/H = 1.38 x 1028 cm - radius of universe
Hubble mass = mH = ħH/c2 = 2.5 x 10-66 cm
Critical density = ρC = 3H2/8πG = 8.5 x 10-30 gm/cm
The equation was:
MH/ mPL = RU/ lPL = (MH/mH)1/2 = (c5/2 ħ GH2)1/2 = N = 6.04x1060
Thus, the ratio, number N, links the micro and macro worlds.

 

[Vanadium 50]

This is numerology. We do not discuss this here.