Systematic Search for Expressions of Dimensionless Constants using the NIST database of Physical Constants (original) (raw)
Systematic Search for Expressions of Dimensionless Constants using the
NIST database of Physical Constants
Michael R. Sheppard
Massachusetts Institute of Technology
| atomic unit of charge density | 1.081 202 300(27) x 1012 C m-3 | to the 80th power |
|---|---|---|
| atomic unit of 2nd hyperpolarizablity | 6.235 380 95(31) x 10-65 C4 m4 J-3 | to the 34th power |
| Newtonian constant of gravitation over h-bar c | 6.708 81(67) x 10-39 (GeV/c2)-2 | singular |
| first radiation constant | 3.741 771 18(19) x 10-16 W m2 | to the -4th power |
| electric constant | 8.854 187 817... x 10-12 F m-1 | to the -108th power |
All multiplied together yields a number which is a dimenionless quantity.
As such, it is independent of any choice of units.
It is a universal constant (...one of many)
1. Reduction of Constants and Equivalent Units
The National Institute of Standards and Technology lists 325 constants on their website as 'Fundamental Physical Constants'. Among the 325 physical constants listed, 79 are unitless in nature (usually by defining a ratio). This produces a list of 246 physical constants with some unit dependence. These 246 physical constants can be further grouped into a smaller set when expressed in standard SI base units. Examples are physical constants with units of Joules or electron-volts which both reduce to [m]2 [kg] [sec]-2 when expressed in standard SI base units. Similarly, measurements of masses expressed in either kilograms or atomic mass units are both equivalent to the SI unit of [kg].
With this reduction to standard SI units, those 246 physical constants reduce to only 55 different expression of units, ranging from simple expression using just one unit, [kg] for masses; to more complicated expressions, as electric charge being officially expressed as [Amp] [sec]; to the largest expression listed in the database of that of the 'atomic unit of 2nd hyperpolarizablity' whose listed units of (C4 m4 J-3) transforms to [m]-2 [kg]-3 [sec]10 [Amp]4.
The reason for using just those 55 unit expressions, which we will label as the Unit Index, is that if a dimensionless constant can be made, for example, with Unit Index #53 (representing some expression of SI units, in this case just [kg]) you can use ANY physical constant that has the same SI base units as that of the Unit Index to create a multitude of dimensionless constants. For example, if you need a unit of mass, you can just as easily use the mass of the electron or the mass of the proton, or any other mass you wish - as long as they have the same Unit Index. (see below)
2. Representation of SI base units as vectors
If you define an array as being the ordered set of {meters, kilograms, seconds, Amps, Kelvin, moles} then those 246 physical constants within NIST's database can be expressed as individual vectors. For that physical constant, the value in the ith position in the vector is equal to the exponent of the unit in the ith position of the ordered array. This creates a one-to-one transformation from aphyscal unit expressions to mathematical vectors in 6-dimensional space.
| Vector | {m, kg, sec, Amp, K, moles} | Physical Meaning |
|---|---|---|
| [0, 0, 1, 1, 0, 0] | [sec]1[Amp]1 | Coulombs |
| [2, 1,-2, 0, 0, 0] | [m]2 [kg]1 [sec]-2 | Joules |
An Excel file is available to view and download listing all 325 physical constants in the NIST database reduced to standard SI units and grouped by similar unit expressions. There were only 55 unique vector representations, representing the 55 unique Unit Index referenced above. Example:
| Unit Index | Physical Meaning | SI Unit representation | Vector representation |
|---|---|---|---|
| #7 | Ohms | [m]2[kg]1[s]-3[Amp]-2 | [2, 1, -3, -2, 0, 0] |
3. Algorithm
Following the Buckingham Pi Theorem, if you can find linear combinations of those vectors that produce a zero vector, then the coefficients for that expression are equal to the exponents of the actual physical constants to produce a dimensionless constant. All units will cancel out since the 'vectors' representing those units - with proper choice of coefficients - will sum, component wise, to all zeros.
The Matlab program which was written for this experiment checked every
55 choose 2 = 1,485
55 choose 3 = 26,235
55 choose 4 = 341,055
55 choose 5 = 3,478,761
combinations of the 55 unique vectors to find those combinations that collectively have a nullity of one with non-zero coefficients - so that only those that produce dimensionless constants with exactly 2, 3,4, or 5 physical constants can be found.
Q: Why nullity of one?
A: If the dimension of the null space is zero, then all the vectors are independent and can not be combined to produce a dimensionless number (except the trivial case with zero for each coefficient). If the dimension of the null space is greater than one, then there exist a smaller subset of your vectors which can be combined into a dimensionless number.
Q: Why non-zero coefficients?
A: With nullity of one and non-zero coefficients we are guaranteed in finding which combination of any exact specified number of physical constants produce a dimensionless constant. If we are trying to find dimensionless constants using five physical constants and one of the coefficients were zero then only four physical constants were needed, i.e. the fifth constant was independent of the other four. Non-zero coefficients guarantees that exactly five physical constants are needed.
4. Explanation of Program Output for small sample
For example, searching for two vectors produce the following 12 combinations
| Program Output | ||||
|---|---|---|---|---|
| Number | Constants | Coefficients | ||
| 1 | 3 | 24 | 1 | 2 |
| 2 | 6 | 7 | 1 | 1 |
| 3 | 14 | 19 | 1 | 1 |
| 4 | 32 | 46 | 1 | 1 |
| 5 | 32 | 49 | 1 | -3 |
| 6 | 32 | 54 | 1 | 3 |
| 7 | 40 | 47 | 1 | 1 |
| 8 | 42 | 53 | 1 | 2 |
| 9 | 46 | 49 | 1 | 2 |
| 10 | 46 | 54 | 1 | -2 |
| 11 | 49 | 54 | 1 | 1 |
| 12 | 50 | 52 | 1 | 1 |
- Number : Dummy index to label the rows
- Constants: Which Unit Indices to use
- Coefficients: Coefficients (exponents) needed to create a zero vector (dimensionless constant)
**EXAMPLE:**Using line #6
| Number | Constants | Coefficients | ||
|---|---|---|---|---|
| 6 | 32 | 54 | 1 | 3 |
Which means that 1 times Unit Index 32 plus 3 times Unit Index 54 should be the zero vector:
| Unit Index | Vector of Unit Index | Coefficients | Vector Produced |
|---|---|---|---|
| 32 | [-3, 0, 0, 0, 0, 0] | 1 | [-3, 0, 0, 0, 0, 0] |
| 54 | [1, 0, 0, 0, 0, 0] | 3 | [3, 0, 0, 0, 0, 0] |
| VECTOR SUM = | [0, 0, 0, 0, 0, 0] |
The vector [-3, 0, 0, 0, 0, 0] represents [m]-3, and [1, 0, 0, 0, 0, 0] represents [m]1. Transformed to using physical units, the coefficient for the vectors given above are now the exponents for the physical constants, and indeed ([m]-3)1([m]1)3 is unitless. This example is true for any choice of physical constants, as long as they have units of [m]-3 and [m]1; one example being Loschmidt constant and Planck Length.
5. Full Results
Below is the table representing the full output of the program. The first set of the txt files are the Unit Indices for the physical constants to be used, while the second set of the txt files are the coefficients that are needed to produce the zero vector as a linear combination of those vectors.
| # of Physical Constants Used | Possible Combinations | Dimensionless Combinations found | Constants | Coefficients | Size of txt files in KB |
|---|---|---|---|---|---|
| 2 | 1,485 | 12 | <2nconst.txt> | <2ncoef.txt> | 1 |
| 3 | 26,235 | 364 | <3nconst.txt> | <3ncoef.txt> | 9 |
| 4 | 341,055 | 10,138 | <4nconst.txt> | <4ncoef.txt> | 306 |
| 5 | 3,478,761 | 343,368 | <5nconst.txt> | <5ncoef.txt> | 27,497 |
The problem with such a large collection of non-trivial possibilities is that it may be hard to separate the meaningful dimensionless constants from the not-so-meaningful. Take for instance one of the most famous dimensionless constants, the Fine-Structure Constant. It uses four physical constants
| Physical Constant | Units | SI Unit representation | Vector representation | Unit Index |
|---|---|---|---|---|
| electric constant | [F]1[m]-1 | [m]-3[kg]-1[sec]4 | [-3,-1, 4, 2, 0, 0] | 4 |
| Planck's constant | [J]1[sec]1 | [m]2[kg]1[sec]-1 | [ 2, 1,-1, 0, 0, 0] | 30 |
| speed of light in vacuum | [m]1[sec]-1 | [m]1[sec]-1 | [ 1, 0,-1, 0, 0, 0] | 44 |
| elementary charge | [Coulomb] | [sec]1[Amp]1 | [ 0, 0, 1, 1, 0, 0] | 45 |
The Fine-Structure Constant was found is the program, but was well hidden among all the other results. Looking at the files for using exactly four Physical Constants line #3055 shows that a dimensionless constant exists using Unit Indices of 4, 30, 44, 45 with coefficients of -1, -1, -1, 2. Those are exactly the physical constants and their exponents which make up the Fine Structure Constant!
| Unit Index | Vector of Unit Index | Coefficients | Vector Produced |
|---|---|---|---|
| 4 | [-3,-1, 4, 2, 0, 0] | -1 | [ 3, 1,-4,-2, 0, 0] |
| 30 | [ 2, 1,-1, 0, 0, 0] | -1 | [-2,-1, 1, 0, 0, 0] |
| 44 | [ 1, 0,-1, 0, 0, 0] | -1 | [-1, 0, 1, 0, 0, 0] |
| 45 | [ 0, 0, 1, 1, 0, 0] | 2 | [ 0, 0, 2, 2, 0, 0] |
| VECTOR SUM = | [ 0, 0, 0, 0, 0, 0] |
6. Unusual Results
The Fine-Structure Constant is an extremely important dimensionless constant used in Quantum Physics. Compare this to two other dimensionless constants the program found using five physical constants, which may be less useful:
Dimensionless Constant which has the largest absolute value for an exponent (Ans=128)
| Newtonian constant of gravitation | 6.674 28(67) x 10-11 m3 kg-1 s-2 | to the 32nd power |
|---|---|---|
| atomic unit of charge density | 1.081 202 300(27) x 1012 C m-3 | to the 80th power |
| Fermi coupling constant | 1.166 37(1) x 10-5 GeV-2 | to the -1st power |
| atomic unit of 2nd hyperpolarizablity | 6.235 380 95(31) x 10-65 C4 m4 J-3 | to the -10th power |
| Rydberg constant times c in Hz | 3.289 841 960 361(22) x 1015 Hz | to the -128th power |
Dimensionless Constant which uses the largest sum of absolute values of the exponents (sum=227)
| atomic unit of charge density | 1.081 202 300(27) x 1012 C m-3 | to the 80th power |
|---|---|---|
| atomic unit of 2nd hyperpolarizablity | 6.235 380 95(31) x 10-65 C4 m4 J-3 | to the 34th power |
| Newtonian constant of gravitation over h-bar c | 6.708 81(67) x 10-39 (GeV/c2)-2 | singular |
| first radiation constant | 3.741 771 18(19) x 10-16 W m2 | to the -4th power |
| electric constant | 8.854 187 817... x 10-12 F m-1 | to the -108th power |
Of course, since each expression is a dimensionless constant any arithmetic using those constants are also dimensionless. You can take one expression to the power of another expression, or even take the n'th root of one expression with the root being yet another expression. Because of this fact, there are an infinite number of combinations that exist.
The output of the program only lists the simplest expressions that can be produced.
Created: July 17, 2007 Extensively Updated: April 28, 2008 [Personal Homepage]