The dimension of physical quantities
This paper is a supporting argument for the article I recently posted about the dimensions of physical quantities. As the article only presents the result of this study, this document will try to provide more information and explanations.
Please read the article first.
As strange as it can seems, my study suggests that the dimension of mass is [M] = L^{7}T^{7} and the electric charge is [Q] = L^{4}T^{3.5}
To put it short, I wanted to create a dimensional analysis matrix, but then I had to justify the initial choice for the dimension of mass and electric charge. I initially intended to use the well known [M]=L^{3}T^{2}, but while doing researches to support it, I discovered that it is based solely on an the assumption that [G] = 1, which not only constitutes an indemonstrable arbitrary choice, but might also appear to be false. While many studies takes it for granted and do not question the validity of this assumption, there is no formal scientific demonstration. In fact, it seems there is nothing in current knowledge that demonstrates clearly and without ambiguities that the dimension of physical quantities can be derived from space and time only.
There is no way that a theory should be blindly accepted or supported with arguments such as "It is well known". Apart from being utterly arrogant, it is more a dogma that a demonstration. If it is so well known, it should be very easy to provide the supporting information. Failure to do so means that we are back in religious mentality as when it was "well known" that the sun was circling the earth at the center of the universe, and scientists of the time had to resort to the epicycle artifice to make the theory fit reality.
I then resorted to the set theory and constructed a matrix with the two sets of Planck space and time constituting the axis.
This means that this matrix represents the Cartesian product, or all combinations, of these two sets. Each cell is a combination of space and time, therefore any value defined from Planck space and time only must appear on this matrix.
Building this matrix with Excel is very easy as the formula for each cell is "=Lp^{x} * Tp^{y}" such as
Then, because the Planck mass is defined from these two sets (our initial hypothesis), the Planck mass must appear on this matrix at the exact position corresponding to its spacetime dimension.
The term dimension should be used with care as it has at least three distinct meanings:
This study is related to the fundamental constitution of physical quantities and states as an initial hypothesis that:
The dimension of all physical quantities is derivable from space and time only.
Meaning not only that the SI system's fundamental values can be reduced to two components (space and time), but also that the "quantum" value (Planck) of all physical quantities can be easily found from their new dimensional expression. This obviously offers new perspectives on the determination and usage of fundamental quantities.
This whole study is performed by using some chosen specific physical quantities such as mass, gravitational constant, Coulomb constant, electric charge, etc …. However, by knowing the relation between all physical quantities, the initial chosen quantities could very well have been different. For example, I heavily use K, the Coulomb constant, but because we know that K = 1/4pi ε and that c^{2} = 1 / µ ε, I could easily use µ or ε in my demonstrations instead of K.
On the same idea, only the Gravitic and the electric domains are mentioned and used in demonstrations. However, because all other quantities can be derived from these ones, adjusting the calculation is trivial.
The only way to progress into the unknown is to walk and construct on solid and proven bases. There is not so much we know about the spacetime dimension of physical quantities, but what we know for sure will help us later to validate our hypothesis.
The initial wellknown equations are:
F = Ma (1)
F = QE (2)
F = GM^{2}/r^{2} (3)
F = KQ^{2}/r^{2} (4)
With F = force (N), M = Mass (Kg), a = acceleration (m/s^{2}), Q = electrical charge (C), E = electric field (V/m) and r = distance between masses or charges (m).
Then, from (1) and (3), we see that
GM = ar^{2} (5)
So
[GM] = L^{3}T^{2} (6)
From the dimensional equality between (3) and (4), we can say that
[KQ^{2}/r^{2}] = [GM^{2}/r^{2}] (7)
So
[K] = [GM^{2}Q^{2}] (8)
From the definition of Planck quantities, we have
Mp^{2} = hc
/ G (9)
So
h / Mp = G Mp /
c (10)
Then, because [GM] = L^{3}T^{2}, it follows that
[h / Mp] = L^{2}T^{1}
(11)
Now, by multiplying both side by GMp, we also have
[hG] = L^{5}T^{3}
(12)
These are important values because we know and can prove that their dimension is only defined from spacetime, so we are absolutely sure they must appear on the matrix without any multiplicative factor and at the right place as we can see
We can then really say that
G * Mp = Lp3 * Tp2 (13)
and
h / Mp = Lp^{2 *
}Tp^{1} (14)
and finally
h * G = Lp5 * Tp^{3}
(15)
We also know for sure the spacetime dimension of the gravitational field, the acceleration ([a] = LT^{2}).
All aspects of the debate around the spacetime dimension can be summarized into a few possibilities depending on the initial assumptions for the dimension of mass and electric charge. The following table presents all possibilities regarding the proposed solutions:

[M] = M 
[M] = ? 
[M] = L^{3}T^{2} 
[M] = L^{7}T^{7} 

[Q^{2}] = ML^{3}T^{2} 





[Q^{2}] = ML 




As can be seen from the grayed areas, rows correspond to different definitions of the electric charge (electrostatic and electromagnetic), while columns are different definitions of the mass.
The first column means that the mass is considered itself as a dimension. It represents the current situation with the SI system and there is no possible further reduction to spacetime. While perfectly coherent and accurate dimensionally, it relativizes everything with respect to this dimension and so, does not help us to find anything as it contradicts our initial hypothesis. But it is still a possibility after all that mass is actually a dimension and so that no Planck value would appear on the matrix.
The second column eventually accepts our initial hypothesis but states that none of the two following proposed solutions are correct and so that the spacetime dimension of mass is still unknown. The second column then represents all possible other solutions (Cartesian product of the space and time sets). In other words, because we know that [GM] = L^{3}T^{2}, then whatever is chosen for one of them, the second will have to be dimensionally coherent. All possibilities are open and there is nothing in current knowledge to help us.
The first and second columns of the table are different in the way that
[G] = M^{1}L^{3}T^{2} (16)
This means that the dimension of G is equal to a portion of mass combined with a portion of space and a portion of time. Mass is then considered as a fundamental constituent of any physical quantity, what is called a dimension.
While in the second column
[G] = [M]^{1}L^{3}T^{2} (17)
This means that the dimension of G is equal to a portion of "the dimension of mass" combined with a portion of space and a portion of time. The expression "the dimension of mass" is generic and does not imply that mass is in itself a dimension, as did the first column of the table. In this case, the dimension of mass could very well be derived from space and time, but in opposition to the two other proposed solutions (see below), it states that the spacetime dimension of mass is still unknown.
The two last columns represent four different and incompatible possibilities (E, F, G and H). The third column corresponds to Maxwell's position, while the last column is my new proposal.
Let us first sort out the two lines of the table and come up with the only one that correctly reflects reality. This will lead to two possibilities that will be analysed and qualified with respect to the assertions and demonstrations they offer to support their claims.
The first line represents the electrostatic version where [Q] = M^{1/2}L^{3/2}T^{1}, while the second line is the electromagnetic version where [Q] = M^{1/2}L^{1/2}. The two solutions are dimensionally incompatible so one has to be true and the other false.
From (1) and (4), we can say that
Q^{2} = Mar^{2} / K (18)
So we are sure that
[Q^{2}] = ML^{3}T^{2}[K]^{1} (19)
Where [K] means the (unknown) dimension of K, whatever it is … do not assume anything.
From there, the electrostatic version implies that [K] = 1 (i.e. K is dimensionless), which, by using (12), leads to
[Q] = M^{1/2}L^{3/2}T^{1} (20)
On the other hand, the electromagnetic version implies that [µ] = 1. But we know also that because [K] = [epsilon]1 and [epsilon] = L^{2}T^{2}[µ], then [K] = [μ] * L^{2}T^{2}, which leads to
[Q] = M^{1/2}L^{1/2} (21)
(20) and (21) are obviously incompatible and Giorgi only hid the problem by relativizing everything with respect to the mass dimension. This leads to the current situation, corresponding to the first column of the table ([M] = M), which is dimensionally coherent but inherently circular.
But, while the incompatibility problem is never studied, it still exist and one of these solutions must be wrong in its initial assumption. Let us mention that they are so incompatible that (20) incorporates a portion of time as a constituent of the electric charge, while (21) does not. Either time is there, or not, not both!
I suggest that the electromagnetic version is true, so, the electrostatic is false. The main reasons behind this position lies into the numerical concordance and the study of units.
Even if dimensions are somehow an abstraction (meta information) from the numerical values, the equality between them usually indicates that the dimensional form completely reflects the numerical form, without any extra dimensionless factor. For example [c] = LT^{1} tells us that the dimension of the speed of light, a velocity, is equal to space over time. But we know that if a dimensionless factor was involved, the dimensional equation would still be the same because a dimensional equation hides the effect of dimensionless factors. Now the fact is that when we do the numerical calculation using the Planck's length and time, we find c = Lp * Tp. There is a perfect concordance between the dimensional and the numerical equation. While numerical concordance is not a full proof in itself because a dimensionless combination of factors could still be hidden, I suggest that the numerical value supports the probability of the dimensional equation to reflect reality.
Now, testing (20) and (21) with real numerical values (Planck based), we immediately see that Qp^{2} = Mp * Lp^{3 * }Tp^{2} is false, while Qp^{2} = Mp * Lp * 1e+7 holds true. The 1e7 portion is invisible into the dimensional equation, indicating that this factor is dimensionless. It comes from the initial choice of μ. Nevertheless, the numerical similarity is startling as Qp = 1,875545870 (47) e18 and (Mp*Lp * 1e+7)^{ 1/2} = 1,87554672106029E18. The deviation between the two values is due to the uncertainty of each value, and the compound effect of multiplication.
Similarly, working with the electron data, we also see that (Me * Re* 1e+7)^{ 1/2} = 1,60217733075514E19 C, and Qe = 1.602 176 487(40) x 1019 C. again, the small discrepancy is due to the uncertainty of the initial values (all from CODATA, except the Planck electric charge Qp, which comes from Wikipedia).
Furthermore, as we know that
K = (c^{2} * μ) / (4 * pi) (22)
It follows that
K = c^{2} * 1e7 (23)
This highlights the correspondence between c2 and K. In this context, the 1e7 term is dimensionally equivalent to μ as it is the result of μ / 4pi. As the 1e7 factor is an arbitrary value and the 4pi part accounts for spatial integration, I suggest that there is a concordance between K and c2, and so that
[K] = L^{2}T^{2} (24)
The study of electromagnetic units is also a clear indication that the electromagnetic version is correct and so that mu is dimensionless. If we consider the magnetomotive force, a magnetic potential, we know that it unit is the "Ampere turn". The "turn" portion is here to reflect the stacking of loops of wire and is dimensionless, so if we consider one loop of wire, the unit of the MMF is Ampere. We also know the relation between the MMF and the magnetic field
MMF = B * l * cos(phi) (25)
It then becomes clear that the unit of the magnetic field B, the Tesla, must correspond to Ampere / meter (this is known, but not very often used). But we also know from the relation B = µ * H that
µ = B / H (26)
which means that
UnitOf (µ) = Tesla / (Ampere / meter) (27)
Now, because we know that the Tesla corresponds to Ampere / meter, we can safely say that µ is dimensionless.
We can now exclude the first line of the table and keep only the electromagnetic line
It leaves only two possibilities "The Maxwell version" (cell F) where [M] = L^{3}T^{2} and "The Hollo version" (cell H) where [M] = L^{7}T^{7}.
The explanation given by Maxwell to support his assertion is :
" If, as in the astronomical system, the unit of mass is defined with respect to its attractive power, the dimensions of [M] are [L^{3}T ^{− 2}]."
But as can easily be seen from (6), it is GM that is equal to L^{3}T^{2}. It becomes very clear that this argument does not hold.
This is the reason why the SI system uses the mass as a fundamental quantity. If it was so clear and proven that [M] = L^{3}T^{2}, then the BIPM could immediately remove the physical mass, the one located in Paris, from fundamental quantities (one of their objectives) or at least, provide other way of determining mass.
But the
definite proof that Maxwell's assumption is false lies in the logic and
mathematics behind it. By proposing [M] = L^{3}T^{2}, Maxwell
is already in the context that the dimension of mass can be derived from space
and time only. In this context, if we study the Cartesian product of the space
and time sets as per the set theory, all physical quantities defined from the
initial sets must be into the resulting set. By creating a matrix whose axis
are the Planck length and time, we provide the Cartesian product of these two
sets. The Planck mass does not show at the (3,2) location (which corresponds
to Lp^{3}Tp^{2}). However, one could argue that the numerical
value of the Planck mass is "hidden" behind the GM product that appears at Lp^{3}Tp^{2}.
While it is highly suspect (smells epicycle), it is still not totally
impossible that the result of a multiplication hides one of the terms if the
other is dimensionless. But what about all other quantities whose spacetime
dimension could then be easily determined, but are not found on the numerical
matrix at the expected locations? Take for example the Dirac constant h
used in the definition of all Planck's values. It must also appear because it
can be defined from the mass which is, per our hypothesis, defined from space
and time. From the definition of the Planck mass, we know that h = GMp^{2}
/ c, so in the Maxwell version, the Dirac constant h becomes
[h] = ML^{2}T^{1} = L^{5}T^{3}
(28)
The numerical concordance should exist and so we should have
h = Lp^{5}Tp^{3}
(29)
This is
obviously not the case. The same can be said for all Planck values. We can
then say that Maxwell's assumption fails because the Cartesian product of the
space and time sets invalidates it. While we correctly find the well known
values such as GM (L^{3}T^{2}), hG () and hG2M
(), no known Planck value appear at the location predicted by the Maxwell
assumption.
So, the "Maxwell version" must be rejected.
My study states that the Planck mass must appear on the numerical matrix because it is defined from the Planck space and time. I found the value 2,17642080049717E+59 at the location (7,7) which corresponds to
Mp = Lp^{7} * Tp^{7} * 1e67 (30)
I can't explain the 1e67 factor, but it is obvious that apart from this factor, the value of the Planck mass is directly equal to Lp^{7}Tp^{7}.
The accuracy
The CODATA value for the Planck mass being Mp = 2.176 44(11) e8 kg, it means, when applying the uncertainty, that the Planck mass is estimated between 2.17633 e8 Kg and 2.17655 e8 Kg.
On the other hand, using CODATA values Lp = 1.616 252 1e35 m and Tp = 5.391 24 1e44 s, we find
Lp^{7}Tp^{7} = 2,176 42 e+59 (31)
Taking the multiplicative factor (1e67) into account, the discovered value is then absolutely compatible with the CODATA value as it falls perfectly into the uncertainty range.
To use the same logic as previously, not only one, but all physical quantities must appear on this matrix. In our context, we see that the dimension of the Dirac constant becomes
[h] = ML^{2}T^{1} = L^{9}T^{8}
(32)
And this time, the numerical
concordance exist because at location (9,8) we find the value
1,0545625705005E+33 which is obviously related to h by the factor 1e67
such as
h = Lp^{9}
* Tp^{8} * 1e67 (33)
Furthermore, all Planck quantities appear at their expected locations as can be seen on the following matrix (some rows were omitted for readability) :
A mere coincidence?
Considering the intrinsic logic of the matrix, the Planck values had to appear and we see that they do. So the expected fact that they appear is not a coincidence, nor is the location where they appear as it has to be dimensionally coherent.
We cannot really say, as I erroneously did in my article, that the fact that all Planck values appear proves it is not a coincidence, because one could point out that because all quantities are defined one from the others, then once the first one is found, all others automatically fall in place. So, for example, because GM = Lp^{3}Tp^{2}, wherever we put Mp, the cell located at Lp^{3}Tp^{2}[Mp]^{1} will automatically be equal to G. So if we find one value, all others appear automatically. This is true, but on the other hand, if what I suggest is true, then the effect will be exactly the same : if the Planck mass is derived from space and time only, it must appear on this matrix (Cartesian product). And wherever it appear, the cell located at Lp^{3}Tp^{2}[Mp]^{1} will again automatically be equal to G.
Then, the last possibility is that Lp7Tp7 is just accidentally close to Mp. In this case, Mp should still be on the matrix somewhere else and all Planck quantities should appear at some other locations. Looking at the matrix, it is clear that it is not the case, so L7T7 is left as the "best" candidate to represent M.
The meaning
One can say that [M] = L^{7}T^{7} "makes no sense". I could eventually agree J. However, what does "making sense" means when it comes to the fundamentals of matter, close to weird quantum physics. The other version [M] = L^{3}T^{2} does not "mean" any more. We understand eventually L3 as a volume, because matter occupies space, but what does "over a squared time" means? It is easy to understand the gradient, a spread of something over distance, or a flow that is a spread of something over time. Nevertheless, something over squared time is nothing that our senses can feel and makes no more "sense" than something over time to the power of anything (7 for example).
As for any theory, its merit is measured from the demonstrations showing the provability of the initial hypothesis. To summarize it all:
If we agree that the dimension of physical quantities can be derived from spacetime only
Then, by definition, all physical quantities are part of the Cartesian product of the space and time sets
If we build a matrix that presents the Cartesian product of Planck space and time sets (Lp^{x} * Tp^{y}, with x and y = 0 to infinity)
Then all Planck values must appear on this matrix
The point 1 is our initial hypothesis. Maxwell says [M] = L3T2, while I say [M] = L7T7 (the dimension of all other quantities can be derived from [M]).
The point 2 has to be true by definition as the Cartesian product represents all combinations of the two initial sets (Lp^{x} and Tp^{y}).
The point 3 is what I did to represent visually this Cartesian product.
The point 4 is the key to all the reasoning.
If point 4 is false, then point 1 must be false also ... and mass is really a dimension in itself.
But if point 1 is true, then point 4 must also be true.
Then we ask ourselves: Do we see the Planck quantities appearing as expected at specific locations predicted by the Maxwell version. The answer is no. So inevitably, the Maxwell version is clearly invalid.
On the other hand, and still in the context of point 1, what my study suggest is what seems to be the "closest" to Planck values. So, although because of the multiplicative factors the concordance is not complete, this study is clearly a better answer to the prerequisites of the theory and could open some very interesting research areas.
I can now safely conclude that:
If the dimension of physical quantities can be derived from spacetime only
Maxwell version is invalid (mathematically impossible)
My version seems closer to reality
Laurent Hollo  2009