# Is the "space of physical quantities" a field of transcendence degree $6$ or $7$ over the rationals / real numbers?

Excuse my naive question and please let me explain it:

In everyday life we experience 3 spatial "dimensions" + time etc. Usually the 3 dimensions are represented by a coordinate system and mathematically as the vector space $$R^3$$. In constrast, the word "dimension" in dimensional analysis has a more objective realism which is based on physical measurements. For instance let $$l$$ be a transcendental number.

Then $$1,l,l^2,l^3,\cdots$$ will be linearly independent over the real / rational numbers, and hence can serve as a basis in a vector space. For instance, instead of saying the point in spatial dimension has coordinates $$(x,y,z)$$ we might as well write:

$$x+y \cdot L + z \cdot L^2$$

where $$L$$ represents a "transcendental number over the reals" and is associated with meter or kilometer etc.

Take for instance the derived unit $$s/t^k$$ for $$k=0,1,2$$ where $$s$$ denotes length and $$t$$ denotes time. Then $$s$$=length, $$s/t =$$ velocity, $$s/t^2=$$ acceleration. If we view these as a basis of a vector space, then this vector space has 3 dimensions. But no one would count subjectively these as dimensions.

On the other hand the seven basic SI-units (or 6 if you do not want to count $$mol$$), could be seen as a transcendental numbers over the rationals or reals, and hence powers of those transcendental numbers could give a basis for vector spaces. Monomials of these transcendent numbers would correspond, as is done in dimensional analysis, to derived units. Adding and subtracting for instance $$LT^{-1}+M$$ would give a point in the "space of physical quantities". As a vector space over the reals this space has infinite dimension but has transcendence degree of $$7$$. Every derived / basic physical quantity measured by SI-Units would correspond to a point in this space / field of transcendence degree $$6$$ (or $$7$$ if you count $$mol$$ which I will not do):

$$\mathbb{R}(T,L,M,I,\Theta,J)$$

Excuse my naive question: Is there any reason from physics to discard this point of view?

Edit: It seems that the main idea of this question can be implemented through the ring (Laurent polynomial ring) $$L:=\mathbb{R}[T,T^{-1},L,L^{-1},M,M^{-1},I,I^{-1},\Theta, \Theta^{-1},J,J^{-1}]$$

second Edit: I was asked to give at least one application of this idea, which I would like to do:

Application:

Let $$k(x,y) = \frac{xy}{x^2+y^2-xy}$$ for $$x \neq 0,y \neq 0, x,y \in \mathbb{R}$$ be a Jaccard-Similarity / positive definite kernel defined on $$\mathbb{R}$$. (This function has the property that given two real numbers $$x,y$$ it will output a number $$k(x,y)$$ which measures how similar $$x,y$$ are: $$0 \le k(x,y) \le 1$$, where $$1$$ is $$100\%$$ similar and $$0$$ is $$0\%$$ similar. Furthermore this function is the dot product of some Hilbert space $$k(x,y) = \left< x,y \right>$$ and has the property that $$k(ac,bc) = k(a,b)$$, which makes it useful when considering change of units in meausrements.)

We can define a similarity and positive definite kernel (which has the same properties as the function $$k$$) on the Laurent polynomial ring $$L$$ as :

$$K(x,y) := \frac{1}{N_x + N_y - N_{xy}} \sum_{X_i^{\alpha_i}=Y_j^{\beta_j}} k(a_i,b_j)$$

for $$x = \sum_{i} a_i X^{\alpha_i},y = \sum_{j} b_j X^{\beta_j}$$ and $$X = (T , L, M, I, \Theta,J)$$, $$\alpha_i, \beta_j \in \mathbb{Z}^6$$ and $$X^{\alpha_i},X^{\beta_j}$$ are multinomials, and $$N_x =$$ number of nonzero $$a_i$$, $$N_y =$$ number of nonzero $$b_j$$, $$N_{xy} =$$ number of $$(X^{\alpha_i} = X^{\beta_j})$$.

Since $$k(ca,cb) = k(a,b)$$ for all $$a,b,c \neq 0$$, we deduce that $$K(c \cdot x,c \cdot y) = K(x,y)$$ for all $$x,y \in L$$, $$c \neq 0, c \in \mathbb{R}$$. Hence this (or any other similarity and positive definite kernel $$k$$ with $$k(ca,cb) = k(a,b)$$. This is to make sure, that rescaling of physical units, does not change the similarity between objects.) gives us a possibility to measure the similarity / inner product of two physical objects $$A,B$$ each of which is defined through measurements $$x = \sum_{i} a_i X^{\alpha_i}$$ and $$y = \sum_{j} b_j X^{\beta_j}$$.

Since there are different possibilities to measure similarites / define positive definite kernels, there should be different possibilities to define equality / similarity between two physical objects $$A$$ and $$B$$.

Hence aposteriori $$L$$ is a Hilbert space by the Aaronszajn-Kolmogorov theorem.

Example:

The meaning of this kernel is to compare two physical objects. For instance $$A = 10 m/s + 1 kg$$, $$B = 9 m/s + 2kg$$, $$C = 1 m/s^2+10kg$$. Then $$K(A,B) = 226/276 = 0.8278$$, $$K(A,C) = 10/91= 0.10989$$, $$K(B,C) = 5/21 = 0.2381$$. Hence $$A$$ is most similar to $$B$$, $$B$$ is most similar to $$A$$, $$C$$ is most similar to $$B$$, and $$A,C$$ are the most dissimilar physical objects in this list.

thir edit: By the reference given in the answer below, one should think about the formal sums in the Laurent polynomial ring as being in two $$T,T^{-1},L,L^{-1}$$ variables: https://arxiv.org/abs/0711.4276.

• To turn the question on it's head, can you suggest any reason to adopt this point of view? Commented Nov 30, 2021 at 8:04
• @J.Murray: It is easier to apply algebraic geometry, invariant theory. For instance the invariance in Noethers Theorem has applications in physics. I could imagine that applying invariant theory would make in this setting would raise maybe interesting questions in physics. On the other hand, suppose someone finds an equation relating some of the 6 units (such as $E=mc^2$ relating energy to mass etc.). Then the adoption would be simply a removal of a unit without changing the point of view very much. Commented Nov 30, 2021 at 8:16
• @J.Murray: I gave a possible application: Measuring the similarity / equality of two physical objects. Commented Dec 1, 2021 at 18:12
• This is related, but not an answer. I just recalled that someone once told me we could study dimensional analysis and similar stuff by means of tensor products. I never took a good look at it, but it seems Terry Tao has written a piece on it. While I believe it doesn't address your question, it seems related to the matter Commented Dec 1, 2021 at 21:31
• also asked here: mathoverflow.net/questions/409663/… in case it is more appropriate Commented Dec 2, 2021 at 6:41

No, it seems like a valid isomorphism. But why would it be meaningful, either? You say "no one would count subjectively these as dimensions" but if it is a valid isomorphism then one would have to count these as dimensions.

• You are right, being consequent one should count those as vector space dimensions. Commented Nov 30, 2021 at 8:08
• (+1) Thank you for your insight in this question :-) Commented Nov 30, 2021 at 8:19

Neither.

Let me first point out a complication: the number of fundamental units is pretty much arbitrary. For example, you are currently considering temperature and energy to be distinct physical quantities, but this is done in the SI just for convenience. Temperature literally is a measure of a system's internal kinetic energy and could just as well be measured directly in terms of energy. More specifically, one can choose the Boltzmann constant to be $$k_b = 1$$ and measure temperature in Joules. We don't always do that because it would lead to quite complicated numbers. Notice, however, that nearly every formula in Thermodynamics or Statistical Mechanics (if not all of them) includes temperature always accompanied by the Boltzmann constant (i.e., $$T$$ only appears in expressions of the form $$k_B T$$). Sometimes this is hidden in the form of the universal constant for ideal gases, $$R = k_B N_A$$, where $$N_A$$ is Avogadro's number. Notice that this particular choice is quite similar to, for example, choosing to measure heights in meters and lengths in kilometers. They aren't really different quantities.

This, of course, poses an issue: how can we then know how many fundamental constants are there? Or what is the transcendence degree of the field you mentioned? (I'll stick to more physical terms, since I'm not that well versed in abstract algebra).

This has been considered in the literature, despite not in the mathematical formulation you are providing. arXiv: 0711.4276 [physics.class-ph], for example, argues that there are only two fundamental quantities, the main reason being that in practice we are only able to measure time intervals and lengths, regardless of the experiment we are performing. Indeed, consider my temperature example. What you actually measure when you check the temperature of a room on a scale thermometer, for example, is not the temperature per se, but rather the height of a fluid column, which is then related to temperature.

While temperature was my first example, it is not the only one. For example, notice that the CGS system fixes some more constants. While $$\epsilon_0$$ is a dimensional quantity in the SI, the CGS system fixes $$\epsilon_0 = \frac{1}{4\pi}$$, and as a consequence there is no fundamental unit of charge or current in CGS. Instead, one can express charges and currents in terms of masses, lengths, and time. See the Wikipedia article on this, for example. The paper I mentioned tackles the case of how to measure masses with clocks and rulers.

Why this point of view is not more spread? I'll jokingly answer that it is due to the ephemeral nature of life. It is not practical to know every bit of Mathematics or Physics that there is out there and in practice everyone must choose what to learn and what not to learn. For example, Differential Geometry gives some beautiful insights on many areas of Physics (e.g. Relativity, Classical Mechanics, Thermodynamics, Electrodynamics and Gauge Theory in general, etc), but it doesn't mean that these insights are likely to be useful to every researcher and/or student, nor that the effort for them will pay off. There was a time when we thought that "representation theory [was] useless", but eventually physicists learned that group and representation theory play an important role in the description of the Universe and we figured out that learning it was worth the effort. However, I believe most physicists do not have a working knowledge of algebraic geometry (to be quite frank I can't understand most of what you wrote on your second edit) and hence it is quite unlikely that "vanilla" physicists would come up with the idea of framing dimensional quantities in terms of algebraic geometry and abstract algebra. Now that you have presented this approach, there is still a chance that many physicists won't want to adopt it simply because it doesn't seem worth the time a priori. This, of course, might change with time.

In summary, the degree you are looking for is $$2$$, unless you want to challenge some of the (quite reasonable, in my opinion) hypotheses made by 0711.4276 [physics.class-ph]. As for why this point of view is not that common among physicists, the reason is likely because most physicists (myself included) lack the mathematical prerequisites necessary to comprehend this approach. Furthermore, it takes time and will for someone to learn new mathematics, and to give a community motivation to learn some new math might be quite non-trivial.

As someone who's been trying to learn differential forms and convince other physicists to do the same, I wish you luck on your endeavor.

• (+1) :-) Thank you for your personal and insightful answer from the physics perspective. Hilbert spaces should be familiar from quantum mechanics. ( And what I do is define a positive definite kernel en.wikipedia.org/wiki/Reproducing_kernel_Hilbert_space . As an application one can measure the similarity between two physical objects $A,B$ defined through physical measurements $x,y$. Commented Dec 1, 2021 at 20:42
• also I like to add that it is a great reference which you provided. The Laurent ring, which is a fancy way of saying that only powers of $T,T^{-1},L,L^{-1}$ are allowed then becomes: $\mathbb{R}[T,T^{-1},L,L^{-1}]$. I will update the question. Commented Dec 1, 2021 at 20:49
• @stackExchangeUser I'm glad you liked it! Hilbert spaces are quite familiar indeed. What I found unusual were terms such as "Jaccard-Similarity", "Aaronszajn-Kolmogorov theorem", or what was the general idea between the kernel construction. Re-reading the construction after reading your comment, I understand the idea is to get an inner-product in the space of physical observables. In this case, some follow-up questions on the applicability would be "What is the physical meaning of this kernel?" or "How is the choice of such a kernel made?", for example Commented Dec 1, 2021 at 21:29
• The meaning of this kernel is to compare two physical objects. For instance $A = 10 m/s + 1 kg$, $B = 9 m/s + 2kg$, $C = 1 m/s^2+10kg$. Then $K(A,B) = 226/276 = 0.8278$, $K(A,C) = 10/91= 0.10989$, $K(B,C) = 5/21 = 0.2381$. Hence $A$ is most similar to $B$, $B$ is most similar to $A$, $C$ is most similar to $B$, and $A,C$ are the most dissimilar physical objects in this list. Commented Dec 2, 2021 at 6:00
• As I have written, there might be other ways to define an inner product, hence there will be different ways to measure similarities between physical objects, depending on the application one has in mind. Commented Dec 2, 2021 at 6:07