Universal constants and natural systems of units in a spacetime of arbitrary dimension

We study the properties of fundamental physical constants using the threefold classification of dimensional constants proposed by J.-M. L\'evy-Leblond: constants of objects (masses etc.), constants of phenomena (coupling constants), and"universal constants"(such as $c$ and $\hbar$). We show that all of the known"natural"systems of units contain at least one non-universal constant. We discuss possible consequences of such non-universality, e.g. dependence of some of these systems on the number of spatial dimensions. In the search for a"fully universal"system of units, we propose a set of constants that consists of $c$, $\hbar$, and a length parameter, discuss its origins and the connection to the possible kinematic groups discovered by L\'evy-Leblond and Bacry. Finally, we make some comments about the interpretation of these constants.


Introduction
The recent reform of the SI system made the definitions of its primary units (except the second) dependent on world constants, such as c, , k B . It thus sharpens a question about the conceptual nature of physical constants in theoretical physics (see, e.g., recent review [1]). This question can be traced back to Maxwell and Gauss, and many prominent scientists made their contribution to the discussion of it. Analysis of fundamental constants could help us to understand the structure of underlying physical theories, as well as to explain this structure. The Gamow-Ivanenko-Landau-Bronstein-Zelmanov cube of physical theories (or simply Gc cube) is possibly the most famous example of such analysis. It has been a popular educational and methodological tool for many decades [2,3] in various forms and generalizations [4][5][6], even though some of the vertices of such cube are still empty.
The physical constants can also serve more practical purposes. Since the end of the XIX century, physicists have been trying to employ various sets of them to invent systems of units that could be "natural" for one or another theory [7]. When constructed, such systems of units allow one to make some qualitative claims about the characteristic behavior of corresponding physical theory, e.g., "the characteristic length of quantum gravity is 10 −33 meters, so measurements are impossible beyond this scale" [8] or "in classical electrodynamics, the characteristic radius of an electron is 10 −15 meters, so at this scale, renormalization might be needed [9]". The possibility to build up characteristic scales out of physical constants and to compare them to each other leads to Dirac's Large Numbers Hypothesis and its numerous variations [2,10].
However, in the construction of such systems and schemes, it is often overlooked that the constants which are serving as their base might have different degrees of "fundamentality". Furthermore, there are many ways to define what constant is more fundamental than another, and many opinions on that [11].
In spite of this, we want to concentrate our attention on the classification of physical constants presented by J.-M. Lévy-Leblond [12]. It was based on their role in the laws and theories (a similar classification was also discussed in [13]). According to Lévy-Leblond, three possible types of constants are: Most of the natural systems of units contain quantities of type A (such as the electron mass in the electrodynamic system of units) or type B (such as gravitational constant G in the Planck units), or both. One might ask about the possible consequences of this inhomogeneity and the properties of such systems of units.
It turns out that dimensional analysis can help to draw some distinction between constants of different types, as well to shed some light on their properties.
Another interesting question is whether it is possible to construct a set of constants that would be universal in the above sense, i.e., contain only those constants that can be considered as universal ones.
This article, which is intended to serve methodical and pedagogical purposes and by no means contain any new physics, is organized as follows. In section 2, we examine the properties of constants of interaction, namely G, e, and Yang-Mills coupling constants g s , g w , and derive their dimensionality in the spacetime with n spatial directions. In section 3, we generalize some well-known natural systems of units, such as Planckian and field-theoretic ones, on the case of n dimensions. It turns out that the dimensionality of space affects the explicit form of their base units. In section 4, we construct a set of "fully universal" constants, dimensions of which are independent of the dimension of space. Besides c and , this set contains some fundamental length. We discuss the origin of these constants and their connection with the structure of the most general kinematic group [14]. In section 5, we review the history of the notion of fundamental length and give some concluding remarks about its interpretation.

Constants of the fundamental interactions
Let us consider the expression for the Newtonian gravitational force between two point masses in n spatial dimensions: which, as it well known, follows from the Gauss theorem and the assumption that gravitational force is long-ranged [15]. If none of the other fundamental constants are present, the dimensionality of the G n itself gives us little information about the physics in an n-dimensional space.
Things become interesting when c enters the game (for a discussion of the gravitational force in n dimensions in the presence of other constants see, e.g., [16,17]). For instance, a simple calculation shows that, if one considers a point body of mass M in a universe with 2 spatial dimensions, it turns out that it is impossible to construct the quantity with the dimension of length out of M, c and G 2 (i.e. 2D gravitational radius). Therefore the metric of spacetime g µν , which ought to be a dimensionless function of the spacetime coordinates x, in fact, can not depend on them, because there is no dimensionless combination containing x. The force between two particles thus is equal to zero, and it seems that the formula (1), which was the starting point of this consideration, is no longer valid in 2+1 dimensions and should not be used for derivation of the dimensionality of G n . However, in the relativistic framework, the dimension of G n in the n + 1-dimensional gravity can also be deduced from the gravitational part of EH action: where κ n = 8πG n /c 4 is the Einstein gravitational constant and R is the scalar curvature of the spacetime. Since the dimension of S is always [S] = ML 2 T −1 and [R] = L −2 , the equation (2) with arbitrary n follows from it immediately. As a side result, we obtain that for classical general relativity with point masses the dimension 2+1 is critical: the metric of spacetime with point masses is flat -the well-known fact [18,19], but obtained in a very straightforward way. The dimension of elementary charge e, which can serve as the electromagnetic coupling constant, can be derived from the Coulomb law: Since two remaining interactions, namely strong and weak ones, are shortranged, the Gauss theorem can not be directly applied to obtain the dimension of their coupling constants. We know, however, that these interactions are described by non-Abelian gauge fields, also known as Yang-Mills fields. Let us write the action of Yang-Mills field interacting with some charged point particles in an arbitrary dimension: where g is a coupling constant, . . n and each of the components of A µ is a matrix whose size depends on the gauge group that is considered. The point particles with Yang-Mills charges may seem peculiar, but as a toy model, they serve well [20], and without them, some lengthy talk about covariant derivatives would be unavoidable. Comparing the dimensions of both terms to the standard dimension of action, we get The dimension of coupling constant g turns out to be the same as e, which is not surprising because EM field is a particular case of Yang-Mills field. As a result, we obtain the dimensions of all four coupling constants in arbitrary dimensions: As can be seen, all of them depend on n. It is tempting to speculate that it occurs due to the local character of all four interactions. Indeed, the actions of local theories can be represented as integrals of Lagrangian density. Since the integration measure depends on n, while the dimension of the action does not, one can conclude that Lagrangian density itself (and therefore coupling constants) must depend on n to compensate the dependence of measure. Note that this is the main formal difference between constants of type A and B: as we saw, the constants of type B depend on the characteristics of the space, whereas constants of type A, in general, do not.

Natural units and the dimension of space
We are now ready to construct n-dimensional generalization of natural units. Let us start with Planck units, which consist of G n , and c: In this case, the dimension n = 1 is critical, which is possibly connected to the fact that in 1 + 1-dimensional spacetime the EH gravitational action (3) is trivial (more precisely, it equals to the so-called Euler characteristics of the spacetime, which is a constant number) [21][22][23].
As we stated in the Introduction, we want to stress that the expressions for Planckian units depend on n since G n does. It is not surprising if the constant G is regarded as belonging to type B, i.e., as the coupling constant and not as a fundamental characteristic of a physical theory. It is possible, though, that after the completion of quantum gravity, G would be promoted to type C (as, e.g., the electron charge e was promoted from type A to type B after the completion of QED). For now, however, G must be associated with type B. Therefore the Planck scale physics is different in different dimensions, and we can't make any general claim about the nature of spacetime without specifying its dimension.
It also occurs in the case of other constants of interaction. For instance, we could attempt to construct some natural units using c, , and coupling constant of one of the other interactions (we denote it as g n ). Such a system corresponds to a quantum theory of some field, so its units can be called QFT units. We obtain that l (n) Again, the dependence on n is present. Note that our space with 3 spatial dimensions represents the critical case here: at n = 3 there is no characteristic length (and therefore energy) scale corresponds to the interaction, so it becomes renormalizable (when n < 3, it is super-renormalizable, and when n > 3, it is non-renormalizable) [24,25]. There are also two remaining possibilities to construct a system of units out of constants of types B and C. The first one is Stoney units [7], based on e, G and c, whose critical dimension is n c = 2, and the second one is some unnamed peculiar system based on G, e and with n c = 4. It is also worth noting that G, e, c and allow one to construct a dimensionless constant which is independent of n: the Barrow-Tipler invariant [16,26]: It is possible to construct many other systems of units, such as electrodynamic (m e , c and e) and atomic (m e , and e) ones, which can also be generalized to the case of arbitrary dimension. While this procedure can shed some light on a theory that possesses such constants, it also shows that none of the theories, in which constants of type B are present, allow the construction of natural units that are independent of the number of spatial dimensions. The usage of type A constants, such as masses of particles, allows one to construct n-independent systems (e.g., m p , c, ) and n-independent dimensionless constants (such as Gm e m p /e 2 [27]). However, one then has to deal with the arbitrariness of the choice of mass scale: why one mass and not the other? One possible reasonable choice might be Higgs mass [28] since it is the Higgs field that gives the mass to all other particles. However, if it turns out that there is more than one Higgs boson, then such a choice would be inconvenient.
Therefore a question arises: Are there enough constants of type C to construct a system of units out of them?
The most popular answer is no, since in the present state of theoretical physics, only c and are recognized as type C constants, and the number of base units in mechanics is three. Another example of a physical constant that could belong to this type is Boltzmann constant k B [12]. However, to introduce it, one has to define the base unit of temperature, raising the number of base units. Therefore one constant is still missing. If it is not a mass of any particle, we could expect that it might be some universal length (or, equivalently, a time interval). It turns out that there is a natural way to include it in the theory alongside c and . We discuss it in the next section.

Kinematic groups and curvature of space
In this section, we are going to show that one candidate on the role of the third constant is a curvature radius of the spacetime.
In classical mechanics, three quantities with nontrivial dimension can be adopted as base units. They are usually chosen to have units of length, mass, and time, although other choices are possible (and even were recently advocated [1]). Let us consider the Galilean group, which is a symmetry group of classical mechanics (a pedagogical explanation of the properties of the Galilean group and its generators can be found, e.g., in [29]. The relation of the Galilean group and other groups to the structure of dimensional quantities was discussed in [30,31]). Its algebra of generators in n + 1 dimensions has the form (we omit indices and Kronecker deltas for the sake of simplicity) [32]: where M, P , K, and H are generators of rotation, spatial translation, boost, and time translation, respectively.
It is known that it can be deformed in many ways to obtain various generalizations of kinematics [14]. It can be shown [33] that after two steps of such deformations the (anti)-de Sitter group appears. The commutation relations of its generators are where R is a parameter with dimension of length, the upper sign in commutators corresponds to AdS case, and lower -to dS one. As was shown in [34], both Galilean and Poincare algebra of n + 1-dimensional flat spacetime can be considered as contractions of an algebra of a certain space with constant curvature. In the case of Poincare to Galilean group contraction such curvature is −1/c 2 and can be considered as a curvature of space of velocities. In the case of (A)-dS to Poincare group the curvature of spacetime is ∓1/R 2 . As can be seen, alongside c, we have another deformation constant R. Moreover, in the case of anti-de Sitter space, there is a so-called R-c duality [35], so the contraction w.r.t. either of two parameters of the anti-de Sitter group can lead to the Poincare group, and the roles of these parameters are completely analogous. We can see this if we replace H with c 2Ĥ in some of the commutation relations (11): We get standard Poincare algebra in the limit R → ∞ in (12) and we get the "second" Poincare algebra in the limit c → ∞, where the roles of the boost and the spatial translation are swapped and the sign of time translation generator was changed. It is also worth noting that, as was discovered by V. A. Fock [36], the most general form of the transformation between the coordinates in the two inertial frames is the fractional linear transformation, which contains both c and R as parameters [37]. The physics in the theory with fractional linear transformations was also investigated in [38].
Finally, if one considers the algebra of quantum operators corresponding to (11), the Planck constant appears in the right side of the commutators of operators. It makes its appearance because the dimension of operators is governed by canonical commutation relations and can not be arbitrary. The role of as a deformation constant was discussed, e.g., in [4].
We can conclude that three constants with nontrivial dimensions have the same "deformation" origin, and their properties are independent of the number of space dimensions, so one can construct a full set of units on the base of them. Such units are closely related to the de Sitter units that were discussed in [16]. The physical sense of these three constants is also similar since all of them allow one to connect different physical notions (a remarkable feature of type C constants [12]). Indeed, c connects the notions of space and time, -coordinate and momentum (or energy and time, etc.), and Rlength and angle. Therefore all these constants are, in some sense, constants of relativity.
There is no need to discuss the properties of c and here. The properties of universal length R, however, deserve some study. We discuss the notion of fundamental length 1 and its interpretations in the last section.

Discussion
In this article, we stated that Planck units, which are widely assumed to be related to the properties of spacetime, are not invariant with respect to change of the dimension of space. It occurs due to the inhomogeneity of the set of physical constants, on which these units are based: G is not a universal constant in the sense of Lévy-Leblond classification. Replacing G with the universal length R, we obtain a homogeneous set and thus have such invariance. The methodological reason for the assumption of universality of R lies in the properties of a possible kinematic group of spacetime, which 1 In the context of this paper, the terms "fundamental length" and "universal length" are, strictly speaking, not synonymous. Indeed, the Planck length is, by no doubt, fundamental, as it is constructed of fundamental constants. However, it is not universal, since one of these fundamental constants, G, is not universal in the sense of the abovementioned classification. In many cases, however, such a distinction is difficult to draw, so we will nevertheless use both terms as synonyms.
might contain a constant length parameter R alongside with constant velocity parameter c, both of which can be considered as curvatures of certain space.
The concept of fundamental length has a very long story, and there are two directions of investigation: a small fundamental length and a large one. Since the existence of small fundamental length could alter the physics on small scales (and high energies), the inclusion of minimal length was initially discussed in the framework of quantum physics [39]. Later these studies gave rise to more specific theories that deal with the concrete realization of fundamental length. The examples of those are non-commutative geometry (its application to the problem discussed here can be found, e.g., in [40] and the references therein) and double-special relativity (the connection of DSR with deformation of operator algebra is discussed, e.g., in [41]).
On the other side, the existence of fundamental length is one of the consequences of the Kaluza-Klein theory (its basic overview can be found in [42]; for a more detailed account see, e.g., [43]). If the description of electromagnetism (or quantum mechanics [44]) in the KK framework is desired, this length ought to be very small. The KK theory can also be treated as one of the predecessors of string theory [45], in which some small fundamental scale α ′ is also present. In the development of string theory, in turn, various brane theories appeared. In many of them, the fundamental length is assumed to be large (as in the Randall-Sundrum model, where it is related to the curvature of five-dimensional bulk spacetime, or in the ADD model, where it plays the role of compactification radius) [46].
The existence of large fundamental length, on the contrary, was initially discussed in the context of general relativity and cosmology. For instance, the two most popular early cosmological models, namely Einstein and de Sitter ones, both have certain characteristic length scales (this is the reason why Friedmann called them "cylindrical" and "spherical" universes respectively) [47]. In the framework of general relativity the cosmological constant introduced by Einstein could be treated either as a spacetime curvature or as a "vacuum energy" [48]. Therefore the question of the existence of large fundamental length had soon become a part of the so-called cosmological constant problem (although there were attempts to connect the cosmological constant with some "atomic" length, see [49]). A brief exposition of the quantum side of this problem can be found in [50]. For a historical review of the cosmological side of the problem, see [51] and the references therein. Since in our universe the quantity R, which is discussed in this paper, has to be, by construction, quite large (in fact, so large that we can not or barely can notice its presence), we can conclude that its role is similar to the role of the cosmological constant.
We want to stress that here we are not claiming that the quantity R must have the value corresponding to the observed density of dark energy. Some researchers still make attempts to solve the cosmological constant problem using similar kinematic considerations (see, e.g., [52]). However, such attempts, in our opinion, are hardly convincing. This problem is not merely about the value of spacetime curvature, but also about its relation to microphysics. Moreover, the nature of the dark energy (which could be treated as an effect of the cosmological constant presence) remains insufficiently clear, especially due to the data appeared in the last few years [53,54]. The reduction of dark energy to the cosmological constant leads to another problem, namely the coincidence problem: why is the value of vacuum energy (i.e. the cosmological constant) is so close to the value of the energy of other matter, which is supposed to be independent of it? [55] Finally, the fundamental length R is somewhat different than fundamental velocity c and fundamental action . We cannot ask why c is so big or is so small (in the assumption that they are true constants and are not affected by any dynamical process), as we have nothing nearly as fundamental to compare them with. However, we can ask why R (if it exists) is so big in comparison to the scales of all fundamental interactions. Such a question, as it was mentioned in the Introduction, would lead to the famous Large Numbers Hypothesis and its variations [10], and can not be solved without some assumptions on dynamics, while in this paper we discuss only the kinematic properties of spacetime.
In other words, we do not know whether R exists. But if it does exist, it could form, together with c and , some set of universal constants (or type C constants, or constants of relativity), and the corresponding system of units would be independent on the number of the spatial dimensions. The search for such a system was the first main goal of this paper.
Secondly, we wanted to note the inhomogeneity of all other known systems of natural units, especially Planckian ones, and to conclude that due to this, they are not so suitable for methodological considerations of the structure of physical theories as it is widely assumed. The constants of interactions: G, e, g s , g w ; and "constants of relativity": c, , R (and k B , whose role in this context was discussed in [12]) play drastically different roles in physical theories and this circumstance needs to be especially underlined in the discussion of various theories and corresponding systems of units.