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Standard and derived Planck quantities: selected analysis and derivations

We provide an overview of the fundamental units of physical quantities determined naturally by the values of fundamental constants of nature. We discuss a comparison between the 'Planck units', now widely used in theoretical physics and the pre-quantum 'Stoney units' in which, instead of the Planck constant, the charge of the electron is used with very similar quantitative results. We discuss some of the physical motivation for these special units, attributed much after they were introduced, and also put forth a summary of the arguments supporting various cases for making specific physical interpretations of the meanings of some of these units. The new aspects we discuss are a possible physical basis for the Stoney units, their link to the Planck units, and also the importance of Planck units for thermodynamical quantities in the context of quantum gravity.

These are the three basic Planck units; they serve as the basis not only for many other possible Planck quantities, but more importantly, they set the scale for quantum gravity phenomena. As we will discuss here, there are many analogs to these units for other physical quantities, as well. We will also focus on the general potential of the Planck units to be used as a self-consistent set of fundamental and universal quantities for the study of physical phenomena, and perhaps as indicators of what may be an underlying set of even more profound physical quantities.
The organization of this paper will be as follows. First, we will make some general remarks about the possible relation between the Planck units, Stoney units and fundamental physics, with the well known and widely accepted speculations about the relation between Planck units and relativistic quantum gravity as the basis. Then we will analyze the known Planck units and their derivatives and, for some of these units, give examples of alternative derivations based on arguments more physically direct than dimensional analysis. Such arguments often support the belief that there is a deeper physical meaning than only one of universal convenience to many of the Planck units, and we will then discuss such interpretations in more detail as they arise.
Next, we will discuss the potential role that some of these units may play in arriving at an intrinsic physical system of units and, in conclusion, we will examine how sufficiently each of these points was addressed and emphasize the need for future work in this area.

Some remarks on the Planck units, Stoney units and fundamental physics
It is widely accepted and remarked, even without a fundamental theory, that the Planck quantities signify natural quantities associated with relativistic quantum gravity.
where is the radius of the compactified dimension and   quantities to be expressed dimensionlessly in terms of them. This allows a discussion on the physics behind these quantities, including fundamental interrelationships. This is reminiscent of the relationships within the Dirac Large Numbers Hypothesis 9 (LNH). In part, the LNH states that large dimensionless quantities like the Eddington numbers at the heart of the LNH should be related to each other through simple algebraic relations with exponents of order unity. Then cosmological quantities got related to microscopic physics, and an evolving cosmology was the basis for discussing temporal variation of fundamental constants. Similarly, Planck quantities can be 'derived' and linked through an underlying physical argument, speculative as it may be, that is plausible and justifiable within a future encompassing theory.
A physical scaling argument for obtaining the Planck mass, other than through dimensional analysis arguments, was presented by de Sabbata 10 in 1994. The argument involves analyzing a superstring's tension, which is essentially the energy (or mass) per unit length, and noting that the maximum mass will occur for a string with length equal to the Planck length and at the point where symmetry breaking occurs. The tension in this regime is found to be .
, / max 2 Then, multiplying this by a length equal to the Planck length, which is thought to be the distance scale at which space-time becomes discontinuous, yields the Planck mass: with the speed of light being to the fifth power under the radical if one is interested in the Planck energy.
In addition to Planck mass and Planck length, we also have the Planck time, which some argue is the smallest meaningful interval of time. Here too the value is typically arrived at via dimensional analysis. However, it may also be synthesized from quantum field theory arguments and other approaches. For instance, it has also been pointed out 11 where n is the normal quantum number and Q is the torsion tensor (and dA is thus a two-form).
Hence, n = 1 corresponds to the minimum possible unit of time and happens to be equal to the Planck time. It follows that there is a limiting 'Planck frequency' defined by which is also important in realizing an ad hoc factor in renormalization in quantum field theories; with a value on the order of 10 48 Hz, it is indeed very large, as required in order for the calculations to converge. See Horowitz and Wald for details of these arguments. One could potentially interpret this quantity as the highest physically possible rate of information flow for which bits may be discerned by a measuring device.

Other representative derived quantities
In a very clear and concise paper, Shalyt-Margolin and Tregubovich 12 derive the fundamental length, time, and inverse temperature, all while generalizing the uncertainty principle for energy and inverse temperature to extremely high-energy regimes. They begin with the generalized form of the position-time uncertainty principle, where Veneziano's limit 13 is employed: They then point out that Adler and Santiago showed 14 in 1999 that the constant term is equal to unity, allowing the equation to be cast in a simple quadratic form: After noting that setting the discriminant in the previous equation equal to zero (13) to obtain the minimum value of the position uncertainty such that it still makes sense to talk about a momentum uncertainty, one finds that . Furthermore, one may also divide the starting equation by the speed of light c and employ the same techniques to find that the minimum sensible time is . The authors also define the Planck momentum They then consider the uncertainty principle between energy and inverse temperature and note that it must be generalized at high energies into the form where for "dimension and symmetry reasons," which can be inverted to yield the maximum sensible temperature: They give an interpretation of the Planck temperature as the regime in which the deformed density matrix ( ) τ ρ (where τ is a parameter describing the deformation effects caused by curved space-time) must be considered to accurately describe thermodynamics in general relativistic space-time instead of the normal density matrix The relation between Planck-scale gravity and Planck-scale thermodynamics has special significance in the context of quantum gravity theories and also some recent work on the possible emergence of gravitational physics from an underlying statistical physics and thermodynamics associated with the space-time 15 . This is the motivation for our treating temperature as a fundamental quantity, and not just energy scaled by the Boltzmann constant.
Indeed, even in classical physics temperature signified both average energy as well as fluctuations, which assume special significance in the link between gravity and statistical mechanics.
A minimum sensible temperature is suggested by de Sabbata as arising from the existence of a minimum sensible acceleration, which also appears in MOND theories 16 and variations 17 thereof. He hypothesizes that the minimum sensible acceleration is given by ,  (22) we have the following for the "Planck number of moles": The important notion of Planck entropy is discussed in the next section. The Planck units are summarized in Table 1, along with representative references.

Discussion
The quantities listed in Table 1 have often been considered in terms of their potential for establishing a natural and unambiguous system of units that derive from the fundamental structure and laws of the physical universe. The first three of the base units, the Planck length,  average of temperature of the universe, and then there will be a dependence on H R ). Thus, we will not further consider the minimum temperature unit in our analysis; similarly, we will also not further consider the minimum energy, mass, or momentum.
Thereafter, in Table 1 η ≈ The "Planck number of moles" is one step removed from this scenario because it involves the value of the ideal gas constant, R.
Most of the rest of the entries in Table 1 are Planckian analogs of electromagnetic units and quantities involving eg., voltage, capacitance and field strengths. We note, however, that one could potentially derive many new quantities involving these concepts, for instance a current of one electron charge per Planck time. However, a current of 1 A would consist of of these units, so it is hard to see what the motivation for derivation and use of such a quantity might be. 62 10 16 . 1 ×

Summary and future potential
We have examined several of the more well known Planck quantities and have arrived at some that are little discussed if not unknown in the literature. We have summarized how others have arrived at alternative derivations for many of these, showing that it is possible to synthesize these quantities via methods other than dimensional analysis. The results were tabulated into an extensive listing of the Planck units and quantities, as per Table 1.
We have also examined the Planck quantities in terms of their potential use for a unified system of natural units and found that many, but not all, of the quantities that we have examined can be classified as either base units or derived units for that purpose. Of particular note is our discussion of Planck temperature as a fundamental unit, and the Boltzmann constant as the Planck entropy, motivated by past and recent work on the connections between statistical mechanics, thermodynamics, gravity, and the physics of space-time. We acknowledge that some of the Planck quantity analogs presented here do not have a direct physical motivation underlying their synthesis, e.g., the electromagnetic analogs fall especially in this category.
However, the interesting exploratory nature of developments in this area, and the general motivation that a new metrological scale would enjoin, nevertheless make it useful to test the limits of the symmetries between the Planck units and the existing classes of anthropocentric ones.