nLab physical unit

under construction

Idea

When formulating a theory of physics in terms of mathematics one typically models the range of certain physical quantities by torsors over some group of transformations.

For instance a wavelength would be identified as an element in the positive real numbers, $\mathbb{R}_{\gt0}$ , being a torsor over the multiplicative group $\mathbb{R}_{\gt0}^\times$ of positive reals.

In order for the “coordination” of the mathematical theory with physical experiment to take place, one needs to choose an identification of this abstract torsor with the (idealized) one that it is supposed to model in nature. Such a choice is equivalent to a choice of unit (in the mathematical sense), hence a choice of element of the torsor. In this context this is then a physical unit.

For instance picking an element in $\mathbb{R}_{\gt0}$ and declaring this to be the length of the path travelled by light in a vacuum in 1/299 792 458 second means defining a physical unit of length (in this example: of the meter).

Notice that choice of unit is also called choice of gauge. This is indeed the same “gauge” as in “gauge theory”, as it is how (Weyl 23) introduced the concept of gauge theory: as a theory in which the choice of unit of length may change along paths in space.

Relation to physical constants

Physical units are often called physical constants. But by definition physical units are arbitrary choices made in the desciption of a physical system. Of course once made, one wants to keep these choices constant, such as to be useful.

The actual constants of nature, when they are “dimensionful”, are instead elements of these abstract torsors. For example, the Planck length belongs to the torsor of length, a torsor over $\mathbb{R}_{\gt0}^\times$. We can describe it by its ratio to another element of the same torsor, such as the meter — the Planck length is approximately $1.616 \cdot 10^{-35} \mathrm{m}$ — but the number appearing in this statement is completely driven by our arbitrary choice of the meter.

An actual constant of nature corresponds to a specific, non-arbitrary real number only when it is a dimensionless quotient of physical quantities. For instance the fine structure constant is the quotient

$$\alpha \coloneqq \frac{e^2}{ (4 \pi \epsilon_0) \hbar c} \in \mathbb{R}$$

where $e$ is the electric charge of the electron, $\hbar$ is Planck's constant, etc. These quantities are such that their dimensionalities cancel out, meaning their quotient is a member not of an abstract torsor but of $\mathbb{R}$ itself, and hence a real number characterizing nature independently of any conventions about how to parameterize it.

For computing such a quotient concretely, one expresses each of the constituent quantities as some real multiple of an appropriate physical unit: e.g. $e$ as a multiple of the coulomb?, $c$ as a multiple of the meter per second, etc. If the units are chosen such that their quotient is unity, then the quotient of the numbers is the actual physical constant.

Units of length in Lagrangian field theory

under construction

Let $\Sigma \simeq \mathbb{R}^{p,1}$ be Minkowski spacetime and let $E \overset{fb}{\to} \Sigma$ be a fiber bundle thought of as a field bundle. Write $\{\phi^a\}$ for local coordinates on the typical fiber of this bundle.

The total space of the corresponding jet bundle $J^\infty_\Sigma(E) \overset{jb^\infty}{\to} \Sigma$ carries an action

$$sc \;\colon\; \mathbb{R}^\times \times J^\infty_\Sigma(E) \longrightarrow J^\infty_\Sigma(E)$$

of the multiplicative group of units $\mathbb{R}^\times$ of the real numbers, given on the induced jet coordinates by

$$\begin{aligned} x^\mu & \mapsto r x^\mu \\ \phi^a & \mapsto \phi^a \\ \phi^a_{,\mu_1, \cdots \mu_k} & \mapsto r^{-k} \phi^a_{,\mu_1 \cdots \mu_k} \end{aligned} \,.$$

Let then

$$\mathbf{L}_{(-)} \;\colon\; \mathbb{R}^n \longrightarrow \mathbf{\Omega}^{p+1}(J^\infty_\Sigma(E))$$

be a smoothly $n$-parameterized collection of Lagrangian densities, equipped with an $R^\times$-action

$$scp \;\colon\; \mathbb{R}^\times \times \mathbb{R}^n \longrightarrow \mathbb{R}^n$$

on $\mathbb{R}^n$.

Observe that the Euler-Lagrange equations induced by a Lagrangian density $\mathbf{L}$ equal those induced by the rescaled Lagrangian $r \mathbf{L}$, and that the presymplectic current $\Omega_{BFV}$ induced by $\mathbf{L}$ scales linearly with $r$ itself. Upon quantization, this rescaling of $\Omega_{BFV}$ may be absorbed in Planck's constant. In conclusion, as long as Lagrangian densities scale homogeneously the rescaled Lagrangian induces the same physics.

Hence we require that the combined scaling action of $\mathbb{R}^\times$ on $J^\infty_\Sigma(E)$ via $sc$ and on the parameters in $\mathbb{R}^n$ via $scp$ is homogeneous on $\mathbf{L}$ in that there exists $dim \in \mathbb{Z}$ such that for every $r \in \mathbb{R}^\times$ we have

$$sc_r^\ast \mathbf{L}_{( scp(-))} = r^{dim} \mathbf{L}_{(-)} \,.$$

Then a parameter $a \colon \mathbb{R}^n \to \mathbb{R}$ such that there exists $w \in \mathbb{Z}$ with

$$scp_r^\ast a = r^{w} a$$

is said to have dimension $[length]^{w}$.

For example the Lagrangian density for the free scalar field

$$\mathbf{L}_{(-)} \;\colon\; \mathbb{R}^1 \longrightarrow \mathbf{\Omega}^{p+1}(J^\infty_\Sigma(\Sigma \times \mathbb{R})$$

given by

$$\mathbf{L}_{m} \;\coloneqq\; \tfrac{1}{2} \left( \eta^{\mu \nu}\phi_{,\mu} \phi_{,\nu} - m^2 \phi^2 \right) dvol$$

is parameterized by the mass $m$. For the Lagrangian to scale homogenously with $r^{p-1}$ the mass parameter has to have dimension $[length]^{-1}$. To indicate this action one writes the mass in the combination $m c / \hbar$, called the inverse Compton wavelength, so that the homogenously scaling collection of Lagrangians is

$$\mathbf{L}_{m c / \hbar} \;\coloneqq\; \tfrac{1}{2} \left( \eta^{\mu \nu}\phi_{,\mu} \phi_{,\nu} - \left( \tfrac{m c}{\hbar} \right) \phi^2 \right) dvol$$

(…)

Examples

• meter, femtometer

• electronvolt, MeV, GeV, TeV

• Boltzmann constant

fundamental scales (fundamental/natural physical units)

• speed of light$\,$ $c$

• Planck's constant$\,$ $\hbar$

• gravitational constant$\,$ $G_N = \kappa^2/8\pi$

• Planck scale

  • Planck length$\,$ $\ell_p = \sqrt{ \hbar G / c^3 }$

  • Planck mass$\,$ $m_p = \sqrt{\hbar c / G}$

• depending on a given mass $m$

  • Compton wavelength$\,$ $\lambda_m = \hbar / m c$

  • Schwarzschild radius$\,$ $2 m G / c^2$

  • depending also on a given charge $e$

    • Schwinger limit$\,$ $E_{crit} = m^2 c^3 / e \hbar$

• GUT scale

• string scale

  • string tension$\,$ $T = 1/(2\pi \alpha^\prime)$

  • string length scale$\,$ $\ell_s = \sqrt{\alpha'}$

  • string coupling constant$\,$ $g_s = e^\lambda$

Related concepts

• comoving time
• natural units

References

For a mathematical description of physical units and the associated “physical dimensions”, including a discussion on how densities can be used to define real and complex powers of physical quantities, see

• Dmitri Pavlov, Answer to Question 402515 on MathOverflow (How do we give mathematical meaning to ‘physical dimensions’?), https://mathoverflow.net/questions/402497/how-do-we-give-mathematical-meaning-to-physical-dimensions/402515#402515.

Additional references:

• Robert Loren Jaffe, Natural Units and the Scales of Fundamental Physics, Course Notes 2007 (pdf, pdf)

Discussion in the context of philosophy of science includes

• Georg Hegel, Book I, third section, first chapter of Science of Logic