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 element in the real line minus its origin, $\mathbb{R}-\{0\}$ , being a torsor over the multiplicative group $\mathbb{R}^\times$ of real numbers.

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 $\mathbb{R}-\{0\}$ and declaring this to be 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).

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 are instead quotients of physical units. 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 expressed in physical units of charge (such as coulomb?s), $\hbar$ is Planck's constant etc. The resulting quotient is then independent of any choices and is hence a real number characterizing nature independently of any conventions about how to parameterize it.

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.

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

• speed of light

• Planck's constant

• gravitational constant

• Compton wavelength

• Schwarzschild radius

• Planck mass

References

Discussion in the context of philosophy of science includes

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