If we know the appropriate physical physical units for the observable \(\Omega\) then we have made significant progress in understanding its behaviour. The Buckingham-Pi theorem tells us how many physical units we need and what we can do with free parameters if we happen to have more physical measurements than what is necessary to model the behaviour of \(\Omega\).

The theorem and its proof:

Let’s suppose we have succeeded in identifying an equation that describes the evolution of \(\Omega\) as a function of \(N\) physical units. Then we have:

\begin{equation} \exists \alpha_i \in \mathbb{Z}, \Omega = \prod_{i=1}^N \omega_i^{\alpha_i} \end{equation}

where \(\omega_i\) are our physical units(ex. Joules).

Now, we also know that each unit of \(\Omega\) may be expressed in terms of the fundamental units \(U = \{u_i\}_{i=1}^k\) so we have:

\begin{equation} \exists \beta_i \in \mathbb{Z}, \Omega = \prod_{i=1}^k u_i^{\beta_i} \end{equation}

and for each physical unit,

\begin{equation} \exists \lambda_{i,j} \in \mathbb{Z}, \omega_j^{\alpha_j} = \prod_{i=1}^k u_i^{\lambda_{i,j} \cdot \alpha_j} \end{equation}

This allows a representation in terms of the system of equations:

\begin{equation} \sum_{j=1}^N \lambda_{i,j} \cdot \alpha_j = \beta_i \end{equation}

so we have

\begin{equation} \Lambda \cdot \vec{\alpha} = \vec{\beta} \end{equation}

Now, in order to translate between different civilisations which might have different metrology institutes, it makes sense to model the problem using dimensionless parameters. This amounts to finding \(\vec{\Delta \alpha}\) such that:

\begin{equation} \Lambda \cdot (\vec{\alpha} + \vec{\Delta \alpha}) = \Lambda \cdot \vec{\alpha} = \vec{\beta} \end{equation}

where the set of all possible \(\vec{\Delta \alpha}\) defines the null-space of \(\Lambda\).

From the rank-nullity theorem we may deduce that:

\begin{equation} \text{dim}(\text{Null}(\Lambda)) = N-k \end{equation}

is the number of dimensionless parameters with which we may describe our physical system.

A useful intuition:

A useful interpretation of the null-space is that it measure the number of physical units that are not dimensionally independent. This number is exactly \(N-k\) and we may use these free parameters to model the physical problem in a dimensionless manner. The reader might also wonder why all units have integer-order dimensions.

The reason for this is that the most fundamental interactions in physics are well-approximated by integer-order partial differential equations such as the Dyson–Schwinger equations.