Duality squared

Physicists tend to assign meaning to equations.  Deeper meaning like

 E=mc^2 means that mass and energy are equivalent”

You recognize Einsteins famous equation, and have maybe heard a physicist make this comparison.

Continuing, adding the momentum term (and switching to units wherein c=1):

     \[ E^2 = m^2+p^2 \]

This relativistic equation represents a huge leap forward in understanding the operation of the universe which relates energy to mass and motion in a particular way.

I wonder about any further meaning …

As it goes, I was dinking around with the notion of the deeper meaning of mathematical symbols.  Math is a language, just like spoken languages, that points to “reality” – it points to some physical thing.   For example, the notion of  2 \times 3 conveys two rows of three things each.

     \[ \bullet \bullet \times \bullet \bullet \bullet  = \begin{matrix} \bullet & \bullet   & \bullet  \\ \bullet & \bullet  & \bullet \end{matrix} \]

So, when you take the mathematical square, in some way,  the mathematical results represents a square.

     \[ 3^2 = 3\times 3 = \bullet \bullet \bullet \times \bullet \bullet \bullet  = \begin{matrix} \bullet & \bullet   & \bullet \\ \bullet & \bullet   & \bullet  \\ \bullet & \bullet  & \bullet \end{matrix} \]

Let’s go back to the relativistic equation relating energy, mass, and motion, and to give it some semblance of reality, let m=3, p=4, and E=5.

     \[ E^2 = m^2 + p^2 \]

     \[ E \cdot E = m \cdot m + p \cdot p \]

      \[ \bullet \bullet \bullet \bullet \bullet \times \bullet \bullet \bullet \bullet \bullet = \bullet \bullet \bullet \times \bullet \bullet \bullet  + \bullet \bullet \bullet \bullet \times \bullet \bullet \bullet \bullet \]

     \[ \begin{matrix} \bullet & \bullet  & \bullet & \bullet& \bullet \\ \bullet & \bullet  & \bullet & \bullet & \bullet \\ \bullet & \bullet  & \bullet & \bullet & \bullet \\ \bullet & \bullet  & \bullet & \bullet & \bullet \\ \bullet & \bullet  & \bullet & \bullet & \bullet \end{matrix} = \begin{matrix} \bullet & \bullet  & \bullet \\ \bullet & \bullet  & \bullet \\ \bullet & \bullet  & \bullet \end{matrix} + \begin{matrix} \bullet & \bullet  & \bullet & \bullet \\ \bullet & \bullet  & \bullet & \bullet \\ \bullet & \bullet  & \bullet & \bullet\\ \bullet & \bullet  & \bullet & \bullet \end{matrix} \]

I don’t know if this means anything, but I happened to notice that this relativistic equation  E^2 = m^2 + p^2 equates the product of two units of energy to the sum of two terms each of which is also a product of two units of something, either mass or momentum.

It’s like the solution to the one of the mysteries of the universe involves duality, squared.


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