Dirac's Algebraic Hack

\[ E^2 = p^2c^2 + m^2c^4 \]

The Relativistic Energy Equation

Einstein's energy-momentum relation, showing energy as a square.

\[ c^2 = a^2 + b^2 \]

Reduced Algebraic Form

Stripping away the constants to see the fundamental geometric structure.

E = \alpha pc + \beta mc^2

The Relationship Dirac Wanted

Dirac needed a linear equation (not squared) to work with quantum wave functions.

E^2 = (\alpha pc + \beta mc^2)^2

The Desired Equation Squared

We square Dirac's linear guess to see if it can match Einstein's original equation.

\[ c^2 = (a + b)^2 \]

Dirac's Desired Form

\[ c^2 = a^2 + b^2 \]

Einstein's Given Form

The Hack: How can \( (a+b)^2 \) ever equal \( a^2 + b^2 \)? Only if \( ab + ba = 0 \).