Why are amplitudes complex?
The reason amplitudes are complex is because a fundamental role is played in quantum theory by symmetry, and the imaginary number in an exponential (e^it) is a very algebraically convenient way to capture an important symmetry.<p>In quantum field theory, the lie algebra elements of certain lie groups (U(1), SU(2), and SU(3)) play a fundamental role.  For example, the 8 quarks correspond to 8 lie algebra elements of SU(3).<p>A lie group is a continuous symmetry group (such as the rotation of a circle which leave it unchanged, as opposed to a finite group like the rotations of a square) and the lie algebra are the &quot;generators&quot; of lie groups in the same sense that finite groups can be generated, such as the symmetry group of rotations of a square can be generated by a single rotation of 90 degree&#x27;s.<p>Here is the real kicker, the imaginary number <i>i</i> is actually equivalent in a very strong sense to the lie algebra element of O(1) (which is just the rotations of a circle) and can also be represented by the matrix:<p>0 -1<p>1  0<p>as can be proven by taking the powers of this matrix and the powers of <i>i</i> and simply checking everything.