There's a lot of quibbling in the literature about the use of complex numbers -- especially, the imaginary component -- in the study of quantics. But it's easy to understand this.

For part of mechanics (hamonic oscillator) and all of optics and electromagnetics, we need the concept of wave. But 3-D wave-in-time has a 2D-projection as rotation on a circle. The math for this (as in AC electricity) is the socalled complex number or vector-of-reals, which (in unit form) is that circle-rotator. Note also the basic coordinate-transformation matrix:

                 ---                  ---
                 |   cos q     - sin q  |
                 |                      |
                 |__ sin q       cos q__|

Its two eigenvalues are complex functions, as shown below. Hence, it should be no mystery that these numbers provide the math for waves. However, for standard measurement, need a scalar-real, so have "Pythagorean"-reduction.

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Given matrix, M:

              |a11   a12|
              |        |
              |a21   a22|
              |__     _|

and identity matrix, I:

              |1   0|
              |     |
              |0   1|
              |__  _|

then we obtain matrix, M - lI:

              |a11 - l   a12|
              |            |
              |a21   a22 - l|
              |__         _|

THEOREM: Scalar l is an eigenvalue of matrix M iff matrix M - lI is singular. That is, if its characteristic polynomial equation has the form: l² - (a₁₁ + a₂₂ )l + (a₁₁a₂₂ - a₁₂a₂₁ .) = 0.

Let M be the matrix for rotating coordinates by angle q:

              |cos q   sin q|
              |             |
              |- sin q cos q|
              |__         __|

Here, a₁₁ = cos q; a₂₁= sin q; a₂₁ = - sin q; a₂₂= cos q.

Substituting these values in the characteristic polynomial equation, l² - (a₁₁ + a₂₂) l + (a₁₁a₂₂ - a₁₂a₂₁) = 0, we have: l² - (2 cos q) l + (cos² q + sin² q) = l² - (2 cos q )l + 1 = 0.

This quadratic equation has the two distinct solutions: l = cos q + i sin q , and l = cos q - i sin q , for i = (- 1)^1/2 -- the two eigenvalues of the ROTATION MATRIX.

These complex functions are part of "Euler's Formula": eⁱ q = cos q + sin q ; e^{- iq} = cos q - sin q. And these apparently were derived from the MacLaurin Series for e^x, for cos q, and for sin q. (At least, this is the usual presentation of Euler's Formula in the literature.) The above derivation shows that Euler's Formula is implicit in the rotation matrix.

More. If, in rotational eigenvalue l = cos q + i sin q, we successively substitute q = 0, p/2, p, 3p/2, the four cardinal points of the compass become the rectangular axes for the complex-plane. (W. R. Hamilton (1805-1865) knew that the complex function is the 2-D rotation operator, motivating his research resulting in the quaternion 3-D rotation operator.) This result is also implicit in the rotation matrix.

And this invokes an enlightening connection with Multivector Theory (a.k.a. Geometric Algebra, Clifford Algebra) via the form of the SPINOR, which ROTATES A VECTOR. (The "complex number" is the simplest form of the spinor.)

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There's also much quibbling about the use of probability amplitudes to obtain a probability measure. But this, also, is easy to understand.

To match the need for a vector to model the wave but reduced to a scalar measure, we need a vector structure that reduces to the scalar probability measure. This vector structure is the "probability amplitude". That is, we have the homology:

	probability amplitude : probability :: wave model : measure

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In recent decades, there is quibbling about the need for "gauge theory". But this, again, is easy to understood.

The trig transformation of these wave model is invariant under multiplication by a factor -- the phase factor. This gives rise to the necessity of considering "gauge fields", which should be called "phase fields".

Dig? Or redig?