But I think I understand DGMP's question now... I would reformulate it - you ask: "Is there any part of physics that can be done only with complex numbers, without any possible replacement (even if it would take to make calculations harder)?"

Well... I'm not really sure. There are many fields of physics using them for comfort (that means that you can avoid them somehow) but there are also many that were derived only through complex algebra and by complex algebra we need to use them on.

Hm, if this is the question I think the answer is that no, no such part of physics exists. Here is my reasoning, which may very well be wrong.

Of course

**i** has no physical meaning, since the square root of -1 has no physical meaning. It is then prudent to ask what we mean by "multiplication". We mean a system of scaling and rotating numbers (or vectors, rather) in a plane. We can say that multiplying by a positive number rotates 360 degrees and scales by some factor. Thus, the product going to face in the same direction as it did before the operation.

A negative number would mean rotation of 180 degrees in the plane and some scaling. Thus, the product is in the opposite direction (positive by negative=negative), while two rotations of 180 degrees would bring it back to the original direction (negative by negative = positive).

Now we can think of some rotation that, when applied twice will bring you to -1 in the plane. That is, some number when multiplied by itself will bring you to negative one, or the square root of negative one. This rotation would have to be 90 degrees, because we want two of these rotations to sum to 180 degrees.

Thinking of it like this, complex numbers represent the rotation, magnitude, and other properties of angles. I think one could avoid using complex numbers when you think of it like this. It of course makes calculations more cumbersome and harder, which is why we use

**i** to begin with.

Just my thoughts, I haven't really ever studied anything like this.