Why are complex numbers important in physics?

[Jamma]

Sorry if this is a question that has been asked before but I've been browsing and not found anything great.

What, specifically, is the advantage of using the field of complex numbers over simply R^2 (the real plane).

I know that neater/less notation may sometimes help, but this would hardly be motivating to pursue the huge area of complex numbers.

I also know that holomorphic functions have very nice properties such as ease of finding the integral around a loop and that holomorphic functions satisfy the Laplace equation. I can see that this will be useful in physics for when all your functions satisfy the Laplace equation.

Basically, are these the only things which make the complex numbers more desirable than just using the real plane? Examples of such things would be nice (but be gentle, I'm a mathematician, not a physicist ;D). Also, aren't they a bit restrictive in the sense that they are only 2-dimensional... can they be used for example in steady fluid flow for 3-dimensions which would be much more complete than having only 2?

I'm also interested in looking at the mathematics of string theory; why do complex numbers have importance here?

 

[mathman]

One place (out of many) where complex numbers come into play is in describing electromagnetic radiation. The radiation is described by complex functions. Whe waves are mixed, the arithmetic is that of complex numbers. Quantum theory descriptions are also using complex valued wave functions.

 

[Jamma]

Thanks for the response, but I know that complex numbers are used in a lot of areas, such as these, in physics. But what I'm really after is why they are; as opposed to numbers from the real plane. Are the points I listed the main/only reasons?

 

[diazona]

I suppose that since imaginary numbers are, well, imaginary, it should technically be possible to formulate physics in terms of real numbers only. But I wouldn't want to work without tools like the complex exponential. Basically, having access to complex numbers makes a lot of calculations easier.

Incidentally, at least I don't think of a complex number as 2-dimensional - as I'm sure you know, the mathematics of complex numbers is slightly different than that of 2-component vectors. So we couldn't just replace all complex numbers with (x, y) ordered pairs without altering the rules of how those ordered pairs get multiplied etc.

 

[Jamma]

Yes, I know this. But as you said, we could introduce arithmetic to these ordered pairs that matches that of complex numbers.

I mean, obviously the reason complex numbers are used because they are easier to deal with; this is inevitable since we can replicate situations such as the functions and relations with these ordered pairs.

The question, basically, is concerning which of these nice properties of complex numbers are most important to aiding to their simplicity for use in certain areas of physics? Are there any other things than those I mentioned above? Why are these things important in areas of physics (I know about steady flow satisfying Laplace equation etc. more complicated examples like why they are useful in string theory would be nice).

 

[Born2bwire]

Compactness of notation, the ease of combining signals of different phase and frequency, the ease in using them with powers and roots, the ability to achieve closed form solutions or approximations by integrating in the complex plane (not to mention the use of things like Jordan's lemma that can allow us to change the path of integration to avoid poles or to make in easier path).

The fact is, I wouldn't want to attempt any of the stuff that I do without using complex numbers.

 

[Jamma]

Ok, thanks. Where exactly would you use complex integration; to find work done or flux? Sorry I'm not a physicist :(

Also, as I said before, what do you do when going into 3 dimensions? I.e. for stable fluid flow; we have nice answers for complex numbers since they represent two dimensional space nicely. But if we introduced a 3rd dimension, how would we proceed? This isn't even to mention the vast number of dimensions you get in other theories. I know that you can just relabel each one as a complex dimension here, but for which reason here would this be useful?

I also understand they are often used half way through the stage of a problem such as finding the solutions to PDEs or in Fourier transforms and then the information will often eventually yield a real number as a result. I understand however that complex numbers aren't just used as an intermediate stage to problems in physics and often have meaning when the final answers aren't real.

 

[Monocles]

Well, for example, some integrals (which of course appear all over the place in physics) are intractable using normal methods, but very easy using calculus of residues, where you can calculate the integral by finding the derivative of a related function (for example).

 

[johnsmi]

One place (out of many) where complex numbers come into play is in describing electromagnetic radiation. The radiation is described by complex functions. Whe waves are mixed, the arithmetic is that of complex numbers. Quantum theory descriptions are also using complex valued wave functions.

Yeah, butthere they are justfor comfort the phasor itself is not the final answer you always multiple by e^j$$\omega$$t and take the real part of all that so it's still just for convenience purposes. Am i wrong?

 

[euquila]

complex numbers crop up a lot in control theory =)
to answer your question... tbh, I always thought that the complex numbers arose from the fact that we needed to include a symbol "i" to represent even roots of a negative number. We conveniently--no, quite necessarily--represent i as 1 angle 90 and can draw meaning out of this 2D space (or higher spaces if you like) depending on our problem. It is necessary to employ this numbering landscape using some syntax or another and I think z = a + ib or e^(jwt) are both great and convenient!

The question of convenience vs necessity is the question of how much time/energy is worth to you.