# Complex numbers sometimes *Required* in Classical Physics?

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1. Nov 22, 2015

### referframe

In general, one thinks of complex numbers as being absolutely required in Quantum Physics but as being optional in Classical Physics. But what about modern classical electromagnetic field theory (gauge theory) in which the electromagnetic field is coupled to the field of charged particles by essentially adding the electromagnetic potential to the (derivative of) complex phase of the field of charged particles (co-variant derivative)? It seems that this is a case in Classical Physics in which complex numbers are required. Comments?

2. Nov 22, 2015

Staff Emeritus
Complex numbers are never required. Anything with complex numbers can be solved with coupled real equations. It may be a lot more work.

3. Nov 22, 2015

### referframe

Well, yes. Complex numbers can always be replaced by any mathematical, 2-dimensional, equivalent of complex numbers. I was referring to the fact that most physical values in classical physics are real-valued and 1-dimensional.

4. Nov 22, 2015

### nasu

1-dimensional? You mean scalar?
There are some scalar quantities but a lot of vector and tensor quantities as well.

5. Nov 22, 2015

### referframe

It seems that Classical Physics has done just fine using real numbers only, be they scalars, vectors or tensors. On the other hand, Quantum Physics needs complex numbers. However, classical field theory that references electromagnetic potentials instead of electromagnetic fields needs the field for the charged particles to be complex. My point is that this seems to be at lease one case in which Classical Physics really does need complex numbers.

6. Nov 23, 2015

Staff Emeritus
You keep saying this. It's not true. You can do QM with purely real numbers. It would be an unholy mess, turning the Schroedinger Equation into 4 coupled differential equations, but you can do it.

7. Nov 23, 2015

### Krylov

I wonder to what degree the fact that $\mathbb{R}$ is not algebraically closed necessitates the use of complex numbers in dynamical problems from physics. For example, every bounded linear operator on a complex Banach space has a non-empty spectrum. This is clearly not the case when we work on a real space instead. For this reason, nobody does operator theory on real spaces, and operator theory in turn is intimately related to physics.

EDIT: So when someone writes
I have my doubts, but this depends on what that person means exactly. Yes, the coupled evolution equations may be real, but the spectrum still contains points with a non-zero imaginary part.

Last edited: Nov 23, 2015
8. Nov 23, 2015

### haushofer

Doesn't e.g. the Hermicity of the operators p and x imply that [x,p] gives imaginary eigenvalues? How would you describe that with only real numbers? I must say this confuses me a bit :P

9. Nov 23, 2015

### haushofer

With other words, what is exactly meant by the statement 'we don't need complex numbers'? 'We can replace all complex numbers by 2-tuples of real numbers on which a product is defined which resembles the way we multiply complex numbers'? Isn't this just the statement that such a space R2 with such a product is isomorphic to the complex plane C?

10. Nov 23, 2015

Staff Emeritus
You can replace complex numbers with pairs of real numbers and the Cauchy-Riemann conditions. Or rather, you can take four ugly equations involving pairs of real numbers and replace them with a single equation using complex numbers. The math is simpler and prettier. But complex numbers are not required. We use them not because they are necessary but because they are very, very convenient.

11. Nov 23, 2015

### Krylov

Except that yes, they are.

As @haushofer already pointed out, you can call a dog a canine, but that doesn't mean it's not a dog anymore.

12. Nov 23, 2015

### Krylov

Saying we don't need complex numbers is similar to saying we don't need fractions because each fraction can be represented as a pair of integers. Very well, good luck with that.

13. Nov 23, 2015

### referframe

In QM, in the Schrodinger Equation, we must choose between one equation using complex numbers or multiple coupled equations using only real numbers. But there are many physical systems in classical physics that can be described by just one equation using all real numbers. However, it seems that there is at least one exception to this rule: Classical electromagnetic gauge field theory in which the electromagnetic 4-potential is coupled to the field of charged particles via the phase of this complex-valued field - no complex phase means nothing to couple with.

14. Nov 23, 2015

### nasu

But Schrodinger equation is just one way to do QM. You don't have to use it, you choose this representation.

15. Nov 23, 2015

### Staff: Mentor

Could you clarify what you mean by other ways of doing QM?

16. Nov 23, 2015

### nasu

Well, there is the matrix formulation of QM which does not use a "wave" to represent states. So there is no "wave" equation.
It does use complex quantities, though.

I just wanted to show that trying to make a general point about QM based on just Schrodinger equation is not a very strong argument.
It is not that complex number or matrices are intrinsic to QM. You can describe the basic principles and concepts without as Feynman famously does in his books. Only when it comes to calculations we need some tools and the most efficient involve complex numbers.

17. Nov 24, 2015

### haushofer

I'd then say we need a complex structure for QM. That you can rewrite this in something real which is isomorphic to that structure is kind of trivial,isn't it? but to me it doesn't make it less complex.

18. Nov 25, 2015

### mpresic

The solution to Euler's equations for the orientation of a Free anaxisymmetric body or orientation for a spherical top with one point fixed under gravity requires Elliptic Functions (Jacobi or Weierstrass). These are doubly periodic functions of a complex variable. It is a mistake to think classical mechanics does not need complex variables.

19. Nov 25, 2015

### nasu

What do you mean by "complex structure"? Do you have a specific meaning for this? Are there other structures besides the "complex"?
What kind of "structure" has the theory of elasticity of solids?
Or the "theory" of motion in 2-D?

20. Nov 25, 2015

### referframe

Interesting. I will check that out. Thanks.