# Physics math without $\sqrt{-1}$

The use of imaginary and complex values comes up in many physics/engineering derivations.

My question is

Is it making the process of derivation easier or is it essential without which it would be impossible to derive some results.

1.It doesn't look like it is mandatory for a Newtonian to gen relativistic results and electrodynamics..

2.Can we say the same thing about quantum mechanics either way for sure?

Could this be a difference in quantum mechanics over the classical picture?

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– Qmechanic♦ Jun 21 '11 at 16:06

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Complex numbers are essential to quantum theory. You can go far with a pair of real numbers, but there's an abstract structure to complex numbers that just seems to be built in to reality. There's a paper or two by David Finkelstein in the 1960s that go into this (my notes are lost or hiding deeply). – DarenW Jun 22 '11 at 5:54

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It bothers me to see sqrt(-1) as a definition of i; it's devoid of operational meaning. I'd rather put it as i^2 = -1. Then i looks like any operator that when applied twice makes some vector point the opposite way, for example, a 90 degree rotation. – DarenW Jun 22 '11 at 5:56

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From a purely mathematical point of view there are many very difficult integrals that can be solved exactly by analytic extension into the complex plane and application of the method of residues. Such integrals appear in many fields of physics. – dmckee♦ Jun 22 '11 at 15:39

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Very often, the shortest route between two real facts is a complex path. – Emilio Pisanty Nov 3 '12 at 2:10
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## 3 Answers

The use of complex numbers is never really essential, but if applicable it is almost always more convenient than the equivalent representation in a 2d real vector space (in fact, one typically learns the formal properties of complex number manipulations by their effect on $(a,b) = a+ib.$)

You mention that complex numbers don't seem necessary for classical electrodynamics, and I agree -- however I can't imagine any clear-minded person forgoing their use. In fact it is in classical E&M that I think complex numbers really exhibit their gracefulness in the description of physical phenomena.

Likewise, as lurscher has mentioned, there are formulations of QM that avoid explicit reference to complex numbers -- they are equivalent mathematical representations, but the manipulations have an added degree of bookkeeping that we had already built into complex numbers.

And that's the rub. Complex numbers are a tool for describing a theory, not a property of the theory itself. Which is to say that they can not be the fundamental difference between classical and quantum mechanics. The real origin of the difference is the non-commutative nature of measurement in QM. Now this is a property that can be captured by all kinds of beasts -- even real-valued matrices.

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also on the same note, you have all the other groups with a higher-than-scalar representation. complex numbers is just the first layer of abstraction you run into.. the groups express symmetries that seem to exist in nature but like you write, if you chose to call them complex numbers, U(1) or simply a system of linked scalar equations it's really up to you. its just a math engine trying to represent experiments. – Bjorn Wesen Jun 21 '11 at 20:03
I don't think it's fair to say that one can always excise complex numbers by implementing a higher degree of book-keeping. Whether one's formulation of QM is the ordinary one or has some two-component wavefunction with an operator with a square of negative identity, one is using the structure of complex numbers, which is the truly mathematically essential part. The rest is just notation. – Stan Liou Nov 3 '12 at 0:07

about 2. check this question about an alternative formalism for Quantum Mechanics with equations where only real probability densities and currents appear. The relevant wikipedia article is this one about Madelung equations.

I don't know any attempts to extend the same to QFT. Since complex residues are the butter and bread of most Feynmann loop diagrams, i would doubt it would be easy, or rewarding

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I agree with @lurscher that it wouldn't be easy or rewarding, and also with what I think is his implied suggestion that it would still be theoretically possible. – Ben Hocking Jun 21 '11 at 17:57
Meromorphic functions of complex theory are just vector fields of point sources in vector calculus.  The residue theorem is just Stokes' theorem.  I don't imagine translating from complex numbers to a real, 2d vector space would be that big a deal. – Muphrid Nov 3 '12 at 4:02

Quantum mechanics necessarily needs complex numbers. Replacing complex numbers with real number is possible, but that would hide a lot of structure and is purely a mathematical trick.

the Feynman amplitude being $e^{i S}$, or the commutation relation indicates that something deep is going on and that can't be understood by treating them as 2 real numbers.

Feynman used to talk about quantum mechanics as a complex extension to classical probability theory.

see

spacetime approach to non-relativistic quantum mechanics and

The Concept of Probability in Quantum Mechanics ( http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.bsmsp/1200500252&page=record)

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