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The use of imaginary and complex values comes up in many physics and engineering derivations. I have a question about that: Is the use of complex numbers simply to make the process of derivation easier, or is it an essential ingredient, without which it would be impossible to derive some results?

I can identify two different settings where the answer may be different:

  1. It doesn't look like it is mandatory for results in Newtonian mechanics, General Relativity and classical 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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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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Emilio -- that's pure poetry! Is it a quote (if so by whom?), or your own words? –  Johannes Nov 3 '12 at 12:20
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6 Answers 6

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. –  BjornW 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
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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
    
I seem to remember that there is a vector-potential-free formulation of QED. That would probably be the best road to eliminating complex numbers, too. –  Jerry Schirmer Nov 5 '13 at 14:01
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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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Good luck trying to do AC circuit theory without complex numbers-- and this is purely classical physics/engineering. Much of my work depends on circuit theory, and the notion of complex impedance. I will not categorically state that circuit theory cannot be done purely in real numbers, but I have never seen it. Maybe examples are found in the 19th century literature. From time to time I have attempted to work simple problems using only real numbers, and have wound up quitting in disgust.

Complex arithmetic works so well, why avoid it?

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To answer part 2 of your question: For quantum mechanics, complex numbers do not only make the process of derivation easier. In fact if two real quantities are combined in QM to form a complex quantity, this is done to emphasize that these two quantities cannot be measured simultaneously, see also my answer here: http://physics.stackexchange.com/a/83219/1648.

So in that sense, complex quantities are an essential ingredient of QM (where not everything can be measured simultaneously), but not in classical mechanics (where simultaneous measurements are not an issue).

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Dear asmaier, it is usually frown upon to directly copy-paste identical answers. (The problem is if everybody start to copy-paste identical answers en mass.) In general in such situations, please consider one of the following options: (i) Delete three of your answers. (ii) Flag for duplicate posts and delete three of your answers. (iii) If you think the four posts are not duplicates, then personalize each answer to address the four different specific questions. –  Qmechanic Nov 2 '13 at 23:14
    
Dear Qmechanic, isn't is also frown upon to copy-paste identical comments? ;-) However I admit, that my answers were too similar. So I tried to follow your suggestion (iii) and personalized my answers to address the specific question in a better way. However I still believe the quote from Dirac is very relevant and important, so I will refer to it in every answer. –  asmaier Nov 3 '13 at 14:49
    
That is a great idea (FWIW, it's fine to quote it in every answer if you want, but cross linking works perfectly). As long as each answer has a modicum of uniqueness (after all, the questions are unique so the answers probably should be too), it's fine :) –  Manishearth Nov 5 '13 at 8:21
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Let's assume the basic nouns of our language to describe the physical world are the members of Lie groups. Okay, this is a pompous-sounding statement and somewhat arbitrary, but my justification is that these objects describe all the continuous symmetries there can be, and almost every clarification of physics using mathematics is done either (1) by viewing a mathematical object from a different standpoint (unification of hitherto seemingly unrelated concepts) or (2) by exploiting symmetries to reduce or get rid of the redundant complexity in a statement. In our continuous manifold descriptions of the physical World, these symmetries are all continuous. So, somewhere in that list of symmetries, we meet $U(1)$, $SU(2)$, $SO(3)$, $U(N)$ and so forth. So we would needfully be doing calculations and simplifcations with these objects when we exploit symmetries of a problem. Whether or not we choose to single out an object like:

$$\left(\begin{array}{cc}0&-1\\1&0\end{array}\right)\in U(1), SU(2), SO(3), U(N) \cdots$$

and give it a special symbol $i$ where $i^2=-1$ is a "matter of taste", so in this sense the use of complex numbers is not essential. Nonetheless, we would needfully still meet this object and ones like it and would have to handle statements involving such objects when describing physics in a continuous manifold - there's no way around this as it belongs to any full description of symmetries of the World. So in this sense, complex numbers, quaternions, octonions and so forth are all there and essential in such description. Notice that complex numbers and their algebra are wonted to almost everyone in physics, quaternions to somewhat fewer physicists and octonions not really to that many. This is simply related to how often the relevant symmetry calculations come up: almost any interesting continuous symmetry involves Lie group objects for which $i^2=-1$ and so we single these out and commit all the rules of their algebra to stop ourselves going outright spare and committed to lunatic asylums writing out their full Lie theoretical representations all the time. Singling out quaternions and doing the same saves some work, but not so much, because quaternions come up in fewer symmetries. By the time we get to octonions, the symmetries wherein they come up are quite seldom, so not that many of us are very adept with their special algebra (me included): we can do the full matrix / Lie calculations without too much pain because we don't do them that often, so we don't notice their octonionhood so readily.

Footnote: One can take "Lie Group members" and "Continuous Symmetries" to be the same by dint of:

  1. The solution to Hilbert's fifth problem by Montgomery, Gleason and Zippin i.e. we don't need the concept of manifold nor the concept of analyticity ($C^\omega$) - these "build themselves" from the basic idea of a continuous topological group;
  2. The classification of all Lie algebras by Wilhelm Killing (whose saw that he could do it, but botched the proof a little) and the great Elie Cartan - so we know what all continuous symmetries look like. Once we have classified all Lie algebras, we can find all possible Lie groups, since every Lie group has a Lie algebra, every Lie algebra can be exponentiated into a Lie group (e.g. through the matrix exponential, since every Lie algebra can be represented as a matrix Lie algebra (Ado's theorem)) and the (global-topological) relationships between Lie groups that have the same Lie algebra is also known.
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