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After years of using complex numbers in every kind of analysis of physical and electrical engineering problems I am starting to wonder: why is this particular algebra so effective in modelling the world? Because from a purely mathematical point of view $\mathbb{C}$ is no more than the field $\mathbb{R^2}$ with the operations given by: $$(x,y) + (a,b) = (x+a, y+b)$$ $$(x,y) * (a,b) = (x*a - y*b,y*a + x*b)$$ Is there a physical motivation for these definitions which might help explain why this particular algebra is so powerful in modelling physical systems?

PS: My friend suggests that perhaps this algebra can be derived to be the algebra that is satisfied by Fourier coefficients. And that it is some kind of implicit Fourier analysis which is actually modelling the physical systems. Can that be true?

EDIT: This question is different from the one about demystifying complex numbers. That one asks for examples for the usefulness of complex numbers. I know they are useful. I would like to know why they are useful if that is an answerable question.

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    $\begingroup$ It is primarily due to the algebraic closure of the complex numbers allowing for ‘polynomial eigenvalue problems’ of operators on $\mathbb{C}$ to always be solved completely in physical settings, but I think this would be more appropriate over at math.stackexchange as a general question about mathematics. $\endgroup$ – Alec Rhea Mar 23 '18 at 5:59
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    $\begingroup$ @FrancoisZiegler The linked question asks for examples for the usefulness of complex numbers. I know they are useful. I would like to know why they are useful if that is an answerable question. $\endgroup$ – Abhijeet Melkani Mar 23 '18 at 6:03
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    $\begingroup$ What about the geometric interpretation of complex multiplication? Multiply the moduli and add the angles. $\endgroup$ – Monroe Eskew Mar 23 '18 at 13:40
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    $\begingroup$ This makes me wonder what kinds of questions might have appeared on MathOverflow had it existed in around 500 BCE, around the time that the Pythagoreans discovered that $\sqrt{2}$ is not rational. "Why the unreasonable applicability of irrational numbers in construction/masonry?" $\endgroup$ – Lee Mosher Mar 23 '18 at 17:39
  • $\begingroup$ The quip about MO existing in 500 BCE made me laugh. Anyway, quaterions are also useful, and they are even stranger than complex numbers. It appears from Wikipedia that Lambek was a fan. $\endgroup$ – James Smith Mar 23 '18 at 18:00
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This is an interesting line of research in modern physics, whether Nature at its most fundamental level is described by real or by complex numbers. Volovik and Zubkov have written about this in Emergent Weyl fermions and the origin of $i=\sqrt{-1}$ in quantum mechanics:

Conventional quantum mechanics is described in terms of complex numbers. However, all physical quantities are real. This indicates, that the appearance of complex numbers in quantum mechanics may be the emergent phenomenon, i.e. complex numbers appear in the low energy description of the underlined high energy theory. We suggest a possible explanation of how this may occur. Namely, we consider the system of multi-component Majorana fermions. There is a natural description of this system in terms of real numbers only. In the vicinity of the topologically protected Fermi point this system is described by the effective low energy theory with Weyl fermions, described by complex numbers.

Majorana fermion: particle described by a real wave function
Weyl fermion: particle described by a complex wave function
Just as a complex number can be represented by two real numbers, a Weyl fermion can be represented by a pair of Majorana fermions. The physics question is then whether unpaired Majorana fermions appear in Nature, since this would imply the fundamental equations are real rather than complex.

So in the quantum world, an answer to the question in the OP "why are complex numbers so effective" is that Weyl fermions, rather than Majorana fermions, appear as fundamental particles.

Notice that such a "real" particle would be its own antiparticle (since the wave function of the antiparticle is obtained by complex conjugation). We do not know of fundamental particles that are their own antiparticle (perhaps the neutrino is one), but they are allowed by the mathematical structure of relativistic quantum mechanics.

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    $\begingroup$ This is some cool stuff, but how exactly does it address the OP? The question asks why complex numbers are so useful for solving physical problems as an engineer, not how they may arise as a low energy limit of some real-number based theory? $\endgroup$ – Alec Rhea Mar 23 '18 at 13:17
  • $\begingroup$ Umm... Photons are their own antiparticles. So are $Z^0$s and $\pi^0$s and maybe gravitons (if they exist). Neutrinos cannot be their own antiparticles due to mismatched chirality. $\endgroup$ – Eric Towers Mar 23 '18 at 16:28
  • $\begingroup$ excuse me, all of this refers to fermions, so bosons are excluded; if a neutrino would be a Majorana fermion then two neutrino's could annihilate each other; this process is searched for (it's called "neutrinoless double beta decay"), but not yet observed. $\endgroup$ – Carlo Beenakker Mar 23 '18 at 17:13
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I think this question can be answered without appealing to quantum mechanics (and the question is more general than quantum mechanics).

In classical physics, quantum physics, electrical engineering, chemistry, etc. you're often describing your the system at hand with differential equations. When solving differential equations, complex numbers pop up all over the place. As alluded to in the quote above, a straightforward solution to many differential equations often makes use of complex analysis, so there's no surprise that they are used and are useful if the tools we're using come from complex analysis.

However, we don't have to limit ourselves to just differential equations. Pick a random polynomial and solve for all the roots. You're far more likely to end up with complex roots than purely real roots. If we describe the world around us in mathematical terms and everything from simple polynomials to differential equations give complex answers more often than purely real answers, it not surprising (to me at least) that complex numbers are going to show up everywhere.

A more boring answer is that complex numbers usually show up whenever phase is important (quantum mechanics, wave mechanics, electrical circuits, etc.) You don't strictly need to use complex numbers, but it sure makes the notation a lot more compact and intuitive. So, we use them because they're convenient.

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  • $\begingroup$ Yes, this should be the accepted answer. $\endgroup$ – Peter Mortensen Mar 23 '18 at 17:54
  • $\begingroup$ So are you saying the introduction of the complex number is only to make the computation easier, convenient and compact, and not that it is strictly necessary? $\endgroup$ – Hans Mar 23 '18 at 19:44
  • $\begingroup$ Exactly - you could define all complex numbers as vectors that have slightly different properties when multiplying them and you'd get the same behavior, but the notation would be (in my opinion) more cumbersome and less intuitive. $\endgroup$ – S. Burt Mar 24 '18 at 23:40
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The most succinct answer to the question why complex numbers are so useful in the analysis of physical and engineering problems was given by Paul Painlevé in 1900: "between two truths of the real domain, the easiest and shortest path quite often passes through the complex domain" (usually this quote is known in Jacques Hadamard's later formulation).

But are complex numbers really essential in physics? The usual argument is based on the Heisenberg's uncertainty principle in the form $[\hat x,\hat p]=i\hbar$. In the words of Paul Dirac (The principles of quantum mechanics, §10, p.35. Stückelberg in his 1960 paper "Quantum Theory in Real Hilbert Space" also provided the similar argument):

One might think one could measure a complex dynamical variable by measuring separately its real and pure imaginary parts. But this would involve two measurements or two observations, which would be alright in classical mechanics, but would not do in quantum mechanics, where two observations in general interfere with one another - it is not in general permissible to consider that two observations can be made exactly simultaneously, and if they are made in quick succession the first will usually disturb the state of the system and introduce an indeterminacy that will affect the second.

This point of view is further extended by Chen Ning Yang in https://www.worldscientific.com/doi/abs/10.1142/9789814449021_0014 (Square root of minus one, complex phases and Erwin Schrödinger). Yang traces the entry of complex numbers into fundamental physics to Schrödinger's 1922 paper in which he had mentioned the possibility of introducing an imaginary factor into Weyl's 1918 gauge theory. The development of this idea by London, Fock and Weyl lead to the gauge theory of electromagnetism. Yang writes:

The importance of the introduction of complex amplitudes with phases into physicists' description of nature was not fully appreciated until the 1970s when two developments took place: (1) all interactions were found to be some form of gauge field; and (2) gauge fields were found to be related to the mathematical concept of fibre bundles (Wu and Yang, 1975), each fibre being a complex phase or a more general phase. With these developments there arose a basic tenet of today's physics: all fundamental forces are phase fields (Yang, 1983). Thus the almost casual introduction in 1922 by Schrödinger of the imaginary unit i has flowered into deep concepts that lie at the very foundation of our understanding of the physical world.

Although the quantum mechanics can be formulated in real Hilbert space, such a formulation is redundant and can be always reformulated in the complex Hilbert space, see https://arxiv.org/abs/1611.09029 (Quantum theory in real Hilbert space: How the complex Hilbert space structure emerges from Poincaré symmetry, by Valter Moretti and Marco Oppio).

A fascinating history of $\sqrt{-1}$ can be found in the book of Paul Nahin "An Imaginary Tale: the story of $\sqrt{-1}$ (here is a review of this book by C O'Sullivan: http://iopscience.iop.org/article/10.1088/0143-0807/20/2/013 )

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The state $|\psi\rangle$ of a single particle in basic quantum mechanics (as opposed to QFT) lives a-priori in a Hilbert space $\mathscr{H}$, and probabilities for observations on the particle correspond to possible eigenvalues of eigenstates for self-adjoint operators $\hat O\in\mathscr{H}\otimes\mathscr{H}^*$, the operator $\hat O$ corresponding to some observational apparatus.

To find these probabilities we find the eigenbasis $\{|\lambda_i\rangle\}_{i<n}\subseteq\mathscr{H}$ generated by $\hat O$ and represent $|\psi\rangle$ in this basis, then act on this representation to obtain the probability coefficients corresponding to each possible eigenstate of $\hat O$: $$\hat O|\psi\rangle=\hat O\sum_{i<n}\langle \lambda_i|\psi\rangle|\lambda_i\rangle=\sum_{i<n}\psi_i\hat O|\lambda_i\rangle=\sum_{i<n}\psi_i\lambda_i|\lambda_i\rangle.$$ The probability that we find the particle in a state corresponding to $|\lambda_i\rangle$ is $|\psi_i\lambda_i|=|\langle\lambda_i|\psi\rangle\lambda_i|$ -- the absolute value taken at the end here is what is typically meant by 'physical quantities are real' when talking about basic quantum mechanics.

The crucial step here is determining the eigenbasis $\{|\lambda_i\rangle\}_{i<n}$ for the operator $\hat O$ corresponding to some observational apparatus -- this is typically a magnetic field generated by a Stern-Gerlach machine detecting spin or some such, but can just as easily be thought of as the human eye observing a photon that was released after the state became excited.

To find this eigenbasis, we typically begin by finding eigenvalues using the standard method. Denote by $k$ the underlying field for $\mathscr{H}$, so $k=\mathbb{R}$ or $k=\mathbb{C}$, and let $\{|e_i\rangle\}_{i<n}$ be the standard basis for $\mathscr{H}$. Then we can form a matrix $M=\{m_{ij}\}_{i,j<n}$ by setting $$m_{ij}=\langle e_i|\hat O|e_j\rangle,$$ and then find the eigenvalues $\{\lambda_i\}_{i<n}$ of $\hat O$ as the zeroes of the polynomial $$\mathfrak{p}=det(M-\lambda I)=\sum_{i<n}c_i\lambda^n$$ with coefficients in $k$. If $k=\mathbb{R}$ then this polynomial may not fully factor which would prevent us from solving for the possible eigenvalues of our operator, thusly preventing us from determining what the possible eigenstates are -- this means we cannot determine what the possible states we might 'see' using $\hat O$ would be. Taking $k=\mathbb{C}$ allows us to ensure that $\mathfrak{p}$ fully factors, yielding a full set of eigenvalues and allowing us to solve for the behavior of the system.

Mathematically speaking the algebraic closure of $\mathbb{C}$ is what made it the preferential choice here. The research Carlo references is fascinating and there is probably a higher level motivation using algebraic geometry and QFT, but I think that this is a big part of why $\mathbb{C}$ is so ubiquitous in physical computations as a base field.

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    $\begingroup$ I get that your argument is that $\mathbb{C}$ is ubiquitous because it is algebraically closed. But your example of QM to argue this is really odd since the eigen-values of self-hermitian operators are always guaranteed to be real! In fact, all observables in QM are real. $\endgroup$ – Abhijeet Melkani Mar 23 '18 at 13:51
  • $\begingroup$ @AbhijeetMelkani This is a good point — I chose QM because it allowed for a relatively quick example of wanting algebraic closure. What is ‘not real’ in this setting is the coordinates of $|\psi\rangle$ in the $\{|\lambda_i\rangle\}_{i<n}$ basis representation, and this essentially allows for things like entanglement to occur which are necessary to satisfy the Bell inequalities. I will think tonight on whether there is a concise way of illustrating this. $\endgroup$ – Alec Rhea Mar 23 '18 at 14:07
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    $\begingroup$ a small comment on this last point: Majorana fermions can be entangled, so we can certainly have entanglement with a real wave function; what is restricted is the relative phase shift in an entangled pair, which can only be 0 or $\pi$ when $\psi$ is real; this is actually useful, it is a way to fight "decoherence" in a quantum computation. $\endgroup$ – Carlo Beenakker Mar 23 '18 at 14:21

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