I have written about similar issues in articles such as Wick rotation, The unbreakable postulates of quantum mechanics, and Zeta-function regularization, among others. But now I would like to promote the complex numbers themselves to the central players of the story.
History of complex numbers in mathematics
Around 1545, Girolamo Cardano (see the picture) was able to find his solution to the cubic equation. He already noticed the quadratic equation "x^2+1=0" as well. But even negative numbers were demonized at that time ;-) so it was impossible to seriously investigate complex numbers.
Cardano was able to additively shift "x" by "a/3" ("a" is the quadratic coefficient of the original equation) to get rid of the quadratic coefficient. Without a loss of generality, he was therefore solving the equations of the type
x3 + bx + c = 0that only depends on two numbers, "b, c". Cardano was aware of one of the three solutions to the equation; it was co-co-communicated to him by Tartaglia (The Stammerer), also known as Niccolo Fontana. It is equal to
x1 = cbrt[-c/2 + sqrt(c2/4+b3/27)] +Here, cbrt is the cubic root. You can check it is a solution if you substitute it to the original equation. Now, using the modern technologies, it is possible to divide the cubic polynomial by "(x - x_1)" to obtain a quadratic polynomial which produces the remaining two solutions once it is solved. Let's assume that the cubic polynomial has 3 real solutions.
+ cbrt[-c/2 - sqrt(c2/4+b3/27)]
The shocking revelation came in 1572 when Rafael Bambelli was able to find real solutions using the complex numbers as tools in the intermediate calculations. This is an event that shows that the new tool was bringing you something useful: it wasn't just a piece of unnecessary garbage for which the costs are equal the expenses and that should be cut away by Occam's razor: it actually helps you to solve your old problems.
Consider the equation
x3 - 15x - 4 = 0.Just to be sure where we're going, compute the three roots by Mathematica or anything else. They're equal to
x1,2,3 = {4, -2-sqrt(3), -2+sqrt(3)}The coefficient "b=-15" is too big and negative, so the square root in Cardano's formula is the square root of "(-15)^3/27 + 4^2/4" which is a square root of "-125+4" or "-121". You can't do anything about that: it is negative. The argument could have been positive for other cubic polynomials if the coefficient "b" were positive or closer to zero, instead of "-15", but with "-15", it's just negative.
Bombelli realized the bombshell that one can simply work with the "sqrt(-121)" as if it were an actual number; we don't have to give up once we encounter the first unusual expression. Note that it is being added to a real number and a cube root is computed out of it. Using the modern language, "sqrt(-121)" is "11i" or "-11i". The cube roots are general complex numbers but if you add two of them, the imaginary parts cancel. Only the real parts survive.
Bombelli was able to indirectly do this calculation and show that
x1 = cbrt(2+11i) + cbrt(2-11i) = (2+i) + (2-i) = 4which matches the simplest root. That was fascinating! Please feel free to verify that (2+i)^3 is equal to "8+12i-6-i = 2+11i" and imagine that the historical characters would write "sqrt(-1)" instead of "i". By the way, it is trivial to calculate the other two roots "x_2, x_3" if you simply multiply the two cubic roots, cbrt, which were equal to "(2+-i)", by the two opposite non-real cubic roots of unity, "exp(+-2.pi.i/3) = -1/2+-i.sqrt(3)/2".
When additions to these insights were made by John Wallis in 1673 and later by Euler, Cauchy, Gauss, and others, complex numbers took pretty much their modern form and mathematicians have already known more about them than the average TRF readers - sorry. ;-)
Fundamental theorem of algebra
Complex numbers have many cool properties. For example, every N-th order algebraic (polynomial) equation with real (or complex) coefficients has exactly "n" complex solutions (some of them may coincide, producing multiple roots).
How do you prove this statement? Using powerful modern TRF techniques, it's trivial. At a sufficiently big circle in the complex plane, the N-th order polynomial qualitatively behaves like a multiple of "x^N". In particular, the complex phase of the value of this polynomial "winds" around the zero in the complex plane N times. Or the logarithm of the polynomial jumps by 2.pi.i.N, if you wish.
You may divide the big circle into an arbitrarily fine grid and the N units of winding have to come from some particular "little squares" in the grid: the jump of the logarithm over the circle is the sum of jumps of the logarithm over the round trips around the little squares that constitute the big circle. The little squares around which the winding is nonzero have to have the polynomial equal to zero inside (otherwise the polynomial would be pretty much constant and nonzero inside, which would mean no winding) - so the roots are located in these grids. If the winding around a small square is greater than one, there is a multiple root over there. In this way, you can easily find the roots and their number is equal to the degree of the polynomial.
Fine. People have learned lots of things about the calculus - and functions of complex variables. They were mathematically interesting, to say the least. Complex numbers are really "new" because they can't be reduced to real diagonal matrices. That wouldn't be true e.g. for "U-complex" numbers "a+bU" where "U^2=+1": you could represent "U" by "sigma_3", the Pauli matrix, which is both real and diagonal.
Complex numbers have unified geometry and algebra. The exponential of an imaginary number produces sines and cosines - and knows everything about the angles and rotations (multiplication by a complex constant is a rotation together with magnification). The behavior of many functions in the complex plane - e.g. the Riemann zeta function - has been linked to number theory (distribution of primes) and other previously separate mathematical disciplines. There's no doubt that complex numbers are essential in mathematics.
Going to physics
In classical physics, complex numbers would be used as bookkeeping devices to remember the two coordinates of a two-dimensional vector; the complex numbers also knew something about the length of two-dimensional vectors. But this usage of the complex numbers was not really fundamental. In particular, the multiplication of two complex numbers never directly entered physics.
This totally changed when quantum mechanics was born. The waves in quantum mechanics had to be complex, "exp(ikx)", for the waves to remember the momentum as well as the direction of motion. And when you multiply operators or state vectors, you actually have to multiply complex numbers (the matrix elements) according to the rules of complex multiplication.
Now, we need to emphasize that it doesn't matter whether you write the number as "exp(ikx)", "cos(kx)+i.sin(kx)", "cos(kx)+j.sin(kx)", or "(cos kx, sin kx)" with an extra structure defining the product of two 2-component vectors. It doesn't matter whether you call the complex numbers "complex numbers", "Bambelli's spaghetti", "Euler's toilets", or "Feynman's silly arrows". All these things are mathematically equivalent. What matters is that they have two inseparable components and a specific rule how to multiply them.
The commutator of "x" and "p" equals "xp-px" which is, for two Hermitean (real-eigenvalue-boasting) operators, an anti-Hermitean operator i.e. "i" times a Hermitean operator (because its Hermitean conjugate is "px-xp", the opposite thing). You can't do anything about it: if it is a c-number, it has to be a pure imaginary c-number that we call "i.hbar". The uncertainty principle forces the complex numbers upon us.
So the imaginary unit is not a "trick" that randomly appeared in one application of some bizarre quantum mechanics problem - and something that you may humiliate. The imaginary unit is guaranteed to occur in any system that reduces to classical physics in a limit but is not a case of classical physics exactly.
Completely universally, the commutator of Hermitean operators - that are "deduced" from real classical observables - have commutators that involve an "i". That means that their definitions in any representation that you may find have to include some "i" factors as well. Once "i" enters some fundamental formulae of physics, including Schrödinger's (or Heisenberg's) equation, it's clear that it penetrates to pretty much all of physics. In particular:
In quantum mechanics, probabilities are the only thing we can compute about the outcomes of any experiments or phenomena. And the last steps of such calculations always include the squaring of absolute values of complex probability amplitudes. Complex numbers are fundamental for all predictions in modern science.Thermal quantum mechanics
One of the places where imaginary quantities occur is the calcuation of thermal physics. In classical (or quantum) physics, you may calculate the probability that a particle occupies an energy-E state at a thermal equilibrium. Because the physical system can probe all the states with the same energy (and other conserved quantities), the probability can only depend on the energy (and other conserved quantities).
By maximizing the total number of microstates (and entropy) and by using Stirling's approximation etc., you may derive that the probabilities go like "exp(-E/kT)" for the energy-E states. Here, "T" is called the temperature and Boltzmann's constant "k" is only inserted because people began to use stupidly different units for temperature than they used for energy. This exponential gives rise to the Maxwell-Boltzmann and other distributions in thermodynamics.
The exponential had to occur here because it converts addition to multiplication. If you consider two independent subsystems of a physical system (see Locality and additivity of energy), their total energy "E" is just the sum "E1+E2". And the value of "exp(-E/kT)" is simply the product of "exp(-E1/kT)" and "exp(-E2/kT)".
This product is exactly what you want because the probability of two independent conditions is the product of the two separate probabilities. The exponential has to be everywhere in thermodynamics.
Fine. When you do the analogous reasoning in quantum thermodynamics, you will still find that the exponential matters. But the classical energy "E" in the exponent will be replaced by the Hamiltonian "H", of course: it's the quantum counterpart of the classical energy. The operator "exp(-H/kT)" will be the right density matrix (after you normalize it) that contains all the information about the temperature-T equilibrium.
There is one more place where the Hamiltonian occurs in the exponent: the evolution operator "exp(H.t/i.hbar)". The evolution operator is also an exponential because you may get it as a composition of the evolution by infinitesimal intervals of time. Each of these infinitesimal evolutions may be calculated from Schrödinger's equation and
[1 + H.t/(i.hbar.N)]N = exp(H.t/i.hbar)in the large "N" limit: we divided the interval "t" to "N" equal parts. If you don't want to use any infinitesimal numbers, note that the derivative of the exponential is an exponential again, so it is the right operator that solves the Schrödinger-like equation. So fine, the exponentials of multiples of the Hamiltonian appear both in the thermal density matrix as well as in the evolution operator. The main "qualitative" difference is that there is an "i" in the evolution operator. In the evolution operator, the coefficient in front of "H" is imaginary while it is real in the thermal density matrix.
But you may erase this difference if you consider an imaginary temperature or, on the contrary, you consider the evolution operator by an imaginary time "t = i.hbar/k.T". Because the evolution may be calculated in many other ways and additional tools are available, it's the latter perspective that is more useful. The evolution by an imaginary time calculates thermal properties of the system.
Now, is it a trick that you should dismiss as an irrelevant curiosity? Again, it's not. This map between thermal properties and imaginary evolution applies to the thermodynamics of all quantum systems. And because everything in our world is quantum at the fundamental level, this evolution by imaginary time is directly relevant for the thermodynamics of anything and everything in this world. Any trash talk about this map is a sign of ignorance.
Can we actually wait for an imaginary time? As Gordon asked, can such imaginary waiting be helpful to explain why we're late for a date with a woman (or a man, to be really politically correct if a bit disgusting)?
Well, when people were just animals, Nature told us to behave and to live our lives in the real time only. However, theoretical physicists have no problem to live their lives in the imaginary or complex time, too. At least they can calculate what will happen in their lives. The results satisfy most of the physical consistency conditions you expect except for the reality conditions and the preservation of the total probabilities. ;-)
Frankly speaking, you don't want to live in the imaginary time but you should certainly be keen on calculating with the imaginary time!
Analytic continuation
The thermal-evolution map was an example showing that it is damn useful to extrapolate real arguments into complex values if you want to learn important things. However, thermodynamics is not the only application where this powerful weapon shows its muscles. More precisely, you surely don't have to be at equilibrium to see that the continuations of quantities to complex values will bring you important insights that can't be obtained by inequivalent yet equally general methods.
The continuation into imaginary values of time is linked to thermodynamics, the Wick rotation, or the Hartle-Hawking wave function. Each of these three applications - and a few others - would deserve a similar discussion to the case of the "thermodynamics as imaginary evolution in time". I don't want to describe all of conceptual physics in this text, so let me keep the thermodynamic comments as the only representative.
Continuation in energy and momentum
However, it's equally if not more important to analytically continue in quantities such as the energy. Let us immediately say that special relativity downgrades energy to the time component of a more comprehensive vector in spacetime, the energy-momentum vector. So once we will realize that it's important to analytically continue various objects to complex energies, relativity makes it equally important to continue analogous objects to complex values of the momentum - and various functions of momenta such as "k^2".
Fine. So we are left with the question: Why should we ever analytically continue things into the complex values of the energy?
A typical laymen who doesn't like maths too much thinks that this is a contrived, unnatural operation. Why would he do it? A person who likes to compute things with the complex numbers asks whether we can calculate it. The answer is Yes, we can. ;-) And when we do it, we inevitably obtain some crucial information about the physical system.
A way to see why such things are useful is to imagine that the Fourier transform of a step function, "theta(t)" (zero for negative "t", one for positive "t"), is something like "1/(E-i.epsilon)". If you add some decreasing "exp(-ct)" factor to the step function, you may replace the infinitesimal "epsilon" by a finite constant.
Anyway, if you perturb the system at "t=0", various responses will only exist for positive values of "t". Many of them may exponentially decrease - like in oscillators with friction. All the information about the response at a finite time can be obtained by continuing the Fourier transform of various functions into complex values of the energy.
Because many physical processes will depend "nicely" or "analytically" on the energy, the continuation will nicely work. You will find out that in the complex plane, there can be non-analyticities - such as poles - and one can show that these singular points or cuts always have a physical meaning. For example, they are identified with possible bound states, their continua, or resonances (metastable states).
The information about all possible resonances etc. is encoded in the continuation of various "spectral functions" - calculable from the evolution - to complex values of the energy. Unitarity (preservation of the total probabilities) can be shown to restrict the character of discontinuities at the poles and branch cuts. Some properties of these non-analyticities are also related to the locality and other things.
There are many links over here for many chapters of a book.
However, I want to emphasize the universal, "philosophical" message. These are not just "tricks" that happen to work as a solution to one particular, contrived problem. These are absolutely universal - and therefore fundamental - roles that the complex values of time or energy play in quantum physics.
Regardless of the physical system you consider (and its Hamiltonian), its thermal behavior will be encoded in its evolution over an imaginary time. If Hartle and Hawking are right, then regardless of the physical system, as long as it includes quantum gravity, the initial conditions of its cosmological evolution are encoded in the dynamics of the Euclidean spacetime (which contains an imaginary time instead of the real time from the Minkowski spacetime). Regardless of the physical system, the poles of various scattering amplitudes etc. (as functions of complexified energy-momentum vectors) tell you about the spectrum of states - including bound states and resonances.
Before one studies physics, we don't have any intuition for such things. That's why it's so important to develop an intuition for them. These things are very real and very important. Everyone who thinks it should be taboo - and it should be humiliated - if someone extrapolates quantities into complex values of the (originally real) physical arguments is mistaken and is automatically avoiding a proper understanding of a big portion of the wisdom about the real world.
Most complex numbers are not "real" numbers in the technical sense. ;-) But their importance for the functioning of the "real" world and for the unified explanation of various features of the reality is damn "real".
And that's the memo.
