# Physical interpretation of complex numbers [duplicate]

Complex numbers are used widely in quantum mechanics and the waveform, is there a physical interpretation of what this means about the structure of the universe? Why is it not used in macro physics?

Do physicists really think that calling imaginary numbers a 90 degree rotation is a good enough answer? It seems to be used in many areas to mean similar things.

Is there an explanation to do with dimensions as I have tried in this conversation a better way to understand them?

• Possible duplicate of What is the need of complex functions in wave analysis? – Brick Nov 5 '19 at 15:20
• I think you should reformulate your question as : "what is the physical interpretation of probability amplitudes?". And the honest answer to that would be : "we don't know." – Undead Nov 5 '19 at 22:32
• They are used for "macro-physics" (depending on what mean by this term) such as to work with AC voltage in electrical systems. – Steeven Nov 6 '19 at 12:39
• @Stian Yttervik- I think the reason for that is that a LOT of people (not the OP on this question) both online and in real life approach physics with some really weird unreal stuff: "what if particles are held together by flows of wavefunction energy" was one I was once asked. The best way of dealing with such people is not to tell them what they are saying doesn't make sense (they will argue back). But to say "Sure, cool theory. Can you predict or calculate something for me with it?". This immediately highlights to them that their "theory" doesn't qualify as science as it is "not even wrong". – Dast Nov 6 '19 at 16:00
• @Dast Yeah, I agree, and some of the things you learn in physics are impossible to learn untill you are familiar with the equations and the only way to get there is to... shut up and calculate. It is self reinforcing. Certainly, if you are able to both have that familiarity with the equations but simultaneously have a mental model of the universe that allows for you to create simplifications and explanation models, you end up being quite revered – Stian Yttervik Nov 6 '19 at 19:15

Complex numbers are used in all of mathematics, and therefore by extension they are used in other fields that require math; not just physics, but also engineering and other fields. Trying to assign a "physical interpretation" to a complex number would be like assigning a physical interpretation to a real number, such as the number 5.

A complex number is just an extension of a real number. Many of us were taught about the "number line" in elementary school, which is just a line that (to quote Wikipedia) serves as an abstraction for real numbers. Being a line, it is 1-dimensional. Complex numbers are the same, except they are 2-dimensional: instead of being described by a 1-dimensional real number line, they are described by a 2-dimensional "complex number plane". Using $$i$$ for the imaginary axis (where $$i^2 = -1$$) is a mathematical convenience that makes the 2-dimensional complex numbers extraordinarily useful.

• Comments are not for extended discussion; this conversation has been moved to chat. – tpg2114 Nov 7 '19 at 15:12
• – OzOz Nov 8 '19 at 1:48

Complex numbers are used in "macro" physics. They are used in analysis of electrical circuits (especially when AC is involved) and in fluid dynamics. Solution of differential equations is simplified if complex numbers are used, as is Fourier analysis. Any scenario that involves periodic or cyclic functions can be modeled using complex numbers.

• they are fundamental to “micro” physics as well since one cannot do quantum mechanics without complex numbers. – ZeroTheHero Nov 5 '19 at 13:40
• @OzOz $x+iy$ is a handy way of combining two real state parameters into a single quantity. For a pendulum these parameters could be position and momentum. For an electrical circuit they could be current and voltage. – gandalf61 Nov 5 '19 at 13:58
• $ix$ has no easy physical interpretation, however $i\frac{d}{dx}$ has (negative momentum). That shows that the complex plane gives us an easy way to merge position and momentum probability distributions into a single function (the wavefunction). – Martin Peschel Nov 5 '19 at 14:00
• You definitely could do QM without complex numbers, @ZeroTheHero. Whether you'd want to do it that way is a different question. Schrodinger equation has a real formulation - one complex equation goes to two real equations. You could do the same for any other field theory, but it probably be a slog. – Brick Nov 5 '19 at 15:23
• $ix$ has a perfectly good physical interpretation, when if $x$ represents a vector in a complex plane: $ix$ represents the vector you get by rotating $x$ clockwise by $90^\circ$. – Lee Mosher Nov 6 '19 at 1:33

The fundamental object in quantum mechanics is the amplitude, which encodes information about how a system transitions from one state to another state. For example, if you are doing a double slit experiment you might care about how an electron transitions from the incoming pre-slit state to a state where it hits a certain location $$x$$ on the detector. For each different outcome state there would be a different amplitude $$\mathcal{M}_x$$.

We care about amplitudes because they can tell us about probabilities. According to the Born rule the probability that the electron ends up at location $$x$$ is given by absolute value of the square of the amplitude, $$P(x) = |\mathcal{M}_x|^2$$.

The probability is a non-negative real number, but what kind of object should represent the amplitude? A positive real number? Any real number? A pair of real numbers? A complex number? Some even more abstract mathematical object?

This paper addresses the question by noticing that since amplitudes correspond to different experiments, and experiments can be chained together in various ways, we have to be able to combine two amplitudes to get a third amplitude, and we have to be able to combine them in at least two distinct ways. The paper then proves that, if you choose to represent amplitudes as pairs of real numbers, the operations that correspond to combining experiments end up acting exactly like complex addition and complex multiplication.

The paper doesn't answer the question of why amplitudes should be pairs of real numbers instead of single real numbers, or triples or something more complex, but it's a good starting point for seeing how complex arithmetic falls out of the logic of quantum experiments.

P.S. Using single real numbers for amplitudes cannot explain the single slit / double slit experiment, where adding a second slit results in zeros in the probability distribution that weren't present in the single slit probability distribution. Using a pair of real numbers (or one complex number) is the next simplest system that can explain this behavior.

• That is very interesting, will take some time to understand. – OzOz Nov 5 '19 at 23:12
• "Using single real numbers for amplitudes cannot explain the single slit / double slit experiment" I don't think so. Please see my answer at physics.stackexchange.com/questions/32422/… – akhmeteli Nov 6 '19 at 3:07
• @akhmeteli My argument is pretty simple. The probabiltiy distribution with one slit is $\psi_1(x)^2$, and has no zeros. The probability distribution with two slits is $(\psi_1(x)+\psi_1(x+a))^2$ where $a$ is the slit separation. The two slit distribution does have zeros. There is no continuous real function that behaves this way when added to a shifted version of itself. Ultimately it is because to pass from positive reals to negative reals you must pass through zero (which isn't true for complex numbers). You can define QM with real wavefunctions, but it can't do the above. – Luke Pritchett Nov 6 '19 at 13:57
• So, to reword this answer, the way to understanding the "physical interpretation of complex numbers" (as asked by OP) is by understanding the physical interpretation of the operations on complex numbers as amplitudes. And then one can realize that the features of the amplitudes give a definition of complex numbers. Sounds neat! – JiK Nov 6 '19 at 14:38
• @LukePritchett : And why do I need to do the above? – akhmeteli Nov 7 '19 at 2:26

Complex number as any number alone does not say anything about physics at all. It has to be bound to some measurement unit(s) or have a well-defined definition in physics.

For example complex refractive index is defined in physics as :

$${\displaystyle {\underline {n}}=n+i\kappa .}$$

Here imaginary part $$\kappa$$ is defined as attenuation coefficient - matterial resistivity to penetration of light waves

EDIT

Complex numbers are used intensively in describing any kind of waves, because you can put wave amplitude and wave phase into a single complex-valued wave amplitude:

$$Z = Ae^{i\phi}$$

So most things which is related to waves can be, at least theoretically, expressed in complex numbers.
For example,- complex refractive index can be traced back to other wave properties in such way : $$\underline{k} = 2\pi \underline{n}/λ_0$$ where $$\underline{k}$$ is complex wavenumber

BONUS

Another reason why complex plane is attractive - you can do more math if you are not bound to real numbers. For example, you can even take a natural logarithm of negative real number : $$\ln(-x) = \ln(x) + \pi \space \textrm{i}$$

which results in complex number ! So, never trust your pocket calculator

• I feel this is getting closer to an actual answer, but it is too specific, is there any way this could be generalized? – OzOz Nov 6 '19 at 7:04
• see edit, I tried to do some sort of generalizations – Agnius Vasiliauskas Nov 6 '19 at 8:32
• Interesting, so a complex number can be used to express any system in which there are two independent properties that can be measured? Which could be a second dimension or a different physical property? – OzOz Nov 6 '19 at 9:30
• @OzOz If we take positive number and multiply it by i we end up with an imaginary one which is like a phase shift of 90 degrees and is completely orthogonal. So there yes, it appears to be just like an independent variable, but then, if we multiply by i again, we now have a phase shift of 180 degrees, which is negation along the original real axis. – Rodney Nov 6 '19 at 14:21
• @OzOz, Re, "[can a] complex number...be used to express any system in which there are two independent properties...?" a two dimensional vector can do that, and the complex plane is a two-dimensional vector space, but some problems (e.g., mathematical descriptions of periodic functions and wave motion) are an especially good fit for the peculiar algebraic properties of complex numbers. – Solomon Slow Nov 7 '19 at 17:27

Complex numbers are just a convenient way of representing a 2-dimensional vector. They are used in all sorts of everyday situations where you have an X and a Y component, or a magnitude and a phase.

• This answer ignores the multiplicative group of the complex numbers. The multiplicative properties of the complex numbers are pretty important, and they do not fall out of just being a 2 dimensional vector. – Yakk Nov 6 '19 at 19:05
• Better a certain kind of 2x2 matrices, not just vectors. – Vladimir F Nov 7 '19 at 22:00

Complex numbers do two obvious things. If you think of them as 2D vectors on a plane, starting at your arbitrary point (0,0), then adding complex numbers is vector addition.

And if you think of them as angles off an arbitrary polar-coordinate angle (0,1), then when you multiply two of them you get the sum of the angles (and the product of the magnitudes).

That can be useful whenever you have something that works like a 2D plane, where you want to do vector addition or addition of angles.

So for example, a pendulum can have kinetic energy and potential energy, and mostly the sum of them is constant. They are two different things so you can represent them on a 2D plane, as a circle whose radius is the total energy. When you convert from one to the other it moves around the circle. You can represent its motion with complex numbers.

You can do that with anything that converts back and forth between two forms, but sometimes it will involve easier complex number math than other times.

Sometimes things fit rotations in 4 dimensions, and then you can use quaternions like you'd use complex numbers for 2 dimensions. You can easily represent elliptical orbits with quaternions -- even easier than you can use them for 3D rotations. For any angle along the orbit, you can get the 3D position and also the time -- how far it's ahead or behind the time it would reach that angle in a circular orbit.

Use the math wherever it fits.

• Agreed, use the maths where it fits, for generations, then one day you have enough places where it fits and then someone comes along and understands it and puts it all into a concise theory like Newton did for calculus. Maybe it but it really feels like the explanations I get to complex numbers are lacking somewhere. – OzOz Nov 6 '19 at 17:22
• I explained the concise theory. You get to do vector addition and rotations, between two things that are independent and so can be considered different dimensions. That's it. It's useful anywhere it makes sense to do either or both of those. – J Thomas Nov 6 '19 at 19:41

If positive real numbers are forwards numbers and negative real numbers are backwards numbers then imaginary numbers are sideways numbers.

In terms of angles, positive real numbers could be thought of as having an angle of 0°, negative numbers have an angle of 180° and sideways or imaginary numbers are at ±90°. This is useful in electrical engineering when quoting impedances. An impedance is the AC version of resistance in a DC circuit. It has a restive component which does not change the phase angle between current and voltage and a reactance which changes the angle between them by ±90°. (The sign depends on whether the reactance is capacitance or inductance.)

If you want to combine the two into one “number” you can use complex numbers where the real part is the resistance and the imaginary part becomes the reactance. Formulas then continue to work just like the simple Ohm’s Law ones using resistance but with complex numbers instead. Both resistance and reactance are taken into account at the same time.

Basically, anywhere you have things that are 90° apart in some way then imaginary numbers could be useful. That could be x and y coordinates or where both sine and cosine waves occur.

So, if you need two dimensional numbers they could be the way to go. For three or more dimensional numbers you would probably move on to tensors.

• +1. You could extend this example with ideas like phase of a varying current shown in a complex number. Adding two complex currents is meaningful (e.g. if they are parallel). Multiplying a complex current by a complex impedance is also meaningful to give a complex voltage, which may be out of phase with the the current. Sometimes you may want to consider your complex numbers in polar form, which will emphases the amplitude and phase – Henry Nov 7 '19 at 14:22

In dealing with sinusoidal functions which are out of phase as in AC circuits or waves, its usually possible to put the equations into a form which resembles the addition of the x components of two or more vectors to get the x component of the resulting vector. The vectors can be thought of as rotating in a 2D plane. Its often more convenient to work with the vectors than with the components. If the vectors are visualized in an xy plane, only the x components are significant. If they are visualized in the complex number plane, they are handily represented by complex functions, but again in most cases, only the real components of the vectors have physical significance.

Sorry for a long story, which only address the title of your question (and not the inner questions).

I remember the first time I was introduced to complex numbers at school. The teacher (of mathematics, not physics) was explaining us how to solve quadratic equations (a.x^2+b.x+c=0). After giving us the method, he ended up with the well known solution for the roots: $$x=\frac{-b\pm\sqrt{b^2-4ac} }{2a}$$

Of course it didn't take long for a bright student to tell the teacher: "Hey, but then what happen if the expression in the square root is negative?" For example solve x^2+1=0, your roots will be: $$x=\frac{\pm\sqrt{-4} }{2}$$

All (or most of) the class understood the conundrum and started to scratch their head as they knew for sure that no number could be squared and retain a negative sign ...

The teacher continued, completely undisturbed, "It's not a problem, we can make tools for that. Let's just use a quantity i defined such as i^2=-1". And he went on to introduce the complex numbers and the rules in the complex plane.

Again it wasn't long before a voice from the baffled audience shouted "So this is a actually a convoluted way to bypass rules you taught us previsouly (like a squared number will always be positive). What use is that? why go to such complexity ? (no pun intended, although I now wonder how the complex numbers got their name from initially).

So the teacher put it this way:

There are many physics equations which follows a quadratic law, or even more complicated laws where the solutions involved square roots of potentially negative numbers, and (before the Complex numbers) the physicians couldn't solve their system fully so they asked mathematicians to define a new domain (larger than the Real domain) where these systems would be solvable. The Complex numbers are the tool mathematicians came up with.

By now my understanding of the complex numbers is a bit deeper, but this simple description still holds true. The Complex numbers are just a mathematical tool. A complex number do not have other physical equivalent than the one you give to them.

Same can be said about the Real numbers. I work with a multi sensor tool which measure 10 different parameters in parallel. The output for anyone is just a list of numbers, only myself knows that:

• the first number represents a Weight, in [N],
• the second is a Moment, in [N.m]
• the third is an acceleration, in [G]
• and so on ...

All different physical dimensions, yet on my screen they're all just numbers, only in my head do I know this one represent this, this one represent that...

For complex numbers, you have 2 components. Each may represent a different physical dimension (electric field and magnetic field for EM). The i part is only the mathematical tool allowing you to handle these numbers in a more graceful form (because you could also describe each components separately with real numbers only, but the equations become real ugly). The i in itself means nothing physically.

• I agree, complex numbers are a mathematical tool. But I don't like the description of a rotation through 90 degrees... I think they are a mathematical tool where you take a main dimension (which you call the real numbers, usually something you can measure), then you put all the other dimensions in the imaginary part. The imaginary part is how all the other dimensions effect the part you have chosen to be real. Earlier in physics there was just one dimension in the imaginary side, now with quantum there are many dims and the maths is complicated because there are multiple dimensions I could be. – OzOz Nov 6 '19 at 13:30

It's a good idea to think of the imaginary numbers as numbers perpendicular to the reals. Multiplying a real by -1 "rotates" it over 180° on the line of real numbers. Multiplying a real by i rotates it over 90° so it lands on the imaginary line. Multiplying again by i rotates it 90 degrees further so it lands on the real axis again. Hence i*i=-1. This is all incredibly basic but it's the way I like to approach complex numbers in more complicated scenarios involving complex exponentials and differential equations etc.

In the end imaginary numbers are no more "unphysical" than negative numbers. Negative numbers extend the line of positive reals by adding some numbers to the left and imaginary numbers extend the reals by adding some numbers perpendicularly. The use of both negative and imaginary numbers could be eliminated from equations but it would make them a lot less convenient.