Physical phenomena with complex number solution

[sujoykroy]

Can anybody give example of a physical phenomenon which can be stated into a polynomial equation (cubic, quintic or whatever) and which has complex number solutions and those complex numbers have some physical interpretation?

The reason for asking such question tagged in physics is that i read from some resources that complex number were came into perception while trying to provide solution for cubic equation like x³ = px + q. While i have no issue in terms of accepting an idea of a thing like a + ib , which is called complex number, i do not understand how these complex numbers are related with physical events. Since, complex numbers are related with cubic equation by origin and each equation is supposed to represent some physical phenomena, i thought it is worth searching for such equation which has complex solution to understand this mysterious number.

 

[Born2bwire]

There was a thread in the EE section where I mentioned one physical aspect that is used.

https://www.physicsforums.com/showthread.php?t=309851&highlight=phase

One of the most relevant aspects is that it allows us to represent the phase of a signal. While we can only measure relative phase for the most part, complex numbers greatly ease the mathematical analysis. Thus, phase is a physical aspect of complex numbers.

 

[jrlaguna]

Be careful... in mathematics, complex numbers are like god, true. This is because everything is more elegant in terms of them. But in physics, all measurements should give a real number in the end. Complex numbers are "only" a computational trick. They can save you many hours of boring calculations...

Hadamard put it this way: sometimes, the shortest path between two points in the real line goes through the complex plane.

In other terms: physics starts with real input data and finishes with real output data, but in between, working with complex numbers is nicer and shorter.

 

[Bob S]

Real transformers and ac electric motors have leakage inductance which cause the impedance to be complex; R + jwL. Thus the voltage is V = IR + jwLI. For the imajinary part, V = L dI/dt.

 

[nasu]

Since, complex numbers are related with cubic equation by origin and each equation is supposed to represent some physical phenomena, i thought it is worth searching for such equation which has complex solution to understand this mysterious number.

This is not true. Why each equation is SUPPOSED to represent some physical phenomenon?
How about x²=-1? What physical phenomenon is represented by this equation?
The math does not need to correspond to physics at all.
It is sort of a miraculous thing that many mathematical tools can be used to describe physical objects. But is not that they have to, from a mathematical point of view.

 

[Bob_for_short]

This is not true. Why each equation is SUPPOSED to represent some physical phenomenon?
How about x²=-1? What physical phenomenon is represented by this equation?
The math does not need to correspond to physics at all.
It is sort of a miraculous thing that many mathematical tools can be used to describe physical objects. But is not that they have to, from a mathematical point of view.

Speaking of phase, the imaginary part represents decrement (or increment) of a wave: exp(-t/tau), for example.Bob.

 

[sujoykroy]

This is not true. Why each equation is SUPPOSED to represent some physical phenomenon?.

I think the very first equations that intrigue the mind of early wise persons were related with physical phenomena. While providing solutions to such equations, they stumbled upon the game of playing with different possibilities of previously discovered concepts. And thus they might think about the idea of square root of negative unity. Even though they didn't find physical explanation of complex number, they moved on to play with it and built a whole lot of rules on it. But, even this game had some universal truth in it, and hence later some physicist found this number to handle and measure some physical phenomena directly or indirectly.

How about x²=-1? What physical phenomenon is represented by this equation?

Place yourself on a giant wall clock, marked from 1-12. When your head is at 12 mark and leg is at 6 mark, consider this position as positive. And when your head is at 6 mark and leg at 12 mark, consider your position as negative, since you are upside down . We can think one step further and say the first situation as +1 and the second one as -1. Now, assume someone can rotate you on this clock, so that your head will go to 3 mark, 6 mark, 9 mark and 12 mark respectively (and your legs move along with head). Now, if you consider the rotation of 3 marks (from 12 to 3 or 3 to 6, or 6 -9 or 9 -12) as an event j, then j² causes you to have -1 and j⁴ causes you to have 1.
So, you can get a equation x² = -1 if you think x as 3 mark rotation in 12-3-6-9-12 direction.

Perhaps this example has many flaws in it than truth. But i think when you try to make sense of everything around you, this attempt does give you a hope.

This example is inspired by David Franklin (https://nrich.maths.org/discus/messages/67613/91931.html?1153696011 , scroll down near the end). Though i had several other examples in mind, but this one was much easier to grasp (and to express).

 

[sujoykroy]

Now, if you consider the rotation of 3 marks (from 12 to 3 or 3 to 6, or 6 -9 or 9 -12) as an event j, then j² causes you to have -1 and j⁴ causes you to have 1.
So, you can get a equation x² = -1 if you think x as 3 mark rotation in 12-3-6-9-12 direction.

I found a falsehood in the definition of event j as moving 3 marks in 12-3-6-9-12 direction. It doesn't satisfy j² = -1, because j² means moving 3 * 3 =9 marks (exceeding 6 marks, the upside down condition) and j⁴ means 81 marks (which will never put your head at 12 mark position).

If we divide 12-3-6 half circle into 36 sub marks or ticks and 6-9-12 half circle into 1260 ticks, and define j as moving 6 ticks in 12-3-6-9-12 direction then j² means 36 ticks(reaching 6 mark) and j⁴ means 36*36 = 1296 = 36 + 1260 ticks (placing head on 12 mark position), then it satisfy x² = -1 equation.

 

[rcgldr]

If we define ... circle then it satisfy x² = -1 equation.

Yes but in this case sqrt(-1) isn't imagnary. and peforming 2 sequential operations should be considered more like addition than multiplication.

Imaginary numbers are just a mathematical way to handle intermediate steps of calculations of real world physics.

I do recall some article claiming that Hawking claimed that space time had only imaginary components until the Big Bang, but I've never found that article again.

 

[sujoykroy]

Yes but in this case sqrt(-1) isn't imagnary. and peforming 2 sequential operations should be considered more like addition than multiplication..

Agree with you to some extent. But, i don't think we should take the term "imaginary" in its literal meaning. Mathematics, more specifically Algebra, is about the playing with relationships among numbers ignoring the interpretation of the numbers. So, imaginary means which doesn't directly effect the numbers of primary world of interest, but influences indirectly, through j²: it has a negative (-) effect. And if j is defined as moving 6 ticks, then j² means moving 6 ticks 6 times, which is more like repeated addition which is termed as multiplication and makes you upside down causing -1, since body posture is primary world of interest in this particular context.

Imaginary numbers are just a mathematical way to handle intermediate steps of calculations of real world physics.

Not agreed with the "are just" phrase, it should be "can be thought of as" instead.
Imaginary number, indeed, is brainchild of pure mathematics, without thinking about any relationship with physical world. But, it certainly has some more truth in it which enabled physicists to use it to solve some physical phenomena. And since, imaginary and real numbers are just numbers, you can dare to interpret it in whatever way as long as it satisfies the properties of imaginary number, as defined by mathematicians.

 

[nasu]

My objection, as shown by the capitalized word, is to the statement that every equation in math must describe a physical system. Like assuming that there is some law of nature that this must be true.

Regarding the example, the equation is not very "physical".
The x in your case is an operator not a variable (or a constant).
then you describe the states of the system by 1, -1 and so on. These are more like labels not real variables.
But anyway, you have the square of an operator being equal to -1 which is the description of the state.
In order to have consistency, you should apply an operator to a state and you'll get another state. You may say that implicitly you apply x^2 to the state 1 and you get state -1.
It will be more like x^2|1>=|-1> to use the symbols from quantum mechanics.
This is not an algebraic equation and does not have complex solutions.

Anyway, the example x^2=-1 was just out of my hat. You may find a real example of physical things described by it (I doubt it) but it won't prove the statement that all equations MUST have a physical counterpart.

Good discussion topic anyway.