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I have a question regarding complex physical quantities. Why do we consider only the real part of a complex physical quantity? Why not the modulus? Since, for $z=a+bi$, we have $|z| = \sqrt{a^2+b^2}$, and so the imaginary part contributes to the modulus, it is not clear to me when to use the modulus or the real part.

For example, the electric (or magnetic) field. And/or the Poyting vector. Below I take texts from Jackson, Classical Electrodynamics:

"Because the diffusion equation is second order in the spatial derivatives and first order in the time, it is convenient to use complex notation, with the understanding that the physical fields are found by taking the real parts of the solutions." (Jackson, 3rd. Edition, page 220)

"Then (6.131) can be written as $$\frac{1}{2} \int_V \mathbf{J^*\!\cdot E}\,d^3x+2i\omega \int_V (w_e-w_m)\, d^3x +\oint_S \mathbf{S\cdot n}\,da \tag{6.134}$$

... It is a complex equation whose real part gives the conservation of energy for the time-averaged quantities and whose imaginary part relates to the reactive or stored energy and its alternating flow." (Jackson, 3rd. Edition, page 265)

"...With the convention that the physical electric and magnetic fields are obtained by taking the real parts of complex quantities, we write the plane wave fields as $$ \mathbf{E}(x,t)=\mathcal{E} e^{ik\mathbf{n\cdot x}-i\omega t} \ \ \ \ (7.8) \\ \mathbf{B}(x,t)=\mathcal{B} e^{ik\mathbf{n\cdot x}-i\omega t}$$" (Jackson, 3rd. Edition, page 296)

Elio Fabri
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Yttria Therbium
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  • In some applications like electronics we use the complex number to indicate the voltage and the phase, so it's really just a way of encoding two quantities into one complex number. Whenever you see the real part of a complex number being used in physics this is probably what is going on. In other applications like quantum mechanics we really are dealing with variables that are complex. To take this any further you will have to tell us in what areas of physics you are using the complex numbers. – John Rennie Jun 17 '21 at 15:02
  • In physics is common to use complex number in their "entirety", for example in quantum mechanics. Your question needs a context to be answerable. – Noumeno Jun 17 '21 at 15:03
  • Related: What is the physical significance of the imaginary part when plane waves are represented as $e^{i(kx-\omega t)}$? – Emilio Pisanty Jun 17 '21 at 18:37
  • We generally consider the things we're interested in, and what we're interested in varies with time and circumstances. This applies even outside of physics. – WillO Jun 17 '21 at 18:41
  • @WillO: I am not trying to solve a specific problem or exam: I just want to discern and learn where/when to use real part or modulus. I gave the example of Jackson's text because It is where I found it more explicit. But when I was studying the propagation of light in a medium with complex permitivitty, it also came to my mind, mainly when I was writing the fortran program. – Yttria Therbium Jun 17 '21 at 19:02
  • @YtrriaTherbium : The only possible answer is that you should concentrate on the part that represents the physical quantity you're interested in. I cannot predict what will interest you. – WillO Jun 17 '21 at 19:30

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There are cases where we use the modulus, imaginary part, argument or some combinations to represent a physical quantity. It depends on how we arrived at using complex numbers in the first place.

Typically one of the first places where complex numbers are encountered when studying physics is when dealing with waves or oscillations as a method of simplifying the algebra, as we can replace trig functions with simpler exponentials. In this case the complex numbers were introduced by observing that the physical quantities we are interested in could be written as the real part of certain complex numbers, so it is the real part that retains the physical significance. We could, in principle, find a way to represent the same phyical quantity as the magnitude of a complex number, but it would likely result in a more complicated calculation.

By Symmetry
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  • We could, in principle, find a way to represent the same phyical quantity as the magnitude of a complex number This might need to be clarified or at least thought through more. A complex number has two "degrees of freedom"; the modulus only has one. So I doubt you could actually use just the modulus in applications where full complex values are used. – BioPhysicist Jun 17 '21 at 15:13