31

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 ...


27

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.


12

In a vector space over the field of complex numbers the notion of complex conjugation is basis dependent. You might say a vector is "real" if its components in some basis are real numbers, but if you change to another basis and the matrix expressing the new basis in terms of the old has complex entries then the "real" vector will have complex components in ...


12

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 ...


10

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 - ...


8

[The following is a half-remembered comment that my doctoral advisor told me some years ago, so I may have garbled it. I welcome corrections in the comments; feel free to tell me I'm full of it as well.] One way to think about a Wick rotation is that the "Euclidean" and "Lorentzian" manifolds (both of which are four-dimensional real manifolds, with a ...


6

The correct definition of the integral is $$ I = \int_{\mathbb{R}^3} d^3\mathbf{k}\int_{-\infty}^{\infty} d k^0 \,\frac{1}{(|\mathbf{k}|^2 - (k^0)^2 - i\varepsilon)^2}\,. $$ The "$+i\varepsilon$" is the so-called $\varepsilon$ prescription which is simply giving you instructions on how to analytically continue this expression. By this I mean that we now ...


6

We can write the first equation as $e^{i2\pi(x/\lambda -ft)}$ where $e^{i2\pi}$ would be 1 making the whole equation 1, i.e., $\psi=1$. First, $(kx-\omega t) = 2\pi(x/\lambda-ft)$ is equal to $2\pi$ only for some values of $x$ and $t$ but not for all values of $x$ and $t$. So it isn't true that $\psi=1$. So where has your logic failed? By the law of ...


6

Oftentimes in physics, we study phenomena that oscillate with a certain frequency, which we'll call $\omega$. These phenomena are usually modeled as the real part of a complex function $e^{i\omega t}$. These oscillations have a tendency to decay exponentially in response to damping, and so are more accurately modeled as $e^{-\beta t}e^{i\omega t}$ for some ...


5

It looks like the author is using a convention where the rapidity, $\phi$, is imaginary. The reason for doing this is to ease your introduction to hyperbolic trigonometry. The reason for doing this is because it makes boosts look exactly like rotation. In 2-dimensions, a rotation matrix has the form $$\left[\begin{array}{c} x' \\ y' \end{array}\right] = \...


5

We need the complex conjugated variable $\bar{u}$ because the Hamiltonian $H(u,\bar{u})$ is typically not holomorphic in $u$. The normalization of the CCR$^1$ $$\{u,\bar{u}\} ~=~-i \tag{1}$$ explains the $\sqrt{2}$-normalization in $$ u=\frac{q+ip}{\sqrt{2}}.\tag{2}$$ Recall that Hamilton's equations can be written in terms of the Poisson bracket $$\dot{u}...


5

Consider a linear operator $A$ on a complex Hilbert space, which we can take as two-dimensional for simplicity, and start off by considering its hermitian conjugate $B=A^\dagger$, which we define as the complex conjugate of the transpose on a given orthonormal basis $\beta$. Thus, if the operator $A$ has a matrix representation $$[A]_\beta=\begin{pmatrix} a &...


5

One example: Complex analysis is used heavily in the proofs of the CPT theorem and spin-statistics theorem in relativistic quantum field theory. The classic book Streater and Wightman, PCT, Spin and Statistics, and All That is filled with complex analysis, such as the "edge of the wedge" theorem described in Section 2-5. This example isn't just some ...


5

There is no unique canonical notion of complex conjugation $C:H\to H$ of vectors in an abstract complex Hilbert space $H$. However, given a notion of complex conjugation $C:H\to H$, it is naturally to demand that it is an antiunitary map $$\forall v,w\in H:~~\langle C(v) | C(w)\rangle~=~\overline{\langle v | w\rangle}.\tag{1}$$ (This is e.g. the case for the ...


5

When we integrate the propagator with respect to $k^0$ (i.e. the energy), we encounter two poles: one at $\omega_{\mathbf{k}}=\sqrt{\mathbf{k}^2+m^2}$ and one at $-\omega_{\mathbf{k}}=-\sqrt{\mathbf{k}^2+m^2}$, where $\mathbf{k}$ is the 3-momentum. In order to regularize this integral, we move these poles slightly off of the real line by adding or ...


5

Using imaginary numbers for current in reactive components just happens to make the maths a lot simpler. In AC circuits there is typically some phase difference between the voltage and the current. Manipulating these quantities without the use of complex numbers, but instead just keeping track of the phase differences (such as the power factor), is a right ...


4

I'm not sure if I truly understand why you're unsure that all the branch currents should be sinusoidal in steady state. First, consider the simple case of just one source and linear circuit elements (and, as usual, stipulate that these linear circuit elements do not vary with time). Each branch current can be written as the output of a linear time-...


4

I believe that you can do all of physics apart from QM without using complex numbers: complex numbers are a convenience (generally because $e^{ix} = \cos x + i \sin x$), but they are only a convenience. However if you want to do QM you either end up using complex numbers or creating mathematical objects which have all the properties of complex numbers: you ...


4

An RLC circuit satisfies$$L\ddot{I}+R\dot{I}+C^{-1}I=\dot{V}.$$To solve this with AC voltage such as $V=V_0\cos\omega t,\,V_0\in\Bbb R$, it's convenient to take the real part of a complex choice of $I$ for the case $V=V_0\exp j\omega t$. Substituting $I=I_0\exp j\omega t,\,I_0\in\Bbb C$ gives$$I_0=\frac{j\omega V_0}{C^{-1}-\omega^2L+j\omega R}.$$The special ...


3

Yes, this is correct (so long as $z$ is a scalar complex number, and not an operator, if course).


3

This is not connected to black holes at all. Complex numbers are 2D objects to you need to be careful when looking at those visualizations as they are trying to represent a mapping from 2D onto 2D, which by definition requires a 4D space. So you are seeing some form of incomplete picture of the mapping and the page does not explain what it is showing. ...


3

You might, or might not, go for $$ |z\rangle= e^{-|z|^2/2} e^{za^\dagger}|0\rangle\qquad \Longrightarrow \\ a^\dagger |z\rangle= e^{-|z|^2/2} \frac{\partial}{\partial z} \left( e^{|z|^2/2} |z\rangle \right)\ . $$


3

Why the integral is ‘easy’ So it is important to understand that the non-convergence in this case is “not so bad” in the sense that if you just do the integral you would first multiply both top and bottom by complex conjugate as $$I=\int_{-\infty}^\infty\frac{d\omega}{2\pi}~\frac{-i\omega - \epsilon}{\omega^2 + \epsilon^2},$$ and then you can argue that ...


3

$\let\s=\sigma \let\pd=\partial \let\dn=\downarrow \let\up=\uparrow \let\dag=\dagger \def\hp{\hat p} \def\hs{\hat s} \def\hx{\hat x} \def\hy{\hat y} \def\hz{\hat z} \def\hI{{\hat I}} \def\hL{{\hat L}} \def\hP{{\skew{1}\hat P}} \def\hT{{\skew{1}\hat T}} \def\ket#1{|#1\rangle} \def\bra#1{\langle#1|} \def\braket#1#2{\langle#1|#2\rangle} \def\mxelm#1#2#3{\...


3

Physics deals with finding models that fit reality well, and then using them as explanations for further observations. Many models have limited domains of applicability and do not make sense outside them because their core assumptions do not work outside. Sometimes a model does produce unexpected predictions outside the normal range and they actually fit ...


3

There's a couple of different points of view on what exactly it means. Wick Rotation and diffusion Even in the Schrodinger equation, Wick rotation has an interesting viewpoint. Namely, that Wick rotating the Schrodinger equation gives us the diffusion equation! More explicitly, the Schrodinger equation (say, with no potential) is given as $$i\partial_t \...


3

The fact that $I_2$, in the second form, vanishes is nothing but a direct application of the so-called Riemann-Lebesgue lemma. No computations are necessary.


3

In my mind, what it means is that, to take the complex conjugate or transpose of a linear operator, it first has to be expressed as a matrix. Expressing it as a matrix means that you have to choose a particular orthogonal basis. It seems a bit confusing to me, because any matrix operator (apart from the identity matrix) expressed in (or translated to) a ...


3

One could say that all numbers are imaginary in that sense, they are abstract concepts that we force on a relation into nature just to allow us to have some descriptive power of our surroundings. The fact that natural numbers seem more "natural" or easy to relate to has nothing to do with what they really are; abstract ideas without physical meaning until we,...


3

The term "imaginary" for referring to the so-called "imaginary numbers" (which, by the way, are only a one-dimensional sliver of the full complex numbers that are actually used here) is a historical artifact that has caused way more confusion than it should, now. Let me say one thing that is crucially important here: There is no ontological difference ...


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