Questions tagged [complex-numbers]

Numbers of the form $\{z= x+ i\,y:\;x,\, y\in\mathbb{R}\}$ where $i^2 = -1$. Useful especially as quantum mechanics, where system states take complex vector values.

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1answer
299 views

Showing $I=\int d^3k\int dk^0\frac{1}{k^4}$ to be logarithmically divergent

Consider a momentum integral of the form $$I=\int d^3k\int dk^0\frac{1}{k^4}$$ where $k^2=(k^0)^2-(\vec{k})^2$ and the integral over $k^0$ runs from $-\infty$ to $+\infty$. This integral is common in ...
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1answer
80 views

Is there a relation between this function and black holes?

I was fiddling with this complex function visualizer and accidentally found this function which looks a lot like the blackhole visualizations that I see on the net: $$ f(z)=(z\bar{z}-1)^z $$ and I'm ...
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Supersymmetry Generator Definition for ${\cal N }= 1$

I am studying SYM $\mathcal{N}$ = 1 in D = 10, and using the bimodular representations for the 32x32 gamma matrices $\Gamma^a$. This means that I work with the off-diagonal 16x16 matrices, which I ...
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1answer
197 views

Why do we regard $z$ and $\bar z$ as independent in CFT? [duplicate]

I have been studying String Theory and CFT for a while, and I am sad to say I do not know why we treat $z$ and $\bar z$ as independent variables, and why we go on to consider the algebra $Vir\oplus\...
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Computation of Wigner Functions

The Wigner function can be computed as the Fourier transform of the Weyl-ordered characteristic function: $$ W(\alpha) = \frac{1}{\pi^2} \int e^{\lambda^* \alpha - \lambda \alpha^*} C_W(\lambda) d^2\...
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1answer
364 views

What is the hermitian of the time reversal operator?

So, I was reading about the time reversal operator, and I came to know that it can be expressed as: $$T = KU$$ where, $U$ is an unitary operator and $K$ is the complex conjugation operator. Now, if I ...
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173 views

Is it possible to have a complex gauge field?

Beyond the obvious fact that the particles in the standard model described by gauge fields do not have an anti-particle pair, is there a reason why a complex gauge field is typically not considered? ...
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26 views

Behaviour of $i$ in polarization identity in complex Hilbert space

I'm working through this video on youtube: https://www.youtube.com/watch?v=fjj6XNRtA40 But there's a step she makes that I don't understand, and when I try to do the proof myself I get stuck. This is ...
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1answer
65 views

Complex conjugate in inner products [duplicate]

When we solve for inner product of $\rvert a \rangle \cdot \rvert b \rangle$ we solve for $\langle a \rvert b \rangle$ where $\langle a \rvert$ is complex conjugate of $\rvert a \rangle$. However this ...
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72 views

Legal values of spin-1/2 field can take: $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$, .. (Grassmann)?

For the spin-1/2 fermion field $\psi$, we may choose it to be a spinor which needs to be Grassmann variable but can also be complex $\mathbb{C}$ Grassmann. (Dirac or Weyl spinor/fermion) We can ...
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1answer
84 views

2-sheeted Riemann surface with 2 branch cuts and Torus

A 2-Sheeted Riemann surface, with 2 branch cuts has a genus 1. A ring torus also has a genus 1 (In fact, section 13.4 of John Terning's book, modern supersymmetry and dynamics and duality claims that ...
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Question on notation for the inner product of complex vectors [duplicate]

Regarding the wiki: https://en.wikipedia.org/wiki/Sesquilinear_form#Hermitian_form you can see that the wiki states that physics defines the inner product for complex vectors as: $$\langle \, \...
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198 views

Solution as the real part of a complex exponential from simple harmonic motion

From the book entitled Classical Mechanics written by John R Taylor, chapter no 5, Simple Harmonic Motion. I'm just citing the lines. $$x(t)=\text{Re }Ce^{i\omega t}=\text{Re }A e^{i(\omega t-\...
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Beamsplitter complex notation

I am trying to understand the formalism presented in this paper about the transmission of a fiber ring resonator. By modelling the coupler as a beamsplitter, the authors write: $$E_{2}=rE_{1}+itE_{3}$...
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2answers
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Convergence Property of Path-Integral

Let the action be $$S= \int \bigg\{ \frac{1}{2} \big(\frac{dX}{dt}\big)^2 - V(X) \bigg\} d\tau$$ and the corresponding Path-Integral $$Z= \int DX(t) e^{iS}.$$ Since the convergence is not clear we ...
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1answer
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Rising/lowering operators and trigonometric functions

I've just started learning about angular momentum and spin theory, and when I came across the definitions of the rising and lowering operators, I noticed the inverse form looks suspiciously like the ...
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1answer
124 views

Path integral calculations $e^{i\omega 0^+}$

When computing correlation functions using the path integral formulation, I often need to compute integrals such as $$ \int_{-\infty}^\infty \frac{d\omega}{2\pi} \frac{1}{i\omega -\epsilon} $$ ...
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1answer
266 views

Complex integration in Peskin and Schroeder

In Peskin and Schroeder, I have a problem with a claim in equation (2.54), which I will rewrite more concisely here. He claims that we have the following equality : $$ \frac{1}{2E_p}e^{-iE_p(x_0)}-\...
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I'm confused about charge density and complex conductivity, complex permitivity!

I'm confused about some notations in electromagnetism. Please help me to solve these questions. I first studied electromagnetism with David cheng's book, but From the book : Lukas Novotny, Bert ...
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2answers
290 views

Can I write a 2-dimensional electric field as an analytic function on the complex plane? [closed]

Let's consider a two-dimensional electric field $\textbf{E}=\textbf{E}(\mathbf x)$, where $\mathbf x\in \mathbb R^2$, and $\textbf{E}$ is a vector representing the direction and strength of the field ...
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676 views

What is the physical interpretation of the Wick rotation?

What is the physical interpretation of the Wick rotation? How is it that we can just propose there's a new time coordinate tau? Are physicists saying time is modeled by an imaginary number? Isn't ...
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Derivation of Holomorphic Ward Identities in Franceso's CFT

In equation 5.37 of francesco's CFT he writes the Ward Identities for traslation symmetry in the language of holomorphic functions. He goes from \begin{equation} \frac{\partial}{\partial x^\mu} \...
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66 views

A. Zee Contour Integral

In A.Zee's book I have come a cross an integral which I found difficult to solve.
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3answers
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Intensity of the resultant of two complex waves [closed]

Suppose I have two waves: $Y_1= a_1e^{i(wt-kx1)}$ and $Y_2= a_2e^{i(wt-kx2)}$ I know by superposition $Y= Y_1+Y_2$ and intensity $(I) = |Y|² $ But how can I solve it. It seems hard for me to find the ...
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245 views

What is the physical interpretation of imaginary lengths?

My question is about the meaning of imaginary lengths, which occur often in the solution to various numerical problems in Physics. Generally imaginary quantities are discarded as nothing but ...
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57 views

2 in the Fermi’s Golden Rule

In the derivation of the Fermi's golden rule many authors expand periodic perturbation in this form $$\hat{V}=\hat{F} e^{-i \omega t}+\hat G e^{i \omega t}$$ However I do not understand the reason. ...
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Imaginary Capacitance and Imaginary Inductance interpretation [closed]

Today I came across with Complex Capacitances and Inductances for the first time: $$L(j\omega)=L_{real}+jL_{imaginary}$$ $$C(j\omega)=C_{real}+jC_{imaginary}$$ So I started looking at their meaning. ...
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LCR circuit , impedance, imaginary [duplicate]

What is the physical interpretation in behind impedance of capacitor and inductor? we know that resistance oppose the current in the same direction of voltage. But how capacitor and impedance oppose ...
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1answer
275 views

Need help understanding KVL in Phasor Notation

I may be missing some crucial observation but I'm getting stumped on this KVL proof for phasor notation. So to prove KVL in phasor notation I'd like to start off with what we know, KVL in a real ...
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237 views

Kramers-Kronig relations in susceptibility

I tried asking this question in the Math Stack Exchange but it got little attention since it was more focused on an application (and the notations provided were a little hard to understand), so ...
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2answers
502 views

Complex conjugated representation and its Young tableaux

This post is an exact copy of one that I posted in Math's site. I do this copy because people there suggested me to do it since, apparentely, in Mathematics and Physics we use different conventions ...
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1answer
115 views

Infinitesimal parameter of Lorentz transformation

I'm working through the SUSY lecture notes by Lambert, and he does something which seems strange to me during the calculation of the Wess Zumino model. He says the spinor $\psi$ has the ...
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2answers
265 views

Complex Potential describing inviscid fluid flow

I'm a Mathematics student, working through a homework sheet for a Fluid Mechanics module. The question is given: Consider the flow described by the complex potential $$w=4z+\frac{8}{z}.$$ ...
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1answer
218 views

Schrödinger Equation with Imaginary Potential

I am trying to solve the following equation (in 1D) and stuck in the middle of the way. Here's the equation: $$i\frac{\partial\psi}{\partial t}=C\cdot\frac{\partial^{2}\psi}{\partial x^{2}}+iD\cdot\...
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1answer
299 views

What is complex frequency? [closed]

I am learning EE, and about complex frequencies, but what is its physical meaning? What is it used for? Why is it? And only happen in the laplace transform?
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2answers
755 views

Euler-Lagrange equations from a complex Lagrangian

I'm looking for generalizations of the Euler-Lagrange equations that would be derived from a complex-valued Lagrangian density. I realize that “minimum” and “maximum” don't have obvious meaning for a ...
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61 views

Four point function with complex momenta?

It is well known that the four-point function $$\int_{\mathbb{R}^{3,1}}\frac {d^4 q}{((q+p_1)^2-i\epsilon)((q+p_2)^2-i\epsilon)((q+p_3)^2-i\epsilon)((q+p_4)^2-i\epsilon)}$$ can be computed using the ...
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1answer
456 views

Orthonormal basis written in Dirac Notation

$\left\{ e _ { i } \right\}$ is an orthonormal basis which has the orthonormal condition as following: $$e _ { i } ^ { T } \cdot e _ { j } = \delta _ { i j }$$ In Dirac Notation where $| i \rangle = | ...
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1answer
179 views

Contour Integration in Schwartz

In Matthew Schwartz's QFT text, on page 39, he has the following contour integral: $$\int_{-\infty}^{\infty}dk\frac{e^{ikr}-e^{-ikr}}{k+i\delta }.\tag{3.63}$$ This can be split into two terms, one ...
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1answer
403 views

Diffusion equation Lagrangian: what is the conjugate field?

Morse and Feshbach state without elaboration that the diffusion equation for temperature or concentration $\psi$ and its "conjugate" $\psi^*$ (quotation marks theirs) has Lagrangian density: $$L=-\...
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Can we get rid of complex number? [duplicate]

Complex number is very weird to me, especially so when they appear in engineering and physics equations. It is possible to represent complex number as real matrix where $i = \begin{pmatrix} 0&1\\...
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1answer
782 views

Derivation for angular acceleration from quaternion profile

Given a profile of unit quaternions $q(t)$ that represents the orientation of a body over time, I like to get the angular acceleration $\dot \omega (t)$. I tried to find a formula myself, but I get ...
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1answer
301 views

Potential must be real for Hamiltonian to be Hermitian?

I have seen a few proofs specify for finite wells, step functions, and harmonic oscillators, that $V$ must be real for $H$ to be Hermitian. Why is that? If we're solving the Schrodinger equation, we ...
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1answer
60 views

Confused over the complex term in the simple harmonic wave equation

I am trying to derive the general equation of Lamb wave. My book says that $$y = A\exp(i(kx−\omega t))$$ is the general equation of simple harmonic wave propagating in +ve $x$ direction. but I am ...
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1answer
145 views

Is it necessary to learn complex analysis in order to learn classical electrodynamics?

I am reading "Classical electricity and magnetism, chapter 1" by Wolfgang K. H. Panofsky and Melba Phillips. I am having little trouble on page 13 and afterwards. It talks about singularity, poles, $2^...
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Why doesn't Feynman take the real part of a complex EM wave before finding the intensity?

This is from The Feynman Lectures vol 2 chapter ch 33 on reflection of complex EM waves on metal. In the above I would expect $\frac{I_r}{I_i} = \frac{Re(E'_0)^2}{Re(E_0)^2}$, rather then $\frac{...
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1answer
109 views

Pure imaginary Schroedinger wave function

I know that the solutions to the time-dependent Schrodinger equation are always linear combinations of the form $$ \Psi(x,t)=\sum_n c_n e^{-iE_nt/\hbar} \psi_n(x) $$ If $ \Psi(x,0) $ is PURELY ...
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1answer
554 views

Action of complex conjugation on Hamiltonian

Consider a finite-dimensional non-relativistic QM system with hamiltonian $H$. Let $K$ denote the complex conjugation operator. What does $K H K$ simplify to, if the system is: (a) spin-zero; (b) spin-...
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46 views

But how exactly do you calculate the Joukowsky Airfoil, within a minimal margin of error?

After reading a fair bit of theory around the uses of conformal mapping to solve for the forces of lift acting on a wing, or a 2D cross section of the wing, in relation to the angle of attack. However ...
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1answer
432 views

Feynman $i\varepsilon$-prescription in path integral by adding an imaginary part to time

It is known that the well-definiteness of the path integral leads to the Feynman's $i\varepsilon$-prescription for the field propagator. I've found many ways of showing this in the literature, but it ...

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