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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Literature request - Dual quaternion dynamics
In my engineering practice, quaternions turned out to be much more practical than trigonometric rotation matrixes. I learned from this book on quaternions and dynamics how to describe rotation and ...
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What is the meaning of this complex derivative with respect to a wave function?
In quantum optimal control papers such as (Loading a Bose-Einstein Condensate onto an Optical Lattice, https://arxiv.org/abs/cond-mat/0209195) and (Introduction to the Pontryagin Maximum
Principle for ...
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Can off-shell particles have imaginary four-momentum components?
I have recently started studying quantum field theory and learned that some particles do not obey the on-shell condition which got me wondering about the physical limitations of this statement. Let me ...
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What material properties are necessary to embed a continuously rotating sphere into static space?
I am deeply intrigued by the properties of rotational connectivity in three-dimensional space, particularly as demonstrated by the Dirac-Belt Trick or Plate Trick. For a related concept, you might ...
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Why are complex coordinates outlawed in physics?
In the case of the Kerr metric, for a high enough angular velocity such that on transformation to Boyer-Lindquist coordinates yields complex coordinates for the event horizon, why is it assumed then ...
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Fermi poles expansion
I want to prove the following formula:
$$\frac{e^{-tE}}{1+e^{-\beta E}} = \frac{1}{\beta}\sum_{k \in \frac{2\pi}{\beta}\mathbb{Z}}\frac{e^{-ikt}}{-ik+E},$$
for $\beta > t > 0$. I know the trick ...
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Why is the gauge field $A_\mu$ real for $\mathrm{U}(N)$ symmetry?
From my notes I have that
The transformation law, $$A_\mu\to MA_\mu M+\frac{i}{g}\left(\partial_\mu M\right)M^\dagger\tag{1}$$ can be realised if $A_\mu$ is an element of the Lie algebra. It can ...
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Computational complexity of approximating partition functions
I would like to understand the computational complexity of approximating the partition function of 2D Ising model with complex external magnetic field and complex couplings for the following cases:
...
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Explanation of complex number in alternating current [duplicate]
Can anyone pls explain how complex numbers are used in alternating current,I was reading about rl circuit and found out they changed voltage $$V \sin \omega t \tag{1}$$ to $V e^{i\omega t}$.
as we ...
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Orientation of the Curve when deriving the Conserved charge in Conformal field theory
I am reading Polchinski Volume 1, and I am stuck at equation for the momentum operator in free scalar field theory in $2D$ euclidian space which is given as :
$$p^\mu=\frac{1}{2\pi\ i}\oint_C \left( ...
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Why is it not possible to describe general Lorentz Boosts using Hyperbolic Quaternions?
The Lorentz Boosts (for 1+1D) can be described by the Split-Complex Numbers. A Lorentz Boost in the direction of $x$ with the rapidity $\alpha$ for a 1+1D-system can be calculated using
$$q \mapsto e^{...
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Why is it reasonable to use Biquaternions for Lorentz-Transformations?
I've read that biquaternions can be used for Lorentz-Transformations using the formula $$q \mapsto e^{\alpha h \mu/2}e^{\phi\epsilon/2} q \overline{e^{\alpha h \mu/2}e^{\phi\epsilon/2}}^{*},$$
$\alpha$...
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Klein Gordon equation 1+1d driven by point time-oscillating source $\cos(\Omega t)\delta(x)$
Consider the classical Klein Gordon equation in 1+1d, $(\partial_t^2-\partial_x^2-a^2)\phi=\cos(\Omega t)\delta(x)$, where $a$ is real. The formal solution is, $$\phi(x,t)=-\frac{1}{2}\int\frac{d\...
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Calculation of the complex field using Huygens sources
I would like to calculate the complex field at the point $\vec{r}=(x, y, z)$ in three-dimensional space. The field is generated by two point sources with position vectors $\vec{s}=(s_{x}, s_{y}, s_{z})...
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One doubt about Dirac notation [closed]
I am encountering an issue with Dirac notation and I hope someone can help me. Thank you in advance.
I know that if $|n\rangle$ and $|m\rangle$ are eigenstates of the time evolution operator $U = e^{-\...
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How to Derive the Time Evolution Equation for Quantum Phase?
In quantum mechanics, the wavefunction $\psi(x,t)$ outputs a complex number that describes the probability amplitude of finding a particle in a particular place and time. The complex number can be ...
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Correct way to take complex conjugate of the Schrodinger equation
There are one or two questions on this site regarding the complex conjugate of Schrodinger equation but they do not clear my doubt.
Question : The Schrodinger equation is $$iħ \frac{∂Ψ}{∂t} = HΨ $$ ...
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Mapping between generalized forces and external torque of a rigid body whose rotation is described by quaternion is not unique(?)
In this paper the mapping between generalized forces and external torques for a rigid body (when the rotation is described by a quaternion) is derived:
$$\textit{F}_Q = 2\textbf{G}^TT'$$
where $\...
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GR and Riemann Surfaces -- does the complex plane have anything to do with it?
I have only the vaguest understanding of Riemann Surfaces -- my sense is that Einstein used them in General Relativity because of their shape.
But Riemann Surfaces I think are not just deformations of ...
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Time derivative of complex conjugate wave function [duplicate]
We have
$$\frac{\partial \Psi}{\partial t} = \frac{i\hbar}{2m} \frac{\partial^2 \Psi}{\partial x^2} - \frac{i}{\hbar}V\Psi$$$$\frac{\partial \Psi^*}{\partial t} = -\frac{i\hbar}{2m} \frac{\partial^2 \...
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Splitting Scalar into Holomorphic and Anti-Holomorphic Parts
I am reading Tong’s string theory lecture notes. On page 78, he splits the 2d free scalar into left- and right-moving parts, seemingly using the classical equation of motion as justification.
Why is ...
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How can the Bloch sphere, built from one complex dimension, specify 2-complex dimensional Pauli spinors?
Two-component spinors can be identified with points on the surface of the Bloch Sphere. The Bloch sphere is constructed from the 1-complex-dimensional complex plane plus the point at infinity.
How ...
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Equation for real/complex $\phi^4$ theory
On wikipedia (see this link), the Lagrangians of the $\phi^4$ equation for real AND complex scalar fields are given. One may derive the Klein-Gordon equation by inserting into the Euler-Lagrange-...
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Is the scalar field in the Yukawa interaction real or complex?
Consider a theory containing a Dirac field $\psi$ and a scalar field $\varphi$ where the only interaction is given by a Yukawa potential
$$
V = -g\bar{\psi}\varphi\psi
$$
I know that real scalar ...
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Lorentz Boosts and Hyperbolic Quaternions
How do you use the Hyperbolic Quaternion formula for a Lorentz Boost? I'm refering to https://en.wikipedia.org/wiki/Hyperbolic_quaternion :
$$t' + x'r = (\cosh(a) t + x\sinh(a)) + (\sinh(a) t + x\cosh(...
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If quaternions are an extension of complex numbers, is there a study of EM wave theory in terms of quaternions?
I find that in standard textbooks and online resources, basic EM wave theory (such as radiation, plane wave solutions, polarization) is discussed by treating fields in terms of complex numbers. Is ...
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Temporal absorption coefficient from complex wavenumber
A complex wavenumber $k=\beta-i\alpha$ can be defined, that when substituted into a time-harmonic solution $e^{i(\omega t - kx)}$ yields $$e^{-\alpha x}e^{i(\omega t - \beta x)}$$
The first negative ...
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Why is there attenuation or amplification of the electric field depending on the sign of angle?(Maxwell's equations in free space)
If you solve Maxwell's equations in free space for a complex $\epsilon$ you get this equation:
$\triangledown^{2}E = -\mu \left |\epsilon\right |e^{i\phi}\omega^{2}E$.
Compared to the equation when $\...
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A question regarding Coulomb sum in two dimension
The following arguments can be found in texts about Laughlin's wavefunction and theta function such as Laughlin's paper "Spin hamiltonian for which quantum hall wavefunction is exact". It is ...
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Intensity and complex electric field
I have come across an issue with the use of complex electric and magnetic fields that I just cannot quite figure out. I will lay out my thought-process and I would like to know if and why it is wrong.
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Difference of $p^0$ and $E_p$
In QFT when I learn about Feynman-propagator, I see such an expression:
$$
\frac{1}{2E_p}e^{-iE_p(x^0-y^0)}=-\frac{1}{2\pi i}\int_Cdp^0\frac{e^{-ip^0(x^0-y^0)}}{(p^0-E_p)(p^0+E_p)}.
$$
I know that ...
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Doran Geo Algebra for Physicists Exercise 2.9 [closed]
In the question says
The Cayley-Klein parameters are a set of four real numbers $\alpha$, $\beta$, $\gamma$ and $\delta$ subject to the normalisation condition $\alpha^2+\beta^2+\gamma^2+\delta^2=1$
...
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Scalar QED amplitudes with BCFW Recursion Relation
(This question comes from exercise 3.5 of Elvang's and Huang's "Scattering amplitudes in Gauge Theory and Gravity" book. This is not for a class, this is to learn a new technique; albeit I ...
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How do reality conditions on complexified Minkowski space induce conjugation on Spinor space
So I am following this script here: https://arxiv.org/abs/1712.02196
I am already stuck at chapter 1.3: I understand for the three cases Lorentzian, Euclidean and Split case that the coordinates need ...
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Complex BCFW-shift of Parke-Taylor amplitude
(This question stems from problem 3.3 of Elvang's and Huang's "Scattering Amplitudes in Gauge Theory and Gravity" book).
Consider the Parke-Taylor amplitude given as
\begin{equation}
A_n[1^- ...
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Why we take only the real part of a solution as the actual motion?
I am taking Analytical Mechanics, and in Goldstein's book, chapter 6 (page 241) about linear oscillations, he says the following:
"... $\eta_i=Ca_ie^{-i\omega t}$ (6.11) ... It is understood of ...
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How does Snell's law work with a complex refractive index?
In order to calculate Fresnel coefficients for layered media, we often need to calculate the angle that light travels inside a material with complex refractive index. Naturally, this is related to the ...
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Complex coordinates $ds^2 = dzdz̄$ in 2d
I have a very elementary question about complex coordinates in two dimensions. When we have a 2D Euclidean space,
$$ds^2 = dx^2 +dy^2$$
and we go to complex coordinates:
$$z = x + iy$$ $$z̄ = x - iy$$
...
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Worldsheet action in the presence of background fields in complex coordinates
We will start with the worldsheet action under massless background fields - the graviton $G_{\mu\nu}$ and Kalb-Ramond field $B_{\mu\nu}$ (we choose to exclude the dilaton $\Phi$ that also appears in ...
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References for multifractal complex measures?
I am learning about multifractal formalism recently. It seems nearly all the work is done for real-valued measures.
Question: I am wondering whether there is study (or even definition!) for ...
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Proving a Grassmann integral identity
How to prove the following identity
$$
\begin{align}
\int {\rm d} \eta_{1} {\rm d} \bar{\eta}_{1} \exp\left(a \left(\bar{\eta}_{1}-\bar{\eta}_{0}\right)\left(\eta_{1}-\eta_{0}\right) + b \left(\bar{\...
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I want to know about origin of non-Hermitian quantum field theory model having two complex scalar fields $\phi_1$ and $\phi_2$
In the paper Symmetries and conservation laws in non-Hermitian field theories by Jean Alexandre, Peter Millington, and Dries Seynaeve, Phys. Rev. D 96, 065027 the authors use this Lagrangian:
$$ L = ...
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Closed form expression of 2D CFT integral
I am currently working on a 2d CFT and am wanting to compute a complex plane integral, making sure I take into consideration potential contact terms as well. The integral in question is
$$ \int_{\...
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Integration over the complex plane and the completeness relation of the coherent states [duplicate]
I am studying some of the properties of coherent states using the book "Introductory Quantum Optics" by C. Gerry & L. Knight. (C. Gerry & L. Knight, Chapter 3, Section 5) And when I ...
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Hermiticity of Majorana Fermions: SYK Model
The SYK Hamiltonian is defined as
$$H = -\frac{1}{4!}\sum_{i,j,k,l=0}^{L-1} J_{ijkl} c^x_{i}c^x_{j}c^x_{k}c^x_{l},$$
where $J_{ijkl}$ is a random all-to-all interaction strength which is normally ...
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On complex impedance representation and Riemann surfaces
We know that a complex number, $z=re^{i\phi}$, can be represented with infinitely many phases, $\phi+2\pi n$, for integer $n$, as can be easily seen from the equivalent picture of a vector on the ...
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Hamiltonian of a complex scalar quantum field
Consider the Lagrangian density of a complex scalar quantum field:
$$
\mathcal{L} = (\partial_\mu\varphi^\dagger)(\partial^\mu\varphi) - m^2\varphi^\dagger\varphi
$$
With the conjugate momenta $\pi^\...
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Gaussian wave packet with complex coefficients [closed]
I am trying to obtain a representation of the momentum-space wavefunction $<p'|\alpha>$
Its position space wavefunction is given as
$$ <x'|\alpha> = N \exp [-(a+ib)x'^2 +(c+id)x'] $$
where ...
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Generalizing the von Mises Criterion to Complex Stress Tensors
I was deriving the von Mises maximum distortion energy criterion:
$${\displaystyle \sigma _{\text{v}}={\sqrt {{\frac {1}{2}}\left[(\sigma _{1}-\sigma _{2})^{2}+(\sigma _{2}-\sigma _{3})^{2}+(\sigma _{...
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How to tell if a composite boson field should be real or complex?
Let's say I have a system with two species of fermions, $f$ and $c$, where $f$'s are neutral but $c$'s are charged. Each of these has its own $U(1)$ related to particle-number conservation.
Now, if I ...
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