Skip to main content

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.

Filter by
Sorted by
Tagged with
3 votes
1 answer
77 views

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-...
Octavius's user avatar
  • 725
1 vote
1 answer
68 views

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 ...
paulina's user avatar
  • 1,560
2 votes
1 answer
80 views

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(...
entiges_Enton's user avatar
1 vote
3 answers
106 views

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 ...
RajaKrishnappa's user avatar
0 votes
0 answers
19 views

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 ...
korokame's user avatar
0 votes
1 answer
39 views

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 $\...
ElectronicsBeginner's user avatar
0 votes
0 answers
11 views

$p(z)$ polynomial with $p(0) \neq 0 \neq p(1)$; $\int_{\partial R} \frac{p(z)}{(z-1)z^2}dz$; $R=[-1,2] \times [-1,3]$ [migrated]

Consider $p(z)$ a polynomial such that $p(0) \neq 0 \neq p(1)$ and the rectangle $R=[-1,2] \times [-1,3]$, calculate $\int_{\partial R} \frac{p(z)}{(z-1)z^2}dz$ The poles $P=\{0,1\}$ are in the ...
J P's user avatar
  • 101
3 votes
0 answers
49 views

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 ...
fdsfsd sd's user avatar
0 votes
1 answer
153 views

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. ...
Hervé Schmit-Veiler's user avatar
-1 votes
1 answer
52 views

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 ...
Gao Minghao's user avatar
1 vote
1 answer
68 views

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 $\alpha^2+\beta^2+\gamma^2+\delta^2=1$ ...
Cro's user avatar
  • 137
1 vote
0 answers
28 views

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 ...
MathZilla's user avatar
  • 704
0 votes
0 answers
14 views

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 ...
Confuse-ray30's user avatar
2 votes
1 answer
36 views

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^- ...
MathZilla's user avatar
  • 704
1 vote
3 answers
137 views

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 ...
A24601's user avatar
  • 13
1 vote
1 answer
88 views

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 ...
cheekylittleduck's user avatar
4 votes
3 answers
108 views

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: $$z = x + iy$$ $$z̄ = x - iy$$ ...
j_stoney's user avatar
1 vote
1 answer
47 views

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 ...
Daniel Vainshtein's user avatar
0 votes
0 answers
9 views

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 ...
MikeG's user avatar
  • 101
3 votes
2 answers
363 views

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{\...
Faber Bosch's user avatar
0 votes
0 answers
46 views

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 = ...
Kawaljeet Kaur 's user avatar
3 votes
3 answers
180 views

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_{\...
NoName's user avatar
  • 63
0 votes
0 answers
18 views

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 ...
Uriel Casco D's user avatar
0 votes
0 answers
54 views

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 ...
Young Kindaichi's user avatar
1 vote
1 answer
49 views

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 ...
user135626's user avatar
4 votes
2 answers
459 views

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^\...
paulina's user avatar
  • 1,560
1 vote
1 answer
60 views

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 ...
raccoon's user avatar
  • 11
0 votes
0 answers
38 views

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 _{...
User198's user avatar
  • 435
1 vote
1 answer
57 views

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 ...
dumbpotato's user avatar
-1 votes
1 answer
87 views

Is complex integration needed when normalizing a wave function?

I have to find $\phi_0$ from following wave function in the momentum space: \begin{equation} \phi(k) = \phi_0 \text{exp}\bigg(-\frac{(k-k_0)^2}{2\kappa^2}\bigg) \end{equation} I know that I have to ...
haifisch123's user avatar
0 votes
0 answers
45 views

Stokes' vectors and quaternions

About a month ago, I was presented in my Optics class (classical and electromagnetic optics, to be more precise) the representation of partially polarised light through Stokes' vectors. Now, it ...
Lagrangiano's user avatar
  • 1,539
1 vote
1 answer
80 views

Kramers-Kronig relations for a Gaussian function

Consider a function of a complex variable $\omega$ which is given by $f(\omega) = e^{-\omega^2/2}$. This function is symmetric, holomorphic everywhere, and vanishes as $|\omega| \rightarrow \infty$. ...
user19642323's user avatar
1 vote
0 answers
136 views

How is Wick rotation an analytic continuation?

Wick rotation is formally described by the transformation $$t \mapsto it.$$ In many place it is stated more rigorously as an analytic continuation into imaginary time. I understand why we do it but ...
CBBAM's user avatar
  • 3,340
1 vote
1 answer
71 views

Planar spin in two-dimensional CFT

I have several questions regarding the definition of planar spin. I was reading the big yellow book (by Di Francesco et. al.) Section 5.1.5 looks a little mysterious. Look at 5.25, which is the two-...
hossein mohammadi's user avatar
2 votes
1 answer
92 views

How can the Rabi frequency be complex?

I've been doing some reading and came across a simple implementation of the Hadamard gate using Rabi oscillations of an atom in a laser field. However, the author mentions that it required the Rabi ...
rb101's user avatar
  • 35
0 votes
2 answers
93 views

Conflicting definitions of vector conjugate in QM

Let $e$ be a finitely matrix representable operator. In physics, specially in quantum mechanics (QM), it is customary to define the conjugate operator $e^{\dagger}$, as the adjoint or the Hermitian ...
physicsrev's user avatar
1 vote
0 answers
69 views

Complex gaussian integral with a complex action and different source terms [duplicate]

I am trying to use the following Gaussian path integral identity $$\int D[\phi_1,\phi_1^*,\cdots,\phi_n,\phi_n^*] \exp(i\int z^\dagger D z+i\int f^\dagger z+z^\dagger g) = \det{D}^{-1}\exp(-i\int f^\...
user1830663's user avatar
0 votes
2 answers
50 views

Possibility of complex EM waves

I'm currently studying Quantum Mechanics, and I have just been presented Schrödinger's (time dependent) equation. Of course, the first solution to said equation I've been taught is that of a (complex) ...
Lagrangiano's user avatar
  • 1,539
5 votes
0 answers
200 views

Do all qubits have complex coefficients or are there “real qubits” as well?

I understand that a qubit is a quantum system with two basic states $|0\rangle$ and $|1\rangle$, so a general pure state of the qubit will be described by a linear combination $\lambda|0\rangle + \mu|...
Gro-Tsen's user avatar
  • 810
0 votes
0 answers
59 views

What is an imaginary gauge potential?

This paper considers a generalised Strum-Liouville equation, that is equations of the form $$ \left[-\frac{d}{dx}p(x)\frac{d}{dx}-\frac{i}{2}\left(\lambda_1(x)\frac{d}{dx}+\frac{d}{dx}\lambda_2(x)\...
bas's user avatar
  • 121
0 votes
0 answers
66 views

Why is psi square a possibility? [duplicate]

Is psi square just an assumption? Or there is a physical reason why they defined like that? My procedure is: It is intuitive for me to think possibility is proportional to energy distribution. ...
user avatar
1 vote
1 answer
86 views

Adjoint and index notation in Weyl field context

In the answer to a question I previously asked, the following manipulation was done but I don’t understand it$.$ $$ (U_{jm}\psi_m)^\dagger=\psi_m^\dagger U_{mj}^\dagger $$ aside from the context from ...
JohnA.'s user avatar
  • 1,713
1 vote
2 answers
138 views

Charge conjugated Dirac equation

I would very much like to understand the motivation behind the correlation between: $(i\partial\!\!/-eA\!\!/-m)\psi=0$ and $(i\partial\!\!/+eA\!\!/-m)\psi_c=0$ when dealing with the derivation of the ...
Xhorxho's user avatar
  • 151
1 vote
0 answers
64 views

Non-Hermitian Hamiltonian in the Heisenberg Picture

I am trying to study a system whose Hamiltonian, after some transformations can be written as \begin{equation} \hat{H} = \hat{N}_1(\omega-i\mu)+\hat{N}_2(\omega +i\mu)+\omega\hat{\mathbb{I}}, \end{...
Jim Charamis's user avatar
1 vote
0 answers
31 views

What is special about conformal field theory in 2d? [duplicate]

In the most of textbooks about CFT, the special case of 2d is noticed in which complex coordinates play important role and it reads some results like the conformal transformation of energy-momentum ...
mon's user avatar
  • 11
1 vote
1 answer
64 views

Beam splitter with complex parameter

A non-polarizing beam splitter can usually be described by a unitary operator such as $U=e^{i\theta(a^\dagger b+b^\dagger a)}$ given a parameter $\theta\in \mathbb R$ and a pair of independent modes $...
Quantastic's user avatar
3 votes
1 answer
852 views

Wave amplitude as a complex number?

In section 1-3 An experiment with waves of The Feynman Lectures on Physics (https://www.feynmanlectures.caltech.edu/III_01.html) it says: "The instantaneous height of the water wave at the ...
ERP's user avatar
  • 167
1 vote
0 answers
91 views

Standard Model irreducible representation notations

Does anyone know what the significance/meaning of the subscripts on the following notations for the irreps. in the Standard Model? an up, down left-hand fermion (quark) pair $(u,d)_L$ is denoted: $(\...
Bijou Smith's user avatar
-1 votes
1 answer
51 views

Multiplying a force vector by rotated unit vector produces strange results [closed]

I'm by no means an expert in math, what I'm trying to do is to Isolate a force aligned with a vehicle in a game (specifically to do a directional friction). the equation is simple I take a vector $v_1$...
LemonJumps's user avatar
1 vote
1 answer
162 views

Application of Cauchy residue theorem to Matsubara sums

For reference, this derivation is closely related to the discussion on pp. 169-173 of Altland and Simons. In quantum field theory (specifically when calculating free fermionic propagators via coherent ...
Jamin's user avatar
  • 75

1
2 3 4 5
21