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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Why is the formula for expected value of nonobservables in quantum mechanics different then in regular statistics? [duplicate]
Specifically, why is the operator “sandwiched in” between $\Psi^*$ and $\Psi$? i.e. Why isn’t the formula just $$\langle \hat{Q} \rangle = \int \hat{Q}\cdot|\Psi|^2 dx = \int \hat{Q}\cdot\Psi \cdot \...
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How or what is the monster depicted in netflixs "a trip to infinity" when computing infinite calculus equations? [closed]
so i just watched netflixs documentary on infinity "a trip to infinity".. fascinating as it is..
in the 3rd segment or chapter theres a diagrammed representation of what is described as the ...
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Why does Wick rotation appear like an ordinary substitution in this example?
I've seen across several posts, that Wick rotation is not an ordinary substitution. Instead we're rotating the contour of integral and analytically continuing time $t$ to include imaginary time $-i\...
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Why is time harmonic follow the form of $e^{-i\omega t}$, not $e^{i\omega t}$? [closed]
In physics, when we solve an PDE or ODE, the solution usually has the form of
\begin{equation}
f=C_+e^{i\lambda x}+C_-e^{-i\lambda x}
\end{equation}
and the "causility" will eliminate one ...
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How can the mass of an unstable composite particle become complex?
To show where the resonances in cross sections come from, one usually considers the exact propagator in the interacting theory, which for a scalar is
$$iG(p^2)=\frac{i}{p^2-m_R^2+\Sigma(p^2)+i\epsilon}...
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Planck constant imaginary instead of imaginary PDE coefficients in the Schrödinger equation
Trying to get a first understanding of QM. The Schrödinger equation in standard form for $\Psi$
$$ i \hbar\frac{\partial }{\partial t} \Psi(x,t)
=\left[-\frac{\hbar^2}{2m}\frac{\partial^2 }{\partial t^...
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Connection between the Beta Function and Residue Theorem?
When we define the bare coupling in Minimal Subtraction we write it as a Laurent series where the analytic part is identified with the finite, renormalized coupling and the nonanalytic part is ...
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Is Dirac theory just a real Clifford algebra?
The gamma matrices $\gamma^\mu$ appearing in the Dirac equation span the Clifford algebra ${\cal Cl}_{1,3}$ over real numbers. They are generators of Clifford algebra in that sense that their products:...
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Why do we use a complex (Schrodinger) equation in quantum mechanics, instead of two interconnected real equations of real functions? [duplicate]
Is it possible to write down two separate differential equations that are interconnected and describe the real and imaginary parts of the wave function, but these equations would still be equivalent ...
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Why is $\bar\phi(-k) = \bar\phi ^*(k)$?
In Peskin and Schroeder chapter 2 p. 20, they claim that for a real field $\phi(x)$, its Fourier transform $\bar{\phi}(k)$ obey
$$\bar{\phi}(-k) = \bar{\phi}^*(k)$$
I am confused as to why this is ...
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Time Reversal symmetry, Quaternions, and spin-1/2 systems
When one has a system with no spin and time reversal symmetry, one can conclude that the Hamiltonian entries (in a particular basis, of course) must all be real. Can something be said about the ...
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Complex Scalar Field propagator from equations of motion
I was going through Chapter 9 of Schwartz's QFT book and one of the results bothers me.
Suppose we have a complex scalar field theory, and we want to find the propagator associated to the complex ...
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How to interpret $\int\mathrm{d}^2z$? [duplicate]
In chapter 6 of Tong's lecture notes on string theory when calculating the Virasoro-Shapiro/4-point Tachyon amplitude he arrives at the integral
\begin{align*}
C(a, b) = \int\mathrm{d}^2z\ |z|^{2a-2}|...
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Condition to the holomorphy of a complex function
In Witten's note https://arxiv.org/abs/1803.04993, during the proof of Reeh-Schlieder theorem, he made an arguement that considering a function
$$g(u)=\langle\chi|\phi(x_1)\dots e^{\mathrm{i}Hu}\phi(...
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Understanding the equivalence of two general solutions of the harmonic oscillator differential equation
The solution of the following differential equation
$$ -kx(t) = m \frac{d^{2}x}{dt^{2}}, $$
with $\omega = \sqrt{k/m}$, is
$$ x(t) = C_{1}e^{-i\omega t} + C_{2}e^{i\omega t}.$$
The real part of this,
$...
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How does the oscillation of the Higgs field contribute to its role as mass-giver?
I read that the Higgs field is a pair of complex numbers at each point of spacetime, and since we know that the Higgs boson has a mass, I'm imagining that these complex numbers oscillate over time at ...
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The general structure of an $x-iy$ substitution in quantum mechanics
In both the algebraic (a la Dirac) solution to the quantum SHO and in the algebraic approach to determining the spectra of the angular momentum operators (generators of rotations), substitutions of ...
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Imaginary part of electric susceptibility
I am reading the book "The Quantum Theory of Light," by Rodney Loudon, and it says on page $24$ that in an environment with electric polarizability $\chi$, the relationship between the ...
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Integration along real axis with singularities
I'm trying to calculate Green function of wave equation
$\begin{align}
\bigg(\nabla^2 - \frac{\partial ^2}{\partial t^2}\bigg)G(\textbf{x},t;\textbf{x'},t')=\delta^3(\textbf{x-x'})\delta(t-t')
\end{...
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Are the equations of the Poynting vector and energy density of an electromagnetic wave only for the real waves?
So, my book says that the Poynting Vector associated to an electromagnetic wave in matter with permeability $\mu$ is $\mathbf{S} = \frac{1}{\mu} \mathbf{E} \times \mathbf{B}$. The thing is, I am ...
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Rotation by 360°, spin-1/2 fermions and quaternions
Rotating a spin-1/2 fermion by 360° multiplies the quantum state by -1.
Representing a continuous 360° rotation as a quaternion is also a multiplication by -1.
Is there a relationship between these ...
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Do commutation relations stay the same when we use different definitions to quantize a free scalar field?
I saw different ways to write the scalar field. For example (Tong p.23):
$$
\phi(x) = \int\frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2E_p}}
\left[a_pe^{ipx}+a_p^\dagger e^{-ipx}\right].\tag{2.18}
$$
And we ...
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Quaternions as rotation generators
The following exercise appears in Geometric Algebra for Physicists by Chris Doran and Anthony Lasenby in section 1.8.
The unit quaternions $i, j, k$ are generators of rotations about their ...
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Evaluating integral that include Fermi distribution function using residue theorem
$$
I=\int_{-\infty}^{+\infty} \frac{d\epsilon}{2\pi} f(\epsilon)
\left[
\frac{1}{(\epsilon-\epsilon_n +i0^+)^2(\epsilon-\epsilon_m +i0^+)} - \frac{1}{(\epsilon-\epsilon_n -i0^+)^2(\epsilon-\epsilon_m ...
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Complex conjugate of angular momentum operator
I am trying to derive relation 5.7.5 in " The Quantum Theory Of Fields " by Steven Weinberg :
$$-J^{(j)^*}_{\sigma,\sigma'} = (-1)^{\sigma-\sigma'} J^{(j)}_{-\sigma,-\sigma'}\tag{5.7.5}$$
...
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If time is denoted by an imaginary number, what does that mean? [closed]
Suppose, the initial velocity of a car is 1m/s, acceleration 2m/s^2, distance travelled (-8)m. What is the time required to reach that distance? This is the first question.
The answer of the first ...
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Integrating infinitesimal transformations in CFT
Consider a conformal transformation $z\mapsto f(z)$ of the complex plane described infinitesimally by the vector field $X\partial+\bar{X}\bar{\partial}$. I was wondering how one can, starting from the ...
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What does the imaginary component of the spherical harmonics tell us about the hydrogen atom orbital?
I just went through a derivation of the spherical harmonics. I totally get how graphing the real part of the spherical harmonics create the shapes of orbitals, but I'm not sure what the imaginary part ...
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Primary Fields under SCTs
Primary fields transform under global conformal transformations like
$$\tilde{\phi}(\tilde{x})=\Omega^{-\Delta}D(R)\phi(x),$$
where $R$ is the rotation associated to the conformal transformation. My ...
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Dropping of terms in an expression in a certain limit in the presence of the complex/imaginary unit
I have the following expression
$$
\frac{\sqrt{\delta_{0}}}{\delta_{0}+i\omega_{s}}
$$
If $\delta_{0}\ll1$, can I simply drop the $\delta_{0}$ in the denominator? Leading to
$$
-i\frac{\sqrt{\delta_{0}...
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Can anyone tell How to write the Real and Imaginary parts of Circular polarisation in terms of Unit vector components?
In my computational simulations, i want to mention the 'real' and 'imaginary' parts of 'Circular' and 'Elliptical' polarised light as an input. Now the syntax of that input is such that i have to give ...
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How do I find the BH curve from the frequency-complex permeability relationship?
I come here after a lot (I mean a lot) of research about complex permeability. I have a ferrite datasheet which give the complex permeability ($\mu'$ and $\mu''$) over frequency. I don't know how to ...
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Prove: Conformal map $w(z)$ transforms a trajectory in potential $U(z)=|dw/dz|^2$ to one in potential $V(z)=-|dz/dw|^2$
This is a conclusion given without proof in the Chinese version of Arnold's Methematical Methods in Classical Mechancs.
Contents related to this conclusion are missing from the English version (I ...
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Purpose of Hermitian adjoints?
During a QM lecture, we went over Hermitian adjoints. While I understand that it is the Hermitian conjugate of an operator, I do not understand what this represents, besides its definition. Also, I do ...
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What does the imaginary part of the exponential of an EM wave represent? [duplicate]
I've read about a bit and some people are saying that either the imaginary part is just forgotten, or that is the orthogonal polarisation.
These don't make sense to me as I thought the imaginary part ...
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Without resorting to QM how do I understand the real world uses of imaginary numbers
I am trying to understand the real world uses of imaginary numbers. I have been told many times that it is not just a convenient tool mathematicians invented but has its places in some fundamental ...
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Complex Cosmological constant
Does a complex cosmological constant ($\Lambda = a + ib,\quad b\neq 0$) exist? If it does exist, what does it represent physically?
For example, we interpret $\Lambda > 0$ as dS space and $\Lambda ...
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Complex Grassmann Dirac Functional - How do we integrate over it?
I'm following the Book of Brian Hatfield (Quantum Field Theory of point particles and Strings), p.192 here: For real Grassmann numbers (and Functionals thereof):
If $\Phi[\psi]$ is a functional, and $\...
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What is the value of $e^{i \frac{\pi}{2} \frac{1}{\sqrt{3}}}$? [closed]
Context:
There is a Unitary Matrix given: $\bf{U}=e^{i\pi\frac{\bf{H}}{2}}$ where $\bf{H}$ is a Hermitian matrix. And
$\bf{H}=\sqrt{3}\begin{bmatrix}\frac{1}{3}&0&\frac{\sqrt{2}}{3}\\0&\...
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Can I switch the convention of QCD by replacing coupling constant $g$ with $-g$?
There are two equivalent conventions in QCD that give two different definitions of the covariant derivative operator: ${D_\mu } = {\partial _\mu } - {\rm{i}}gA_\mu ^\alpha {T_\alpha }$ and ${D_\mu } = ...
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Eigenfunctions of $C\psi(x) = \psi^*(x)$
I am considering the complex conjugation operator $C\psi(x) = \psi^*(x)$.
As $C^2$ is the identity operator, I've deduced that the possible eigenvalues of the $C$ are $\pm 1.$ To determine the ...
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3
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What is the angle in a complex inner product?
The dot product in 2D or 3D Euclidean space between vectors $\mathbf{a}$ and $\mathbf{b}$ with magnitudes $|\mathbf{a}|$ and $|\mathbf{b}|$ can be written as
$$
\cos\theta = \frac{\mathbf{a}\cdot\...
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Can I see $Ψ^∗(x,t)$ as a linear functional which can be aplied on wavefunction $\Psi(x,t)$?
Let's say I have a wavefunction $\Psi(x,t) = A e^{i(kx−ωt)}$. Now I complex conjugate it which gives me $\Psi^∗(x,t)$.
My first question is: Does $\Psi^∗(x,t)$ live in the dual of a Hilbert space?
My ...
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Finding complex potential function
Let $\tilde{\chi} = z^{2} -iz + 3$ and use the following theorem to define a new complex potential function $\chi$ such that the unit circle is a streamline:
Theorem: Let $\tilde{\chi}$ be a complex ...
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Regularisation of supersymmetric two-point function
In four-dimensional Minkowski space, one of the building blocks for two-point functions in CFTs is the squared modulus of the separation between the two points $x_1$ and $x_2$,
$$x_{12}^2 = x_{12}^ax_{...
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Linear Response theory
I was reading through Lecture of of Prof Patrick Lee on Linear Response Theory. I have found the following relation and could not understand why is it true:
$$\Im \left\{\frac{1}{x + i\eta}\right\} = -...
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Can the QFT path integral be re-expressed using a real, positive-definite function of the action? [duplicate]
This question is based on my rather shaky grasp of QFT, so if I'm missing a key concept then just let me know!
If you're deriving the Schrodinger equation from the path integral as Feynman did, then ...
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Connection between Pauli matrices and Quaternions? [duplicate]
reading this sentence from wikipedia:
The Pauli matrices (after multiplication by i to make them anti-Hermitian) also generate transformations in the sense of Lie algebras: the matrices iσ1, iσ2, iσ3 ...
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Reality of Hermitian Operators
I know that Hermitian Operators have all their eigenvalues real. But are the operators themselves real too? So if A is an Hermitian Operator, should it be real, and thus, A = A* be true?
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Hermitian conjugates of Dirac equation
I am given the following Dirac equations:
$$
(\gamma^{\mu}p_{\mu} - m)u_s(p) = 0\;\;\;\;\;\;\;\;\;eqn\;(1)
$$
$$
(\gamma^{\mu}p_{\mu} + m)v_s(p) = 0\;\;\;\;\;\;\;\;\;eqn\;(2)
$$
where u are the ...
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