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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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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Can magnetic Susceptibility be complex?

I have seen that magnetic permeability can be complex at high frequencies. Since there is a lag between $B$ and $H$, their ratio $\mu$ can be complex. If so can the magnetic susceptibility be also ...
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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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Eigenvalue problem of $L_z$

From Shankar's QM book pg. 313, the eigenvalue problem for $L_z=XP_y-YP_x$ in polar coordinates is $$-i\hbar \frac{\partial \psi(\rho,\phi)}{\partial \phi}=l_z\psi(\rho,\phi)$$ since $L_z=-i\hbar\frac{...
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Harmonic Oscillator with imaginary frequency

What is an physical interpretation of these harmonic oscillators: $$\ddot{x}+i\cdot x=0$$ and $$\ddot{x}-1\cdot x=0.$$ I assume that the system satisfies this second order DE $$\ddot{x}+\omega^2\cdot ...
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Kasner-Arnold theorem energy role in Needham's "Visual Complex Analysis" (VCA)

In "Visual Complex Analysis" (chapter 5.X.6) Tristan Needham writes In general, positive energy orbits in either the attractive or repulsive field $F \propto r^A$ map to attractive orbits ...
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Why is the complex time not considered in the solution?

In the solution, why is the the complex time not considered and real and positive time is considered? Is it because complex time does not exist or there is some other reason?
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Energy minimization of complex periodic scalar field with modulus constraint

I'm looking for an elegant method to (in general numerically) minimize an energy functional $E(\psi)$ for a complex field $\psi(x,y,z)$ which I know will be periodic in $z$ direction (due to self ...
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Derivative of a real harmonic signal

simple question I can't figure out: $s(t) = A\cos(\omega t + \phi) = \mathfrak{R}[A e^{jwt}]$ is the temporal function of a real harmonic signal. I don't get how the derivative is still an imaginary ...
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Help with an integral in Peskin & Schroeder - QFT

In chapter 2, page 27, eq. 2.51, P&S solves the following integral - $$ \frac{4\pi}{8\pi^3} \int _0 ^\infty dp \ \frac{p^2 \ \ \ e ^{-it\sqrt{p^2 + m^2}}}{2\sqrt{p^2 + m^2}}.\tag{2.51}$$ My ...
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Imaginary numbers in AC circuits

I've heard/read multiple times that "the use of imaginary numbers in ac circuits simplyfy calculations". My questions is: how is the calculations simplified? (exaple calculations?) And what ...
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Electric current through the resistor of an AC $(L)RC$-circuit

I have a question regarding more complex circuits. For the sake of simplicity, suppose we have an AC source with an alternating voltage $U=U_0\cos{(\omega t)}$ and the angular frequency $\omega$ tied ...
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Why do we prefer to use $i$ with generators in Lie Algebra [duplicate]

I am reading A. Zee. Group theory in a Nutshell for Physicists and for some reason, he prefers to write the generators with an $i$ near them For example, a rotation can simply be described as: $$e^{\...
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On symmetry of Lorentz matrix

For Lorentz transformations, if we put $x^1=x$ and $x^2=ct$ and restrict ourselves to $2D$ we get $$x'=\gamma(x-\beta ct) \tag{1} $$ $$ct'=\gamma(ct-\beta x) \tag{2} $$ The matrix associated with this ...
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What do imaginary voltage/potential map onto in the physical world?

Follow up to this question: Why is there no physical interpretation to non-real potentials in classical electromagnetism?. Is there anything that the imaginary quantities in math map onto in the ...
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Why is there no physical interpretation to non-real potentials in classical electromagnetism?

In Griffiths 4ed, he solves Laplace's equation and omits one of the solutions because it is imaginary. What are the grounds on which imaginary solutions or values can be marked as "non-physical&...
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Intuition Behind for Complex Vector Spaces $ \langle v | Tv \rangle = 0$ for all $v \in V$ implies $T=0$

I have seen the mathematical proof for the statement, but intuitively, I know that the real vector space is a subspace of the complex. As such, I incorrectly presume that this theorem should hold, ...
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Is the Schrödinger equation the heat equation with imaginary constants?

Playing around with the Schrödinger equation, I separated the time partial derivative this way: $$\frac{\partial \Psi}{\partial t}=\frac{i\hbar}{2m}\frac{\partial^2\Psi}{\partial x^2}-\frac{i}{\hbar}V\...
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A contradiction in Nonrelativistic Quantum Field Theory

Reference : "Field Quantization" by W.Greiner & J.Reinhardt, Edition 1996. In the above reference as concerns the Hamilton density $\:\mathcal H\:$ and the Hamiltonian $\:H\:$ of the ...
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Gaussian Grassmann integral with complex/bosonic source term

I'm interested in solving the following multi-dimensional integral $$ \int d \theta d \bar{\theta} e^{-\bar{\theta}M \theta +\Lambda \theta + \bar{\theta} J } $$ where $\theta$ is a $N$-dimensional ...
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Wick rotation of real-space trajectories

In CFT 2D, we often like to wick rotate the minkowski theory, to end up with an euclidean theory. Often, we then use complex coordinates to parametrize the plane, to make things easier. For instance, ...
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How to map a complex plane Mobius transformation to 1+1D Minkowski real plane?

So consider the $(x,t)$ plane endowed with the minkowski metric, namely: $$ds^2 = dx^2-dt^2.$$ It is well known that we can Wick rotate the time coordinate to get to the Euclidean metric. This can be ...
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Are Grassmann numbers always "under the hood" if we deal with fermionic ladder / field operators?

For the set of all fermionic field operators $\Psi(x) | x \in \mathbb{R}^{3 +1}$, we won't find a $|\phi \rangle$ that is an eigenstate to the complete set of field operators, unless we make use of ...
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Confusion in group theory at the example of the Lie algebra $su(2)$

Please excuse me for this unstructured question: I have some problems with understanding group theoretical aspects relevant to physics and would like to make one of my confusions clear by discussing ...
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Property of a wavefunction

We know that a wavefunction can't be completely real, because then it would have some complex expectation values for some operators. If we let $\psi$ be a real wavefunction, then $$\langle P\rangle= \...
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Properties of time reversal operation

During a course of mine time-reversal symmetry was introduced as an anti-linear operator. One property the lecturer pointed is that \begin{equation} \langle\phi| A\psi\rangle \neq \langle A^\dagger \...
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Complex $\Phi$ to prove the group velocity of a wavepacket

I am reading a lecture note (author: Pr. Barton Zwiebach from the physics department of M.I.T., Course: 8.04 'Quantum Physics I', title: de Broglie Wavelength and Galilean Transformations, Phase ...
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Non-Hermitian solution for $SU(2)$ Lie algebra?

In QFT textbooks, representations for Lorentz group is constructed from $A$-spin & $B$-spin discussion. The Lie albegra of Lorentz group is $[J_i, J_j] = i\epsilon_{ijk} J_k,~[J_i, K_j] = i\...
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How should I use the complex permittivity of a material?

Here: $$\epsilon = \epsilon' + j *\epsilon''$$ I understand that the first part ($\epsilon'$) is the relative permittivity of a material, while the second part $\epsilon'' = \frac{\sigma}{\epsilon_0\...
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The Partition Function of $0$-Dimensional $\phi^{4}$ Theory

My question is related with this question. Several years ago, I posted an answer to the question, and the author of the reference removed the link permanently, now I have no clue what's going on. In ...
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Arbitrarity of $i$ in the propagator

My question is simple: how arbitrary can the factor in front of the propagator be? What I mean by that is, if we call the wave operator $K$ and the propagator $G$, I've seen different books use ...
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Why is it always $\psi= A \sin kx + B \cos kx$ to solve a wave function?

Why is it always $\psi= A \sin kx + B \cos kx$ to solve a wave function instead of the one with $e^{ikx}$? Both are the solutions but the one with $e^{ikx}$ is seldom used.
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1D bound state for a real potential

The prof says: "for 1Dimensional bound states with a real potential, the wave function is real, up to a phase". The proof goes like this: 1D bound states are never degenerated. So $\Psi_{...
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What are the spaces in which quantum fields belong and how does that affect the hermitian conjugate of $\partial_{\mu}$?

In this post, I claimed in my answer that $\partial_{\mu}^{\dagger}=\partial_{\mu}$. The reason I claimed that is because $D^{\dagger}_{\mu}=\partial_{\mu}-iqA_{\mu}$ and I assumed that $$(\partial_{\...
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Questions on the Zakharov-Shabat inverse scattering paper

I am trying to work through the Zakharov and Shabat paper on inverse scattering for the nonlinear Schrodinger equation (PDF). I am stuck on section 2. Problem 1. I need to know how to reconstruct $\...
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2D rotation dynamics/control systems as a complex number

I have a dynamic system (it's a rocket in a 2D plane), that I'd like to model the orientation of using complex numbers to remove the need for trig functions in my ode. I'm having trouble defining the ...
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Applying Kramers-Kronig to the real part of conductivity results in a real and imaginary term

Following Tinkham's Introduction to Superconductivity, the complex conductivity in the two-fluid approximation is defined as \begin{equation} \sigma_i(\omega)\equiv\sigma_{1i}(\omega)-i\sigma_{2i} \...
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Real and Imaginary part for particle polarisability

I am trying to determine the real and imaginary parts from the following equation for the polarisability of particles $\Lambda_{MG}(\lambda)$: $$\Lambda_{MG}(\lambda) = \frac{\epsilon(\lambda) - \...
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Why don't we chose $\det(H)$ in winding number density?

Hello i have a short question on the winding number in chiral systems. If we have a chiral system described by a hamiltonian like this $$ H = \left [ \begin{array}{cc} 0 & K \\ K^{\dagger} & 0 ...
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Why are the electromagnetic wave equation is represented in a complex number? [duplicate]

The wave function $$ f(x,t) = A \sin (\vec{k}\cdot\vec{x} - \omega t+ \phi) $$ Can graphically describe the linear 2d wave propagation. Why this equations is written in this form: $$ f(x,t) = A [\cos ...
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What is the fundamental reason for the imaginary unit in Heisenberg's commutator relations?

The well known Heisenberg commutator relation $$[p,q]=\cfrac{\hbar}{i} \cdot \mathbb{I}$$ introduces the imaginary unit $i$ into quantum mechanics. I ask for the deeper reason: Why does the ...
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Separating a complex number into real and imaginary equations

I'm trying to find motion equations in a rotating frame, using the Cauchy-Riemann equations. I've currently arrive at the function $$\eta(t)=e^{-i\Omega t}\left[ x_0+v_{x,0}t + i(v_{y,0}+\Omega x_0)t ...
6 votes
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186 views

Why does time reversal symmetry requires a real Hamiltonian?

I have some problems understanding the consequences of time reversal symmetry. If Hamiltonian $H$ is symmetric under time reversal, it satisfies: $$ \mathcal{T} H \mathcal{T}^{-1} = H \quad \mathrm{...
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Anti-holomorphic contribution to 2d conformal algebra

I am reading Ginsparg's notes on 2D-CFT, and I am deeply confused about why Ginsparg states after (1.8) that the conformal algebra for 2d Euclidean space consists of two copies of the Witt algebra. My ...
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On the physics of representations of the Lorentz group over real vs. complex vector spaces

After reviewing this question as well as this one, I am left with some confusion, mainly about the nature of complex and real representations of the Lorentz group and how we do physics. I understand ...
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