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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53 views

Square root of -1 vs many-worlds [closed]

I am not sure if this is a dublicate. When I see the suggested questions about the subject they seem to take the concept more or less for granted. Sometimes we may benefit from seemingly unrealistic ...
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Why do physicists like to put the imaginary unit $\:i=\sqrt{-1}\:$ everywhere?

There are many disagreements of convention between mathematicians and physicists, but a recurring theme seems to be that physicists tend to insert unnecessary factors of $i = \sqrt{-1}$ into ...
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Eigenvalues of antiunitary operators

I have sometimes come across the statement that antiunitary operators have no eigenvalues. For example, on page 34 in the book "Topological Insulators and Topological Superconductors" by ...
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$Z=\int D[\phi]e^{iS(\phi)}$; is $Z$ real or complex?

Is the result of $\int D[\phi]e^{iS(\phi)}$ real or complex? If it is complex, how does the expectation value for a field, given as the following works? $$ \langle F\rangle = \frac{\int D[\phi] F(\phi)...
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How to explain the minus (negative) sign before Kinetic energy operator in Hamiltonian operator?

The Hamiltonian operator is normally written in this form: \begin{align} \large H_\mathrm{operator} = \ & \large \frac{-\hbar^2}{2m}\frac{\partial^2}{\partial x^2} &+ \quad & \large V(x) \\...
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Linear combination of group generators

In Matthew Robinson's book Symmetry and the Standard Model he explains that we have generators for rotations $J$ and for boosts $K$. To analyse the group structure, we will look at $N^\pm = J \pm i K$ ...
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I can't understand this paragraph on vector spaces. Can anyone help demistify this? [migrated]

So far from what i understand,a vector space is essentially an abstraction of the idea/properties of a vector that can be applied freely to many things. This paragraph is right after talking about the ...
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75 views

A Complex Field Meaning

Here are some conceptual questions I have been thinking about but I either am not sure about the answer or do not know if my thoughts are correct. Some might seem silly, but I ask nevertheless. Any ...
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115 views

Imaginary and real parts of a ground state wave function

Here're key-frames from a time evolution of a quantum harmonic oscillator in it's ground state (source): So, I do understand that such a state may equally well describe grand variety of systems, but ...
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Does anybody know of a good source to learn about “Imaginary Time” in depth?

I have done plenty of popular science reading about this pet idea of Hawking's, but I'm planning on working on a more in-depth project involving it for a class. Can anyone direct me to any scientific ...
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Imaginary component of a wavefunction - simple meaning

Is it safe to say that imaginary part of wavefunction (one that "located" in a Complex plane around it's real component) - always represents some physical entity, that cannot be ...
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What is the origin of the overall phase of a state ket?

In, Sakurai's textbook on quantum mechanics, he mentions the "Overall phase" of a state ket while discussing the spin half system. What is the origin of this phase term? Is it because the ...
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Showing that a two-dimensional Euclidean CFT ghost action is hermitian

At the end of chapter 6 in Polchinski's String Theory book he says that the $c$ ghost is anti-hermitian. With that information, I tried to show that the action for the $bc$ system \begin{equation} S= \...
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Complex physical quantities

I have a question regarding complex physical quantities. Why do we consider only the real part of a complex physical quantity? Why not the modulus? Since, for $z=a+bi$, we have $|z| = \sqrt{a^2+b^2}$, ...
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Inner product of $\langle \phi | \psi \rangle$ gives a complex value - why/meaning?

Say I have the following two wave-functions: $$ |\psi\rangle= \pmatrix{a_1+i b_1\\ a_2+ib_2 } $$ $$ |\phi\rangle= \pmatrix{c_1+i d_1\\ c_2+id_2 } $$ Since these are unit vectors of the Hilbert space, ...
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Complex fourier transformation function [duplicate]

can the FT function can be a complex function ? and if yes what does it mean because in all cases i came across till now the FT function only had a real part
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Meaning of complex Fourier transformation function

what does it mean, if the Fourier transformation function of an electromagnetic wave is complex? I know that normally the FT function $f(k)$ shows the wavenumbers that are involved in the wave. but i ...
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Wave equation and skin-depth

I initially posted this question to the math stack overflow, but realized that it is really a physics question This is a follow-up question to one of my unanswered questions https://math....
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Solution to differential equation

If I have a differential equations of the form $$\frac {d^2y}{dt^2}=\alpha^2y$$ Assuming the roots of the characteristic equation is complex the solution to the differential equation is: $$y=C_1e^{j\...
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Why not just complex conjugate bras and kets instead of Hermitian conjugate?

I read that one equation involving bras, kets and operators, implies another equation (its transpose conjugate), analogous to how one equation involving complex numbers implies its complex conjugate ...
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Electric field/intensity for complex envelope

When calculating the propagation of a pulse I can either use an envelope-based or a carrier-based approach. For the carrier-based approach I can define my (focused) pulse with the central frequency $\...
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(Non-)Hermiticity of Dirac operator

I have a Dirac operator given by \begin{equation} D\!\!\!/[A, A^{5}]=\gamma^\mu D_\mu=\gamma^\mu (\partial_{\mu} - {\rm i} A_{\mu} - {\rm i} \gamma_{5} A_{\mu}^{5}), \end{equation} where $A_{\mu}$ ...
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How the Lorentz algebra bracket relations change during complexification

There have been a lot of questions in the past about the subtleties of the Lorentz algebra. In particular the usage of the real Lie algebra $\mathfrak{so}(1,3;\Bbb R)$ and it's complexification which ...
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Mysterious path integral divergence after Hubbard-Stratonovich transformation

Let us consider common gaussian path integral over some complex random field $\displaystyle \Psi (\mathbf{r})$: \begin{equation*} N=\int D\Psi ^{*} D\Psi \ \exp\left( -\int d^{n} r\ \Psi ^{*}\hat{K} \...
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Residue of the Fermi Distribution Function

In the "Lecture notes on many-body theory" by Michele Fabrizio, it is stated: How we do show that the Fermi distribution function $f(z)$ has residue $-T$? In the examples on Wikipedia, the ...
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How to decide if a mass matrix is real or complex?

In the lagrantian in particle physics, how to decide if a mass matrix is real or complex? As I know, a Dirac field is complex while a Majorana field is real. Does a complex mass matrix correspond to ...
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Complex time theories with spacetime $\mathbb{R}^3\times\mathbb{C}$

Are there any well-developed (string?..) theories assuming that, what we perceive as a (3+1) Minkowskian manifold, is a projection/compactification of a 5-dim spacetime, locally obtained via ...
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Gauge theory on complex manifolds

recently I'm pondering on defining gauge theory on complexified space. However, I found that it is hard to make electromagnetism (which will be the simplest gauge theory) work well on complex ...
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Deriving the complex dynamic modulus in a maxwell model

In the context of the Maxwell model for viscoelastic materials, how would one derive the complex dynamic modulus? It is defined as the ratio of the stress to the strain.
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Dielectric constant and conductivity in Maxwell's macroscopic equation

In my experimental physics lecture we are looking at the Maxwell equations in matter (macroscopic Maxwell equations) and there is a point where we jump from $$ \nabla \times \vec B = \mu_0 \biggl( \...
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Is the time dimension naturally linked to the real axis?

The real number axis is asymmetric against zero: for instance, multiplication of two negative or two positive numbers will produce a positive number, a square root of a negative number is not real, ...
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Gaussian integral with respect to Grassmann variables

Let $A$ be an antisymmetric matrix of even dimension $n$ and $\theta$ be a column vector consisting of $n$ Grassmann variables $\theta_i$. Then the solution of the integral $$\int d\theta_1\dots d\...
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In the Wick rotated path integral, are the paths functions of an imaginary time variable?

Consider the following action: \begin{equation} S=\int_{-\infty}^{\infty}[\frac{1}{2} \dot x^2(t)-V(x(t))]dt. \end{equation} I promoted $t$ to a complex parameter, and calculate the action over the ...
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Expression for the causal retarded potential for $t<0$ must give $0$ but my calculation produces a nonzero result. What's the mistake?

This question was previously asked here in the Mathematics StackExchange but using a slightly different notation. But I did not find the answer I was looking for or rather got two very different ...
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QED Lagrangian - real or complex?

I'm confused about the use of complex numbers in the QED Lagrangian: $$\mathcal{L}=\bar{\psi}(i\gamma^{\mu}\partial_{\mu} - m)\psi - \frac{1}{4}F_{\mu\nu}F^{\mu\nu}-e\bar{\psi}\gamma^{\mu}A_\mu\psi.$$ ...
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Solving coupled propogation equations for EM waves

I have recently come across a set of partial differentials that describe the propogation of two coupled EM fields, in a 2D system currently being investigated. In their most general form they are $\...
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Does swapping “$i$” with “$-i$” change a physical theory?

Mathematically speaking, "$i$" and "$-i$" are the two roots of the equation $x^2+1=0$ and it seems to me at least that there is no obvious way of distinguishing between them. Thus, ...
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Do we necessarily need real number for quantum? [closed]

In the quantum mechanics, one asked if the complex number was necessary? A typical answer was that it was not, or that it's simple direct product of real numbers. However, consider rational number to ...
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Complex exponential method of solving differential equations

In the twenty third Feynman lecture, the solution of the following differential equation is discussed: $$ \frac{d^2 x}{dt^2} + \frac{kx}{m} = \frac{F}{m}$$ AFter 'complexifying' this differential ...
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How to find the input impedance of a woodwind instrument? (playing frequency of a woodwind instrument)

I'm trying to reproduce the model described in this paper https://hal.archives-ouvertes.fr/file/index/docid/683477/filename/clarinette-logique-8.pdf. The logical clarinet is a succession of 18 ...
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The use of complex fields in electromagnetism

Virtually all treatments of electromagnetic wave propagation, and in particular of monochromatic plane waves, use basic complex analysis to simplify calculations. I am comfortable with these ...
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Derivative of a complex potential for the $\lambda \Phi^{4}$-model

A charged scalar particle is described by a complex field $\Phi(x) = \phi_{1}(x)+i\phi_{2}(x)$. Consider a Lagrangian of the $\lambda \Phi^{4}$-model whose potential in the Euclidean action is given ...
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Phasor transformation to sinus or cosinus?

In my EM waves lecture, our lecturer somehow explained the way we make phasor transformation of a particular function such as $$A\cos(\omega t-( \alpha +\beta z))u_{y}$$ converted into phasor form of $...
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Kramers-Kronig relations for geometric series

Suppose $\phi(z)$ is an analytic function in the upper complex plane, so it satisfies the Kramers-Kronig relations, i.e. \begin{equation} \Re\phi(w) = \frac{1}{\pi}\int \frac{\Im\phi(x)}{x-w} dx \end{...
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Quantum Mechanics Griffiths Problem [closed]

I was doing problem no. $4.4$ from Griffiths Third Edition. I cannot understand one thing related to the solution of the normalization of $Y_2^1$. $$Y_2^1 = -\sqrt{\frac{15}{8\pi}}e^{i\phi}sin\theta ...
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Imaginary part of complex non-Hermitian Hamiltonians

I have just started studying the $PT$-symmetric and non-hermitian hamiltonians. But I am not able to interpret the imaginary part of the hamiltonian. If Hamiltonian is basically the total energy of ...
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Are gaussian integrals also valid for complex constants? [duplicate]

In the WP article about propagators, there is an integral solved as: $$K(x,x';t)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}dk\,e^{ik(x-x')} e^{-\frac{i\hbar k^2 t}{2m}}=\left(\frac{m}{2\pi i\hbar t}\right)...
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Real and imaginary parts of derivative of complex velocity in fluids

Just by chance, I noticed that for an ideal fluid which has a complex velocity say $w=u-iv$, that its derivative with respect to $z=x+iy,$ when written out in real and imaginary parts, looks like $$\...
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110 views

Confusion About Complex and Scalar Fields

I have been reading QFT for the Gifted Amateur, on page 110, it reads: "For the case of the real scalar field...it is Hermitian. For our complex scalar field there is no reason why the field ...
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Problems deriving the Quantum Hamilton-Jacobi equation

This is my first question at Physics SE so please be kind. I am well versed in the etiquette over at Math SE, but not so much here. Anyway, I thought this question was better suited to this site ...

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