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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221 views

Why isn't the Segal-Bargmann space used more often in Quantum Mechanics?

The Stone-von Neumann Theorem states that a Hilbert space on which is defined an irreducible set of operators which satisfy the exponentiated canonical commutation relations is unitarily equivalent to ...
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75 views

Why allow only one singularity at $z =0$ when defining the local conformal algebra in 2D?

The reference I'm using for this question is the review "Applied CFT" by Paul Ginsparg. In section 1.2 (Conformal algebra in 2 dimensions) he argues that if the metric is the Euclidean one $g_{\mu\nu}=...
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Notation on four-vectors using imaginary spacelike components [duplicate]

Can one just change the notation of four vectors so as instead of having $$ X^{\mu} =(X^0, \vec{X})$$we define $$ X^{\mu}=(X^0,i\vec{X})?$$ This way we could use the Euclidean metric instead of $$g^{\...
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EM wave: Why do you take the imaginary part of complex $H(z,t)$?

The following is an electromagnetic wave traveling in +z direction: $$\vec{E}(z,t) = 1.0~e^{-\alpha z} e^{j(2\pi f t - \beta z)}~\hat{y}$$ $$\vec{H}(z,t) = - (2.28\times 10^3)~ e^{-\alpha z}~e^{j(2\...
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Go from trigform to complex form

I'm given the expression for a current as $$\mathbf{I}=I_0\sin\omega t \ \hat{z}\tag1$$ and I want to write it in complex form. My professor just writes $$\mathbf{I}=I_0\cos(\omega t-\pi/2) \ \hat{...
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Can I get any leads on a development of an octonion generalization of the usual angular momentum operators? [closed]

I'm working through, developing the formalism for an octonion generalization of angular momentum operators, but the whole time I'm thinking that this has probably been done elsewhere. My searches ...
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1answer
369 views

Variation and functional derivative of a complex function

I have learnt that to get the functional derivative, we must carry out the variation. The functional derivative is the thing next to the direction the variation is taken. For example for some real ...
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1answer
147 views

A more intuitive formulation of time-dependent Schrödinger equation? [closed]

Is there an intuitive reason why the time-dependent Schrödinger equation is formulated as: $i{\hbar}\frac{\text{d}}{\text{dt}}\Psi=H\Psi$ and is that the most intuitive way to look at it? Or are ...
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1answer
65 views

Converting a function of a wave into complex form [closed]

I have an electromagnetic wave with the magnetic field $$\mathbf{B}=B_0\sin\left(\omega\left(t-\frac{a_zz}{c}\right)\right) \ \hat{x}.$$ I'm asked to write this in complex form. I know ...
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Explain Imaginary Time and Temperature [duplicate]

I was amazed to learn that we can use Imaginary unit iota into physical quantities like time and Temperature but how exactly? The explanation was not something I would say stellar so I am hoping can ...
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Propagator of a real scalar field does not give an unambiguous result

The propagator of a real scalar field is basically a Green function. Its evaluation requires specifying the contour which seems arbitrary. The integral depends on the choice of the contour. For four ...
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3answers
304 views

How to draw quiver plot for complex-valued electric field?

I have a matrix of complex numbers for the electric field inside a medium. Since I want to draw the quiver plot of these elements, it will be completely different if I only use the absolute part. Then ...
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110 views

How to show that imaginary part of complex power is reactive power?

Suppose I have a voltage across an element $V(t) = A \cos(\omega t)$ and the current through it given by $I(t) = B \cos(\omega t + \phi)$. The instantaneous power is $$P(t) = V(t)I(t) = AB \cos(\...
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1answer
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'Complex dimensions' in a metric

In Special Relativity the metric is (with $\eta=\text{diag}(1, -1, -1, -1)$) $$ \text{d}s^2 = \text{d}t^2 - \text{d}\mathbf{x}^2. $$ What sets time apart from space in this equation is the "$-$" in ...
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Physical interpretation of complex numbers, part 2

I walk in the x direction, if I walk twice as fast, 2x, if I walk backwards, -x. What about ix? If I say that I walk an imaginary distance ix then this means in physics and maths that I walk ...
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Physical interpretation of complex numbers [duplicate]

Complex numbers are used widely in quantum mechanics and the waveform, is there a physical interpretation of what this means about the structure of the universe? Why is it not used in macro physics? ...
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1answer
119 views

Method of pole shifting (feyman's trick) in Scattering theory vs contour deformation trick

I am studying Scattering theory but I am stuck at this point on evaluating this integral $G(R)={1\over {4\pi^2 i R }}{\int_0^{\infty} } {q\over{k^2-q^2}}\Biggr(e^{iqR}-e^{-iqR} \Biggl)dq$ Where $ ...
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2answers
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In light of Wick rotations is position and time on the same footing in QFT?

Taken from here Wick rotation connects statistical mechanics to quantum mechanics by replacing inverse temperature $1/(k_{B}T)$ with imaginary time $it/ℏ$ But I was under the impression position ...
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1answer
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Is complex conjugation operator Hermitian?

I wonder whether the complex conjugation operator, defined on a wavefunction as $$ C \psi(x) = \psi^*(x), $$ is Hermitian? On one hand, its eigenvalues are not necessarily real. On the other ...
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Is there any "real" use of complex analysis in quantum mechanics? [closed]

After learning some quantum mechanics, I see a lot of applications of complex numbers. However, I have not yet seen any application of complex analysis. The full name for complex analysis is "...
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473 views

Question about the Dirac adjoint and Feynman slash notation

I was trying to prove the identity $\overline{\displaystyle{\not}{a}\displaystyle{\not}{b}\dots \displaystyle{\not}{p}} = \displaystyle{\not}{p}\dots \displaystyle{\not}{b}\displaystyle{\not}{a}$. On ...
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What significance do field-operators have, if they don't correspond to observables because of non-hermicity?

Since field-operators are not always hermitian (for example in case of a complex scalar field, or the dirac-field), they don't (in the quantum-mechanical sense) correspond to observables. Does that ...
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1answer
152 views

How to take derivative with respect to Lagrangian of complex field?

Basics: The Lagrangian in field theory was written as $$\frac{\partial \mathfrak{L}}{\partial \varphi}=\partial_\mu(\frac{\partial\mathfrak{L}}{\partial(\partial_\mu\varphi)})$$. Question 1: Is $\...
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1answer
49 views

Complex number representation of a wave

There are some aspects to waves I am confused, for instance in Chapter 11. Fraunhofer Diffraction. The incoming electric fields can be partially expressed as $e^{i(kr-\omega t)}$. I have two ...
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4answers
245 views

Are Imaginary Numbers Really “Imaginary?” [duplicate]

I find the naming convention of “Imaginary” misleading, as it does give a sense that the quantity is merely an abstract construct used to mitigate the difficulties of performing some mathematical ...
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1answer
72 views

Isometry of Riemann sphere?

The complex metric on the Riemann sphere is given in the Wikipedia article to be $$ds^2=\frac{4}{(1+\zeta\bar \zeta)^2}d\zeta d\bar \zeta$$ while the sphere should be mapped to itself under $SL(2,\...
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Proving that $$ [\phi(\vec{x}, 0), \phi(\vec{x}, t)] \sim e^{-i m t}-e^{+i m t} $$ in QFT

So far, I get the following (for the left term in the integral, $d$=3): \begin{equation} \begin{aligned} \Delta_{+}(x) &= \int \frac{\mathrm{d} \vec{p}^{d}}{(2 \pi)^{d} 2 e(\vec{p})} \exp (-i t e(...
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2answers
158 views

Complex numbers in physics [duplicate]

Can someone please explain the origin of complex numbers in physical values. For instance, denoting a plane wave with Euler's identity and also the complex relative permittivity? Thank you.
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1answer
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Can there be an **essential topic** in physics which cannot be archimedean? [closed]

In physics it seems everything is explained with $\mathbb R$ or $\mathbb C$ typed entitites. Is there anything in or that would be in future in physics that would need the utility of $p$-adics in an ...
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"Imaginary-time" argument in high energy physics

In many-body physics, there are many "imaginary-time" techniques, such as Matsubara Green's function, imaginary-time path integral and others. It seems that these concepts are frequently used in ...
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1answer
81 views

Notation of physical dimension of complex values

I can't find a good answer on the proper way to write the physical units for complex numbers. $$ \begin{align} z &= 707 \text{ mV} + 0.707\mathrm{i} \tag{1} \\ z &= (707 + 707\mathrm{i}) \...
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Gravity and complex numbers

In the lectures of Gary Gibbons on Supergravity held 2009@DAMTP http://www.damtp.cam.ac.uk/research/gr/members/gibbons/gwgPartIII_Supergravity.pdf it is remarkable that when he introduces spinors he ...
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4answers
261 views

Sinusoidal currents in AC circuits

(I had edited this question a number of times but did not receive a satisfactory answer. So I am re-wording this question yet again. The earlier version of this question can be found here : https://...
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3answers
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As $SL(2,\mathbb{C})$ is a double cover of the Lorentz group, is $SL(2,\mathbb{Z})$ a discrete subgroup of the Lorentz group?

The group $SL(2,\mathbb{C})$, the group of $2 \times 2$ complex matrices with determinent $1$, is a double cover of the Lorentz group. (These transformations can be understood as Mobius ...
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5answers
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Complex conjugate and transpose "with respect to a basis"

In my quantum mechanics notes, my teacher described the complex conjugate and transpose of a linear operator X as "with respect to an orthogonal basis." What does it mean to take a transpose or ...
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1answer
182 views

General solution of a particle in a box

The general solution of particle in a 1D infinite potential well(width L) is given by: $$\psi(x,t)=\sum_n a_{n}.\sqrt{\frac{2}{L}}.\sin\bigg(\frac{n\pi x}{L}\bigg).\exp\bigg(-\frac{ iE_n.t}{\hbar}\...
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1answer
604 views

Is a Wick rotation a change of coordinates?

My understanding is that a Wick rotation is a change of coordinates from $(t,x) \rightarrow (\tau , x)$ where $\tau = i t$. In the $(t,x)$ coordinate system, the Minkowski metric has components $ \...
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1answer
77 views

Solution of a differential equation in physics

In physics when we solve the differential equation, in some cases we get two part of the solution, one is real and another is imaginary. Some cases we consider that the real part have some physical ...
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1answer
248 views

Complex conjugation between particle/ anti-particle?

I am just a first year undergrad so I don't really know much about particle physics or the underlying mathematics. So I'm very sorry if the following question may be just stupid :D So I noticed that ...
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2answers
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Integral evaluation

I am reading this paper where I have encountered the following integral: $$ I_2 = \lim_{a\rightarrow \infty}\int_0^\infty e^{-\beta k^2} \frac{\cos(2ka)}{\kappa^2 + k^2}dk.$$ where $\beta>0$ is ...
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1answer
309 views

General wave function equates to one after equating Euler's identity. Why?

Am typing on my phone, so apologies for any mistakes. Basically we know a general wave function taken as an example to be $\psi = e^{i(kx-\omega t)}$ where $k=2\pi/\lambda$ and $\omega=2\pi f$. Euler'...
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121 views

The imaginary time [duplicate]

Some people work with the interpretation that the time basis vector has magnitude sqrt(-1) to justify the negative sign in a -+++ Minkowski metric signature. I came across a Youtube comment that ...
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1answer
368 views

Complex rabi frequency

I am learning quantum optics. In the book written by Scully includes complex rabi frequency, when dealing with $\Delta$ level configuration, the Hamiltonian including laser-atom interacting concludes ...
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1answer
99 views

Complex Scalar Fields and Killing Vectors

In a stationary and axisymmetric spacetime, there are two Killing vectors, say $\zeta^\mu$ and $\xi^\mu$, one timelike and one space like. I understand that for a real scalar field, $\phi$, that ...
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1answer
262 views

The complex form of Hamilton canonical equations

I found an excerpt on page 171 of "The variational principles of mechanics" written by Cornelius Lanczos stated that If, however, the conjugate variables $q_k$, $p_k$ are replaced by the complex ...
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1answer
816 views

How imaginary part of susceptibility is measure of dissipation?

In linear response theory, we focus only imaginary part of the generalized susceptibility and consider it a measure of dissipation in the system. Can someone throw some light at it that what is meant ...
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1answer
95 views

The Proof of $\cos\phi=\gamma$ Equation in Special Relativity [closed]

In the Introductory Special Relativity book, by W. G. V. Rosser, page 182, Section 7.3, the author is defining the 4-vector methods using complex numbers. In his derivation, he writes the following ...
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5answers
374 views

Why can't quantum field theory be complex instead of imaginary?

In the following question 1, the author claims that a QFT is defined as: $$Z[J] \propto \int e^{iS[\phi]+J.\phi} D[\phi]$$ Then uses this definition to explore the possibility of formulating a QFT ...
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1answer
75 views

Step in the derivation of complex wave notation [duplicate]

I'm reading Hecht's Optics and I have a problem understanding a step in the derivation of the complex notation of waves He writes that the wave equation for a harmonic wave can be written as $\Psi(x,...
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2answers
834 views

Wick rotation: still trouble in getting how it works

I'm preparing my second exam in QFT and I still have trouble in getting the Wick rotation and its analytic continuation. I know that this topic have been discussed a lot in previous threads, but I ...

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