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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44 views

Were the original Maxwell quaternion equations different than Heaviside's vector equations in any of their predictions?

Gibbs vector calculus removed scalar quaternion parts of equations thus predicting no scalar waves that transfer energy. Did Maxwell have a different take on similar predictions.
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What paths are allowed in the Fourier form of the Dirac Delta distribution?

In this form of the Dirac Delta distribution $$\delta(x) = \frac{1}{2 \pi i}\int_{- i \infty}^{i \infty}e^{-\omega x} d\omega$$ can $\omega(t)$ be evaluated over any path (that starts at $\omega(-\...
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Origin of the factor of $i$ in the photon propagator

I'm following Peskin and Schroeder and am having trouble tracking down a particular factor of i that is persistently used in the definition of Green's functions. For example, equation 9.52 states that ...
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41 views

What is the old (50's) convention on Dirac gamma matrices?

What were the standard relations for gamma matrices in the mid 50's, when 4-vectors where represented by $(x_1, x_2, x_3, ict)$? In particular the values of $\gamma^\mu\gamma^\nu$ , the definition of $...
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Can someone help me understand this equation that finds derivative of quaternion when angular velocity is known?

I found the following equation in an IEEE paper: $\frac{dq}{dt} = \frac{1}{2}\text{Q}\omega$ where $q$ is the quaternion that rotates a vector from the body frame to the world, $q=\left[q(1), q(2), ...
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Why complex numbers are used in electronics? [duplicate]

The impedance of a capacitor or an inductor is imaginary. How do we know these quantities are imaginary?
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How to find polar and rectangular form of $z$ if $z^2$ is given? [migrated]

For example, if you have a question: $$z^2 = 3(9 + √8i + 16i^2)$$ how would you find its polar and rectangular form? the general equation is $z = a+bi$ but in this equation simplifying it would give $...
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139 views

Cauchy-Schwarz inequality in Shankhar's Quantum Mechanics

I'm trying to understand proof of this inequality. But I have some problems. So, Shankar starts a proof with definition a new vector $|z \rangle$: $$ |z \rangle = |v\rangle - \frac{\langle w|v \...
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52 views

Evaluating integral for Friedel oscillation using branch cuts

I am finding some difficulties understanding the following problem. I have the following logarithm for which I have to identify branch cuts: $$\lim_{\epsilon\rightarrow0}\ln{\frac{(p+2p_F)^2+\epsilon^...
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37 views

Complex notation convention for time variation

In describing plane wave EM variation, some textbooks use the complex notation $\exp(i\omega t)$, while others use $\exp(-i\omega t)$. Is there a motivation for chosing one or the other convention, ...
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45 views

What is the complex conjugate of two fermionic fields coupled? $(\bar{\psi} \chi)^{\ast} =$ ____?

Suppose $\psi$ and $\chi$ are fermionic fields, and suppose I want to calculate the hermitian conjugate of the operator $\bar{\psi} \chi$ $$ h.c.\ \mathrm{of}\ \bar{\psi} \chi = (\bar{\psi} \chi)^{\...
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Confused about choosing a rotation formalism for maximum simplicity

Say we have two vectors $v$ and $w$, operations of sum, difference and vector product, and normalization operations. Thus: $e_x(t) = \frac{v(t)+w(t)}{\left\| v(t)+w(t)\right\| }$, $e_y(t) = \frac{v(t)...
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What is the relationship between the complex frequecy of a RLC circuit and the half power frequency of it?

First of all, I would like to apologize for possible mistakes in the writing of this text, since English is not my mother tongue and it is my first time in this blog. Secondly, and more important, ...
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49 views

Functional Cauchy-Riemann equation?

I have a question on the complex scalar field theory. Say you have a $U(1)$ invariant Lagrangian of the form $$\int d^4 x |\partial\phi|^2 -m^2 |\phi|^2 + \frac{g}{4}(|\phi|^2)^2.$$ Then if you ...
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Mode expansion of string

On Polchinski, the mode expiation of an open string is given as $$X^i(\tau,\sigma)=x^i+\frac{p^i}{p^+}+i \sqrt{2\alpha'}\sum_{n\neq 0} \frac{1}{n}\alpha_n^ie^{-\frac{\pi i n c \tau}{\ell}}\cos\frac{\...
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Wick's rotation for the harmonic oscillator. Explicit computation

I'm stuck in this computation; it shouldn' be difficult but it's always better to check with a lot of details these things. Consider the propagator for the harmonic oscillator Ashok p.55 bottom of the ...
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1answer
61 views

Why does $\phi=\phi^*$ imposed on complex scalar field Lagrangian miss out $1/2$ factors?

If we require the reality condition $\phi=\phi^*$ on the Lagrangian for a complex scalar field is $$\mathcal{L}=(\partial^\mu\phi^*)(\partial_\mu\phi)-m^2(\phi^*\phi),$$ two degrees of freedom $\phi$ ...
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3answers
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What is the meaning of purely imaginary physical measures?

I would like to know what is the meaning of working with a physical measure that is purely imaginary. What I mean with "purely imaginary" is $ Z - Re(Z)$ where $Z$ is a complex number. Furthermore, ...
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Convert propagators from Euclidean to Minkowski spacetime

I'm looking for a rule to "convert" the propagators of a quantum field theory formulated in Euclidean spacetime into those of the same theory but in Minkowski spacetime (with the $\operatorname{diag}(-...
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80 views

How could a relation of the form $t=-i\tau$ hold with both $t$ and $\tau$ being real?

The physical time is a real quantity. But in quantum field theory, whenever we find oscillatory exponentials in time and we cannot literally take the limit $t\to \infty$, we make a change of variable, ...
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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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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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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
91 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
61 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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297 views

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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2answers
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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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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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'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
66 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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Method of pole shifting in Scattering theory [closed]

I asked a question Related to scattering theorey in Mathematics stalk exchange but no one answer to it. I read in a QFT book that this trick of shifting the poles is called Feynman's trick. https://...
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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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183 views

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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239 views

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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145 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
82 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
39 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
170 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
35 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
92 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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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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