# 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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### Literature request - Dual quaternion dynamics

In my engineering practice, quaternions turned out to be much more practical than trigonometric rotation matrixes. I learned from this book on quaternions and dynamics how to describe rotation and ...
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### What is the meaning of this complex derivative with respect to a wave function?

In quantum optimal control papers such as (Loading a Bose-Einstein Condensate onto an Optical Lattice, https://arxiv.org/abs/cond-mat/0209195) and (Introduction to the Pontryagin Maximum Principle for ...
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### Can off-shell particles have imaginary four-momentum components?

I have recently started studying quantum field theory and learned that some particles do not obey the on-shell condition which got me wondering about the physical limitations of this statement. Let me ...
1 vote
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### What material properties are necessary to embed a continuously rotating sphere into static space?

I am deeply intrigued by the properties of rotational connectivity in three-dimensional space, particularly as demonstrated by the Dirac-Belt Trick or Plate Trick. For a related concept, you might ...
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### Why are complex coordinates outlawed in physics?

In the case of the Kerr metric, for a high enough angular velocity such that on transformation to Boyer-Lindquist coordinates yields complex coordinates for the event horizon, why is it assumed then ...
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### Fermi poles expansion

I want to prove the following formula: $$\frac{e^{-tE}}{1+e^{-\beta E}} = \frac{1}{\beta}\sum_{k \in \frac{2\pi}{\beta}\mathbb{Z}}\frac{e^{-ikt}}{-ik+E},$$ for $\beta > t > 0$. I know the trick ...
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### Why is the gauge field $A_\mu$ real for $\mathrm{U}(N)$ symmetry?

From my notes I have that The transformation law, $$A_\mu\to MA_\mu M+\frac{i}{g}\left(\partial_\mu M\right)M^\dagger\tag{1}$$ can be realised if $A_\mu$ is an element of the Lie algebra. It can ...
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1 vote
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### Computational complexity of approximating partition functions

I would like to understand the computational complexity of approximating the partition function of 2D Ising model with complex external magnetic field and complex couplings for the following cases: ...
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### Explanation of complex number in alternating current [duplicate]

Can anyone pls explain how complex numbers are used in alternating current,I was reading about rl circuit and found out they changed voltage $$V \sin \omega t \tag{1}$$ to $V e^{i\omega t}$. as we ...
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### Why is it reasonable to use Biquaternions for Lorentz-Transformations?

I've read that biquaternions can be used for Lorentz-Transformations using the formula $$q \mapsto e^{\alpha h \mu/2}e^{\phi\epsilon/2} q \overline{e^{\alpha h \mu/2}e^{\phi\epsilon/2}}^{*},$$ $\alpha$...
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### Splitting Scalar into Holomorphic and Anti-Holomorphic Parts

I am reading Tong’s string theory lecture notes. On page 78, he splits the 2d free scalar into left- and right-moving parts, seemingly using the classical equation of motion as justification. Why is ...
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### How can the Bloch sphere, built from one complex dimension, specify 2-complex dimensional Pauli spinors?

Two-component spinors can be identified with points on the surface of the Bloch Sphere. The Bloch sphere is constructed from the 1-complex-dimensional complex plane plus the point at infinity. How ...
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### Equation for real/complex $\phi^4$ theory

On wikipedia (see this link), the Lagrangians of the $\phi^4$ equation for real AND complex scalar fields are given. One may derive the Klein-Gordon equation by inserting into the Euler-Lagrange-...
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### Is the scalar field in the Yukawa interaction real or complex?

Consider a theory containing a Dirac field $\psi$ and a scalar field $\varphi$ where the only interaction is given by a Yukawa potential $$V = -g\bar{\psi}\varphi\psi$$ I know that real scalar ...
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### Integration over the complex plane and the completeness relation of the coherent states [duplicate]

I am studying some of the properties of coherent states using the book "Introductory Quantum Optics" by C. Gerry & L. Knight. (C. Gerry & L. Knight, Chapter 3, Section 5) And when I ...
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### Hermiticity of Majorana Fermions: SYK Model

The SYK Hamiltonian is defined as $$H = -\frac{1}{4!}\sum_{i,j,k,l=0}^{L-1} J_{ijkl} c^x_{i}c^x_{j}c^x_{k}c^x_{l},$$ where $J_{ijkl}$ is a random all-to-all interaction strength which is normally ...
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We know that a complex number, $z=re^{i\phi}$, can be represented with infinitely many phases, $\phi+2\pi n$, for integer $n$, as can be easily seen from the equivalent picture of a vector on the ...