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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100 views

How does a metric of the form $\mathrm{d}z \mathrm{d}\bar z$ work, if $z$ and $\bar z$ are not independent? [duplicate]

My question is motivated by 2D CFT where one works in "complex coordinates". The question is the following: Suppose I am in 2D flat Euclidean space, i.e. $$\mathrm{d}s^2 = \mathrm{d}x^2 + \mathrm{d}y^...
2
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3answers
157 views

How is complex permittivity measured?

Complex relative permittivity is defined as $$\epsilon_r = \frac{\epsilon(\omega)}{\epsilon_0}=\epsilon_r^{\prime}(\omega) + i\epsilon_r^{\prime\prime}(\omega) = \epsilon_r^{\prime}(\omega) + \frac{i\...
0
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1answer
80 views

Causality and response functions

Referring to David Tong's notes on Electromagnetism, page 29 (of the PDF, numbered 183), section 7.5.4; It is proved that the frequency domain response function (in this case describing the ...
2
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2answers
177 views

What is the need of complex functions in wave analysis?

It is commonly known that waves can be express in terms of sine or cosine function. But when I study further, I seen that for analyising the waves, it is common to use complex functions in the form $$...
2
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2answers
284 views

How many parameters are needed to specify a quantum state?

We have a spin state \begin{align} \ |{\Psi}\rangle=a_u|{U}\rangle+a_d|{D}\rangle \end{align} where $|U\rangle$ and $|D\rangle$ are $up$ and $down$ basis vectors, and $a_u$,$a_d$ are their complex ...
3
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2answers
102 views

Complex / real representations of the Lorentz group

In Michele Maggiore's book "A Modern Introduction to QFT" he describes the spinorial representations of the Lorentz group as The representations are in general complex. I always thought the ...
7
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1answer
146 views

Why complexify in order to construct Dirac representation?

Suppose we have a theory is covariant under the Spin group Spin(2n-1; 1). We consider the real vector space $V = R^{2n-1,1}$, which naturally comes with a Lorentzian inner product. On this vector ...
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0answers
92 views

Complex Space Time - Mathematical Foundations [duplicate]

I am really curious as to what the current research is in complex space time. Because in "The theory of Everything". Stephen Hawking does talk about imaginary time. Is there any mathematical ...
1
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1answer
281 views

Concrete example of the application of complex analysis in electrostatics [closed]

I've heard complex analysis can be useful in solving electrostatics problems, but despite doing some research I was unable to find any concrete examples. Would anyone be able to provide a simple ...
3
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1answer
182 views

The derivation of the irreducible representations of the Lorentz group

I took the way of classification of Lorentz group representations from Sexl, Urbantke, Relativity, groups and particles (Germ. ed. 1975). But I don't understand it as I outline in the following: In ...
-1
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2answers
356 views

What do “ℜe” and “A*” mean?

What do "$\mathfrak{Re}$" and "A*" mean in the following equation (taken from James Binney and David Skinner's QM lecture notes, equation 1.12), \begin{align} p(S\text{ or }T)&=\left|A\left(S\text{...
4
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3answers
2k views

Born Interpretation of Wave Function

I have just started Griffiths Intro to QM. I was studying Born's interpretation of Wave function and it says that the square of the modulus of the wave function is a measure of the probability of ...
1
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0answers
68 views

Does Quantum Mechanics need imaginary numbers? [duplicate]

In quantum mechanics, we assume wavefunctions are complex valued, and that probability amplitudes are given by the modulus of the wavefunction squared. This formalism can correctly explain ...
1
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1answer
213 views

Time-harmonic Maxwell's equations

My question is about the time-harmonic or frequency domain (differential) form of Maxwell's equations. If one solves these and obtain the complex vector field $E$, are the real and imaginary parts ...
4
votes
1answer
596 views

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

So, the definition of QFT in terms of path integrals is that the partition function is: $$Z[J] \propto \int e^{iS[\phi]+J.\phi} D[\phi]$$ But does it have any meaning if instead of this $U(1)$ ...
2
votes
1answer
365 views

What is the difference between real orbital & complex orbital?

While reading Atomic orbitals, I came before these two terms. The 'real orbital' is given here: Real orbitals An atom that is embedded in a crystalline solid feels multiple preferred axes, ...
0
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2answers
32 views

Complex charge of a Hertzian Dipole

Im breaking my head over the solution of an old exam problem: Given is the following hertzian dipole: Given are also the equations: $I(t) = \Re\left(I_0 e^{j\omega t}\right)$, $Q(t) = \Re\left(\...
2
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1answer
79 views

Are the pion fields in chiral perturbation theory complex or real fields?

The chiral perturbation theory Lagrangian is written $$\mathcal{L}_2=\frac{f_{\pi}^2}{4}Tr(D_{\mu}U^{\dagger}D^{\mu}U)$$ where $$U=e^{i\sqrt{2}\Phi/f}$$ and $$\Phi= \begin{pmatrix} \frac{1}{\sqrt{2}}\...
3
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2answers
385 views

Can temperature be a complex number?

Is it possible for a temperature to be a complex number? I want to say "no" but I can't be so sure. If it is possible I would like to know of an example. I found an interesting article which treats ...
4
votes
1answer
358 views

What is the purpose of the imaginary portion of the wave function?

I recently watched this video. I'm trying to learn about the origin of the wave function and therefore understand its use in the Schrödinger Equation. However at the end of the video I understood up ...
3
votes
2answers
569 views

Why does the magnitude squared of the wave function give us the probability density? [duplicate]

My question doesn't go much beyond the title: Why does $$\left | \psi \left ( x,t \right ) \right |^{2}$$ give us the probability density of something appearing at a certain location? I understand ...
5
votes
2answers
137 views

Elementary question about endpoint singularities

In George Sterman's book "An Introduction to Quantum Field Theory", on pages 413-414, there is a description of the endpoint singularity. One begins with the function $$ I(w) ~=~ \int_{\zeta_a}^{\...
6
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2answers
168 views

$su(1,1) \cong su(2)$?

The three generators of $su(2)$ satisfy the commutation relations $$ [J_0 , J_\pm] = J_\pm , \quad [J_+, J_- ] = +2J_0 .$$ The three generators of $su(1,1)$ satisfy the commutation relations $$ [...
3
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3answers
254 views

Why are complex fields in the Lagrangian?

I know that a complex field has twice the number of degrees of freedom of a real field, and that fields (in QFT) aren't observables so we don't really care if they are real. But why the need for ...
3
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1answer
117 views

Path integral measure Fourier transformation for case of real field

Let's have $$ Z[J] = \int D \varphi e^{iS[\varphi , J]}, $$ where $\varphi$ denotes real scalar field. Let's make Fourier transform, $$ \varphi (x) = \int e^{iqx}\varphi (q), \quad \varphi^{*} (q) = \...
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1answer
88 views

How to separate k into real and imaginary parts?

In $k^2 - \frac{\omega^2}{c_o^2} + (\tau_{\alpha} i \omega)^{\alpha} k^2 = 0$, $k$ is the wavenumber, $\omega$ is angular frequency, others are constants. How can I separate the wavenumber $k$ into ...
0
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3answers
77 views

Use of Imaginary Angles in Physics

I am studying higher algebra. I have learnt that as well as for reals, trigonometric functions can also be defined for complex numbers, by means of power series. Such as: $$\sin(z)=z-\frac{z^3}{3!}+\...
0
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1answer
302 views

Metal Refractive index

I'm working on Fresnel equation for calculation of reflection of a light (532 nm) on Iron. I've got a question: Is metals refractive index always a real number or it can be a complex number?
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1answer
90 views

QM: Why is there a minus sign on the Energy operator when using complex conjugate?

I understand how they get the first equation. But I have no idea why there is a minus sign on the second equation: This is from a derivation for the probability density current found here: http://...
3
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0answers
229 views

Why is imaginary time “outdated”? [closed]

I was looking at reviews for Sakurai's Quantum Mechanics textbook, and some mentioned it being outdated, specifically mentioning his use of imaginary time. Is this idea deliberately avoided in modern ...
2
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3answers
111 views

Can we define the zero potential at an imaginary point?

Consider a force field defined as $$\vec{F}(x) = \left(\frac{A}{x^2}-B\right)\hat{i}\space$$ where $A, B$ are positive constants. We want to get the potential energy function for this field. We can ...
2
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1answer
156 views

$SU(2)$ generators and creation annihilation operators

The algebraic method to find the irreducible representation of the $SU(2)$ group makes use of the operators: $$J_z\\J_+=\frac{1}{\sqrt{2}}(J_x+iJ_y)\\J_-=\frac{1}{\sqrt{2}}(J_x-iJ_y)$$ In the book ...
4
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2answers
115 views

Complex dimensional analysis

Does complex numbers have physical dimensions? Is it sensible to talk about the dimensional analysis of $Z$ where $Z$ is the impedance of a mechanical oscillatory system? Or is it the $|Z|$ which has ...
0
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2answers
91 views

Hermitian Metric and Geodesics

Why isn't general relativity developed with a Hermitian metric and a theory of complex valued paths and geodesics? The concept of arc length and geodesic suffers under a pseudo-Riemannian metric. My ...
1
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1answer
105 views

Variation over complex function in Ginzburg-Landau theory

When deriving the Ginzburg-Landau equations, we minimize the following free energy over the complex function $\psi$: $$F = \int dV \left \{\alpha |\psi|^2 + \frac{\beta}{2}|\psi|^4 + \frac{1}{2m^*} \...
0
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2answers
200 views

Physical Interpretation of the Residue

In my complex analysis course (taught by the physics department no less) we've obviously paid close attention to the residue for solving our problems, but there has been no attempt to try and attach ...
0
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2answers
200 views

Properties of Hodge Duality

So we know that Hodge duality works this way $$⋆(dx^i_1 \wedge ... \wedge dx^i_p)= \frac{1}{(n-p)!} \epsilon^{i_1..i_p}_{i_{p+1}..i_n} dx^{i_{p+1} } \wedge dx^{i_n}$$ where $p$ represents the $p$ in ...
1
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1answer
314 views

What is the advantage of using exponential function over trigonometric function in analyzing waves?

A.P.French in his book Vibrations and Waves writes: . . . Why should the exponential function be such an important contribution to the analysis of vibrations? The prime reason is the special ...
0
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1answer
214 views

Finding the Direction of Measurement Given the Spin State

I am currently trying to gain a fuller understanding of the meaning of various spin states and their relation to the direction of measurement by a Stern-Gerlach device. I came across two spin-${1 \...
3
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2answers
436 views

Complex numbers in quantum mechanics and in special relativity

Is there a physical relation between the use of complex numbers for the wavefunction in (non-relativistic) quantum mechanics and in special relativity (as formulated in the setting of Minkowski space)?...
0
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1answer
326 views

Is imaginary time a fifth dimension? [duplicate]

I've read that by introducing the concept of imaginary time, the dimension of time can be treated like a spatial dimension mathematically. Assuming, without imaginary time, one considers the universe ...
1
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1answer
148 views

Complex comjugate of Schrodinger equation: paradox in matrix form?

We can take the complex conjugate of schrodinger equation, and obtain $$ -\frac{\hbar^2 }{2m}\frac{\partial^2\psi}{\partial x^2} + V(x)\psi = i \hbar \frac{\partial \psi}{\partial t} $$ $$ -\frac{\...
1
vote
2answers
89 views

A question over the reality of $\sin x$

Harmonic functions are in widespread use in physical descriptions of natural real phenomena. I am just wondering therefore how we can define $\sin(x)$ to be part of a real physical quantity (with ...
4
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3answers
181 views

Root of $i$, which one to take?

The propagator of a free particle in 1d is $$ K(x_b, t_b; x_a, t_a ) = \sqrt{\frac{m}{2\pi i \hbar (t_b-t_a)}} \exp \left [ \frac{i m (x_b-x_a)^2}{2 \hbar (t_b-t_a)} \quad \right ] .$$ It looks nice....
3
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1answer
109 views

Why is the value of the action integral in general relativity the same on all regions that are homologous?

In their famous paper Action integrals and partition functions in quantum gravity, Gibbons and Hawking argue that in order to avoid the singularity of a Schwarzschild black hole you can complexify ...
2
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5answers
4k views

Complex Conjugate of Wave Function

I've been reading through Griffiths QM book, and the only thing bugging me is they never fully described what $\Psi^* $ should be for any given function. I know it's the complex conjugate at the same ...
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0answers
77 views

Topological implications of symbolic represenation of the relativity

I have seen in the online Stanford Encyclopedia of Philosophy in the entry on Copenhagen Interpretation of Quantum Mechanics that Niels Bohr had argued that the theory of relativity is not a literal ...
5
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1answer
232 views

Faraday's law: sin and cos?

I am looking at this paper (Hanna S. A., Varhue W. J. and Titcomb S., IEEE Trans. on Instrumentaion and Measurement, Vol. 58, No. 1, 2009). They claim that the voltage generated in a loop of $N$ turns ...
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0answers
31 views

Flow in the strip $0 < x < \pi/2$, $y > 0$ [closed]

Could anyone please explain how to show that a complex valued function represents a flow? For example, how does one show that $\Phi(z) = \sec^2 z$, where $z = x + iy$ with $x, y \in \mathbb{R}$, ...
4
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1answer
297 views

Imaginary time is to inverse temperature what imaginary entropy is to …?

The Wick-Rotation rotates imaginary time into inverse temperature (as can be seen from its "rotating" the Schrödinger equation into the heat equation). Now since entropy is temperature's conjugate, I ...