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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66 views

Why is imaginary time “outdated”? [on hold]

I was looking at reviews for Sakurai's Quantum Mechanics textbook, and some mentioned it being outdated, specifically mentioning his use of imaginary time.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 something most people think? I can't ...
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83 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 ...
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43 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 ...
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51 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 ...
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49 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 ...
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1answer
37 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^*} ...
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51 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 ...
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66 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 ...
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1answer
105 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 ...
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1answer
77 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 ...
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2answers
135 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 ...
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1answer
178 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 ...
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1answer
99 views

Complex comjugate of Schrodinger equation: paradox in matrix form?

We can take the complex conjugate of schrodinger equation, and obtain $$ -\frac{\hbar^2 }{2m}\frac{\partial^2\psi}{\partial x^2} + V(x)\psi = i \hbar \frac{\partial \psi}{\partial t} $$ $$ ...
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79 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 ...
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3answers
163 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 ...
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1answer
70 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 ...
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4answers
252 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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66 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 ...
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133 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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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 $x, y \in \mathbb{R}$, ...
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1answer
104 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 ...
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46 views

Can we take transport equation of imaginary quantity?

In the RANS equation we approximate the nonlinear fluctuating terms to eddy viscosity times strain rate. Then by using turbulence models like Spalart-Allmaras etc, we take the transport equation of ...
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191 views

What is the physical meaning of complex eigenvalues?

I understand the mathematical origin of complex eigenvalues, and that complex eigenvalues come in pairs. But what is the meaning of the imaginary part? In particular I refer to an acoustic problem ...
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137 views

Motivating Complexification of Lie Algebras?

What is the motivation for complexifying a Lie algebra? In quantum mechanical angular momentum the commutation relations $$[J_x,J_y]=iJ_z, \quad [J_y,J_z] = iJ_x,\quad [J_z,J_x] = iJ_y$$ become, on ...
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51 views

Set of orthotogonal complex functions [closed]

Show that the functions $e^{in\pi x/l}$, n = 0, ±1, ±2, ..., are a set of orthogonal functions on $(-l, l)$ using: $A(x)$ and $B(x)$ are orthogonal on $(a,b)$ if $$\int^b_a A^*(x)B(x)dx = 0$$ ...
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136 views

Difficulty evaluating a complex integral on Griffiths

This actually a question from Griffiths QM. (Q2.21) I have difficulty understanding integrals involving imaginary components. In this example, it looks like the first term (encircled in red) explodes ...
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1answer
83 views

Complex theory in physics [duplicate]

I'm a physics graduate and usually encounter with complex numbers in physics. For example, in electrical engineering, Why do we express capacitive reactance as an imaginary number..? Can't it be ...
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1answer
83 views

Why can we leave off half of the general solution?

In these pdf notes, it says at the bottom of the first page and beginning of the second: [...] whose solution is: $$\Psi(\theta) = c_1 e^{i\omega\theta} + c_2 e^{-i\omega\theta}$$ Since we are ...
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2answers
228 views

QFT Hilbert spaces over other rings than the complex numbers $\mathbb{C}$

I would like some help evaluating a physics theory recently proposed by a physics professor at the College of Dupage. I think the theory is utterly wrong, for very simple reasons. If an amateur ...
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26 views

Confusion regarding the trial solution taken in the mathematical treatment of forced oscillations, at steady state

In the text-book that I am currently using, it is given that in case of forced oscillations, the periodic external driving force is a complex-driving force, and is generally of the form $F_0e^{jwt}$. ...
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72 views

Tensors of rotations about an arbitrary vector in C^2

I'm trying to solve the following equation: $$e^{-i\theta/2 \sigma_{\vec{i}}^A} \otimes e^{-i\theta/2 \sigma_{\vec{i}}^B} |\Psi\rangle_{AB} = e^{i\phi} |\Psi\rangle_{AB} $$ where $e^{i\phi}$ should ...
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1answer
153 views

Complex Quantum Wave [closed]

Can the complex nature of quantum wave arise from the fact that particle is represented as wave packet in spatial frequency and particle's total energy is represented as wave packet in time frequency? ...
5
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3answers
607 views

What does the Schrodinger Equation really mean?

I understand that the Schrodinger equation is actually a principle that cannot be proven. But can someone give a plausible foundation for it and give it some physical meaning/interpretation. I guess ...
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1answer
109 views

Is it okay to Wick rotate to give the negative of the Euclidean metric? Also, could we make the space-like coordinates imaginary instead?

There are 2 parts to my question: 1) Say we choose the metric signature to be (-+++), as in the Wikipedia page. Then the invariant interval in Minkowski space is written: $ds^{2} = -(dt^{2}) + ...
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4answers
262 views

Complex integration by shifting the contour

In section 12.11 of Jackson's Classical Electrodynamics, he evaluates an integral involved in the Green function solution to the 4-potential wave equation. Here it is: $$\int_{-\infty}^\infty dk_0 ...
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2answers
369 views

Physical meaning of imaginary part of Electric field?

As far as I know (or I thought I knew), if we have an electric field $$\mathbf{E}=\mathbf{E_0}\cos(\omega t - kx),$$ we can define it as the real part of $$\mathbf{E}=Re(\mathbf{E_0}e^{i(\omega t - ...
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1answer
38 views

Is real voltage always the real part of complex voltage?

If I have a complex voltage $V_z$ is real voltage $V$ (i.e. the voltage used in the normal ohms law and the voltage we normally talk about) always given by $V=Re(V_z)$. Does the same apply to current? ...
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1answer
107 views

Integral over a product of two Green's functions

Need some help here on a frequently encountered integral in Green's function formalism. Forgive me since I am a junior student. I have an integral/summation as a product of a retarded and advanced ...
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165 views

Complex Conjugate of Wave Function's Derivative

I am reading Griffiths QM textbook and I got confused by the following identity: How to prove from $$\frac{\partial \Psi}{\partial t} = \frac{i\hbar}{2m} \frac{\partial^2 \Psi}{\partial x^2} - ...
6
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3answers
389 views

Is quantum tunneling related to imaginary time?

I was studying for my exam and looking at the chapter which talks about Potential-energy graphs. Let's take this as an example: My book states that: "If the object is in $B$ and has a total energy ...
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2answers
214 views

Unknown letter ℑ used in an equation

I need to write by hand the equation from the attached snapshot but I really don't know what letter is that seen in the front of square brackets [ . Can anyone help ...
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1answer
72 views

Complex semi-definite programming

I'm doing some calculations and I want to simulate them in python or matlab (or whatever). However I use hermitian matrices and I don't really manage to find a library which enables me to calculate ...
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2answers
132 views

Quantum Mechanical States

What can be the precise answer to the question that Quantum states are complex and infinite dimensional. Why is this so? Is it because they belong to the complex Hilbert space? Even if they ...
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2answers
129 views

Can speed be defined in the complex plane?

This question cropped up while I was playing with the equation for time dilation. If I set the speed to be $i$ (imaginary unit) the answer from the equation still makes sense, but does that matter if ...
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3k views

What is imaginary time? [duplicate]

I am not professional physicist; but I am curious about Stephen Hawking's "imaginary time". It would be better to elaborate exactly what it is. I am not confused because of the word "imaginary" but I ...
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60 views

Finding the total space that an oscillating body has gone through via complex analysis

I was solving my homework and I got to an exercise that stated: An harmonic oscillating body has an equation of $$y(t) = A \sin(t)$$ Find the total space that the body has travelled during $t \in ...
4
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3answers
146 views

Special relativity and imaginary coefficient of the time coordinate

I read somewhere that part of Minkowski's inspiration for his formulation of Minkowski space was Poincare's observation that time could be understood as a fourth spatial dimension with an imaginary ...
6
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2answers
225 views

Decoupling of Holomorphic and Anti-holomorphic parts in 2D CFT

This maybe a very naive question. I have just started studying CFT, and I am confused by why we have two separate parts of everything in CFT (operator algebras and hilbert space), the holomorphic ...
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1answer
191 views

Unstable states and imaginary (complex) energy?

I came across the notion of complex energy while studying instanton method to study the unstable state. Unstable states are those which have energy with an imaginary part. But as we know Hamiltonian ...
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1answer
156 views

A question about a complex integration in Peskin's QFT textbook

In page 27 (2.52), the integration is $$\int_{-\infty}^{\infty}dp \frac{p e^{ipr}}{\sqrt{p^2+m^2}}$$ He says that there are two branch cuts starting from $\pm im$ But I learn in complex analysis ...