Representation Definition and 722 Threads

So, we know that if g is a Lie algebra, we can take the cartan subalgebra h ⊂ g and diagonalize the adjoint representation of h, ad(h). This generates the Cartan-Weyl basis for g. Now, let G be the Lie group with Lie algebra g. Is there a way to diagonalize the adjoint representation Ad(T) of...

Homework Statement I know that we can write ## \int_{-\infinity}^{\infinity}{e^{ikx}dx}= 2\pi \delta (k) ## But is there an equivalent if the interval which we are considering is finite? i.e. is there any meaning in ##\int_{-0}^{-L}{e^{i(k-a)x}dx} ## is a lies within 0 and L? Homework...

Homework Statement Compute the antiparticle spinor solutions of the free Dirac equation whilst working in the Weyl representation.Homework Equations Dirac equation $$(\gamma^\mu P_\mu +m)v_{(p)}=0$$ Dirac matrices in the Weyl representation $$ \gamma^\mu= \begin{bmatrix} 0 & \sigma^i \\...

Homework Statement Considering the function $$f(x) = e^{-x}, x>0$$ and $$f(-x) = f(x)$$. I am trying to find the Fourier integral representation of f(x). Homework Equations $$f(x) = \int_0^\infty \left( A(\alpha)\cos\alpha x +B(\alpha) \sin\alpha x\right) d\alpha$$ $$A(\alpha) =...

Hi guys :) I'm just wondering if anyone knows of a book that has the Dirac equation solved in the Weyl basis in it? I'd like to check my method to make sure I'm on the correct lines. Thanks

Hello, I Have a particle with wavefunction Psi(x) = e^ix and would like to find its Hilbert space representation for a period of 0-2pi. Which steps should I follow? Thanks!

Hey! :o I am looking the following exercise: A medium-sized company has $n = 3$ manufacturing departments. Faults in the production process can occur in these departments. We have the following events: \begin{align*}&A=\{"\text{All departments work without faults}"\} \\ &B=\{"\text{ no...

Homework Statement Show that this is a valid representation of the Dirac Delta function, where ε is positive and real: \delta(x) = \frac{1}{\pi}\lim_{ε \rightarrow 0}\frac{ε}{x^2+ε^2} Homework Equations https://en.wikipedia.org/wiki/Dirac_delta_function The Attempt at a Solution I just...

Part 1. The second part of my journey to the manifold ##SU(2)## deals with some representations. We start with some bases and cite the classification theorem of representations of the three-dimensional, simple Lie algebra. Continue reading ...

I am trying to find a series representation for the following expression $$\int_{i=0}^\infty {x^{\frac{2n-1}{2}}(b+x)^{-n}}e^{\left(-{\frac{x^2}{2m}}+\frac{x}{p}\right)} dx$$ ; b,m,n,p are constant. Is there a name for this function? I found a series representation for $$\int_{i=0}^\infty...

After reading some books on Group Theory, I have two questions on group representations (Using matrix representation) with the second related to the first one: 1 - Can we always find a diagonal generator of a group? I mean, suppose we find a set of generators for a group. Is it always possible...

A spin 1/2 particle is represented by a spinor while its position is represented by a three-vector. What object should we use to represent such particle if we want to consider both features? That is, what object should we use if we want to consider both spin and space position? It seems there's...

I'm reading a paper on physics where it's said it can be shown that every irreducible representation of ##SU(2)## is equivalent to the one which uses the Ladder Operators. I am a noob when it comes to this subject, but I'd like to know whether or not the proof is easy to carry out.

Is Sphere a more generalized form of Cone i.e. formed by 2 dimensional rotation to 360° of a cone? Or is Cone a more generalized form of Sphere since sphere can be formed by rotating about Z axis a zero eccentric planar intersection of a cone? @fresh_42 @FactChecker @WWGD

When solving problems, particularly in optics, it is often that we represent the wave-function as a complex number, and then take the real part of it to be the final solution, after we do our analysis. u(\vec{r},t)=Re\{U(\vec{r},t)\}=\frac{1}{2}\left(U+U^*\right) Here U is the complex form of...

@fresh_42 @FactChecker After thinking, I understood that the answer for this question might make the complex numbers comprehensible for me. My question in detail is as follow Let the equation of a sphere with center at the origin be ##Z1²+Z2²+Z3² = r²## where Z1 = a+ib, Z2 = c+id, Z3 = s+it...

If the square root as two coordinate axes in the complex plane, does the cubic root has 3 coordinate axes and so on for nth root? @vanhees71 Can you please explain this?

Can ##sqrt(-i)## be expressed as a complex number z = x + iy with unitary power?

Hello! I am reading some representation theory/Lie algebra stuff and at a point the author says "the states of the adjoint representation correspond to generators". I am not sure I understand this. I thought that the states of a representation are the vectors in the vector space on which the...

Hello! I am reading some representation theory (the book is Lie Algebra in Particle Physics, by Georgi, part 1.17) and the author solves a problem of 3 bodies connected by springs forming a triangle, aiming to find the normal modes. He builds a 6 dimensional vector formed of the 3 particles and...

If I have a lagrangian which has terms of the form ##\Psi^{\dagger}_\mu \Psi^\mu## then I can decompose the n complex ##\Psi## fields into 2n real fields by ##\Psi_\mu = \eta_{2\mu+1} + i\eta_{2\mu}##. When I look at the lagrangian now it seems to have SO(2n) symmetry from mixing the 2n real...

In Chapter 70 of Srednicki's QFT he discusses what he calls the index of a representation T(R) defined by Tr(TaR TbR) = T(R)δab I think other places call this a killing form, but I may be mistaken. In any case he discusses reducible representations R = R1⊕R2. He then states (eqn 70.11) that...

Hello! I understand that the vector formed of the scalar and vector potential in classical EM behaves like a 4-vector (##A^\nu=\Lambda^\nu_\mu A^\mu##). Does this means that the if we make a vector with the 3 components of B field and 3 of E field, so a 6 components vector V, will it transform...

Hello all, Given a complex number: \[z=r(cos\theta +isin\theta )\] I wish to find the polar representation of: \[-z,-z\bar{}\] I know that the answer should be: \[rcis(180+\theta )\] and \[rcis(180-\theta )\] but I don't know how to get there. I suspect a trigonometric identity, but I...

I am working on the integral representation of the Euler-Mascheroni constant and I can't seem to understand why the first of the two integrals is (1-exp(-u))lnu instead of just exp(-u)lnu. It is integrated over the interval from 1 to 0, as opposed to the second integral exp(-u)lnu which is...

$\textrm{a. find the power series representation for $g$ centered at 0 by differentiation}\\$ $\textrm{ or Integrating the power series for $f$ perhaps more than once}$ \begin{align*}\displaystyle f(x)&=\frac{1}{1-3x} \\ &=\sum_{k=1}^{\infty} \end{align*} $\textsf{b. Give interval of convergence...

Hello everyone. Iam just Learning about State space representation and controller design and have a fundamental question about the difference between classical Control theory and modern Control theory. I understood the state space is of advantage when dealing with MIMO systems or non-linear...

Hi everybody; I have plotted a matrix with sea surface temperature's correlation with another variable, the size is (360x180x12); using to plot : figure for i=1:12 subplot(4,3,i);imagescnan(loni,lati,squeeze(double(r4_sat(:,:,i)))'),colorbar; end Now, I want to plot over it another matrix of...

Ive been working through calculus this year and will be into next year, and as nearly every time I open my calculus book I have found something new and mysterious. This time it's something called parametric representation. It isn't clearly explained what this means or how you go about...

Homework Statement I have a vector V with components v1, v2in some basis and I want to switch to a new (orthonormal) basis a,b whose components in the old basis are given. I want to find the representation of vector V in the new orthonormal basis i.e. find the components va,vb such that |v⟩ =...

In some Yang-Mills theory with gauge group ##G##, the gauge fields ##A_{\mu}^{a}## transform as $$A_{\mu}^{a} \to A_{\mu}^{a} \pm \partial_{\mu}\theta^{a} \pm f^{abc}A_{\mu}^{b}\theta^{c}$$ $$A_{\mu}^{a} \to A_{\mu}^{a} \pm...

I've been working my way through Peskin and Schroeder and am currently on the sub-section about how spinors transform under Lorentz transformation. As I understand it, under a Lorentz transformation, a spinor ##\psi## transforms as $$\psi\rightarrow S(\Lambda)\psi$$ where...

I am told that ##\varphi_g (x) = g x g^{-1}## is a group action of G on itself, called conjugacy. However, I am a little confused. I thought that a group action was defined as a binary operation ##\phi : G \times X \rightarrow X##, where ##G## is a group and ##X## is any set. However, this...

I had heard of adinkras but didn't realize that they were meant to play this role. Nor did I realize that the representation theory of supersymmetry is mathematically underdeveloped.

Hi everybody. I am currently doing a metheorological study of Angola's climate. I can draw the country's shape without much trouble. >> worldmap angola >> load coastlines >> plotm(coastlat,coastlon) But once I have the shape drawn things start to go downhill. I have a vector with the...

Homework Statement Using the irreducible representation of ##su(2)##, with ##j=\frac{5}{2}##, calculate ##J_z##, ##exp(itJ_z)## and ##J_x##. Homework EquationsThe Attempt at a Solution There seem to be loads of irreducible representations of ##su(2)## online, but no reference at all to a...

In E6, the product 27 x 27 contains the (conjugate) 27. In SU(3), something similar happens with 3 x 3, which decomposes as 3 + 6. I was wondering, how usual is this? Do we have some lemmas telling when a product N x N is going to "recover" the original N, or its conjugate, inside the sum?

Hello Forum, My understanding is that the state of the system is ##|\Psi>##. We can take the inner product between the state ##|\Psi>## and the eigenstates of the position operator ##\hat{x}##: $$<x|\Psi>=\Psi(x)$$ The function ##\Psi(x)## is the wave function we are initially introduced to in...

Homework Statement ## r_{A} (n) = ## number of solutions of ## { \vec{x} \in Z^{m} ; A[\vec{x}] =n} ## where ##A[x]= x^t A x ##, is the associated quadratic from to the matrix ##A##, where here ##A## is positive definite, of rank ##m## and even. (and I think symmetric?) I am solving for the...

Hello Forum, The state of a quantum system is indicated by##\Psi## in Dirac notation. Every observable (position, momentum, energy, angular momentum, spin, etc.) corresponds to a linear operator that acts on ##\Psi##.Every operator has its own set of eigenstates which form an orthonormal basis...

I understand that the Fourier transform to obtain the representation of a wavefunction in k space is $$ \phi(k) =\frac{1}{2\pi}\int{dx \psi(x)e^{-ikx} } $$ and that $$p=\bar{h} k$$ But why then is $$\phi(p) =\frac{\phi(k)}{\sqrt{\bar{h}}} $$ Many thanks in advance :)

[mentor note: thread moved from non-hw forum to here hence no homework template] Can someone explain to me how it is that $$\sum_{n=a}^b (2n+1)=(b+1)^2-a^2$$ I thought it would be $$\sum_{n=a}^b (2n+1)=(2a+1)+(2b+1)$$ but I am clearly very wrong. I would greatly appreciate any help.

##\theta(\tau, A) = \sum\limits_{\vec{x}\in Z^{m}} e^{\pi i A[x] \tau } ## ##=\sum\limits^{\infty}_{n=0} r_{A}(n)q^{n} ##, where ## r_{A} = No. [ \vec{x} \in Z^{m} ; A[\vec{x}] =n]## where ##A[x]= x^t A x ##, is the associated quadratic from to the matrix ##A##, where here ##A## is positive...

The hydrogen is placed in the external magnetic field: $$ \textbf{B}=\hat{i}B_1 cos(\omega t) + \hat{j} B_2 sin(\omega t) + \hat{k} B_z ,$$ Using the relation ## H = - \frac{e\hbar}{2mc} \mathbf \sigma \cdot \mathbf B ##, then I got the form $$ H = H_0 + H' , $$ where $$ H'= - \frac{e...

$\tiny{206.r2.11}$ $\textsf{find the power series represntation for $\displaystyle f(x)=\frac{x^7}{3+5x^2}$ (state the interval of convergence), then find the derivative of the series}$ \begin{align} f(x)&=\frac{x^7}{3}\implies\frac{1}{1-\left(-\frac{5}{3}x^2\right)}&(1)\\...

Homework Statement Bead on spoke: constant speed ##u## along spoke it starts at center at ##t=0## angular position is given by ##\theta=\omega t##, where ##\omega## is a constant Homework Equations ## \frac{d\hat r}{dt} = \dot \theta \hat \theta ## (1) ## \frac{d\hat \theta}{dt} = -\dot...

I am having some difficulties in understand the convention of probability like P(A|E) ... And I am not able convert the questions into this for .I can solve those questions but can someone help me to understand this topic .And also multiplication theorem .

$\tiny{s4.12.9.13}$ $\textsf{find a power series reprsentation and determine the radius of covergence.}$ $$\displaystyle f_{13}(x) =\frac{1}{(1+x)^2}=\frac{1}{1+2x+x^2}$$ $\textsf{using equation 1 }$ $$\frac{1}{1-x} =1+x+x^2+x^3+ \cdots =\sum_{n=0}^{\infty}x^n \, \, \left| x \right|<1$$...

Is pixel a representation of intensity for light at a given location?

Just a couple of quick questions on index notation, may be because of the way I'm thinking as matrix representation: 1) ##V^{u}B_{kl}=B_{kl}V^{u}## , i.e. you are free to switch the order of objects, I had no idea you could do this, and don't really understand for two reasons...