Transformation Definition and 1000 Threads

\begin{align} \psi_L \rightarrow (1-i \vec{\theta} . \frac{{\vec\sigma}}{2} - \vec\beta . \frac{\vec\sigma}{2}) \psi_L \\ \psi_R \rightarrow (1-i \vec{\theta} . \frac{{\vec\sigma}}{2} + \vec\beta . \frac{\vec\sigma}{2}) \psi_R \end{align} I really cannot evaluate these from boost and rotation...

The three events; t = 0s, x=0 t=1.3s, x=1.3 light seconds t=2.6s, 1.3 light seconds

> Let ##C## be the disk with radius 1 with center at the origin in ##R^2##. > Consider the following linear transformation: ##T: (x,y) \to (\frac{5x+3y}{4},\frac{3x+5y}{4})## > > What is the lowest number such that ##T^{n}(C)## contains at lest ##2019## points ##(a,b)##, with a and b integers.So...

Let us say we have data which is for simplicity in N tables. All the tables have the same number of rows and columns. The columns ##A_i## have for all tables the same meaning (say measured quantaties like pressure, temperature) where the first 3 columns is the position in space. Again for...

Hello, Today I started to think about why graphs, of the same equation, look different on the Cartesian plane vs. the polar grid. I have this visualization where every point on the cartesian plane gets mapped to a point on the polar grid through a transformation of the grids themselves...

$$S'^{12} = R^{1k}R^{2l}S^{kl}, S'^{12} = R^{11}R^{21}S^{11} + R^{11}R^{22}S^{12} + R^{12}R^{21}S^{21} + R^{12}R^{22}S^{22}, S'^{12} = R^{11}R^{21}(S^{11}-S^{22}) + (R^{11}R^{22} + R^{12}R^{21})S^{12}$$ Is this enough to say that (S12, S11 − S22) transform like a doublet? To be pretty...

In physics is usually defined that in cylindrical coordinates ##\varphi \in [0,2 \pi)##. In relation with Deckart coordinates it is usually written that \varphi=\text{arctg}(\frac{y}{x}). Problem is of course because arctg takes values from ##-\frac{\pi}{2}## to ##\frac{\pi}{2}##. What is the...

I need to find the matrix transformation of y = \frac{1}{x} onto y = \frac{-1}{3x-1}-2 I think its \begin{bmatrix} x'\\ y' \end{bmatrix} =\begin{bmatrix} 3 & 0 \\ 0 & -1 \end{bmatrix} \begin{bmatrix} x\\ y \end{bmatrix} + \begin{bmatrix} -1\\ -2 \end{bmatrix}

I have two questions regarding Fourier transformation. First of all is it ok to call Fourier transformation operator, or it should be distinct more? For instance, if I wrote F[f(x)]=\lambda f(y) is that eigenproblem, regardless of the different argument of function ##f##? Could I call ##F##...

I got stuck in this calculation, I can't collect everything in terms of ##dx^{\mu}##. ##x'^{\mu}=\frac{x^{\mu}-x^2a^{\mu}}{1-2a_{\nu}x^{\nu}+a^2x^2}## ##x'^{\mu}=\frac{x^{\mu}-g_{\alpha \beta}x^{\alpha}x^{\beta}a^{\mu}}{1-2a_{\nu}x^{\nu}+a^2g_{\alpha \beta}x^{\alpha}x^{\beta}}##...

I would like to investigate a function that sends ##f(x)## to ##f(x) - \frac{1}{c}f(x^{c})##, or a function ##g## such that ##g(f(x)) = f(x) - \frac{1}{c}f(x^{c}).## Since symmetries produced by groups are used in physics, I thought there might be someone here who could help explain what the...

On the Yale University Prof Shankar Youtube vid 'Lorentz Transformation' Prof Shankar writes up on the board that x = ct and then x prime = c t prime. It is the basis of all that follows. But i don't understand. at x = 0, t = 0 and x prime = 0 and t prime = 0. He's got that written up...

##A^{x'} = T(A^{x})##, where T is a linear transformation, in such way maybe i could express the transformation as a changing of basis from x to x' matrix: ##A^{x} = T_{mn}(A^{x'})##, in such conditions, i could say det ##T_{mn} \neq 0##. But how to deal with, for example, ##(x,y) -> (e^x,e^y)## ?

Let me define ##L_{x;v}## as the operator that produce a Lorentz boost in the ##x##-direction with a speed of ##v##. This operator acts on the components of the 4-position as follows $$L_{x;v}(x) =\gamma_{v}(x-vt),$$ $$L_{x;v}(y) =y,$$ $$L_{x;v}(z) =z,$$ $$L_{x;v}(t)...

In the Hamiltonian formalism, the space-time transformation are realized via canonical transformation, and the transformations are generated by Poisson brackets of certain functions of phase-space variables. In Newtonian mechanics, Galilean boosts are generated by the sometimes called dynamic...

The graph of y=x^2 was transformed to the graph of y=-3(x+5)-2. The point (-3, 9) lies on the graph of y=x^2. Determine its image point after the transformations.

As the subject title states, I am wondering how would one go about transforming Cartesian coordinates in terms of spherical harmonics. To my understanding, cartesian coordinates can be transformed into spherical coordinates as shown below. $$x=\rho \sin \phi \cos \theta$$ $$y= \rho \sin \phi...

##\mathcal{P}## is a point in Minkowski spacetime ##M##, and ##\varphi_1: U \in M \mapsto \mathbb{R}^4## and ##\varphi_2: U \in M \mapsto \mathbb{R}^4## are two coordinate systems on the spacetime. A scalar field is a function ##\Phi(\mathcal{P}): M \mapsto \mathbb{R}##, and we can define...

This question is beyond my level of understanding, nonetheless I feel it can’t be right. I have been studying Geometric algebra and was thinking about (6-component) bivectors in spacetime, (specifically the electromagnetic field and 4D-angular-momentum). The conventional perspective is to...

I got a bit confused, and hoped someone could clarify a few things. As far as I am aware, a change of basis is an identity transformation ##I_V## on the vector space (pg. 113) and we can write the relationship between the components of some vector ##v## in the different bases ##\beta## and...

According to this this the Darboux transformation preserves the discrete spectrum of the Haniltonian in quantum mechanics. Is there a proof for this? My best guess is that it has to do with the fact that $$Q^{\pm}$$ are ladder operators but I'm not sure.

The Lorentz transformations of electric and magnetic fields (as given, for example in Wikipedia) are $$ \begin{align*} \bar{\boldsymbol{E}}_{\parallel} & =\boldsymbol{E}_{\parallel}\\ \bar{\boldsymbol{E}}_{\perp} &...

Hello again. I am sorry I got another problem when learning QFT regarding the Lorentz transformation of derivatives. In David Tong's notes, he says I am not sure how to transform the partial derivatives. Explicitly, should ##\frac {\partial} {\partial x ^{\mu}}## transform to ##\frac...

The problem is given in the summary. My attempt: Assume that ##\psi^\prime (x^\prime)## is a solution of the Dirac equation in the primed frame, given the transformation ##x\mapsto x^\prime :=\Lambda^{-1}x## and ##\psi^\prime (x^\prime)=S\psi(x)##, we have $$ \begin{align*} 0&=(\gamma^\mu...

I'd like to get some feedback on the following argument. Gallilean mechanics, with the Gallilean transformation laws, is a perfectly consistent theory. Special relativity, with the Lorentz transformation laws, is another perfectly consistent theory. The question is - can we have some physical...

This approach is seeming intuitive to me as I can visualize what's going on at each step and there's not much complex math. But I'm not sure if I'm on the right track or if I'm making some mistakes. Here it is: ##A## has set up a space-time co-ordinate system with some arbitrary event along his...

I have to derive the Lorentz time transformation given the equation for gamma and the equation for the Lorentz space transformation. I started by using relevant equations from the Space derivation done in class (also the one that Ramamurti Shankar does). Here is a picture of what I have tried...

The components of a vector ##v## are related in two coordinate systems via ##v'^\mu = \frac{\partial x'^\mu}{\partial x^\sigma}v^\sigma##. When evaluating this at a specific ##x'(x_0) \equiv x'_0##, how should we proceed? ##v'^\mu(x'_0) = \frac{\partial x'^\mu}{\partial...

a) I think I got this one (I have to thank samalkhaiat and PeroK for helping me with the training in LTs :) ) $$\eta_{\mu\nu}\Big(\delta^{\mu}_{\rho} + \epsilon^{\mu}_{ \ \ \rho} +\frac{1}{2!} \epsilon^{\mu}_{ \ \ \lambda}\epsilon^{\lambda}_{ \ \ \rho}+ \ ...\Big)\Big(\delta^{\nu}_{\sigma} +...

I'm a bit confused about the condition given in the description of the symmetry transformation of the filed. Usually, given any symmetry transformation ##x^\mu \mapsto \bar{x}^\mu##, we require $$\bar\phi (\bar x) = \phi(x),$$ i.e. we want the transformed field at the transformed coordinates to...

kindly note that this solution is NOT my original working. The problem was solved by my colleague. I have doubts with the ##k## value found. Is it not supposed to be ##k=0.5?## as opposed to ##k=2?##. From my reading on scaling, the graph shrinks when ##k## is greater than ##1## and conversely.

This exercise was proposed by samalkhaiat here Given the defining property of Lorentz transformation \eta_{\mu\nu}\Lambda^{\mu}{}_{\rho}\Lambda^{\nu}{}_{\sigma} = \eta_{\rho \sigma}, prove the following identities (i) \ (\Lambda k) \cdot (\Lambda x) = k \cdot x (ii) \ p \cdot...

In Griffith's Introduction to Electrodynamics, chapter 12, he discusses how electromagnetic fields transform when we move from one inertial reference frame to another. On page 553, he claims He then considers how the electric field inside a conductor made up of two parallel rectangular plates...

For the given joint pdf of X and Y $$f(x,y) = 12xy(1 - y); 0 < x < 1; 0 < y < 1$$ Let $Z = XY^2$ and $W = Y$ be a joint transformation of (X,Y) Sketch the graph of the support of $(Z,W)$ and describe it mathematically. I'm not very sure how to describe (Z,W). First, I draw the graph of the...

Dear reader, there is a physics problem where I couldn't understand what the solutions. It is about the lorentz transformation of a bilinear spinor matrix element thing. So the blue colored equation signs are the parts which I couldn't figure out how. There must be some steps in between which...

I am trying to solve the following problem from my textbook: Formulate the vector field $$ \mathbf{\overrightarrow{a}} = x_{3}\mathbf{\hat{e_{1}}} + 2x_{1}\mathbf{\hat{e_{2}}} + x_{2}\mathbf{\hat{e_{3}}} $$ in spherical coordinates.My solution is the following: For the unit vectors I use the...

I'm reading the article https://www.researchgate.net/publication/267938119_ON_THE_GALILEAN_COVARIANCE_OF_CLASSICAL_MECHANICS (pdf link here), in which the authors want to establish the transformation rule for momentum, assuming only that ##\vec{F}=d\vec{p}/dt## and notwithstanding the relation...

I think this should be t'= Lorentz factor* (1+v/c)t, but that doesn't make sense to me.

Hello, I know question surrounding this topic have been asked again before, however the more I search for the answer of my question, the more confused I become since each book, paper and thread uses its own formulation. I am trying to figure out what is a conformal transformation and as a...

Hi all, I can't find a single thing online that translates a cartesian velocity vector directly to spherical vector coordinate system. If I am given a cartesian point in space with a cartesian vector velocity and I want to convert it straight to spherical coordinates without the extra steps of...

Are Lorentz transforms actual "rotations" in the commonly understood sense, or a non-intuitive formal mathematical operation?

Hey! :o When it is given that a signal $x(t)$ has a real-valued Fourier transformation $X(f)$ then is the signal necessarily real-valued? I have thought the following: $X_R(ω)=\frac{1}{2}[X(ω)+X^{\star}(ω)]⟺\frac{1}{2}[x(t)+x^{\star}(−t)]=x_e(t) \\ X_I(ω)=\frac{1}{2i} [X(ω)−X^{\star}(ω)]⟺...

My work seems to be wrong somehow since my answer is wrong, and I need your help to find out which part is going wrong for me :/ My work: 1. 2. 3. 4. 5. 6. (10/7)/(10/7+7) = 0.7627A, which is not the answer. May I consult where is the problem in solution? THanks

We have 4-tensor of second rank. For example energy-momentum tensor ##T_μν## , which is symmetric and traceless. Then ##T_{μν}=x_μx_ν+x_νx_μ## where ##x_μ## is 4-vector. Every 4- vector transform under Lorentz transform as (12,12). If we act on ## T_{μν}## , by representation( with...

I have one project in my mind bu tI have no eductaion in this topic. please help. question is: if there is container of water. and we have certain number of solar rays getting into this container how does process of water warm up look like comparing: 1. direct sun rays into water vs 2. on...

I kinda know how to do this problem, it is just that I hit a sign problem. If I take the partial derivative of the coordinate transformation with respect to ##x'^\mu##, I get writing it first in the inverse form, ##x^\alpha = x'^\alpha - \epsilon^\alpha## ##\frac{\partial x^\alpha}{\partial...

I wanted to make a derivation of the Lorentz transformation : $$x'=Ax+Bt\\t'=Dx+Et$$ The conservation of the quadratic form ##c^2t'^2-x'^2## yields the equations: $$A^2-B^2/c^2=1\\D^2-E^2/c^2=-1/c^2\\AD=BE/c^2$$ Hence ##B=c\sqrt{E^2-1}##,##D=\sqrt{E^2-1}/c##,##A=\pm E##. The speed of the...

The given Lagrangian is: ##L = \frac 1 2 m_1 ( \dot x_1^2 + \dot y_1^2 + \dot z_1^2) + \frac 1 2 m_2 ( \dot x_2^2 + \dot y_2^2 + \dot z_2^2) + G \frac{m_1 m_2}{\sqrt{(x_1-x_2)^2 + (y_1-y_2)^2 + (z_1-z_2)^2}}##Please note: I have been inspired by the post...

I have read many GR books and many posts regarding the title of this post, but despite that, I still feel the need to clarify some things. Based on my understanding, the contravariant component of a vector transforms as, ##A'^\mu = [L]^\mu~ _\nu A^\nu## the covariant component of a vector...

What is the difference between special relativity and the Lorentz transformation? Aren't they basically the same thing? Also, I was wondering what about matter makes spacetime curve?