Representation Definition and 722 Threads

For $ \text{Re} (a) >0$ and $\text{Re} (s)>1$, the Hurwitz zeta function is defined as $ \displaystyle \zeta(s,a) = \sum_{n=0}^{\infty} \frac{1}{(a+n)^{s}} $. Notice that $\zeta(s) = \zeta(s,1)$. So the Hurwitz zeta function is a generalization of the Riemann zeta function. And just like the...

Hi, A bit is a fundamental unit of information, classically represented as a 0 or 1 in your digital computer. I now number 100 is written in classical bits 0 and 1 as 1100100.Then How to represent 100 in qbits. cheers!

Here is another integral representation of $\zeta(s)$ that is valid for all complex values of $s$. It's similar to the first one, but a bit harder to derive.$ \displaystyle \zeta(s) = 2 \int_{0}^{\infty} \frac{\sin (s \arctan t)}{(1+t^{2})^{s/2} (e^{2 \pi t} - 1)} \ dt + \frac{1}{2} +...

Why do we use phasor representation in physics..For example,why we need maxwells equation in phasor form as well??

Hi everyone! I would like to ask you a very basic question on the decomposition 3\otimes\bar 3=1\oplus 8 of su(3) representation. Suppose I have a tensor that transforms under the 8 representation (the adjoint rep), of the form O^{y}_{x} where upper index belongs to the $\bar 3$ rep and the...

In Helmholtz original thesis On integrals of the hydrodynamical equations, which express vortex-motion, he mentioned in the first section that the change undergone by an arbitrary infinitesimal volume of water under the time dt is composed of three different motions. One of them is an expansion...

Show that $\displaystyle \zeta(s) = \frac{2^{s-1}}{1-2^{1-s}} \int_{0}^{\infty} \frac{\cos (s \arctan t)}{(1+t^{2})^{s/2} \cosh \left( \frac{\pi t}{2} \right)} \ dt $The cool thing about this representation is that it is valid for all complex values of $s$ excluding $s=1$. This integral is...

Consider the complete graph with 5 vertices, denoted by K5. C. Find an isomorphic representation (graph) of K5. Give the isomorphism mappings. Can someone please tell me if this is correct? One dot on graph = K1 One dot on graph = K2 One dot on graph = K3 One dot on graph = K4 One dot on...

Hi, I'm getting a bit confused about the adjoint representation. I learned about Lie algrebras using the book by Howard Georgi (i.e. it is very "physics-like" and we did not distinguish between the abstract approach to group theory and the matrix approach to group theory). He defines the...

Hi, According to what I understood, when a terminal is connected to a line, it causes electrons to flow in one direction. And so for a single phase transmission line in parallel if connected to supply, then in one side electrons will be flowing in one direction and the same electrons will be...

Any given representation (some matrices of some algebra) will be reducible if there exists a singular, but nonzero matrix S that commutes with all elements of the representation; conversely, if no such S exists, then the representation is irreducible. This is Schur's Lemma. My question is...

Well, i´m trying to understand this: I´ve got a representation of SU(2)_L\otimes U(1)_Y such that the left lepton doublets can be represented as (2, -1) and lepton singlets rights as (1, -2). Then I can be left antiparticles bilinear representations as (2,1)\times(2,1) or...

I am reading the article Mirela Vinerean: http://www.math.kau.se/mirevine/mf2bess.pdf On page 6, I have a question about e^{\frac{x}{2}t} e^{-\frac{x}{2}\frac{1}{t}}=\sum^{\infty}_{n=-\infty}J_n(x)e^{jn\theta}=\sum_{n=0}^{\infty}J_n(x)[e^{jn\theta}+(-1)^ne^{-jn\theta}] I think there is a...

Hi, I am quite naive in Particle Physics, and I have a question that Can Elementary Particles be related with irreducible representation? Could we say scalar, vector, and spinor are irreducible representation of SO(3)? Thanks a lot! I also wish I could have some reference on...

Hello, I'm reading Zee's book 'Quantum Field Theory in a Nutshell', the chapter about Lorentz group representations at the moment. In the end of the chapter there is suggested an exercise - "Show by explicit computation that (1/2,1/2) is indeed the Lorentz vector". And I just can't figure it...

Hello! I'm trying to derive the general matrix form of a lorentz boost by using the generators of rotations and boosts: I already managed to get the matrices that represent boosts in the direction of one axis, but when trying to combine them to get a boost in an arbitrary direction I always...

Homework Statement Find the 3D representation of what I think are the commutators [T_a,T_b] for the SU(2) groupHomework Equations I think the generators(X_i) in SU(2) group are the 3 Pauli matrices, which are 2X2 matrices... I assume I need to find the matrices for these generators as 3x3...

Hey guys, How come a representation \rho of a group G is always equivalent to a unitary representation of G on some (inner product) space V ? Can anyone provide a good source (book, preferably) which states a proof? Thanks

Hello everyone, Say I have two irreducible representations \rho and \pi of a group G on vector spaces V and W. Then I construct a tensor product representation \rho \otimes \pi : G\to \mathrm{GL}\left(V_1 \otimes V_2\right) by \left[\rho \otimes \pi \right] (g) v\otimes w = \rho (g) v...

Hi everyone, I'm having some trouble with the concept of the direct sum and product of representations. Say I have two representations \rho_1 , \rho_2 of a group G on vector spaces V_1, V_2 respectively. Then I know their direct sum and their product are defined as \rho_1 \oplus \rho_2 : G...

representation of linear operator using "series"? I was looking into the progression of quantum states with respect to time. From what I understood the progression of a state ## \left|\psi(t)\right> ## is given by: $$ \left|\psi(t)\right> = U(t)\left|\psi(0)\right> $$ I'm not sure if that's...

Hello everyone, I'm reading a bit about the Wigner D matrix, defined by \mathscr{D}\left(\hat{n},\phi \right) = \exp[-\frac{i \phi}{\hbar}\vec{J}\cdot \hat{n}]. Now I'm wondering : is the map \pi : \text{SO(3)} \to \text{GL}\left( \mathscr{H} \right) given by R\left(\hat{n},\phi...

Here is the question: Here is a link to the question: Help with this power series representation? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Ok this question may be kinda stupid, but here goes. Do any surfaces exist for which a parametric form is possible, but cannot be described explicitly due to their highly irregular shape? (Or vice-versa)

Homework Statement For the power series representation of, f(x)=1+x1−x which is 1+2∑from n=1 to inf (x^n), Where does the added 1 in front come from? How do I get to this answer from ∑n=0 to inf (x^n)+∑n=0 to inf (x^(n+1)) Homework Equations The Attempt at a Solution I arrived at ∑n=0 to inf...

Edit: Never mind. Got it. Homework Statement f(x)=\frac { x }{ { (2-x) }^{ 2 } } Homework Equations The Attempt at a Solution I tried finding the first derivative, the second derivative, and so on, but it just keeps getting more complicated, so I suspect I have to use binomial series. The...

Homework Statement Find the Taylor series of f(x) = x2ln(1+2x2) centered at c = 0. Homework Equations Taylor Series of f(x) = ln(1+x) is Ʃ from n=1 to ∞ of (-1)n-1xn/n The Attempt at a Solution I have worked the problem to (-1)n4nx2n/n I am not sure where to go from here...

A = \langle q_f(t) \mid q_i(t) \rangle = \langle q_{f,H} \mid e^{iH(t_0-t)} e^{-iH(t-t_0)} \mid q_{i,H} \rangle = \langle q_{f,H} \mid q_{i,H} \rangle This means that A is time-independent, and depends only on the reference point ##t_0##. How is it possibly? From Schoedinger picture it...

I have a question about the formalism of quantum mechanics. For some operator A... \langle x |A|\psi\rangle = A\langle x | \psi \rangle Can this be derived by sticking identity operators in or is it more a definition/postulate. Thanks.

Most physicists are familiar with the representation of vectors perpendicular to the plane (\otimes and \odot) which look like the fletching or head of an arrow, commonly used to represent magnetic fields, currents and so on. Can anyone tell me the name of this representation? Does it even...

Here is the question: Here is a link to the question: Find the first five non-zero terms of power series representation centered at x=0 for the function below.? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Homework Statement Find the standard matrix representation for each of the following linear operators: L is the linear operator that reflects each vector x in R2 about the x1 axis and then rotates it 90° in the counterclockwise direction. Homework Equations The Attempt at a Solution So my...

I am very confused by the treatment of Peskin on representations of Lorentz group and spinors. I am confronted with this stuff for the first time by the way. For now I just want to start by asking: If, as usual Lorentz transformations rotate and boost frames of reference in Minkowski...

Homework Statement Plane: 4x−2y+10z =16. Homework Equations The Attempt at a Solution So I've used two parameters, "u" and "v" with x = u and y = v Re-arranging z in terms of "u" and "v": z = 1.6 - 0.4x + 0.2y Hence r(t) = (u , v , 1.6 - 0.4 x + 0.2y) Is this correct?

Whenever you see representations of gravity in terms of relativity, you see a planet sitting on a 2d surface of fabric (space) and it is making an indentation, almost as if there another source of gravity pulling it downwards against the fabric. I think this is a poor representation. I mean...

p=\hbar k So ##dp=\hbar dk## How to define Fourier transform from momentum to coordinate space and from wave vector to coordinate space? I'm confused. Is there one way to do it or more equivalent ways?

Coordinate representation ##\psi=\psi(x)## Momentum representation ##\psi=\psi(p)## Fourier transform \psi(p)=\frac{1}{\sqrt{2\pi\hbar}}\int^{\infty}_{-\infty}\psi(x)e^{-\frac{ipx}{\hbar}}dx I'm confused with this ##\hbar##? Why not...

Hi there, eI have two numbers: z1 = 2 + i z2 = exp(iδ) * z1 i are complex numbers and δ is a real number. I need to answer a question - what does the graphical representation of z2 have in relation to the graphical representation of z1. Thanks for any help!

Homework Statement Use differentiation to find a power series representation for f(x)=1/(1+x)2 Homework Equations The Attempt at a Solution 1/(1-x) = \sum(x)n 1/(1-(-x)) = \sum(-x)n Deriving 1/(1-(-x)) -1/(1-(-x))2= \sumn(-x)n-1 from n=1 to infinity indexing it from...

Can anybody please tell me how to represent a neutral antimatter body such as a planet or a star in the classical formulation of special and general relativities? Thanks.

[EDIT]: Correct answer for this problem is 1/2, not 4 as I thought before; that means the result for the explicit representation was correct. Still I don't understand how to treat the case with the parametric representation. Greetings, I need to evaluate $$\iint_{S}\mathbf{F}\cdot\mathbf{n}\...

Homework Statement write a power series representation of the following: \frac{x}{15x^2 +1} Homework Equations the formula \frac{1}{1-x} = 1 + x + x^2 + ... = \sum_{n=0}^{∞} x^n The Attempt at a Solution we can rewrite the summnd like \frac{x}{15} \left(...

Heyy! Please check the attachment and explain why are they arranged (the matrix entries) the way they are? I mean, what is the rule for building up a matrix? In other words, why did not we start with <-1/2, -1/2 l S^2 l -1/2, -1/2 > and placed it as being the first entry? Thanks.

Hi guys, I was studying the proof below and just can't figure out the the first highlighted step leads to the second and I was wondering if you guys can help me to fill that in. (: Thank you so much for your help in advance guys!

We were told in lectures that the time independent Schrodinger equation can be applied to wavefunctions, i.e. \frac{hbar^2}{2m}\frac{d^2U}{dx^2}+V(x)U=EU where U is the wavefunction bra x ket psi. I don't understand why this is valid, as wavefunctions are probability amplitudes, and operators...

In Mark Srednicki's book "Quantum Field Theory" He says that a tensor field B^{αβ} with no particular symmetry can be written as :- B^{αβ} = A^{αβ} + S^{αβ} + (1/4) g^{αβ} T(x) Equn. 33.6 where A - Antisymmetric, S = symmetric and T(x) = trace of B^{αβ} . Is there any reason...

Homework Statement I am trying to represent the function 2x/(1+x2) as a series and determine its radius of convergence. Homework Equations The Attempt at a Solution Using the series for ln(1-x) I have come up with the following series: Ʃ(n=1,infinity) [2*(-1)n*x2n-1], with radius...

I have a question that is a little hard to explain, since i don't know the name of this method, but I'll try my best, if anyone knows the name please do tell me. So let's say we have three numbers, 1 2 3 (in this order) and we have a container for this numbers: C123 and we have some...

The Fourier series of a delta train is supposedly (1/T) + (2/T ) Ʃcos(nωt) ... where T is period and ω=2*Pi/T ...but when I plot this, it doesn't give me just a spike towards positive infinity, but towards negative infinity as well (see attached pic), so this does not seem to converge to the...

What do we mean by 'Equivalent Projective representation"? I know that we say two representations R and R' of a group G is equivalent if there exists a unitary matrix U such that URU^(-1)=R'. But what do we mean by equivalent projective rerpesentations? I've heard of the theorem that the...