Representation Definition and 722 Threads

For some reason, diffeomorphism invariance seems to be treated like a second-class citizen in the land of symmetries. In nonrelativistic quantum mechanics, we consider Galilean invariance so important that we form our Hilbert space operators from irreducible representations of the Galilei...

Hi there. I have a silly doubt about the entropy of mixing for ideal gases The entropy of mixing is this (Eq. [1]): S_{mix}=\sum_j N_j s_{j0}+\left (\sum_j N_j c_j \right) R \ln{\frac{T}{T_0}}+\sum_j N_j R \ln {\frac{V}{N_jv_0}} Now I don't know what identity the book uses to rewrite this on...

Well, I have a doubt about something that I've found in the book. It's really silly, but it's been bothering me for a while, so perhaps you can help me to understand this. As you should know, the fundamental equation of a system can be represented in the entropic representation as a function...

I am familiar with the representation theory of finite groups and Lie groups/algebra from the mathematical perspective, and I am wondering how quantum mechanics/quantum field theory uses concepts from representation theory. I have seen the theory of angular momentum in quantum mechanics, and I...

Alright, so excuse my ignorance, but I have no idea why the choice he uses for boosts is "convenient" Just to make sure everyone is on the same metric etc etc. Weinberg uses (-,+,+,+) with gamma defined traditionally and God-given units He requires that transformations..(oh my,,,how...

Greetings all, I am the newest addition of the PF community and let me express my excitement of finally tracking down this intelectual family and hopefully making myself eventually an equal member of it. Now, I will not waste too much of your time let us get down to this PFbuisness=pleasure...

Homework Statement When determining the two's complement of a decimal number, one converts the decimal to binary, finds the complement of each individual digit, adds one, and there we have it. For example, -4=1100--->0011+1=0100=+4 However, when I apply the same logic for -99, I don't...

Please teach me this: What is the adjoint representation in Lie group? Where is the vector space that the ''elements of the group'' act on in this representation(adjoint representation)? Thank you very much for your kind helping.

Can you please give an example for a finite group with a three dimensional pseudo real representation? I can find examples of finite groups with 2, 6 and 8 dimensional pseudo real representations, but couldn't find any with a three dimensional pseudo real rep. Is there some theorem that states...

Hello! I am confused about this circuit representation. I am not familiar with how it works. I was under the impression charge cannot flow across a capacitor (a wire is required). So how does charge move in the following circuit? Not across the battery - since the charge is supposed to be...

Homework Statement Hi, This question is about Lorentzgroups. In my course of Relativity, we've seen a very little about representations of complete Lorentz groups but there are two little exercises, which we can do, but I do not understand what should be checked, not even how to start this...

Homework Statement Determine [T]β for linear operator T and basis β T:((x1; x2]) = [2x1 + x2; x1 - x2] β = {[2; 1], [1; 0]} Homework Equations Now that would be MY question :rolleyes: The Attempt at a Solution Well the answer is [1, 1; 3, 0], but i have no idea what I'm even...

Howdy. I'm interested in finding connections between graph theory and particle physics, particularly where particles are represented as nodes on a graph. It's my understanding that this is not a unique idea. Would anyone know where I can find some of these models? I'm trying to figure out if...

I was wondering if there was something similar to the 8 fold way representation used on mesons for the fermions?

Homework Statement Write down the time independant Schrodinger eqn in the momentum representation for a particle with mass m when the potential is given by V(x) = \frac{1}{2} \gamma x^2 Given that a possible solution is given by \Phi(p) = e^{\frac{-Bp^2}{2}} determine B and the...

If I want to derive the matrix representation for operator Q in the |S1=1/2 ,m1> |S2=1/2 ,m2 > basis, where |Si,mi> are common eigenstates of S2 , Si,z for the ith particle. And I do it in this way: <↑↑|Q|↑↑> <↑↑|Q|↑↓> <↑↓|Q|↓↑> <↑↑|Q|↓↓> <↑↓|Q|↑↑> <↑↓|Q|↑↓> <↑↓|Q|↓↑> <↑↓|Q|↓↓> ...

Homework Statement Fin the Taylor series about x = 0 for: f(x) = 1 / (1-x2)2 Homework Equations g(x) = 1 / (1-x2) The Attempt at a Solution Differentiating g(x), the series representation of g'(x) is Ʃn(x2)n-1 Since f(x) = g'(x)/2x f(x) = Ʃ(n/2)x2n-1 I'm pretty sure that's right...

Homework Statement I'm practicing for my finals this coming week and I'm confused about these 2 problems. Homework Equations The Attempt at a Solution For e). I followed my notes and came up with D=U-1AU..since its asking for some basis to standard basis. And once computed, its the diagonal...

Write a power series representation of xln(1+x2) My first instinct was to attempt to take the second derivative and then find the summation and then integrate but that approach seemed to be a dead end. Basically, the x thrown in there confuses me and you can't split the function into two...

Homework Statement See my blog http://blog.yahoo.com/_HYUKNALCF4ULVTXY2JYSRAWDEA/articles/173228/index?bb=0 Homework Equations The Attempt at a Solution My attempt is also on the blog http://blog.yahoo.com/_HYUKNALCF4ULVTXY2JYSRAWDEA/articles/173228/index?bb=0

Find the length of the curve given by the parametric representation... Homework Statement Calculate the length of the curve given by the parametric representation r(t) = t2(cos t; sin t; cos 2t; sin 2t) for 1≤ t ≤+1: Homework Equations The Attempt at a Solution I know that...

Homework Statement The raising and lowering angular momentum operators, J-hat(subscript +), J-hat(subscript -) are defined in terms of the Cartesian components J-hat(subscript x), J-hat(subscript y), J-hat(subscript z) of angular momentum J-hat by J-hat(+)=J-hat(x)+iJ-hat(y) and...

Hi, I stumbled upon the following two functions which have the same freq domain representation, 1/(j.pi.f) 1.signum(t) = u(t) - u(-t) 2. 2u(t) what is the reasoning behind them having the same f domain representation?

I've been browsing through MTW recently and I found something that puzzles me: They claim that if you have two form, call it \mathbf{T}, it's value, say \mathbf{T}(\mathbf{u} , \mathbf{v} ) can be represented geometrically as follows: take two vectors \mathbf{u} and \mathbf{v}; the surface...

I've done some more work on my http://homepage.mac.com/lpetrich/Science/SemisimpleLieAlgebras.zip package, adding decompositions of representation powers. It was only recently that I discovered that they are called "plethysms". Rep powers can be decomposed by symmetry types, a feature that can...

x²+2xsin(xy)+1=0

I have a doubt from Dirac’s book which states that , Pr =-iħ∂/∂qr This equation was concluded after stating that the linear operator iħ∂/∂qr satisfy the same commutation relations with the q’s and with each other as p’s do. Next it is stated that(Dirac, p-92) “ This possibility enables us to...

Hi guys! I'm having some problems in understanding the direct products of representation in group theory. For example, take two right weyl spinors. We can then write\tau_{0\frac{1}{2}}\otimes\tau_{0\frac{1}{2}}=\tau_{00}\oplus\tau_{01} Now they make me see that...

(At least, I think it's simple.) Disclaimer: I'm approaching this subject from the vantage point of a chemist, so be careful with how much lingo/jargon/rigor you lay on me :redface: The claim is that if you have two representations of a group, \Gamma_1 and \Gamma_2, with bases \{ f_i \} and \{...

Homework Statement See Attachment Homework Equations None I can think of The Attempt at a Solution I'm fairly certain that phi_yx is zero Also I tried factoring out the cos and splitting up the equation into it's respective components, but to no avail. Am I even going about this correctly?

Hi everybody, I have a question: We know that the geometrical representation of 1st order derivative is the slope of a function. Then what is the geometrical representation of derivatives having order more than 1? I mean what does it actually represent in a function? Please some body clear my...

Give a parametric representation of the following surfaces in terms of the given parameter variables: a) The first octant portion of the sphere (x^2) + (y^2) + (z^2) = 16 in terms of the spherical variables theta and phi. b)The graph of the function z = (x^3) - sqrt(y) in terms of the...

After you find the irreducible representation of a molecule, how do you determine the number of IR active bands? For example, in methane, the pointgroup is Td , then you found that the irreducible representation is A1 + T2. How do you find the number of IR active and Raman active bands...

I am lecturing out of R.Bartle, The Elements of Integration and Lebesgue Measure, for the first time. In the most recent lecture I got stuck not being able to argue an inequality on page 107. I cannot post the text here, sorry. But if anyone has the book, can you also explain how to derive the...

http://planetmath.org/encyclopedia/InducedRepresentation.html The thing I don't get is the definition of a group element g's action on a vector \sigma v. In the link it defines the action as g(\sigma v):=\tau (hv), where \tau is the unique left coset of G/H containing gg_{\sigma}(i.e. such...

Homework Statement Find the power series representation for the function f(x)=x/(x^2-3x+2) and determine the interval of convergence. Homework Equations The Attempt at a Solution First I separate into partial fractions 2/(x-2) - 1/(x-1) 2/(x-2) = sum n=0 to infinity (x/2)^n...

The equation is z = y^3. I know how to do normal planes and spheres, but I don't know what to set for r(u,v) when it comes to paraboloid cylinders.

Homework Statement The spin of an electron points along the +x direction. a) What is this state in the representation where +z= \left( {\begin{array}{c} 1 \\ 0 \\ \end{array} } \right) -z= \left( {\begin{array}{c} 0 \\ 1 \\ \end{array} } \right) b)...

Homework Statement In circuits, when we represent a battery we note by a long vertical line(positive) followed by a short vertical line(negative). But in some circuits, we follow a long line followed by a short line, again a long line and a short line with dots between them. What does this...

I'm having difficulty understanding this concept of uniqueness. What's the precise definition of it? Let say we have some direct sum decomposition, (1)Are \left( {\begin{array}{*{20}{c}} {{R_1}} & 0 \\ 0 & {{R_2}} \\ \end{array}} \right) and \left( {\begin{array}{*{20}{c}}...

I was wondering if there is a representation of gamma matrices unitarily equivalent to the standard representation for which Dirac Spinors with positiv energy and generic momentum have only the first two component different prom zero. Anyone can help me?

Edit : Removing my personal theory as I realized they are not tolerated or allowed on these forums.

Hi all, I have been thinking long and hard and trying to rationalize the reason for squaring an equation. I still don't understand why we do it. It's mainly in physics that I don't get it. I understand full well and accept that to get the area of a circle you multiply pi by r^2. But why...

I have read that, in QFT, unlike QM, there is no position probability density function because position is not considered an observable. Then how is a position measurement represented/modeled in QFT? As always, thanks in advance.

Homework Statement In a given basis \{ e_i \} of a vector space, a linear transformation and a given vector of this vector space are respectively determined by \begin{pmatrix} 2 & 1 & 0 \\ 1 & 2 & 0\\ 0&0&5\\ \end{pmatrix} and \begin{pmatrix} 1 \\ 2 \\3 \end{pmatrix}. Find the matrix...

In group representation theory, identity element's representation matrix is unit matrix. But why? Could you give me an exact explanation? Thany you.

The definition of the Euler - Mascheroni constant, \gamma, is given as \gamma = \lim_{n\rightarrow\infty}\sum_{k=1}^{n}\frac{1}{k} - \ln(n) or equivalently in integral form as \gamma = \int_{1}^{\infty}\frac{1}{\left\lfloor x\right\rfloor} - \frac{1}{x}\ dx I saw a seeming related integral...

Hi, I am working through Maggiore's QFT book and have a small problem that is really bothering me. It involves finding the representation of the Poincare algebra generators on fields. I always end up with a minus sign for my representation of a translation on fields compared to Maggiore...

I am 99% sure it is not, but I would like to hear that from someone else to be more serene.

Homework Statement Prove that a representation of a finite group G is faithful if and only if its image is isomorphic to G. Homework Equations The Attempt at a Solution