Invariant Definition and 406 Threads

Hi, from Srednickis QFT textbook, we know the following coupling of Lorentz group representations: (2,1)\otimes (2,1) = (1,1)_A \oplus (3,1)_S, which yields \epsilon_{a b} as an invariant symbol. Generalising, we can look at (2,1)\otimes (2,1) \otimes (2,1) \otimes (2,1) = (1,1) \oplus...

Can anyone help my to see why my solution is wrong because the in the solutions it says that it is not time invariant

Homework Statement Prove that the electromagnetic wave equation:  (d^2ψ)/(dx^2) + (d^2ψ)/dy^2) + (d^2ψ)/(dz^2) − (1/c^2) * [(d^2ψ)/(dt^2)]= 0 is NOT invariant under Galilean transformation. (i.e., the equation does NOT have the same form for a moving observer moving at speed of...

second order pde -- on invariant? What the meaning for a second order pde is rotation invariant? Is all second order pde are rotation invariant? or only laplacian?

A Theorem in our textbook says... If R is a PID, then every finitely generated torision R-module M is a direct sum of cyclic modules M= R/(c_1) \bigoplus R/(c_2) \bigoplus ... \bigoplus R/(c_t) where t \geq 1 and c_1 | c_2 | ... | c_t . There is an example from our textbook that I...

It is often stated that the Kronecker delta and the Levi-Civita epsilon are the only (irreducible) invariant tensors under the Lorentz transformation. While it is fairly easy to prove that the two tensors are indeed invariant wrt Lorentz transformation, I have not seen a proof that there aren't...

Homework Statement I nedd some help to write a left-invariant and right invariant metric on SU(2) Homework Equations The Attempt at a Solution

Homework Statement Suppose two "small" particles of equal mass m collide, annihilate each other, and create another particle of mass M > 2m . (Note that the final state is just that one "big" particle, nothing else.) If one of the small particles is initially at rest, what must be the minimum...

If X is a left invariant vector field, then L_x \circ x_t = x_t \circ L_x , where xt is the flow of X and Lx is the left translation map of the lie group G. In order to show this, I am trying to show that x_t = L_x \circ x_t \circ L_x^{-1} by showing that L_x \circ x_t \circ L_x^{-1}...

Homework Statement Let W be a 1-dimensional subspace of V that is A-invariant. Show that every non zero vector in W is a eigenvector of A. [A element of Mn(F)] The Attempt at a Solution We know W is A-invariant therefore for all w in W A.w is in W. W is one dimensional which implies to...

I can't generally map Loop Invariant method for proving the correctness of an Algorithm. Take the case of an Insertion Sort http://csnx.groups.allonline.in/pool/Introduction%20to%20Algorithms-Cormen%20Solution.pdf in the above link, at the page 2-3, they have proved the correctness of...

A friend of mine was reading Penrose's new book on CCC; I do not want to discuss this story here but a rather interesting detail which could be relevant w/o the whole CCC stuff. SR and GR rely on (global and local) Lorentz invariance. From these symmetries one can derive invariant mass M² and...

Homework Statement Show that the quantity T = c^2(Δt)^2 - (Δx)^2 - (Δy)^2 - (Δz)^2 is invariant under a change of frame Homework Equations Lorentz transformations Δx' = \gamma(Δx - vΔt) Δt' = \gamma(Δt - vΔx/c^2) Δy' = Δy Δz' = Δz The Attempt at a Solution I know that the way to do...

Hi the problem that I am dealing with is to check whether y[n]=T{x[n]}=x[kn] time invariant or not? My solution is I said z(n)=T{x[n-A]}=x[k(n-A)] and y[n-A]=x[k(n-A)] and because y[n-A]=z(n) so it is time invariant but solution is saying that it is time varying because...

Show the diffusion equation is invariant to a linear transformation in the temperature field $$ \overline{T} = \alpha T + \beta $$ Since $\overline{T} = \alpha T + \beta$, the partial derivatives are \begin{alignat*}{3} \overline{T}_t & = & \alpha T_t\\ \overline{T}_{xx} & = & \alpha T_{xx}...

Is the local propagation of an entity at invariant speed a sufficient condition for its stress-energy tensor, independently of its explicit mathematical form, to be trace-free, or to have null covariant divergence, or both in curved space-time? In the book “An Introduction To Mechanics”, by...

I'm reading wald page 85, and he defines a stress-energy tensor for the linearized gravitational field. he mentions that it not gauge invariant as a problem. but isn't that a general property of any tensor (except scalars). so any stress-energy tensor will not be gauge invariant (change of...

Power, defined as P = dE/dt is Lorentz invariant according to http://farside.ph.utexas.edu/teaching/em/lectures/node130.html, Eq. 1645 But, considering another equation for the power, P = q E v, where E and v are electric field and velocity vectors, respectively; this is obviously not the...

"In a Lorentz invariant theory in d dimensions a state forms an irreducible representation under the subgroups of SO(1,d-1) that leaves its momentum invariant." I want to understand that statement. I don't see how I should interpret a state as representation of a group. I have learned that...

Homework Statement I have decay of Higgs to fermion and antifermion and I need to find out the invariant, averaged amplitude. And I wrote down the Feynman diagram, and calculated everything and I came to this part: \langle|M|^2\rangle=\frac{g_w^2}{4}\frac{m_f^2}{m_w^2}(4p_1\cdot...

In our world where 'c' is large, most people intuitively understand Galilean addition of velocities at everyday speeds i.e. if someone stands 40m behind me and rolls a ball towards me at 10m/s (assuming we are not moving relative to each other) it takes 4s to reach me. If we repeat the...

Homework Statement Hi everyone, I have a physics assignment that asks: Prove that the square of relativistic four-momentum for a massive particle is a relativistic invariant under Lorentz transformations. Can anyone help me to work on the problem? I'm always lost in the class ever since my...

Hi all, I am trying to prove the invariant form for the exterior derivative http://en.wikipedia.org/wiki/Exterior_derivative#Invariant_formulations_of_grad.2C_curl.2C_div.2C_and_Laplacian by following these notes...

Hey everybody, I have to give a talk in our seminar on invariant theory of Lie Groups. And I'm now looking for easy applications of invariant theory to quantum physics. I want to present them to motivate the discussion. I would be lucky if someone of you has an idea where I can found...

Homework Statement Transfer functions of the continuous compensation links are given as follows. Find the impulse transfer functions of the digital compensation links using the impulse invariant method. \frac{a}{s+a} I don't know how to solve the problem correctly :cry: Homework...

Hi, Dear all, Facing problem to understand strain energy function invariant terms A typical strain energy function consist of strain invariant can be defined as followed W(I1,I4)=C0+C1(I1-3)(I4-1)+C2(I1-3)^2+C3(I1-4)^2+C4(I1-3)+C5(I4-1), I1 and I4 are so called invariants of Green's strain...

Weinberg in his 1st book on QFT writes in the paragraph containing 2.5.12 that we may choose the states with standard momentum to be orthonormal. Isn't that just true because the states with any momentum are chosen to be orthonormal by the usual orthonormalization process of quantum mechanics...

I've been trying to find out some info about CP violation in the lepton sector at a basic (ie. a fresh postgraduate) level. We can take the neutrino mixing matrix U in its standard parametrization: \left( \begin{array}{ccc} c_{12}c_{13} & s_{12}c_{13} & s_{13}e^{-i\delta} \\ -s_{12}c_{23}...

Homework Statement I understand the premise of Noether's theorem, and I've read over it in as many online lectures as I can find as well as in An Introduction to Quantum Field Theory; Peskin, Schroeder but I can't seem to figure out how to actually calculate it. I feel like I'm missing a...

Homework Statement Find the eigen values of the following mapping and determine if there are invariant lines. (2 -4) (-3 3) is the mapping. Homework Equations det (L-λI)=0 The Attempt at a Solution L-λI= (2-λ -4) (-3 3-λ) det(L-λI)=0=ac-bd=(3-λ)(2-λ)-12 ...

Hi guys, I'm reading Shankar and he's talking about the Variational method for approximating wave functions and energy levels. At one point he's using the example V(x) = λx^4, which is obviously an even function. He says "because H is parity invariant, the states will occur with alternating...

I'm afraid I need help again... First, these two things are shown: 1) Let v \in T_{\bar p}\mathbb{CP}^n, ||v|| = 1. Then: R(w, v)v = w \forall w \in (\mathbb Cv)^\perp 2) Let v \in T_{\bar p}\mathbb{HP}^n, ||v|| = 1. Then: R(w, v)v = w \forall w \in (v\mathbb H)^\perp Afterwards...

Homework Statement A particle of mass m moving at speed \frac{3}{5}c collides with an identical particle at rest, and forms a new particle of mass M which moves off at speed v. Find v.Homework Equations E-P invariant: E_1^2-p_1c^2=E_2^2-p_2^2c^2=\mathrm{const.} Momentum...

Homework Statement Find all Invariant Probability Measures for P (Markov Chain) E = {1,2,3,4,5} The screenshot below has P and my attempted solution. I am wondering if it acceptable to have infinitely many answers ("all" seems to indicate that is acceptable). Basically, I had too many unknowns...

Why is c invariant for all FORs? Special R took it as a postulate but it did not explain it,,,

Hi, I'm having trouble getting a loop invariant expression for this algorithm: Majority(A): c = 1 m = A[0] for i = 1 to len(A) - 1: if c == 0: m = A[i] c = 1 else if A[i] == m: c = c + 1 else: c = c - 1 return...

Homework Statement Show that I = log(u)-u+2log(v)-v is an invariant of the following system \dot{u}=u(v-2) \dot{v}=v(1-u) Homework Equations The Attempt at a Solution The question was given on a homework assignment, but I have very little idea what it is asking for and even...

Hi all! Long story short, my QFT class recently covered gauge equivalence in QED, and this discussion got me thinking about more general gauge theory. I spent last weak reading about nonabelian symmetries (in the context of electroweak theory), and I like to think I now have a grasp on the...

Does anybody know what interpretation the invariant corresponding to the global U(1) invariance of the Dirac Lagrangian is? I have always had it in my head that it's charge, but then I realized that uncharged free particles such as neutrinos satisfy this equation too! Any thoughts much...

the discrete time system defined by y[n]= x[n] ^ 2 Is it time varying ? I proceeded as follows x[n] → x[n]^2 x[n+a] → x[n+a]^2 so y[n+a] = x[n+a]^2 So according to me it is time invariant Am i right ?

Hey all, Since first learning about Emmy Noether's proof that time invariant laws of physics imply conservation of energy, I can't shake the idea that this is the argument against the notion of free will. Here is my argument: By Noether's first theorem, whenever the laws are invariant in...

Please teach me this: Why is Casimir operator T^{a}T^{a} be an invariant of the coresponding Lie algebra? I know that Casimir operator commutes with all the group generators T^{a}. Thank you very much for your kind helping.

Why in deriving invariant distance in space time we use Pythagorean Theorem with a negative sign?

Let's say that we have a particle flying through space, at a collision course with a planet. As seen from an observer on this planet, the particle has an enormous energy, and its wavelength is just slightly bigger than the Planck length. As the particle falls down the gravitational well of the...

When I was learning translational symmetry I saw that for translation invariance, i.e [T,H]=0 the momentum P needs to be conserved [P,H]=0. This momentum is actually the generator of small translations defined as T:x→x+ε. Now, I was solving some problems and I met one which is...

Hello PF community! I'm having trouble with what strikes me as an inconsistency within conservation of energy, conservation of momentum, and the four-momentum invariant equation (E2-p2c2 = m2c4). For the sake of this question, I'll be using non-relativistic mass--i.e. mass is the same in all...

... I think I've misunderstood their definition of an invariant. The pth power of the trace function seems to be homogeneous of degree p rather than linear: I(\lambda A) = J((\lambda A)\otimes (\lambda A)) = (\text{Tr}\, \lambda A)^2 =\lambda\lambda...

Hi all, I was following Nakahara's book and I really got my mind stuck with something. I would appreciate if anybody could help with this. The Lie derivative of a vector field Y along the flow \sigma_t of another vector field X is defined as L_X...

Varying Gravitational Field - Invariant Tetrahedron?? Classical Theory of Fields, Landau Lifgarbagez, page 246: "Strictly speaking, the number of particles should be greater than four. Since we can construct a tetrahedron from any six line segments, we can always, by a suitable definition of...

Homework Statement So I'm having some difficulty with my QFT assignment. I have to solve the following problem. In three spacetime dimensions (two space plus one time) an antisymmetric Lorentz tensor F^{\mu\nu} = -F^{\nu\mu} is equivalent to an axial Lorentz vector, F^{\mu\nu} =...