Recent content by knowwhatyoudontknow

I am looking at the point group C_3v described shown here. I am trying to understand the block diagonalization process. The note says that changing the basis in the following way will result in the block diagonal form. What is the rationale for choosing the new basis. Is it...

For transformations, A and B are similar if A = S-1BS where S is the change of basis matrix. For Lie groups, the adjoint representation Adg(b) = gbg-1, describes a group action on itself. The expressions have similar form except for the order of the inverses. Is there there any connection...

New to group theory. I have 3 questions: 1. Tensor decomposition into Tab = T[ab] +T(traceless){ab} + Tr(T{ab}) leads to invariant subspaces. Is this enough to imply these subreps are irreducible? 2. The Symn representations of a group are irreps. Why? 3. What is the connection between...

Intuitively, the Rindler wedge is timelike in Minkowski coordinates and an object crossing the horizon enters a spacelike region. This seems at odds with my understanding of the light cone where the 2 regions are reversed. I think this may be related to the signature of the metric but I'm not...

I found this: https://www.physicsforums.com/threads/gravitational-length-contraction-and-dilation.404153/. Post# 9. What bearing does this have on the current discussion?

OK, I think I have it. Maybe my language was a little sloppy at times. I meant to say looks like a 2D object from the point of view of a distant observer - not that it becomes a 2d object (actually disappears). 1, 2, 3, 3(a) and 3(b) explain things perfectly and clear up my confusion...

The original intent of my question was to reconcile the idea of time dilation and length contraction near a black hole. Tidal forces stretch things in a radial direction whereas a stationary observer at a distance sees a progressive slowing of the object until it get stuck at the horizon, in a...

Thank you all for the spirited conversation! My 50000 foot take away is that length contraction as measured by an observer at a distance is ambiguous due the effects of curvature. However, length contraction may make sense in the context of a local comoving observer where the effects of...

In GR, a free falling object when viewed by a distant observer appears to be length contracted and slows down as it approaches the event horizon of a black hole. The length contraction piece, however, seems counterintuitive. I would have thought that the leading edge of the object would...

Thank you for your responses. I think I have a much better (but maybe not complete) understanding now. I have been watching this video: . At 23:00, Prof. Susskind writes: dΩ32 = dr2 + r2dΩ22 for flat space. Which is still a little confusing to me regarding interpretation.

A 1-sphere exists in 2D space. It is a circle in flat space. A 2-sphere is a 1-sphere embedded in 3D space. Its surface is non flat and 2 dimensional. A 3-sphere is a 2-sphere embedded in 4D space. Its surface is non flat and 3 dimensional. What does this last sentence mean in...

Yes. I agree. Bad choice of words on my behalf. Indeed, the mixed state is not a singlet state.

OK. I think my problem is understanding that states can be pure-separable, pure-entangled, mixed-separable and mixed-entangled. If I treat the singlet as a pure state it cannot be factored and therefore is entangled. If I treat it as a 50%/50% ensemble it is a mixed state that is also...

Let me focus on 4 in the hope it will clarify 1, 2 and 3. Math aside, it seems I can write the singlet as either a pure state as a result of superposition or a mixed state if I consider it as a statistical ensemble with P = 0.5. Also, if I assume that superposition gives me a pure state, the ρ...

I have some basic questions about mixed states and entanglement. 1. Do mixed states always imply that the states are entangled and vice versa? 2. Can mixed states ever be separable? 3. Does interference have anything to do with entanglement? In terms of Density Matrices, ρ = |ψ><ψ|: 4...