Recent content by jambaugh

The surface of a 4-ball is a 3-sphere. Here's a narrative to help you "visualize" it. Imagine you are on a tiny smooth planet (2-sphere). You set down a wading pool made of super-elastic bubble plastic on the north poll. You begin filling it with water and instead of overflowing its floor...

We are often a bit sloppy with the notation ##\mathbb{C}^n## could mean the n-dimensional complex vector space with no specific inner product or to the Hilbert space with a given Hermitian inner product (since they are all equivalent). Similarly with ##\mathbb{R}^n##. The author should give...

Under special linear transformations they should behave identically. However, under reflections I believe they behave differently. E.g. under inversion in the x-direction ##\hat{x}\wedge \hat{y}\to -\hat{x}\wedge \hat{y}## while ##(\hat{z}•)\to (\hat{z}•)##. (Notation...

No, Bell utilizes an RAA hypothesis of local causality and that probability acts as a measure over a state manifold and thus is additive. The local causality is just one of many ways to invoke the hypothesis that a system can be factored and interactions between components can be prevented...

In the treatments I've seen the x and k are c-numbers i.e. parameters and not operators. All operators are expressed in terms of the creators and annihilators. Typically however the creators and annihilators are themselves indexed e.g. by momentum k. Note from the texts how e.g. the momentum...

It might be better phrased as "non localized correlations". Remember that even a single particle in a momentum eigen-"state" is a non-localized quantum system. A positive detection at a given point is correlated with the null detection at all other points in space. That's not shocking. The...

When attending a quantum computing conference in Italy many years ago we referred to this "no FTL signalling" theorem as the "No Bell Telephones" theorem. In my opinion, with no observable supraluminal causation, we should dispense with assertions about hidden supraluminal causation (just like...

The tensor product inside the parentheses are also non-commuting. When two spaces are distinct it is a choice of convention which factor we decide will go on the left and which on the right in applications. I believe that the seeming commutativity your are referencing is merely one playing...

There are several \mathbb{Z}_2's in O(p,n) you must be careful which one you are considering. Let me think out loud and see where this goes: Simple example: ISO(3) = \mathbb{R}^3 \rtimes SO(3). The action of the rotation group(s) on the translation subgroup (normal subgroup) is to rotate it...

I'm still unclear as to what you mean by "reaction force". The piston exerts a force on the fluid and the fluid exerts a force on the piston. Assuming no acceleration these are equal (equal and opposite reaction forces). That force will be independent of the surface geometry of the fluid...

Ah, My apologies for not understanding your question. Where are you implementing the Finite Element simulation? Is it for the fluid or for the piston? If the piston then I think my answer, still works in that my "polygon approximation" is equivalent to a simple finite element decomposition...

I don't think your characterization of O(3,1) as \mathbb{Z}_2\rtimes SO(3,1) isn't quite correct. There is no direct action of SO(3,1) on \mathbb{Z}_2. Rather, I believe your semi-direct product should be reversed as it is SO(3,1) that is the normal subgroup. Recall that two inversions will...

Note that the force applied to the surface by pressure is a vector normal to that surface. Try it with a simple polygon, figure the vectors and notice how the horizontal components all cancel(edit) and the vertical components are exactly the same as would occur if the surface had been flat...

Since this is a combination of a row operation and a column operation you can express it as the product of a matrix on the left and another on the right. As it happens since these matrices are row/column swaps of identity matrices they are unipotent (they square to the identity) or said another...

I'd vote it "wild speculation" at best. But it makes great click-bait so you'll see it often on pop science websites and news feeds. Now if you want to worry about something more pressing, Andromeda will collide with the Milky Way and probably the central black holes will merge. I afraid that...