Recent content by cianfa72

Yes, let me say the "process" of sliding the linked pair of fibers occurs along the rays of the stereographic projection from the north pole of the 3-sphere to the hyperplane ##x_4 = 0## in R^4.

Consider the Hilbert space on ##\mathbb C## of square integrable functions of one variable ##L^2(\mathbb R)##. As claimed in a recent thread in QM subforum, there exists a "good" Hilbert basis for it given by the set of Hermite functions defined on ##\mathbb R##. This set is countable hence...

Ok, so basically the first step employs the stereographic projection from the north pole of the 3-sphere to R^3 (i.e. ##x_4 = 0##) -- all this occurs within R^4.

Ok, so why aren't these Hermite functions employed in quantum physics ? (Or rather, I've never seen them).

No. By lifting one of them in the fourth dimension out of the 3-sphere in R^4, they can be separated.

Ok, furthermore such Hermite functions (that are defined on the entire R, are measurable w.r.t. the Lebesgue measure on R and the square of any of them has Lebesgue integral equals to 1) are members of the Hilbert space of square integrable functions and constitute a maximal orthonormal set...

Sorry, I'm not sure what that link refers to. Does the part in bold above refer to Hermite functions ?

Ok, so in the context of Newtonian physics let's pick the frame in which I am at rest. What if this isn't inertial? To do dynamic w.r.t. it, one is forced to add inertial forces appearing to act on all objects (including me) in this frame. I'm not sure whether from a dynamic perspective I've...

Apart from this, can we explicitly exhibit an Hilbert basis (i.e. a maximal orthonormal set) consisting of square integrable functions over ##\mathbb R## ?

The idea could be "lift" one the two linked loops in R^3 along the fourth dimension to separate it from the other. Yes, move it along the third dimension "outside" of the 2-sphere inside R^3. It should be analogous to two linked circles in R^2: they intersect. To separate them, lift one of...

Sorry, I've not idea for this.... :rolleyes:

Can you explain why they can always be separated in the "ambient" ##\mathbb R^4## ?

I'm not sure whether what follows fits better in a math subforum. Take the Hilbert space of square integrable functions of one variable (the relevant notion of integrability is Lebesgue integral and to get an Hilbert space one needs to consider the equivalence classes of functions that may only...

Ah ok, so here we are taking just a kinematic viewpoint/description, no dynamic is involved.

Yes, however the person who says "I'm stationary" must measure for themself zero proper acceleration.