Recent content by cianfa72

You mean a detector aligned along the ##x## axis in Alice lab and another detector aligned along the ##z## axis in Bob lab (each of them measure just their own entangled particle).

Ah ok, so the key point is that, in the last claim, EPR reject a definition of reality for quantities that depends upon the process of measurement carried out on the first system/particle. Therefore both quantities P and Q are "elements of reality" in contrast with uncertainty principle applied...

Ok, since according to the math of QM there is not a common eigenvector/eigenstate for two non-commuting observable operators. However I still don't have a crystal clear understanding of what EPR said. I believe EPR did not reject the uncertainty principle, so Alice cannot simultaneously...

Sorry, maybe I misinterpreted the paradox. The point is that, starting from Alice spin measurement on her particle, the spin along ##z## and ##x## axes for Bob's particle should have both definite values (before Bob performs any spin measurement).

He can't since the measurement spin operators along ##z## axis and along ##x## axis for the entire entangled system actually do not commute, right ?

Hi, I was reading about the EPR paradox in Bohm simplified formulation. From my understanding the paradox is that Bob is actually able to get a value for the positron's spin along both the ##z## and ##x## axes. Since electron and positron are entangled, he get the value of spin along ##z##...

Note that Carroll'book, instead, uses only northwest/southeast indices.

From this page any separable Hilbert space is isomorphic to ##\mathcal l^2##. I believe that holds true only for infinite dimensional separable Hilbert spaces (i.e. with a infinite countable orthonormal Hilbert basis), otherwise for finite dimensional Hilbert space over ##\mathbb C## it should...

You mean an orthonormal system of elements that is complete - i.e. an (orthonormal) Hilbert basis. For non-separable Hilbert spaces there is not a countable Hilbert basis, however any vector ##v## in ##H## can be given as a convergent series of basis's elements.

Yes, ok. The linear span of an Hilbert basis is, by definition, dense in the Hilbert space ##H##. Any vector ##v## in ##H## can be given as countable linear combination of Hilbert basis's elements in the sense of convergent series to ##v## according the norm induced by inner product. Does the...

I think my doubt is somehow related to the concept of Hamel/algebraic basis. For example in ##L^2([a,b])## there is an orthonormal Hilbert basis dense in it, however it is not an Hamel basis since its linear span is not the entire space.

Another point related to this. Consider for instance the Hilbert space of ##L^2(\mathbb R)## Lebesgue square-integrable functions on ##\mathbb R##. One can show it is separable i.e. there is a countable basis of orthonormal vectors/functions (these basis's elements are dense in ##L^2(\mathbb...

Ah ok, by induction one can prove the sum is an element of the set only for a finite number of elements.

Hi, a doubt about the definition of vector space. Take for instance the set of polynomals defined on a field ##\mathbb R ## or ##\mathbb C##. One can define the sum of them and the product for a scalar, and check the axioms of vector space are actually fullfilled. Now the point is: if one...

This because we are looking at the coincidence of events (whether they are the same spacetime point or not).