Genady

This question belongs to another forum 😉

When you take second derivatives of metric, you get rates of its deviation from the flat spacetime.

Curvature is encoded in second derivatives of metric. So, any tensor that depends on second derivatives of metric, describes curvature in some way. Einstein tensor describes curvature in a way that can be related to the spacetime's physical contents.

Let me try to explain it with a bit of algebra. In an expanding homogeneous isotropic universe, a distance between any two points - let's call them, galaxies - is proportional to a number, \(a(t)\), called scale factor, which increases with time, \(t\). So, for example, if a distance between some two galaxies at some moment is \(D\) then later, when \(a(t)\) is twice as large, the distance between these two galaxies is \(2D\). Thus, this distance increases with time as \(a(t)D\). If the universe is finite, then there is a largest distance in it, which, just like any other distance, is proportional to \(a(t)\). Let's call it, \(a(t)L\). The only way for the \(a(t)L\) to become infinitely large is that \(a(t)\) becomes infinitely large. But, if \(a(t)\) becomes infinitely large, then distance between any two galaxies, \(a(t)D\), becomes infinitely large. IOW, all galaxies become infinitely far from each other. We of course know that it isn't so. Thus, either the universe was finite and remains finite, or it was infinite to start with.

The "diagonal argument" can be expressed purely algebraically, without lists, rows, diagonals, and any other "visual aids". I think that such algebraic presentation would eliminate a lot of confusion.

A really large number is finite by definition.

No, in a continuous process it cannot. What is "number TREE3"?

Here is an example: a superposition of two Hamiltonian eigenstates, \(\psi_1\) and \(\psi_2\), with the energies \(E_1\) and \(E_2\): \(\frac 1 {\sqrt 2}(e^{-iE_1t}\psi_1+e^{-iE_2t}\psi_2)\). The probability is squared modulus of this function, which includes a time component, \((E_1-E_2)t\).

I just try to clarify the question because I don't know what they were referring to in the thing you've read. Let's consider an example, a particle in a Hamiltonian eigenstate \(\psi(x)\) with energy \(E\). It evolves in time as \(e^{-iEt}\psi(x)\). The probability density for it to be in position \(x\) is \(\bar {\psi}(x) \psi(x)\). This outcome does not depend on time and thus doesn't change if the sign of time is flipped.

You mean, like in Newtonian mechanics?

As QFT obeys SR, there is direction of time in the QFT as much as it is in the SR.

The domain of the function \(\frac 1 {1-t}\) excludes \(1\), as the domain of the function \(\frac 1 x\) excludes \(0\): (Domain of a function - Wikipedia)

Mathematically speaking, it cannot reach infinite size, but it rather increases unboundedly as \(t \rightarrow 1\) from below. At \(t=1\), the formula is undetermined: mathematically speaking, \(\frac 1 0\) is undefined.

I did not miss that point, but my counterpoint was that it does not work mathematically. It reaches the infinity and momentarily changes to \(- \infty \) and stays negative after that. Distance cannot do this. P.S. You want the size to reach infinity and to stay infinite. Try another formula.

The universe was much more homogeneous just after the Big Bang, as is evidenced by the isotropy of the CMB radiation. It became less homogeneous with time, presumably because of the gravitational clumping. OTOH, expansion works against clumping, and an accelerated expansion even more so. Does anybody know what will happen in the future to the "scale of homogeneity"? and becomes negative after that.

I can't see a reason that connects what comes before and what comes after this phrase. Poetry, perhaps, but not a reason.

Of course, I "see" multilinear machines taking in vectors and producing numbers. I also "see" equivalence classes of indexed collections of numbers with certain transformations being the equivalence relations.

Sure, this is right. Then the question is, how much interbreeding would create a stable inherited set throughout the entire population. The OP question is still open, I think. 1-4% of what? Do we have on average 1 neanderthal gene? 5? 100? Are they genes that we share with neanderthal but with nothing else? Are they "genes" or long chunks of DNA? How long? Is this just a pop-science number?

However, (All modern humans have Neanderthal DNA, new research finds | CNN) It is not clear to me how DNA acquired by a direct descent differs from DNA shared because of a common ancestry. Another question regarding direct descent is about the amount, 1-4%. We have this amount of DNA directly from our grandparents of 5-6 generations back, which is too recent for neanderthals.

Should not this percentage depend on the length of DNA chunks that are compared? In the extreme case, if we compare pieces of one nucleotide long, any organism on Earth shares 100% of its DNS with any other organism on Earth; they all are AT and CG.

I've never seen the SR kind of spacetime diagrams used in QM or in QFT. But a rest frame of a particle or a system is often used and selected in such a way that makes calculations easier. In such a frame, some momentum is zero, which simplifies formulas.

I don't say or imply anything like that and don't have any idea how it seems so. I say that the states, which are spinors, evolve in spacetime. Just like a scalar such as temperature can evolve in spacetime but is not the same as spacetime. But you don't need to go to spinors to express your concern. Long before Dirac, in the good old Schrödinger equation, the wave functions are complex-valued functions and they represent particle states in a complex vector space. In case of the Hamiltonian observable basis, the states, complex functions, are eigenstates, while the energies, real scalars, are eigenvalues. I don't think there is any problem in this distinction.

In the Dirac equation, the evolving objects are spinors. They evolve in spacetime. They constitute states in a spinor space. The equation does not pick any specific basis in this space to expand the spinors as superposition. We are free to choose such a basis and thus such expansion is arbitrary.

I refer to eigenstates of any observable. We can expand any state in any basis. This makes the notion of "states in a superposition" arbitrary.

Yes, but I refer to a coordinate basis of the space of states rather than the geometric 3D space.