Can two streamlines in a fluid be tangential to each other not crossing each other?

A streamline cannot cross other one in steady state of a fluid but can they be tangent to each other ?

What I mean is consider fluid in steady state. if let us say particle A has its own path in fluid and at point B it has velocity x. But let us say another particle C has its own path and has to cross ppoint B it must also have same velocity and hence the two streamlines are tangent at B. Is this possible otlr is it only valid if particle A and C both have the exact same trajectory fro the beginning ?

quantum field theory - Why cannot fermions have non-zero vacuum expectation value?

In quantum field theory, scalar can take non-zero vacuum expectation value (vev). And this way they break symmetry of the Lagrangian. Now my question is what will happen if the fermions in the theory take non-zero vacuum expectation value? What forbids fermions to take vevs?

fluid dynamics - Why does the Vlasov equation not consider higher order time derivatives?

The Vlasov equation is given by: $$ \frac {df}{dt} = \frac {\partial f}{\partial t} + \vec{v} \cdot \frac {\partial f} {\partial \vec{x}} + \vec{a} \cdot \frac {\partial f} {\partial \vec{v}} $$

But why are higher order time-derivatives such as $ \frac {\partial \vec{a}} {\partial t} $ or $\frac {\partial f} {\partial \vec{a}}$ not included? Are they simply considered small or are they absolutely zero for some reason?

cosmology - Is the Big Bang defined as before or after Inflation?

Is the Big Bang defined as before or after Inflation? Seems like a simple enough question to answer right? And if just yesterday I were to encounter this, I'd have given a definite answer. But I've been doing some reading while writing up my thesis and I'm finding conflicting definitions of the Big Bang.

Everyone agrees that in standard Big Bang cosmology, the Big Bang is defined as the singularity; the moment in time when the scale factor goes to zero. Okay, but when you include the theory of inflation, it gets a bit murky.

So here's what I mean by conflicting definitions. As an example, in The Primordial Density Perturbation by Lythe and Liddle, they define the Big Bang as the beginning of the era of attractive gravity after inflation. However, Modern Cosmology by Dodelson defines the Big Bang as coming before inflation; it effectively uses the old definition that the Big Bang is the moment when the scale factor approaches zero.

This contradiction is evident in multiple places. When doing a google search for it, one can find many persuasive explanations for both definitions. All definitions agree that we cannot any longer define it as the singularity where $a=0$. But every one makes sense in its own way and so, I become more and more confused about which is right the more of them I read.

The argument for the Big Bang coming after is that inflationary theory diverges from the standard Big Bang cosmology around $10^{-30}s$ before we'd expect to run into the singularity, when inflation ended, and that we have no evidence to anything coming before that, thus the big bang is now defined as the initial conditions for the hot, expanding universe that are set up by and at the end of inflation.

The argument for the Big Bang coming before seems to be that inflation is still a period where the scale factor grows and as such, the Big Bang can be defined as the closest value to zero (which is before inflation), or rather, the earliest time as the scale factor approaches zero. This essentially seems to be based on saying "well, we defined it as the moment when the scale factor was smallest before inflation was added. Why would we not continue to have that as the definition after inflation is added?"

The former argument has merit because it defines the start of the epoch where the universe is describable (practically) by the standard Big Bang cosmology. But the latter argument has merit because of its simplicity and that it uses the spirit of the original definition; the smallest scale factor and the moment when the expansion of the universe seems to begin.

Thus, my root question: Which definition is correct? Do we say the Big Bang came before or after inflation?

^{P.S. I realize that asking this here only serves to add one or more additional persuasive arguments to an already crowded debate. However, this is Physics.SE, so I figure whatever we decide here can be definitive. Even if we can't find a truly correct answer, this can set the record straight, or at least, firmly crooked.}

momentum - Collisions between an object and a wall

Is momentum conserved when an object bounces back against a wall? The wall doesn’t move, but the object moves in the opposite direction. Assume this is an ideal, elastic collision.

If, initially, the momentum of the system came from the ball, and after the collision, the momentum of the system was also from the ball, but in opposite directions, then momentum would not have been conserved?

I have a feeling this is wrong, so anybody clarify this for me?

Answer

Is momentum conserved when an object bounces back against a wall? The wall doesn’t move, but the object moves in the opposite direction. Assume this is an ideal, elastic collision.

I have a feeling this is wrong, so anybody clarify this for me?

Yes, it is wrong. You are wrong if you think that in an ideal, elastic collision the velocity of the bouncing object is exactly the same.

And also you are wrong if you think that the wall doesn't move. You can't see it move but it wouldn't only if it had infinite mass, which is impossible.

Suppose a mass of 1 kg hits a wall of 10, 000 kg at $v_0=10$ m/s. It has momentum 1*10 = 10 kg m/s. The formula for an elastic collision tells you that the wall will absorb momentum $ \frac {20}{10 001} = 2$ g m/s and the ball will keep $v'=9.9980002$ m/s.

quantum field theory - Derivation of total momentum operator QFT

The expansion of the Klein Gordon field and conjugate momentum field are

$\hat{\phi}(x) = \int \frac{d^3k}{(2 \pi)^3} \, \frac{1}{ \sqrt{2 E_{k}}} \left( \hat{a}_{k} + \hat{a}^{\dagger}_{-k} \right) e^{i k \cdot x}$

and

$\hat{\pi}(x) = \int \frac{d^3k}{(2 \pi)^3} \, \sqrt{\frac{E_{k}}{2}} \left( \hat{a}_{k} - \hat{a}^{\dagger}_{-k} \right) e^{i k \cdot x}$

The total momentum in the classical field is given by

$P^{i} = -\int d^3x \, \pi(x) \, \partial_{i} \phi(x)$

which is promoted to an operator

$\hat{P}^{i} = -\int d^3x \, \hat{\pi}(x) \, \partial_{i} \hat{\phi}(x)$

We are told that putting the expansion into this should give

$\int \frac{d^3p}{(2\pi)^3} \, p^i \left( \hat{a}_{p}^{\dagger} \hat{a}_{p} + \frac12 (2\pi)^3 \delta^{(3)}(0) \right)$

which I do get, however I also get terms involving $\hat{a}_{p}\hat{a}_{-p}$ and $\hat{a}^{\dagger}_{p} \hat{a}^{\dagger}_{-p}$. How in the world can these terms possibly go away? Have a made a mistake in the math (I am prone to mistakes at the moment.. quite tired) or is there some way that those terms cancel?

Answer

The two integrands $p^i\hat{a}_{{\bf p}}\hat{a}_{-{\bf p}}$ and $p^i\hat{a}^{\dagger}_{{\bf p}}\hat{a}^{\dagger}_{-{\bf p}}$ are antisymmetric wrt. ${\bf p} \leftrightarrow -{\bf p}$. Hence the corresponding integrals $\int d^3p(\ldots )$ are zero.

quantum mechanics - Can atoms have nonzero dipole moments?

Let's make the question easier by considering two-level atoms(with spin states, i.e. spin up $|\uparrow\rangle$ and spin down $|\downarrow\rangle$). An article I recently read claims that atoms do not have dipole moments when they are in energy eigenstates (i.e. when you put it into a magnetic field in z direction).

I was thinking if it's an eigenstate, since the spin cannot be pointing in z direction (remember it's actually pointing in a cone area), how could it not have a dipole moment?