Gravitation and Decoherence : New Scientist Article

[zonde]

The expectation value we are computing is a diagonal element of what might be called the transition or scattering matrix, where given a state $|i\rangle$ in the past, we want to compute the amplitude that we find the system in the state $|f\rangle$ in the future.

I looked up scattering matrix and sure operation looks similar. But as you say the two states are past state and future state. But The_Duck applied that operation to two components of past state (future state should be decohered state). So if operation would be carried out using scattering matrix then we would have to have coherent two component past wavefunction (unitary?) evolving into incoherent future wavefunction.

 

[zonde]

I may have got this wrong, but I would appreciate help finding the flaw in this logic:

1. A reference frame moving along a geodesic is an inertial frame.
2. A molecule beam follows a parabola in space, and a geodesic in space-time
3. So this molecule's proper reference frame is inertial.
4. If the molecule is dumbbell shaped (say), both parts move in the same inertial frame
5. Hence both parts share the same proper time and their degrees of freedom evolve together

Can't see flaw in this.

I'm pretty sure this is not accurate enough to describe this system, but I am not advanced enough understand this completely. Perhaps someone could provide us both with more details?

I can give you an argument from the other side. Using Rindler coordinates you can verify that you can get differential time dilation even in flat spacetime region by means of acceleration.
Maybe this can convince you that the opposite is right too (getting rid of differential time dilation in free fall).

 

[fzero]

I looked up scattering matrix and sure operation looks similar. But as you say the two states are past state and future state. But The_Duck applied that operation to two components of past state (future state should be decohered state). So if operation would be carried out using scattering matrix then we would have to have coherent two component past wavefunction (unitary?) evolving into incoherent future wavefunction.

Once the Hamiltonian is specified, and it's eigenstates have been found, then they can be used to write down the states of the system at any time, pure or mixed.

Suppose we prepare the system to be in the state $|\psi_1\rangle$ at time $t_1$, then $|\psi(t_1)\rangle = |\psi_1\rangle$. Physically $|\langle \psi(t_1)|\psi(t_2)\rangle |^2$ is the probability that the system is found in the state $|\psi_1\rangle$ at time $t_2$. This is a perfectly meaningful computation. As The_Duck argued, this probability decreases with time.

It's also just toy model, so it's not going to be exactly the calculation that you might want to see done. Also decoherence is something that is supposed to be derived for the original model, not assumed.

 

[zonde]

Once the Hamiltonian is specified, and it's eigenstates have been found, then they can be used to write down the states of the system at any time, pure or mixed.

Suppose we prepare the system to be in the state $|\psi_1\rangle$ at time $t_1$, then $|\psi(t_1)\rangle = |\psi_1\rangle$. Physically $|\langle \psi(t_1)|\psi(t_2)\rangle |^2$ is the probability that the system is found in the state $|\psi_1\rangle$ at time $t_2$. This is a perfectly meaningful computation. As The_Duck argued, this probability decreases with time.

It's also just toy model, so it's not going to be exactly the calculation that you might want to see done. Also decoherence is something that is supposed to be derived for the original model, not assumed.

Thanks for bearing with me.

So as I understand it, complex phase is specified for each Hamiltonian (energy?) eigenstate. So if we have many energy eigenstates they overlap with different complex phases at different times, right?
And this is similar to white light as opposed to monochromatic light. White light has many frequencies and we can't observe interference. But then again we say that white light is not coherent right at the start.

 

[jerromyjon]

time dilation even in flat spacetime region by means of acceleration.

I thought acceleration doesn't cause time dilation?

White light has many frequencies and we can't observe interference.

Skip to 3:32 and you'll see it...

 

[zonde]

I thought acceleration doesn't cause time dilation?

Yes, that's right.