A Quantum mechanics and frame dragging

[kelly0303]

Hello! Is there any work which looks at the frame dragging effect due to a rotating quantum object (e.g. in an eigenstate which is a spherical harmonic)? I would appreciate any reference on that.

 

[Vanadium 50]

You mean like frame dragging from a hydrogen atom? Its hard enough to see frame draging from a planet.

 

[PeterDonis]

Hello! Is there any work which looks at the frame dragging effect due to a rotating quantum object (e.g. in an eigenstate which is a spherical harmonic)? I would appreciate any reference on that.

Frame dragging is a phenomenon in classical GR, which is not a quantum theory and does not treat quantum objects.

 

[kelly0303]

Frame dragging is a phenomenon in classical GR, which is not a quantum theory and does not treat quantum objects.

I know we don't have a full theory trying to combine both, but I was wondering if there are any papers trying to look at this phenomenon (e.g. Hawking radiation is using both GR and QFT, without us having a full theory combining both, so I was wondering if anyone tried to look at this phenomenon in particular).

 

[kelly0303]

You mean like frame dragging from a hydrogen atom? Its hard enough to see frame draging from a planet.

I am interested more theoretically than actually measuring it. I had in mind more like, for example, a molecule rotating, but a hydrogen atom works, too (but in that case is the electron doing most of the spinning, not the nucleus, so the effect would be even smaller).

 

[renormalize]

I am interested more theoretically than actually measuring it.

Frame-dragging at a large distance $r$ from a slowly-rotating object of mass $M$ and angular-momentum $L$ is a consequence of the off-diagonal terms in the weak-field Lense-Thirring metric tensor:$$ds^{2}\approx\left(1-\frac{2M}{r}\right)dt^{2}-\frac{4L^{2}\sin^{2}\theta}{r}dt\,d\phi-\left(1+\frac{2M}{r}\right)\left[dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta\,d\phi^{2}\right)\right]\tag{1}$$In the same spirit as doing quantum field theory in curved spacetime, where the classical Einstein field equations are sourced by the expectation value of the quantum energy-momentum tensor, you could estimate the (tiny) effect of frame-dragging that arises from a distant quantum system with angular momentum $\mathbf{J}$ by substituting $L^{2}\rightarrow\left\langle J^{2}\right\rangle =j(j+1)\,\hbar^2$ or $L\rightarrow\left\langle J_{z}\right\rangle =m_{j}\,\hbar\,$ into (1).

 

[weirdoguy]

but in that case is the electron doing most of the spinning

Nope, that's not how it works. Electron is not spinning in any classical sense.

 

[kelly0303]

Nope, that's not how it works. Electron is not spinning in any classical sense.

What do you mean by classical? The wavefunction of hydrogen does have a spherical harmonics, which corresponds to a given angular momentum.

 

[Vanadium 50]

You cannot paint a red X on an electron and count how often it passes you as the electron rotates. Quantum spin is not classical spin.

 

[Vanadium 50]

OP, you really, really need to put in some numbers.

We cannot see the gravitational field from a single electron. Technology is about 30 orders of magnitude away.

The frame-dragging effect is, for an electron, about 10 orders of magnitude smaller than the ordinary gravitational force.

An electron has a HUGE amount of angular momentum per unit mass: 3 maybe 4 orders of magnitude larger than anything macroscopic.

So you are something like 44 orders of magnitude away from where you want to be.

 

[Nugatory]

What do you mean by classical? The wavefunction of hydrogen does have a spherical harmonics, which corresponds to a given angular momentum.

It corresponds to a given angular momentum, but that doesn't mean that anything is spinning. Angular momentum is a property of a system. All systems that include a spinning object will have some angular momentum, but many systems that do not will also have angular momentum. The bound electron that we're talking about in this thread is an example.

For another example, consider a free electron which has intrinsic angular momentum - it's a spin-1/2 particle. But when we try computing that angular momentum as if it were a spinning object... $\vec{r}\times\vec{p}$ is zero because it's a point particle and $\vec{r}=0$.

 

[kelly0303]

OP, you really, really need to put in some numbers.

We cannot see the gravitational field from a single electron. Technology is about 30 orders of magnitude away.

The frame-dragging effect is, for an electron, about 10 orders of magnitude smaller than the ordinary gravitational force.

An electron has a HUGE amount of angular momentum per unit mass: 3 maybe 4 orders of magnitude larger than anything macroscopic.

So you are something like 44 orders of magnitude away from where you want to be.

I am not interested in the way to measure it, just how you'd go about calculating it theoretically. Also, as I mentioned above, I had in mind a rotating molecule, not an electron (still a small effect, but given that the nuclei are rotating it should be many orders of magnitude higher).

 

[kelly0303]

It corresponds to a given angular momentum, but that doesn't mean that anything is spinning. Angular momentum is a property of a system. All systems that include a spinning object will have some angular momentum, but many systems that do not will also have angular momentum. The bound electron that we're talking about in this thread is an example.

For another example, consider a free electron which has intrinsic angular momentum - it's a spin-1/2 particle. But when we try computing that angular momentum as if it were a spinning object... $\vec{r}\times\vec{p}$ is zero because it's a point particle and $\vec{r}=0$.

But I mentioned in the initial post that I am only interested in orbital, not intrinsic spin, this is why I mentioned spherical harmonic (not sure if spherical harmonics can also describe spin, sorry if that is the case, but I meant only orbital angular momentum which has an easier classical counter part than the intrinsic spin).

 

[weirdoguy]

I am only interested in orbital, not intrinsic spin

Still, it does not work the way you think it does. Even with non-zero orbital angular momentum, nothing is spinning in an atom.

 

[kelly0303]

Still, it does not work the way you think it does. Even with non-zero orbital angular momentum, nothing is spinning in an atom.

I am not sure I understand. People are searching experimentally for quantum spin-gravity interactions. So there must be at least attempts to describe a quantum spinning object in gravity. I don't see why doing the jump from there and asking about frame dragging would be necessary so difficult (of course I wouldn't be able to derive that, at least not quantum mechanically, but the people doing these calculations I assume would be able to).

 

[Nugatory]

People are searching experimentally for quantum spin-gravity interactions. So there must be at least attempts to describe a quantum spinning object in gravity.

Quantum spin is a thing. "Quantum spinning objects" are not, especially in the context of this thread where the "objects" are atoms and smaller.

 

[kelly0303]

Quantum spin is a thing. "Quantum spinning objects" are not, especially in the context of this thread where the "objects" are atoms and smaller.

I am not sure I understand what you mean. As I said, people are searching experimentally for quantum spin interacting with gravity. Whether you think of it as actually rotating or not doesn't matter too much, the point is that there is an interaction. My question is simply if this interaction has an associated frame dragging and if so, has anyone calculated it?

 

[PeterDonis]

people are searching experimentally for quantum spin interacting with gravity.

Can you give some specific references?

 

[Vanadium 50]

People are searching experimentally for quantum spin-gravity interactions.

Who?

I am going to go out on a limb and say nobody is looking for the sort of effect you describe, at least not seriously. There are people looking at the effect of gravity on spin (the reverse of what you suggested) and people looking at the effect of classical angular momentum as a source of gravity, but nobody is looking for a 0.00000000000000000000000000000000000000001% effect.

 

[kelly0303]

Who?

I am going to go out on a limb and say nobody is looking for the sort of effect you describe, at least not seriously. There are people looking at the effect of gravity on spin (the reverse of what you suggested) and people looking at the effect of classical angular momentum as a source of gravity, but nobody is looking for a 0.00000000000000000000000000000000000000001% effect.

@Vanadium 50 @PeterDonis I don't work in this field so I don't know the whole extent of these experiments, but this is a relatively recent paper looking at the interaction between spin and gravity.

@Vanadium 50 as I already said several times, I didn't claim that people are looking for frame dragging at the quantum level. I asked if anyone developed any theoretical formalism for it. Also, I am not sure what is the difference between the effect of gravity on spin and the effect of spin on gravity. Based on the paper I referenced, the hamiltonian (Eq. 1) doesn't seem to favor spin over gravity or the other way around. All the physics of the interaction is contained in the coupling constant characterizing this interaction. The way you interpret this interaction seems to only depend on the question you are trying to ask.

For example, in the hyperfine interaction, you have the electron spin dot product with the nuclear spin. You can think of it as the magnetic field of the electron acting on the nucleus, or the magnetic field of the nucleus acting on the electron, but in the end you still get the same result i.e. same hyperfine structure constant. Why is the Hamiltonian in Eq. 1 different i.e. why can't I think of it as the spin acting on gravity, rather than the gravity acting on spin?

 

[Vanadium 50]

but this is a relatively recent paper looking at the interaction between spin and gravity.

It is not, however, looking at frame-dragging, which is the title of this thread.

I didn't claim that people are looking for frame dragging at the quantum level. I asked if anyone developed any theoretical formalism for it

And you have Message #6. We are now at #21.

I know you don't care about experiment, but scientists do. People are unlikely to spend their time improving calculations beyond #6 unless there is a reason to. The title of this thread is "quantum mechanics and frame dragging". The earth has quantum number l = 10⁶⁸ or so. Its frame dragging is a few parts per trillion.

Most people's reaction to the effect being zero to 40+ decimal places is not "we need a better calculation!"

 

[PeterDonis]

this is a relatively recent paper looking at the interaction between spin and gravity.

Actually it is looking at the theoretically predicted absence of interaction between spin and gravity, and finding that experimentally it is indeed absent, at least to within the accuracy of the experiments.

And as @Vanadium 50 has said, it is not looking at frame dragging. But the theoretical prediction in the paper would also predict the absence of any frame dragging effect.

So it looks like the answer to your original question is "no".

And with that, this thread is now closed.