Thursday, April 19, 2012

Schrödinger meets Newton

In January, we discussed semi-classical gravity: Classical general relativity coupled to the expectation value of quantum fields. This theory is widely considered to be only an approximation to the still looked-for fundamental theory of quantum gravity, most importantly because the measurement process messes with energy conservation if one were to take it seriously, see earlier post for details.

However, one can take the point of view that whatever the theorists think is plausible or not should still be experimentally tested. Maybe the semi-classical theory does in fact correctly describe the way a quantum wave-function creates a gravitational field; maybe gravity really is classical and the semi-classical limit exact, we just don't understand the measurement process. So what effects would such a funny coupling between the classical and the quantum theory have?

Luckily, to find out it isn't really necessary to work with full general relativity, one can instead work with Newtonian gravity. That simplifies the issue dramatically. In this limit, the equation of interest is known as the Schrödinger-Newton equation. It is the Schrödinger-equation with a potential term, and the potential term is the gravitational field of a mass distributed according to the probability density of the wave-function. This looks like this

Inserting a potential that depends on the expectation value of the wave-function makes the Schrödinger-equation non-linear and changes its properties. The gravitational interaction is always attractive and thus tends to contract pressureless matter distributions. One expects this effect to show up here by contracting the wave-packet. Now the usual non-relativistic Schrödinger equation results in a dispersion for massive particles, so that an initially focused wave-function spreads with time. The gravitational self-coupling in the Schrödinger-Newton equation acts against this spread. Which one wins, the spread from the dispersion or the gravitational attraction, depends on the initial values.

However, the gravitational interaction is very weak, and so is the effect. For typical systems in which we study quantum effects, either the mass is not large enough for a collapse, or the typical time for it to take place is too long. Or so you are lead to think if you make some analytical estimates.

The details are left to a numerical study though because the non-linearity of the Schrödinger-Newton equation spoils the attempt to find analytical solutions. And so, in 2006 Carlip and Salzmann surprised the world by claiming that according to their numerical results, the contraction caused by the Schrödinger-Newton equation might be possible to observe in molecule interferometry, many orders of magnitude off the analytical estimate.

It took five years until a check of their numerical results came out, and then two papers were published almost simultaneously:
• Schrödinger-Newton "collapse" of the wave function
J. R. van Meter
arXiv:1105.1579 [quant-ph]
• Gravitationally induced inhibitions of dispersion according to the Schrödinger-Newton Equation
Domenico Giulini and André Großardt
arXiv:1105.1921 [gr-qc]
They showed independently that Carlip and Salzmann's earlier numerical study was flawed and the accurate numerical result fits with the analytical estimate very well. Thus, the good news is one understands what's going on. The bad news is, it's about 5 orders of magnitude off today's experimental possibilities. But that's in an area of physics were progress is presently rapid, so it's not hopeless!

It is interesting what this equation does, so let me summarize the findings from the new numerical investigation. These studies, I should add, have been done by looking at the spread of a spherical symmetric Gaussian wave-packet. The most interesting features are:
• For masses smaller than some critical value, m less than ~ (ℏ2/(G σ))1/3, where σ is the width of the initial wave-packet, the entire wave-packet expands indefinitely.
• For masses larger than that critical value, the wave-packet fragments and a fraction of the probability propagates outwards to infinity, while the rest remains localized in a finite region.
• From the cases that eventually collapse, the lighter ones expand initially and then contract, the heavier ones contract immediately.
• The remnant wave function approaches a stationary state, about which it performs dampened oscillations.
That the Schrödinger-Newton equation leads to a continuous collapse might lead one to think it could play a role for the collapse of the wave-function, an idea that has been suggested already in 1984 by Lajos Diosi. However, this interpretation is questionable because it became clear later that the gravitational collapse that one finds here isn't suitable to be interpreted as a wave-function collapse to an eigenstate. For example, in this 2002 paper, it was found that two bumps of probability density, separated by some distance, will fall towards each other and meet in the middle, rather than focus on one of the two initial positions as one would expect for a wave-function collapse.

Erik said...

Hey Bee,
I am new to your blog, but I definitely love it!
I am an undergraduate applied physicist from the Netherlands and I will attend a graduate programme in theoretical physics soon.
This "Schrödinger meets Newton" intrigues me, have you got any recommendations for books on the subject?
Erik

Bee said...

Hi Erik,

Thanks for the kind words. I don't know if there's any book which covers the SN equation. But the two papers that I mentioned above (and references therein) are a good start. They are very readable actually, so give it a try. Whereabouts in the Netherlands are you? Best,

B.

Ben John said...
This comment has been removed by a blog administrator.
Georg said...

""the lighter ones contract initially and then contract, the heavier ones contract immediately""

Dunkel, oh Sabine, ist dieses Satzes Sinn. :=)
Georg

Bee said...

Danke, oh Georg, ein Loch war im Eimer. I've fixed that.

Erik said...

Hi Bee,
Sure, I've scanned the articles, but I have an exam on condensed matter physics coming up tomorrow, so a thourough read will have to wait :) I am originally from a place close to Maastricht (actually quite close to Aachen, which you probably know). At the moment I'm living in Eindhoven, where I study applied physics. Next year (or the year after that) I'll be going to the Utrecht Institute for theoretical physics..
Erik

Uncle Al said...

arxiv:1105.1579 "by solving Eq. (3) with the Green's function" Physical chirality is excluded. "sigma = 4.4x10^(-9) m ...mass is given in units of 3.33x10^(-17) kg (2x10^10 u)"

4.47 nm diameter is 4.68x10^(-26) m^3. One vaccinia virus masses 9.5x10^(-18) kg in 2.5x10^(-25) m^3. That looks like 3.5 vaccinia viruses in the volume of 0.187 virus, or 18.7X denser atom packing overall. Kinda iffy, that.

arxiv:1105.1921
"Proper orthochroneous Galilei Transformations" Physical chirality is excluded. "...for a mass of 1011 u...observed with a grating period of 0.5 nm...observe a loss of interference with a coherence time of approximately 300 ms." Graphite layer spacing is 0.67079 nm. C_84 MW = 1008.9 (Sigma-Aldrich, product #482986). Unlikely but not impossible.

Robert L. Oldershaw said...

With sincere apologies for repeating myself, but I am hoping that open-minded readers will please consider the following argument.

Whenever someone states, as if it were a fact, that 'gravitational interactions within the atom are weak', I feel that it is scientifcally required of me to point out that this is decidedly an untested assumption.

The value of G has never been measured WITHIN an atom or subatomic particle.

Strong gravity theories, which are as defendable as the weak gravity theories, yield a completely different approach to understanding: what a wavefunction is, what the fine structure constant represents, what Planck's constant is, and why the conventional Planck scale makes little physical sense.

Bottom line: our assumptions about the value of G within atomic and subatomic particles has never been measured, so alternatives should be considered.

Science - yes! Dogma - No!

RLO
Discrete Scale Relativity

Bee said...

Robert,

You do not only repeat yourself, you also continue to ignore my reply. Particle collisions are now routinely testing interactions down to a thousands of a fermi, that's well below any subatomic distances. No deviation from the standard model has been found, the gravitational interaction has not played any role. Best,

B.

Phillip Helbig said...

Also note that RLO predicted substructure for the electron at a level well above that which is now routinely probed. This falsification of a "definitive prediction" of his theory hasn't stopped him from peddling said theory. What use is a "definitive prediction" then?

Matti Pitkanen said...

I realized that the proposed equations cannot describe single particle. It does not make sense to assume that a solution of Schrodinger equation -in this case non-linear - describes something behaving like classical distribution of matter which is many particle state.

The equations could however describe order parameter for many particle system such as condensate of Cooper pairs.

Phil Warnell said...

Hi Bee,

Thanks for a nice piece outlining the features and virtues of an equation regarding an approach I wasn’t familiar with before. I find this particularly interesting as I think you’re aware I like this idea of looking at the older concepts to consider what elements of them might be fundamental and those which can be discarded, as I find that’s how truly meaningful physics so often gets done. That is this idea that one theory will have to be overthrown as to conform with another I’ve long thought to be misguided as thinking we should be rather looking for the quality resident in each before we rush to such decision.

”…in relativity, movement is continuous, causally determinate and well defined, while in quantum mechanics it is discontinuous, not causally determinate and not well-defined. Each theory is committed to its own notions of essentially static and fragmentary modes of existence (relativity to that of separate events connectible by signals, and quantum mechanics to a well-defined quantum state). One thus sees that a new kind of theory is needed which drops these basic commitments and at most recovers some essential features of the older theories as abstract forms derived from a deeper reality in which what prevails is unbroken wholeness.”

-David Bohm, “Wholeness and Implicate Order” Introduction page XVIII (1980)

Best,

Phil

Plato Hagel said...

Hi Bee,

In a "phenomenological way" could you apply what you are saying to "something real" so as to get a sense of the meeting other then the way yo are saying it is taking place?

So I guess the The
gravitational-wave spectrum
is definitely out of the picture?:)

Best,

Robert L. Oldershaw said...

Bee,

And you continue to completely ignore the distinction of:

1. WITHIN BOUND atomic systems

2. BETWEEN UNBOUND atomic systems

Only #2 has been measured and this this measurement is not relevant to what I am talking about, as I have explained before several times and in several places.

Helbig,

I have answered your misleading criticism regarding the structure of the electron at several places, and you know it. Why do you ignore my clear answer that undercuts your criticism?

RLO
Discrete Scale Relativity

Bee said...

Robert: If you have anything scientific to say, write a paper, publish it, win a Nobel prize. Good luck. All further off-topic comments will be deleted. Have a nice weekend,

B.

Marcos said...

Does it even makes sense to use Quantum Mechanics with a non-linear Schrödinger equation? Is there a non-linear version of it? (Yep, I don't know that much of physics)

I mean, if the equation is non-linear, how can you reach conclusions like that the wave packet tends to spread?

Bee said...

Well, as I wrote in my earlier post, there are good reasons to think the SN equation doesn't actually make sense. In any case, the Schroedinger equation is normally linear, in case that's what you're asking. The SN equation is a non-linear modification and is, as I wrote, in that not particularly pleasant. Best,

B.

Fizeg said...

Hi!

Great blog, I can't say the same about the papers in question.

I think there's quite easy way to see that there are problems with their credibility. We can in some situations consider electromagnetic field in the same semiclassical way, however nobody (at least as far as I know) perform absolutely the same calculations with electric potential instead of gravitational because obviously we can no more treat this field as purely classical object even if loop contribution is neglible.

Fizeg said...

By the way, there is huge difference between the semiclassical gravity and these SN equations. In semiclassical gravity the source is quantum field and in SN equations it is one particle. And while equations on expectation value of this field (and we talk then not about one particle but about quantum soup of particles) quite possibly take this form (I have serious doubts without honest computation) and after some little playing with formulas it seems to me that the second equation (with instead of Phi) is true even in this case, the first SN equation on wavefunction is a big lie.

Fizeg said...

Oops, there should be expectation value of Phi before "instead of Phi". My hope to write here without using codes of symbols is destroyed :'(

Jim said...

Matti: The above equations are for a single particle. The appearance of probability density playing the role of gravitational mass density here comes from the assumption that the expectation value of the stress-energy tensor couples to gravity:
<T00>=<psi|mdelta(x-x')|psi>=m|psi|^2

Marcos: Weinberg has previously investigated the possibility of a nonlinear Schrödinger equation, in general. Any nonlinearity has to be small, at least in the context of all experiments to date. But such is the case for the SN equation.

Fizeg: I think your implication is correct, that the SN equation is not a valid approximation of a conventional kind of quantum gravity theory. It does, however, represent a particular theory in which gravity is not quantized. I suspect this particular theory will eventually be experimentally falsified, but to my knowledge that hasn't happened yet.

Marcos said...

Thanks Bee and Jim.

That should teach me to not jump the preliminar section.

Arun said...

Dear Bee,

Does this mean that whatever the real theory of quantum gravity is, it has no meaningful limit where there is a non-relativistic quantum source of gravity?

Bee said...

Dear Arun,

No, that's not what it means.

There's a perfectly fine non-relativistic limit as long as the background curvature remains small because then you can use the "naive" perturbative quantization of gravity as an effective theory. That should cover everything we can plausibly observe on Earth, including interfering molecules. The effect described above doesn't exist in this case, because each possible quantum path of the particle has its own graviational field and the gravitational field in that case exists in superpositions as well. The thing with the SN equation is that you don't allow such quantum superpositions of the metric because you take seriously that gravity is classical.

Best,

B.

Phillip Helbig said...

@RLO:

Let me quote from the abstract of your paper (in a refereed journal, by the way): "Two definitive predictions are also pointed out: (1) the model predicts that the electron will be found to have structure with radius of about 4 x 10 to the -17th cm".

This is a definitive prediction (your words) which has been experimentally ruled out. End of story.

Do you debate that it has been ruled out experimentally? Or do you claim that, despite your very clear working in the abstract, that your theory has moveable goalposts? If the former, end of story. If the latter, it's not a very useful theory.

Bee said...

Philip, please, that was addressed to you too: It's off-topic. If you want to discuss Robert's theory of something or other, please do it elsewhere. Thanks,

B.

Arun said...

Dear Bee,

So what would the Schrodinger equation (non-relativistic, but with quantum source) look like when taking the appropriate limit of the naive perturbative quantization of gravity?

The small metric perturbations should presumably all wind up in the potential term of the Schrodinger equation. How the metric dynamically changes because of the quantum source is less clear to me, i.e., what is the equivalent of the Poisson or Einstein equation? It has to involve energy-mass density and so presumably is quadratic in the wave function. I mean to say, how do you avoid making it look like the model whose equations you have provided in your article?

Bee said...

Dear Arun,

Well, it's quantum mechanics, it doesn't actually include interactions. You can either make a mean field approximation for the background and end up with the Schroedinger equation with a gravitational potential for the background. Or you include additional graviton-interactions, for which you need the perturbative quantization. Best,

B.

Arun said...

Sigh, the realm where the only physical thing is the S-matrix....