Quantum equation describes galaxy mechanics

Using perturbation theory to describe the orbits of smaller bodies around large ones in space required Batygin to posit all objects in each specific orbit as a single entity and “smear” them into the form of a concentric ring, or wire. In the model, each such ring exhibited the same gravitational force as the combined individual objects, but uniformly distributed.

In such an approach, the solar system, for instance, would be represented by the sun, followed by a wire ring for each planet, plus others for the asteroid belt and Kuiper belt. Computer simulations representing millions of years showed that these rings behaved in ways that closely mirrored the behaviour of the real composite disc surrounding the sun.

Batygin then started refining the model, realising that he could portray any astrophysical system as a centre surrounded by ever more numerous, but ever thinner, wires until, inevitably, the wires blended into a single plane.

“Eventually, you can approximate the number of wires in the disk to be infinite, which allows you to mathematically blur them together into a continuum,” he says. “When I did this, astonishingly, the Schrödinger equation emerged in my calculations.”

This was a surprise, because the equation was thought to be only applicable to phenomena occurring on a quantum scale. It is used to describe one of the most bizarre aspects of quantum mechanics – the way in which subatomic particles behave simultaneously like particles and waves, a condition known as “wave-particle duality”.

“This discovery is surprising because the Schrödinger equation is an unlikely formula to arise when looking at distances on the order of light-years,” says Batygin.

CosmosMagazine.com, March 5, 2018 - Quantum equation describes galaxy mechanics.

and, Wikipedia: Schrödinger equation.

He was modeling a galaxy around it's central black hole (BH). He used a form of mathematics that allowed him to model large systems. A "wire" (this is a thought experiment in some ways, this is how Cosmos described it, so...), he could place smaller bodies in orbit around the BH. The "blurring" he is describing is making each ring continuous across the surface of the disk (they "blur" into each other). What that mathematically allows you is to use calculus for continuous functions, or, integration! When his perturbation model was subjected to the calculus, oddly enough, the Schrödinger equation pops out!

This is like the concepts of fractals where there is a repetition as scale grows. But this is not is not in some irrational space that makes Mandelbrot sets so interesting to look at.

This could all be due to statistics which make made Einstein nervous of Schrödinger's probability wave (where he famously quipped, "[The Old One] does not toss dice with the universe"). Both equations have the same roots.

I think that it is really cool that while looking at the super massive interactions within a galaxy we find the 1926 equation that has survived test after test of quantum mechanics!

And look ma! No gravity needed! (I wonder what that means?)



Ideas, thoughts, too much math?, too much science?

"What I have presented in this paper is a framework," Batygin said. "I have attacked one particular problem with it, which is the problem of disk rigidity — the extent to which the disk can remain gravitationally rigid under external perturbations. There's a broad range of additional applications that I'm looking into at the moment."

One example is the evolution of the disk of debris that eventually formed our solar system, Batygin said. Another is the dynamics of rings around extrasolar planets. And a third is the disk of stars surrounding the black hole at the center of the Milky Way, which itself is highly bent.

Laughlin noted that the work should be particularly helpful in improving researchers' understanding of newborn star systems because they're harder to observe from afar, and researchers currently can't simulate their development from start to end.

space.com - Massive Space Structures Have Surprising Connection to Quantum Mechanics Math.

Nice to hear from the guy himself! I probably should have done a bit more homework.

I see. The big news is "disc rigidity leads to the Schrödinger equation" but that the entire methodology is what is important because it can further applied to other systems of equations. And you have a general idea of what you should be seeing.

Plus I always like it when I hear news of cosmologists and quantum physicists getting together!

This sounds like a major advancement. Basic gravity physics tells you that the sum of all the gravity forces on the surface of a planet like the Earth can be approximated through integration as a single point at the centre, since forces from different directions on the surface cancel out leaving only the downward forces.

This would seem to extend the idea by integrating all the possible positions that the planet could be around the star. Even if there is some precession due to the elliptic orbit of the planet, it would still end up as a infinitely flat disc plane with given enough observations. Then all those points at the same distance from the star could be joined up as infinitely small rings.

With atoms, electrons are moving around so fast, that we can't predict where they are at any time, so can just use a probability cloud. Hopefully they can unify everything together.

a reply to: stormcell


The thing that gets me is the combination of two different aspects of math: discrete and continuous. In all honesty, they should be like lemonade and chocolate, not very good with each other. Then, using mathematical methods everybody uses, you end up at one after stretching the other one to infinity!??

I think I said it right there: simplicity in infinity.

That alone is a wild concept to let sink in. Then add unification on there? Now that one is a good two beer problem to think about!

Ask a spaceman is new to me.... fascinating. S&F

Nothing more interesting than Quantum Mechanics.

Quantum gravity research will likely benefit from this.

I have a math class now so I can't get deep into it. Suffice to say I will be back.

a reply to: TEOTWAWKIAIFF

This is what I have been waiting for I can't wait to look further!

originally posted by: projectvxn
Quantum gravity research will likely benefit from this.

I doubt it.

Fundamental Equation of Quantum Physics Also Describes Rings and Disks in Space

Also, no, this is not the fundamental link between general relativity and quantum mechanics that particle physicists are hunting for.

However it might turn out to be helpful for studying planetary formation.

a reply to: TEOTWAWKIAIFF

Another example of the power of mathematics to describe the physical world.
Electron tunneling for example was predicted by a solution of Shroedinger's equation. It wasn't intuitive at all but proved to be correct.

a reply to: Arbitrageur

It's a little strange to plug in gravitational variables that lead to mega structures like galaxies and get Schrodinger's equation.

I'm not saying this is a bridge between general relativity and quantum mechanics, only that it might be an avenue to explore.

01:50 in the second video... So do protons move or are they stationary...ie the chair, curious ideas all. That analogy escapes me.

a reply to: Deluxe


I've had some time to have a couple beers and think about this some more.

The "smearing" of dimensions of an entire galaxy down to a lower dimension (@all: math use of the term meaning "number of variables"), encodes time into it. I think that is the bridge between the galaxy model and Schrodinger's equation. They use the galaxy perturbation model to evolve a galaxy over time; Schrodinger's equation is the quantum evolution of energy over time. Saying anything meaningful about evolution of systems over time should involved the oddly placed Schrodinger's equation (if you look deep enough).

At least that is my best guess at this moment! I will have to go see if I can make sense of the math and ponder some more.

Thanks to all that have replied! This is an interesting topic and if an entire system of approaches can be made to other problems... that is even better!



PS - Nerd alert! I ran home yesterday to watch the series finale of Star Wars: Rebels!! Sorry to post and run... but, priorities.

The General time-dependent Schrödinger equation

Ĥ | ψ (r,t)

H-hat is the Hamiltonian operator and is cross multiplied (?? That is what that symbol normally means. Probably means “operates on”) the Psi wave function where r, is the position vector, and t is time. In all honesty, Schrödinger invented it! He was looking at Maxwell and what de Broglie had suggested and made the choice to use a wave mathematics instead of linear algebra. He treated atoms as waves so used the math. Psi is the name he gave his collection of frequency, amplitude, and wavelength. The Hamiltonian churns the wave forward and backward in time because it is measure of energy in the system at time t. The Hamiltonian also has Plank’s constant and i (sqrt -1), making the whole math a real function outside of the sine and cosine stuff we all suffered through school (it is a bit beyond my comfort zone).

In physical cosmology, cosmological perturbation theory is the theory by which the evolution of structure is understood in the big bang model. It uses general relativity to compute the gravitational forces causing small perturbations to grow and eventually seed the formation of stars, quasars, galaxies and clusters.

Wikipedia – Cosmological perturbation theory.

Why did Schrödinger equation (SE) show up? My guess is that both perturbation theory and SE include change over time. When spreading a galaxy disk to lower dimensions to model (evolve) over time, you can treat gravity and time as continuous which, as I mentioned, allows you to do calculus. It is math, so why not? And poof! SE shows up.

That using general relativity gets you to SE is kind mind boggling. Kind of the way that an infinite sum leads to a specific number! I know it can be done (had to do it many times in homework and tests) but a casual understanding of either would make this announcement one of those, “Huh,” moments.

Disclaimer: The actual paper is pay-walled so I went with the broad cosmological version (there are others).

I knew if I talked about it that sooner or later the good folks at Quanta magazine would hit the topic as well.

And they did! Very readable article about complex curves and mirror symmetry. It explains what they discovered and are still discovering about how this mirror symmetry math works.

One method arose in the mathematical field of algebraic geometry. Here, mathematicians study polynomial equations — for example, x^2 + y^2 = 1 — by graphing their solutions (a circle, in this case). More-complicated equations can form elaborate geometric spaces. Mathematicians explore the properties of those spaces in order to better understand the original equations. Because mathematicians often use complex numbers, these spaces are commonly referred to as “complex” manifolds (or shapes).

The other type of geometric space was first constructed by thinking about physical systems such as orbiting planets. The coordinate values of each point in this kind of geometric space might specify, for example, a planet’s location and momentum. If you take all possible positions of a planet together with all possible momenta, you get the “phase space” of the planet — a geometric space whose points provide a complete description of the planet’s motion. This space has a “symplectic” structure that encodes the physical laws governing the planet’s motion.

Symplectic and complex geometries are as different from one another as beeswax and steel. They make very different kinds of spaces. Complex shapes have a very rigid structure. Think again of the circle. If you wiggle it even a little, it’s no longer a circle. It’s an entirely distinct shape that can’t be described by a polynomial equation. Symplectic geometry is much floppier. There, a circle and a circle with a little wiggle in it are almost the same.

“Algebraic geometry is a more rigid world, whereas symplectic geometry is more flexible,” said Nick Sheridan, a research fellow at Cambridge. “That’s one reason they’re such different worlds, and it’s so surprising they end up being equivalent in a deep sense.”

Quantamagazine.org - Mathematicians Explore Mirror Link Between Two Geometric Worlds.

a reply to: TEOTWAWKIAIFF

Thanks for the link.
I just wish I could understand most of it.
Mathematics never ceases to amaze me.
I also find it wild that sometimes the math precedes the physics and in other cases the physics creates the math and the mathematicians have to catch up.

a reply to: Deluxe


I have found a non-technical blog posting explaining a couple concepts including universality.

Terrance Tao (terrytao.wordpress.com) - A second draft of a non-technical article on universality.

The concept is that like the Bell Curve, there seems to be some mathematical description that applies. He starts with voting, exit polls, and predicting a winner. The sample is representative of the larger, complete population of voters, and the error of the sample is "3%" regardless of sample size. That error rate (whatever the actual number is, he just chose "3%") is because of statistics and known functions. For the universality in the math article, it is the distribution that is showing similarities between energy states in particles (quantum physics) and movement of galaxies. The distribution is like a UPC code, it shows up everywhere. The question is: why?



In a follow up article, there has been headway made in proving the "mirror symmetry" and the Strominger-Yau-Zaslow (SYZ) Conjecture.

"Basic" intro to SYZ conjecture, arxiv.org, arXiv:1710.05894 (PDF) - The SYZ conjecture via homological mirror symmetry.
My eyes began to water at trying to remember 6-dimensional Calubi-Yau manifolds! It is notes taken at a particular symposium so it is rather technical. What is important to take away is the mapping from defined math space to another which can be described using a different function that has nothing to do with the math used in the original.

The question was asked, "What cohomolgy was being used?", and I think we finally have an exact answer to that!

In the early 1990s, physicists were trying to figure out the details of string theory. They wanted to explain the physical world as a product of tiny, vibrating strings woven through an additional six dimensions of space. They tried to understand what the geometry of those six dimensions might be. The first option came from the mathematical field of algebraic geometry; a second one came from the mathematical field of symplectic geometry. To the trained mathematical eye, the two could hardly have seemed more different.

And yet, the physicists noticed some strange similarities between them. In particular, when they performed a calculation on one space, they generated numbers that matched the numbers they generated when they performed a very different type of calculation on the other side. “Two things that looked, in principle, unrelated, magically were equal,” said Denis Auroux, a mathematician at the University of California, Berkeley.

...

In this new work, the four mathematicians use techniques from a field called tropical geometry. Using those techniques, they prove that these “special pieces” explain why numbers on opposite sides of the mirror differ by exactly a factor of zeta values. So far, their proof holds for many cases of mirror symmetry. The authors are waiting until they’ve been able to prove even more cases before they make the proof public.

Quantamagazine.org - Three Decades Later, Mystery Numbers Explained.

There is a paper due to be released which may take math in a whole new direction!

Being able to define a function in two different forms using a transitional mapping function (and probably noting something like, "Assuming continuity...") will help enormously. But it is that next step, of describing how universality arises by its own math that will be transformative. My bet is that is where the solution to the Riemann Hypothesis lies.