# Struggles With The Continuum – Part 2

Last time we saw that that nobody yet knows if Newtonian gravity, applied to point particles, truly succeeds in predicting the future. To be precise: for four or more particles, nobody has proved that almost all initial conditions give a well-defined solution for all times!

The problem is related to the continuum nature of space: as particles get arbitrarily close to other, an infinite amount of potential energy can be converted to kinetic energy in a finite amount of time.

I left off by asking if this problem is solve by more sophisticated theories. For example, does the ‘speed limit’ imposed by special relativity help the situation? Or might quantum mechanics help, since it describes particles as ‘probability clouds’, and puts limits on how accurately we can simultaneously know both their position and momentum?

We begin with quantum mechanics, which indeed does help.

### The quantum mechanics of charged particles

Few people spend much time thinking about ‘quantum celestial mechanics’—that is, quantum particles obeying Schrödinger’s equation, that attract each other gravitationally, obeying an inverse-square force law. But Newtonian gravity is a lot like the electrostatic force between charged particles. The main difference is a minus sign, which makes like masses attract, while like charges repel. In chemistry, people spend a lot of time thinking about charged particles obeying Schrödinger’s equation, attracting or repelling each other electrostatically. This approximation neglects magnetic fields, spin, and indeed anything related to the finiteness of the speed of light, but it’s good enough explain quite a bit about atoms and molecules.

In this approximation, a collection of charged particles is described by a wavefunction ##\psi##, which is a complex-valued function of all the particles’ positions and also of time. The basic idea is that ##\psi## obeys Schrödinger’s equation

$$ \frac{d \psi}{dt} = – i H \psi $$

where ##H## is an operator called the Hamiltonian, and I’m working in units where ##\hbar = 1##.

Does this equation succeeding in predicting ##\psi## at a later time given ##\psi## at time zero? To answer this, we must first decide what kind of function ##\psi## should be, what concept of derivative applies to such funtions, and so on. These issues were worked out by von Neumann and others starting in the late 1920s. It required a lot of new mathematics. Skimming the surface, we can say this.

At any time, we want ##\psi## to lie in the Hilbert space consisting of square-integrable functions of all the particle’s positions. We can then formally solve Schrödinger’s equation as

$$ \psi(t) = \exp(-i t H) \psi(0) $$

where ##\psi(t)## is the solution at time ##t##. But for this to really work, we need ##H## to be a self-adjoint operator on the chosen Hilbert space. The correct definition of ‘self-adjoint’ is a bit subtler than what most physicists learn in a first course on quantum mechanics. In particular, an operator can be superficially self-adjoint—the actual term for this is ‘symmetric’—but not truly self-adjoint.

In 1951, based on earlier work of Rellich, Kato proved that ##H## is indeed self-adjoint for a collection of nonrelativistic quantum particles interacting via inverse-square forces. So, this simple model of chemistry works fine. We can also conclude that ‘celestial quantum mechanics’ would dodge the nasty problems that we saw in Newtonian gravity.

The reason, simply put, is the uncertainty principle.

In the classical case, bad things happen because the energy is not bounded below. A pair of classical particles attracting each other with an inverse square force law can have arbitrarily large *negative* energy, simply by being very close to each other. Since energy is conserved, if you have a way to make some particles get an arbitrarily large *negative* energy, you can balance the books by letting others get an arbitrarily large *positive* energy and shoot to infinity in a finite amount of time!

When we switch to quantum mechanics, the energy of any collection of particles becomes bounded below. The reason is that to make the potential energy of two particles large and negative, they must be very close. Thus, their difference in position must be very small. In particular, this difference must be accurately known! Thus, by the uncertainty principle, their difference in momentum must be very poorly known: at least one of its components must have a large standard deviation. This in turn means that the expected value of the kinetic energy must be large.

This must all be made quantitative, to prove that as particles get close, the uncertainty principle provides enough positive kinetic energy to counterbalance the negative potential energy. The Kato–Lax–Milgram–Nelson theorem, a refinement of the original Kato–Rellich theorem, is the key to understanding this issue. The Hamiltonian ##H## for a collection of particles interacting by inverse square forces can be written as

$$ H = K + V $$

where ##K## is an operator for the kinetic energy and ##V## is an operator for the potential energy. With some clever work one can prove that for any ##\epsilon > 0##, there exists ##c > 0## such that if ##\psi## is a smooth normalized wavefunction that vanishes at infinity and at points where particles collide, then

$$ | \langle \psi , V \psi \rangle | \le \epsilon \langle \psi, K\psi \rangle + c. $$

Remember that ##\langle \psi , V \psi \rangle## is the expected value of the potential energy, while ##\langle \psi, K \psi \rangle## is the expected value of the kinetic energy. Thus, this inequality is a precise way of saying how kinetic energy triumphs over potential energy.

By taking ##\epsilon = 1##, it follows that the Hamiltonian is bounded below on such

states ##\psi##:

$$ \langle \psi , H \psi \rangle \ge -c . $$

But the fact that the inequality holds even for smaller values of ##\epsilon## is the key to showing ##H## is ‘essentially self-adjoint’. This means that while ##H## is not self-adjoint when defined only on smooth wavefunctions that vanish at infinity and at points where particles collide, it has a unique self-adjoint extension to some larger domain. Thus, we can unambiguously take this extension to be the true Hamiltonian for this problem.

To understand what a great triumph this is, one needs to see what could have gone wrong! Suppose space had an extra dimension. In 3-dimensional space, Newtonian gravity obeys an inverse square force law because the area of a sphere is proportional to its radius squared. In 4-dimensional space, the force obeys an inverse *cube* law:

$$ F = -\frac{Gm_1 m_2}{r^3} . $$

Using a cube instead of a square here makes the force stronger at short distances, with dramatic effects. For example, even for the classical 2-body problem, the equations of motion no longer ‘almost always’ have a well-defined solution for all times. For an open set of initial conditions, the particles spiral into each other in a finite amount of time!

The quantum version of this theory is also problematic. The uncertainty principle is not enough to save the day. The inequalities above no longer hold: kinetic energy does not triumph over potential energy. The Hamiltonian is no longer essentially self-adjoint on the set of wavefunctions that I described.

In fact, this Hamiltonian has *infinitely many* self-adjoint extensions! Each one describes *different physics*: namely, a different choice of what happens when particles collide. Moreover, when ##G## exceeds a certain critical value, the energy is no longer bounded below.

The same problems afflict quantum particles interacting by the electrostatic force in 4d space, as long as some of the particles have opposite charges. So, chemistry would be quite problematic in a world with four dimensions of space.

With more dimensions of space, the situation becomes even worse. In fact, this is part of a general pattern in mathematical physics: our struggles with the continuum tend to become worse in higher dimensions. String theory and M-theory may provide exceptions.

Next time we’ll look at what happens to point particles interacting electromagnetically when we take special relativity into account. After that, we’ll try to put special relativity and quantum mechanics together!

### For more

For more on the inverse cube force law, see:

• John Baez, The inverse cube force law, *Azimuth*, 30 August 2015.

It turns out Newton made some fascinating discoveries about this law in his *Principia*; it has remarkable properties both classically and in quantum mechanics.

The hyperbolic spiral is one of 3 kinds of orbits possible in an inverse cube force; for the others see:

• Cotes’s spiral, *Wikipedia*.

The picture of a hyperbolic spiral was drawn by Anarkman and Pbroks13 and placed on Wikicommons under a Creative Commons Attribution-Share Alike 3.0 Unported license.

Is there any previous work, any attempt on trying to discretize space, or even space time and trying to work the laws of physics, dynamics, classical mechanics, quantum mechanics, and relativity in such a discretized space? I supposse that in the limit of the grid space tending to zero one could get the usual mechanics in continuum space or space-time, but I would like to know what kind of physical predictions the discretized space would make, and the possibility to observe experimentally some of those predictions.

Hi – I'll be a bit naive here, forgive me, but couldn't it be that our mathematics is not representing reality, but just a model of it? Nature could be not-continuous by itself. I cannot think of a "real" particle being "infinitely close" to another one… And even quantization, or discretizations used in computerized models, aren't real themselves, but just convenient models of natures, useful to make good predictions only up to a certain "extent".I'm afraid we still don't have mathematical models that really represent nature completely (will ever we?) and the difficulties in having converging solutions under certain circumstances might be the result of using inadequate mathematics.Sorry if my point is only loosely defined, but I wonder if this or similar arguments has ever been raised before (as I guess).Thanks John for the excellent presentation of these concepts, anyhow.

It doesn't seem to me that anything is ever continuous; continuous is always an approximation, even in classical mechanics. Take a baseball. Suppose you integrate numerically the simple case where there is just gravity, no air resistance. Can you really make your timestep arbitrarily small? How about if you integrated at a femtosecond timescale? In principle you could obtain the entire motion, but in practice each step would, individually, produce no information about the system, since the change in position would be, relative to the characteristic scales of the problem, zero during each step. There is arguably a minimum time step, which is the smallest step that produces a nonzero change in state relative to the characteristic scales.Space is also always discretized in terms of the characteristic scales of the system.

I'm new here, but I thinking you keep breaking down physical objects, you will reach point where breaking it down further will yield no information, but it can be still be theorotically broken down. Would this be discrete or continuous?

And he has a point! The real numbers are a bit more man made than the integers. After all from rationals to reals there is a choice, you can complete them to get the reals or any of the p-adic numbers. May be one shouldn't take reals over any of the p-adics. May be the way to go is to work with the adeles.

Is the problem with your Newtonian example due to the implied instantaneous-action-at-a-distance in the fundamental equation (which means the result isn't fully conservative)? At some point you have to modify the equations to allow for changes in the gravitational system to propagate, like Heaviside did in the 1890s, and then you get significantly different results in extreme cases.One final note. Your reply to the God-made/Man-made joke was uncomfortable but you have to remember that discussing the continuum is really the physics equivalent to a "religious" question. I'm an atheist but when I step back, ignore all the equations/models/theories, and look around … I wonder "what in the hell is all this stuff anyway, none of it makes sense". It seems like the only thing we can do is perpetually oscillate between experimental and theoretical advances, never reaching a "ta da, we're done!" moment.

The closest I've encountered about grains of space is in loop quantum gravity where space seems to be broken into tiny pieces, maybe a googol of them in a teaspoon. But as I understand it, the pieces are not arranged in a regular grid; they are constantly rearranging themselves, and all their possible arrangements get quantum-superposed so they are all smudged together into something that feels kind of continuous.Is this image even approximately a correct view of the theory?

[I]”At any time, we want ψ to lie in the Hilbert space consisting of square-integrable functions of all the particle’s positions. We can then formally solve Schrödinger’s equation as

ψ(t)=exp(−itH)ψ(0)

where ψ(t) is the solution at time t. But for this to really work, we need H to be a self-adjoint operator on the chosen Hilbert space. The correct definition of [URL=’https://en.wikipedia.org/wiki/Self-adjoint_operator#Self-adjoint_operators’]‘self-adjoint’[/URL] is a bit subtler than what most physicists learn in a first course on quantum mechanics. In particular, an operator can be superficially self-adjoint—the actual term for this is [URL=’https://en.wikipedia.org/wiki/Self-adjoint_operator#Symmetric_operators’]‘symmetric’[/URL]—but not truly self-adjoint.”[/I]

Is this because we want the movement of a system in that space to be perfectly reversible? Is Reimannian continuity (if that’s the right term) really about wanting to assume things are equally

able or likelyto go in any direction in their phase-space from any point?Basically I am confused here by similar but different terms “Continuity”, “Reversibility”, “Symmetry”, “Commutativity”, “Hermitian-ness” and “Self-Adjoint-ness”. I think the last two may be synonymous, but are these terms just stronger/weaker versions of the same idea, precluding any preferred “direction of the grain” in the phase space?

Thanks for the awesome articles by the way.

Thanks!

I suggest that you look up “Continuity”, “Reversibility”, “Symmetry”, “Commutativity”, “Hermitianness” and “Self-Adjointness” on Wikipedia. They all mean very different things – except for the last two, which are closely related. In my post I was using “symmetric” in a specific technical sense, closely akin to “hermitian”, which is quite different from the general concept of “symmetry” in physics. That’s why I included a link to the definition.

Learning the precise definitions of technical terms is crucial to learning physics. You’re saying a lot of things that don’t make sense, I’m afraid, so I can’t really comment on most of them. That sounds rude, but I’m really hoping a bit of honesty may help here.

Anyway: we need the Hamiltonian H to be self-adjoint for the time evolution operator exp(-itH) to be unitary. And we need time evolution to be unitary for probabilities to add up to 1, as they should.

Sure, lots.

Yes, for example when they compute the mass of the proton, they discretize spacetime and use lattice gauge theory to calculate the answer – nobody knows any other practical way. But you try to make the grid spacing small to get a good answer.

You’d mainly tend to lose isotropy – that is, rotation symmetry. Homogeneity – that is, translation symmetry – will still hold for discrete translations that map your grid to itself. People have looked for violations of isotropy, but perhaps more at large scales than microscopic scales.

What seems cool to me is how cleverly chosen lattice models of fluid dynamics can actually do a darn good job of getting approximate rotation symmetry. For example, in 2 dimensions a square lattice is no good, but a hexagonal lattice is good, thanks to some nice math facts.