A new theory of quantum gravity in superspace

In 1915, Einstein published his theory of general relativity for the first time, correct and complete. The theory was a masterpiece in both deduction from a few principles such as special relativity and the equivalence principle and also an inference that space and time were ultimately the cause of gravity which we had, hitherto, known as a force.

It also bolstered Einstein’s philosophical notion that change was an illusion. Things didn’t move, they simply existed as geodesics within a grander stage. Like a modern day Zeno, Einstein wanted people to think about space and time as a single entity, motionless and crystalline.

This put general relativity at odds with quantum mechanics. In that theory, time was given a special significance, different from space. While a particle’s position was given by the wavefunction, and thus had the status of an operator, time was merely a parameter. Everything in quantum mechanics evolves in time which makes it a successor of Newton’s concept of time as an eternal absolute.

Progress was made in marrying quantum mechanics up with special relativity. Thus was born quantum field theory. Now time was returned to its proper place on the same level as space, but not quite because in quantum theory we still cannot predict what will happen based on the current state of the universe. 

Because we cannot predict, John Wheeler proposed, we have to let go of Einstein’s concept of space and time merged together into a single spacetime.

Instead of spacetime, we have to turn to a much larger space of potential spatial geometries. This is superspace.

Superspace is an infinite dimensional space where every point in it is a spatial geometry. In classical general relativity, as space evolves in time, a path wends its way through superspace in much the same way that, in spacetime, a particle wends its way through that.

Yet, neither spacetime nor superspace really exist. Instead, they are concepts that help us to understand the dynamics of either a point particle or an infinite collection of points.

And like point particles, we can describe how space evolves through superspace using the equations for classical dynamics. 

The Hamilton-Jacobi equation is key.

The HJ equation is equivalent to both Hamilton’s equations of motion as well as the principle of least action but it is a single equation rather than a system of equations.

The dynamics of space through superspace is described by the Einstein-Hamilton-Jacobi equation. In 1969, Gerlach showed that this equation is equivalent to the ordinary, 4D Einstein field equations, the same ones that Einstein published in 1915 about spacetime.

The reason these are the same is because we can build spacetime out of points in superspace sort of like how you make a mummy with strips of cloth. Each point in superspace is a complete, 3D spatial “slice” of spacetime. Yet, because we can choose which slicings we want to have, much like how we can choose how to slice a loaf of bread, they may overlap when combined into spacetime. But that’s ok. They all contribute to the same spacetime.

This is the basis for geometrodynamics, the study of the dynamics of space. Quantum gravity theories such as Loop Quantum Gravity continue to make use of this concept.

But we aren’t restricted to thinking about the dynamics of space. What if, instead, we accepted the existence of spacetime yet allowed spacetime itself to evolve in a 5th dimension?

In this case, spacetime traces out a path through a larger superspace, a superspace made of 4D spacetimes instead of 3D spaces as points.

The reason we might want to propose such an idea is that we can show that classical dynamics in a 5th dimension can be averaged into quantum field theory in 4 dimensions. This is often called stochastic or chaotic quantization and has been known since the 1950s.

If spacetime were to evolve in a 5th dimension, it could explain quantum gravity.

The key to understanding that is the Einstein-Hamilton-Jacobi equation and that is because of its connection to quantum wave mechanics.

The Hamilton-Jacobi equation has value not only because it is a single equation but because it connects classical mechanics, the behavior of balls and rocket ships, and the propagation of waves. The Hamilton-Jacobi equation is in fact the classical limit of Schroedinger’s equation.

In addition, if you solve the Hamilton-Jacobi equation, you get a phase function, S, that you can insert into a wave representation. This is a semi-classical approximation because you are taking the classical dynamics and fitting them into a quantum wavefunction, a probability distribution.

You can think of a single particle, with classical dynamics, moving through spacetime. Now, you make a semi-classical approximation by taking the solution to its HJ equation, which is just classical dynamics, and putting it into a probability amplitude. This probability amplitude is a wavefunction and solves Schroedinger’s equation. It doesn’t represent every possible thing that can happen, after all, it just follows the classical trajectory. But the wavefunction is spread out. The particle no longer has a definite location.

To give a location back to the particle, you take many solutions to the HJ equation, all with similar but not identical inputs in initial velocity and position but such that the solutions, the outputs, which are phase functions, all agree with one another. You superimpose them on one another. Adding these together causes constructive interference at the classical location of the particle. Now the particle has a location again and follows its classical path.

This is how you build up a classical trajectory from solutions to a quantum equation.

John Wheeler and others, in the 1960s, tried to do this for space in superspace. This gives us a semi-classical quantum gravity. It isn’t completely quantum because it hasn’t been quantized properly. We’re just superimposing individual waves.

We know that we can average stochastic or chaotic classical trajectories through a 5th dimension to arrive at a true quantum theory. Can we do it in superspace as well and can the EHJ help?

Yes and yes, so imagine just a one dimensional spacetime as an example. This is really just time I suppose. It’s just a strand of cooked spaghetti, a string, that I, for simplicity, will pin at two points, one in the past, let’s call it this spacetime’s Big Bang, and one in the future, which we’ll just call the Gnab Gib.

This string of spaghetti now evolves in an additional dimension (not a 5th dimension here but a 2nd), which isn’t really a dimension, but rather it is a parameter for change. It is its trajectory through superspace.

The spaghetti’s superspace is infinite dimensional, but for simplicity let’s imagine a chain instead of spaghetti. The chain has a finite number of links. The connections between the links are the variables here so the number of dimensions of the superspace is the number of links minus one. Let’s say it has 3 links so the number of dimensions of the superspace is 2, a flat plane.

A stochastic quantization of this would just have the chain moving in an additional dimension with a bunch of random noise. Averaging over all that noise (and doing something called a Wick rotation from imaginary to real time) gives you the statistics you need to calculate quantum physics.

Instead of doing that, however, we are just going to look at 2D superspace and, taking the solution to the chain’s HJ function, we are going to put it directly into a probability amplitude. This becomes a function on the superspace that tells me the likelihood of finding the chain in any given configuration. This is a semi-classical approximation.

Now I will let that probability function evolve in superspace. If I let it evolve for long enough and add up all the probability functions, and then average, I will, in most cases, get something approximating the quantum field quantization. If I let it evolve forever and add up an infinite number, then the equivalence will be exact. And I don’t have to add random noise at all because I am working in probabilities, which represent noise as abstract distributions.

This is pretty easy to show mathematically. It’s almost trivial. From a computational point of view, there is no reason to do it. It makes things harder.

From a theory perspective, however, it makes a big deal whether we propose (a) that spacetime is following a noisy classical trajectory in an unseen dimension or (b) that spacetime is a wave in a superspace, propagating through configurations in an unseen dimension, and only appears classical because of constructive interference of these configurations.

The first proposes that quantum gravity is entirely a trick of classical dynamics and there is no wavefunction while the second proposes that the wavefunction is real and fundamental, quantum effects are a trick of classical dynamics, and classical measurements are a trick of wave interference.

There is, however, one benefit to the second interpretation. The first interpretation has the universe evolving noisily. That noise has energy and so it would create additional gravity. In fact, there would be so much noise that it would overwhelm everything else in the universe. It would create an enormous cosmological constant problem.

The second interpretation has the universe evolving without extra noise. Instead, its fundamental nature is as a wavefunction. It has no definite form except that which emerges from constructive interference. That means that there is no extra energy after all and no huge cosmological constant. I count that a big win.

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Misner, Charles W., Kip S. Thorne, and John Archibald Wheeler. Gravitation. Macmillan, 1973.