If you are into Physics then you might have heard about Loop Quantum Gravity. It is often considered as a viable candidate of the grand unified theory but what exactly is it? What are the loops? And can it defeat string theory in our quest for a Theory of Everything?
The long-sought dream of Physicists is to connect quantum physics with Einstein’s general theory of relativity. Our search for a unified theory is quite old, and we’ve talked about it quite a bit already. Generally, string theory is associated with the theory of everything. But string theory isn’t the only game in town. There may be another way to reconcile the physics of the tiny and the gigantic. Another way to quantize gravity without dealing with tiny strings or extra dimensions. You guessed it, that other way would be loop quantum gravity, and we’re going to find out exactly what it is.
Why is it so hard to combine General Relativity and Quantum Mechanics?
You might have heard how it’s a near-impossible task to unify the laws of Physics. Einstein himself spent up to thirty years of his life trying to unify Physical laws but didn't succeed. So what is it that makes the quest of a unified theory so hard to achieve? Well, there are many things.
One of the major reasons is that general relativity breaks down when we quantize massive bodies like the black hole. Apart from this, there are more fundamental conflicts: background independence and the problem of time. For the most part, we are going to be concerned with background independence as this is what Inspired Loop Quantum Gravity.
So what is background independence?
Quantum Mechanics, along with other theories, describe how stuff behaves on background coordinates. Like actors on a stage, where the actors are particles and wavefunctions and the stage is the coordinates of space and time. In quantum mechanics that stage is flat and static and isn’t influenced by the actors.
In short: quantum mechanics is NOT background-independent.
But General relativity has to be background independent - because that’s what the equations do - they change the background. They describe how the presence of mass and energy warp the fabric of spacetime. Our background coordinate system itself becomes a dynamic entity.
The theory of Quantum Loop Gravity
Why is it so hard to quantize General Relativity?
QLG tries to quantize general relativity with no strings attached. Along with this, it also preserves the background independence already inherent to GR.
But why is quantizing general relativity so difficult in the first place? The challenge gets us to the fundamentals of what a quantum theory actually is.
In classical physics, we have variables like position, time, momentum, energy. Some of these - say, position and time - also form our background coordinate system. But in quantum mechanics, things aren’t so straightforward.
Certain properties have an in-built uncertainty and only take concrete values after measurement. They exist in a fuzzy space of possibilities called a wave function. The wave function describes the distribution of possible positions and momenta of, say, a particle. These can then be resolved into concrete, measured values by acting on the wavefunction with so-called position and momentum operators. The wave function and operators are fundamentally tied to the coordinate system. There are other ways to formulate quantum mechanics, like quantum field theory, but these ultimately have the same issue.
But it gets worse actually. In quantum mechanics, time is treated completely separately from other variables. There is no “time wave function” or “time operator”. This is completely at odds with general relativity. General relativity treats time as just another dimension. This is the “problem of time” that I mentioned, and it’s strongly connected to background independence. A quantum theory of gravity needs to fix both of these issues - but we’re going to focus on background independence for now.
The position and momentum of quantum mechanics describe the location on a spatial coordinate system and the change in that location over time. That makes it highly background-dependent. The equations of quantum mechanics let you calculate changing properties of a particle-like its position or momentum - relative to the background coordinate system. The equations of general relativity let you calculate the changing shape of the coordinate system itself, encapsulated in the metric.
So instead of thinking about the quantum fuzziness of position and momentum, we can think about the quantum fuzziness of the metric itself. So probably there’s an equation that describes the quantum evolution of the geometry of space. Well, there is - or at least an attempt at one. It's called the Wheeler-DeWitt equation.
Wheeler-DeWitt equation
Wheeler-DeWitt equation is based on something called the ADM Formalism. ADM formalism is the decomposition of four-dimensional space-time into layers of time on a three-dimensional surface. For example, a cake has multiple layers and each layer is described by a particular value. As you move up the layers, you move upwards in layers of time. It's a coordinate system that describes where you are in the metric of space. As you move through this coordinate system, the geometry of space changes.
So the Wheeler-DeWitt equation quantizes these - turns them into quantum operators. The result is a quantum equation for the fabric of space. A contender for a theory of quantum gravity.
The Wheeler-DeWitt equation was promising but turned out to be unsolvable. Which makes it not so useful, and impossible to verify as correct. So perhaps this whole path of using abstract coordinates is a dead end, or perhaps we just haven’t gone down it far enough. That’s what loop quantum gravity does - it takes us down the abstraction rabbit hole. It takes us past our space of metrics into a space of something called connections. And these connections are going to give us our loops.
What are connections
Connections are mathematical functions that tell you how something, like a vector, changes as it moves between two points in space. As you move the base of a vector along a path in curved space, the vector rotates. And the amount of rotation encodes information about the changes in geometry along the path. If connections contain all the information about spacetime, then we can represent spacetime with these connections instead of with regular coordinates.
Einstein himself tried to rewrite general relativity in terms of these parallel transport vector connections, but the result was a mess. The breakthrough came in the 80s when Abey Ashketar tried a different type of connection and rewrote general relativity in terms of these different connections. In this formalism, the “space of metrics” looks like a space of fields in quantum field theory.
The theory of Quantum Loop Gravity
What are the loops in Quantum loop gravity?
Lee Smolin and Carlo Rovelli realized they could solve the Wheeler-DeWitt equation by representing spatial metrics using Ashketar's connections. But they needed one more trick - one layer deeper in abstraction.
They evaluated these connections over closed loops – so each point connected back to itself. They realized it was possible to define any geometry of 3-D space out of a sort of weave of these closed loops. Each loop looked like an elementary closed circuit of the gravitational field. So now you have a space of loops with which you can construct the fabric of space. That space of loops can be quantized rather neatly in a background-independent way.
After all, there IS no background until these now-quantum loop states build it. The result, of course, is loop quantum gravity. It’s general relativity – our modern theory of gravity – cast in terms of very abstract building blocks.
The big success of loop quantum gravity is that it manages to combine general relativity and quantum mechanics in their currently accepted forms, without taking away their most important foundational principles. And without adding big assumptions – like the existence of strings or extra dimensions or supersymmetry.
The theory has some other successes, for example, the theory seems to predict Hawking radiation and black hole entropy consistent with Hawking and Bekenstein’s equations.
Problems with Quantum Loop Gravity
However, there are also many who identify serious, fundamental issues with the theory. While LQG has background independence in terms of different 3-D spatial geometries, it’s not actually clear that this independence extends to 4-D spacetime. And connected to this, QLG doesn’t solve the problem of time. Some researchers think that the method and foundations are sound and that the current criticisms and shortcomings can be resolved with more research and extensions to the existing formalism.
QLG also proposes certain experiments. It seems to predict that the speed of light should vary slightly depending on the energy of the photon. For example, high-energy gamma rays travel a bit slower than low-energy radio waves due to the way they propagate through the graininess of a loop quantum gravity spacetime.
This was tested in 2009 by looking for differences in the arrival time of light from a gamma-ray burst nearly a billion light-years away. If there was any difference it was barely measurable, and that doesn't look great for loop quantum gravity.
The bottom line
At this point, we don’t know if Quantum loop gravity is THE theory. Even though it does give promising results, it does have its shortcomings. Loop quantum gravity is an intriguing alternative to the more popular string theory. Both currently live deep in their respective theoretical rabbit holes. Only time would reveal the fundamental blocks of nature.