The String Theory is one of the most controversial and brilliant ideas even though no physicists have proven it. The heart of this theory is the thread of an idea that has run through physicists for many centuries. It is all about the different forces, interactions, particles, and manifestations of reality tied together. It is more than just our four independent fundamental forces; strong, electromagnetic, weak, and gravitational.
So far, string theory is the best theory to match the quantum theory of gravitation. Even though there is no experimental evidence for the string theory, there are some theoretical reasons why it could be true. It was in 2015, Ed Witten, the top living string theorist wrote a piece on what every physicist should know about this theory.
What comes to your mind when you see the way two massive bodies gravitate? And according to Newton’s laws, it is identical to the way how electrically charged particles attract or repel. The way a pendulum oscillates is almost identical to the way a mass on a spring moves back-and-forth. So, gravitational waves, water waves, and light waves basically share similar features. Most of us do not realize that the quantum theory from a particle and how you would approach a quantum theory of gravity are analogous.
How quantum field theory works is that you take a particle and then you perform a mathematical “sum over histories”. Here, you cannot calculate the particles, where they are and how they got to be there, because there is an inherent and fundamental quantum uncertainty to nature. You can calculate the quantum state of a single particle by adding up all the possible ways it could have arrived at its present state and appropriately weighted.
How to work with gravitation? So, you need to change the story. The General relativity is not about particles, but the curvature of space-time.
It is quite difficult to work in three spatial dimensions. When a physic problem is appearing, commonly we try and solve the simpler version first. If we go down to one dimension, of course, everything becomes very simple. So, the only possible one-dimensional surfaces are an open string, which two separate and unattached ends. Or, it can be a closed string with two ends is attached to form a loop. Additionally, the spatial curvature, which is so complicated in three dimensions, then becomes trivial.
As long as you can define a momentum vector, it will be the main dimension that matters. It is because the extra degrees of freedom a particle gains from being in multiple dimensions are not quite crucial. It does not play an important role. In one dimension, quantum gravity is like a free quantum particle in any dimensions.
Next, we can incorporate the interactions and shift from a free particle with no cross-sections or scattering amplitudes to one that plays a physical role, combined with the Universe. Graphs in quantum gravity allow us to describe the physical concept of action. So, we need to write down all the combinations of such graphs and sum over them to complete the analogy.
After that, we need to move from one spatial dimension to 3+1 dimensions, in which the Universe offers three spatial dimensions and one time dimension. But this upgrade for a theory about gravity can be challenging. There should be a better approach in the opposite direction.
We do not have to calculate how a single particle or a zero-dimensional entity behaves in any dimensions. But, we could calculate how a string, open or closed behaves. After that, we can look for analogies to get a more complete theory of quantum gravity.
Doing these steps are more than just working with the points and interactions, but more about the surfaces, membranes, and others. You just start getting with behavior that becomes the root of the spacetime curvature in the Universe as General Relativity.
1D quantum gravity provided us the quantum field theory for particles in the curved spacetime but did not describe gravitation, yet. There was no correspondence between operators or functions to represent quantum mechanical properties, forces, states, and how particles and their properties evolve by the time. There should be a missing ingredient. However, if we move from point-like particles to string-like entities, of course, we will see the correspondence.
Once you upgraded particles to strings, there is a real operator-state correspondence. A fluctuation in the spacetime metric represents a state in the quantum mechanical description of a string’s properties. In a short way, from string theory, you can get a quantum theory of gravity in spacetime.
Moreover, doing this means you will get quantum gravity unified with other particles and forces in spacetime, the ones that correspond to other operators in the field theory of the string. Also, there is also the operator that describes the spacetime geometry’s fluctuation, and the other quantum states of the string. The string theory gives you a working quantum theory of gravity. However, the string theory is the path to quantum gravity. There is a big hope of string theory. These analogies should hold up at all scales.
So, what is the biggest challenge in string theory? It is all about how you can get Einstein gravity and 3+1 dimensions from the 10-dimensional Brans-Dicke theory.