One of the most famous and important open problems in physics is implementation of gravity, more precisely General Relativity (GR), into Quantum Mechanical (QM) framework. GR & QM are not compatible to each other for various reasons.
The GR is based on a smooth spacetime whereas QM is discrete & probabilistic in nature. There are 2 key points in QM.
1. Physical observables are mathematical operators.
2. Hermitian operators (such as position & momentum) cannot be measured simultaneously. The products of their errors is equal or larger than h’/2. This limits our ability to measure physical the observables. (h’ is a reduced Planck’s constant). There is also an uncertainty relation between Energy & Time. But here is a catch. Time ‘t’ is just a parameter; not an operator. Ordinary quantum mechanics treats time as absolute & external to the system. In Relativity Theory, space and time are relative.
(Note: Operators are non-commutative. [x, p]= [xp — px]= ih’.)
There were some early attempts to merge QM & RT since there is no wave equation compatible with both. In fact Schrodinger too delved into the merger. The Relativity theory, the energy equation famously known as E = mc2, relates energy and mass. In fact, based on the conservation law, energy we required is twice the mass in order to create a particle and its anti-particle. Hence neither the particle number nor its type is fixed. This is in direct conflict with the quantum mechanics in non-relativistic domain.
The famous Schrodinger’s equation which describes the dynamics of the quantum system needs the number of particles and its type to be absolutely fixed, meaning it cannot handle changing particles or new types particles appearing and annihilating as relativity allows. As mentioned earlier Schrodinger himself could not agree with his own relativistic version of his equation as it gave negative probabilities and negative energy. Hence he finally discarded.
A similar equation was derived by Klein & Gordon (KG) which used ‘Φ’ in place of Schrodinger’s wave function. Dirac, however, solved some of the issues of KG equation allowing negative energy state.
Finally, in order to make RT & QM compatible, some notions are discarded & some “redefined”. The wave functions in Schrodinger’s equation & KG equation are considered, not as wave functions, but as FIELDS which are operators that create & annihilate particles. The newly promoted fields satisfy the rules of operators by transitioning to continuum. This relation holds within the casualty. I.e. in quantum, position ‘x’ is an operator while time ‘t’ is a parameter. Whereas in relativity they both have a similar footing. So instead of considering time ‘t’ as an operator, position ‘x’ is treated as a parameter. By doing so, the absolute, external time in QT is substituted by SRT’s flat Minkowski spacetime. Thus was born Quantum Field Theory.
In contrast, time in General Relativity ( GT) is dynamic. Hence unifying both QM & GR leads to a new concept of time. The unification forces us to express GRAVITY in the QM framework and leads into the so-called “problem of time”, meaning the time is absolute in QM while it is dynamic in GR.
In pursuit of winning the race, there are many theories put forward to unify QM & GR. AdS/CFT correspondence (in tuned with quantum informatics) is one of my most favorite approaches. In AdS/CFT, AdS stands for Anti de Sitter toy universe and CFT for Conformal Field Theory. As the name goes, it relates quantum field theory with conformal invariance, living in our flat 4D world and string theory living in the solution of AdS5XS5, a curved 5D space with the property of light signal sent to infinity but comes back in finite time.
The flat 4D space living at the boundary of the AdS5XS5 encodes the information of the higher dimension. This is similar to the holography where 3D object is encoded in 2D hologram. In this picture, gravity in the higher dimension vanishes in the 2D boundary. This method fuses quantum information with quantum gravity comfortably. Recently several groups work on quantum spacetime and AdS/CFT to study the quantum gravity in detail.