Theory of Everything
Boson. Furthermore, quantum mechanics suggests is that all matter has an associated wave function which governs the physical properties of the matter which is given by the Schrödinger equation:
∂ ∂𝑡
𝑖ℏ
Ψ(𝑟, 𝑡) = 𝐻̂Ψ(𝑟, 𝑡)
This equation is a differential equation and is essentially a statement of the conservation of energy for a particle. The momentum of a particle can be calculated using the wavelength of the wave function, and the modulus squared of this wave function describes probability density function of the particle. From this fact, we can derive the Heisenberg uncertainty principle which states that the momentum and position of a particle cannot be measured with an infinite accuracy simultaneously (they are non- commutable). This is because, in order for position to be measured with high accuracy, there must be a single distinct peak in the probability density function of the particle. However, as the Hamiltonian ( 𝐻̂ ) is a linear operator, this wave function can be represented as the sum of many other wave functions using a Fourier transformation. As each of the constituent wave functions have different wavelength and therefore correspond to a different momentum, the momentum is known to a higher uncertainty. These ideas cause conflict between the two frameworks as, firstly, general relativity is defined by the stress-energy tensor at a point. 6 However, by the Heisenberg uncertainty principle of quantum mechanics, this is impossible, as the stress-energy tensor has information about the momentum at a point in space-time which are non-commutable quantities. Moreover, for gravity to be represented like other forces in quantum mechanics, the gravitational field must be quantized into a boson referred to as the graviton. This is an issue because the gravitational field, by definition, is a continuous quantity, whereas the graviton is discrete. Furthermore, if such a particle did exist and represented the curvature of space-time due to energy, the particle itself would have energy and would therefore curve space-time itself, generating more gravitons. This self-reproducing nature of gravitons would lead to the infinite reproduction of gravitons and therefore infinite energy. Although this same issue occurs in QED when an electron interacts with itself due to its charge affecting the local electric field, it is circumvented using perturbation theory and renormalization which simplifies the interaction with a series of adjustments after a weak disturbance to the system. However, in order for renormalization to be successful, highly precise measurements of relevant quantities are required but, owing to the inability to measure precisely the gravitational field at a point, we say that gravity is non-renormalizable and therefore the infinite energies that are calculated cannot be renormalized to sensible values unlike in QED. Finally, quantum mechanics is not background independent (it assumes a flat space travelling through time) and treats time independent of space which is contrary to the teachings of general r elativity’s mutable space-time and its background independent equations. 7 One proposed solution to the problems with quantum gravity is known as loop quantum gravity. 8 Loop quantum gravity addresses the issue of background independence by proposing to describe the quantum evolution of space-time by describing the change over time of 3d space slices taken out of the 4d space-time in a similar way to how wave functions describe the motion of particles.
6 See Odenwald: What is it about quantum mechanics that is incompatible with general relativity? 7 The Schrödinger equation treats time and space independently proven by how there are both temporal derivatives and spatial derivatives in the Hamiltonian operator contrary to the teachings of relativity. 8 See Loop Quantum Gravity explained (2019).
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