(untitled) (London Journal of Research In Science: Natural and Formal, 2025)
The main idea I want to discuss is the possibility that quantum-mechanical de Sitter space admits a holographic description. In spirit, such a description would be a boundary-like quantum system that makes no explicit reference to gravity, yet somehow encodes the physics of the bulk. Why focus on de Sitter space? Because de Sitter space is the “elephant in the room”: enormously large, highly symmetric, cosmologically relevant, and almost certainly closely related to the universe we inhabit. But unlike Anti–de Sitter space, de Sitter space has no natural boundary, which makes holography in this setting significantly more challenging. Despite this difficulty, researchers have explored many different perspectives. Discussions about de Sitter holography often resemble the story of the “Blind Men and the Elephant,” where each observer touches a different part and reaches a different conclusion. Some approaches emphasize dS/CFT, others TT̄ deformations, the swampland program, dS/dS duality, or matrix-theory–like constructions. None of these perspectives is clearly wrong, but none gives a complete picture. In this paper I present a set of “fragmentary circumstantial evidence’’ suggesting that certain aspects of de Sitter space may be described by a type of matrix theory. This idea was originally proposed by several theorists; here I revisit their arguments and add some additional clues.The framework I adopt is static-patch holography. A static patch is the region seen by an observer located at its center, which I call the “pode.” By symmetry, there is another static patch on the opposite side of the space, whose center I call the “anti-pode.” At time t = 0, spatial slices of de Sitter space resemble a sphere, with the pode at one end, the anti-pode at the other, and the cosmological horizon in the middle. At other times, one can naturally identify two horizons—one from each static patch. The basic hypothesis is that all physics inside a single static patch can be described by a holographic theory that is essentially quantum mechanics without gravity. De Sitter space behaves as a thermal system with temperature proportional to the inverse of the de Sitter radius, meaning larger de Sitter spaces are colder.The horizon carries an entropy proportional to its area. Defining this thermodynamics already assumes that the static patch is described by a unitary quantum system with a Hilbert Made with Xodo PDF Reader and Editor Made with Xodo PDF Reader and Editor space, a Hamiltonian, and a set of symmetry generators that form the algebra of de Sitter space. In AdS, holographic degrees of freedom live at the asymptotic boundary. De Sitter space has no such boundary, and the static patch itself has none either. One might try to place the holographic degrees of freedom near the pode or anti-pode, but this fails because a small surface near the pode does not have enough area to encode the entire static patch. The only viable location is the stretched horizon. Therefore the holographic degrees of freedom must correspond to disturbances of the horizon itself. Operators that create excitations near the pode are complicated from the holographic viewpoint, while simple operators correspond to local changes of the horizon. This parallels what happens in AdS: excitations far from the holographic degrees of freedom appear as complex operators.In a thermal de Sitter background, rare Boltzmann fluctuations can move enough degrees of freedom from the horizon to the region near the pode, assembling macroscopic objects such as black holes. Such a configuration can be described by the Schwarzschild–de Sitter geometry, which contains two horizons: a small black hole horizon and a larger cosmological horizon. For sufficiently small black holes, the spacetime contains two identical black holes—one near the pode and one near the anti-pode. The spatial slice at t = 0 is a sphere with two small black hole horizons near the poles and the cosmological horizon at the equator. Assuming this configuration arises as a fluctuation, its probability is determined by the difference between the entropy of pure de Sitter space and the entropy of the configuration containing the black holes. The probability is exponentially suppressed, behaving like the exponential of minus the entropy difference. For small black holes, this entropy difference is proportional to the product of the de Sitter radius and the black hole mass, making such fluctuations extremely rare. This behavior is precisely what one expects from a finite quantum system with a horizon, and it strengthens the idea that the holographic description of the static patch may resemble a matrix-theory-type construction.