2018/01/30 by Jason Pollack, Ashmeet Singh, Pollack, Jason +1
Physics and Astronomy · #Cosmology and Gravitation Theories #Dark Matter and Cosmic Phenomena #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #High Energy Physics - Theory (hep-th) #Quantum Mechanics and Applications #Quantum Physics (quant-ph)
paper · pdf · doi:10.48550/arxiv.1801.10168
openalex publication_date 2018/01/30 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
Field theories place one or more degrees of freedom at every point in space.\nHilbert spaces describing quantum field theories, or their finite-dimensional\ndiscretizations on lattices, therefore have large amounts of structure: they\nare isomorphic to the tensor product of a smaller Hilbert space for each\nlattice site or point in space. Local field theories respecting this structure\nhave interactions which preferentially couple nearby points. The emergence of\nclassicality through decoherence relies on this framework of tensor-product\ndecomposition and local interactions. We explore the emergence of such lattice\nstructure from Hilbert-space considerations alone. We point out that the vast\nmajority of finite-dimensional Hilbert spaces cannot be isomorphic to the\ntensor product of Hilbert-space subfactors that describes a lattice theory. A\ngeneric Hilbert space can only be split into a direct sum corresponding to a\nbasis of state vectors spanning the Hilbert space; we consider setups in which\nthe direct sum is naturally decomposed into two pieces. We define a notion of\ndirect-sum locality which characterizes states and decompositions compatible\nwith Hamiltonian time evolution. We illustrate these notions for a toy model\nthat is the finite-dimensional discretization of the quantum-mechanical\ndouble-well potential. We discuss their relevance in cosmology and field\ntheory, especially for theories which describe a landscape of vacua with\ndifferent spacetime geometries.\n