The smallest interacting universe

Abstract

Abstract We study a mechanism by which the most basic structures of quantum physics can emerge from the most meager of starting points, a Hilbert space, lacking any preassigned structure such as a tensor decomposition, and a loss function. In a simple toy model of the universe, we hypothesize a fundamental loss functional for the combined Hamiltonian and quantum state, and then minimize this loss functional by gradient descent. We find that this minimization gives rise to a co-emergence of locality, i.e. a tensor product structure simultaneously respected by both the Hamiltonian and the state, suggesting that locality can emerge by a process analogous to spontaneous symmetry breaking. We discuss the relevance of this program to the arrow of time problem.In our toy model, we interpret the emergence of a tensor factorization as the appearance of individual degrees of freedom within a previously undifferentiated (raw) Hilbert space. Earlier work [5, 6] looked at the emergence of locality in Hamiltonians only, and in that context found strong numerical confirmation of the hypothesis that raw Hilbert spaces of dim = n are unstable and prefer to settle on tensor factorization when n is not prime, expressing, for example, n = pq, and in [6] even primes were seen to “factor” after first shedding a small summand, e.g. 7 = 1 + 2 · 3. This was found in the context of a rather general potential functional F on the space of metrics {gij} on $$ \mathfrak{su} $$ su (n), the Lie algebra of symmetries. This emergence of qunits through operator-level spontaneous symmetry breaking (SSB) may help us understand why the world seems to consist of myriad interacting degrees of freedom. But understanding why the universe has an initial Hamiltonian H0 with a many-body structure is of limited conceptual value unless the initial state, ∣ψ0〉, is also structured by this tensor decomposition. Here we adapt F to become a functional on {g, | ψ0〉} = (metrics) × (initial states), and find SSB now produces a conspiracy between g and ∣ψ0〉, where they simultaneously attain low entropy by jointly settling on the same qubit decomposition. Extreme scaling of the computational problem has confined us to studying ℂ4 breaking to ℂ2 ⊗ ℂ2 and ℂ8 breaking to ℂ2 ⊗ ℂ4 or ℂ2 ⊗ ℂ2 ⊗ ℂ2.