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 {g ij } on su $$ \mathfrak{su} $$ (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 H 0 with a many-body structure is of limited conceptual value unless the initial state, ∣ψ 0〉, is also structured by this tensor decomposition. Here we adapt F to become a functional on {g, | ψ 0〉} = (metrics) × (initial states), and find SSB now produces a conspiracy between g and ∣ψ 0〉, 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.

Keywords

• Effective Field Theories
• Spontaneous Symmetry Breaking