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Does the Hamiltonian determine the tensor product structure and the 3d space?

Ovidiu Cristinel Stoica

Abstract

It was proposed that the tensor product structure of the Hilbert space is uniquely determined by the Hamiltonian's spectrum, for most finite-dimensional cases satisfying certain conditions. I show that any such method would lead to infinitely many tensor product structures. The dimension of the space of solutions grows exponentially with the number of qudits. In addition, even if the result were unique, such a Hamiltonian would not entangle subsystems. These results affect the proposals to recover the 3d space from the Hamiltonian.

Does the Hamiltonian determine the tensor product structure and the 3d space?

Abstract

It was proposed that the tensor product structure of the Hilbert space is uniquely determined by the Hamiltonian's spectrum, for most finite-dimensional cases satisfying certain conditions. I show that any such method would lead to infinitely many tensor product structures. The dimension of the space of solutions grows exponentially with the number of qudits. In addition, even if the result were unique, such a Hamiltonian would not entangle subsystems. These results affect the proposals to recover the 3d space from the Hamiltonian.
Paper Structure (3 sections, 1 theorem, 15 equations)

This paper contains 3 sections, 1 theorem, 15 equations.

Table of Contents

  1. Introduction
  2. Proof of non-uniqueness
  3. Discussion: TPS, entanglement, and emergent spacetime

Key Result

Theorem 1

No method to obtain the TPS from the Hamiltonian's spectrum can give a unique result. The number of additional continuous parameters needed to fix the TPS uniquely is exponential in the number of qudits.

Theorems & Definitions (2)

  • Theorem 1
  • proof