Many-Body Schrödinger Operators, Conditions and Notation

W. Hunziker

Charles Alexandre Bédard

Vedral claims that the Schrödinger picture can describe quantum systems as locally as the Heisenberg picture, relying on a product notation for the density matrix. Here, I refute that claim. I show that the so-called `local factors' in the product notation do not correspond to individual systems and therefore fail to satisfy Einsteinian locality. Furthermore, the product notation does not track where local gates are applied. Finally, I expose internal inconsistencies in the argument: if, as is also stated, Schrödinger-picture locality ultimately depends on explicit bookkeeping of all operations, then the explanatory power of the product notation is de facto undermined.

Charles Alexandre Bédard

In his recent paper, Vlatko Vedral claims that the Schrödinger picture can describe quantum systems as locally as the Heisenberg picture, relying on a product notation for the density matrix. Here, I refute that claim. I show that the so-called ‘local factors’ in the product notation do not correspond to individual systems and therefore fail to satisfy Einsteinian locality. Furthermore, the product notation does not track where local gates are applied. Finally, I expose internal inconsistencies in the argument: if, as is also stated, the Schrödinger-picture locality ultimately depends on the explicit bookkeeping of all operations, then the explanatory power of the product notation is de facto undermined.

Hiroshi Isozaki

Teschl, Gerald

Xing Min Wang

Following the Dirac vector bracket notation (VBN), we proposed the probability bracket notation (PBN) in our previous paper. We mentioned that under the special Wick rotation (imaginary time), a stationary Schrodinger equation in the Hilbert space transforms into the master equation of a microscopic probabilistic process (MPP) in the probability space. In this article, we first study the MPP of the system of a single particle, we show that the energy expectation of the MPP eventually approaches the lowest energy level in its initial condition and its von Neumann entropy finally vanishes. Then we explore the MPP for the quantum system of identical particles in the Fock space, we recover the expected occupation number of particles and the grand partition function in quantum statistics by connecting time with temperature (the Wick-Matsubara relation). We also reproduce the internal energy of an ideal gas in thermodynamics by using the relation. To address the entropy issue and relate the PBN with research topics of statistics in the literature, we express the density operators and the von Neumann entropy in the probability space by using the PBN. The Wick-Matsubara relation plus the PBN might provide a new way of microscopic probability modeling.