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Non-Cartesian Systems : an Open Problem

Elemér Elad Rosinger

Published Dec 9, 2010• 3 citations
OpenAlex

Abstract

The following open problem is presented and motivated : Are there physical systems whose state spaces do not compose according to either the Cartesian product, as classical systems do, or the usual tensor product, as quantum systems do ?

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Further on Non-Cartesian Systems

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A class of non-Cartesian physical systems, [7], are those whose composite state spaces are given by significantly extended tensor products. A more detailed presentation of the way such extended tensor products are constructed is offered, based on a step by step comparison with the construction of usual tensor products. This presentation clarifies the extent to which the extended tensor products are indeed more general than the usual ones. ”History is written with the feet...” Ex-Chairman Mao ”Science in not done scientifically, since it is mostly done by non-scientists... ” Anonymous 1 1. Preliminaries As seen in [7], there is a natural division of physical systems in Cartesian

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Composite systems and tensor products

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Introduction A composite system is one involving more than one particle, or a particle with internal degrees of freedom in addition to its center of mass. In classical mechanics the phase space of a composite system is a Cartesian product of the phase spaces of its constituents. The Cartesian product of two sets A and B is the set of (ordered) pairs {( a, b )}, where a is any element of A and b is any element of B . For three sets A, B , and C the Cartesian product consists of triples {( a, b, c )}, and so forth. Consider two classical particles in one dimension, with phase spaces x 1 , p 1 and x 2 , p 2 . The phase space for the composite system consists of pairs of points from the two phase spaces, that is, it is a collection of quadruples of the form x 1 , p 1 , x 2 , p 2 , which can equally well be written in the order x 1 , x 2 , p 1 , p 2 . This is formally the same as the phase space of a single particle in two dimensions, a collection of quadruples x, y, p x , p y . Similarly, the six-dimensional phase space of a particle in three dimensions is formally the same as that of three one-dimensional particles. In quantum theory the analog of a Cartesian product of classical phase spaces is a tensor product of Hilbert spaces. A particle in three dimensions has a Hilbert space which is the tensor product of three spaces, each corresponding to motion in one dimension. The Hilbert space for two particles, as long as they are not identical, is the tensor product of the two Hilbert spaces for the separate particles.

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There's no cloning in symplectic mechanics

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The fact that you can’t clone a quantum system is closely related to the fact that the tensor product in the category of Hilbert spaces is non-Cartesian. At the end of his 2008 classical mechanics course, John Baez pointed out that the tensor product in the category of Poisson manifolds is also non-Cartesian, which should mean there’s a classical analogue of the no-cloning theorem [4]! This turns out to be true in symplectic mechanics, and I don’t see any reason it shouldn’t be true in more generalized settings as well. 1.2 Symplectic mechanics In symplectic mechanics, physical systems are represented by symplectic manifolds, and physical processes are represented by symplectomorphisms. If M and N are symplectic manifolds with symplectic forms ω and σ, respectively, the product manifold M × N has a natural symplectic form Ω given by Ω[(u, s), (v, t)] = ω(u, v) + σ(s, t), where u and v are tangent vectors at the same point in M, and s and t are tangent vectors at the same point in N [3, page 7]. I’ll denote the resulting symplectic manifold M ⊗ N to emphasize that it’s not a Cartesian product in the category of symplectic manifolds. I’ll often write a point in M ⊗ N as an ordered pair of points in M and N, just as I previously wrote a tangent vector on M ⊗ N as an ordered pair of tangent vectors on M and N. 2

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Characterization of Entanglement via Non‐Existence of a Subquantum Random Field

Andrei Yu. Khrennikov

Abstract Any pure state of a compound system with the state space determines a kind of covariance operator acting in the Cartesian product . If this operator is positively defined, then it determines a random field valued in . In this case compound quantum system can be treated as a classical random field system whose configuration space is not tensor, but Cartesian product space. It happens that and a subquantum process exists if and only if quantum state is not entangled. The technical framework used in this note is already presented by von Neumann.

Annalen der Physik•Jul 22, 2024•3 citations
A class of non-Cartesian physical systems, [7], are those whose composite state spaces are given by significantly extended tensor products. A more detailed presentation of the way such extended tensor products are constructed is offered, based on a step by step comparison with the construction of usual tensor products. This presentation clarifies the extent to which the extended tensor products are indeed more general than the usual ones. ”History is written with the feet...” Ex-Chairman Mao ”Science in not done scientifically, since it is mostly done by non-scientists... ” Anonymous 1 1. Preliminaries As seen in [7], there is a natural division of physical systems in Cartesian
Introduction A composite system is one involving more than one particle, or a particle with internal degrees of freedom in addition to its center of mass. In classical mechanics the phase space of a composite system is a Cartesian product of the phase spaces of its constituents. The Cartesian product of two sets A and B is the set of (ordered) pairs {( a, b )}, where a is any element of A and b is any element of B . For three sets A, B , and C the Cartesian product consists of triples {( a, b, c )}, and so forth. Consider two classical particles in one dimension, with phase spaces x 1 , p 1 and x 2 , p 2 . The phase space for the composite system consists of pairs of points from the two phase spaces, that is, it is a collection of quadruples of the form x 1 , p 1 , x 2 , p 2 , which can equally well be written in the order x 1 , x 2 , p 1 , p 2 . This is formally the same as the phase space of a single particle in two dimensions, a collection of quadruples x, y, p x , p y . Similarly, the six-dimensional phase space of a particle in three dimensions is formally the same as that of three one-dimensional particles. In quantum theory the analog of a Cartesian product of classical phase spaces is a tensor product of Hilbert spaces. A particle in three dimensions has a Hilbert space which is the tensor product of three spaces, each corresponding to motion in one dimension. The Hilbert space for two particles, as long as they are not identical, is the tensor product of the two Hilbert spaces for the separate particles.
The fact that you can’t clone a quantum system is closely related to the fact that the tensor product in the category of Hilbert spaces is non-Cartesian. At the end of his 2008 classical mechanics course, John Baez pointed out that the tensor product in the category of Poisson manifolds is also non-Cartesian, which should mean there’s a classical analogue of the no-cloning theorem [4]! This turns out to be true in symplectic mechanics, and I don’t see any reason it shouldn’t be true in more generalized settings as well. 1.2 Symplectic mechanics In symplectic mechanics, physical systems are represented by symplectic manifolds, and physical processes are represented by symplectomorphisms. If M and N are symplectic manifolds with symplectic forms ω and σ, respectively, the product manifold M × N has a natural symplectic form Ω given by Ω[(u, s), (v, t)] = ω(u, v) + σ(s, t), where u and v are tangent vectors at the same point in M, and s and t are tangent vectors at the same point in N [3, page 7]. I’ll denote the resulting symplectic manifold M ⊗ N to emphasize that it’s not a Cartesian product in the category of symplectic manifolds. I’ll often write a point in M ⊗ N as an ordered pair of points in M and N, just as I previously wrote a tangent vector on M ⊗ N as an ordered pair of tangent vectors on M and N. 2
Abstract Any pure state of a compound system with the state space determines a kind of covariance operator acting in the Cartesian product . If this operator is positively defined, then it determines a random field valued in . In this case compound quantum system can be treated as a classical random field system whose configuration space is not tensor, but Cartesian product space. It happens that and a subquantum process exists if and only if quantum state is not entangled. The technical framework used in this note is already presented by von Neumann.