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arXiv:hep-th/9702160

Nonlinear modification of quantum mechanics

Peter Leifer

Published Feb 22, 1997• 6 citations
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Abstract

In order to prevent ``unavoidable'' break-down of the ``peaceful coexistence'' between foundations of quantum theory and relativity I propose a new type of a quantum gauge theory (superrelativity). This differs from ordinary gauge theories in the sense that the affine connection of this theory is constructed from first derivatives of the Fubini-Study metric tensor in the projective Hilbert space of the pure quantum states CP(N). That is we have not merely analogy with general relativity but this construction should presumably provide a unification of general relativity and quantum theory.

DOI
10.48550/arxiv.hep-th/9702160

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In order to prevent unavoidable'' break-down of the peaceful coexistence'' between foundations of quantum theory and relativity I propose a new type of a quantum gauge theory (superrelativity). This differs from ordinary gauge theories in the sense that the affine connection of this theory is constructed from first derivatives of the Fubini-Study metric tensor in the projective Hilbert space of the pure quantum states CP(N). That is we have not merely analogy with general relativity but this construction should presumably provide a unification of general relativity and quantum theory.
State-dependent gauge principle invoked to realize the relativity to a measuring device, has been proposed. Self-consistent global (cosmic) potential forms the state space of the fundamental field and its connection, agreed with Fubini-Study metric of CP(N−1)CP(N-1)CP(N−1), serves as state-dependent gauge potential. In this framework the linearity of the ordinary quantum mechanics appears as a `tangent approximation' to the totally nonlinear underlying pre-dynamical `functional' field theory on CP(N−1)CP(N-1)CP(N−1).
A new type of gauge quantum theory (superrelativity) has been proposed. This differs from ordinary gauge theories in sense that the affine connection of our theory is constructed from first derivatives of the Fubini-Study metric tensor. That is we have not merely analogy with general relativity but this construction should presumably provide a unification of general relativity and quantum theory. Here we shall discuss the physical meaning of metric properties of the projective Hilbert space and manifestation of its nontrivial physical character.
Principle of ``Superrelativity'' has been proposed in order to avoid the contradiction between principle of relativity and foundations of quantum theory. Solutions of a newly derived non-linear Klein-Gordon equation presumably may be treated as primordial nonlocal elements of quantum theory. It is shown that in the framework of CP(N-1) model supplementary elements which are non-local in spacetime but local in the projective Hilbert space permit us to avoid at least one of the main difficulties of quantum theory-the necessity to relate the ``reality'' of a quantum state with a measuring process. In the framework of superrelativity the geometry of the projective Hilbert space (Fubini-Study metric and connection) together with the non-linear wave equation are full and closed quantum scheme.
A fundamental starting point in quantum information theory is the consideration of the von Neumann entropy and its generalization to relative quantum entropies. A particular feature of quantum relative entropies is their relation via their Hessian to monotonic Riemannian metrics on the dense set of invertible mixed quantum states (Lesniewski and Ruskai 1999). These metrics are also known as quantum Fisher information metrics and provide a direct link to quantum estimation theory (Helstrom 1969). Quantum Fisher information metrics which are extendable to pure states coincide all with the Fubini Study metric of the projective Hilbert space of complex rays. This theses outlines possible advantages of an inverse approach to quantum information theory, by starting with the Fubini Study metric rather then with the von Neumann entropy. This is done in a first step by associating to the Fubini Study metric a covariant and a contra-variant structure on the punctured Hilbert space as being available in the geometric formulation of quantum mechanics. While the contra-variant structure leads to a quantum version of the Cramér-Rao inequality for general 1-dimensional submanifolds of pure states, the covariant structure provides alternative entanglement monotones by identifying an inner product on the pullback tensor fields on local unitary group orbits of quantum states. It is shown in the case of two qubits that these monotones yield a more efficient estimation of entanglement than standard measures from the literature as those associated with the linearization of the von Neumann entropy.