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Quantum Studies Mathematics and Foundations

On nonlocal complexified Schrödinger equation and emergence of discrete quantum mechanics

Rami Ahmad El‐Nabulsi

Published Jul 14, 2016• 10 citations
Publisher
DOI
10.1007/s40509-016-0080-z

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The general problem of finding exactly soluble quantum systems is considered. It is argued that discrete space quantum mechanics emerges in a natural way as an avenue of approach. Discrete space quantum mechanics is formulated and applied to one-dimensional quantum systems with emphasis on single-channel models. It is found that a large variety of systems are exactly soluble in the sense that they only require the inversion of a finite-dimensional matrix. The interaction may in general be both nonlocal and non-time-reversal-invariant. The analytic structure of the resolvent is worked out in detail for a simple class of examples. It is shown that a slightly modified version of the usual continuous space Schrödinger equation may in principle be solved exactly for any finite range local potential by writing the solution in terms of corresponding discrete space solutions. It is also shown that from an algebraic viewpoint the models constructed are realizations of generalized versions of the Weyl relations.
This paper presents a compact formal framework in which the standard Schrödinger equation emerges as the short-delay limit of a time-symmetric nonlocal effective theory. Starting from a symmetric delayed action containing both backward and forward temporal shifts, the derivation shows that local quantum dynamics appears naturally at leading order, while higher even-order terms remain as memory-type corrections generated by the underlying nonlocal structure. A central mathematical feature of the construction is that the symmetric temporal averaging cancels all odd-order contributions identically, yielding a particularly clean effective expansion. The paper does not claim a fully rigorous nonlocal variational foundation; instead, it isolates the structural mechanism by which local quantum mechanics can arise from a reversible delayed effective description. It further proposes a two-layer interpretation: retarded or causal dynamics may be regarded as ontologically fundamental, whereas a time-symmetric short-delay expansion governs the effective local theory. In this sense, standard quantum mechanics is interpreted not as the deepest starting point, but as the leading local approximation of a deeper delayed nonlocal framework.
Abstract The authors show that a large class of 2‐adic Schrödinger equations is the scaling limit of certain continuous‐time quantum Markov chains (CTQMCs). Practically, a discretization of such an equation gives a CTQMC. As a practical result, new types of continuous‐time quantum walks (CTQWs) on graphs using two symmetric matrices are constructed. The transport between nodes in one direction is described by one matrix, while the transport between nodes in the opposite direction. This construction includes, as a particular case, the CTQWs constructed using adjacency matrices. The final goal of this work is to contribute to the understanding of the foundations of quantum mechanics (QM) and the role of the hypothesis of the discreteness of space. The connection between 2‐adic QM and CTQWs shows that 2‐adic QM has a physical meaning. 2‐Adic QM is a nonlocal theory because the Hamiltonians used are nonlocal operators, and consequently, spooky actions at a distance are allowed. However, this theory is not a mathematical toy. The violation of Bell's inequality is experimentally confirmed, which implies that realism is allowed by this theory. The authors pointed out several new research problems connected with the foundations of QM.