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Quantum mechanics of real systems. The Heisenberg commutation relations

Людвиг Дмитриевич Фаддеев, O. Yakubovskiĭ

Published Mar 9, 2009• 0 citations
Publisher
DOI
10.1090/stml/047/10

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This paper analyzes the modified canonical Heisenberg commutation relations or GUP, from a standard Hamiltonian point of view. For a one-dimensional system, a such modified canonical Heisenberg commutation relation is defined by the commutator between a position x^ and a momentum operator p^ (called the deformed momentum), which becomes a function F of the same operators: x^,p^=F(x^,p^), that is, the Heisenberg algebra closes itself in general in a nonlinear way. The function F also depends on a parameter that controls the deformation of the Heisenberg algebra in such a way that for a null parameter value, one recovers the usual Heisenberg algebra x^,p^0=iℏI. Thus, it naturally raises the following questions: What does a relation of this type mean in Hamiltonian theory from a standard point of view? Is the deformed momentum the canonical variable conjugate to the position in such a relation? Moreover, what are the canonical variables in this model? The answer to these questions comes from the existence of two different phase spaces: The first one, called the non-deformed phase (which is obtained for control parameter value equal to zero), is defined by the Cartesian x^ coordinate and its non-deformed conjugate momentum p^0, which satisfies the standard quantum mechanical Heisenberg commutation relation. The second phase space, the deformed one, is given by the deformed momentum p^ and a new position coordinate y^, which is its canonical conjugate variable, so y^ and p^ also satisfy standard commutation relations. We construct a classical canonical transformation that maps the non-deformed phase space into the deformed one for a specific class of deformation functions F. Additionally, a quantum mechanical operator transformation is found between the two non-commutative phase spaces, which allows the Schrödinger equation to be written in both spaces. Thus, there are two equivalent quantum mechanical descriptions of the same physical process associated with a deformed commutation relation.
We derive the structural modifications and physical implications of the Heisenberg Uncertainty Principle under the framework of the Three-Dimensional Complex Necessity (3DCN) principle. In standard quantum mechanics, the uncertainty relation is bounded by a constant scalar value derived from the axiomatic commutation relation [x, p] = i*hbar. By replacing the static imaginary unit i with the dynamically locked universal complex structure operator J_U, we prove that the commutator deforms into a state-dependent operator equation [x, p] = J_U*hbar. We demonstrate that the lower bound of quantum uncertainty is no longer invariant, but fluctuates based on the local geometry of the real phase space substrate and the energy density of the flavor-locking mechanism. This deformation naturally regularizes sub-Planckian spatial measurements, establishing a fundamental physical cutoff that prevents infinite information density localized in real space, thereby avoiding gravitational collapses at the quantum level.
Recent advances, particularly in the area of quantum optics, have caused heightened interest in the fundamental limitations on the achievable accuracy that may be obtained in measuring quantum mechanical systems. We present a quantum correlation result and a noise commutation relation for correlated quantum systems. The measurement of quantum system observables requires the correlation of these micro-observables and the measuring apparatus (usually a macroscopic system). The noise commutation relation is applied to quantum measurements leading to a generalized Heisenberg uncertainty relation. The generalized Heisenberg uncertainty relation yields a lower bound on the inherent unavoidable extra noise in quantum measurements, which is due to the measuring process itself. To indicate the utility of the noise commutation relation, two different applications in optics are discussed. The first, an application of the generalized Heisenberg uncertainty relation, indicates how a model-independent noise limit for balanced homodyne detection can be obtained. Another application is discussed in which the noise commutation relation is applied to develop a model-independent lower bound for the inherent noise of a quantum optical linear amplifier.