A Proof for Poisson Bracket in Non-commutative Algebra of Quantum Mechanics
نویسنده
چکیده
The widely accepted approach to the foundation of quantum mechanics is that the Poisson bracket, governing the non-commutative algebra of operators, is taken as a postulate with no underlying physics. In this manuscript, it is shown that this postulation is in fact unnecessary and may be replaced by a few deeper concepts, which ultimately lead to the derivation of Poisson bracket. One would only need to use Fourier transform pairs and KramersKronig identities in the complex domain. We present a definition of Hermitian time-operator and discuss some of its basic properties. c © Electronic Journal of Theoretical Physics. All rights reserved.
منابع مشابه
Classical and quantum q-deformed physical systems
On the basis of the non-commutative q-calculus, we investigate a q-deformation of the classical Poisson bracket in order to formulate a generalized q-deformed dynamics in the classical regime. The obtained q-deformed Poisson bracket appears invariant under the action of the q-symplectic group of transformations. In this framework we introduce the q-deformed Hamilton’s equations and we derive th...
متن کاملAlgebraic and geometric aspects of generalized quantum dynamics.
We briefly discuss some algebraic and geometric aspects of the generalized Poisson bracket and non–commutative phase space for generalized quantum dynamics, which are analogous to properties of the classical Poisson bracket and ordinary symplectic structure. \pacs{} Typeset using REVTEX 1 Recently, one of us (SLA) has proposed a generalization of Heisenberg picture quantum mechanics, termed gen...
متن کاملOn Quantization of the Semenov-tian-shansky Poisson Bracket on Simple Algebraic Groups
Let G be a simple complex factorizable Poisson algebraic group. Let U (g) be the corresponding quantum group. We study the U (g)-equivariant quantization C [G] of the affine coordinate ring C[G] along the Semenov-Tian-Shansky bracket. For a simply connected group G, we give an elementary proof for the analog of the Kostant–Richardson theorem stating that C [G] is a free module over its center.
متن کاملOn zeroth Poisson homology in positive characteristic
A Poisson algebra is a commutative algebra with a Lie bracket {, } satisfying the Leibniz rule. Such algebras appear in classical mechanics. Namely, functions on the phase space form a Poisson algebra, and Hamilton’s equation of motion is df dt = {f, H}, where H is the Hamiltonian (energy) function. Moreover, the transition from classical to quantum mechanics can be understood in terms of defor...
متن کاملNon-associative algebras associated to Poisson algebras
Poisson algebras are usually defined as structures with two operations, a commutative associative one and an anti-commutative one that satisfies the Jacobi identity. These operations are tied up by a distributive law, the Leibniz rule. We present Poisson algebras as algebras with one operation, which enables us to study them as part of non-associative algebras. We study the algebraic and cohomo...
متن کامل