SpringerBriefs in mathematical physics

Algebraic Quantum Mechanics

J.-B. Bru, W. de Siqueira Pedra

Published Dec 1, 2016• 0 citations

Publisher

DOI: 10.1007/978-3-319-45784-0_2

Related papers

A Review of The Algebraic Approaches to Quantum Mechanics. Appraisals on Their Theoretical Relevance

Antonino Drago

arXiv:2102.00861Jan 29, 20210 citations

Algebraic Quantum Mechanics

Nicolaas P. Landsman

Compendium of Quantum PhysicsJan 1, 200941 citations

Algebraic Formulation of Quantum Mechanics

Ajit Kumar

Cambridge University Press eBooksOct 20, 20180 citations

On the operator origins of classical and quantum wave functions

Xerxes D. Arsiwalla, David Chester, Louis Hirsch Kauffman

Abstract We investigate operator algebraic origins of the classical Koopman–von Neumann wave function $$\psi _{KvN}$$ ψ KvN as well as the quantum-mechanical one $$\psi _{QM}$$ ψ QM . We introduce a formalism of Operator Mechanics (OM) based on a noncommutative Poisson, symplectic, and noncommutative differential structures. OM serves as a pre-quantum algebra from which algebraic structures relevant to real-world classical and quantum mechanics follow. In particular, $$\psi _{KvN}$$ ψ KvN and $$\psi _{QM}$$ ψ QM are both consequences of this pre-quantum formalism. No a priori Hilbert space is needed. OM admits an algebraic notion of operator expectation values without invoking states. A phase space bundle $${{\mathcal {E}}}$$ E follows from this. $$\psi _{KvN}$$ ψ KvN and $$\psi _{QM}$$ ψ QM are shown to be sections in $${{\mathcal {E}}}$$ E . The difference between $$\psi _{KvN}$$ ψ KvN and $$\psi _{QM}$$ ψ QM originates from a quantization map interpreted as “twisting” of sections over $${{\mathcal {E}}}$$ E . We also show that the Schrödinger equation is obtained from the Koopman–von Neumann equation. What this suggests is that neither the Schrödinger equation nor the quantum wave function are fundamental structures. Rather, they both originate from a pre-quantum operator algebra. Finally, we comment on how entanglement between these operators suggests emergence of space; and possible extensions of this formalism to field theories.

arXiv:2211.01838Quantum Studies Mathematics and FoundationsDec 19, 20236 citations

Linear algebra and quantum algorithm

BongJu Kim

arXiv:2008.08905Aug 14, 20200 citations

Algebraic structure of quantum mechanics

K. Heyde, John L. Wood

Institute of Physics eBooksApr 1, 20200 citations

SpringerBriefs in mathematical physics

Algebraic Quantum Mechanics

J.-B. Bru, W. de Siqueira Pedra

Published Dec 1, 2016• 0 citations

Publisher

DOI: 10.1007/978-3-319-45784-0_2

Related papers

A Review of The Algebraic Approaches to Quantum Mechanics. Appraisals on Their Theoretical Relevance

Antonino Drago

I review the various algebraic foundations of quantum mechanics. They have been suggested since the birth of this theory till up to last year. They are the following ones: Heisenberg-Born-Jordan (1925), Weyl (1928), Dirac (1930), von Neumann (1936), Segal (1947), T.F. Jordan (1986), Morchio and Strocchi (2009) and Buchholz and Fregenhagen (2019). Three cases are stressed: 1) the misinterpretation of Dirac foundation; 2) von Neumann conversion from the analytic approach of Hilbert space to the algebraic approach of the rings of operators; 3) the recent foundation of quantum mechanics upon the algebra of perturbation Lagrangians. Moreover, historical considerations on the go-and-stop path performed by the algebraic approach in the history of QM are offered. The level of formalism has increased from the mere introduction of matrices till up to group theory and C*-algebras. But there was no progress in approaching closer the foundations of physics; therefore the problem of discovering an algebraic formulation of QM organized as a problem-based theory and making use of no more than constructive mathematics is open.I review the various algebraic foundations of quantum mechanics. They have been suggested since the birth of this theory till up to last year. They are the following ones: Heisenberg-Born-Jordan (1925), Weyl (1928), Dirac (1930), von Neumann (1936), Segal (1947), T.F. Jordan (1986), Morchio and Strocchi (2009) and Buchholz and Fregenhagen (2019). Three cases are stressed: 1) the misinterpretation of Dirac foundation; 2) von Neumann conversion from the analytic approach of Hilbert space to the algebraic approach of the rings of operators; 3) the recent foundation of quantum mechanics upon the algebra of perturbation Lagrangians. Moreover, historical considerations on the go-and-stop path performed by the algebraic approach in the history of QM are offered. The level of formalism has increased from the mere introduction of matrices till up to group theory and C*-algebras. But there was no progress in approaching closer the foundations of physics; therefore the problem of discovering an algebraic formulation of QM organized as a problem-based theory and making use of no more than constructive mathematics is open.

arXiv:2102.00861Jan 29, 20210 citations

Algebraic Quantum Mechanics

Nicolaas P. Landsman

Compendium of Quantum PhysicsJan 1, 200941 citations

Algebraic Formulation of Quantum Mechanics

Ajit Kumar

Cambridge University Press eBooksOct 20, 20180 citations

On the operator origins of classical and quantum wave functions

Xerxes D. Arsiwalla, David Chester, Louis Hirsch Kauffman

Abstract We investigate operator algebraic origins of the classical Koopman–von Neumann wave function $\psi _{KvN}$ ψ KvN as well as the quantum-mechanical one $\psi _{QM}$ ψ QM . We introduce a formalism of Operator Mechanics (OM) based on a noncommutative Poisson, symplectic, and noncommutative differential structures. OM serves as a pre-quantum algebra from which algebraic structures relevant to real-world classical and quantum mechanics follow. In particular, $\psi _{KvN}$ ψ KvN and $\psi _{QM}$ ψ QM are both consequences of this pre-quantum formalism. No a priori Hilbert space is needed. OM admits an algebraic notion of operator expectation values without invoking states. A phase space bundle ${{\mathcal {E}}}$ E follows from this. $\psi _{KvN}$ ψ KvN and $\psi _{QM}$ ψ QM are shown to be sections in ${{\mathcal {E}}}$ E . The difference between $\psi _{KvN}$ ψ KvN and $\psi _{QM}$ ψ QM originates from a quantization map interpreted as “twisting” of sections over ${{\mathcal {E}}}$ E . We also show that the Schrödinger equation is obtained from the Koopman–von Neumann equation. What this suggests is that neither the Schrödinger equation nor the quantum wave function are fundamental structures. Rather, they both originate from a pre-quantum operator algebra. Finally, we comment on how entanglement between these operators suggests emergence of space; and possible extensions of this formalism to field theories.Abstract We investigate operator algebraic origins of the classical Koopman–von Neumann wave function $$\psi _{KvN}$$ ψ KvN as well as the quantum-mechanical one $$\psi _{QM}$$ ψ QM . We introduce a formalism of Operator Mechanics (OM) based on a noncommutative Poisson, symplectic, and noncommutative differential structures. OM serves as a pre-quantum algebra from which algebraic structures relevant to real-world classical and quantum mechanics follow. In particular, $$\psi _{KvN}$$ ψ KvN and $$\psi _{QM}$$ ψ QM are both consequences of this pre-quantum formalism. No a priori Hilbert space is needed. OM admits an algebraic notion of operator expectation values without invoking states. A phase space bundle $${{\mathcal {E}}}$$ E follows from this. $$\psi _{KvN}$$ ψ KvN and $$\psi _{QM}$$ ψ QM are shown to be sections in $${{\mathcal {E}}}$$ E . The difference between $$\psi _{KvN}$$ ψ KvN and $$\psi _{QM}$$ ψ QM originates from a quantization map interpreted as “twisting” of sections over $${{\mathcal {E}}}$$ E . We also show that the Schrödinger equation is obtained from the Koopman–von Neumann equation. What this suggests is that neither the Schrödinger equation nor the quantum wave function are fundamental structures. Rather, they both originate from a pre-quantum operator algebra. Finally, we comment on how entanglement between these operators suggests emergence of space; and possible extensions of this formalism to field theories.

arXiv:2211.01838Quantum Studies Mathematics and FoundationsDec 19, 20236 citations

Linear algebra and quantum algorithm

BongJu Kim

In mathematical aspect, we introduce quantum algorithm and the mathematical structure of quantum computer. Quantum algorithm is expressed by linear algebra on a finite dimensional complex inner product space. The mathematical formulations of quantum mechanics had been established in around 1930, by von Neumann. The formulation uses functional analysis, linear algebra and probability theory. The knowledge of the mathematical formulation of QM is enough quantum mechanical knowledge for approaching to quantum algorithm and it might be efficient way for mathematicians that starting with mathematical formulations of QM. We explain the mathematical formulations of quantum mechanics briefly, quantum bits, quantum gates, quantum discrete Fourier transformation, Deutsch's algorithm and Shor's algorithm.In mathematical aspect, we introduce quantum algorithm and the mathematical structure of quantum computer. Quantum algorithm is expressed by linear algebra on a finite dimensional complex inner product space. The mathematical formulations of quantum mechanics had been established in around 1930, by von Neumann. The formulation uses functional analysis, linear algebra and probability theory. The knowledge of the mathematical formulation of QM is enough quantum mechanical knowledge for approaching to quantum algorithm and it might be efficient way for mathematicians that starting with mathematical formulations of QM. We explain the mathematical formulations of quantum mechanics briefly, quantum bits, quantum gates, quantum discrete Fourier transformation, Deutsch's algorithm and Shor's algorithm.

arXiv:2008.08905Aug 14, 20200 citations

Algebraic structure of quantum mechanics

K. Heyde, John L. Wood

Institute of Physics eBooksApr 1, 20200 citations

Abstract We investigate operator algebraic origins of the classical Koopman–von Neumann wave function

\psi _{KvN}

ψ KvN as well as the quantum-mechanical one

\psi _{QM}

ψ QM . We introduce a formalism of Operator Mechanics (OM) based on a noncommutative Poisson, symplectic, and noncommutative differential structures. OM serves as a pre-quantum algebra from which algebraic structures relevant to real-world classical and quantum mechanics follow. In particular,

\psi _{KvN}

ψ KvN and

\psi _{QM}

ψ QM are both consequences of this pre-quantum formalism. No a priori Hilbert space is needed. OM admits an algebraic notion of operator expectation values without invoking states. A phase space bundle

{{\mathcal {E}}}

E follows from this.

\psi _{KvN}

ψ KvN and

\psi _{QM}

ψ QM are shown to be sections in

{{\mathcal {E}}}

E . The difference between

\psi _{KvN}

ψ KvN and

\psi _{QM}

ψ QM originates from a quantization map interpreted as “twisting” of sections over

{{\mathcal {E}}}

E . We also show that the Schrödinger equation is obtained from the Koopman–von Neumann equation. What this suggests is that neither the Schrödinger equation nor the quantum wave function are fundamental structures. Rather, they both originate from a pre-quantum operator algebra. Finally, we comment on how entanglement between these operators suggests emergence of space; and possible extensions of this formalism to field theories.