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

States in quantum mechanics

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Springer eBooks

Quantum Mechanics: States and Operators

Roman Schmied

Published Sep 28, 2019• 0 citations

Publisher

DOI: 10.1007/978-981-13-7588-0_2

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The Quantum Mechanical Time Reversal Operator.

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The analysis of the reversibility of quantum mechanics depends upon the choice of the time reversal operator for quantum mechanical states. The orthodox choice for the time reversal operator on QM states is known as the Wigner operator, T*, where * performs complex conjugation. The peculiarity is that this is not simply the unitary time reversal operation, but an anti-unitary operator, involving complex conjugation in addition to ordinary time reversal. The alternative choice is the Racah operator, which is simply ordinary time reversal, T. Orthodox treatments hold that it is either logically or empirically necessary to adopt the Wigner operator, and the Racah operator has received little attention. The basis for this choice is analysed in detail, and it is concluded that all the conventional arguments for rejecting the Racah operator and adopting the Wigner operator are mistaken. The additional problem of whether the deterministic part of quantum mechanics should be judged to be reversible or not is also considered. The adoption of the Racah operator for time reversal appears prima facie to entail that quantum mechanics is irreversible. However, it is concluded that the real answer to question depends upon the choice of interpretation of the theory. In any case, the conventional reasons for claiming that quantum mechanics is reversible are incorrect.The analysis of the reversibility of quantum mechanics depends upon the choice of the time reversal operator for quantum mechanical states. The orthodox choice for the time reversal operator on QM states is known as the Wigner operator, T*, where * performs complex conjugation. The peculiarity is that this is not simply the unitary time reversal operation, but an anti-unitary operator, involving complex conjugation in addition to ordinary time reversal. The alternative choice is the Racah operator, which is simply ordinary time reversal, T. Orthodox treatments hold that it is either logically or empirically necessary to adopt the Wigner operator, and the Racah operator has received little attention. The basis for this choice is analysed in detail, and it is concluded that all the conventional arguments for rejecting the Racah operator and adopting the Wigner operator are mistaken. The additional problem of whether the deterministic part of quantum mechanics should be judged to be reversible or not is also considered. The adoption of the Racah operator for time reversal appears prima facie to entail that quantum mechanics is irreversible. However, it is concluded that the real answer to question depends upon the choice of interpretation of the theory. In any case, the conventional reasons for claiming that quantum mechanics is reversible are incorrect.

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In the previous studies and literatures, quantum mechanics representations are a pure state representation, while in this paper beginning with reasonable ordering of operators in quantum mechanics and the normal distribution thought in probability statistics, we propose the concept of quantum mechanics mixed state representation firstly,and prove its completeness and orthogonality. The advantages of mixed state representation is that it can reflect a variety of expression of quantum mechanics operators and the corresponding operators ordering rules.In the previous studies and literatures, quantum mechanics representations are a pure state representation, while in this paper beginning with reasonable ordering of operators in quantum mechanics and the normal distribution thought in probability statistics, we propose the concept of quantum mechanics mixed state representation firstly,and prove its completeness and orthogonality. The advantages of mixed state representation is that it can reflect a variety of expression of quantum mechanics operators and the corresponding operators ordering rules.

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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

States in quantum mechanics

L. D. Faddeev, O. Yakubovskiĭ

Student mathematical libraryMar 9, 20092 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.