Lecture notes in physics

Quantum Mechanics. Algebraic Structure and States

Franco Strocchi

Published Oct 30, 2007• 0 citations

Publisher

DOI: 10.1007/978-3-540-73593-9_11

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1 Quantum Mechanics. Algebraic Structure and States

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The bicomplex algebra as a minimal extension of the scalar algebra of quantum mechanics

Ralf Otte

Mar 24, 20260 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

Quantum Mechanics: Algebraic Point of View

Yair Shapira

Springer eBooksNov 15, 20220 citations

Quantum Mechanics: States and Operators

Roman Schmied

Springer eBooksSep 28, 20190 citations

Lecture notes in physics

Quantum Mechanics. Algebraic Structure and States

Franco Strocchi

Published Oct 30, 2007• 0 citations

Publisher

DOI: 10.1007/978-3-540-73593-9_11

Related papers

1 Quantum Mechanics. Algebraic Structure and States

Franco Strocchi

Lecture notes in physicsMay 10, 20050 citations

Algebraic structure of quantum mechanics

K. Heyde, John L. Wood

Institute of Physics eBooksApr 1, 20200 citations

The bicomplex algebra as a minimal extension of the scalar algebra of quantum mechanics

Ralf Otte

Quantum mechanics is formulated on complex Hilbert spaces, where the complex numbers constitute the underlying scalar algebra. In this work, the structural question is investigated whether this scalar structure is algebraically minimal or whether a consistent extension exists that preserves the analytical infrastructure of the theory. We formulate this problem as a minimality question for finite-dimensional real, commutative and associative algebras that contain the complex numbers as a subalgebra and at the same time possess a nontrivial ideal that is isomorphic to C as a real algebra and carries a complex-like exponential structure. It is shown that such structures necessarily require at least four real dimensions. The bicomplex algebra B=C ⊗R C realizes these requirements in minimal dimension and is unique up to isomorphism under the stated conditions. For the extended state space M=H ⊗C B one obtains a canonical decomposition M≅H⊕H, so that the bicomplex scalar extension structurally corresponds to two parallel complex quantum sectors. The Born rule remains unchanged. The present work provides a structural analysis of possible extensions of the scalar algebra underlying quantum mechanics.Quantum mechanics is formulated on complex Hilbert spaces, where the complex numbers constitute the underlying scalar algebra. In this work, the structural question is investigated whether this scalar structure is algebraically minimal or whether a consistent extension exists that preserves the analytical infrastructure of the theory. We formulate this problem as a minimality question for finite-dimensional real, commutative and associative algebras that contain the complex numbers as a subalgebra and at the same time possess a nontrivial ideal that is isomorphic to C as a real algebra and carries a complex-like exponential structure. It is shown that such structures necessarily require at least four real dimensions. The bicomplex algebra B=C ⊗R C realizes these requirements in minimal dimension and is unique up to isomorphism under the stated conditions. For the extended state space M=H ⊗C B one obtains a canonical decomposition M≅H⊕H, so that the bicomplex scalar extension structurally corresponds to two parallel complex quantum sectors. The Born rule remains unchanged. The present work provides a structural analysis of possible extensions of the scalar algebra underlying quantum mechanics.

Mar 24, 20260 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

Quantum Mechanics: Algebraic Point of View

Yair Shapira

Springer eBooksNov 15, 20220 citations

Quantum Mechanics: States and Operators

Roman Schmied

Springer eBooksSep 28, 20190 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.