Quantum spin systems versus Schroedinger operators: A case study in  spontaneous symmetry breaking

Ven, Christiaan J. F. van de; Groenenboom, Gerrit C.; Reuvers, Robin; Landsman, Klaas

doi:10.21468/scipostphys.8.2.022

Cited by 14 publications

(22 citation statements)

References 45 publications

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“…The exact characteristic equation of the full Hamiltonian for the combined atomphoton system is [8] The corresponding eigenvector, up to normalization but otherwise without approximation, is [5] (1)…”

Section: Hamiltonian Structure Of Quantum Decay; Decay Finalitymentioning

confidence: 99%

“…The extensive decoherence program [4] also deals with density matrices rather than single measurement trials, and as such is able to explain away interference terms, but is unable to describe the occurrence of individual outcomes. There have been attempts [5] to construe single quantum measurement trials as examples of spontaneous symmetry breaking, but their scenarios have so far been severely limited and the conclusions drawn by them are so far not backed up by a dynamical analysis. I myself formulated a resonance mechanism to explain the position-space Born rule in an idealized model of a particle detector composed of two-level atoms [6].…”

Section: Introductionmentioning

confidence: 99%

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The First Droplet in a Cloud Chamber Track

Schonfeld

2021

Found Phys

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In a cloud chamber, the quantum measurement problem amounts to explaining the first droplet in a charged-particle track; subsequent droplets are explained by Mott’s 1929 wave-theoretic argument about collision-induced wavefunction collimation. I formulate a mechanism for how the first droplet in a cloud chamber track arises, making no reference to quantum measurement axioms. I look specifically at tracks of charged particles emitted in the simplest slow decays, because I can reason about rather than guess the form that wave packets take. The first visible droplet occurs when a randomly occurring, barely-subcritical vapor droplet is pushed past criticality by ionization triggered by the faint wavefunction of the emitted charged particle. This is possible because potential energy incurred when an ionized vapor molecule polarizes the other molecules in a droplet can balance the excitation energy needed for the emitted charged particle to create the ion in the first place. This degeneracy is a singular condition for Coulombic scattering, leading to infinite or near-infinite ionization cross sections, and from there to an emergent Born rule in position space, but not an operator projection as in the projection postulate. Analogous mechanisms may explain canonical quantum measurement behavior in detectors such as ionization chambers, proportional counters, photomultiplier tubes or bubble chambers. This work is important because attempts to understand canonical quantum measurement behavior and its limitations have become urgent in view of worldwide investment in quantum computing and in searches for super-rare processes (e.g., proton decay).

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Section: Hamiltonian Structure Of Quantum Decay; Decay Finalitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The First Droplet in a Cloud Chamber Track

Schonfeld

2021

Found Phys

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“…By convention, we take e 1 such that σ 3 e 1 = e 1 , and σ 3 e 2 = −e 2 . It is not difficult to show that [25,26]…”

Section: Proofmentioning

confidence: 99%

“…A typical example of a mean-field quantum spin system is the Curie-Weiss model (see, for example,[1,6,11,[25][26][27] and references therein) 3. The same result holds for an arbitary unital C * -algebra B playing the role of the matrix algebra M k (C) in the above setting[14].…”

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confidence: 94%

Bulk-boundary asymptotic equivalence of two strict deformation quantizations

Moretti

Ven

2020

Lett Math Phys

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The existence of a strict deformation quantization of $$X_k=S(M_k({\mathbb {C}}))$$ X k = S ( M k ( C ) ) , the state space of the $$k\times k$$ k × k matrices $$M_k({\mathbb {C}})$$ M k ( C ) which is canonically a compact Poisson manifold (with stratified boundary), has recently been proved by both authors and Landsman (Rev Math Phys 32:2050031, 2020. 10.1142/S0129055X20500312). In fact, since increasing tensor powers of the $$k\times k$$ k × k matrices $$M_k({\mathbb {C}})$$ M k ( C ) are known to give rise to a continuous bundle of $$C^*$$ C ∗ -algebras over $$I=\{0\}\cup 1/\mathbb {N}\subset [0,1]$$ I = { 0 } ∪ 1 / N ⊂ [ 0 , 1 ] with fibers $$A_{1/N}=M_k({\mathbb {C}})^{\otimes N}$$ A 1 / N = M k ( C ) ⊗ N and $$A_0=C(X_k)$$ A 0 = C ( X k ) , we were able to define a strict deformation quantization of $$X_k$$ X k à la Rieffel, specified by quantization maps $$Q_{1/N}:/ \tilde{A}_0\rightarrow A_{1/N}$$ Q 1 / N : / A ~ 0 → A 1 / N , with $$\tilde{A}_0$$ A ~ 0 a dense Poisson subalgebra of $$A_0$$ A 0 . A similar result is known for the symplectic manifold $$S^2\subset \mathbb {R}^3$$ S 2 ⊂ R 3 , for which in this case the fibers $$A'_{1/N}=M_{N+1}(\mathbb {C})\cong B(\text {Sym}^N(\mathbb {C}^2))$$ A 1 / N ′ = M N + 1 ( C ) ≅ B ( Sym N ( C 2 ) ) and $$A_0'=C(S^2)$$ A 0 ′ = C ( S 2 ) form a continuous bundle of $$C^*$$ C ∗ -algebras over the same base space I, and where quantization is specified by (a priori different) quantization maps $$Q_{1/N}': \tilde{A}_0' \rightarrow A_{1/N}'$$ Q 1 / N ′ : A ~ 0 ′ → A 1 / N ′ . In this paper, we focus on the particular case $$X_2\cong B^3$$ X 2 ≅ B 3 (i.e., the unit three-ball in $$\mathbb {R}^3$$ R 3 ) and show that for any function $$f\in \tilde{A}_0$$ f ∈ A ~ 0 one has $$\lim _{N\rightarrow \infty }||(Q_{1/N}(f))|_{\text {Sym}^N(\mathbb {C}^2)}-Q_{1/N}'(f|_{_{S^2}})||_N=0$$ lim N → ∞ | | ( Q 1 / N ( f ) ) | Sym N ( C 2 ) - Q 1 / N ′ ( f | S 2 ) | | N = 0 , where $$\text {Sym}^N(\mathbb {C}^2)$$ Sym N ( C 2 ) denotes the symmetric subspace of $$(\mathbb {C}^2)^{N \otimes }$$ ( C 2 ) N ⊗ . Finally, we give an application regarding the (quantum) Curie–Weiss model.

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“…The general and common concept behind spontaneous symmetry breaking is based on the idea that if a collection of quantum particles becomes larger, the symmetry of the system as a whole becomes more unstable against small perturbations [52,50]. A similar statement can be made for quantum systems in their classical limit, where sensitivity against small perturbations now should be understood to hold in the relevant semi-classical regime, meaning that a certain parameter (e.g.…”

Section: Introductionmentioning

confidence: 99%

The classical limit and spontaneous symmetry breaking in algebraic quantum theory

Ven¹

2021

Preprint

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In this paper an overview on some recent developments on the classical limit and spontaneous symmetry breaking (SSB) in algebraic quantum theory is given. In such works, based on the theory of C *algebras, the concept of the classical limit has been formalised in a complete algebraic manner. Additionally, since this setting allows for commutative as well as non-commutative C * -algebras, and hence for classical and quantum theories, it provides an excellent framework to study SBB as an emergent phenomenon when transitioning from the quantum to the classical world by turning off a semi-classical parameter. We summarise the main results and show that this algebraic approach sheds new light on the connection between the classical and the quantum realm, where particular emphasis is placed on the role of SSB in Theory versus Nature. To this end a detailed analysis is carried out and illustrated with three different physical models: Schrödinger operators, mean-field quantum spin systems and the Bose-Hubbard model.

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Quantum spin systems versus Schroedinger operators: A case study in spontaneous symmetry breaking

Cited by 14 publications

References 45 publications

The First Droplet in a Cloud Chamber Track

The First Droplet in a Cloud Chamber Track

Bulk-boundary asymptotic equivalence of two strict deformation quantizations

The classical limit and spontaneous symmetry breaking in algebraic quantum theory

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