Abstract
In this short note we study Spin-Boson Models from the Quasi-Classical standpoint. In the Quasi-Classical limit, the field becomes macroscopic while the particles it interacts with, they remain quantum. As a result, the field becomes a classical environment that drives the particle system with an explicit effective dynamics.
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Notes
- 1.
We use a somewhat unorthodox notation for the second quantization functor. We denote by \(\mathcal {G}_{\varepsilon }^{\mathrm {s}}(\mathfrak {h})=\bigoplus _{n\in \mathbb {N}} \mathfrak {h}_n\) the symmetric Fock space over \(\mathfrak {h}\) in which the canonical creation and annihilation operators have \(\varepsilon \)-dependent commutation relations:
$$\displaystyle \begin{aligned} [a_{\varepsilon}(f),a_{\varepsilon}^{*}(g)]= \varepsilon\langle f , g \rangle_{\mathfrak{h}};. \end{aligned}$$The second quantization of an operator A on \(\mathfrak {h}\) is written thus as
$$\displaystyle \begin{aligned} \mathrm{d}\mathcal{G}_{\varepsilon}(A)= \sum_{i,j=0}^{\infty}A_{ij}a^{*}_{\varepsilon,i}a_{\varepsilon,j}\;, \end{aligned}$$with \(A_{ij}=\langle e_i , A e_j \rangle _{\mathfrak {h}}\), and \(a^{\sharp }_{\varepsilon ,k}=a_{\varepsilon }^{\sharp }(e_k)\), with \(\{e_k\}_{k\in \mathbb {N}}\) an O.N.B. of \(\mathfrak {h}\). The quasi-classical parameter \(\varepsilon \) clearly plays the role of a semiclassical parameter for the (Segal) field \(\varphi _{\varepsilon }(f)= a^{*}_{\varepsilon }(f)+ a_{\varepsilon }(f)\): as \(\varepsilon \to 0\), the field becomes a classical commutative observable [see 15, for a gentler and more detailed introduction to the quasi-classical scaling], and [11,12,13,14] for other recent papers concerning the quasi-classical regime.
- 2.
We denote by \(\mathcal {C}_0^{\infty }\big (\mathrm {d}\mathcal {G}_{\varepsilon }(1)\big )\) the Fock space vectors with a finite number of particles (i.e., for which the k-particle components are all zero for \(k> \underline {k}\), for some \( \underline {k}\in \mathbb {N}\)).
- 3.
A cylindrical measure is a finitely additive measure that is \(\sigma \)-additive on any subalgebra of cylinders generated by a finite number of vectors.
- 4.
\(\mathfrak {U}_{t,s}(z)\) is the unique solution of \(i\partial _t\mathfrak {U}_{t,s}(z) = \mathfrak {H}(z) \mathfrak {U}_{t,s}(z)\) and \(\mathfrak {U}_{t,t}(z)=\mathbf 1\).
- 5.
We restrict our attention only to the limits \(\nu =1\) and \(\nu =0\), since they encode all different and well-defined outcomes that one could obtain for the effective dynamics. In fact, every choice of \(\nu (\varepsilon )\) such that \(\lim _{\varepsilon \to 0}\varepsilon \nu (\varepsilon )= \lambda >0\) would amount in a rescaling of the field dispersion relation, while any choice such that either \(\lim _{\varepsilon \to 0}\varepsilon \nu (\varepsilon )= \lambda = \infty \) or such that the limit does not exist would prevent an explicit definition of the effective dynamics in the limit \(\varepsilon \to 0\).
- 6.
We plan to investigate the non-Markovian character of the quasi-classical effective dynamics in an upcoming paper.
- 7.
If \(\mathfrak {S}\) has compact resolvent (and it is bounded from below), \(\mathfrak {A}=\mathfrak {S} + \lvert \inf \sigma (\mathfrak {S}) \rvert _{ }^{}+1\) would be a natural choice, and the associated condition (2) would mean that mass is not lost if one restricts to states with \(\varepsilon \)-uniformly-bounded Spin kinetic energy.
- 8.
The scalar convergence \(\xi _{\varepsilon }\underset {\varepsilon \to 0}{\longrightarrow } \mu \) is perfectly analogous to the quasi-classical one, and could be seen as a particular case of it where the additional degrees of freedom are trivial. Let us remark again that for the scalar case—and thus also for the unentangled quasi-classical states considered here – (1) is sufficient to guarantee that \(\mu (\mathfrak {h})=1\).
- 9.
Quasi-classical convergence is the pointwise convergence of Fourier transforms in weak-* topology, i.e. when tested with compact operators. Since \(\mathfrak {sk}_{r+}\) is compact, we have pointwise convergence of \(\hat {\Upsilon }_{\varepsilon }(\tau )\) traced together with \(\mathfrak {sk}_{r+}\).
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Correggi, M., Falconi, M., Merkli, M. (2023). Quasi-Classical Spin Boson Models. In: Correggi, M., Falconi, M. (eds) Quantum Mathematics I. INdAM 2022. Springer INdAM Series, vol 57. Springer, Singapore. https://doi.org/10.1007/978-981-99-5894-8_3
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