Thermal Equilibrium Distribution in Infinite-Dimensional Hilbert Spaces
| Autor: | Roderich Tumulka |
|---|---|
| Rok vydání: | 2020 |
| Předmět: |
Unit sphere
Quantum Physics Weak convergence 010308 nuclear & particles physics Operator (physics) Mathematical analysis Hilbert space FOS: Physical sciences Statistical and Nonlinear Physics Mathematical Physics (math-ph) Gaussian measure 01 natural sciences Measure (mathematics) symbols.namesake 0103 physical sciences symbols 010307 mathematical physics Uniqueness Quantum Physics (quant-ph) Mathematical Physics Probability measure Mathematics |
| Zdroj: | Reports on Mathematical Physics. 86:303-313 |
| ISSN: | 0034-4877 |
| DOI: | 10.1016/s0034-4877(20)30085-9 |
| Popis: | The thermal equilibrium distribution over quantum-mechanical wave functions is a so-called Gaussian adjusted projected (GAP) measure, $GAP(\rho_\beta)$, for a thermal density operator $\rho_\beta$ at inverse temperature $\beta$. More generally, $GAP(\rho)$ is a probability measure on the unit sphere in Hilbert space for any density operator $\rho$ (i.e., a positive operator with trace 1). In this note, we collect the mathematical details concerning the rigorous definition of $GAP(\rho)$ in infinite-dimensional separable Hilbert spaces. Its existence and uniqueness follows from Prohorov's theorem on the existence and uniqueness of Gaussian measures in Hilbert spaces with given mean and covariance. We also give an alternative existence proof. Finally, we give a proof that $GAP(\rho)$ depends continuously on $\rho$ in the sense that convergence of $\rho$ in the trace norm implies weak convergence of $GAP(\rho)$. Comment: 12 pages LaTeX, no figures |
| Databáze: | OpenAIRE |
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