Quasi-Equivalence of Gaussian States and Energy Estimates for Functions of Modular Hamiltonians

Abstract To compare two Gaussian states of the Weyl-CCR algebra of a free scalar QFT we study three closely related perspectives: (i) quasi-equivalence of the GNS-representations, (ii) differences of the total energy (on some Cauchy surface), and (iii) differences between functions of the modular Hamiltonians. (For perspective (ii) we will only consider real linear free scalar quantum fields on ultrastatic spacetimes.) These three perspectives are known to be related qualitatively, due to work of Araki and Yamagami, Verch and Longo. Our aim is to investigate quantitative relations, including in particular estimates of differences between functions of modular Hamiltonians in terms of energy differences. E.g., for a suitable class of perturbations of the Minkowski vacuum state of a massive free scalar field, which have a positive energy density and a finite total energy E on some inertial time slice, the modular Hamiltonian K satisfies $$\Big \Vert \frac{1}{\cosh \left( \frac{K}{2}\right) }\Big \Vert _{\text {HS}}^2\le 8\frac{E}{m}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mo>‖</mml:mo> </mml:mrow> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>cosh</mml:mo> <mml:mfenced> <mml:mfrac> <mml:mi>K</mml:mi> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mfenced> </mml:mrow> </mml:mfrac> <mml:msubsup> <mml:mrow> <mml:mo>‖</mml:mo> </mml:mrow> <mml:mrow> <mml:mtext>HS</mml:mtext> </mml:mrow> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mo>≤</mml:mo> <mml:mn>8</mml:mn> <mml:mfrac> <mml:mi>E</mml:mi> <mml:mi>m</mml:mi> </mml:mfrac> </mml:mrow> </mml:math> .