A KMS-like state of Hadamard type on Robertson-Walker spacetimes and its time evolution.

*(English)*Zbl 0937.83052A quantum field propagating in a generic space-time \((M,g)\) admits, due to the lack of a non-trivial isometry group of \((M,g)\), no distinguished vacuum state. A partial solution to this problem is given in the free field case by the “Hadamard condition” which distinguishes not a single state but at least a local folium (=quasiequivalence class) of “physically relevant” states. In some special cases however there are alternative approaches possible and useful. E.g. in Robertson-Walker space-times we can consider the class of “adiabatic vacuum states” which was originally introduced by L. Parker [Quantized fields and particle creation in expanding universes. I, Phys. Rev., II. Ser. 183, 1057-1068 (1969; Zbl 0186.58603)] to get an optimal definition of physical particles.

To discuss questions about quantum statistical mechanics in an expanding universe, the author considers in this paper quasi-free states on the CCR-algebra of the Klein-Gordon equation (in a Robertson Walker space-times) and combines the definitions of adiabatic vacua and quasifree KMS states to get “adiabatic KMS-states”, which are interpreted physically as approximations to thermal equilibrium states. In this context it is shown that an adiabatic KMS state reduces to an adiabatic vacuum state if the temperature becomes zero, and to a (quasi free) KMS state if the space-time is static. The paper follows at this point closely previous work of C. Lueders and J. E. Roberts [Local quasiequivalence and adiabatic vacuum states. Commun. Math. Phys. 134, No. 1, 29-63 (1990; Zbl 0749.46045)] and W. Junker [Hadamard states, adiabatic vacua and the construction of physical states for scalar quantum fields on curved spacetime. Rev. Math. Phys. 8, No. 8, 1091-1159 (1996; Zbl 0869.53053)].

In the rest of the paper the author considers some non-trivial properties of the new class of states. First he discusses the question whether an adiabatic KMS state is a Hadamard state or not. The proof of the corresponding theorem suffers however from a subtle error in the work of Junker (loc. cit.). Therefore adiabatic vacua are in contrast to the statement of the author not Hadamard states but they belong only to the corresponding local folium. Fortunately this is sufficient to consider adiabatic KMS states as physically relevant states.

The last topic considered in this paper is the time-evolution of adiabatic KMS states. The author claims that the time-evolution derived from the Klein-Gordon equation maps adiabatic KMS states to adiabatic KMS, such that the inverse temperature associated to the states is proportional to the scale parameter in the metric. This statement is based on an analysis of the Klein-Gordon dynamics on the classical phase space which contains unfortunately another, more severe error. The above mentioned statement about time evolution can therefore only be considered as a conjecture. This is quite unfortunate, since the physical idea behind the definition of adiabatic KMS states strongly suggests such a behavior.

Apart from the technical problems the paper suffers from, this is an interesting work containing some new ideas which are worth to be discussed further.

To discuss questions about quantum statistical mechanics in an expanding universe, the author considers in this paper quasi-free states on the CCR-algebra of the Klein-Gordon equation (in a Robertson Walker space-times) and combines the definitions of adiabatic vacua and quasifree KMS states to get “adiabatic KMS-states”, which are interpreted physically as approximations to thermal equilibrium states. In this context it is shown that an adiabatic KMS state reduces to an adiabatic vacuum state if the temperature becomes zero, and to a (quasi free) KMS state if the space-time is static. The paper follows at this point closely previous work of C. Lueders and J. E. Roberts [Local quasiequivalence and adiabatic vacuum states. Commun. Math. Phys. 134, No. 1, 29-63 (1990; Zbl 0749.46045)] and W. Junker [Hadamard states, adiabatic vacua and the construction of physical states for scalar quantum fields on curved spacetime. Rev. Math. Phys. 8, No. 8, 1091-1159 (1996; Zbl 0869.53053)].

In the rest of the paper the author considers some non-trivial properties of the new class of states. First he discusses the question whether an adiabatic KMS state is a Hadamard state or not. The proof of the corresponding theorem suffers however from a subtle error in the work of Junker (loc. cit.). Therefore adiabatic vacua are in contrast to the statement of the author not Hadamard states but they belong only to the corresponding local folium. Fortunately this is sufficient to consider adiabatic KMS states as physically relevant states.

The last topic considered in this paper is the time-evolution of adiabatic KMS states. The author claims that the time-evolution derived from the Klein-Gordon equation maps adiabatic KMS states to adiabatic KMS, such that the inverse temperature associated to the states is proportional to the scale parameter in the metric. This statement is based on an analysis of the Klein-Gordon dynamics on the classical phase space which contains unfortunately another, more severe error. The above mentioned statement about time evolution can therefore only be considered as a conjecture. This is quite unfortunate, since the physical idea behind the definition of adiabatic KMS states strongly suggests such a behavior.

Apart from the technical problems the paper suffers from, this is an interesting work containing some new ideas which are worth to be discussed further.

Reviewer: Michael Keyl (Braunschweig)

##### MSC:

83F05 | Cosmology |

81T20 | Quantum field theory on curved space or space-time backgrounds |

47L90 | Applications of operator algebras to the sciences |

58J45 | Hyperbolic equations on manifolds |