×

On the adiabatic limit of Hadamard states. (English) Zbl 1374.81068

Summary: We consider the adiabatic limit of Hadamard states for free quantum Klein-Gordon fields, when the background metric and the field mass are slowly varied from their initial to final values. If the Klein-Gordon field stays massive, we prove that the adiabatic limit of the initial vacuum state is the (final) vacuum state, by extending to the symplectic framework the adiabatic theorem of Avron-Seiler-Yaffe. In cases when only the field mass is varied, using an abstract version of the mode decomposition method we can also consider the case when the initial or final mass vanishes, and the initial state is either a thermal state or a more general Hadamard state.

MSC:

81T20 Quantum field theory on curved space or space-time backgrounds
35S05 Pseudodifferential operators as generalizations of partial differential operators
35S35 Topological aspects for pseudodifferential operators in context of PDEs: intersection cohomology, stratified sets, etc.
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Avron, J., Seiler, R., Yaffe, L.: Adiabatic theorem and applications to the quantum hall effect. Commun. Math. Phys. 110, 33-49 (1987) · Zbl 0626.58033
[2] Benini, M., Dappiaggi, C., Murro, S.: Radiative observables for linearized gravity on asymptotically at spacetimes and their boundary induced states. J. Math. Phys. 55, 082301 (2014) · Zbl 1298.83044
[3] Bär, C., Ginoux, N., Pfäffle, F.: Wave Equation on Lorentzian Manifolds and Quantization. ESI Lectures in Mathematics and Physics. EMS Publishing House, Zurich (2007) · Zbl 1118.58016
[4] Cheeger, J., Gromov, M.: Bounds on the von Neumann dimension of \[L^2\] L2-cohomology and the Gauss-Bonnet theorem for open manifolds. J. Differ. Geom. 21, 1-34 (1985) · Zbl 0614.53034
[5] Chernoff, P.: Essential self-adjointness of powers of generators of hyperbolic equations. J. Funct. Anal. 12, 401-414 (1973) · Zbl 0263.35066
[6] Dappiaggi, C., Drago N.: Constructing Hadamard states via an extended Möller operator, arXiv:1506.09122 (2015) to appear in Lett. Math. Phys. 106 (2016) · Zbl 1362.81073
[7] Derezinski, J., Gérard, C.: Scattering Theory of Classical and Quantum N-particle Systems, Texts and Monographs in Physics. Springer, Berlin (1997) · Zbl 0899.47007
[8] Drago N., Faldino F., Pinamonti N.: On the stability of KMS states in perturbative algebraic quantum field theories. arXiv:1609.01124 · Zbl 1386.81111
[9] Drago N., Hack,T.-P., Pinamonti N.: The generalized principle of perturbative agreement and the thermal mass. http://arxiv.org/abs/1502.02705 · Zbl 1362.81064
[10] Dappiaggi, C., Moretti, V., Pinamonti, N.: Rigorous steps towards holography in asymptotically at spacetimes. Rev. Math. Phys. 18, 349-416 (2006) · Zbl 1107.81040
[11] Finster, F., Murro, S., Röken, C.: The fermionic projector in a time-dependent external Potential: mass oscillation property and hadamard states. Preprint arXiv:1501.05522 [math-ph] (2015) · Zbl 1345.81033
[12] Fulling, S.A., Narcowich, F.J., Wald, R.M.: Singularity structure of the two-point function in quantum field theory in curved spacetime. Commun. Math. Phys. 3, 257-264 (1978) · Zbl 0401.35065
[13] Fulling, S.A., Narcowich, F.J., Wald, R.M.: Singularity structure of the two-point function in quantum field theory in curved spacetime, II. Ann. Phys. 136, 243-272 (1981) · Zbl 0495.35054
[14] Fewster, C.J., Verch, R.: The necessity of the Hadamard condition. Class. Quantum Gravity 30, 235027 (2013) · Zbl 1284.83057
[15] Gonnella, G., Kay, B.S.: Can locally hadamard quantum states have nonlocal singularities? Class. Quantum Gravity 6, 1445 (1989) · Zbl 0678.53082
[16] Gérard, C., Wrochna, M.: Construction of Hadamard states by pseudodifferential calculus. Commun. Math. Phys. 325, 713-755 (2013) · Zbl 1298.81214
[17] Gérard, C., Wrochna, M.: Construction of Hadamard states by characteristic Cauchy problem. Anal. PDE 9, 111-149 (2016) · Zbl 1334.83020
[18] Gérard, C., Oulghazi, O., Wrochna, M.: Hadamard states for the Klein-Gordon equation on Lorentzian manifolds of bounded geometry. Preprint arXiv:1602.00930 [math-ph] (2016) · Zbl 1364.35362
[19] Georgescu, V., Gérard, C., Häfner, D.: Resolvent and propagation estimates for Klein-Gordon equations with non-positive energy. J. Spectr. Theory 5, 113-192 (2015) · Zbl 1326.35212
[20] Hollands, S., R.M, Wald: Existence of local covariant time ordered products of quantum fields in curved space-time. Commun. Math. Phys. 231, 309-345 (2002) · Zbl 1015.81043
[21] Kato, T.: Perturbation Theory for Linear Operators, Classics in Mathematics. Springer, Berlin (1995) · Zbl 0836.47009
[22] Khavkine, I., Moretti V.: Algebraic QFT in curved spacetime and quasi-free Hadamard states: an introduction. arXiv:1412.5945 (2014) · Zbl 1334.81081
[23] Radzikowski, M.J.: Micro-local approach to the Hadamard condition in quantum field theory on curved space-time. Commun. Math. Phys. 179, 529-553 (1996) · Zbl 0858.53055
[24] Radzikowski, M.J.: A local to global singularity theorem for quantum field theory on curved space-time. Commun. Math. Phys. 180, 1-22 (1996) · Zbl 0874.58079
[25] Reed, M., Simon, B.: Methods of Modern Mathematical Physics II: Fourier Analysis, Self-adjointness. Academic Press, Cambridge (1975) · Zbl 0308.47002
[26] Roe, J.: An index theorem on open manifolds I. J. Differ. Geom. 27, 87-113 (1988) · Zbl 0657.58041
[27] Sanders, K.: Thermal equilibrium states of a linear scalar quantum field in stationary spacetimes. Int. J. Mod. Phys. A 28, 1330010 (2013) · Zbl 1268.82005
[28] Schmid, J., Griesemer, M.: Kato’s theorem on the integration of non-autonomous linear evolution equations. Math. Phys. Anal. Geom. 17, 265-271 (2014) · Zbl 1309.34118
[29] Wald, R.M.: Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics, Chicago Lectures in Physics. University of Chicago Press, Chicago (1994) · Zbl 0842.53052
[30] Wrochna M., Zahn J.: Classical phase space and Hadamard states in the BRST formalism for gauge field theories on curved spacetime. arXiv:1407.8079 [math-ph] · Zbl 1447.81165
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.