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DTSTAMP:20260424T063036Z
UID:1705418100@ist.ac.at
DTSTART:20240116T161500
DTEND:20240116T171500
DESCRIPTION:Speaker: Eugene Lytvynov\nhosted by Lorenzo Dello Schiavo\nAbst
 ract: A quasi-free state over the algebra of the canonical anticommutation
 s relations (CAR) is a state with respect to which the moments of the fiel
 d (Segal-type) operators are calculated similarly to the expectation of a 
 Gaussian random field (when one additionally takes into account the sign o
 f  a partition). An important subclass of quasi-free states is given by g
 auge-invariant states. For a given representation of the CAR\, one formall
 y defines its particle density as the product of the creation and annihila
 tion operators at point\, and again formally the smeared particle density 
 is a family of commuting Hermitian operators. For a class of quasi-free st
 ates\, we show that its particle density can be rigorously realised as a f
 amily of commuting self-adjoint operators and its joint spectral measure i
 s a determinantal point process\, i.e.\, a point process whose correlation
  functions are determinants built upon a correlation kernel $K(x\,y)$. In 
 the case of a gauge-invariant quasi-free state\, the correlation kernel $K
 (x\,y)$ is Hermitian. We also consider the particle-hole transformation in
  the continuum as a certain Bogoliubov transformation of a gauge-invariant
  quasi-free state\, which leads to a non-gauge-invariant quasi-free state.
  For the corresponding particle density\, the joint spectral measure is a 
 determinantal point process with a correlation kernel $K(x\,y)$ that is J-
 Hermitian. The latter means that the integral operator with integral kerne
 l $K(x\,y)$  is self-adjoint with respect to an indefinite inner product.
LOCATION:Heinzel Seminar Room (I21.EG.101)\, Office Building West\, ISTA\, 
 ISTA
ORGANIZER:birgit.oosthuizen-noczil@ist.ac.at 
SUMMARY:Eugene Lytvynov: Determinantal point processes and quasi-free state
 s on the CAR algebra 
URL:https://talks-calendar.ista.ac.at/events/4508
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