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DTSTAMP:20240806T022317Z
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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