In this paper the connection between quantum field theories on flat noncommutative space(-times) in Euclidean and Lorentzian signature is studied for the case that time is still commutative. By making use of the algebraic framework of quantum field theory and an analytic continuation of the symmetry groups which are compatible with the structure of Moyal space, a general correspondence between field theories on Euclidean space satisfying a time zero condition and quantum field theories on Moyal Minkowski space is presented (“Wick rotation”). It is then shown that field theories transferred to Moyal space(-time) by Rieffel deformation and warped convolution fit into this framework, and that the processes of Wick rotation and deformation commute.

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In case that
$\alpha ^\mathcal {E}$
αE
is not strongly continuous, we could pass to a representation of the Euclidean net on a Euclidean Hilbert space
$\mathcal {H}^\mathcal {E}$
HE
as indicated in Sec. II. Making use of the warped convolution setting outlined above, we could then work with a unitary representation U of
$\mathbb {R}^d$
Rd
on
$\mathcal {H}^\mathcal {E}$
HE
and deformations of operators in von Neumann algebras. But for technical convenience, we stick to the assumption of strongly continuous
$\alpha ^\mathcal {E}$
αE
here.
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