Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Playing with functions of positive type, classical and quantum   A function of positive type can be defined as a positive functional on a
convolution algebra of a locally compact group. In the case where the group is
abelian, by Bochner's theorem a function of positive type is, up to
normalization, the Fourier transform of a probability measure. Therefore,
considering the group of translations on phase space, a suitably normalized
phase-space function of positive type can be regarded as a realization of a
classical state. Thus, it may be called a function of classical positive type.
Replacing the ordinary convolution on phase space with the twisted convolution,
one obtains a noncommutative algebra of functions whose positive functionals we
may call functions of quantum positive type. In fact, by a quantum version of
Bochner's theorem, a continuous function of quantum positive type is, up to
normalization, the (symplectic) Fourier transform of a Wigner quasi-probability
distribution; hence, it can be regarded as a phase-space realization of a
quantum state. Playing with functions of positive type, classical and quantum,
one is led in a natural way to consider a class of semigroups of operators, the
classical-quantum semigroups. The physical meaning of these mathematical
objects is unveiled via quantization, so obtaining a class of quantum dynamical
semigroups that, borrowing terminology from quantum information science, may be
called classical-noise semigroups.

21
The categories of this paper are: adap-org
4
The categories of this paper are: cond-mat
34
The categories of this paper are: chem-ph
17
The categories of this paper are: q-fin
27
The categories of this paper are: comp-gas
16
The categories of this paper are: stat
33
The categories of this paper are: plasm-ph
1
The categories of this paper are: cs
22
The categories of this paper are: chao-dyn
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The categories of this paper are: eess
32
The categories of this paper are: atom-ph
3
The categories of this paper are: physics
12
The categories of this paper are: nucl-th
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The categories of this paper are: hep-th
9
The categories of this paper are: nlin
37
The categories of this paper are: supr-con