The closed-time path formalism acts on the complex-time plane in such a way that the contour goes along the real axis and then returns. For the real-time formalism there are two approaches: the closed-time path formalism and Thermofield Dynamics (TFD) formalism. In this formalism, the propagators have a non-trivial dependence on temperature. For example, Matsubara approach is an imaginary-time formalism, which is based on replacing the time, t, with a complex time, \(i\tau \). These formalisms are based on ideas that use real-time or imaginary-time transformations as a basis. To discuss quantum field theory at finite temperature, in particular QED theory, there are some formalisms that introduce these thermal effects into the usual zero-temperature theories. Thus, it is necessary and interesting to analyze the Compton scattering subject to finite temperature, looking at the consequences and for the asymptotic cases. Furthermore, it is inherent to the environment that temperature and, consequently, thermal effects on physical systems are fundamental and natural characteristics presents in any real physical process. To make a more accurate and complete description of this scattering process, it is necessary to use some QED concepts, starting from the theory of the Standard Model. However, despite being a topic discussed in all studies of basic physics, its formulation is made, in almost all cases, from the classical fundamentals to finite temperature. In summary, the higher the energy the less accurate is the determination of the initial polarisation via Compton scattering and the optimal scattering angles diverge from 90°.Compton scattering is a very important process with regard to Quantum Electrodynamics (QED), mainly because it brings with it a historical fact: one of the particular phenomena that brought about the birth of quantum mechanics. in a scan over the azimuthal angle Φ the maxima and minima differ by 69%. This visibility is depending solely on the incoming energy k i of the photon and on the scattering angle \(\tilde,1)=0.69\), i.e. Firstly, the scattering of a single photon in a scintillator is formulated in terms of an envelope function and a term in front of the polarisation interference, an a priory visibility or interference contrast. In detail the Klein-Nishina formula 17 is reformulated in the open quantum formalism. If this step is taken, observables sensitive to entanglement may become visible in living beings along with all the well-known benefits of a standard PET-scan. This paper shows how the entanglement can be witnessed and provides a concise quantum information theoretic framework for describing high energetic photons undergoing Compton scattering processes. The new prototype J-PET ( Jagelonian- Positron- Emission Tomograph) 5, 6, 7, 8, 9, 10 is based on plastic scintillators 11, 12 that shall be a key technology of a new generation of low cost and total-body scan PETs and, in addition, has shown in providing all key elements to detect the positronium 13, 14, 15 and the Compton-scattered gammas 16. However, the theoretically predicted entanglement in those gammas has never been observed, because the energies are around the mass of an electron (511 keV) and for such high energetic photons standard optical polarizers do not work. One such may be based on detecting cancer via the various types of entanglement manifesting in the two- or three-photon states of the decay process of positronium 1, 2, 3, 4, a bound state of an electron and its antiparticle. Moreover, new technologies based on entanglement are currently emerging. No doubt manifestations of entanglement are fascinating phenomena that have been witnessed for numerous physical systems at low and high energies.
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