I would just like to correct a typographical error in a numerical factor of (`12.7) in the draft (click HERE for the previous draft). The following now goes with the right numerical factor mentioned and compare this to that in the uncorrected draft to see where the mistake is. (I haven’t corrected the said draft yet.)
(12.7)
^{3}}} e^{-i \vec{k}^{\prime} \cdot \vec{x}^{\prime}} \int \frac{d^{3} \vec{x}}{\sqrt{(2 \pi)^{3}}} e^{i \vec{k} \cdot \vec{x}} G_{x^{\prime} x} = \\ \\ \\ \frac{1}{2(2 \pi)} \int d k^{0} \frac{e^{i k^{0}\left(x^{\prime 0}-x^{0}\right)}}{\left(k^{0}\right)^{2}-\left(\vec{k} \cdot \vec{k} %2B M^{2} %2B i \epsilon\right)} \delta^{3}\left(\vec{k}^{\prime}-\vec{k}\right)_{\vec{x}^{\prime}} &bg=ffffff&s=2.5)
^{2}} G\left( x^{\prime} - x \right) &bg=ffffff&s=2.5)
(12.9)
}}{\left(k^{0}\right)^{2}-\left(\vec{k} \cdot \vec{k} %2B M^{2} %2B i \epsilon\right)}=\frac{i \pi}{\omega(\vec{k})} e^{i k^{0}\left(x^{\prime 0}-x^{0}\right)} &bg=ffffff&s=2.5)
at
(12.10)
 = \sqrt{\vec{k} \cdot \vec{k} %2B M^{2}} &bg=ffffff&s=2.5)
(13.1)
\right| \vec{k}\right\rangle\right|_{J=0}= \\ \\ \\ \exp \left( %2D \frac{i}{\hbar}\left(T_{\text {out}} %2D T_{\text {in}}\right) E_{0}^{0}\right) e^{i k^{\circ}\left(x^{\prime 0}%2D x^{0}\right)} \delta^{3}\left(\vec{k}^{\prime} %2D \vec{k}\right)_{\vec{x}^{\prime}} &bg=ffffff&s=2.5)
In this workout I am not concerning myself yet about time ordering in the field operators involved in the calculations.
This is just to post here the last updating of the draft with a title One Scalar Particle Scattering Into One Scalar Particle. With this updating this workout number 1 is brought to its tentative conclusion but there is still room for necessary improvement.
Or click https://www.slideshare.net/alfredie44/one-particle-toonepartlcescatteringsqrdcpy1 to view the draft at my slideshare.
Besides continuing some workouts in matching up wave solutions involved in elementary approaches to Hawking radiation, I also spent half of the day reviewing my notes that contain some basic workouts in scalar-to-scalar scattering and particularly, my workouts in which I have set up a scattering of one scalar particle into supposedly two scalar particles.
In quantum field theory set up, this is given starting with raising a one-particle state of spatial momentum from the vacuum state through a raising or creation operator. Note that since we are dealing with scalar fields, our raising and annihilation operators are those for bosonic fields such as scalar fields (bosons of spin zero).
We may then think of this state as the initial state and then evolve this state into another one particle state with the application of a unitary operator, the time evolution operator. What comes next is the projection of this evolved one particle state on a two-particle state of different spatial momenta. That is, forming a scattering matrix in which an initial one particle state of some initial spatial momentum scatters into a two-particle state with two final spatial momenta.
The result in calculating such scattering process is dependent on the presence of sources – setting these sources to zero would result into a null (zero) scattering matrix. For the scattering matrix to be non-zero, the sources are necessary since they provide the needed extra spatial momenta that must be carried away from these sources to the vertex where all spatial momenta are summed up.
Supposedly in the set up one scalar particle scatters into two scalar particles but at the scattering vertex four spatial momenta are to be summed up instead of just three. The fourth spatial momentum needed at the vertex is provided by a source at some space-time point and this momentum is carried by an additional scalar propagator.
The said scattering matrix can be expanded into large (infinite) number of terms as contained in the Euler (Taylor/Maclaurin) expansion and first of such terms depicts what has been mentioned above. This is shown in the picture.
}^{\prime}, \vec{k}^{\prime}\left|U\left(T_{\text {out}}, T_{i n}\right)\right| \vec{k}\right\rangle= 1s t %2B 2n d %2B \cdots &bg=ffffff&s=2.5)
^{3}} e^{ %2D i\left(\vec{k} \prime_{(1)} %2B \vec{k}^{\prime}\right) \cdot \vec{x}^{\prime}} \int \frac{d^{3} \vec{x}}{(2 \pi)^{3}} e^{i \vec{k} \cdot \vec{x}} \int d^{4} y \frac{G\left(x^{\prime} %2D y\right)}{(2 \pi)^{2}} J(y) \frac{G\left(x^{\prime} %2D x\right)}{(2 \pi)^{2}}
&bg=ffffff&s=2.5)
In the picture, there are two Green’s functions ( &bg=ffffff&s=0)
 &bg=ffffff&s=0)
In the picture it is to be noticed that at the single final spatial point where two propagators converge, we are to integrate over this point or region and this implies a Dirac-delta function at this point or region where we are to sum up the said four spatial momenta.
}^{\prime} %2B \vec{k}^{\prime}%2B \vec{P}%2D \vec{P}_{(1)}^{\prime}\right)_{\vec{x}^{\prime}}=\int \frac{d^{3} \vec{x}^{\prime}}{(2 \pi)^{3}} e^{ %2D i\left(\vec{k}_{(1)} ^{\prime} %2B \vec{k}^{\prime} %2B \vec{P} %2D \vec{P}_{(1)}^{\prime}\right) \cdot \vec{x}^{\prime}} &bg=ffffff&s=2.5)
Here are my workout notes presenting my practice calculations on scattering processes involving scalar fields. What I have done in this presently posted draft is to consider one scalar particle scattering into one scalar particle and such scattering process has a corresponding scattering matrix.
This is just an initial draft and there might be typos and even errors in concept and calculations lurking in the draft. So the readers especially those who are still new in this same subject area are encouraged to do some checking or workouts of their own and compare the details and results to those in the draft.
Here is one video I am sharing here. This is from Prof. Flournoy’s lectures dealing on some calculations in QED.
Early this morning, operators of the CERN Control Centre turned off the LHC, ending the very successful 2nd run of the world’s most powerful particle accelerator.
Scientists will now begin #upgradingLHC, CERN's accelerator complex and the experiments.
➡️ https://t.co/wry48RsrAr pic.twitter.com/EEvvOGT5Jg
— CERN (@CERN) December 3, 2018
The greater task now to follow from (21.22) is to carry out the indicated differentiations
(21.23)
})} \right ) \left ( \displaystyle \prod_{i=0}^{2} \langle J_{x} G_{xy}J_{y} \rangle_{i} \right ) &bg=ffffff&s=2.5)
… so why must the Kaluza-Klein scheme fail, I’ve just learned that through this video.
… for the meanwhile some updates from CERN
The #LHC has begun colliding heavy ions and CMS has started the heavy-ion data taking run. What are we going to study with these collisions? https://t.co/2EuLciq36x pic.twitter.com/jxDlBkCN5a
— CMS Experiment CERN (@CMSexperiment) November 9, 2018
The submission of this paper has been cancelled nevertheless, this is still to be accomplished.
^{3} \hbar}{i} e^{ %2D \frac{i}{\hbar}\left(T_{o u t} %2D T_{i n}\right) E_{0}^{0}} \frac{\sqrt{2 \omega\left(\vec{k} \prime_{(1)}\right.} )}{\sqrt{\hbar}} \frac{\sqrt{2 \omega\left(\vec{k}^{\prime}\right)}}{\sqrt{\hbar}} \frac{\sqrt{2 \omega(\vec{k})}}{\sqrt{\hbar}} \int \frac{d V\left(\vec{x}^{\prime}\right)}{\sqrt{(2 \pi)^{3}}} \times &bg=ffffff&s=2.5)
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