infinitesimal reflection

[AntonioLao]

It has been proved in plane geometry that a clockwise (CW) 2-axis reflection of a radius vector is equivalent to a counterclockwise (CCW) continuous rotation about the axial intersection. Moreover, the angle of rotation is twice the vertical angle of intersection in the quadrant where the radius vector is located. For orthogonal systems, where the vertical angles are all 90°, the angles of rotation are all 180°. Consequently, orthogonal systems represent perfect symmetry for both transformation of discrete reflections and continuous rotations where and when CW and CCW direction are indistinguishable.

On the other hand, for infinitesimal vertical angles, the positive angles of CW infinitesimal continuous rotation are equivalent to twice the negative angles of CCW infinitesimal quantized reflection. If the axis for a particular infinitesimal rotation is taken to be the time axis then infinitesimal quantized reflections can only be defined if and only if there are orthogonal systems of spatial axes, or at the least two of these spatial axes must be orthogonal. Continuous groups of infinitesimal rotations are called Lie groups of all gauge invariance symmetries. These are used for QED as U(1) symmetry, for electroweak as the product of U(1) and SU(2) symmetry, and for QCD as SU(3) symmetry. However, all these gauge transformations depend only on one direction of infinitesimal time reflection (T), which indicates a broken chiral symmetry for parity (P), broken charge conjugation symmetry (C), and broken CP symmetry. All these were verified by previous experiments in high energy interactions of elementary particles. Nevertheless, CPT invariance symmetry would require a doubly orthogonal infinitesimal time reflections between positive past and negative future or negative past and positive future where and when the time axes of past and future must be always orthogonal.

[Profpat]

How about a positive/negative past and a positive/negative future and screw the CPT symmetry, and we'll decide if the now is positive or negative.

[AntonioLao]

How about a positive/negative past and a positive/negative future and screw the CPT symmetry

You are correct, without the former, the latter is meaningless.

[Profpat]

Good let's hope we both have a wonderful and positive future.

[AntonioLao]

it all depends on how we choose the next time we reach a fork on the road specially ones with infinite prongs of possibility.

[mkirkpatrick]

We are all on the rıght road for each of us.may we meet up now and then prehaps here
on the forum and reflect a spell.If these words come out odd lookıng ıt ıs because I am usıng a Turkısh keyboard whıch ıs dıfferent to the usual one I use.

regards mıchael