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Dec 27, 2021, 12:50:55 AM12/27/21


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The CPT theorem appeared for the first time, implicitly, in the work of Julian Schwinger in 1951 to prove the connection between spin and statistics.[3] In 1954, Gerhart Lüders and Wolfgang Pauli derived more explicit proofs,[4][5] so this theorem is sometimes known as the Lüders–Pauli theorem. At about the same time, and independently, this theorem was also proved by John Stewart Bell.[6] These proofs are based on the principle of Lorentz invariance and the principle of locality in the interaction of quantum fields. Subsequently, Res Jost gave a more general proof in the framework of axiomatic quantum field theory.

Efforts during the late 1950s revealed the violation of P-symmetry by phenomena that involve the weak force, and there were well-known violations of C-symmetry as well. For a short time, the CP-symmetry was believed to be preserved by all physical phenomena, but in the 1960s that was later found to be false too, which implied, by CPT invariance, violations of T-symmetry as well.

Derivation of the CPT theorem
Consider a Lorentz boost in a fixed direction z. This can be interpreted as a rotation of the time axis into the z axis, with an imaginary rotation parameter. If this rotation parameter were real, it would be possible for a 180° rotation to reverse the direction of time and of z. Reversing the direction of one axis is a reflection of space in any number of dimensions. If space has 3 dimensions, it is equivalent to reflecting all the coordinates, because an additional rotation of 180° in the x-y plane could be included.

This defines a CPT transformation if we adopt the Feynman–Stueckelberg interpretation of antiparticles as the corresponding particles traveling backwards in time. This interpretation requires a slight analytic continuation, which is well-defined only under the following assumptions:

The theory is Lorentz invariant;
The vacuum is Lorentz invariant;
The energy is bounded below.
When the above hold, quantum theory can be extended to a Euclidean theory, defined by translating all the operators to imaginary time using the Hamiltonian. The commutation relations of the Hamiltonian, and the Lorentz generators, guarantee that Lorentz invariance implies rotational invariance, so that any state can be rotated by 180 degrees.

Since a sequence of two CPT reflections is equivalent to a 360-degree rotation, fermions change by a sign under two CPT reflections, while bosons do not. This fact can be used to prove the spin-statistics theorem.

Consequences and implications
The implication of CPT symmetry is that a "mirror-image" of our universe — with all objects having their positions reflected through an arbitrary point (corresponding to a parity inversion), all momenta reversed (corresponding to a time inversion) and with all matter replaced by antimatter (corresponding to a charge inversion) — would evolve under exactly our physical laws. The CPT transformation turns our universe into its "mirror image" and vice versa. CPT symmetry is recognized to be a fundamental property of physical laws.

In order to preserve this symmetry, every violation of the combined symmetry of two of its components (such as CP) must have a corresponding violation in the third component (such as T); in fact, mathematically, these are the same thing. Thus violations in T symmetry are often referred to as CP violations.

The CPT theorem can be generalized to take into account pin groups.

In 2002 Oscar Greenberg published an apparent proof that CPT violation implies the breaking of Lorentz symmetry.[7] If correct, this would imply that any study of CPT violation also includes Lorentz violation. However, Chaichian et al later disputed the validity of Greenberg's result.[8] Greenberg replied that the model used in their paper meant that their "proposed objection was not relevant to my result".[9]

The overwhelming majority of experimental searches for Lorentz violation have yielded negative results. A detailed tabulation of these results was given in 2011 by Kostelecky and Russell
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Thermodynamics of Hell
A thermodynamics professor had written a take home exam for his graduate students. It had one question:

"Is hell exothermic or endothermic? Support your answer with a proof."
Most of the students wrote proofs of their beliefs using Boyle's Law or some variant. One student, however, wrote the following:
First, we postulate that if souls exist, then they must have some mass. If they do, then a mole of souls can also have a mass. So, at what rate are souls moving into hell and at what rate are souls leaving?

I think that we can safely assume that once a soul gets to hell, it will not leave. Therefore, no souls are leaving.

As for souls entering hell, lets look at the different religions that exist in the world today. Some of these religions state that if you are not a member of their religion, you will go to hell. Since, there are more than one of these religions and people do not belong to more than one religion, we can project that all people and all souls go to hell.

With birth and death rates as they are, we can expect the number of souls in hell to increase exponentially.

Now, we look at the rate of change in volume in hell. Boyle's Law states that in order for the temperature and pressure in hell to stay the same, the ratio of the mass of souls and volume needs to stay constant.

Thus, there are two possibilities:

If hell is expanding at a slower rate than the rate at which souls enter hell, then the temperature and pressure in hell will increase until all hell breaks loose.
If hell is expanding at a rate faster than the increase of souls in hell, than the temperature and pressure will drop until hell freezes over.
If we accept the postulate given to me by a young lady during my first year, "It will be a cold night in Hell before I sleep with you", then number 2 above is not true and so Hell is exothermic.

A Response from a Professor of Classics at U. Iowa
Now, in Dante's description, upper Hell has regions of intense heat/falling flame, etc, while the very bottom is in fact a frozen lake. This would suggest that while upper hell is largely static, lower hell is rapidly expanding and/or accommodates fewer souls. Since the bottom contains Satan and the worst sinners, the latter explanation (fewer souls) would seem to apply, whereas simply being in the wrong religion would be accommodated above, hence the more intense heat.
The student is clearly right.
Robert Ketterer
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Matter and antimatter: the two arrows of time
Massimo Villata
Antiparticles may be interpreted as ordinary particles travelling backwards in time and the two descriptions are considered equivalent, at least in special relativity and relativistic quantum mechanics. It is suggested that, vice versa, the discovery of antimatter should be the confirmation that our world is "endowed" with two opposite time-arrows and such a description could be more useful and convenient from the point of view of the understanding of the world itself, at least for a simple reason: whenever phenomena are observed from a "reference frame" from which the world appears more symmetric, it is easier to understand the physical laws which regulate it. If, in the future, it is possible to discover how a macroscopic system of antimatter behaves, it will be also possible to confirm (or not) the "reality" of the two arrows of time.
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