Decades ago, Wolfgang Pauli has proved the CPT-theorem. G. Lüders and J. Bell did the same thing almost independently in 1954 but all of them had to use Pauli's 1940 spin-statistics theorem.

Every Lorentz-invariant quantum field theory must be microscopically invariant under the simultaneous replacement of particles by antiparticles (C for charge conjugation), the reflection of shapes and positions in the mirror (P for parity), and the exchange of past and future (T for time reversal). The combined action is called the CPT-conjugation.

The separate symmetries may fail - although in the real world, the weak nuclear interactions are the only known processes that break C, P, and T (which is the same, because of the CPT-theorem, as PT, CT, and CP). All other forces preserve all the binary symmetries I listed. The experimental proof of a P- and C-violation in 1957 was a shock. Although people should have been much wiser seven years later, even the 1964 experimental proof of CP-violation was another shock. ;-)

The C and P symmetries are broken by the very existence of neutrinos which are always left-handed - and antineutrinos are always right-handed. A mirror image of a left-handed neutrino would be a right-handed neutrino but it doesn't exist: that's why the parity (P) is broken even at the level of the spectrum of weakly-interacting particle physics. The charge conjugation symmetry (C) is broken for the same reason.

The spectrum itself is invariant under CP - the CP-partner of a left-handed neutrino is a right-handed antineutrino that exists, too. However, even the CP-symmetry itself is broken by some terms in the Lagrangian, namely by the CP-odd phase in the CKM matrix (and perhaps new terms that wait to be discovered).

As of today, the CPT-symmetry is absolutely valid according to all known experimental facts.

And as I have mentioned, Pauli has showed that a CPT-violation can only occur in Lorentz-violating theories. That's why the proper vacua of string theory - all of them - exactly preserve both Lorentz invariance as well as CPT-symmetry. That's the main reason why theory - and indeed, string theory is the only known theory that can derive the fields and terms in an effective field theory from a more fundamental starting point - also says that the CPT-symmetry should be exact.

By the way, Pauli's proof may be morally paraphrased in the following way due to your humble correspondent: the reversal of the signs of coordinates "t" and "z" may be visualized as a rotation by "pi" in the Euclideanized "t_E-z" plane, or, if you wish, a boost of the Minkowski spacetime by the rapidity "pi.i".

Geometrically, this operation looks like a PT-conjugation. However, the time-reversal also changes all the charges of the particles so it's actually a CPT. All Lorentz-invariant theories must be invariant under this boost of the "t,z" coordinates by the "pi.i" rapidity and this transformation preserves the Hermiticity of all Hermitian fields, too. So it must be a symmetry of a Lorentz-invariant theory.

**New experiments**

However, you may want to be a heretic :-) and claim that string theory could be imperfect, after all. If that's so, you would generically predict the Lorentz symmetry and CPT-symmetry to be broken. How much broken can they be? If you pretend that you are a complete idiot for a while and you deny all of theoretical physics, i.e. you deny string theory, the best limits are given by the previous experiments. You never learn you lesson from the previous experiment, either. So as a complete idiot, you assume that the preservation of the symmetries in all previous experiments was just a coincidence. ;-)

Let's look at a specific Lorentz-violating effect. The mass term of a Dirac field - imagine a neutron - may be modified as follows:

L = - ψThe first term is a normal mass term and the second term uses an extra vector field "b" that has to be CPT-odd because of the extra "gamma_5" matrix inside the term. Note that the "b" field has a dimension of mass, much like the actual mass coefficient "m" in the first term.^{†}γ^{0}(m + b_{μ }γ^{5 }γ^{μ}) ψ

Of course, a nonzero value of the spatial components of "b" also breaks the rotational symmetry. But you may imagine that there are various values of the vector "b" whose direction is given by objects and fields in the vicinity of the fermion - e.g. by the motion of the Solar System through the Universe (which creates spatial components out of the temporal one). The CPT-symmetry actually implies that even in all such cases, the effective "b" still has to be exactly zero.

A month ago, a new experimental paper by authors from Princeton University was published in

*Physical Review Letters*:

New limit on Lorentz- and CPT-violating neutron spin interactions (full text PDF)Brown, Smullin, Kornack, and Romalis have played with the spin of K and He-3 atoms in various magnetic fields etc. They could determine the upper limit on "b" which is 30 times better than the previous best limit. This "b" is always smaller than 10^{-33} GeV or so, they say - recall that in the real world, it is exactly zero.

This improvement by a factor of 30 is impressive. But these authors have figured out that the main systematic effects that limit the sensitivity of their gadget is the Earth's rotation combined with the gravity. They plan to repeat the experiment on the South Pole. That could easily improve the bound to 10^{-36} GeV. This is a very interesting number because at that point, the experiment would become sensitive to hypothetical coefficients "b" that are suppressed by two powers of the Planck mass, and not just one. Note that "m_{neutron}/m_{Planck}" is approximately equal to "1 GeV / 10^{18} GeV" which is "10^{-18}" or so. If you square it, you get 10^{-36}. For neutron physics, the natural unit is a GeV, its mass.

The fundamentally Lorentz-violating theories that people have been able to propose so far imply large violations proportional to one inverse Planck mass: those are safely excluded - the extra numerical coefficient is smaller than 10^{-15} while the fundamentally Lorentz-breaking theories predict numbers of order one. However, you could speculate that the leading violations could be miraculously canceled - although no one knows why - and only CPT-violating terms that include "1/m_{Planck}^2" survive. But even this more refined speculations may soon be killed experimentally.

Recall that the Fermi satellite has also recently excluded the "1/m_{Planck}" violations of the Lorentz-symmetry with an extra numerical coefficient of order one, by showing that some pulses from extremely distant (billions of light years) fast events arrive almost simultaneously.

Thanks to Olda Klimánek's scinet.cz for the tip.

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