No. It simply means, for example,  that the decay rates of particle states, that are related by charge conjugation and/or parity, aren't, necessarily,  equal. It does not imply that the weak interaction evolution operator isn't unitary. 

It is a popular belief that time is linear which also means that it has a direction in which it flows. But when you consider time as a periodic phenomenon (quantized), then there is no question of arrow of  time. I think CP and T violation in weak sector can be resolved by including velocity vector in the CPT symmetry. Velocity vector automatically takes care of the arrow of time.

No. It simply means, for example,  that the decay rates of particle states, that are related by charge conjugation and/or parity, aren't, necessarily,  equal. It does not imply that the weak interaction evolution operator isn't unitary. 

While there does appear to be ``an arrow of time'', for ``macroscopic'' phenomena, it's not true that this follows from CP violation in the weak interactions. The weak interactions are described, within the Standard Model, by a unitary theory, which means that there isn't any preferred direction of time. CP violation means that eigenstates of CP are not eigenstates of the weak interaction Hamiltonian-and vice versa.  This is standard quantum mechanics, in fact, where energy eigenstates, that aren't eigenstates of some other operator, mix during time evolution.  Indeed, were the weak interaction Hamiltonian not unitary, this would imply the existence of additional degrees of freedom, not the existence of an arrow of time-which, for instance, is one way to understand how the non-unitary behavior of Fermi theory is corrected by the dynamics of the W and Z gauge bosons, whose mass is defined by the interactions with the Higgs boson. 

Mainz, Germany

Dear Brahmachari,

This strikes me as a very engaging question and that it would be useful to clarify the various positions that have been taken on it. First seeing the question appear on RG, I immediately thought of a passage I'd recently read in Sean Carroll's book, From Eternity to Here, The Quest for the Ultimate Theory of Time (2010), p. 139:

At the end of the day, all the would-be symmetries C, P and T are violated in Nature, as well as any combination of two of them together. The obvious next step is to inquire about the combination of all three CPT. In other words, if we take some process observed in nature, switch all the particles with their antiparticles, flip right with left, and run in backward in time, do we get a process that obeys the laws of physics? ...

As far as any experiment yet performed can tell, CPT is a perfectly good symmetry of Nature. And its more than that; under certain fairly reasonable assumptions about the laws of physics, you can prove that CPT must be a good symmetry--this result is known imaginatively as the "CPT theorem." Of course, even reasonable assumptions might be wrong, and neither experimentalists nor theorists have shied away from exploring the possibility of CPT violation. But as far as we can tell, this particular symmetry is holding up.

---end quotation.

This passage perhaps shifts the focus just a bit, though violation of T symmetry is stipulated (regarding kaon decay). However, everyone I have read on this subject seems to discount the importance of the violation of T symmetry--more or less on the grounds that it could not account for temporal asymmetry on a massive scale.

How are such arguments to be evaluated? Are the relevant results still too much up in the air--in spite of Cronin and Fitch?

H.G. Callaway                               

Mohamed I have a mild complain:  It is very difficult for us to read complete papers if you just cut and paste, why don't you simply give the reference of your article and the relevant link for us then we can download the pdf file. Those files are also printer friendly so that one can get them printed in a printable format.

I was, by the way thinking about the Carnot cycle, and the thermodynamic definition of entropy. There, I need to get equations for an ideal gas. For adiabatic expansion and isothermal expansion. At the end of each Carnot cycle, the entropy increases. It takes a finite amount of time to complete a Carnot cycle, therefore entropy increases with time but not continuously, at the end of each Carnot cycle, in discrete jumps. Whenever heat energy is converted to work, a finite amount of energy is lost in a complete cycle, leading to the increase of entropy, which defines a direction for time. Reference: Thermodynamics by E. Fermi (Dover Publications), 1937.

In a class of interesing theories CP-Violation is spontaneous, therefore the Lagrangian does have CP-symmetry and hence T-symmetry by CPT theorem.

Another interesting consequence of all spontaneouly broken quantities is that they tend to vanish at higher and higher energy scales. Ultimately the symmetry is restored above  the high energy scale at which it is brokien sontaneously.

Therefore the absence of T-invariance violation in such theories at high energies.

However CP is *explicitly* broken by the flavor structure of the weak interactions, since the phase of the CKM matrix cannot be eliminated by field redefinitions, doesn't vanish by any dynamical mechanism and does turn out to be sufficient to describe CP violation in the weak interactions-so there isn't any CP symmetric configuration to begin with, in order for spontaneous breaking to make sense. (Indeed the whole point of Kobayashi and Maskawa's construction, generalizing Cabibbo's work on flavor mixing, was that, with, at least, three families, the CP violating phase was inevitable.).  It's a non-trivial issue to realize *spontaneous* breaking of CP in a grand unified theory, so as to obtain the pattern observed in the Standard Model; that's, still, an open problem.

High scale theories where CP is conserved have to contain manifest left-right symmetric gauge theories rather than the SM gauge theories of Weinberg-Salam-Glashow type.

CP conservation doesn't mean that the fermion representation isn't complex, i.e. the fermions are not chiral, so more is needed. Indeed, not even left-right symmetric theories necessarily preserve CP-it depends on the couplings (example: the theta term in Yang-Mills theories in general and  in QCD, in particular, breaks CP.) Cf. for instance, the paper by Haber and Surujon, ``Group-theoretic Condition for Spontaneous CP Violation'', http://arxiv.org/abs/1201.1730 

For a recent discussion on CP-violation in spontaneously broken manifest left-right symmetric models  a possible reference is PR D91(2015)071901 {arxiv: 1408.6382].

I am little confusing. The original question was if we have a time arrow in particle physics due K0 system. I think the answer for the question is yes.

there is confusing point about microscropical time arrow and macroscopical time arrow. The macrioscropical time arrow existe due entropy. the microscopical due T violation,

The problem is that at the microscopic system there are reverse reactions. If forward and reverse reactions are in equilibrium then the T asymmetry, which is generated is also erased out. The T violating reaction should therefore be out of equilibrium, meaning forward reaction rate and backward reaction rates has are be different. This is in a sense closely linked  to Sakharov's conditions for Baryogenesis. This was pointed out to me by Prof. E. A. Paschos a few days ago in a different thread of discussion in Researchgate.

The question asked by  Dr. Brahmachari  on the arrow of time attracted  my attention in the mid-ninenties. I asked myself  does CP violation in K-meson decays determine the arrow of time ? I went as far as to send a letter to Stephen Hawking and I am attaching his answer.

  To determine the direction of time in decays of particles you need a measurable quantinty.

1.One such quantity  is the difference

               Γ{K(long) --> π(- ) μ(+) ν } -  Γ{ K(long) --> π(+ )μ(-) ν},

implying the symmetric production  excess of μ(+)  over μ(-) in the universe. There are other processes as well. Now if you had an proton-proton accelerator producing K° and K°(bar) pairs symmetrically (associated production) you can measure the difference of numbers  μ(+) - μ(-) and the number will be different from zero and increasing contnuously giving you a direction of time. This assumes the construction and running of an accelerator, that determines an initial condtion.

2. In the universe we also need a measurable quantity. When the universe is in equilbrium the inverse processes ( recombinations) also take place and wash out the difference.

3. Thus we need a deviation from equilibrium. This is provided by the expansion of the Universe and therefore an intitial condition in the Universe.

Conclusion: CP violation in particle decays is not enouph to determine the direction  of the future and we need all thee conditions of Sacharov to create an asymmetry and an arrow of time.

I agree with the answer of Stephen Hawking in the attachment and thank him for taking the time to answer me.

Hi, not concerning to the answere but I leave you a link of a recent paper for some experimental measurement  wich is related to CPT  invariance https://cds.cern.ch/record/2033777/files/publicnote.pdf hope it be of interest.