**Microscopic processes involving particles proceed differently if forced to go backwards**

*One of them is Babar the Elephant. Don't ask me which one – I would guess it's the daddy. Instead, I can offer you Peter F.'s elephant who can paint an elephant with a flower.*

Physical Review Letters just published a paper

Observation of Time Reversal Violation in the B0 Meson System (arXiv, July 2012)by the BaBar collaboration at Stanford's SLAC that directly proves the violation of T, or the time-reversal symmetry. Even though the result isn't new anymore, the publication was an opportunity for some vibrations in the media:

Stanford press releaseThe T-violation is equivalent to the CP-violation, via the CPT-theorem, as I discuss below, but comments about the discovered "microscopic arrow of time" weren't just a new sexy way to describe experiments looking for CP-violation. They have actually seen the T-violation "directly". Physicists have known what would happen in this experiment for decades; but they actually performed it for the first time now (the detailed idea behind this experiment has been around since the late 1990s when the long experiment was actually getting started).

Ars Technica, Google News

What did they do?

They studied B-mesons – the same particles whose decays were recently claimed to send supersymmetry to the hospital. Mesons are particles constructed out of 1 quark and 1 antiquark, roughly speaking, and "B" means that bottom quarks and/or antiquarks are involved. The high frequency of the letter "B" in "BaBar" has the same reason. In fact, "BaBar" is \(B\bar B\) [bee-bee-bar] as pronounced by Babar the Elephant.

The BaBar Collaboration looked for various processes in which \(B^0\) and \(\bar B^0\), two "flavor eigenstates" of the neutral B-mesons, transform either to \(J/\psi K_L^0\) (called \(B_+\)) or \(c\bar cK_S^0\) (called \(B_-\)). And if I simplify things just a little bit, statistics applied to 468 million entangled \(B\bar B\)-pairs produced in \(\Upsilon (4S)\) decays showed that some "asymmetries" that should be zero if T were a symmetry were decidedly nonzero.

One may say that the transformation of \(B^0\) into \(B_-\) was detectably faster than the inverse process. We often talk about 2-sigma or 3-sigma "bumps" and 5 standard deviations is a threshold for a "discovery". So you may want to ask how many sigmas these BaBar folks actually have to claim that the microscopic laws have an inherent arrow of time. Their signal is actually 14 sigma or 17 sigma, depending on the details, so the probability of a false positive is something like \(10^{-43}\). Compare it with the 10-percent risk of false positives tolerated in soft scientific disciplines.

Note that one doesn't need an exponentially huge amount of data. If most of the errors are statistical in character, you only need about a 10 times greater dataset to go from 5 standard deviations to 15 standard deviations. Just 10 times more data and the risk of a false positive drops from \(10^{-6}\) to \(10^{-43}\).

**Reminding you of C, P, T, CP, CPT, and all that**

Our bodies (and many other things) are "almost" left-right symmetric. For a long time, physicists believed (and most laymen probably still believe) that the fundamental particles in Nature had to be left-right-symmetric as well, and behave in a left-right symmetric manner, too. And if some particles (such as amino acids) are left-right-asymmetric (look like a screw), there must exist their mirror images with exactly the same masses and other properties.

This assumption seemed to be satisfied by all the phenomena known before the 1950s. But experiments in the 1950s showed that this left-right symmetry that physicists call "parity" and denote "P" is actually violated in Nature. There are particles that are spinning much like the wheels of a car going forward – but they prefer to shoot to the left side, not right side, for no apparent reason. While \(SO(3)\) is a symmetry, \(O(3)\) is not.

Left-right-asymmetric physics (physicists say "chiral" physics, referring to the word "kheir/χειρ" for a "hand" because the left hand and the right hand differ which is why we use various hands for various right-hand rules) is easily constructed using spinors, mathematical objects generalizing vectors that may be described as "square roots of vectors". In particular, in 3+1 dimensions, or any even total dimensionality, one may write down equations for "Weyl [chiral] spinors" that will force a particle to resemble a left-handed screw, or right-handed screw, but forbid its opposite motion.

And indeed, all the neutrinos are left-handed while the antineutrinos – and it's the antineutrino that you get by a decay of a neutron – are always right-handed. Nature has an inherent left-right asymmetry built into it. Note that this correlation also violates the C symmetry, or the charge-conjugation symmetry that replaces particles by antiparticles. If you act with C on a (possible) left-handed neutrino, you get a left-handed antineutrino which is not allowed.

For a decade, people thought that a more sophisticated symmetry, CP, that you obtain by the simultaneous action of P and C is obeyed by Nature. If you mirror-reflect all the objects and particles

*and*replace all particles by their antiparticles, you should get another allowed state, one that has the same mass/energy and behaves in the "same way".

However, in the 1960s, even this CP-symmetry was found to be violated. The spectrum of allowed objects is pretty much CP-symmetric in Nature and in all Lagrangian quantum field theories we may write down but the pairs related by CP behave differently. The complex phase in the CKM matrix is the only truly established source of CP-violation we know in Nature. New physical effects such as supersymmetry implies that new sources of CP-violation probably exist. They're probably also badly needed to obtain the high matter-antimatter antisymmetry that had to exist when the Cosmos was young, before almost everything annihilated, so that we're still here. But no clear proofs of other sources of CP-violation are available at this moment although some hints of discrepancies exist.

So C and P are not symmetries; they are violated even by the spectrum of allowed objects. CP is allowed by the spectrum of allowed objects but the dynamics (especially mixing angles etc.) imply that it is not an exact symmetry. As you can see, the CP-violation is even weaker than the C-violation and the P-violation.

But there is a combination of operations that has to be a symmetry in every relativistic quantum field theory, the CPT-symmetry. This fact was proved by Wolfgang Pauli and is called the CPT-theorem. The CPT operation does C and P at the same moment and it also performs the time reversal – it reverts the direction of the arrow of time.

Note that among C,P,T, only T is an "antilinear operator" which means that \[

T\ket{\lambda \psi} = \lambda^* T\ket\psi

\] including the asterisk which means complex conjugation (that's the reason of the prefix, anti-). Various combinations of C,P,T are linear or antilinear depending on whether T is included. Note that the complex conjugation is needed for the time reversal already in ordinary non-relativistic quantum mechanics because the complex conjugation is the only sensible way to change \(\exp(+ipx/\hbar)\) to \(\exp(-ipx/\hbar)\) i.e. to change the sign of the momentum \(p\) – and the velocity \(v=dx/dt\) – which is needed for particles to evolve backwards.

Why the CPT has to be a symmetry in quantum field theory – and almost certainly is an exact symmetry in string theory as well? It's because it may be interpreted as the "rotation of the spacetime by 180 degrees" which is a symmetry because it belongs to the Lorentz group analytically extended to complex values of the parameters (which is allowed).

Work in the momentum space and extend the time coordinate to imaginary values\[

t\to \tau = it.

\] Analytically continue all fields or Green's functions and amplitudes (as functions of the momenta, to be kosher, because only as functions of the momenta, the functions are holomorphic) to the imaginary values of the time component. Now, the 4-dimensional spacetime with points \((x,y,z,\tau)\) becomes a Euclidean 4-dimensional space.

The rotations between \(z\) and \(\tau\) are nothing else than \(tz\)-boosts extended to imaginary values of the "boost rapidity". By the analyticity, if the ordinary real boosts are symmetries, so must be the imaginary boosts. The imaginary rapidity is nothing else than the ordinary angle. Take the angle to be \(\pi\). This will revert the sign of both \(\tau\) and \(z\) – which means that it will perform both P and T. Now, if you analytically continue it back, the effect is clearly nothing else than the flipping of signs of \(t\) and \(z\), so you naively get the PT transformation and prove it is a symmetry because it is just a \(\pi\)-rotation.

However, you actually get a CPT transformation. Purely geometrically, by looking at the shape of the world lines, you can't distinguish PT from CPT because C only acts "internally" and doesn't change the shape of the world lines etc. The reason why the rotation by 180 degrees is CPT and not just PT is that the reflection of T also reverts the "arrow" on all the world lines, and particles moving backwards in time are actually antiparticles. (I could formulate an equivalent argument more mathematically and convincingly, but it's enough here, I hope.)

So CPT is always a symmetry. If you replace all particles by antiparticles; change their configuration to its mirror image; and invert the sign of all the velocities, then the subsequent evolution in time will look like exactly the evolution of the original system backwards in time (reflected in space as well and enjoying the inverted labels for all particles/antiparticles).

Because CP isn't a symmetry and CPT is a symmetry, T – which is a convolution of the CP and CPT transformations, a convolution of a symmetry and a non-symmetry – clearly refuses to be a symmetry, too. That's also why they could directly detect a violation of T in the BaBar experiment.

**This has nothing to do with the arrow of time in statistical physics**

*Thank God, even Sean Carroll knows and acknowledges this fact.*

I must emphasize that these effects are only large enough in special systems interacting via the weak interactions and they're weak, anyway. In reality, we know the "arrows of time" that have been discussed many times on this blog. We forget but rarely "unforget", eggs break but not unbreak, we mostly get older but not younger, the heat goes from warmer bodies to cooler ones but not vice versa, friction slows downs vehicles but doesn't speed them up, and so on. Decoherence produces nearly diagonal density matrices out of pure and coherent states but the opposite process – emergence of quantum coherence out of decoherent chaos – doesn't occur.

These manifestations of the "arrow of time" have nothing whatsoever to do with the violation of T that was discussed at the beginning of the article and that was experimentally demonstrated by BaBar. The microscopic BaBar-like T-violation is neither necessary nor sufficient a condition for the existence of the arrow of time in thermodynamics etc.

Even if you had microscopically time-reversal-symmetric laws of physics, they would produce time-reversal-asymmetric macroscopic laws with friction and the second law. It's because the origin of all these "macroscopic asymmetries" is in the logical arrow of time – the fact that the probabilities of evolution between ensembles of microstates have to be averaged over initial states but summed over final states, so the initial states and final states have to be treated differently, because of the basic laws of logic and probability calculus.

Again, the microscopic T-violation isn't a necessary condition for the entropy to increase and for other signs of the arrow of time in the macroscopic world around us.

The opposite relationship is also wrong; the microscopic T-violation wouldn't be sufficient for the macroscopic one, either. If you tried to deny the existence of the logical arrow of time, the BaBar-like T-violation in the microscopic laws of physics wouldn't be sufficient to produce the "huge" asymmetries between the processes that go in one direction and those that (usually don't) proceed in the opposite direction simply because the microscopic T-violation is far too weak and doesn't have a "uniform arrow" that would give the future its futureness and award the past with its pastness, anyway.

I plan to dedicate some article to statistical physics in a foreseeable future again. Right now, one must emphasize that the experimental detection of the T-violation is a detection of an asymmetry in the fundamental equations of physics that apply when the initial state and the final state are fully specified and known – so ignorance, the main prerequsite needed for thermodynamics to emerge, is absent.

## snail feedback (18) :

Possible explanation of violation:

t2-t1=dt

because

mt2-mt1=dtm

et2-et1=dte

m and e not constants during the evolution of the Universe.that confirms the existence of generation of particles.

Hi Lubos - I look forward to reading what you write about stat physics and time. I watched the Feynman lecture about the direction of time with my 13 year old son and we found it fascinating. I recall that some people claim that the requirement of increasing entropy is what drives time in one direction - that it is a more fundamental explanation. I have never understood this - how does one necessary follow from the other ? Why couldn't they both be independent facts ?

is there any relation between CPT and exchange symetry?

Thanks for this article Lumo,

it was very pleasant reading.

The argument involving the worldlines to see why PT and CPT can not be destinguished, I have not yet seen stated elsewhere and I like this.

I look forward to read some nice reminders about statistical physics here on TRF too :-)

Hi Lubos,

good luck with a new article on arrow of time/statistical mechanics, but I don't think you can beat the deniers (determinists) with logic - they bascially have it right in their minds, IF the universe is deterministic then it could just as well go backwards as forwards (entropy wise).

You can only defeat them by arguing that the universe doesn't evolve determistically

Dear James, misunderstandings in statistical physics - which may be even classical - and misunderstandings of quantum physics are a priori two different things. However, I agree with you that they're linked in the way you mention.

In both cases, these people believe in some totally wrong and totally naive "realism" which means that they believe that physics should describe "how things are". But quantum physics is a tool to (more generally) "say valid statements about Nature" and indeed, this was really the case in classical statistical physics, too. A statement about thermodynamics inevitably works with ignorance and ensembles and propositions about such things inseparably have a logical arrow of time incorporated in them.

That's why I have often called Boltzmann the "forefather of quantum mechanics" and why the misunderstanding of the second law of thermodynamics and of the postulates of quantum mechanics usually co-exist.

Hi Lubos,

I enjoyed your article; thanks! I am having trouble with believing that this is an ironclad test of time reversal violation. It seems that another hypothesis is that it shows an asymmetry between matter and antimatter. For instance, the branching ratios for decay are different for matter and antimatter. Sure, the theory says that an antiparticle traveling backward in time is exactly the same as a particle traveling forward. But what if that is not true? In other words, we are assuming a symmetry between matter and antimatter, but what if no perfect theory has this symmetry?

Another hypothesis, valid even if there is symmetry between matter and antimatter, is that the Upsilon has memory of the fact that it was created by matter (the accelerator beam is made of matter).

How do they discard these hypotheses? I have found nothing on this.

Best regards, Dan

Thanks, Dan, for your interest. The only generally valid relationship between matter and antimatter is the CPT-theorem that holds for all relativistic quantum field theories and almost certainly for its extensions - string theory is the only example worth mentioning.

What I wrote about the antiparticles' being particles moving backwards in time was a sketch of a rigorous proof of the CPT-theorem. I would even claim that a good physicist who has never heard about the CPT-theorem could use my sketch to rediscover the CPT-theorem and write its pretty rigorous version himself. So theoretically, it's an indisputable, rigorously proved result that antimatter evolving backwards and in the mirror obeys the same rule as matter evolving forward without a mirror.

The CPT-theorem also holds according to all the experiments and observations we know. The previous sentence is equivalent to the fact that all the effects violating the past-future symmetry boil down to the same effects and same terms in the Lagrangian as the observationally known effects displaying a difference between matter and antimatter (when the mirroring of the space is added as well). The CPT-invariant theories (mostly quantum field theories which are enough) are simply keep on describing all the known empirical data but if the CPT-theorem were significantly wrong, e.g. as wrong as the CP-symmetry or even P-symmetry, a discrepancy would already have been discovered.

So, first, I want to make clear, when I criticize the description of the experiment as a "direct observation of time reversal violation", I am not talking about the specific time-reversal asymmetry in the standard model. I am not assuming CPT. I don't think you can invoke a symmetry (CPT) that explains an experiment supposedly on time reversal only. You either directly measure something, or you don't. You can't invoke a correspondence between particles and antiparticles, even if it has worked well in describing all experiments. That standard would allow numerous theories consistent with all known observations (vis. Copernicus). You can't demonstrate a thing that your theory, which is incomplete, says is equivalent. You can't assume anything, like local interactions for example. Local interactions lead to singularities in field theory, as far as I am aware, and running of the coupling constants, and renormalization is great for getting physical results but has no known physical origin with local interactions. I might be completely wrong on this point.

For this experiment invoking the hypothesis that the Upsilon has memory that it was created by matter, as I have done, is a bit unfair, because it would make the measurement of time reversal asymmetry impossible. But a philosopher might argue that it is indeed unprovable. Notwithstanding this argument, I think most everybody would agree that time reversal is defined as interchanging initial and final states. Defined, not understood to be equivalent in terms of an existing incomplete theory.

In the present context, I think the following are plausible hypotheses fully consistent with theory and experiment. I may not be correct. What if there were a CPT eigenvalue that was conserved, such that the entire observable universe was in a condensate of one or the other of the CPT eigenstates. Call it left. The one particle CPT eigenstates are degenerate, left and right. This quantum number determines the direction of beta decay of Cobalt 60 for instance. The universe is more symmetric than it appears Now which way does this hypothetical, unobserved quantum number transform with respect to CPT? Well it could transform either way, right, depending on the actual symmetry of the universe? What if this quantum number determines the result of the babar experiment?

I think that you must be saying, there is no superior theory, written much differently in terms of the mathematics, without anything like CPT symmetry, but with a time coordinate, which is symmetric with respect to t<->-t, there is a clear symmetry under a global operation that interchanges t and -t , but in which the Upsilon has memory of the fact that it was created by matter, or has a left eigenvalue, which causes the babar result. There is no possibility. In quantum mechanics you can define an effective Hamiltonian in a reduced space that is a function of energy, H(E_i)Psi_i = E_i Psi_i, and in the time dependent version there is a memory (non-markovian) term. How about that? What if the observable universe is actually a subset of the full universe, such that there is such an effective Hamiltonian, for instance?

Also, the universe is understood to be asymmetric with respect to matter/antimater, with the observed abundance of the former. Some would argue that that unambiguously disqualifies any theory that assumes any such symmetry.

Sorry, Dan, your comments make no sense. You are not only "not assuming" the Standard Model: you are deliberately overlooking the evidence supporting the Standard Model - which is all the evidence. So your comments can't have anything to do with the reality.

The Standard Model (like many field theories of this kind) implies that C, P, CP, T is broken (in well-defined ways) but CPT is preserved. As long as the Standard Model's predictions hold for everything, I may say the same about Nature: Nature breaks C, P, CP, and T, but it preserves CPT. You can't falsify CPT just by your desire to arrogantly and stupidly pretend that no observations exist.

Remember we are talking about PROVING something, not showing that everything is consistent with known observations.

Anyhow I thought we were having a civil conversation but clearly I am wrong. I'll go ahead and insult you back below.

>So your comments can't have anything to do with the reality.

You did not follow my reference to Copernicus. The geocentric theory fit all observations until it didn't. You are being an arrogant scientist, not me.

If you assume that the center of the universe is the earth, then you can prove that mars travels in a highly unphysical way as it circles the earth, through direct observation. You would be wrong.

Peace.

Just to make it CRYSTAL:

Before the Copernican revolution one could have made a DIRECT OBSERVATION of Mars moving around the earth in a loopy way. Lots of folks did and they were wrong.

Dear Dan, it's not true that we are talking just about proving something. I am not talking just about "proving" something all the time especially because in this situation - and in most situations in science - it's just not possible to "prove" something. It's not how science works. If you need someone else to say the same thing, watch e.g. this video:

http://www.youtube.com/watch?v=wLaRXYai19A

The time reversal symmetry has nothing to do with Copernicus. Have you ever tried to read your own comments and imagine how incredibly obnoxious, stupid crackpot you are in the eyes of others?

Lubos,

You can directly observe parity violation in a laboratory. You can set up the magnetic coil, collect the beta ray, observe that things are left-right asymmetric.

Parity violation means setting up an experiment that is left-right symmetric and observing a left-right asymmetry in the result. Parity violation has been proven to exist. Fair statement? I think so.

Table salt has been proven to be a compound of sodium and chlorine. Fair statement? I think so.

So yes, in science we can prove things. If you are a philosopher, you can put all forms of proof in question. I'm not talking about that. We all agree on what salt, sodium, and chlorine are. Given that agreement, we can do an experiment that shows the former is composed of the latter.

Has time reversal violation been proven to exist -- been directly observed -- been shown to happen -- like parity violation?

No. Arguably, it cannot.

However, the man on the street would say that reversing initial and final states is basically time reversal. Start talking to him about antiparticles actually being particles traveling backwards in time, and he'll want to know what you're selling. At the very least, he'll want you to prove it. And you can't prove that an antiparticle is the same thing as a particle traveling backwards in time, no matter their equivalence in your theory with CPT.

Google scholar the terms "Memory kernel" and "chiral condensate."

GEOCENTRIC MODEL <--> CPT SYMMETRY

You are pretending to be dense, right?!?

i think that the violation of PT must to exist,independently of cpt.the asymmetry of left-right handed to the rotational invariance,implies the existence of speed of light as CONSTANT AND LIMIT TO UNIVERSE.AND THE NON-EXISTENCE OF ANTiMATTER.THE ANTIPARTICLES ARE PRODUCTS OF VIOLATION OF SYMMETRY IN THE SPACETIME CONTINUOS,PERMITING THAT THE SPACETIME IN THE QUADRIDIMENSIONAL MANIFOLDS BE LATTICES.IT IS DISCRETES AND NON TOTALLY SMOOTH IN ALL IT STRUCTURE.

THE VIOLATION OF PT DOES APPEAR THE ANTIPARTICLES THAT ARE ENERGY STATES LOCALLY BUNDLEDLED IN THE VACCUM.THE ASYMMETR IN SPACETIME THAT IS SEEN IN THE SPACE AND TIME, SEPARATELY,OCCUR THAT THE TRANSFORMATIONS OF MASS INTO ENERGY AND VICEVERSA AREN"T SYMMETRICS WITH THE INCRESE OF SPEED IN RETATIONS TO OTHERS INERTIAL SYSTEMS IN RELATIVE MOTIONS

THEN T is BROKEN TOGHETER TO P,CONNECTING SPACE AND TIME IN SPACETIME CONTINUOS.

the antimatter doesnt exist in the universe same in the earliest universe.then the violation of pt is maximal,compareted to the violation weak of cp,then the conservation of cp for stronger interactions is in hidden dimension generated by fields-PT- symmetry breaking that implies the junction of space and time into spacetime continuos in 4-dimensional manifolds

and generating the speed of light as constant due to

the imaginary part of the connection of space and time,with reversing P with rotation of 2pi,doing have a time reversal time as double of the initial time

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