Arrow of Time

Last month (Aug 09), the world of physics was enthralled by a mathematical work that led to an explanation to the arrow of time. The paper was by Lorenzo Maccone, one of the biggest names in quantum physics in this generation. The physicists in the universities (at least big universities) around the world have a hard copy of this paper in their hands, gearing themselves up for lunch time debates on its merits and demerits. As a result, now we have a paper that provides a counter-argument for Maccone's proposal. Moreover this paper says that quantum explanation given by Maccone actually complicates the arrow of time rather than resolving it. This paper is actually written by professors from Imperial College, London. Before taking a deep-dive into it, let's look at what arrow of time means.

The term Arrow of time was first coined by English Astronomer, Arthur Eddington. The name rings any bell? Yes, he is the same Eddington who studied the 1919 solar eclipse to confirm the theory of relativity. The mathematical equations of physics are always time symmetric. That is, whatever happened in a period of time can be reversed without affecting any physical law. It's like rewinding your VCR. But the crystal jar that I broke last week, did not rebuild itself. And it is ridiculous to think that it would.

When we look at the real world, if we leave things alone it always goes towards chaos. In the words of thermodynamics, entropy always increases in the real world as per the second law of thermodynamics. In the movie Godzilla, when running through Manhattan, Godzilla dashes against Chrysler building and it falls down and breaks. It is possible and realistic, if we make proper assumption about the strength of Godzilla and weakness of the building. If we look at it reverse, Godzilla runs through the rubble and it forms into a tower and moves up the top of the building. Does it sound realistic? No. But in a universal, completely mathematical sense, any object can return to lower entropy eventually. So arrow of time is this phenomenon that we always move forward in time in reality with a monotonous increase in entropy. At quantum level, the evolution of quanta is given by Schrodinger's equation, which is time-symmetric. Here, the term entropy is not about heat, but it is about information.

Lets look at some quantum level example. I have built a particle accelerator at the back of my house. I power it up and it generates a quantum particle. But how would I know whether it did generate a particle or not? Copenhagen interpretation says that unless I observe that particle, I cannot say. In other words, unless I observe the particle, I cannot assume that the particle exists, because I don't have the information about its existence. I decide to test the accelerator and I use a Geiger-Muller counter and it ticks. Wow! Now I know, my accelerator works. But what I did by observing its presence is that I collapsed the wave function of that particle. So I know that I generated a particle in past, because it leaves a trace of information.

The time-symmetry of maths and the second law of nature can be looked at as a paradox. If there is a time-symmetric dynamics, as in thermodynamics, it should be impossible for me to have a working system that is time-asymmetric. But everybody knows that the second law works. This paradox is called Loschmidt's paradox. Maccone in his work has made an attempt to answer this Loschmidt's paradox.

"How wonderful that we have met with a paradox. Now we have some hope of making progress." -Niels Bohr

Now let us get back to the particle-accelerator-in-my-backyard example to explain what Maccone says. Assume that my Geiger-Muller counter ticks, but I have a selective amnesia and I forget the count immediately after I heard it. Or in other words, what if somebody shot down the neuron of my brain, in which I saved the information that I heard a count? OK. My example is bad. Let me make it more classical. What if the crystal jar that I broke has comeback from shards, but I just don't remember the happening of that event? So the point is, the activity that results in a decrease in entropy is accompanied by the erasure of its observer's memory. At quantum-level, some instantaneous memory reduction is observed, but never at macroscopic-level because of decoherence. Here is the finding in Maccone's own words:

... any decrease in entropy of a system that is correlated with an observer entails a memory erasure of said observer, in the absence of reservoirs (or is a zero-entropy process for a super-observer that keeps track of all the correlations). That might seem to imply that an observer should be able to see entropy-decreasing processes when considering systems that are uncorrelated from her.

Quick Note: That particle observation problem that I took as an example is one process in quantum physics that is considered to be irreversible. It is called Born Rule. Even that would be reversible according to this paper, if somebody shoots the neuron of the observer at the instance of observation, as I said earlier.

At this point, if you are really confused, shocked and finds it all illogical and bulls**t, please wait for a moment and read this quote before we go to the counter-argument:

"No, no, you're not thinking; you're just being logical." - Niels Bohr.

The counter-argument by Jennings et al starts with the claim made by Maccone, as given in the quote. So during an entropy-decreasing event, the mutual information about the event is destructed as the part of the information. The important point here is what kind of mutual information has to be nullified as a part of memory erasure. Classical mutual information. Is there another kind? Oh.. yes! It is quantum mutual information. So the proof made by Maccone holds good only if a reduction in classical mutual information correlates with a reduction in quantum mutual information.

Information is usually represented as I(A:B), that is information that system A gets due to an event that happened in system B, observed by system A. So if the reduction in quantum mutual information happens without any change to classical mutual information, the earlier proof holds no good. The counter-argument does not stop there. If there is a reduction in quantum mutual information with no harm done to classical mutual information, the situation gets worse. A mathematical proof is made for a case in which, an entropy decreasing event erases the quantum information, but actually increase the classical information. What do the authors mean by this? It's possible for some people to remember observing the crystal jar reforming itself. But nobody does. So the Loschmidt's paradox still remains unsolved. I am not aware whether Maccone has responded to this yet. Or may be, like Walter Ritz and Albert Einstein, Maccone and the Imperial college professors may agree to disagree.