An Email Conversation with Prof. Edward Witten

When I proposed the linear version of the Schrodinger equation, one of the things that I tried was to find a connection with supersymmetry. Since this equation is the equivalent of the Dirac equation and introduces spin in the non-relativistic limit it can lead to formally introducing supersymmetry in non-relativistic quantum mechanics. This is actually an interesting problem to work on for someone who is interested in theoretical problems (although it wont help you get a postdoc). I spent some time on this problem but due to a busy teaching schedule and job uncertainties I have not been able to give this problem its due time. Another reason is the lack of evidence for supersymmetry. In any case I do feel that there should be a connection, at least mathematically.

This is what several luminaries like Prof. Juan Maldacena and Prof. Warren Siegel suggested I do when I emailed them. I then emailed Prof. Ed Witten for suggestions. Prof. Ed Witten is a well known string theorist and it was a pleasure to have this conversation with him. Here you go, following is the unmodified email exchange between us:

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Me:
Dear Prof. Witten,

In two recently published papers I have introduced an equation in non-relativistic quantum mechanics:

http://link.springer.com/article/10.1007%2Fs10701-015-9944-z
http://www.ijqf.org/archives/3480

The square of this equation gives the Schrodinger equation. I have seen several of your papers on non-relativistic susy. Prof. Warren Siegel and Prof. Juan Maldacena have suggested I connect the approach in the above two papers with non-relativistic susy. I have tried but not succeeded yet. I would really appreciate if you can guide me in this matter and suggest how to seek a possible connection of the two approaches.

Thank you.

Best Regards,


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Prof. Ed. Witten:

Dear Dr. Ajaib,

Have you included a potential in your constructions?
I.e. do you get a Schrodinger equation with a potential?
Supersymmetric quantum mechanics leads to special potentials
so if you were getting a potential, that would probably make it obvious
whether one could compare to supersymmetric quantum mechanics


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Me:

Thank you for your reply Prof. Witten

I did include a potential. For example, in section 3 of the paper I referenced earlier (http://arxiv.org/pdf/1502.04274.pdf) I solved potential step problem. Sorry but I am not sure how this makes the connection obvious.

Best Regards,

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Prof. Ed. Witten:

Dear Dr. Ajaib,

I am not sure if there is a connection. If you can incorporate a completely general potential
then what you have does not correspond to supersymmetric quantum mechanics, in which only
certain rather special potentials are possible.


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Me:

I see. Thank you so much for this insight I really appreciate it.

Best Regards,

Are the Paradoxes in Special Relativity Really Resolved?

A paradox in a theory appears to contradict the theory itself but might be resolved if looked at carefully. Physicists in the early days would take the occurrence of a paradox as a chance to find something new, as Neils Bohr famously said, "How wonderful that we have met with a paradox. Now we have some hope of making progress." These days physicist would go to any limit in trying to resolve these paradoxes in theories in a way that always shows that the theory is perfect and no need to be questioned. The paradoxes in special relativity are a classical example of this [1].

A well known paradox in special relativity is the twin paradox. I once investigated the twin paradox because I was not able to wrap my head around the reason for the asymmetric aging of the two twins. One of the main postulates of the theory of relativity is that frames of references are equivalent so if one of the twins leaves the planet and comes back they both should have aged by the same amount because they are moving relative to each other. The equations of relativity do not predict any asymmetry between two references moving relative to each other.

Now lets see how physicists resolve this paradox. One explanation is that since the twin who leaves the planet accelerates, especially when he turns back, the postulates of relativity do not hold and the frames are not equivalent. Furthermore, you have to take general relativity into account for the part of the journey where he turns back. There are two reasons I do not find this explanation satisfying. One is that if equivalence of frames in relativity breaks because of acceleration, no two frames of references should be equivalent since every moving frame starts from zero velocity and accelerates to attain a non-zero velocity. The second reason is that the two references were equivalent for at least half of the journey so there should be some implications of this when the twin returns. I have even seen a paper which says that it is actually the twin on the planet, and not the one travelling, who is going to be younger because he stays in an inertial frame of reference. 

Another paradox is the train and tunnel paradox. Lets say that a train is moving towards a tunnel which is a bit smaller than the train. An observer is sitting on the tunnel and wants to trap the train. If the train is moving close to the speed of light it is length contracted for the stationary observer and much smaller than the tunnel. Therefore, as soon as the train disappears in the tunnel the observer will close the gates of the tunnel and trap the train. From the trains perspective, however, the tunnel is much smaller so it is impossible to trap the train in it. Even if the observer closes the door it might chop the train into pieces but its impossible to trap the train. Both of these cases have very different outcomes. In the first case the train gets trapped, when it slows down becomes longer and breaks within the tunnel because the gates are closed (assuming that the gates are sturdy enough). In the second case the train splits into pieces and never gets trapped. This paradox is typically resolved using the relativity of simultaneity. Again, I do not find these explanations completely satisfying because the fact that the outcomes of the two observers are completely different implies that this paradox cannot be completely resolved.

Look, dont get me wrong, I am not saying that relativity is a wrong theory, no way. There is a lot of experimental evidence that agrees with the predictions of relativity, such as the lifetime of the muons. The point that I am trying to make here is that we should sometimes allow for explanations of these paradoxes that indicate that there might be loop holes in the theory. Physicists never question the theory of relativity because it is considered too sacred by the physics community and anyone who questions these ideas is considered a crackpot. This attitude of physicists feels more like religion than science where they would not even open the door to question theories unless experimental evidence is found. If experimental evidence is found, they loose their minds and would do any thing to explain it (like what happened in the case of ultra-relativistic neutrinos from OPERA). We should have an atmosphere where people, especially young physicists, should be able to question these theories and not fear being ridiculed. 

[1] Here is a video that describes paradoxes in relativity. https://www.youtube.com/watch?v=kGsbBw1I0Rg

Is Schrodinger Equation Exclusively Non-Relativistic?

One of the aims of this blog is to raise questions that, in my view, have never been raised before. So the reader can disagree with what I am discussing here but the main purpose is to make people think from different angles.

When I started rethinking about the foundational ideas in physics one of the things that bothered me was the way we take the non-relativistic limit of the relativistic dispersion relation to prove that Schrodinger's equation (SE) is the non-relativistic limit of the Dirac and Klein Gordon (KG) equation.  Since all these equations are partial differential equations (PDEs) I started learning how PDEs are studied by mathematicians. I tried to find whether this transition from Klein Gordon to Schrodinger equation can be performed with partial differential equations and it turned out that the answer was no (as far as I went in studying PDEs). I spent a long time trying to understand how mathematicians dissect PDEs to study them and see if this can give us more information about these fundamental equations in physics. This whole effort led to a paper in which I tried to make a connection of physics with numerical analysis [1] (This is one of my most viewed papers on researchgate and has been cited several times).

An important concept that I learned from this was regarding the domain of dependence of a differential equation. It turns out that the Schrodinger equation is a parabolic PDE and these equations allow for physical information to travel with infinite speed [2]. I discussed in the paper on how to implement causality in the Schrodinger equation using the explicit method. But this insight appears to indicate that the Schrodinger equation might not be an exclusively non-relativistic equation. It does allow for solutions that are non-relativistic but this may not be true in general. This might be the reason that entanglement is possible which allows for instant information transfer between entangled quantum mechanical states. Later, I came to the same conclusion from a completely different angle, i.e., Lorentz violation, which I will explain below.

I wrote the numerical analysis paper in 2013 and then continued on my journey in trying to tackle some fundamental issues in physics. In 2015, I proposed an equation that was the equivalent of the Dirac equation, i.e., the linear form of Schrodinger equation. In the follow up paper I showed that the equation I proposed can be obtained from the non-relativistic limit of the Dirac equation [4]. Later, however, I realized that there is completely different way of obtaining the linear form of the Schrodinger equation from the Dirac equation, which is by adding enhanced Lorentz violating terms to it [5]. In other words, Schrodinger equation does not obey Lorentz symmetry and apparently violates it maximally. This may be the reason that its domain of dependence allows for information to travel with infinite speed.

The above discussion is to encourage readers (especially physicists) to think more about this. I am not saying that what I have concluded is the right answer but it does appear to indicate that this might be a missing link in our understanding of fundamental physics.

[1] https://arxiv.org/pdf/1302.5601.pdf

[2] A good reference is Numerical Methods for Engineers and Scientists by Joe D. Hoffman, Ch 10.

[3] https://link.springer.com/article/10.1007%2Fs10701-015-9944-z

[4] https://arxiv.org/pdf/1511.07901.pdf

[5] https://arxiv.org/pdf/1403.7622.pdf

When the Problems in the Physics Academic Culture Started

I recently visited Millersville University, where my wife is an Assistant Professor, and saw a poster which reminded me of an important turn in the history of physics. This poster describes how there was a boom in physics research after World War II [1]. In particular, I want to highlight the following part which describes how the government took over the field after the war:

"Far from closing down, weapons laboratories developed into permanent national institutions devoted to both military and civilian research. For the first time, the federal government undertook the systematic support of basic science."[1]

I frequently talk to my friends about the current status of physics in societies and in my view this is the point where all the mess in physics started. Before the wars, physicists were mostly isolated academics trying to understand how the Universe works. But, after the wars, various countries in the world realized that physics can be used to maintain their power and hegemony in the world. I remember reading Frank Close's book in which he describes the post war scenario as follows:

"Once the war ended, the stature of physics—which had produced the atomic bomb—changed utterly. The U.S. government poured money into research; physicists had become heroes; Einstein—previously described as a mathematician—was now reinvented as a physicist." pg 37, The Infinity Puzzle.

I think we are still living in this messy post war scenario where there is a great deal of government intervention in physics. After the war, particle physics went through a boom as well, because, it was the obvious next step to nuclear physics, which enabled the creation of the bomb. These days, the main motivation of funding most of the research in physics is whether it helps in making more weapons or not. There were obviously pros and cons of this intervention at that time since physicists got the money they needed to think more about the mysteries of the Universe but in my view the cons are turning out to be the true legacy of this intervention.

With the extensive government involvement in physics, the mess is getting worse with time. I have seen professors shifting their interests and trying to work primarily to get funding, not for the physics. This is the reason for the "publish or die" culture and why physics is going through a phase I call "the codification of physics". I have seen various groups in physics departments trying to introduce codes in their groups to get swift publications. This culture of "publish or die" is being transferred through out the world because US is one of the leaders in science. Part of this is the reason you cannot get an Assistant Professor job in the US after completing your PhD unless you can get funding. Time will tell how long will this post world war mess continues but for now there is no sign of alleviation.

[1] https://www.aps.org/publications/apsnews/199808/century.cfm

Falsifiability in Modern Science

It appears that science and, in particular, particle physics has reached a point where one of the most foundational requirements of science is coming under spot light. For any theory to be science it has to be falsifiable. In other words, we should be able to devise an experiment to test whether the theory is correct or not. This modern view of science was put forward by Karl Popper. 

I am not sure about other fields of science but in physics this question is very relevant these days. There are fields of physics, such as condensed matter and plasma physics that relate to low energy phenomena but then there is particle physics which addresses phenomena at high energies. In particle physics, string theory, supersymmetry and grand unification are examples of theories that appear to be very elegant mathematically but have not yielded any experimental signatures. The late Stephen Hawking wrote in his book "A Brief History of time" that to truly test grand unified theories we would have to build an accelerator the size of our Universe. To test string theory we need an even bigger accelerator. Experiments have been conducted to test the indirect signatures of these theories but to date no evidence have been found. For example, one of the predictions of grand unified theories is proton decay which the Super-K experiment has been testing but no signatures have been found. 

So the question arises whether we should call these theories science or not and should we continue investing money and effort in trying to test these theories. I think we have come across a decisive point in the history of science as we did at the time of Ludwig Boltzmann. During Boltzmann's time, the popular approach in science was that of the "positivist" [1]. Positivism was based on Ernst Mach's philosophy that we cannot introduce any variable in physics that cannot be directly tested or observed. In Boltzmann's famous equation "S=k log W", the variable "W" [2] is not directly observable so he was confronted by the scientists of his time. It is said that the opposition to his ideas was one of the reasons he went into depression and committed suicide. Luminaries like Einstein and Heisenberg were deeply influenced by Mach's philosophy which is why there is so much emphasis on observables in their theories. So a lesson that we can learn from this part of history is that we may have to broaden the definition of science as Karl Popper did.

The fact that most of the senior particle physicists have spent their lifetimes on these theories makes things even more complicated. So, it appears that we are dealing with not only a complicated philosophical question but a difficult social issue as well. They continue to emphasize on making bigger experiments to test these theories. But is it worth it? I recently wrote about the proposal of an even bigger collider here.

So, are we reaching an era where we need a post modern view of science? If we are not able to truly test these theories should we reject them as science and instead invest in theories that are accessible with our current technologies. These are tough question that the scientific community has to think about.  Maybe when we get to the level II or III civilization on the Kardashev scale, we can think of directly testing these theories but for now let's spend on what we can afford.

[1] I remember reading a great concise article about logical positivism in Roger Bowley and Mariana Sanchez's book on Statistical Mechanics which I strongly recommend.

[2] The variable W measures the number of accessible states in a system. So in a way it is a measure of entropy of the system.

Benefits of the Flipped Classroom Method

I have been working as a full time instructor for around 5 years now. I have used different methods and strategies for effectively teaching my classes. These strategies, for instance, include spending the whole class time giving a power point lecture, using clickers to make student answer a few multiple choice questions, taking some time in between every lecture to assign students a problem to work on, starting a youtube channel, etc. Recently, I have used a strategy at CCU (where I am currently a VAP) that seems to be working more than any other strategy I have used.

One of the issues with millennials is that most of them cannot focus on anything for more than a few minutes. They always get distracted by their cell phones, laptops and other gizmos they carry. The only way to make them focus is to create an environment in which they feel they have been challenged to complete a task and they are working in a group. I have recently been experimenting on the flipped classroom method more and have noticed that the students are more involved than ever. I have been using the following strategy: 

1. The students are given the chapter as a reading assignment. 

2. I dont like flipping the classroom completely because it is almost impossible to make sure that every student reads the chapter (we can have a reading quiz at the start to make sure they read but they usually dont like this idea!). Therefore, the first half an hour (approximately) of the class is typically an interactive board lecture in which the students are continuously asked questions which reminds them to read the chapter. This lecture focuses on the concepts they need to solve the problems on the worksheet. They are continuously reminded that the exam is going to look similar to the worksheet so they take the problems on the worksheet seriously.

3. They are then given the worksheet and they solve it individually but discuss the concepts with each other while solving. The students are only allowed to leave as groups so students who finish quickly cannot leave until everyone in their group is done. This encourages them to discuss the problems more.

4. I continuously move around the classroom to make sure I am ready to answer any questions they have. I try to notice students who appear to be left behind from the herd or show signs of confusion  so that they can be pushed in the right direction.

I have seen the following benefits of this approach:

1. Students who are quick in working out the problems help their peers and that group usually finishes earlier. The students or groups who are struggling stay longer and interact with the professor more.

2. The professor interacts with the students more and this overcomes the communication barrier between the student and the professor.

3. Its always tricky but I try to make sure that the worksheets are not really long so that their is always some time at the end of the class to interact with students who are struggling. The office hours never work for all the students (and most of them only show up close to the exam) so some class time at the end left for student interaction is very useful.

I should mention that I am in a work environment that's conducive to this strategy as well, i.e., my classes are around two hours long and the classrooms are designed for group work. So I might have to modify the strategy if my work environment changes. I am not sure what the outcome of the whole strategy is going to be but from an instructors point of view the students appear to be very involved in the class. 

Finally, I am not claiming that this is the best method to teach. I am quite sure that, as I move ahead in my career, I will find newer and better ways to teach and learn more from my colleagues but I have seen firsthand as to why the flipped classroom method enhances student learning.

The Problems in Science These Days

The scientific community is composed of human beings and suffers from problems similar to social systems, such as religious or political systems, being implemented by humans around the world in different forms. This might sound a bit of an exaggeration but let me put forth the characteristics of an established belief or political system and discuss how all of these characteristics are found in the contemporary scientific community. I have a career in science and have spent most of my scientific career trying to propose new ideas. I have succeeded sometimes but not always, however, with time I have gained a better insight on how the community really functions.

Any enforced belief or political system managed by human beings has the following salient features:

1. An atmosphere in which you cannot openly question established ideas.

2. A system to implement established ideas.

3. People who question established ideas are ridiculed or have to suffer in some way.

4. A group of people who play the role of policing and monitor people who question these ideas.

The current system in academia is mostly designed to teach what is established in science and not to question it. For instance there are no courses in academia that train students to question established theories and make sure their bias is not involved in judging these theories. There should be courses in universities, especially at the graduate level to teach students to be creative, just as we teach kids to be creative in art classes.

Furthermore, there are loop holes or short comings in any theory of science, even the established ones. When professors teach these theories they tend to not touch on these loop holes either because they do not know of these themselves or because they do not want to present these theories as incomplete.

The second point mentioned above is the presence of a system to implement established ideas. The journals in science mostly play this role. In order to establish a career in science we have to make sure that we continue publishing. However, the journals make sure that they publish ideas that either agree with established theories or deviate slightly from them. The famous preprint website Arxiv is a great example of how established theories are enforced. I have personal experience of being on their watch list. At some point in my career they would always put my papers on hold. They have a very secretive system of reviewing papers which they do not usually make public. The famous nobel laureate, Prof. Brian Josephson has been trying to confront them for a while now. There is a website called Vixra established by Philip Gibbs to try to present an alternative to arxiv. There is also a website that presents unpleasant experiences of different people with Arxiv.

I wrote a blog earlier in which the last two points are made evident. The last two points are especially relevant these days with the announcement of a plan for a new collider by CERN (about which I wrote here). Sabine Hossenfelder, who is not in favor of this plan is being criticized and in some cases ridiculed (see Lubos Motl’s blog) for saying something that senior physicists see as an attack on their plans to make progress in science. Similarly, people writing in favor of this plan are being celebrated.

If you disagree with what I have discussed above then there is a good chance you don't know about what happened to Ludwig Boltzmann, one of the most celebrated physicists in the history of science. He was frequently ridiculed by his fellow scientists because his theories were considered incorrect by them. This is understood to be one of the reasons behind his suicide.  His theories, however, laid the foundations of modern statistical mechanics. Boltzmann's time was much long ago, the scientific community is much bigger today and worst of all their is a lot of involvement by politicians (who always make things worse). Time will tell if things change for the better or worse but we need to think of strategies to cleanse the scientific community of these problems otherwise science will end up just being another belief system.