A Universe within a Universe

Have you ever thought or read about a Universe that had different physical laws, where the speed of light was different, where faster of light travel was possible. If you did, then that's great, but let me tell you that we do not have to go out of the Universe to see what happens if the physical laws were different. Condensed matter systems are a great example of what can happen if the physical laws of our Universe are different.

It is well known that the properties of particles change in a condensed matter system. For example, electrons can have a different effective mass in various materials, they can travel faster than the speed of light in that material (the origin of the so-called Cherenkov radiation), light has a different speed in various materials, etc. There is a lot of progress being made in condensed matter physics. Recently, a research group observed magnetic monopoles in a class of topological materials.

Condensed matter lies at the core of many ideas in particle physics as well. The Higgs mechanism is one such example. In fact, in my view, particle physicists have just modeled the whole Universe on a condensed matter system. In a simplistic picture, a condensed matter system can be thought of a region where there is an effective potential. Due to this potential the properties of particles, such as their effective mass, can change. So particle physicists by introducing the Higgs field (which is done by adding a potential in the Lagrangian) have essentially made the Universe a big condensed matter system. So, in my view, what they have done with the Higgs mechanism is nothing non-trivial.

Another point that I would like the reader to think about is the assumptions under which Einstein's theory of special relativity was formulated. It was based on light, in particular, the concepts time dilation and length contraction. The question is, do these concepts change in condensed matter systems? I always think what would happen if we lived in a condensed matter system where faster than speed of light is possible. What would happen to the theory of relativity in such a system? This is again a philosophical way of thinking which is essentially forbidden in physics and academia in general. Classical physicists during Einstein's time used to think about ideas philosophically because they were typically well versed in philosophy. These days physicists are mostly like materialistic robots who live in a bubble of ideas they studied during their PhD. Most of them stay in that bubble their whole life and if you talk to them about these ideas their response is sometimes of fascination but most of the cases their brains cannot process a philosophical approach. They just want to take their favorite equation, twiddle it and publish a new paper.

Therefore, I view condensed matter systems as terrestrial demonstrations of the multiverse (although I am not a big fan of the theory of multiverse that the particle physicists have proposed). Note that this is not the standard way physicist would refer to condensed matter systems (there is a reason I call myself a non-conformist). This view of mine has evolved from my research in Lorentz violation.

Can Lorentz Violation Unify Particle Physics and Condensed Matter Physics?

Physics today featured a great article that discusses unification in physics. I especially want to highlight the unification of phenomenon in condensed matter and particle physics. In my view, one of the main reasons that unification of particle physics and condensed matter physics is not taking place is that the latter involves phenomena that involves Lorentz violating operators, and particle physicists abhor any mention of this (Read more about why this is the case here). 

I published a paper (arXiv:1206.2530) in 2012 showing that the well known Rashba interaction in condensed matter physics can be obtained from Lorentz violation. In a follow up paper (arXiv:1403.7622) I pointed out various terms in condensed matter physics that are already there in Lorentz violation but these terms are well enhanced in condensed matter physics. The fact that we have enhanced Lorentz violation indicates that we might be missing something in our understanding of Lorentz invariance. But, again, particle physicists are so obsessed with Lorentz symmetry that they will never consider this as a possible path of unification between the two fields. Another issue is that condensed matter physicists are usually unaware of this possible connection as well.

So in my opinion, there is clear indication that the unification might be possible but physicists from both fields are not very inclined to take an interdisciplinary approach.

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:

--------------------------------------------------------------------------------------------------------------------------

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,


--------------------------------------------------------------------------------------------------------------------------


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


--------------------------------------------------------------------------------------------------------------------------


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,

--------------------------------------------------------------------------------------------------------------------------

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.


--------------------------------------------------------------------------------------------------------------------------


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.