Category Archives: Physics

A “higher graded” version of supersymmetry and superspace

board In a preprint On a ℤ₂ⁿ-Graded Version of Supersymmetry I construct a “higher” graded version of the extended supersymmetry algebras and construct the corresponding generalisation of Minkowski superspace.

Supersymmetry is a powerful non-classical symmetry that relates bosons and fermions. A geometric understanding of this can be found under the umbrella of “superspace” methods, which rely on the theory of supermanifolds. At a basic level, one starts with Minkowski space-time and then appends to this anticommuting spinor coordinates. By anticommuting we mean that

θ1 θ2 = – θ2 θ1

The fact that we append object that anticommute is deeply tied to that fact that quasi-classically, fermionic fields require us to use such weird things. This is really a form of the Pauli exclusion principle.

However, from a mathematical point of view, there is no reason why we cannot append spinors with more exotic relations between them. Indeed, people have considered “non-anticommuting superspaces” inspired by the way string theory should modify space-time on the smallest scales. In the preprint, I consider a very mild version of this non-anticommutativity by appending spinors that commute (i.e., the order does not matter) up to a sign given by a ℤ₂ⁿ-grading.

This leads to spinors that square to zero (as they should), yet commute amongst themselves! This is very different from the standard theory of supermanifolds and supersymmetry. In fact, we are immediately reminded of Green-Volkov parastatistics. I comment on this in the preprint, though parastatistical versions of “superspace” were not my main motivation with this work.

It seems that just about everything can be generalised to this higher graded setting using the theory of ℤ₂ⁿ-geometry, which is itself a new and developing piece of mathematics. In particular, a higher graded version of Minkowski superspace is given and the corresponding supersymmetry transformations are explored in the preprint.

Can one disprove special relativity with high school mathematics?

Is it possible using mathematics that is not much beyond high school mathematics to prove that special relativity is wrong? And what does that even mean?

The mathematics of special relativity
It is more-or-less true that Einstein’s original works on special relativity do not really use any highbrow mathematics. In a standard undergraduate introduction to the subject no more than linear algebra is really used: vector spaces, matrices and quadratic forms.

So, as linear algebra is well-founded, one is not going to find some internal inconsistencies in special relativity.

Moreover, today we understand special relativity to be based on the geometry of Minkowski space-time. Basically, this is Euclidean with an awkward minus sign in the metric. Thus, special relativity, from a geometric perspective, is as well-founded as any thing in differential geometry.

So one is not going to mathematically prove that special relativity is wrong in any mathematical sense.

On to physics…
However, the theory of special relativity is falsifiable in the sense of Popper. That is, taking into account the domain of validity (ie., just the situations you expect the theory to work), experimental accuracy, statistical errors etc. one can compare the theoretical predictions with what is measured in experiments. If the predictions match the theory well, up to some pre-described level, then the theory is said to be ‘good’. Otherwise the theory is ‘bad’ and not considered to be a viable description of nature.

In this sense, using not much more that linear algebra one could in principle calculate something within special relativity that does not agree well with nature (being careful with the domain of validity etc). Thus, one can in principle show that special relativity is not a ‘good’ theory by finding some mismatch between the theory and observations. This must be the case if we want to consider special relativity as a scientific theory.

Is special relativity ‘good’ or ‘bad’?
Today we have no evidence, direct or indirect, to suggest that special relativity is not a viable description of nature (as ever taking into account the domain of validity). For example, the standard model of particle physics has at its heart special relativity. So far we have had great agreement with theory and experiment, the electromagnetic sector is extremely well tested. This tells us that special relativity is ‘good’.

Even the more strange predictions like time dilation are realised. For example the difference in the life-time of muons as measured at rest and at high speed via cosmic rays agrees very well with the predictions of special relativity.

Including gravity into the mix produces general relativity. However, we know that on small enough scales general relativity reduces to special relativity. Any evidence that general relativity is a ‘good’ theory also indirectly tells us that special relativity is ‘good’. Apart from all the other tests, I offer the discovery of gravitational waves as evidence that general relativity is ‘good’ and thus special relativity is also ‘good’.

The clause
The important thing to remember is that the domain of validity is vital in deciding if a theory is ‘good’ or ‘bad’. We know that physics depends on the scales at which you observe, so we in no way would expect special relativity be a viable description across all scales. For example, when gravity comes into play we have to consider general relativity.

On the very smallest length scales, outside of what we can probe, we expect the nature of space-time to be modified to take into account quantum mechanics. Thus, at these smallest length scales we would not expect the description of space-time using special relativity to be a very accurate one. So, no one is claiming that special relativity, nor general relativity is the final say on the structure of space and time. All we are claiming is that we do have ‘good’ theories by the widely accepted definition.

Are all claims that relativity is wrong bogus?
Well, one would have to examine all claims carefully to answer that…

However, in my experience most objections to special relativity are based on either philosophical grounds or misinterpreting the calculations. Neither of these are enough to claim that Einstein was completely wrong in regards to relativity.

The Polish and Welsh contributions to the discovery of gravitational waves

I just want to acknowledge the contributions of two teams to the discovery of gravitational waves. These groups are only part of the wider community and I highlight them for purely personal reasons.


The Polish group

The Virgo-POLGRAW group,  lead by   Prof. Andrzej Królak at IMPAN.


The Welsh group

The Cardiff Gravitational Physics Group,  and within that the Data Innovation Institute lead by Prof Bernard F Schutz.




On the physics of chocolate

Researchers at Technische Universität München, Germany, have reported that molecular dynamics can be used to gain new insights into the chocolate conching [1].

Chocolate conching is the stage of manufacturing where aromatic sensation, texture and mouthfeel are developed.

This work seems to be the first to attempt to properly understand the role of lecithins in chocolate production.

Physics, helping to build a tasty more palatable world.

[1] M Kindlein, M Greiner, E Elts and H Briesen, Interactions between phospholipid head groups and a sucrose crystal surface at the cocoa butter interface, 2015 J. Phys. D: Appl. Phys. 48 384002.

Chocolate physics: how modelling could improve mouthfeel, IOP website.

The 2nd Conference of the Polish Society on Relativity

I will be attending the 2nd conference of the Polish Society of Relativity which will celebrate 100 years of general relativity.

The conference is in Warsaw and will be held over the period 23-28 November 2015.

The invited speakers include George Ellis, Roy Kerr, Roger Penrose and Kip Thorne. I am a little excited about this.

Registration is now open and you can follow the link below to find out more.

Polskie Towarzystwo Relatywistyczne

Quantum gravity

The subject of a quantum theory of gravity is interesting, technical and very difficult. However, there are three basic principles that we expect such a theory to obey.

Creating a full quantum theory of gravity seems to be out of our reach right now. String theory comes close, but the full theory here is not understood. Loop quantum gravity also offers a good picture, but again technicalities spoil achieving the goal.

I am no expert in quantum gravity, but I thought it maybe interesting to outline three basic ‘rules’. The full quantum theory of gravity should be:

  1.  Renormalisable (maybe not perturbatively) or finite.
  2. Background independent.
  3. Reducible to general relativity (plus small corrections) in a sensible classical limit.

As a warning, I will not be too technical here, but will use some standard language from quantum field theory.

The standard methods of quantum field theory are to expand the theory about some fixed configuration, usually the vacuum, and consider small fluctuations about this reference configuration. However, in doing so some techniques are needed to remove the appearance of infinite values of things you would like to measure in the lab. These methods are collective known as ‘perturbative renormalisation’. For example, we know that the quantum theory of electrodynamics can be handled properly using these methods.

However, general relativity as described by Einstein is not amenable to methods of perturbative renormalisation. Well, this is true if we want a full theory. What one can do is consider quantum general relativity as an effective theory. That is we accept that at some energy scale the theory will breakdown, but as long as we are not at that scale the theory is okay. By adding a ‘cut-off’ we can understand quantum general relativity using Feynman diagrams to ‘one-loop’ and calculate graviton scattering amplitudes and so on.

Interestingly, there is some evidence that general relativity or something close to it is nonperturbatively renormalisable; this is known as asymptotic safety. With no details, the idea is that quantum general relativity is not ‘sick’ and well-defined, just not as a perturbative theory like quantum electrodynamics. This is fascinating as it means that a proper quantum theory of gravity may not be a theory of gravitons after all! Recall that small ripples in the electromagnetic field are quantised and understood to be photons. Maybe it is not really possible to describe quantum gravity in a similar way where small ripples in space-time are quantised.

Alternatively, a full theory of quantum gravity could be finite. That is we can employ perturbative methods, but do not need renormalisation techniques. Amazingly, we know of supersymmetric Yang-Mills theories that are finite. Moreover, superstring theory is also finite (I am unsure as to how rigours the proof are here, but the string community generally accept this as fact). It maybe possible that the full theory of quantum gravity is finite from the start. This suggests that looking at supersymmetric theories of gravity is a good idea, but by no means the only thing one can think about.

In short, any full quantum theory of gravity must allow us to calculate things we can hope to measure.

Background independence
This means that the theory should not depend on any chosen background geometric fields. In particular, this is taken to mean that the theory should not require some chosen background metric.

String theory as it stands fails on this. However, string theory is usually employed using perturbation theory and so some classical background is chosen, often 10-d flat space-time.

Loop quantum gravity seems better in this respect, but it has other problems.

In short, any full quantum theory of gravity should not require us to fix the geometry (and maybe topology) from the start.

Reduce to general relativity
General relativity has been so successful in describing classical gravitational phenomena. It is tested to some huge degree of accuracy and so far no deviations from it’s predictions have been found. General relativity is a good theory within the expected domains of validity.

Thus, any quantum theory of gravity must in some classical limit reduce to general relativity, up to small corrections. These quantum corrections must be small enough as not to be seen already in astrophysics and cosmology.

If a quantum theory of gravity cannot be shown to reduce to general relativity in some limits (there maybe several ways of doing this) then we cannot be sure that we really have a quantum theory of gravity.

Today we know that string theory gives us general relativity + small corrections. In essence this is because the spectra of closed string theory contains a spin-2 boson, via rather general arguments we know that this has to be the graviton and the field equations are essentially the Einstein field equations. (Remember this is all in perturbation theory).

Recovering general relativity from loop quantum gravity has yet to be done. This I would say is a sticking point right now.

In short, any full quantum theory of gravity must reproduce the phenomena of general relativity is some classical limit(s).

The original review of general relativity

It has now been 99 years, to the day (20/03/2015) since Einstein published his original summary of general relativity [1].

Before that he had published some incomplete works that have the wrong field equation, but the key ideas were in place by 1914. The core idea is that space-time is dynamical and interacts with the matter and energy.

It is hard to believe that this theory of gravity has stood the test of time so well. We know for various reasons that general relativity cannot be the complete picture, but nature just refuses to give us hints on what could be the more complete theory.

[1] A. Einstein, Die Grundlage der allgemeinen Relativitätstheorie, Annalen der Physik 354 (7), 1916, 769-822.

Supersymmetry and mathematics

CERN Prof Beate Heinemann, from the Atlas experiment at CERN had said that they may detect supersymmetric particles as early as this summer. But what if they don’t?

What if nature does not realise supersymmetry? Has my interest in supermathematics been a waste of time?


We hope that we’re just now at this threshold that we’re finding another world, like antimatter for instance. We found antimatter in the beginning of the last century. Maybe we’ll find now supersymmetric matter

Prof Beate Heinemann [1]

In nature there are two families of particles. The bosons, like the photon and the fermions, like the electron. Bosons are ‘friendly’ particles and they are quite happy to share the same quantum state. Fermions are the complete opposite, they are more like hermits and just won’t share the same quantum state. In the standard model of particle physics the force carriers are bosons and matter particles are fermions. The example here is the photon which is related to the electromagnetic force. On the other side we have the quarks that make up the neutron & proton and the electron, all these are fermions and together they form atoms.

Supersymmetry is an amazing non-classical symmetry that relates bosons and fermions. That is there are situations for which bosons and fermions can be treated equally. Again note the very different ‘lifestyle’ of these two families. If supersymmetry is realised in nature then every boson will have a fermionic partner and vice versa. In one swoop the known fundamental particles of nature are (at least) doubled! Moreover, the distinction between matter and forces becomes blurred!

A little mathematics
Without details, the theory of bosons requires the so called Canonical Commutation Relation or CCR. Basically it is given by

\([\hat{x},\hat{p}] = \hat{x} \hat{p} – \hat{p} \hat{x} = i \hbar \).

Here x ‘hat’ is interpreted as the position operator and p ‘hat’ the momentum. The right hand side of this equation is a physical constant called Planck’s constant (multiplied by the complex unit, but this is inessential). The above equation really is the basis of all quantum mechanics.

The classical limit is understood as setting the right hand side to zero. Doing so we ‘remove the hat’ and get

\(xp- px =0 \).

Thus, the classical theory of bosons does not require anything beyond (maybe complex) numbers. Importantly, the order of the multiplication does not matter here at all, just think of standard multiplication of real numbers.

The situation for fermions is a little more interesting. Here we have the so called Canonical Anticommutation Relations or CAR,

\(\{\hat{\psi}, \hat{\pi} \} = \hat{\psi} \hat{\pi} + \hat{\pi} \hat{\psi} = i \hbar\).

Again these operators have an interpretation as position and momentum, in a more generalised setting. Note the difference in the sign here, this is vital. Again we can take a classical limit resulting in

\(\psi \pi + \pi \psi =0\).

But hang on. This means that we cannot interpret this classical limit in terms of standard numbers. Well, unless we just set everything to zero. Really we have taken a quasi-classical limit and realise that the description of fermions in this limit require us to consider ‘numbers’ that anticommute; that is ab = -ba. Note this means that aa= -aa =0. Thus we have nilpotent ‘numbers’, that is non-zero ‘numbers’ that square to zero. This is odd indeed.

Supermathematics and supergeometry
In short, supermathematics is all about the algebra, calculus and geometry one can do when including these anticommuting ‘numbers’. The history of such things can be traced back to Grassmann in 1844, pre-dating the applications in physics. Grassmann’s interests were in linear algebra. These odd ‘numbers’ (really the generators of) are usually referred to as Grassmann variables and the algebra they form a Grassmann algebra.

One of my interests is in doing geometry with such odd variables, this is well established and a respectable area of research, if not very well represented. Loosely, think about simple coordinate geometry in high school, but now we include these odd numbers in our description. I will only reference the original paper here [2], noting that many other works evolved from this including some very readable books.

What if no supersymmetry in nature?
This would not mean the end of research into supermathematics and its applications in both physics & mathematics.

From a physics perspective supersymmetry is a powerful symmetry that can vastly simplify many calculations. There is an industry here that works on using supersymmertic results and applying them to the non-supersymmetric case. This I cannot see simply ending if supersymmetry is not realised in nature, it could be viewed as a powerful mathematical trick. In fact, similar tricks are already mainstream in physics in the context of quantising classical gauge theories, like the Yang-Mills theory that describes the strong force. These methods come under the title of BRST-BV (after the guys who first discovered it). Maybe I can say more about this another time.

From a mathematics point of view supergeometry pushes what we know as geometry. It gives us a workable stepping stone into the world of noncommutative geometry, which is a whole collections of works devoted to understanding general (usually associative) algebras as the algebra of functions on ‘generalised spaces’. The motivation here also comes from physics by applying quantum theory to space-time and gravity.

Supergeometry has also shed light on classical constructions. For example, the theory of differential forms can be cast neatly in the framework of supermanifolds. Related to this are Lie algebroids and their generalisations, all of which are neatly described in terms of supergeometry [3].

A very famous result here is Witten’s 1982 proof of the Morse inequalities using supersymmetric quantum mechanics [4]. This result started the interest in applying physics to questions in topology, which is now a very popular topic.

In conclusion
Supermathematics has proved to be a useful concept in mathematics with applications in physics beyond just ‘supersymmetry’. The geometry here pushes our classical understanding, provides insight and answers to questions that would not be so readily available in the purely classical setting. Supergeometry, although initially motivated by supersymmetry goes much further than just supersymmetric theories and this is independent of CERN showing us supersymmetry in nature or not.

[1] Jonathan Amos, Collider hopes for a ‘super’ restart, BBC NEWS.

[2] F. A. Berezin and D. A. Leites, Supermanifolds, Soviet Math. Dokl. 6 (1976), 1218-1222.

[3] A Yu Vaintrob, Lie algebroids and homological vector fields, 1997 Russ. Math. Surv. 52 428.

[4] Edward Witten, Supersymmetry and Morse theory, J. Differential Geom. Volume 17, Number 4 (1982), 661-692.

Breakthrough Prize in Fundamental Physics Symposium Videos

Higgs event The videos from the Breakthrough Prize in Fundamental Physics Symposium are now available to watch, follow the link below. The symposium was held on the 10th November at Stanford University and co-hosted by UC-San Francisco and UC-Berkeley.

There was a panel discussion with Adam Riess, Brian Schmidt, Saul Perlmutter and Yuri Milner, and individual 20-minute talks from Nima Arkani-Hamed, Juan Maldacena, Andrei Linde, Stephen Shenker, Alexei Kitaev, Patrick Hayden, John Preskill, Nathan Seiberg, Joe Polchinski and Uros Seljak.

Breakthrough Prize in Fundamental Physics Videos 2015

Why there is no equivalence principle for electromagnetic theory

Quite regularly one will come across a website, blog or some thread on a forum that says the gravity is just electromagnetism. For sure they are not the same. However, I am not sure what would constitute the ‘nail in the coffin’ for all these clams.

With this in mind, I am not going to try to debunk every such claim. However, I think the reason for this seeming equivalence comes from the static limit. In this limit it is true that there are many formal similarities between Newtonian gravity and electrostatics. In fact there are many formal similarities in the full classical theories, both are very geometrical in nature, but I won’t go into details here.

I just wanted to point out one very clear difference between gravity and electromagnetism that can be seen in this static limit. That is the lack of a generalisation of the equivalence principal for electromagnetic theory. This principal in gravity is very important and one that I will comment on in due course.

The static limit
I am guessing that we have all seen Coulombs law for electrostatics and Newton’s law for gravity. Let me just write them down

\( F = k \frac{qQ}{r^{2}}\),

where \(k\) is Coulombs constant, it is a measure of the strength of the electrostatic force and \(q\) and \(Q\) are the electric charges of two point particles. The above expression is the electrostatic force between two such charged particles.

Similarly we have Newton’s law of gravity

\( F = G \frac{mM}{r^{2}}\),

where \(G\) is Newton’s constant which measures the strength of the gravitational force and \(m\) and \(M\) are the masses of two point particles. The above expression measures the gravitational force between these particles.

These expressions for the forces should be seen as the static non-relativistic limit. I just mean that as long as the particles are moving slow enough then the change in the fields can be viewed as instantaneous. This is okay for many applications, but it is not the full picture. However, it is the one you see at high school.

The formal similarities at this level are clear. You just need to swap constants and interchange charge and mass. But this does not mean they are the same, and there is a subtle issue here. Before that we need Newton’s law of motion

Newton’s second law
Newton’s second law tells us that the force exerted on a particle is proportional to the acceleration of that particle. Moreover, the constant of proportionality is the (inertial) mass.

\(F = m a \).

That is all we will need.

The gravitational equivalence principal
Let us think of the particle of mass \(m\) as a test particle. That is we will think of how it is moving in the gravitational field generated by the particle \(M\) and that it does not generate a gravitational field of its own. This approximation is good for small objects moving in the gravitational field of big objects; say planets around a star or satellites in orbit around the Earth.

Now we can examine how the small mass is influenced by the big mass. We should just equate the two expressions due to Newton

\( ma = G \frac{mM}{r^{2}} \),

for which we can solve for the acceleration

\(a = G \frac{M}{r^{2}}\).

We notice a very amazing thing. The small mass cancels from both sides of the equation. (We assume that gravitational mass and inertial mass are the same. This seems to be case in nature.)

This means that the motion of the test particle does not care about any of the intrinsic properties of that particle. The only things it does care about is the initial position and velocity. This is one form of the equivalence principal which has it’s roots in the experimental work of Galileo- acceleration of a test particle due to gravity is independent of the mass being accelerated.

Thus, really all the information about the test particles motion is encoded in the gravitational field alone. All test particles whatever their mass will behave the same. This is the clue that gravity can be formulated very intrinsically in terms of space-time geometry alone; this leads to general relativity which is not the subject for today.

The electromagnetic version
Now let us play the same game with electrostatics…

\( ma = k \frac{q Q}{r^{2}} \),

where we think of the test particle \((m, q)\) moving in the electric field generated by the particle \((M,Q)\). Now solving for the acceleration gives us

\(a = k \left(\frac{q}{m}\right) \frac{Q}{r^{2}} \).

Now we see the difference. The motion of the test particle does depend on the intrinsic properties of that particle, namely the charge-mass ratio. There is no similar statement like for gravitational physics; there is no equivalence principle.

Closing remarks
Everything above is done in a particular limit of the full classical theories. The same picture is true when we examine the motion of test particles in general relativity and the Lorentz force in electromagnetic theory. I have chosen these limits because I think this is clear and also the source of the instance that gravity is just electromagnetic theory. I have pointed out one clear and explicit difference.

One can do the same with Coulomb’s law for the magnetic force. Although magnetism is a bit more complicated we can examine the situation for point-like poles. This is okay for small enough poles that are well separated. You will reach the same conclusion that there is no equivalence principal in this situation. Thus, gravity is not magnetism either.