Monthly Archives: February 2012

General

H5N1 and the ethos of research

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flasks

There has been a lot of controversy about research work done on the H5N1 bird flu virus. In particular there is some discussion about if the results should be openly published.

The fear

The fear is that the experiments based on mutations of the H5N1 bird flu virus could be used by terrorist groups. The experiments show how the virus could mutate into a form that would spread easily amongst humans.

There are two papers submitted to Nature, one written by Kawaoka (Wisconsin-Madison) and the other by Fouchier (Erusmus Medical Centre). You can read a Q&A from Nature publishing group here.

The US National Security Advisory Board for Biotechnology (NSABB) has asked Nature to remove some of the information that could be useful to terrorists. World Health Organization in Geneva has stated that more discussion is needed.

There is also the opinion that the dissemination of the full work is important if we want to tackle the natural threat of bird flu.

I am far from an expert in this area and cannot offer an informed professional opinion about this specific issue. I suggest people start by reading the BBC report.

The wider issue

The problem is that not disseminating the work in full goes against the ethos of modern science. Progress is made only by sharing ideas and results. The scientific community will pick up on parts of a given work and further develop them. This is how progress is made and prevents scientists “re-discovering the wheel” every time then engage in research.

Science is there for the benefit of wider society. This is not just in practical medical or engineering terms, but also culturally. Couple this with the fact that most fundamental science is paid for by public tax money, scientists have a moral duty to disseminate their work.

But…

The debate really starts when scientific work can clearly be perverted and used for harm. One has to think about the greater good.

It is an unfortunate fact that human conflicts drive science and invention. We must reconcile our position with the assertion that all scientists and engineers involved in “war work” are developing terrible things for a greater good, at least as they see it.

So that said, we still have the issue of scientific censorship by governments and other agencies. Generically, scientists will want to publish their work, with the greater good in mind.

Now what?

We need academic and scientific freedom. However, that does not mean a “free for all” attitude and we do have many safeguards about conducting ethical research.

It could be possible that some research, principally the details are just so sensitive that they do pose a real threat. Scientists, governments and society have to think about this.

Maybe we need a wider debate about ethical dissemination.

Physics

The experimental status of general relativity

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It is now folklore that general relativity is well tested and that there are no experiments that disagree with the predictions. This goes back to the early days, Einstein calculated the perihelion of Mercury   accurately in 1915  and Eddington in 1919 proved that the bending of light around the Sun is in agreement with general relativity.

Since then there has been many different experiments aimed at testing different aspects of the theory. These include detailed analysis of the time delays of messages to spacecraft all the way to studies of binary pulsars.

What I was not aware of is just how accurate general relativity is.

The Eötvös Experiment

One of the founding pillars of general relativity is the weak equivalence principle. It basically says that the passive gravitational mass is the same as the active inertial mass. This idea is much older then general relativity and Newton was the first to do experiments testing this.

Eötvös used a two equal masses of different composition on a torsion balance to test this principle. More details can be found here. The Eötvös parameter is defined as

\(\eta = 2 \frac{|a_{1}-a_{2}|}{|a_{1}+ a_{2}|}\),

which is the fractional ration of the accelerations of the two masses. If gravity couples differently to the different materials then this should show up as a non-zero value of this parameter.

Eötvös was able to get this parameter down to \(10^{-9}\), so clearly very small.

The Eöt-Wash group at Washington using modern techniques have brought this value to \(\eta \approx 10^{-13}\).

Local Lorentz Invariance

General relativity also requires that locally we have Lorentz invariance. The breaking of Lorentz invariance would imply some universally preferred rest frame. One way to test this is to look at the speed of light. So let us define

\(\sigma = c^{-2} -1\).

Units have been picked here so that the “usual speed of light” is one. So in general relativity \(\sigma =0\) locally.

Examine very carefully the energy levels of atoms and how this changes due to our orientation in the Universe one can test Lorentz invariance of the electromagnetic sector.

Such test give \(\delta \approx 10^{-22}\).

There has been a bit of interest in examining the potential for Lorentz violating in extensions of the standard model. These tend to have motivation from quantum gravity where it is expected that local Lorentz invariance will be broken.

Other tests

Other tests, both direct and indirect have been preformed and all give good agreement with general relativity. This includes:

This is both reassuring and frustrating for theoretical physics. The lack of experimental direction on what replaces general relativity at the quantum level has, in my opinion, not helped the quest for quantum gravity. But that is another story.

For more details of the experimental tests of general relativity see [1,2].

References

I won’t give references to the original material, see the following for details:

[1] Clifford M. Will. The Confrontation between General Relativity
and Experiment. Living Rev. Relativity, 9, (2006), 3.

[2] S G Turyshev. Experimental tests of general relativity: recent progress and future directions. Phys.-Usp. 52 1, 2009.

Astronomy

Essential amateur astronomy equipment

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Planisphere and a torch with red filter.

I thought I would share with you some of the basic equipment that I consider to be essential when starting amateur astronomy.

In reality you need no more than your eyes, but you will soon want to be more methodical than just looking up at night.

My bare minimum equipment is as follows:

  1. Planisphere: an instrument that can be adjusted to display the stars at a given date and time. This is very useful for planning your sessions as well as navigating your way around the night sky.
  2. Torch: fit the torch with a red filter. I use a mini Maglite that can be fitted with a purpose build plastic filter. Other methods include painting the bulb red or attaching a red plastic sweet wrapper over the end of the torch fastening it with elastic bands. Red light effects your night-vision the least. You need a torch for checking star charts and making your observation notes.

  3. Cartes du Ciel: this is a free planetarium programme for your computer. There are versions for Linux, Mac OS X and Windows. Cartes du Ciel allows you to make star charts that can help you plan your sessions. It is quite easy to use and quite accurate.

    If you have a laptop you can take Cartes du Ciel outside with you, it has a night vision mode.

    The power of planetarium programmes is that they allow you to track the planets and the Moon. The planisphere cannot do that.

  4. A sky atlas: charts of the constellations, locations of deep sky objects etc. I use Philip’s Atlas of the Universe by Patrick Moore. This book has nice charts of the constellations. There are plenty of other good books that have very usable charts.
  5. A compass : You will soon learn which way is North by the position of the stars. Until then a compass will help orientate yourself. I always have a compass to hand while observing, just in case.

Other things to be aware of is the weather and the temperature drop at night. Dress appropriately, take a woolly hat and gloves.

Astronomy

My Telescope: Bresser Skylux NG 70-700 refractor

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My Bresser Skylux NG 70-700 refractor I own a small telescope, the Bresser Skylux NG 70-700 refractor. I have used it to view the moons of Jupiter, the phases of Venus, the Andromeda galaxy and the Orion Nebula. The most impressive, and I think what the telescope is best for is observing the Moon.

For your enjoyment, I have taken a few pictures of the telescope.

The telescope has an aperture of 70mm and a focal length of 700mm.

I changed the original finder scope to a laser finder. The original finder scope I found to be difficult to use, it has poor focus and thus I was unable to find anything but the very brightest stars using it.

The mount is a EQ3 mount. I found the mount easy to use and quite steady.
As I have already stated, I changed the finder scope to a laser finder. This does not magnify the sky, but places a clear red dot in the viewer. I find this much easier to use than the more traditional cross-hair finder scope.

I have used this telescope for basic astrophotography. In particular I have taken fairly good pictures of the Moon directly through the eyepiece using a rather modest digital camera.

I tend to couple this with a Moon filter which helps with the contrast and helps stop the features on the Moon getting washed out. You can find one of my Moon pictures in an older post here.

The telescope came with three eyepieces: 20mm, 12mm, 4mm. This gives magnifications of 35x-262x. This covers “sensible” magnifications for this telescope. Sometimes cheaper poorly made telescopes come with eyepieces that are unsuitable, this is not the case with the Skylux NG 70-700.

Although the telescope is quite portable I tend to use it in my back garden, which is far from a dark sky site.

This telescope was a Christmas present from my wife back in 2008.

Physics

Pure Energy?

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People often talk about “pure energy” in rather an informal way. In truth there is no proper notion of pure energy. Loosely one often means photons when the term pure energy is used. For example, you may come across statements like: when matter and antimatter collide they annihilate producing pure energy.

Energy is a property of “stuff”; that is a physical system. A configuration of a physical system will have a property that we can indirectly measure, which we call energy. One cannot have energy as some independent “thing”.

As an analogy, you may talk about the colour of a car. Lets say a red car. Being red is a property of the car. One cannot talk about “red” as some notion independent of the objects we see as red in colour. Red does not exist by itself.

So, what is energy?

Informally energy is understood as a property of one physical system that allows it to preform work on another physical system. In essence this means that energy is the property associated with movement or change. It is the “doing” property.

Not that the above tell you what energy is. However, this is not really a problem as physics tends not to deal with the metaphysical notion of existence and what is. Physics deals with mathematically modelling nature. With that in mind, one should keep close the quote (I paraphrase) I believe is due to Feynman:

Energy is a number we can calculate at different points in time and find the same value.

That is energy is something we can calculate, given some configuration of a physical system and (given some technical stuff) we see that the energy does not change in time. That is, it is conserved.

A little more mathematically

We have a very powerful and beautiful theorem that relates symmetry and conservation laws.

Noether’s first theorem: Any differentiable symmetry of the action of a physical system has a corresponding conserved charge.

This theorem is at the heart of modern physics and is based on the calculus of variations.

What does this theorem mean?

Theorems as theorems are by their very nature technical. But we can informally understand some consequences of this statement quite easily.

If the mathematical description of the physical system does not alter upon changes in time then there is a conserved quantity that we call energy.

This is as close to answering the question what is energy? as you can really get. Energy is the quantity that is associated with a physical system not explicitly depending on time.

The caveat here is that the physical system not depend explicitly on time. This is generally reasonable. From a physical perspective this seems natural, any experimental outcome should not depend on when you preform the experiment. Because you get the same result today as you will tomorrow, energy is conserved.

Back to pure energy

I hope I have explained that the notion of “pure energy” is not well founded. Energy is a number that is associated with physics not changing on when you preform your experiments.

Noether’s first theorem makes this association with time and energy explicit. Other common conserved quantities exist:

Symmetry Conserved Quantity
Translations in Time Energy
Translations in Space Linear Momentum
Rotations in Space Angular Momentum

In the same way nobody talks about “pure angular momentum” as some thing in its own right, no one should use the term “pure energy”.

Physics

The trampoline of gravity

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courtesy of gifmania.co.uk Analogies are an important part of understanding, as well as the popularisation of physics.

However, analogies are analogies and at some point always fail to capture the full picture of what is going on. More than that, taking analogies too seriously can lead to misunderstandings.

Einstein’s general relativity basically tells us that massive objects bend the space-time they are sat in and that this is the origin of gravity. To really understand this one has to pull apart the Einstein field equations in all their tensorial beauty. I won’t do that here and now.

A common analogy here is that of a heavy bowling ball placed on a trampoline. The bowling ball deforms the elastic trampoline surface, it sags, and this is similar to how a massive object, say a star, bends the space-time around it.

One can now “model” photons or test particles by using light balls, say ping-pong balls. The point is that these near weightless balls will not deform the trampoline’s elastic surface. When the bowling ball is not on the trampoline the light balls move in straight lines when given a light initial push. When the bowling ball is on the trampoline the light balls no longer follow straight lines, but curved paths. These light balls are attracted to the bowling ball: thus we have gravity!

This is a great analogy for light rays or photons in general relativity. Light is bent around massive objects like stars. If you have access to a trampoline and some heavy and light balls, play around and experiment for yourself.

However, This analogy seems to be the principle source of misunderstandings and even scepticism of general relativity for the untrained.

Conceptionally the analogy breaks down because the trampoline does not represent the three dimensional space we inhabit, or rather a time slice of our four dimensional world. All we have is an embedding of a two dimensional geometry in our three dimensional flat world.

The trouble is that the space-time of general relativity does not require any such embedding in a higher dimensional flat space. It is of course true that mathematically we can always find (isometric) embeddings in higher dimensional spaces of the geometries found in general relativity, but this does not imply that nature uses such things.

The other issue is that the trampoline analogy really misses the curvature of space-time and only highlights space curvature. The ping-pong balls move about the sheet being “attracted” to the bowling ball because of the depression in the elastic sheet. The trouble is that in general relativity test particles, say photons, move in the “straightest possible path” in four dimensions, including time. This fact is missed by the analogy.

So however useful this analogy is, and I say it is useful, it cannot really describe general relativity. Objections, philosophically or otherwise to general relativity cannot be founded on the trampoline analogy.

The great man himself. Moreover, there is plenty of direct and indirect experimental verification that general relativity is a good model of gravity. This fact seems rather inescapable: there are no consistent repeatable experiments that, taking into account the domain of applicability and experimental errors, that suggest that general relativity is not a good model. I may say more about this another time.

In short, love analogies, use analogies, tell other people about analogies, however remember they are analogies and no replacement for mathematical models.

Book Reviews

Quantum Field Theory A Modern Introduction by M. Kaku

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Quantum Field Theory: A Modern Introduction

Quantum field theory is a many faceted subject and represent our deepest understanding of the nature of forces and matter. Quantum field Theory A Modern Introduction by Michio Kaku gives a rather wide overview of many essential ideas in modern quantum field theory.

The readership is graduate students in theoretical physics who already have some exposure to quantum mechanics and special relativity.

The book is divided into three parts.

Part 1 Quantum Fields and Renormalization

Chapter 1 gives a historic overview of quantum field theory. Topics here include: a review of the strong, weak and gravitational interaction, the idea of gauge symmetry, the action principle and Noether’s theorem.

Symmetries and group theory are the subjects of Chapter 2. Topics include: representations of U(1), SO(2), SO(3) and SU(2), spinors, the Lorentz group, the Poincare group and supersymmetry.

Chapter 3 moves on to the quantum theory of spin-0 and spin 1/2 fields. The emphasis here is on canonical quantisation. Topics covered here include: the Klein-Gordon field, propagator theory, Dirac spinors and Weyl neutrinos.

Quantum electrodynamics is the topic of Chapter 4. Again the emphasis is on canonical quantisation. Topics include: Maxwell’s equations, canonical quantisation in the Coulomb gauge, Gupta-Bleuler quantisation and the CPT theorem.

Chapter 5 describes the machinery of Feynman diagrams and the LSZ reduction formula. Topics here include: cross sections, propagator theory, the LSZ reduction formulas, teh time evolution operator, Wick’s theorem and Feynman rules.

The final chapter of part 1, Chapter 6 describes the renormalization of quantum electrodynamics. Topics here include: nonrenormalizable & renormalizable theories, the renormalization of phi-4 theory, regularisation, the Ward-Takahashi identites and overlapping divergences. The renormalization of QED is then broken down into fours steps.

Part 2 Gauge Theory and the Standard Model

Chapter 8 introduces path integrals which are now fundamental in particle theory. Topics here include: path integrals in quantum mechanics, from first to second quantisation, generators of connected graphs, the loop expansion, integration over Grassmann variables and the Schwinger-Dyson equations.

Chapter 9 covers gauge theory. Topics here include: local symmetry, Faddeev-Popov gauge fixing, the Coulomb gauge and the Gribov ambiguity.

The Weinberg-Salam model is the subject of Chapter 10. Topics here include: broken symmetries, the Higgs mechanism, weak interactions and the Coleman-Weinberg mechanism.

Chapter 11 discusses the standard model of particle physics. Topics here include: the quark model, QCD, jets, current algebra, mixing angles & decays and the Kobayashi-Maskawa matrix.

Chapter 12 discusses anomalies and the Ward identities. Topics here include: the Ward-Takahashi identity, the Slavonov-Taylor identities, BRST symmetry & quantisation, anomalies and Fujikawa’s method.

Chapter 12 covers the remormalization of gauge theories. Topics include: counterterms, dimensional regularization and BPHZ renormalization.

The modern perspective of QFT is based on Wilson’s renormalization group. Chapter 14 introduces the reader to this concept in the context of QCD. Topics here include: deep inelastic scattering, neutrino sum rules, the renormalisation group, asympptotic freedom and the Callan-Symanzik relation. The renormalization of QCD is presented via renormalization groups methods.

Part 3 Nonperturbative Methods and Unification

Chapter 15 introduces lattice gauge theory which allows questions in quantum field theory to be numerically tackled on computers. Topics here include: the Wilson lattice, scalars & fermions on the lattice, the strong coupling approximations, Monte Carlo simulations and the renormalization group.

Topological objects in field theory are the topic of Chapter 16. Topics include: solitons, monopoles, instantons & tunneling and Yang-Mills instantons & the theta vacua.

Chapter 17 discusses phase transitions and critical phenomena. Topics covered include: critical exponents, the Ising model, the Yang-Baxter relations, the mean-field approximation and scaling & the renormalisation group.

The idea of unification is the subject of Chapter 18. Topics include: unification & running coupling constants, SU(5), anomaly cancellation, the hierarchy problem, SO(10), technicolor, preons & subquarks and supersymmetry and strings.

Chapter 19 discusses quantum gravity. This chapter is about attempting to construct a perturbative theory of quantum general relativity. Topics include: the equivalence principle, vierbeins & spinors, GUTs & cosmology, the cosmological constant, Kaluza-Klein theory and counter terms in quantum gravity.

Supersummetry is the subject of Chapter 20. Topics covered here include: supersymmetric actions, superspace methods, Feynman rules, nonrenormalization theorems, finite field theories, super groups and supergravity.

Chapter 21 introduces the superstring. Topics include: quantisation of the bosonic string, teh four superstring theories, higher loops, string phenomenology, light-cone string field theory and the BRST action.

The book contains exercises.

Paperback: 804 pages
Publisher: OUP USA; New Ed edition (6 Oct 1994)
Language English
ISBN-10: 0195091582
ISBN-13: 978-0195091588

Research work

Low dimensional contact supermanifolds

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I have been interested in contact structures on supermanifolds. I though it would be useful, and fun to examine a low dimensional example to illustrate the definitions. Let \(M\) be a supermanifold. We will understand supermanifolds to be “manifolds” with commuting and anticommuting coordinates.

For manifolds there are several equivalent definitions. The one that is most suitable for generalisation to supermanifolds is the following:

Definition A differential one form \(\alpha \in \Omega^{1}(M)\) is said to be a contact form if

  1. \(\alpha\) is nowhere vanishing.
  2. \(d\alpha\) is nondegenerate on \(ker(\alpha)\)

This needs a little explaining. First we have to think about the grading here. Naturally, any one form decomposes into the sum of  even and odd parts. To simplify things it makes sense to consider homogeneous structures, so we have even and odd differential forms. Due to the natural grading of differentials as fibre coordinates on antitangent bundle a Grassmann odd  form will be known as an even contact structure and vice versa. The reason for this will become clearer later.

 

The definition of a nowhere vanishing one form is that there exists vector fields \(X \in Vect(M)\) such that \(i_{X}\alpha =1\). Again, via our examples this condition will be made more explicit.

The kernel of a one form is defined as the span of all the vector fields that annihilate the one  form.  Thus we have

\(ker(\alpha) = \{X \in Vect(M)| i_{X}( \alpha)=0 \}\).

The condition of nondegeneracy  on \(d\alpha\) is that \(i_{X}(d \alpha)=0\) implies that  \(X=0\). That is there are non-nonzero vector fields in the kernal of the contact form that annihilate the exterior derivative of the contact form.

On to simple examples. Consider the supermanifold \(R^{1|1}\) equipped with natural coordinates \((t, \tau)\). here \(t\) is an even or commuting coordinate and \(\tau\) is an odd or anticommuting coordinate.

 

I claim that the odd one form \(\alpha_{0} = dt + \tau d \tau\) is an even contact structure.

First due to our conventions, \(dt\) is odd and \(d \tau\) is even,  so the above one form is homogeneous and  odd.

Next we see that if we consider \(X = \frac{\partial}{\partial t}\) then the nowhere vanishing condition holds. Maybe more intuitively we see that considering when \(t = \tau =0\)  the one form does not vanish.

The kernel is given by

\(ker(\alpha_{0}) = Span\left \{   \frac{\partial}{\partial \tau} {-} \tau \frac{\partial}{\partial t} \right \}\).

That is we have a single odd vector field as a basis of the kernel. That is we have one less even vector field as compared to the tangent bundle. Thus we have a codimension \((1|0)\) distribution.

Then \(d \alpha_{0}= d \tau d \tau\) , so it is clear that the nondegeneracy condition holds.

Thus I have proved my claim.

This is just about the simplest even contact structure you can have.

The odd partner to this is given by

\(\alpha_{1} = d \tau {-} \tau dt\)

This is clearly an even one form that is nowhere vanishing. The kernel is given by

\(ker(\alpha_{1}) = Span\left\{ \frac{\partial}{\partial t} {-} \tau \frac{\partial}{\partial \tau}  \right\}\).

Thus we have a codimension \((0|1)\) distribution.

The nondegeneracy condition also follows directly.

For those who know a little contact geometry compare these with the standard contact structure on \(R^{3}\).

There is a lot more to say here, but it can wait.  For those of you that cannot wait, see Grabowski’s preprint arXiv:1112.0759v2 [math.DG]. Odd contact structures are also discussed in my preprints arXiv:1111.4044 and arXiv:1101.1844.

Book Reviews

Topology and geometry for physicists, by C. Nash & S. Sen

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Topology and Geometry for Physicists

Geometry and topology are now a well established tools in the theoretical physicists tool kit. Topology and geometry for physicists by C. Nash & S. Sen gives a very accessible introduction to the subject without getting bogged down with mathematical rigour.

Examples from condensed matter physics, statistical physics and theoretical high energy physics appear throughout the book.

However, one obvious topic missing is general relativity. As the authors state, good books on geometry & topology in general relativity existed at the time of writing.

The first 8 chapters present the key ideas of topology and differential geometry.

Chapter 1 discusses basic topology. Topics include homomorphisms, homotopy, the idea of topological invariants, compactness and connectedness. The reader is introduced to “topological thinking”.

Manifolds are the subject of Chapter 2. Topics include: the definition of manifolds, orientablilty, calculus on manifolds and differential structures.

Chapter 3 discusses the fundamental group. Topics include: the definition of the fundamental group, simplexes, triangulation and the fundamental group of a product of spaces.

Chapter 4 moves on to the homology group. Topics include: the definition of homology groups, relative homology, exact sequences, the Kunneth formula and the Poincare-Euler formula.

The higher homotopy groups are the subject of Chapter 5. Topics covered include: the definition of higher homotopy groups, the abelian nature of higher homotopy groups and the exact homotopy sequence.

The de Rham cohomology of a manifold is the subject of Chapter 6. Topics include: Poincare lemma, calculation of de Rham cohomology for simple examples, the cup product and a comparison of homology with cohomology.

Chapter 7 presents the core concepts of differential geometry. Topics here include: fibre bundles, sections, the Lie derivative, connections on bundles, curvature, parallel transport, geodesics, the Yang-Mills connection and characteristic classes.

Chapter 8 outlines Morse theory. Topics include: the Morse inequalities and the Morse lemma. Connection with physics is established via symmetry breaking selection rules in crystals.

The next two chapters look at application in physics of some of the ideas presented earlier in the book.

Defects and homotopy theory is the subject of Chapter 9. Topics include: planar spin in 2d, ordered mediums and the stability of defects theorem.

Chapter 10 discusses instantons and monopoles in Yang-Mills theory. Topics here include: instantons, instanton number & the second Chern class, instantons in terms of quaternions, twistor methods, monopoles and the Aharanov-Bohm effect.

Paperback: 311 pages
Publisher: Academic Press Inc; New edition edition (Jun 1987)
Language English
ISBN-10: 0125140819
ISBN-13: 978-0125140812

The book has also been reprinted by Dover Books in 2011.

Paperback: 311 pages
Publisher: Dover Publications Inc.; Reprint edition (17 Feb 2011)
Language English
ISBN-10: 0486478521
ISBN-13: 978-0486478524

Astronomy

Moon picture

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moon pic

Here is a picture of the Moon I took on the 3rd February 2012. The picture was taken using my 7MP Advent digital camera (“point and click”) directly through the eyepiece of my Bresser Skylux NG 70-700 retractor. I used a Moon filter and 20mm eyepiece.

The results are ok. I will post more as I take them.