CERN is putting plans in place to build a successor to the Large Hadron Collider (LHC). Possible options for the next generation of colliders will be discussed at the University of Geneva next week.

There are plans for a massive circular collider – with a circumference of 80–100 km – that would accelerate protons to energies of about 100 TeV! The LHC has a 27 km circumference and can collide protons with energies up to about 7 TeV.

Link
CERN kicks off plans for LHC successor.

Introduction

We all know Newton’s second law “F=ma”. In words, we only have acceleration when an external force is applied. But could he have some how chosen not acceleration but say something higher order ?

Acceleration is the rate of change of velocity, which itself is the rate of change of position. Thus acceleration is second order in derivatives with respect to time. This is not just a feature of Newtonian mechanics but is rather general and found throughout the fundamental laws of nature.

There are some rather general results using the Hamiltonian formalism that tells us that theories with equations of motion that are higher order than two are unstable. In particular the energy is not bounded from below and this can lead to problems classically and quantum mechanically. This is the famous Ostrogradski instability of (non-degenerate) Lagrangian theories with higher order derivative terms.

Note that such theories are still of interest as effective theories, but they cannot be seen as fundamental.

I won’t say anything more here about this.

So the question is can we understand simply why Newton could not really have picked anything of higher order in “F=ma”?

Notation: I will use Newton’s dot notation for the first and second order derivative with respect to time. For the n-th order derivative (n>2) I will use \(x^{(n)}(t)\).

Order three

Let us just pick a different form of Newton’s law as

\(F = N x^{(3)} (t) \),

that is let us suppose the force is proportional to the third order derivative of the position. (That would be the rate of change of acceleration). Here N is some property of the particle analogous to mass.

Now let us think about the motion of a free particle. So set the force term equal to zero and see what happens. Solving our “higher order” Newton’s law with no force is simple. We have

\(x(t) = c_{3} t^{2} + c_{2}t + c_{1}\).

The constants here are set by our initial conditions;

\(x(0) = c_{1}\), \(\dot{x}(0) = c_{2}\) and \(\ddot{x}(0) = 2 c_{3}\).

So what do we notice? The velocity as a function of time is given by

\(v(t) := \dot{x}(t) = \ddot{x}(0)t + \dot{x}(0)\).

This means that even if we set the initial velocity to zero the isolated free particle will speed up! Remember this is without any forces acting on the particle.

Newton’s first law says (in part) that “a particle at rest will remain at rest unless acted upon by an external force”. This higher order form of the second law is inconsistent with the first.

Worse than this, there are no forces here and so no potentials. The particle just speeds up all by itself and so clearly we lose conservation of energy. The particle can gain kinetic energy, defined as usual, but at no loss of potential energy! We would have to abandon our usual notion of conservation of energy in simple mechanics!

Higher order again
The same arguments work for higher order terms. The particle will just speed up by itself in violation of the first law and conservation of energy.

Modification of Newton’s law

Let us consider a slightly different situation in which “F = ma” becomes

\(F = M \ddot{x}(t) + N x^{(3)}(t)\).

That is we will add a higher order term to Newton’s law. Again, let is consider the case with no force term. We want to solve the equations of motion

\(\ddot{x}(t) + \frac{N}{M} x^{(3)}(t) =0\).

Here we assume that M is not equal to zero and is positive. For now we make no assumption at all about N.

One can directly solve the equations of motion

\(x(t) = \left( \frac{N}{M}\right)^{2} c_{1} e ^{-(M/N) t} + c_{2}t + c_{3}\).

Again we notice that the velocity is not constant and so we do not have conservation of energy in this situation either. But let us have a look at the particles trajectory for different ratios M/N.

In the above we have set M/N = 2 and all the constants to 1. The purple line is what we expect from F=ma. Note that we have a quick decay to the classical case. This itself signifies we do not have conservation of energy as we have no mechanism for the loss in kinetic energy, it just happens!

In the above we have set M/N =1. This case is very similar to the previous case.

Now in the above we have set M/N = 0.5. Again this is very similar to the previous cases.

We have a decay so that after some period of time everything looks the same as the standard Newtonian case. However, we still have to violate the first law to achieve this.

Now what happens if we let N be negative?

Here we see we have a runaway situation in which the particle just keeps on speeding up! Even if initially the trajectory is very close to the standard one after some time it just blows-up. Again this is in violation of the first law and conservation of energy.

Lower order

What about lower order laws?

Well if we had \(F \propto x\) then when there are no forces we simply have \(x =0\). Everything not in motion would have to sit at x=0. Meaning we cannot have any extended objects that are not in motion. This cannot be consistent with our Universe.

What about \(F \propto \dot{x}\)? Again let us set the forces to zero and we see that the solution is just \(x(t) = x_{0}\), some constant. However, this does not sit comfortably with our notion of relativity. Different inertial observers will not agree on the value of $latexx_{0}$. Thus if we don’t want to introduce absolute space we cannot allow this lower order form.

Conclusion
So as Newton wanted his first law to be true, have a good notion of statics and did not want to introduce absolute space he could have only have picked “F = ma”.

The Physics World award for the 2013 Breakthrough of the Year goes to “the IceCube South Pole Neutrino Observatory for making the first observations of cosmic neutrinos”. Nine other achievements are highly commended and cover topics ranging from nuclear physics to nanotechnology

Links
IceCube homepage

Physics world

The Fundamental Physics Prize Foundation announced the 2014 winners of the Physics Frontiers Prizes and New Horizons in Physics Prizes on the 5th of November 2013.

2014 Physics Frontiers Prize
The laureates of the 2014 Physics Frontiers Prize are:

• Joseph Polchinski, KITP/University of California, Santa Barbara, for his contributions in many areas of quantum field theory and string theory. His discovery of D-branes has given new insights into string theory and quantum gravity, with consequences including the AdS/CFT correspondence.
• Michael B. Green, University of Cambridge, and John H. Schwarz, California Institute of Technology, for opening new perspectives on quantum gravity and the unification of forces.
• Andrew Strominger and Cumrun Vafa, Harvard University, for numerous deep and groundbreaking contributions to quantum field theory, quantum gravity, string theory and geometry. Their joint statistical derivation of the Bekenstein-Hawking area-entropy relation unified the laws of thermodynamics with the laws of black hole dynamics and revealed the holographic nature of quantum spacetime.

Laureates of the 2014 Frontiers Prize now become nominees for the 2014 Fundamental Physics Prize. Those who do not win it will each receive $300,000 and will automatically be re-nominated for the next 5 years.

2014 New Horizons in Physics Prize
The laureates of 2014 New Horizons in Physics Prize are:

• Freddy Cachazo, Perimeter Institute, for uncovering numerous structures underlying scattering amplitudes in gauge theories and gravity.
• Shiraz Naval Minwalla, Tata Institute of Fundamental Research, for his pioneering contributions to the study of string theory and quantum field theory; and in particular his work on the connection between the equations of fluid dynamics and Albert Einstein’s equations of general relativity.
• Vyacheslav Rychkov, CERN/Pierre-and-Marie-Curie University, for developing new techniques in conformal field theory, reviving the conformal bootstrap program for constraining the spectrum of operators and the structure constants in 3D and 4D CFT’s.

The New Horizons Prize is awarded to up to three promising researchers, each of whom will receive $100,000.

2014 Fundamental Physics Prize
The winner of the 2014 Fundamental Physics Prize will be announced on December 12, 2013 in San Francisco, along with the winners of the 2014 Breakthrough Prize in Life Sciences.

Link
Fundamental Physics Prize

François Englert and Peter W. Higgs were jointly awarded the Nobel Prize in Physics 2013 for the theory of how fundamental particles acquire mass on the 8th october. Both Englert (with the deceased Robert Brout) and Higgs independently in 1964 proposed a mechanism for the elementary particles in to acquire mass; the so called Higgs mechanism.

On 4th July 2012, the ATLAS and CMS experiments at CERN’s Large Hadron Collider announced they have observed a new particle in the mass region around 126 GeV. This fits well with the predicted Higgs particle. However further work is needed to determine if this particle is exactly the Higgs as proposed in the standard model or something a little more exotic.

Peter Higgs. Photo by G-M Greuel	François Englert. Photo by Pnicolet

I am overwhelmed to receive this award… I would also like to congratulate all those who have contributed to the discovery of this new particle.

Peter Higgs talking to the BBC.

The standard model
Today the standard model, with the Higgs sector, is one of the key stones in our understanding of the fundamental forces of nature. The standard model describes electromagnetism and the strong and weak nuclear forces. The other key stone of fundamental physics is Einstein’s general relativity which describes the gravitational force.

Links
The 2013 Nobel prize in physics

Peter Higgs-facts

François Englert – Facts

Nominations are now being taken for the 2014 fundamental physics prize. The nominations are open to the public as well as the scientific community.

There are two prize categories:

• The New Horizons in Physics Prize, a $100,000 prize awarded to young researchers.

• The Fundamental Physics Prize, a $3,000,000 award given to researchers making transformative advancements in the field, with special attention to recent developments.

You can make your nomination by following this link.

Nominations will be accepted until October 15, 2013 and the three winners will be announced in November. Read more by following the link below.

Link
ONLINE NOMINATIONS FOR 2014 FUNDAMENTAL PHYSICS PRIZE