Why is there a conserved quantity for every continuous global symmetry?

One of the most important theorems in physics is the theorem that states:

"To every differentiable global symmetry of a physical system there corresponds a conservation law"

This theorem is known as the first Noether's theorem, in honor of the great mathematician Emmy Noether, who proved it in 1915 in the context of classical mechanics (both relativistic and non-relativistic, but not quantum). By the way, Noether is part of the group of leading scientists and professors in her field who lost their jobs due to the intolerance of the Nazis when they came to power, as they immediately passed a law preventing Jews and Communists from working in universities and public institutions. This happened before the Holocaust and the Second World War. It is important to remember this so that it does not happen again.

This relationship between symmetries and conservation laws established by Noether is one of the most powerful ideas that human beings have ever had. Conservation laws are a very useful tool for finding out how the quantities of a physical system change over time. Knowing that there are physical quantities that do not change allows us to write equations where the unknowns are the quantities that do change. We can then use the quantities that do not change to find out how the other quantities change.

On the other hand, the symmetries of a physical system are related to its aesthetic aspect. For example, a sphere is beautiful because, no matter how you rotate it, it remains the same. Noether's theorem thus relates beauty to usefulness in physics in a certain way. Pragmatism and aesthetics go hand in hand.

However, for the physics student, it is not immediately evident that a continuous symmetry implies a conserved quantity. Apparently, they are two things that have nothing to do with each other. What is the reason for this relationship?

However, today we know that the world is not classical, but quantum, and that classical mechanics is nothing more than an approximation of the behavior of physical systems in a certain limit. Therefore, Noether's original proof does not serve us for the fundamental laws of nature. Does Noether's theorem still hold in quantum mechanics?

These are the two questions we are going to answer in this post.

Preparation for the International Physics Olympiad (IPhO): Relativity

The IPhO Syllabus includes:

• 2.5 Relativity: Principle of relativity and Lorentz transformations for the time and spatial coordinate, and for the energy and momentum; mass-energy equivalence; invariance of the spacetime interval and of the rest mass. Addition of par­allel velocities; time dilation; length contraction; relativ­ity of simultaneity; energy and momentum of photons and relativistic Doppler effect; relativistic equation of motion; conservation of energy and momentum for elas­tic and non-elastic interaction of particles.

Members of the Spanish National Team can download the notes here:

• Introducción a la relatividad especial

To know more:

• Taller de Relatividad para estudiantes de bachillerato.

Some cognitive conflicts involving Relativity:

• Cognitive Dissonances: Relativity

It is also useful to follow the IPhO's Study Guide by Siim Ainsaar:

• A Glimpse into the Special Theory of Relativity

Here you can find the solutions to some of the problems:

• [IPho2006Theory2] Problem Solution
• [IPhO2003Theory3] Problem Solution
• [IPhO1994Theory1] Problem Solution

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Does the expansion of space apply in the solar system?

The prevailing cosmological model for the universe accounts for the fact that the universe expanded from a very high density and high temperature state, and that nowadays the expansion is even accelerating. This is an expansion of space, that is, the increase of the distance between two distant parts of the universe with time. It is an intrinsic expansion whereby the scale of space itself changes. This is different from other examples of expansions and explosions in that, as far as observations can ascertain, it is a property of the entirety of the universe rather than a phenomenon that can be contained and observed from the outside.

By NASA/WMAP Science Team - Original version: NASA; modified by Ryan Kaldari, Public Domain, https://commons.wikimedia.org/w/index.php?curid=11885244

Since it is an intrinsic expansion, it is natural to think that the planets in our solar system are expanding with time, as universe is. Moreover, our measurement devices should be expanding too. But, taking into account that a measurement is the assignment of a number to a characteristic of an object by comparing with other objects, why were we able to measure the expansion of the universe if our devices are expanding too?

Does the expansion of space apply to the objects inside our solar system?

Please, explain your reasoning. You can post your attempted answers in the comment box below. Please, do not use Facebook or Twitter to give your answers.

Why are cosmic ray muons decaying more slowly than predicted?

Muons are unstable elementary particles. They are heavier than electrons and neutrinos but lighter than all other matter particles. They decay via the weak interaction. A muon decays most commonly to an electron, an electron antineutrino, and a muon neutrino:

The mean lifetime, τ = 1/Γ, of the muon is (2.1969811±0.0000022 ) µs. That means that every 2.19698 µs the population of muons is reduced by a factor e=2.71828.

An experiment compared the population of cosmic-ray-produced muons at the top of a mountain, whose height is 2 km, to that observed at sea level. Those muons were traveling at 0.95c, where c is the speed of light, so they arrive to the sea level t=7 µs later. At the top of the mountain the measured population was No=563 muons per hour. Therefore, according to the decay law, the expected population of muons at the sea level should be:

$$ N=N_0 e^{-\frac{t}{\tau}}=23$$

muons. Nevertheless, 413 muons where measured, so the muon sample at the sea level was only moderately reduced! The muons were decaying about 10 times slower!

Are you able to explain this anomaly? Try it!

Please, explain your reasoning. You can post your attempted answers in the comment box below. Please, do not use Facebook or Twitter to give your answers.

Can an object exceed the speed of light if we push it for enough time?

The second Newton's law of motion establishes that, in an inertial reference frame, the vector sum of the forces F on an object is equal to the mass m of that object multiplied by the acceleration vector a of the object. That means that if we apply a constant force on a body without friction, then the object will move with constant acceleration, increasing its speed by the same amount every second.

The acceleration is proportional to the force, so if the net force is 100 Newtons and the mass is 2 kilograms, the acceleration will be 50 meters per second every second. But if the force is 2 N, the body will increase its speed by 1 m/s every second. Notice that this is not a huge acceleration. Nevertheless, if we keep pushing and wait for 300000000 seconds (9.5 years) the object will move faster that light.

But we know that nothing can exceed the speed of light. This is a well-established law of nature whose confirmation has become routine in current particle accelerators.

Try to find the solution to this contradiction!

Please, explain your reasoning. You can post your attempted answers in the comment box below. Please, do not use Facebook or Twitter to give your answers.