Friday, March 6, 2015

Noether's Theorem For every conservation law, there must exist a continuous symmetry, if space had the structure of a crystal, then moving the origin of coordinates from a nucleus to

http://gfif.udea.edu.co/emmynoether/noeth.htm

V. Noether's Theorem and Conservation Laws.

gfif.udea.edu.co/emmynoether/noeth.htm
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Symmetries of the Laws of Physics and Noether's Theorem ... For every conservation law, there must exist a continuous symmetry. ... For example, if space had the structure of a crystal, then moving the origin of coordinates from a nucleus to a ...

[PDF]Noether's Theorem - UCSD Department of Physics

www-physics.ucsd.edu/students/courses/.../CH07.pdf
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In general, whenever the system exhibits a continuous symmetry, there is an associated conserved charge. ... result, known as Noether's Theorem. ... where ζ is the continuous parameter. ... As another example, consider the potential ...... system, then integrating the conservation law ∂µ Jµ over the spatial region Ω gives. dΛ.

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Emmy Noether Intro

Emmy Noether

Symmetries of the Laws of Physics and Noether's Theorem

In 1905, a mathematician named Emmy (Amalie) Noether* proved the following theorem:

For every continuous symmetry of the laws of physics, there must exist a conservation law.

For every conservation law, there must exist a continuous symmetry.

Thus, we have a deep, deep connection between a symmetry of the laws of physics, and the existence of a corresponding conservation law. In presenting Noether's theorem at this level we usually state it without proof. A simple proof can be given if the student is familiar with the action principle. However, it is better to motivate the result through examples. (For a proof, check out Mathematics of Symmetry and Physics.)

Conservation laws, like the conservation of energy, momentum and angular momentum (these are the most famous), are studied in high school. We now see from Noether's theorem that they emerge from symmetry concepts far deeper than Newton's laws. We will also learn that there are many other conservation laws in physics. Finally, we will give some idea of how this theorem plays out in the quantum theory domain.

Now, as we have stated above, it is an experimental fact about the nature that the laws of physics are invaraint under spatial translations. This is a strong statement. For example, if space had the structure of a crystal, then moving the origin of coordinates from a nucleus to a void would change the laws of nature within the crystal. The hypothesis that space is translationally invariant is equivalent to the statement that one point in space is equivalent to any other point, i.e. the symmetry is such that translations of any system or, equivalently, the translation of the coordinate system, does not change the laws of nature. Equivalently, the laws (and equations that express these laws) are invariant to translations (translational symmetry).

Now comes the amazing result of mathematician Emmy Noether, whose theorem, in this case, states:

The conservation law corresponding to space translational symmetry is the Law of Conservation of Momentum.

So, we learn in senior physics class that the total momentum of an isolated system remains constant. The ith element of the system has a momentum in Newtonian physics of the form:

p_i = m v_i

and the total momentum is just the sum of all of the elements,

P_total = p₁ + p₂ + ... + p_N for a system of N elements. Noether's theorem states that P is conserved, i.e., it does not change in time, no matter how the various particles interact, because the interactions are determined by laws that don't depend upon where the whole system is located in space!

Note that momentum is, and must be, a vector quantity (hence the little arrow over the stuff in the equations). Why? Because momentum is associated with translations in space, and the directions you can translate (move) a physical system form a vector! So, if you remember the Noether theorem, you won't forget that momentum is a vector when taking an SAT test!

Turning it around, the validity of the Law of Conservation of Momentum as an observational fact, via Noether's theorem, supports the hypothesis that space is homogeneous, i.e., possessing translational symmetry. The more we verify the law of conservation of momentum, and it has been tested literally trillions of times in laboratories all over the world, at all distance scales, the more we verify the idea that space is homogeneous!

We note that Noether's Theorem further assures us that for any translationally invariant physical system there is always something called ``momentum'' and it is always conserved. The exact formula for the momentum depends upon what we are studying. For the Newtonian particle it takes the form mv, while for the relativistic particle, mv/(1- v²/c²)^1/2, and for the electromagnetic wave E times

B/4

c, etc. Note that in each case the momentum is always a vector (3 component) object corresponding to the three translational directions of space.

The experimental evidence also favors very strongly the homogeneity of time, i.e. any point on the time axis is as good as any other point, i.e., the laws of physics are invariant under tanslations in time. What conservation law then follows by Noether's Theorem?

Surprise! It is nothing less than the law of conservation of energy:

The conservation law corresponding to time translational symmetry is the Law of Conservation of Energy.

Since the constancy of the total energy of a system is extremely well tested experimentally, this tells us that nature's laws are invariant under time translations.

Here is an example of how time invariance and energy conservation are interrelated. Consider a water tower that can hold a mass M of water and has a height of H meters. Assume that the gravitational constant, which determines the acceleration of gravity, is g, on every day of the week except Tuesday, when it is a smaller value g'< g. Now, we run water down from the water tower on Monday through a turbine (a fancy water wheel) generator which converts the potential energy MgH to electrical current to charge a large storage battery. We'll assume 100% efficiencies for everything, because we are physicists. This is Monday's job.

Now on Tuesday we pump the water back up to H, using the battery power that we accumulated from Monday's job to run the pump. But now the g' value is smaller than g and the work done is Mg'H, which is now much less than the energy we got from Monday's job. This leaves us with M(g-g')H extra energy still in the battery, which we can sell to a local power company to live on until next Monday. This is a perpetual motion machine! It produces energy for us, and we can convert that to cash. It does not conserve energy because we cooked up false laws of physics, in this case gravity, that are not time translationally invariant!!! Hence, we violated a precept of Noether's Theorem. (Can you come up with similar cute example of violating momentum conservation by making the laws of physics spatially inhomogeneous?)

We also live in a world where the laws of physics are rotationally invariant:

The conservation law corresponding to rotational symmetry is the Law of Conservation of Angular Momentum.

Conservation of angular momentum is often demonstrated in lecture by what is usually called ``the 3 dumbbell experiment". The instructor stands on a rotating table,

his hands outstretched, with a heavy dumbbell in each hand (who is the third dumbbell?). A student starts the lecturer rotating on his table. He turns slowly, then brings his hands (and dumbbells) close to his body. His rotation speed (angular velocity, omega

) increases substantially. What is kept constant is the angular momentum, J, the product of I, the moment of inertia, times the angular velocity \omega. Hence, J = I omega

By bringing his dumbbells in close to his body, the moment of inertia I, is decreased (roughly speaking, the moment of inertia is I₀, the Professor's body's moment, plus the two dumbell masses 2M times the length of his arms squared, r², or I = I₀ + 2Mr²; in the experiment, r is initially the extended length of his arms, and the finally is approximately 0, the retracted length of the arms). But J, the angular momentum, must be conserved, so omega

must increase. Skaters do this trick all the time.

Atoms, elementary particles, etc., all have angular momentum and in any reaction, the final angular momentum must be equal to the initial angular momentum. Like our planet earth, particles spin and execute orbits and both motions have associated angular momentum. Data over the past 70 or so years confirms conservation of this quantity on the macroscopic scale of people and their machines and on the microscopic scale of particles. And now, (thanks to Emmy) we learn that these data imply that space is isotropic; —there is no preferred direction. All directions in space are equivalent. Incidently, the conservation of angular momentum is encoded into Kepler's third law of planetary motion, and in some sense represents the first exact statement of a conservation law (Archimedes anticipated energy conservation).

These translational and rotational symmetries of space and time need not have existed. That they do is the way nature is. We are learning some of the actual properties of the concepts we use to describe the world: space and time. It didn't have to be this way. For example, if all of space were constructed like the insides of a crystal, then all directions would not be equivalent, and continuous translational symmetry would be lost.

What about the Lorentz invariance? What is the conserved quantity associate with it? Actually, the devil gets into this one; we find that the conserved quantity is actually 0. Fortunately, 0 is conserved, so there is a technical conservation law here, it just isn't a useful one. Nonetheless, Lorentz invariance has profound effects, like the fact that mass is equivalent to energy through E=mc², which we'll prove below, using Noether's Theorem.

There are more abstract symmetries which do not involve space or time coordinates. An example is the symmetry of an assemblage of positive and negative charges. We can define an operation that tests for symmetry as follows: `` change the signs of all the charges." In this case the appearance of the system changes. For example, if we have a Hydrogen atom, the nucleus is a positively charged proton and the electron orbiting the nucleus is negatively charged. After performing our operation, we are left with a system containing a negatively charged nucleus and a positively charged electron orbiting. The ``invariance'' under this operation is the statement that both systems have identical physical properties.

Such a system could in principle be constructed. Positive electrons were discovered in 1932 (positrons) and negative protons were discovered in 1955. These are examples of ``antimatter.'' An anti-Hydrogen atom could be constructed from the anti-proton and the positron; in fact, it has been made in a clever particle acelerator experiment, and some of its properties have been observed. Matter-anti-matter symmetry implies taht the mass, spins, binding energies, excited states, etc. of both real Hydrogen and anti-Hydrogen must be identical. The charge-reversal symmetry is another example of a discrete symmetry. We will have more to say about anti-matter, and ultimately we can glimpse why anti-matter must exist from the basic symmetry principles of space and time embodied in Einstein's Theory of Relativity.

*(pronounced like ``mother'' with an ``n"; born, 1882, she had a great deal of trouble finding permanent employment in male-dominated European universities, and had to flee the rise of Naziism she spent her last few years at Bryn Mawr; she died in 1935)