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When Einstein examined her work, he wrote to Hilbert:

**"Yesterday I received from Miss Noether a very interesting paper on invariants. I'm impressed that such things can be understood in such a general way. The old guard at Göttingen should take some lessons from Miss Noether! She seems to know her stuff."**

**Emmy Noether**

The solution to this problem began a hundred years ago with the work of the greatest woman mathematician and physicist you've never heard of --

**Emmy Noether**:

Noether was brought to Göttingen in 1915 by David Hilbert and Felix Klein, who wanted her expertise in invariant theory to help them in understanding general relativity....Hilbert had observed that the conservation of energy seemed to be violated in general relativity, due to the fact that gravitational energy could itself gravitate. Noether provided the resolution of this paradox, and a fundamental tool of modern theoretical physics, with her

For illustration, if a physical system behaves the same, regardless of how it is oriented in space, the physical laws that govern it are rotationally symmetric; from this symmetry, Noether's theorem shows the angular momentum of the system must be conserved....As another example, if a physical experiment has the same outcome at any place and at any time, then its laws are symmetric under continuous translations in space and time; by Noether's theorem, these symmetries account for the conservation laws of linear momentum and energy within this system, respectively.

Noether's theorem has become a fundamental tool of modern theoretical physics, both because of the insight it gives into conservation laws, and also, as a practical calculation tool. Her theorem allows researchers to determine the conserved quantities from the observed symmetries of a physical system.

**first Noether's theorem**....She solved the problem not only for general relativity, but determined the conserved quantities for every system of physical laws that possesses some continuous symmetry.For illustration, if a physical system behaves the same, regardless of how it is oriented in space, the physical laws that govern it are rotationally symmetric; from this symmetry, Noether's theorem shows the angular momentum of the system must be conserved....As another example, if a physical experiment has the same outcome at any place and at any time, then its laws are symmetric under continuous translations in space and time; by Noether's theorem, these symmetries account for the conservation laws of linear momentum and energy within this system, respectively.

Noether's theorem has become a fundamental tool of modern theoretical physics, both because of the insight it gives into conservation laws, and also, as a practical calculation tool. Her theorem allows researchers to determine the conserved quantities from the observed symmetries of a physical system.

*[symmetric]*in time.

Suppose this were not the case. Suppose the Gravitational Constant varied in time -- that it was greater on Monday than it was on Wednesday. Then you could drop an object on Monday and carefully store the energy it produced by its fall. On Wednesday, when gravity was weaker, you could use that energy to raise the object to a greater height than it had on Monday. The following Monday, you drop the object again and store its energy, which will be greater than it was before. Repeating this every Monday, you will gain more and more energy, essentially creating a perpetual motion machine, and negating the Law of the Conservation of Energy. Thus, the Law of the Conservation of Energy arises from the time-symmetry of physical processes.

Noether's Theorem roughly says :

**If a system has a continuous symmetry property, then there are corresponding quantities whose values are conserved in time.**

Some of these symmetries are quite abstract, yet they produce very concrete Conservation Laws.

For example, a particle's quantum wave-function in complex space is represented by a complex vector for each point in that space. Each vector is defined by a length and a rotation ("phase shift").

The square of the length of each vector is the probability of finding the particle in a particular region of space. The probability is not affected by rotations in complex space. This means that if all the vectors in the complex space, at every point in that space, undergo the same identical phase shift, the state of the particle in physical space does not change -- the probabilities stay the same. The quantum wave is

**with respect to "rotations in complex space," or, symmetric to "global" phase change -- one might say that the entire space "rotates" by the same amount everywhere -- this is called**

*symmetric***.**

*global "gauge" symmetry*The quantum wave is symmetric with respect to global rotations in complex space. Noether's Theorem may be used to determine the Conservation Law associated with this very abstract symmetry in quantum mechanics.

It turns out to be the Law of the Conservation of Electric Charge !!

So what happens if the vectors in the wave function are rotated by

*different*amounts at each point in complex space? Only changes in the lengths of the vectors change measurable properties of the particle. However, hidden beneath what is measurable, the wave function changes. The phase shifts cause the wave function to "interfere" with itself, just like waves in water, where ripples from different sources can "boost" each other, or cancel out.

In this case, the wave function is

**symmetric with respect to "local" rotations in complex space -- where, from a global standpoint, the rotations are all higgledy-piggledy.**

*not*However, Conservation of Electric Charge requires global phase symmetry! How can one restore this symmetry under these myriads of local phase shifts? This seemingly herculean task is accomplished by a field of force which pervades space. Wonderfully, this field of force turns out to be the electromagnetic field !!

The laws of electromagnetism, which students of physics struggle so hard to understand, are simply the consequence of a simple symmetry principle : the laws of motion for a quantum particle must be the same under arbitrary rotations in complex space at different points in space. The electromagnetic field exists for one reason, and one reason only -- to restore local "gauge" symmetry.

Symmetries pervade all of physics. Yet the world around us looks to be anything but symmetric. Why is this?

In general, symmetries spontaneously break as the temperature drops. Consider a container of water. In its liquid state, it is highly symmetric. Cool it down, and at a certain point, the symmetry spontaneously breaks, the molecules line themselves in lattices in which directions are distinguishable, cracks and fissures appear.

The electromagnetic force and the weak force in physics appear wildly different. Yet, a Nobel Prize was won by theorists who, by following certain symmetry principles, managed to work out how these two forces, at sufficiently high temperatures, become a single, unified electro-weak force.

It now appears that if we ran the universe backwards, like a film, it would become hotter and hotter, and simpler and simpler. Forces and particles would become more and more one thing. It seems that the universe, in its earliest moments was in a state of maximum possible symmetry.

Everywhere, in every direction, from every conceivable point of view, it would look the same. It would conform to every conceivable symmetry, concrete and abstract.

But the state of maximum symmetry appears to be the state of a completely empty void.

The symmetries which are the root of the fundmental laws of physics are nothing more than the symmetries of the void.

For this reason it has been said that,

**"the laws of physics are the laws of nothing"**-- or, if you prefer, the laws of Nothing.

Why did Nothing turn into Something?

**"Because Something is more stable than Nothing."**The universe, in its most symmetrical phase turns out to be unstable. "Nothing" spontaneously breaks down, moving to states of less energy, as the Law of Entropy dictates.

The universe has no net electrical charge, as one would expect if it had evolved from Nothing.

The positive mass-energy of all the matter in the universe appears to be exactly balanced by the negative potential-energy of all the gravitational fields of that matter. The sum-total of all the energy of the universe appears to be zero. The universe, still, in a sense, is nothing.

The laws of physics appear to be the way they are, because -- they could not be anything less than complete, total symmetry. One is driven to the conclusion that the universe is simply structured, re-arranged Nothing.

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