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--- basic_tools:symmetry [2018/03/30 10:03]
jakobadmin [Concrete]
+++ basic_tools:symmetry [2019/01/24 10:23] (current)
jakobadmin [Why is it interesting?]
@@ Line 1: / Line 1: @@
 ====== Symmetry ======
+//see also [[advanced_tools:internal_symmetry]] and [[advanced_tools:gauge_symmetry]] //
 <tabbox Intuitive>
@@ Line 54: / Line 54: @@
-see http://axelmaas.blogspot.de/2010/05/symmetries.html
+for explicit examples see http://axelmaas.blogspot.de/2010/05/symmetries.html
+----
+**Active Vs. Passive Transformations**
+[{{ :basic_tools:activepassivetrafos.png?nolink&400 |[[http://physics.openmetric.org/quantum/quantum-mechanics-beginners.html#fundamental-concepts|Source]]}}]
+----
+  * The symmetry of Classical Mechanics is the Galilei group. The canonical structure of the phase space is characterized by the symplectic groups $Sp(2N)$.
+  * The symmetry of Electrodynamics and Special Relativity is the Poincare group. (However, take note that the Maxwell equations are also invariant under the larger conformal group.) The Poincare group becomes the Galilei group in the limit when everything moves slow compared to the speed of light. In addition, electrodynamics has an $U(1)$ gauge symmetry.
+  * The symmetry of Quantum Mechanics and Relativistic Field Theory is the double cover of the Poincare group.
+  * The symmetry of particle physics is the $SU(3)_C\times SU(2)_L \times U(1)_Y$ gauge symmetry.
+  * The symmetry of general relativity is the diffeomorphism group.
 ----
@@ Line 80: / Line 94: @@
 <cite>Seven Science Quests by Sudarshan</cite>
 </blockquote>
+<blockquote>In physics we use many different types of symmetries, but they have one thing in common: they are potent unifying principles because they explain how things that once appeared very different actually belong together, connected by a symmetry transformation.
+<cite>Lost in Math by Sabine Hossenfelder</cite>
+</blockquote>
+<blockquote>For the physicist, a symmetry is an organizing principle that avoids unnecessary repetition. Any type of pattern, likeness, or order can be mathematically captured as an expression of symmetry. The presence of a „symmetry always reveals a redundancy and allows simplification. Hence, symmetries explain more with less.
+For example, rather than telling you today’s sky looks blue in the west and the east and the north and the south and the southwest, and so on, I can just say it looks blue in every direction. This independence on the direction is a rotational symmetry, and it makes it sufficient to spell out how a system looks in one direction, followed by saying it’s the same in all other directions. The benefit is fewer words or, in our theories, fewer equations.
+The symmetries that physicists deal with are more abstract versions of this simple example, like rotations among multiple axes in internal mathematical spaces. But it always works the same way: find a transformation under which the laws of nature remain invariant and you’ve found a symmetry. Such a symmetry transformation may be anything for which you can write down an unambiguous procedure—a shift, a rotation, a flip, or really any other operation that you can think of. If this operation does not make a difference to the laws of nature, you have found a symmetry.
+ [...] The symmetry requirement therefore limits the possible laws we can write down. The logic is similar to coloring a mandala. If you want the color fill to respect the symmetry of the design, you have fewer options than when you ignore the symmetry.
+<cite>Lost in Math by Sabine Hossenfelder</cite>
+</blockquote>
+<blockquote>„One moonlit night, we walked over the Hainberg Mountain, and he [Werner Heisenberg] was completely enthralled by the visions he had, trying to explain his newest discovery to me. He talked about the miracle of symmetry as the original archetype of creation, about harmony, about the beauty of simplicity, and its inner truth.“
+<cite>Inner Exile: Recollections of a Life with Werner Heisenberg by Elisabeth Heisenberg</cite>
+</blockquote>
 <blockquote>”Symmetry pervades the
@@ Line 100: / Line 134: @@
 </blockquote>
+<blockquote>As far as I see, all a priori statements in physics have their origin in symmetry.
+<cite>H. Weyl</cite></blockquote>
+<blockquote>The most important lesson that we have learned in this century is that the secret
+of nature is symmetry. <cite>D. Gross</cite></blockquote>
+<blockquote>Today we realize that symmetry principles . . . dictate the form of the laws of nature.
+<cite>D. Gross</cite></blockquote>
+<blockquote>Symmetry principles have moved to a new level of importance in this century and
+especially in the last few decades: there are symmetry principles that dictate the
+very existence of all the known forces of nature. <cite>S. Weinberg</cite></blockquote>
+<blockquote>To a remarkable degree, our present detailed theories of elementary particle inter-
+action can be understood deductively, as consequence of symmetry principles . . ..
+<cite>S. Weinberg</cite></blockquote>
+<blockquote>. . . profound guiding principles are statements of symmetry.
+<cite>F. Wilzcek</cite></blockquote>
+<blockquote>If you can identify Nature’s complete symmetry group, you will know everything”
+is what became a pivotal dogma.
+<cite>G.’t Hooft</cite></blockquote>
 <tabbox Global Vs. Local, Continous vs. Discrete>

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