Mathematical Concept of Symmetry (Hermann Weyl)
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1 May-June 2006 FANTOM Study Week Gent 2007
2 Mathematical Concept of Symmetry (Hermann Weyl) Toolbox: Transformation groups Triangle Rotation Triangle R.P. Feynman: An object is called symmetric, if one can do something with it, without, at the end, when one is finished with the procedure, having changed it.
3 Architecture (1) mirror symmetry rotary symmetry translatory symmetry
4 different perspective and low energy apparent symmetries can be observed May-June 2006 FANTOM Study Week Gent 2007
5 The Noether-Theorem Symmetry Conservation Law Laws of physics are independent of... Emmi Noether ( ) origin of time axis energy conseravtion origin of spatial axes momentum conseravtion orientation of spatial axes angular mom. conserv.
6 Symmetry in Physics Simplification of problem solutions Example: The double pendulum Recognition of system properties Example: Conserved quantities and invariances Classification Examples: Crystals, Elementary Particles, Theories Unification of the fundamental forces Example: Electro-Magnetism, Grand Unification Symmetry = fundamental law of nature!!??
7 Problem Solution: Double Pendulum pendulum x initial excursion time 0 x simple oscillation
8 Problem Solution: Double Pendulum double pendulum x initial excursion in x time y initial excursion in y y 0 0 x Symmetry of the system: time x y complicated oscillation
9 Problem Solution: Double Pendulum double pendulum x initial excursion in x time y initial excursion in y y 0 0 x symmetrical excursion same side: simple oscillation time
10 Problem Solution: Double Pendulum double pendulum x initial excursion in x time y initial excursion in y y 0 0 x symmetrical excursion opposite side: simple oscillation time
11 Double Pendulum Symmetry Trick = + L L/2 L/2 L/2 L/2 complicated nonsymmetric oscillation = sum of simple symmetric oscillations
12 May-June 2006 FANTOM Study Week Gent 2007
13 May-June 2006 FANTOM Study Week Gent 2007
14 gauge Transformation: Symmetries and Forces α constant: Global Symmetries -> Conservation Laws α varies in space and time: Local Symmetries -> Forces
15 May-June 2006 FANTOM Study Week Gent 2007
16 Symmetries are related to Space and Time. Translations and Rotations in Space and Time can have any arbitrary value and therefore called continuous. A physics law is symmetric under space time transformations, if it does not change, i.e. the law is invariant. May-June 2006
17 May-June 2006
18 Local Symmetries Forces fundamental interactions Global Symmetries Conservation Laws energy momentum electric charge.. May-June 2006
19 May-June 2006
20 May-June 2006
21 May-June 2006 Conserved apparently in daily life
22 e - e + Time Time May-June 2006
23 Forces and Elementary Particles Electromagnetic Force Hadrons Strong Force Neutrinos Weak Force
24 Spatial Reflection P - Parity P-reflected neutrino not realised in nature!!
25 Time Reversal T - Parity But: T-reversed weak processes not exactly at equal rate!!
26 Charge Conjugation C - Parity C-conjugated neutrino not realised in nature!!
27 Symmetry Violation of Weak Force Symmetry Force P C T CP CPT gravitational electromagnetic strong weak Weak Force: mysterious symmetry violations? => universe made of matter while no antimatter is left
28 start P C T matter anti-matter end mirror image time time anti-particle May-June 2006 e + e - particle from H.W. Wilschut
29 Very powerful Theorem, unfortunately often only seen in connection with one of its consequences: Particle-Antiparticle properties May-June 2006 P. Lueders J. Schwinger, Phys. Rev. 82, 914 (1951); G. Lueders, Dansk. Mat. Fys. Medd. 28, 17 (1954),vAnn.Phys. 2, 1 (1957);W. Pauli, in: Niels Bohr and the Development of Physic, McGraw-Hill (1955), Nuovo Cimento 6, 204 (1957)
30 Unification of the Forces Electro-Magnetic Force (1) electric charge electric force-field moving electric charge Current electric current
31 Unification of the Forces Electro-Magnetic Force (2) Iron magnetic bodies magnetic force-field S N electric current Iron magnetic field
32 Unification of the Forces Electro-Magnetic Force (3) varying magnetic field electric force-field S N varying electric fields varying magnetic fields electromagnetic waves
33 Unification of the Forces Electro-Magnetic Gauge Symmetry Gauge Symmetry: electric ground potential is arbitrary global re-gauging lokal re-gauging to Volts to Volts e.m.-field, current pulse Volts Ground: 0 Volts
34 Deep Insight Nature possesses a local gauge symmetry The complete theory of electromagnetic forces is a necessary consequence of this local gauge symmetry! The local gauge symmetry is a! fundamental law of nature
35 Unification of the Forces The Electro-Weak Force electromagn. force gauge symmetry: weak force gauge symmetry:????? electroweak force gauge symmetry:
36 Deep Insight The electromagnetic und weak forces are just two aspects of a more fundamental force of nature, the electroweak force The complete theory of the electroweak force is a necessary consequence of an extended local gauge symmetry This extended local gauge symmetry is a yet more fundamental law of nature
37 The Grand Unification of Forces the electro-weak-strong force electroweak force gauge symmetry: strong force gauge symmetry: electroweakstrong force gauge symmetry:?
38 The Unification of all Forces the theory of everything electroweakstrong force gauge symmetry: gravitational force gauge symmetry: Poincaré-group gauge theory of all forces supersymmetry?? superstring theories??
39 How Symmetric is Nature? natural phenomena are only partially symmetric (including human products) asymmetries are often hints for deeper (and more beautiful) structures, and not just signs of disorder the deepest basis of the laws of nature seems to be a perfect and magnificent Symmetry
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