102
1 Symmetry and Physics
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Symmetry and Physics
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1. Origin
2. Greeks
3. Copernicus & Kepler
4. 19th century
5. 20th century
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1. Origin of Concept of Symmetry
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5
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Painting
Sculpture
Music
Literature
Architecture
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8
9
10
11
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2. Greeks
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Harmony of the Spheres
Dogma of the Circles
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3. Copernicus (1473-1543)
Kepler (1571-1630)
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Six planets:
Saturn, Jupiter, Mars,Earth, Venus, Mercury
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Mysterium Cosmographicum
1596
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One of the methods now to find reasons of some observed regularity:
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(a) Choose some mathe-matical regularity resulting from symmetry require-ments.
(b) Match it to observed regularity.
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•Discussed why snow flakes are 6-sided
•Albertus Magnus: +1260
•In China: -135
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But no effort to try to explain why.
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4. 19th Century
Groups and Crystals
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Galois (1811-1832)
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Concept of groups is the mathematical representation of concept of symmetry.
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Symmetry
and invariance
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A 90° rotation is called a 4-fold rotation.
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It will be denoted by 4.
It is an invariant element of the graph.
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40
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3 dimensional 230 (1890)
2 dimensional 17 (1891)
4 dimensional 4895 (~1970)
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5. 20th Century
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5.1 Symmetry applied to concepts of space and time
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Special Relativity
1905
Lorentz Symmetry
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General Relativity
1916
Very Large Symmetry
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5.2 Symmetry applied to atomic, nuclei, particle properties
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Quantum Numbers, spin, parity
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Great importance in most branches of physics 1920
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Symmetry = Invariance
Conservation Laws
(Except for discrete symmetry in classical mechanics)
Other Consequences
Quantum Numbers
Selection Rules
(In quantum mechanics only)
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5.3 Symmetry applied to structure of interactions (forces).
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Maxwell Equations have,
beyond Lorentz Symmetry,
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Another symmetry:
Gauge Symmetry
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In 1915-1916 Einstein published his general relativity, making gravity a geometrical theory. He then emphasized that EM should also be geometricized.
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H. Weyl (1885 – 1955) took up the challenge and proposed in 1918 a geometrical theory of EM.
62Hermann Weyl (1885-1955)
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Levi–Civita and others have developed the idea of “parallel transport”
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.A
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On a curved surface, the parallel transported vector may not come back to its original direction.
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Weyl asked, if so
“Why not also its length?”
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“Warum nicht auch seine Länge?”
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B
A
x dexp
A
B
.
.
Proportionalitätsfaktor
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And pointed out that some changes inleaves his theory invariant, while the EM vector potential has similar properties.
,A
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So he put
eA
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Connecting EM with
geometry
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Masstab InvarianzMeasure InvarianceCalibration InvarianceGauge Invariance
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Weyl submitted his paper to the Prussian Academy. The editors, Planck and Nernst, asked for the opinion of Einstein:
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With his penetrating physical intuition, Einstein objected.
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A B
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Einstein’s postscript:
“the length of a common ruler (or the speed of a common clock) would depend on its history.”
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QM came to the rescue.
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1926-1927
Fock, London
)d(expdexp xiAxA
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Proportionality Factor
Phase Factor
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Gauge Theory
Phase Theory
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With gauge phase,
how about Einstein’s objection?
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Phase difference at B
A B
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1959 Aharonov-Bohm
A B
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Chambers used a tapered magnetic needle instead of a long solenoid and claimed he had seen the A-B effect.
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But the leaked flux
from his needle
caused objection.
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Finally in the mid 1980s, Tonomura et. al. quantitatively proved the A-B effect. Thus introducing experimentally topology into fundamental physics.
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Weyl’s idea was generalized in 1954
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Searching for a Principle for Interaction
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First Motivation:
Many new particle. How do they interact?
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Second Motivation:
“the electric charge serves as a source of electromagnetic field; an important concept in this case is gauge invariance ...”
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“We have tried to generalize this concept of gauge invariance to apply to isotopic conserva-tions.”
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Third Motivation:
“It is pointed out that the usual principle of invariance under isotopic spin rotation is not consistent with the concept of localized fields.”
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Maxwell Non AbelianGauge Theory
,, bbF kjijk
i,
i,
i bbcbbF
JF ,ikji
jki
, JFbcF
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Beautiful and Unique Generalization.But too much symmetry to agree with experiments in 1954 to late 1960s.
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Symmetry Breaking
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Algebraic Symmetry.
But broken symmetry in observation.
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Symmetry Dictates
Interaction
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Symmetry Invariance
—
Conservation Laws
Gauge Symmetry
Symmetry Dictates Interaction
Other Consequences
QuantumNumbers
Selection Rules
StrongForce
︴︴︴︴︴︴︴︴︴︴︴︴︴︴︴︴︴
︴︴︴︴︴︴︴︴︴︴︴︴︴︴︴︴︴ Electromagnetic
Force
Weak Force
GravityForce
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Usual Symmetry Gauge Symmetry
Equation Equation
Sol. Sol. Sol. Sol. Sol. Sol.
Different Physics Same Physics
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Supersymmetry 1973
Supergravity 1976
Superstrings 1984
