SYMMETRIES in PHYSICS

SYMMETRIES in PHYSICS
Jerzy DUDEK
Institute for Subatomic Research
University of Strasbourg I
2nd October 2007
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Part I
Symmetry and Groups: Historical Aspects
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
The Origin and the Meaning of Word: Symmetry
The word symmetry (συµµɛτρια) originates from the Greek
language: συµ (’together’) and µɛτρων (’measure’)
The implied meaning: ’measured together’, well proportioned
The meaning has evolved in time into: beauty, unity, harmony
In sciences the first meaning of the word symmetry was that
related to proportions
Todays meaning as the equality of elements under geometrical
transformations (translations, rotations, reflexions) arrived only
towards the end of the Renaissance
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
The Origin and the Meaning of Word: Symmetry
The word symmetry (συµµɛτρια) originates from the Greek
language: συµ (’together’) and µɛτρων (’measure’)
The implied meaning: ’measured together’, well proportioned
The meaning has evolved in time into: beauty, unity, harmony
In sciences the first meaning of the word symmetry was that
related to proportions
Todays meaning as the equality of elements under geometrical
transformations (translations, rotations, reflexions) arrived only
towards the end of the Renaissance
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
The Origin and the Meaning of Word: Symmetry
The word symmetry (συµµɛτρια) originates from the Greek
language: συµ (’together’) and µɛτρων (’measure’)
The implied meaning:
’measured together’, well proportioned
The meaning has evolved in time into: beauty, unity, harmony
In sciences the first meaning of the word symmetry was that
related to proportions
Todays meaning as the equality of elements under geometrical
transformations (translations, rotations, reflexions) arrived only
towards the end of the Renaissance
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
The Origin and the Meaning of Word: Symmetry
The word symmetry (συµµɛτρια) originates from the Greek
language: συµ (’together’) and µɛτρων (’measure’)
The implied meaning:
’measured together’, well proportioned
The meaning has evolved in time into: beauty, unity, harmony
In sciences the first meaning of the word symmetry was that
related to proportions
Todays meaning as the equality of elements under geometrical
transformations (translations, rotations, reflexions) arrived only
towards the end of the Renaissance
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
The Origin and the Meaning of Word: Symmetry
The word symmetry (συµµɛτρια) originates from the Greek
language: συµ (’together’) and µɛτρων (’measure’)
The implied meaning:
’measured together’, well proportioned
The meaning has evolved in time into: beauty, unity, harmony
In sciences the first meaning of the word symmetry was that
related to proportions
Todays meaning as the equality of elements under geometrical
transformations (translations, rotations, reflexions) arrived only
towards the end of the Renaissance
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Symmetry: Plato (428 - 347 BC) and even earlier ...
The symbol of ’beauty in symmetry’ are five Platonic Figures
Platonic Figures: Three-dimensional polyhedra whose faces are
identical planar regular convex polygons
The allowed polygons are either equilateral triangles, or squares
or regular pentagons
There exist only five regular convex (=platonic) polyhedra:
tetrahedron, cube, octahedron, icosahedron & dodecahedron
As it seems, neolithic people from Scotland have developed the
five Platonic solids about 1000-3000 years before Plato (stone
models in Ashmolean Museum, Oxford)
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Symmetry: Plato (428 - 347 BC) and even earlier ...
The symbol of ’beauty in symmetry’ are five Platonic Figures
Platonic Figures: Three-dimensional polyhedra whose faces are
identical planar regular convex polygons
The allowed polygons are either equilateral triangles, or squares
or regular pentagons
There exist only five regular convex (=platonic) polyhedra:
tetrahedron, cube, octahedron, icosahedron & dodecahedron
As it seems, neolithic people from Scotland have developed the
five Platonic solids about 1000-3000 years before Plato (stone
models in Ashmolean Museum, Oxford)
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Symmetry: Plato (428 - 347 BC) and even earlier ...
The symbol of ’beauty in symmetry’ are five Platonic Figures
Platonic Figures: Three-dimensional polyhedra whose faces are
identical planar regular convex polygons
The allowed polygons are either equilateral triangles, or squares
or regular pentagons
There exist only five regular convex (=platonic) polyhedra:
tetrahedron, cube, octahedron, icosahedron & dodecahedron
As it seems, neolithic people from Scotland have developed the
five Platonic solids about 1000-3000 years before Plato (stone
models in Ashmolean Museum, Oxford)
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Symmetry: Plato (428 - 347 BC) and even earlier ...
The symbol of ’beauty in symmetry’ are five Platonic Figures
Platonic Figures: Three-dimensional polyhedra whose faces are
identical planar regular convex polygons
The allowed polygons are either equilateral triangles, or squares
or regular pentagons
There exist only five regular convex (=platonic) polyhedra:
tetrahedron, cube, octahedron, icosahedron & dodecahedron
As it seems, neolithic people from Scotland have developed the
five Platonic solids about 1000-3000 years before Plato (stone
models in Ashmolean Museum, Oxford)
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Symmetry: Plato (428 - 347 BC) and even earlier ...
The symbol of ’beauty in symmetry’ are five Platonic Figures
Platonic Figures: Three-dimensional polyhedra whose faces are
identical planar regular convex polygons
The allowed polygons are either equilateral triangles, or squares
or regular pentagons
There exist only five regular convex (=platonic) polyhedra:
tetrahedron, cube, octahedron, icosahedron & dodecahedron
As it seems, neolithic people from Scotland have developed the
five Platonic solids about 1000-3000 years before Plato (stone
models in Ashmolean Museum, Oxford)
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Models of Platonic Figures
Symmetry in Physics: A Short History
Group Theory: First Concepts
Recall: Platonic Figures = Polyhedra whose
faces are identical regular convex polygons
Allowed polygons are equilateral triangles, or
squares or regular pentagons
Plato
Tetrahedron, Cube, Octahedron, Icosahedron, Dodecahedron
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Models of Platonic Figures
Symmetry in Physics: A Short History
Group Theory: First Concepts
Recall: Platonic Figures = Polyhedra whose
faces are identical regular convex polygons
Allowed polygons are equilateral triangles, or
squares or regular pentagons
Plato
Tetrahedron, Cube, Octahedron, Icosahedron, Dodecahedron
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Models of Platonic Figures
Symmetry in Physics: A Short History
Group Theory: First Concepts
Recall: Platonic Figures = Polyhedra whose
faces are identical regular convex polygons
Allowed polygons are equilateral triangles, or
squares or regular pentagons
Plato
Tetrahedron, Cube, Octahedron, Icosahedron, Dodecahedron
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Models of Platonic Figures
Symmetry in Physics: A Short History
Group Theory: First Concepts
Recall: Platonic Figures = Polyhedra whose
faces are identical regular convex polygons
Allowed polygons are equilateral triangles, or
squares or regular pentagons
Plato
Tetrahedron, Cube, Octahedron, Icosahedron, Dodecahedron
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Fascination by Symmetry of Platonic Figures: Kepler
A. Many natural scientists and philosophers were
fascinated by symmetry of Platonic figures
B. One of the first ones in the modern times was
Johannes Kepler
Kepler’s Planetary System
Johannes Kepler
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Fascination by Symmetry of Platonic Figures: Kepler
A. Many natural scientists and philosophers were
fascinated by symmetry of Platonic figures
B. One of the first ones in the modern times was
Johannes Kepler
Kepler’s Planetary System
Johannes Kepler
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Fascination by Symmetry of Platonic Figures: Kepler
A. Many natural scientists and philosophers were
fascinated by symmetry of Platonic figures
B. One of the first ones in the modern times was
Johannes Kepler
Kepler’s Planetary System
Johannes Kepler
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Kepler ”Theory” of Planetary Constellation [Failure]
The Earth orbit is the measure of all orbits
Around it we circumscribe the dodecahedron
The Mars orbit is circumscribed around the dodecahedron
Around the Mars orbit we circumscribe the tetrahedron
The Jupiter orbit is circumscribed around the tetrahedron
Around of the Jupiter orbit we circumscribe the cube
The Saturn orbit lies in the sphere surrounding the cube
In the Earth orbit the regular icosahedron is inserted
The orbit entered in it is the Venus orbit
In the Venus orbit the octahedron is inserted
The orbit entered in it is the Mercury orbit
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Kepler ”Theory” of Planetary Constellation [Failure]
The Earth orbit is the measure of all orbits
Around it we circumscribe the dodecahedron
The Mars orbit is circumscribed around the dodecahedron
Around the Mars orbit we circumscribe the tetrahedron
The Jupiter orbit is circumscribed around the tetrahedron
Around of the Jupiter orbit we circumscribe the cube
The Saturn orbit lies in the sphere surrounding the cube
In the Earth orbit the regular icosahedron is inserted
The orbit entered in it is the Venus orbit
In the Venus orbit the octahedron is inserted
The orbit entered in it is the Mercury orbit
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Kepler ”Theory” of Planetary Constellation [Failure]
The Earth orbit is the measure of all orbits
Around it we circumscribe the dodecahedron
The Mars orbit is circumscribed around the dodecahedron
Around the Mars orbit we circumscribe the tetrahedron
The Jupiter orbit is circumscribed around the tetrahedron
Around of the Jupiter orbit we circumscribe the cube
The Saturn orbit lies in the sphere surrounding the cube
In the Earth orbit the regular icosahedron is inserted
The orbit entered in it is the Venus orbit
In the Venus orbit the octahedron is inserted
The orbit entered in it is the Mercury orbit
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Kepler ”Theory” of Planetary Constellation [Failure]
The Earth orbit is the measure of all orbits
Around it we circumscribe the dodecahedron
The Mars orbit is circumscribed around the dodecahedron
Around the Mars orbit we circumscribe the tetrahedron
The Jupiter orbit is circumscribed around the tetrahedron
Around of the Jupiter orbit we circumscribe the cube
The Saturn orbit lies in the sphere surrounding the cube
In the Earth orbit the regular icosahedron is inserted
The orbit entered in it is the Venus orbit
In the Venus orbit the octahedron is inserted
The orbit entered in it is the Mercury orbit
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Kepler ”Theory” of Planetary Constellation [Failure]
The Earth orbit is the measure of all orbits
Around it we circumscribe the dodecahedron
The Mars orbit is circumscribed around the dodecahedron
Around the Mars orbit we circumscribe the tetrahedron
The Jupiter orbit is circumscribed around the tetrahedron
Around of the Jupiter orbit we circumscribe the cube
The Saturn orbit lies in the sphere surrounding the cube
In the Earth orbit the regular icosahedron is inserted
The orbit entered in it is the Venus orbit
In the Venus orbit the octahedron is inserted
The orbit entered in it is the Mercury orbit
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Kepler ”Theory” of Planetary Constellation [Failure]
The Earth orbit is the measure of all orbits
Around it we circumscribe the dodecahedron
The Mars orbit is circumscribed around the dodecahedron
Around the Mars orbit we circumscribe the tetrahedron
The Jupiter orbit is circumscribed around the tetrahedron
Around of the Jupiter orbit we circumscribe the cube
The Saturn orbit lies in the sphere surrounding the cube
In the Earth orbit the regular icosahedron is inserted
The orbit entered in it is the Venus orbit
In the Venus orbit the octahedron is inserted
The orbit entered in it is the Mercury orbit
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Kepler ”Theory” of Planetary Constellation [Failure]
The Earth orbit is the measure of all orbits
Around it we circumscribe the dodecahedron
The Mars orbit is circumscribed around the dodecahedron
Around the Mars orbit we circumscribe the tetrahedron
The Jupiter orbit is circumscribed around the tetrahedron
Around of the Jupiter orbit we circumscribe the cube
The Saturn orbit lies in the sphere surrounding the cube
In the Earth orbit the regular icosahedron is inserted
The orbit entered in it is the Venus orbit
In the Venus orbit the octahedron is inserted
The orbit entered in it is the Mercury orbit
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Kepler ”Theory” of Planetary Constellation [Failure]
The Earth orbit is the measure of all orbits
Around it we circumscribe the dodecahedron
The Mars orbit is circumscribed around the dodecahedron
Around the Mars orbit we circumscribe the tetrahedron
The Jupiter orbit is circumscribed around the tetrahedron
Around of the Jupiter orbit we circumscribe the cube
The Saturn orbit lies in the sphere surrounding the cube
In the Earth orbit the regular icosahedron is inserted
The orbit entered in it is the Venus orbit
In the Venus orbit the octahedron is inserted
The orbit entered in it is the Mercury orbit
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Kepler ”Theory” of Planetary Constellation [Failure]
The Earth orbit is the measure of all orbits
Around it we circumscribe the dodecahedron
The Mars orbit is circumscribed around the dodecahedron
Around the Mars orbit we circumscribe the tetrahedron
The Jupiter orbit is circumscribed around the tetrahedron
Around of the Jupiter orbit we circumscribe the cube
The Saturn orbit lies in the sphere surrounding the cube
In the Earth orbit the regular icosahedron is inserted
The orbit entered in it is the Venus orbit
In the Venus orbit the octahedron is inserted
The orbit entered in it is the Mercury orbit
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Kepler ”Theory” of Planetary Constellation [Failure]
The Earth orbit is the measure of all orbits
Around it we circumscribe the dodecahedron
The Mars orbit is circumscribed around the dodecahedron
Around the Mars orbit we circumscribe the tetrahedron
The Jupiter orbit is circumscribed around the tetrahedron
Around of the Jupiter orbit we circumscribe the cube
The Saturn orbit lies in the sphere surrounding the cube
In the Earth orbit the regular icosahedron is inserted
The orbit entered in it is the Venus orbit
In the Venus orbit the octahedron is inserted
The orbit entered in it is the Mercury orbit
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Kepler ”Theory” of Planetary Constellation [Failure]
The Earth orbit is the measure of all orbits
Around it we circumscribe the dodecahedron
The Mars orbit is circumscribed around the dodecahedron
Around the Mars orbit we circumscribe the tetrahedron
The Jupiter orbit is circumscribed around the tetrahedron
Around of the Jupiter orbit we circumscribe the cube
The Saturn orbit lies in the sphere surrounding the cube
In the Earth orbit the regular icosahedron is inserted
The orbit entered in it is the Venus orbit
In the Venus orbit the octahedron is inserted
The orbit entered in it is the Mercury orbit
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Paper on Snow-Crystal Symmetry (N o 1 in History)
In the winter of 1609 Kepler got caught by the snow-storm on
one of the bridges in Prague
In 1611 he published an article ’Strena seu de nive sexangula’
examining the planar hexagonal symmetry 1
Examples of Hexagonal Symmetry 2
1 ’Strena seu de nive sexangula’ - ’On the Six-Cornered Snowflakes’;
2 From Kenneth K. Librecht, Caltech
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Paper on Snow-Crystal Symmetry (N o 1 in History)
In the winter of 1609 Kepler got caught by the snow-storm on
one of the bridges in Prague
In 1611 he published an article ’Strena seu de nive sexangula’
examining the planar hexagonal symmetry 1
Examples of Hexagonal Symmetry 2
1 ’Strena seu de nive sexangula’ - ’On the Six-Cornered Snowflakes’;
2 From Kenneth K. Librecht, Caltech
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Paper on Snow-Crystal Symmetry (N o 1 in History)
In the winter of 1609 Kepler got caught by the snow-storm on
one of the bridges in Prague
In 1611 he published an article ’Strena seu de nive sexangula’
examining the planar hexagonal symmetry 1
Examples of Hexagonal Symmetry 2
1 ’Strena seu de nive sexangula’ - ’On the Six-Cornered Snowflakes’;
2 From Kenneth K. Librecht, Caltech
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Multiple Examples of the Hexagonal Symmetry
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
What Kepler Could Not Know ... [1]
The Secret of Hexagonal Symmetry of Snow Crystals?
Figure: The water molecules in an ice crystal form a hexagonal lattice.
Each red ball represents an oxygen atom, the grey sticks represent two
hydrogen atoms.
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
What Kepler Could Not Know ... [2]
Other Forms of Symmetry: 12-Fold and Cylindrical
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
What Kepler Could Not Know ... [3]
Empirical Systematics of Nakaya
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
What Kepler Could Not Know ... [4]
Crystal Growing, Humidity and Temperature
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Observations from a Certain Time-Perspective:
Despite all (human) efforts the understanding of nature is a
long process
There were many hypotheses and scientific quarrels about this
particular and many other puzzles of nature in the past ...
It is a good advice to remeber the question:
Are we sure?
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Observations from a Certain Time-Perspective:
Despite all (human) efforts the understanding of nature is a
long process
There were many hypotheses and scientific quarrels about this
particular and many other puzzles of nature in the past ...
It is a good advice to remeber the question:
Are we sure?
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Observations from a Certain Time-Perspective:
Despite all (human) efforts the understanding of nature is a
long process
There were many hypotheses and scientific quarrels about this
particular and many other puzzles of nature in the past ...
It is a good advice to remeber the question:
Are we sure?
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Symmetry: How Did It All Begin?
Kepler (1611) - observes hexagonal symmetry of snow crystals
Steno (1669) - observes that inclination angles of faces of the
quartz crystals are the same → independent of the crystal size
De Lisle (1783) - angles between crystal faces identify crystals
Haüy (1784) - studies the symmetries: rotations & reflections
Wollaston (1809) - first goniometer to measure crystal angles
Hausmann (1821) - introduces the spherical trigonometry
Hessel (1830) - shows existence of 32 polyhedral symmetries
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Symmetry: How Did It All Begin?
Kepler (1611) - observes hexagonal symmetry of snow crystals
Steno (1669) - observes that inclination angles of faces of the
quartz crystals are the same → independent of the crystal size
De Lisle (1783) - angles between crystal faces identify crystals
Haüy (1784) - studies the symmetries: rotations & reflections
Wollaston (1809) - first goniometer to measure crystal angles
Hausmann (1821) - introduces the spherical trigonometry
Hessel (1830) - shows existence of 32 polyhedral symmetries
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Symmetry: How Did It All Begin?
Kepler (1611) - observes hexagonal symmetry of snow crystals
Steno (1669) - observes that inclination angles of faces of the
quartz crystals are the same → independent of the crystal size
De Lisle (1783) - angles between crystal faces identify crystals
Haüy (1784) - studies the symmetries: rotations & reflections
Wollaston (1809) - first goniometer to measure crystal angles
Hausmann (1821) - introduces the spherical trigonometry
Hessel (1830) - shows existence of 32 polyhedral symmetries
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Symmetry: How Did It All Begin?
Kepler (1611) - observes hexagonal symmetry of snow crystals
Steno (1669) - observes that inclination angles of faces of the
quartz crystals are the same → independent of the crystal size
De Lisle (1783) - angles between crystal faces identify crystals
Haüy (1784) - studies the symmetries: rotations & reflections
Wollaston (1809) - first goniometer to measure crystal angles
Hausmann (1821) - introduces the spherical trigonometry
Hessel (1830) - shows existence of 32 polyhedral symmetries
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Symmetry: How Did It All Begin?
Kepler (1611) - observes hexagonal symmetry of snow crystals
Steno (1669) - observes that inclination angles of faces of the
quartz crystals are the same → independent of the crystal size
De Lisle (1783) - angles between crystal faces identify crystals
Haüy (1784) - studies the symmetries: rotations & reflections
Wollaston (1809) - first goniometer to measure crystal angles
Hausmann (1821) - introduces the spherical trigonometry
Hessel (1830) - shows existence of 32 polyhedral symmetries
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Symmetry: How Did It All Begin?
Kepler (1611) - observes hexagonal symmetry of snow crystals
Steno (1669) - observes that inclination angles of faces of the
quartz crystals are the same → independent of the crystal size
De Lisle (1783) - angles between crystal faces identify crystals
Haüy (1784) - studies the symmetries: rotations & reflections
Wollaston (1809) - first goniometer to measure crystal angles
Hausmann (1821) - introduces the spherical trigonometry
Hessel (1830) - shows existence of 32 polyhedral symmetries
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Symmetry: How Did It All Begin?
Kepler (1611) - observes hexagonal symmetry of snow crystals
Steno (1669) - observes that inclination angles of faces of the
quartz crystals are the same → independent of the crystal size
De Lisle (1783) - angles between crystal faces identify crystals
Haüy (1784) - studies the symmetries: rotations & reflections
Wollaston (1809) - first goniometer to measure crystal angles
Hausmann (1821) - introduces the spherical trigonometry
Hessel (1830) - shows existence of 32 polyhedral symmetries
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
The Idea and Some Properties of Permutations
Joseph-Louis Lagrange (1736-1813)
J. L. Lagrange
A. Lagrange studied (among many other things)
algebraic equations
B. He introduced the notion of permutations in
relation to solutions of these equations ...
C. ... however, he did not develop it further - this
will come only later ...
During the years 1772-1785 Lagrange created a series of memoirs about
’Differential Equations’, he developed calculus of variations; in 1788 he
published his ’Mécanique analytique’ including the ’Multiplier method’
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
The Idea and Some Properties of Permutations
Joseph-Louis Lagrange (1736-1813)
J. L. Lagrange
A. Lagrange studied (among many other things)
algebraic equations
B. He introduced the notion of permutations in
relation to solutions of these equations ...
C. ... however, he did not develop it further - this
will come only later ...
During the years 1772-1785 Lagrange created a series of memoirs about
’Differential Equations’, he developed calculus of variations; in 1788 he
published his ’Mécanique analytique’ including the ’Multiplier method’
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
The Idea and Some Properties of Permutations
Joseph-Louis Lagrange (1736-1813)
J. L. Lagrange
A. Lagrange studied (among many other things)
algebraic equations
B. He introduced the notion of permutations in
relation to solutions of these equations ...
C. ... however, he did not develop it further - this
will come only later ...
During the years 1772-1785 Lagrange created a series of memoirs about
’Differential Equations’, he developed calculus of variations; in 1788 he
published his ’Mécanique analytique’ including the ’Multiplier method’
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
The Idea and Some Properties of Permutations
Joseph-Louis Lagrange (1736-1813)
J. L. Lagrange
A. Lagrange studied (among many other things)
algebraic equations
B. He introduced the notion of permutations in
relation to solutions of these equations ...
C. ... however, he did not develop it further - this
will come only later ...
During the years 1772-1785 Lagrange created a series of memoirs about
’Differential Equations’, he developed calculus of variations; in 1788 he
published his ’Mécanique analytique’ including the ’Multiplier method’
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
The Concept of Permutation Group
Symmetry in Physics: A Short History
Group Theory: First Concepts
Evariste Galois (1811-1832)
E. Galois
A. One of the most adventurous figures in the
world of mathematics of XIX century
B. His works were judged incomprehensible by the
contemporary - among others by Poisson ...
C. ... they were correctly interpreted by Liouville,
published in 1846, 24 years after his death ...
Figure: A fragment of the original text written the night before the duel
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
The Concept of Permutation Group
Symmetry in Physics: A Short History
Group Theory: First Concepts
Evariste Galois (1811-1832)
E. Galois
A. One of the most adventurous figures in the
world of mathematics of XIX century
B. His works were judged incomprehensible by the
contemporary - among others by Poisson ...
C. ... they were correctly interpreted by Liouville,
published in 1846, 24 years after his death ...
Figure: A fragment of the original text written the night before the duel
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
The Concept of Permutation Group
Symmetry in Physics: A Short History
Group Theory: First Concepts
Evariste Galois (1811-1832)
E. Galois
A. One of the most adventurous figures in the
world of mathematics of XIX century
B. His works were judged incomprehensible by the
contemporary - among others by Poisson ...
C. ... they were correctly interpreted by Liouville,
published in 1846, 24 years after his death ...
Figure: A fragment of the original text written the night before the duel
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
The Concept of Permutation Group
Symmetry in Physics: A Short History
Group Theory: First Concepts
Evariste Galois (1811-1832)
E. Galois
A. One of the most adventurous figures in the
world of mathematics of XIX century
B. His works were judged incomprehensible by the
contemporary - among others by Poisson ...
C. ... they were correctly interpreted by Liouville,
published in 1846, 24 years after his death ...
Figure: A fragment of the original text written the night before the duel
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Another Young Genius
Symmetry in Physics: A Short History
Group Theory: First Concepts
Niels Abel (1802-1829)
A. Another tragic figure; he left behind important
mathematical ideas despite his very short life
B. Today we find in literature Abel’s integral
equations, Abelian functions, Abel theorem
about convergence; commutative groups are
called Abelian
N. Abel
Abel died in the age of 26 of tuberculosis and malnutrition; only two
days after his death the letter offering him a teaching position in Berlin
arrived in Norway ...
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Another Young Genius
Symmetry in Physics: A Short History
Group Theory: First Concepts
Niels Abel (1802-1829)
A. Another tragic figure; he left behind important
mathematical ideas despite his very short life
B. Today we find in literature Abel’s integral
equations, Abelian functions, Abel theorem
about convergence; commutative groups are
called Abelian
N. Abel
Abel died in the age of 26 of tuberculosis and malnutrition; only two
days after his death the letter offering him a teaching position in Berlin
arrived in Norway ...
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Another Young Genius
Symmetry in Physics: A Short History
Group Theory: First Concepts
Niels Abel (1802-1829)
A. Another tragic figure; he left behind important
mathematical ideas despite his very short life
B. Today we find in literature Abel’s integral
equations, Abelian functions, Abel theorem
about convergence; commutative groups are
called Abelian
N. Abel
Abel died in the age of 26 of tuberculosis and malnutrition; only two
days after his death the letter offering him a teaching position in Berlin
arrived in Norway ...
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Another Young Genius
Symmetry in Physics: A Short History
Group Theory: First Concepts
Niels Abel (1802-1829)
A. Another tragic figure; he left behind important
mathematical ideas despite his very short life
B. Today we find in literature Abel’s integral
equations, Abelian functions, Abel theorem
about convergence; commutative groups are
called Abelian
N. Abel
Abel died in the age of 26 of tuberculosis and malnutrition; only two
days after his death the letter offering him a teaching position in Berlin
arrived in Norway ...
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Further Development in Studies of Permutations
Augustin Louis CAUCHY (1789-1857)
A. L. Cauchy
A. One of the most successful mathematicians of
his times, pioneered studies of infinite series,
differential equations, probability ...
B. He undertook the studies of the properties of
permutations essential for the development of
group theory
It was Arthur CAYLEY who, after CAUCHY, follows the study of groups
of permutations - the only groups known at that time ...
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Further Development in Studies of Permutations
Augustin Louis CAUCHY (1789-1857)
A. L. Cauchy
A. One of the most successful mathematicians of
his times, pioneered studies of infinite series,
differential equations, probability ...
B. He undertook the studies of the properties of
permutations essential for the development of
group theory
It was Arthur CAYLEY who, after CAUCHY, follows the study of groups
of permutations - the only groups known at that time ...
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Further Development in Studies of Permutations
Augustin Louis CAUCHY (1789-1857)
A. L. Cauchy
A. One of the most successful mathematicians of
his times, pioneered studies of infinite series,
differential equations, probability ...
B. He undertook the studies of the properties of
permutations essential for the development of
group theory
It was Arthur CAYLEY who, after CAUCHY, follows the study of groups
of permutations - the only groups known at that time ...
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Further Development in Studies of Permutations
Augustin Louis CAUCHY (1789-1857)
A. L. Cauchy
A. One of the most successful mathematicians of
his times, pioneered studies of infinite series,
differential equations, probability ...
B. He undertook the studies of the properties of
permutations essential for the development of
group theory
It was Arthur CAYLEY who, after CAUCHY, follows the study of groups
of permutations - the only groups known at that time ...
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Arrival of the Definition of Abstract Groups
Arthur CAYLEY (1821-1895)
A. Defines for the first time the concept of an
abstract group
B. Introduces the idea of Group Multiplication
Table
C. Observes that matrices (and quaternions)
form groupes
A. Cayley
A. CAYLEY, graduated from Cambridge, spent several year as lawyer,
later became professor of pure mathematics at Cambridge. With over
900 publications one of the most active mathematicians of his times.
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Arrival of the Definition of Abstract Groups
Arthur CAYLEY (1821-1895)
A. Defines for the first time the concept of an
abstract group
B. Introduces the idea of Group Multiplication
Table
C. Observes that matrices (and quaternions)
form groupes
A. Cayley
A. CAYLEY, graduated from Cambridge, spent several year as lawyer,
later became professor of pure mathematics at Cambridge. With over
900 publications one of the most active mathematicians of his times.
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Arrival of the Definition of Abstract Groups
Arthur CAYLEY (1821-1895)
A. Defines for the first time the concept of an
abstract group
B. Introduces the idea of Group Multiplication
Table
C. Observes that matrices (and quaternions)
form groupes
A. Cayley
A. CAYLEY, graduated from Cambridge, spent several year as lawyer,
later became professor of pure mathematics at Cambridge. With over
900 publications one of the most active mathematicians of his times.
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Arrival of the Definition of Abstract Groups
Arthur CAYLEY (1821-1895)
A. Defines for the first time the concept of an
abstract group
B. Introduces the idea of Group Multiplication
Table
C. Observes that matrices (and quaternions)
form groupes
A. Cayley
A. CAYLEY, graduated from Cambridge, spent several year as lawyer,
later became professor of pure mathematics at Cambridge. With over
900 publications one of the most active mathematicians of his times.
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Arrival of the Definition of Abstract Groups
Arthur CAYLEY (1821-1895)
A. Defines for the first time the concept of an
abstract group
B. Introduces the idea of Group Multiplication
Table
C. Observes that matrices (and quaternions)
form groupes
A. Cayley
A. CAYLEY, graduated from Cambridge, spent several year as lawyer,
later became professor of pure mathematics at Cambridge. With over
900 publications one of the most active mathematicians of his times.
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Abstract Groups - Today’s Formulation
One of the most powerful tools to study transformation properties
and symmetries in physics is the theory of groups.
Definition (Group, CAYLEY)
Abstract elements g ∈ G form a group under the operation ”◦” if:
1 o For any g 1 , g 2 ∈ G the ’product’ g 1 ◦ g 2 ≡ g ∈ G
2 o For any g 1 , g 2 , g 3 ∈ G we have (g 1 ◦ g 2 ) ◦ g 3 = g 1 ◦ (g 2 ◦ g 3 )
3 o There exists a neutral element e ∈ G: ∀ g ∈ G : e ◦ g = g
4 o For any g ∈ G we find inverse g −1 ∈ G such that g ◦ g −1 = e
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Abstract Groups - Today’s Formulation
One of the most powerful tools to study transformation properties
and symmetries in physics is the theory of groups.
Definition (Group, CAYLEY)
Abstract elements g ∈ G form a group under the operation ”◦” if:
1 o For any g 1 , g 2 ∈ G the ’product’ g 1 ◦ g 2 ≡ g ∈ G
2 o For any g 1 , g 2 , g 3 ∈ G we have (g 1 ◦ g 2 ) ◦ g 3 = g 1 ◦ (g 2 ◦ g 3 )
3 o There exists a neutral element e ∈ G: ∀ g ∈ G : e ◦ g = g
4 o For any g ∈ G we find inverse g −1 ∈ G such that g ◦ g −1 = e
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Abstract Groups - Today’s Formulation
One of the most powerful tools to study transformation properties
and symmetries in physics is the theory of groups.
Definition (Group, CAYLEY)
Abstract elements g ∈ G form a group under the operation ”◦” if:
1 o For any g 1 , g 2 ∈ G the ’product’ g 1 ◦ g 2 ≡ g ∈ G
2 o For any g 1 , g 2 , g 3 ∈ G we have (g 1 ◦ g 2 ) ◦ g 3 = g 1 ◦ (g 2 ◦ g 3 )
3 o There exists a neutral element e ∈ G: ∀ g ∈ G : e ◦ g = g
4 o For any g ∈ G we find inverse g −1 ∈ G such that g ◦ g −1 = e
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Abstract Groups - Today’s Formulation
One of the most powerful tools to study transformation properties
and symmetries in physics is the theory of groups.
Definition (Group, CAYLEY)
Abstract elements g ∈ G form a group under the operation ”◦” if:
1 o For any g 1 , g 2 ∈ G the ’product’ g 1 ◦ g 2 ≡ g ∈ G
2 o For any g 1 , g 2 , g 3 ∈ G we have (g 1 ◦ g 2 ) ◦ g 3 = g 1 ◦ (g 2 ◦ g 3 )
3 o There exists a neutral element e ∈ G: ∀ g ∈ G : e ◦ g = g
4 o For any g ∈ G we find inverse g −1 ∈ G such that g ◦ g −1 = e
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Abstract Groups - Today’s Formulation
One of the most powerful tools to study transformation properties
and symmetries in physics is the theory of groups.
Definition (Group, CAYLEY)
Abstract elements g ∈ G form a group under the operation ”◦” if:
1 o For any g 1 , g 2 ∈ G the ’product’ g 1 ◦ g 2 ≡ g ∈ G
2 o For any g 1 , g 2 , g 3 ∈ G we have (g 1 ◦ g 2 ) ◦ g 3 = g 1 ◦ (g 2 ◦ g 3 )
3 o There exists a neutral element e ∈ G: ∀ g ∈ G : e ◦ g = g
4 o For any g ∈ G we find inverse g −1 ∈ G such that g ◦ g −1 = e
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Abstract Groups - Today’s Formulation
One of the most powerful tools to study transformation properties
and symmetries in physics is the theory of groups.
Definition (Group, CAYLEY)
Abstract elements g ∈ G form a group under the operation ”◦” if:
1 o For any g 1 , g 2 ∈ G the ’product’ g 1 ◦ g 2 ≡ g ∈ G
2 o For any g 1 , g 2 , g 3 ∈ G we have (g 1 ◦ g 2 ) ◦ g 3 = g 1 ◦ (g 2 ◦ g 3 )
3 o There exists a neutral element e ∈ G: ∀ g ∈ G : e ◦ g = g
4 o For any g ∈ G we find inverse g −1 ∈ G such that g ◦ g −1 = e
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Abstract Groups - Today’s Formulation
One of the most powerful tools to study transformation properties
and symmetries in physics is the theory of groups.
Definition (Group, CAYLEY)
Abstract elements g ∈ G form a group under the operation ”◦” if:
1 o For any g 1 , g 2 ∈ G the ’product’ g 1 ◦ g 2 ≡ g ∈ G
2 o For any g 1 , g 2 , g 3 ∈ G we have (g 1 ◦ g 2 ) ◦ g 3 = g 1 ◦ (g 2 ◦ g 3 )
3 o There exists a neutral element e ∈ G: ∀ g ∈ G : e ◦ g = g
4 o For any g ∈ G we find inverse g −1 ∈ G such that g ◦ g −1 = e
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Very Special Role of the Group of Permutations
One of the most powerful tools to study finite group properties is
the Theorem of Cayley.
Theorem (CAYLEY)
Every group G composed of n elements is necessarily isomorphic
with an n-element sub-group of the group S n
Conclusion:
It is sufficient to examine the group structure of the permutation
group - the structure of all other ones is the same !!!
Example: Group S 3
j„
df . 1, 2, 3
S 3 =
1, 2, 3
« „
1, 2, 3
,
2, 1, 3
« „
1, 2, 3
,
1, 3, 2
« „
1, 2, 3
,
3, 2, 1
« „
1, 2, 3
,
2, 3, 1
« „
1, 2, 3
,
3, 1, 2
«ff
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Very Special Role of the Group of Permutations
One of the most powerful tools to study finite group properties is
the Theorem of Cayley.
Theorem (CAYLEY)
Every group G composed of n elements is necessarily isomorphic
with an n-element sub-group of the group S n
Conclusion:
It is sufficient to examine the group structure of the permutation
group - the structure of all other ones is the same !!!
Example: Group S 3
j„
df . 1, 2, 3
S 3 =
1, 2, 3
« „
1, 2, 3
,
2, 1, 3
« „
1, 2, 3
,
1, 3, 2
« „
1, 2, 3
,
3, 2, 1
« „
1, 2, 3
,
2, 3, 1
« „
1, 2, 3
,
3, 1, 2
«ff
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Very Special Role of the Group of Permutations
One of the most powerful tools to study finite group properties is
the Theorem of Cayley.
Theorem (CAYLEY)
Every group G composed of n elements is necessarily isomorphic
with an n-element sub-group of the group S n
Conclusion:
It is sufficient to examine the group structure of the permutation
group - the structure of all other ones is the same !!!
Example: Group S 3
j„
df . 1, 2, 3
S 3 =
1, 2, 3
« „
1, 2, 3
,
2, 1, 3
« „
1, 2, 3
,
1, 3, 2
« „
1, 2, 3
,
3, 2, 1
« „
1, 2, 3
,
2, 3, 1
« „
1, 2, 3
,
3, 1, 2
«ff
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Groups: Historical Aspects
Symmetry in Physics: A Short History
Group Theory: First Concepts
Very Special Role of the Group of Permutations
One of the most powerful tools to study finite group properties is
the Theorem of Cayley.
Theorem (CAYLEY)
Every group G composed of n elements is necessarily isomorphic
with an n-element sub-group of the group S n
Conclusion:
It is sufficient to examine the group structure of the permutation
group - the structure of all other ones is the same !!!
Example: Group S 3
j„
df . 1, 2, 3
S 3 =
1, 2, 3
« „
1, 2, 3
,
2, 1, 3
« „
1, 2, 3
,
1, 3, 2
« „
1, 2, 3
,
3, 2, 1
« „
1, 2, 3
,
2, 3, 1
« „
1, 2, 3
,
3, 1, 2
«ff
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Part II
Symmetry and Spontaneous Symmetry Breaking
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Identical Particles in 3D-Spaces and Permutations
Given system with n identical particles ↔ H = H (x 1 , x 2 , . . . x n )
Particles being identical - Hamiltonian must be totally symmetric
P ij H(x 1 , . . . x i , . . . x j , . . . x n ) P −1
ij
= H(x 1 , . . . x j , . . . x i , . . . x n )
and H(x 1 , . . . x i , . . . x j , . . . x n ) = H(x 1 , . . . x j , . . . x i , . . . x n )
so that
P ij H P −1
ij
= H → [P ij , H] = 0, ∀ i ≠ j ≤ n
Implications:
1 o Operators H and P can be diagonalised simultaneously
2 o H |Ψ〉 = E |Ψ〉 → P ij Ĥ P −1
ij
(|P ij Ψ〉) = E (|P ij Ψ〉)
... consequently, if |Ψ〉 is an eigenstate of H so is |P ij Ψ〉
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Identical Particles in 3D-Spaces and Permutations
Given system with n identical particles ↔ H = H (x 1 , x 2 , . . . x n )
Particles being identical - Hamiltonian must be totally symmetric
P ij H(x 1 , . . . x i , . . . x j , . . . x n ) P −1
ij
= H(x 1 , . . . x j , . . . x i , . . . x n )
and H(x 1 , . . . x i , . . . x j , . . . x n ) = H(x 1 , . . . x j , . . . x i , . . . x n )
so that
P ij H P −1
ij
= H → [P ij , H] = 0, ∀ i ≠ j ≤ n
Implications:
1 o Operators H and P can be diagonalised simultaneously
2 o H |Ψ〉 = E |Ψ〉 → P ij Ĥ P −1
ij
(|P ij Ψ〉) = E (|P ij Ψ〉)
... consequently, if |Ψ〉 is an eigenstate of H so is |P ij Ψ〉
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Identical Particles in 3D-Spaces and Permutations
Given system with n identical particles ↔ H = H (x 1 , x 2 , . . . x n )
Particles being identical - Hamiltonian must be totally symmetric
P ij H(x 1 , . . . x i , . . . x j , . . . x n ) P −1
ij
= H(x 1 , . . . x j , . . . x i , . . . x n )
and H(x 1 , . . . x i , . . . x j , . . . x n ) = H(x 1 , . . . x j , . . . x i , . . . x n )
so that
P ij H P −1
ij
= H → [P ij , H] = 0, ∀ i ≠ j ≤ n
Implications:
1 o Operators H and P can be diagonalised simultaneously
2 o H |Ψ〉 = E |Ψ〉 → P ij Ĥ P −1
ij
(|P ij Ψ〉) = E (|P ij Ψ〉)
... consequently, if |Ψ〉 is an eigenstate of H so is |P ij Ψ〉
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Identical Particles in 3D-Spaces and Permutations
Given system with n identical particles ↔ H = H (x 1 , x 2 , . . . x n )
Particles being identical - Hamiltonian must be totally symmetric
P ij H(x 1 , . . . x i , . . . x j , . . . x n ) P −1
ij
= H(x 1 , . . . x j , . . . x i , . . . x n )
and H(x 1 , . . . x i , . . . x j , . . . x n ) = H(x 1 , . . . x j , . . . x i , . . . x n )
so that
P ij H P −1
ij
= H → [P ij , H] = 0, ∀ i ≠ j ≤ n
Implications:
1 o Operators H and P can be diagonalised simultaneously
2 o H |Ψ〉 = E |Ψ〉 → P ij Ĥ P −1
ij
(|P ij Ψ〉) = E (|P ij Ψ〉)
... consequently, if |Ψ〉 is an eigenstate of H so is |P ij Ψ〉
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Identical Particles in 3D-Spaces and Permutations
Given system with n identical particles ↔ H = H (x 1 , x 2 , . . . x n )
Particles being identical - Hamiltonian must be totally symmetric
P ij H(x 1 , . . . x i , . . . x j , . . . x n ) P −1
ij
= H(x 1 , . . . x j , . . . x i , . . . x n )
and H(x 1 , . . . x i , . . . x j , . . . x n ) = H(x 1 , . . . x j , . . . x i , . . . x n )
so that
P ij H P −1
ij
= H → [P ij , H] = 0, ∀ i ≠ j ≤ n
Implications:
1 o Operators H and P can be diagonalised simultaneously
2 o H |Ψ〉 = E |Ψ〉 → P ij Ĥ P −1
ij
(|P ij Ψ〉) = E (|P ij Ψ〉)
... consequently, if |Ψ〉 is an eigenstate of H so is |P ij Ψ〉
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Identical Particles in 3D-Spaces and Permutations
Given system with n identical particles ↔ H = H (x 1 , x 2 , . . . x n )
Particles being identical - Hamiltonian must be totally symmetric
P ij H(x 1 , . . . x i , . . . x j , . . . x n ) P −1
ij
= H(x 1 , . . . x j , . . . x i , . . . x n )
and H(x 1 , . . . x i , . . . x j , . . . x n ) = H(x 1 , . . . x j , . . . x i , . . . x n )
so that
P ij H P −1
ij
= H → [P ij , H] = 0, ∀ i ≠ j ≤ n
Implications:
1 o Operators H and P can be diagonalised simultaneously
2 o H |Ψ〉 = E |Ψ〉 → P ij Ĥ P −1
ij
(|P ij Ψ〉) = E (|P ij Ψ〉)
... consequently, if |Ψ〉 is an eigenstate of H so is |P ij Ψ〉
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Identical Particles in 3D-Spaces and Permutations
Given system with n identical particles ↔ H = H (x 1 , x 2 , . . . x n )
Particles being identical - Hamiltonian must be totally symmetric
P ij H(x 1 , . . . x i , . . . x j , . . . x n ) P −1
ij
= H(x 1 , . . . x j , . . . x i , . . . x n )
and H(x 1 , . . . x i , . . . x j , . . . x n ) = H(x 1 , . . . x j , . . . x i , . . . x n )
so that
P ij H P −1
ij
= H → [P ij , H] = 0, ∀ i ≠ j ≤ n
Implications:
1 o Operators H and P can be diagonalised simultaneously
2 o H |Ψ〉 = E |Ψ〉 → P ij Ĥ P −1
ij
(|P ij Ψ〉) = E (|P ij Ψ〉)
... consequently, if |Ψ〉 is an eigenstate of H so is |P ij Ψ〉
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Identical Particles in 3D-Spaces and Permutations
Given system with n identical particles ↔ H = H (x 1 , x 2 , . . . x n )
Particles being identical - Hamiltonian must be totally symmetric
P ij H(x 1 , . . . x i , . . . x j , . . . x n ) P −1
ij
= H(x 1 , . . . x j , . . . x i , . . . x n )
and H(x 1 , . . . x i , . . . x j , . . . x n ) = H(x 1 , . . . x j , . . . x i , . . . x n )
so that
P ij H P −1
ij
= H → [P ij , H] = 0, ∀ i ≠ j ≤ n
Implications:
1 o Operators H and P can be diagonalised simultaneously
2 o H |Ψ〉 = E |Ψ〉 → P ij Ĥ P −1
ij
(|P ij Ψ〉) = E (|P ij Ψ〉)
... consequently, if |Ψ〉 is an eigenstate of H so is |P ij Ψ〉
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Identical Particles Must Be either Fermions or Bosons!
From definition of the permutation operator it follows that P 2
ij
= 1
while P ij Ψ = p ij Ψ → P 2
ij Ψ = p 2
ij Ψ ↔ p 2
ij = 1 → p ij = ±1
In the particle-number representation we may write for short
df .
Ψ 1,2, ... n = Ψ(x 1 , x 2 , . . . x n )
Conclusion: We thus Discover the Pauli Principle !
1 o P ij Φ n1 , ... n i , ...n j , ... n n
= +Φ n1 , ... n j , ...n i , ... n n
→ Bosons
2 o P ij Ψ n1 , ... n i , ...n j , ... n n
= −Ψ n1 , ... n j , ...n i , ... n n
→ Fermions
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Identical Particles Must Be either Fermions or Bosons!
From definition of the permutation operator it follows that P 2
ij
= 1
while P ij Ψ = p ij Ψ → P 2
ij Ψ = p 2
ij Ψ ↔ p 2
ij = 1 → p ij = ±1
In the particle-number representation we may write for short
df .
Ψ 1,2, ... n = Ψ(x 1 , x 2 , . . . x n )
Conclusion: We thus Discover the Pauli Principle !
1 o P ij Φ n1 , ... n i , ...n j , ... n n
= +Φ n1 , ... n j , ...n i , ... n n
→ Bosons
2 o P ij Ψ n1 , ... n i , ...n j , ... n n
= −Ψ n1 , ... n j , ...n i , ... n n
→ Fermions
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Identical Particles Must Be either Fermions or Bosons!
From definition of the permutation operator it follows that P 2
ij
= 1
while P ij Ψ = p ij Ψ → P 2
ij Ψ = p 2
ij Ψ ↔ p 2
ij = 1 → p ij = ±1
In the particle-number representation we may write for short
df .
Ψ 1,2, ... n = Ψ(x 1 , x 2 , . . . x n )
Conclusion: We thus Discover the Pauli Principle !
1 o P ij Φ n1 , ... n i , ...n j , ... n n
= +Φ n1 , ... n j , ...n i , ... n n
→ Bosons
2 o P ij Ψ n1 , ... n i , ...n j , ... n n
= −Ψ n1 , ... n j , ...n i , ... n n
→ Fermions
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Identical Particles Must Be either Fermions or Bosons!
From definition of the permutation operator it follows that P 2
ij
= 1
while P ij Ψ = p ij Ψ → P 2
ij Ψ = p 2
ij Ψ ↔ p 2
ij = 1 → p ij = ±1
In the particle-number representation we may write for short
df .
Ψ 1,2, ... n = Ψ(x 1 , x 2 , . . . x n )
Conclusion: We thus Discover the Pauli Principle !
1 o P ij Φ n1 , ... n i , ...n j , ... n n
= +Φ n1 , ... n j , ...n i , ... n n
→ Bosons
2 o P ij Ψ n1 , ... n i , ...n j , ... n n
= −Ψ n1 , ... n j , ...n i , ... n n
→ Fermions
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Identical Particles Must Be either Fermions or Bosons!
From definition of the permutation operator it follows that P 2
ij
= 1
while P ij Ψ = p ij Ψ → P 2
ij Ψ = p 2
ij Ψ ↔ p 2
ij = 1 → p ij = ±1
In the particle-number representation we may write for short
df .
Ψ 1,2, ... n = Ψ(x 1 , x 2 , . . . x n )
Conclusion: We thus Discover the Pauli Principle !
1 o P ij Φ n1 , ... n i , ...n j , ... n n
= +Φ n1 , ... n j , ...n i , ... n n
→ Bosons
2 o P ij Ψ n1 , ... n i , ...n j , ... n n
= −Ψ n1 , ... n j , ...n i , ... n n
→ Fermions
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Identical Particles Must Be either Fermions or Bosons!
From definition of the permutation operator it follows that P 2
ij
= 1
while P ij Ψ = p ij Ψ → P 2
ij Ψ = p 2
ij Ψ ↔ p 2
ij = 1 → p ij = ±1
In the particle-number representation we may write for short
df .
Ψ 1,2, ... n = Ψ(x 1 , x 2 , . . . x n )
Conclusion: We thus Discover the Pauli Principle !
1 o P ij Φ n1 , ... n i , ...n j , ... n n
= +Φ n1 , ... n j , ...n i , ... n n
→ Bosons
2 o P ij Ψ n1 , ... n i , ...n j , ... n n
= −Ψ n1 , ... n j , ...n i , ... n n
→ Fermions
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Arrival of the Theory of Continuous (Lie) Groups
Sophus LIE (1842-1899)
A. Introduces new type of groups whose elements
are continuous functions of some parameter(s)
S. LIE
B. Most important for physics applications are
groups of matrices
C. Examples: GL n , SL n , U n , SU n , O n , Sp 2n ...
Distinguished role in physics play the space-time Lie groups: Rotation,
Lorentz and Poincaré, dimensions 3, 6 and 10, respectively, as well as
the unitary and special unitary groups together with all their sub-groups
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Arrival of the Theory of Continuous (Lie) Groups
Sophus LIE (1842-1899)
S. LIE
A. Introduces new type of groups whose elements
are continuous functions of some parameter(s)
B. Most important for physics applications are
groups of matrices
C. Examples: GL n , SL n , U n , SU n , O n , Sp 2n ...
Distinguished role in physics play the space-time Lie groups: Rotation,
Lorentz and Poincaré, dimensions 3, 6 and 10, respectively, as well as
the unitary and special unitary groups together with all their sub-groups
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Arrival of the Theory of Continuous (Lie) Groups
Sophus LIE (1842-1899)
S. LIE
A. Introduces new type of groups whose elements
are continuous functions of some parameter(s)
B. Most important for physics applications are
groups of matrices
C. Examples: GL n , SL n , U n , SU n , O n , Sp 2n ...
Distinguished role in physics play the space-time Lie groups: Rotation,
Lorentz and Poincaré, dimensions 3, 6 and 10, respectively, as well as
the unitary and special unitary groups together with all their sub-groups
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Arrival of the Theory of Continuous (Lie) Groups
Sophus LIE (1842-1899)
S. LIE
A. Introduces new type of groups whose elements
are continuous functions of some parameter(s)
B. Most important for physics applications are
groups of matrices
C. Examples: GL n , SL n , U n , SU n , O n , Sp 2n ...
Distinguished role in physics play the space-time Lie groups: Rotation,
Lorentz and Poincaré, dimensions 3, 6 and 10, respectively, as well as
the unitary and special unitary groups together with all their sub-groups
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
About Special Unitary Groups: SU n
Groups SU n ↔ n × n unitary matrices U ↔ [det(U) = 1]
It turns out that these matrices can be expressed as
U = exp [ ∑ ]
p αβ ĝ αβ , pαβ ↔ real parameters
αβ
Constant operators (generators) ĝ αβ satisfy
[ ĝ αβ , ĝ γδ ] = δ βγ ĝ αδ − δ αδ ĝ γβ
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
About Special Unitary Groups: SU n
Groups SU n ↔ n × n unitary matrices U ↔ [det(U) = 1]
It turns out that these matrices can be expressed as
U = exp [ ∑ ]
p αβ ĝ αβ , pαβ ↔ real parameters
αβ
Constant operators (generators) ĝ αβ satisfy
[ ĝ αβ , ĝ γδ ] = δ βγ ĝ αδ − δ αδ ĝ γβ
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
About Special Unitary Groups: SU n
Groups SU n ↔ n × n unitary matrices U ↔ [det(U) = 1]
It turns out that these matrices can be expressed as
U = exp [ ∑ ]
p αβ ĝ αβ , pαβ ↔ real parameters
αβ
Constant operators (generators) ĝ αβ satisfy
[ ĝ αβ , ĝ γδ ] = δ βγ ĝ αδ − δ αδ ĝ γβ
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
N-Body Systems with Two-Body Interactions
Consider an N-particle system with a two-body Hamiltonian
Ĥ = ∑ αβ
h αβ c α + c β + 1 ∑ ∑
v αβ;γδ c α + c +
2
β c δ c γ
αβ
γδ
Introduce operators
ˆN αβ
df .
= c + α c β
It is easy to verify that
[ ˆN αβ , ˆN γδ ] = δ βγ ˆN αδ − δ αδ ˆN γβ thus ˆN αβ ↔ ĝ αβ
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
N-Body Systems with Two-Body Interactions
Consider an N-particle system with a two-body Hamiltonian
Ĥ = ∑ αβ
h αβ c α + c β + 1 ∑ ∑
v αβ;γδ c α + c +
2
β c δ c γ
αβ
γδ
Introduce operators
ˆN αβ
df .
= c + α c β
It is easy to verify that
[ ˆN αβ , ˆN γδ ] = δ βγ ˆN αδ − δ αδ ˆN γβ thus ˆN αβ ↔ ĝ αβ
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
N-Body Systems with Two-Body Interactions
Consider an N-particle system with a two-body Hamiltonian
Ĥ = ∑ αβ
h αβ c α + c β + 1 ∑ ∑
v αβ;γδ c α + c +
2
β c δ c γ
αβ
γδ
Introduce operators
ˆN αβ
df .
= c + α c β
It is easy to verify that
[ ˆN αβ , ˆN γδ ] = δ βγ ˆN αδ − δ αδ ˆN γβ thus ˆN αβ ↔ ĝ αβ
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
N-Body Hamiltonians and the SU n -Generators
N-Body Hamiltonians are functions of SU n -group generators
Two-body interactions lead to quadratic forms of ˆN αβ = c + α c β
,
three-body interactions to the cubic forms of ˆN αβ , etc.
Hamiltonians of the N-body systems can be diagonalised within
bases of the irreducible representations
Solutions can be constructed that transform as the SU n -group
representations thus establishing a link H ↔ SU n
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
N-Body Hamiltonians and the SU n -Generators
N-Body Hamiltonians are functions of SU n -group generators
Two-body interactions lead to quadratic forms of ˆN αβ = c + α c β
,
three-body interactions to the cubic forms of ˆN αβ , etc.
Hamiltonians of the N-body systems can be diagonalised within
bases of the irreducible representations
Solutions can be constructed that transform as the SU n -group
representations thus establishing a link H ↔ SU n
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
N-Body Hamiltonians and the SU n -Generators
N-Body Hamiltonians are functions of SU n -group generators
Two-body interactions lead to quadratic forms of ˆN αβ = c + α c β
,
three-body interactions to the cubic forms of ˆN αβ , etc.
Hamiltonians of the N-body systems can be diagonalised within
bases of the irreducible representations
Solutions can be constructed that transform as the SU n -group
representations thus establishing a link H ↔ SU n
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
N-Body Hamiltonians and the SU n -Generators
N-Body Hamiltonians are functions of SU n -group generators
Two-body interactions lead to quadratic forms of ˆN αβ = c + α c β
,
three-body interactions to the cubic forms of ˆN αβ , etc.
Hamiltonians of the N-body systems can be diagonalised within
bases of the irreducible representations
Solutions can be constructed that transform as the SU n -group
representations thus establishing a link H ↔ SU n
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
N-Body Hamiltonians and More General Observations
Consequences in Terms of Chains of Sub-Groups
Group SU n has numerous sub-groups: U m and SU m with m < n,
similarly O m , SO m and in particular R(3) and all the point groups
More Generall Intellectual Challenges
If a group is a symmetry group of a physical system - so are its
sub-groups! Do we know their physical meaning? ...
Consequences?... Interpretation?
... but even if not ...
The group theory offers a guaranteed guidance through the jungle
of possibilities!
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
N-Body Hamiltonians and More General Observations
Consequences in Terms of Chains of Sub-Groups
Group SU n has numerous sub-groups: U m and SU m with m < n,
similarly O m , SO m and in particular R(3) and all the point groups
More Generall Intellectual Challenges
If a group is a symmetry group of a physical system - so are its
sub-groups! Do we know their physical meaning? ...
Consequences?... Interpretation?
... but even if not ...
The group theory offers a guaranteed guidance through the jungle
of possibilities!
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
N-Body Hamiltonians and More General Observations
Consequences in Terms of Chains of Sub-Groups
Group SU n has numerous sub-groups: U m and SU m with m < n,
similarly O m , SO m and in particular R(3) and all the point groups
More Generall Intellectual Challenges
If a group is a symmetry group of a physical system - so are its
sub-groups! Do we know their physical meaning? ...
Consequences?... Interpretation?
... but even if not ...
The group theory offers a guaranteed guidance through the jungle
of possibilities!
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Subgroup Structure Can Be Very, Very Rich ...
32 Point Groups: Subgroups
.
D 4h
T h
O T d
D 6h
C 4h
D 4
D 2d
C 4v
D 2h
T D 6
C 6h
C 6v
D 3d
D
3h
S 4
C 4
D 2
C 2h
C 2v
C 6
C 3i
D 3
C 3v
C 3h
O h
C C C C
i 2 s 3
Figure: Richness of the sub-group
structures at the end of chain...
C
1
.
Dashed lines indicate that the
subgroup marked is not invariant
Trivial groups are denoted here
C 1 ≡ {1I}, C s ≡ {1I, ˆσ},
C i ≡ {1I, ˆπ}
Here we show the structure only
at the very end of the SU n chain
- it helps imagining how rich the
full group structure is ...
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Point Groups - Examples from Molecular Physics
Group C 3v - Molecule: [NH 3 ] Group C 3h - Molecule: [B(OH) 3 ]
Group D 3d - Molecule: [C 2 H 6 ] Group D 3h - Molecule: [C 2 H 6 ]
From J. Goss, University of Newcastle
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
The Highest Symmetries in Molecular Physics
Group T d - Molecule: [CH 4 ] Group O h - Molecule: [SF 6 ]
Group D 6d - Mol.: [Cr(C 6 H 6 ) 2 ] Group I h - Molecule: [C 60 ]
From J. Goss, University of Newcastle
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Strong Interactions and Sub-Atomic Physics
Except for the stellar objects (for which N → ∞) the subatomic
objects are mesoscopic: N max ∼ 10 2 ≪ Avogadro Number ∼ 10 23
Standard Models of Nuclear Interactions Symmetries
1 o Mean-Field Methods SU n , SO n , point groups
2 o Nuclear Shell-Model SU n , SO n , point groups
3 o Nuclear Cluster Models point groups
4 o Compound (Interacting) Bosons group algebras
5 o Collective Nuclear Models point groups
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Strong Interactions and Sub-Atomic Physics
Except for the stellar objects (for which N → ∞) the subatomic
objects are mesoscopic: N max ∼ 10 2 ≪ Avogadro Number ∼ 10 23
Standard Models of Nuclear Interactions Symmetries
1 o Mean-Field Methods SU n , SO n , point groups
2 o Nuclear Shell-Model SU n , SO n , point groups
3 o Nuclear Cluster Models point groups
4 o Compound (Interacting) Bosons group algebras
5 o Collective Nuclear Models point groups
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Strong Interactions and Sub-Atomic Physics
Except for the stellar objects (for which N → ∞) the subatomic
objects are mesoscopic: N max ∼ 10 2 ≪ Avogadro Number ∼ 10 23
Standard Models of Nuclear Interactions Symmetries
1 o Mean-Field Methods SU n , SO n , point groups
2 o Nuclear Shell-Model SU n , SO n , point groups
3 o Nuclear Cluster Models point groups
4 o Compound (Interacting) Bosons group algebras
5 o Collective Nuclear Models point groups
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Strong Interactions and Sub-Atomic Physics
Except for the stellar objects (for which N → ∞) the subatomic
objects are mesoscopic: N max ∼ 10 2 ≪ Avogadro Number ∼ 10 23
Standard Models of Nuclear Interactions Symmetries
1 o Mean-Field Methods SU n , SO n , point groups
2 o Nuclear Shell-Model SU n , SO n , point groups
3 o Nuclear Cluster Models point groups
4 o Compound (Interacting) Bosons group algebras
5 o Collective Nuclear Models point groups
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Strong Interactions and Sub-Atomic Physics
Except for the stellar objects (for which N → ∞) the subatomic
objects are mesoscopic: N max ∼ 10 2 ≪ Avogadro Number ∼ 10 23
Standard Models of Nuclear Interactions Symmetries
1 o Mean-Field Methods SU n , SO n , point groups
2 o Nuclear Shell-Model SU n , SO n , point groups
3 o Nuclear Cluster Models point groups
4 o Compound (Interacting) Bosons group algebras
5 o Collective Nuclear Models point groups
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Strong Interactions and Sub-Atomic Physics
Except for the stellar objects (for which N → ∞) the subatomic
objects are mesoscopic: N max ∼ 10 2 ≪ Avogadro Number ∼ 10 23
Standard Models of Nuclear Interactions Symmetries
1 o Mean-Field Methods SU n , SO n , point groups
2 o Nuclear Shell-Model SU n , SO n , point groups
3 o Nuclear Cluster Models point groups
4 o Compound (Interacting) Bosons group algebras
5 o Collective Nuclear Models point groups
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Strong Interactions and Sub-Atomic Physics
Except for the stellar objects (for which N → ∞) the subatomic
objects are mesoscopic: N max ∼ 10 2 ≪ Avogadro Number ∼ 10 23
Standard Models of Nuclear Interactions Symmetries
1 o Mean-Field Methods SU n , SO n , point groups
2 o Nuclear Shell-Model SU n , SO n , point groups
3 o Nuclear Cluster Models point groups
4 o Compound (Interacting) Bosons group algebras
5 o Collective Nuclear Models point groups
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Symmetry vs. Asymmetry - Historical Perspective [1]
Principle of Sufficient Reason (LEIBNIZ)
If there is no sufficient reason for things to happen - then the
original situation does not change
The underlying idea: the presence of symmetry guarantees that
one choice is as good as any other - thus nothing happens
Pierre CURIE (’Sur la symétrie dans les phénomènes physiques’)
For a phenomenon to take place - an absence of symmetry is
needed - the asymmetry gives the origin to phenomena
If one phenomenon is a ’cause’ and another ’result’ - the symmetry
of cause must be higher than the symmetry of the result
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Symmetry vs. Asymmetry - Historical Perspective [1]
Principle of Sufficient Reason (LEIBNIZ)
If there is no sufficient reason for things to happen - then the
original situation does not change
The underlying idea: the presence of symmetry guarantees that
one choice is as good as any other - thus nothing happens
Pierre CURIE (’Sur la symétrie dans les phénomènes physiques’)
For a phenomenon to take place - an absence of symmetry is
needed - the asymmetry gives the origin to phenomena
If one phenomenon is a ’cause’ and another ’result’ - the symmetry
of cause must be higher than the symmetry of the result
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Symmetry vs. Asymmetry - Historical Perspective [1]
Principle of Sufficient Reason (LEIBNIZ)
If there is no sufficient reason for things to happen - then the
original situation does not change
The underlying idea: the presence of symmetry guarantees that
one choice is as good as any other - thus nothing happens
Pierre CURIE (’Sur la symétrie dans les phénomènes physiques’)
For a phenomenon to take place - an absence of symmetry is
needed - the asymmetry gives the origin to phenomena
If one phenomenon is a ’cause’ and another ’result’ - the symmetry
of cause must be higher than the symmetry of the result
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Symmetry vs. Asymmetry - Historical Perspective [1]
Principle of Sufficient Reason (LEIBNIZ)
If there is no sufficient reason for things to happen - then the
original situation does not change
The underlying idea: the presence of symmetry guarantees that
one choice is as good as any other - thus nothing happens
Pierre CURIE (’Sur la symétrie dans les phénomènes physiques’)
For a phenomenon to take place - an absence of symmetry is
needed - the asymmetry gives the origin to phenomena
If one phenomenon is a ’cause’ and another ’result’ - the symmetry
of cause must be higher than the symmetry of the result
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Symmetry vs. Asymmetry - Historical Perspective [1]
Principle of Sufficient Reason (LEIBNIZ)
If there is no sufficient reason for things to happen - then the
original situation does not change
The underlying idea: the presence of symmetry guarantees that
one choice is as good as any other - thus nothing happens
Pierre CURIE (’Sur la symétrie dans les phénomènes physiques’)
For a phenomenon to take place - an absence of symmetry is
needed - the asymmetry gives the origin to phenomena
If one phenomenon is a ’cause’ and another ’result’ - the symmetry
of cause must be higher than the symmetry of the result
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Symmetry vs. Asymmetry - Historical Perspective [1]
Principle of Sufficient Reason (LEIBNIZ)
If there is no sufficient reason for things to happen - then the
original situation does not change
The underlying idea: the presence of symmetry guarantees that
one choice is as good as any other - thus nothing happens
Pierre CURIE (’Sur la symétrie dans les phénomènes physiques’)
For a phenomenon to take place - an absence of symmetry is
needed - the asymmetry gives the origin to phenomena
If one phenomenon is a ’cause’ and another ’result’ - the symmetry
of cause must be higher than the symmetry of the result
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Symmetry vs. Asymmetry - Historical Perspective [2]
Caricature - A Classical (Macro) Example: Buridan’s Ass
Figure: Buridan’s ass, having two strictly identical bundles of carrots on
both sides has no reason to select one of the two - and dies of starvation
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Symmetry vs. Asymmetry - Historical Perspective [3]
... already Aristotle (384-322 BC)
In his ’De Caelo’ Aristotle asks how a dog faced with the choice
of two equally tempting meals could rationally choose one ...
French philosopher Jean BURIDAN, XIV century: an ass faces
starvation when it is unable to choose between two equally
appetising piles of hay ...
Remarks:
The ideas of Leibniz and Curie do not necessarily correspond to the
present day view - today the spontaneous symmetry breaking effects
are given much more attention
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Symmetry vs. Asymmetry - Historical Perspective [3]
... already Aristotle (384-322 BC)
In his ’De Caelo’ Aristotle asks how a dog faced with the choice
of two equally tempting meals could rationally choose one ...
French philosopher Jean BURIDAN, XIV century: an ass faces
starvation when it is unable to choose between two equally
appetising piles of hay ...
Remarks:
The ideas of Leibniz and Curie do not necessarily correspond to the
present day view - today the spontaneous symmetry breaking effects
are given much more attention
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
An Alternative: Spontaneous Symmetry Breaking
Aren’t there ”Some Miracles” Hidden Somewhere?
1 There exist phenomena not involving any obvious symmetry
2 There exist transitions from less symmetric to more symmetric
situations
3 Let us turn to a somewhat misleading picture of s. symmetry
breaking
Weinberg’s Caricature of the Spontaneous Symmetry Breaking
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
An Alternative: Spontaneous Symmetry Breaking
Aren’t there ”Some Miracles” Hidden Somewhere?
1 There exist phenomena not involving any obvious symmetry
2 There exist transitions from less symmetric to more symmetric
situations
3 Let us turn to a somewhat misleading picture of s. symmetry
breaking
Weinberg’s Caricature of the Spontaneous Symmetry Breaking
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
An Alternative: Spontaneous Symmetry Breaking
Aren’t there ”Some Miracles” Hidden Somewhere?
1 There exist phenomena not involving any obvious symmetry
2 There exist transitions from less symmetric to more symmetric
situations
3 Let us turn to a somewhat misleading picture of s. symmetry
breaking
Weinberg’s Caricature of the Spontaneous Symmetry Breaking
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
An Alternative: Spontaneous Symmetry Breaking
Aren’t there ”Some Miracles” Hidden Somewhere?
1 There exist phenomena not involving any obvious symmetry
2 There exist transitions from less symmetric to more symmetric
situations
3 Let us turn to a somewhat misleading picture of s. symmetry
breaking
Weinberg’s Caricature of the Spontaneous Symmetry Breaking
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Symmetry: Exact, Approximate, Spontaneously Broken
Consider an exact symmetry defined by transformations ĝ ∈ {G}:
ĝ H ĝ −1 = H ↔ Hψ = Eψ → ĝHĝ −1 (ĝψ) = E(ĝψ)
Conclusion: ψ and ĝψ are solutions. There are two possibilities:
1 o An n-fold degenerate spectrum: ĝψ p α = ∑ k≤n ψk α D α kp (ĝ)
Matrices Dkp α (ĝ) form an n-dimensional representation of {G}
2 o Non-degenerate spectrum: ĝψ p α = D α (ĝ) ψ k q
Numbers D α (ĝ) form one-dimensional representations of {G}
In both cases the index α enumerating irreducible representations
plays a role of the label (quantum number) characterising symmetry
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Symmetry: Exact, Approximate, Spontaneously Broken
Consider an exact symmetry defined by transformations ĝ ∈ {G}:
ĝ H ĝ −1 = H ↔ Hψ = Eψ → ĝHĝ −1 (ĝψ) = E(ĝψ)
Conclusion: ψ and ĝψ are solutions. There are two possibilities:
1 o An n-fold degenerate spectrum: ĝψ p α = ∑ k≤n ψk α D α kp (ĝ)
Matrices Dkp α (ĝ) form an n-dimensional representation of {G}
2 o Non-degenerate spectrum: ĝψ p α = D α (ĝ) ψ k q
Numbers D α (ĝ) form one-dimensional representations of {G}
In both cases the index α enumerating irreducible representations
plays a role of the label (quantum number) characterising symmetry
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Symmetry: Exact, Approximate, Spontaneously Broken
Consider an exact symmetry defined by transformations ĝ ∈ {G}:
ĝ H ĝ −1 = H ↔ Hψ = Eψ → ĝHĝ −1 (ĝψ) = E(ĝψ)
Conclusion: ψ and ĝψ are solutions. There are two possibilities:
1 o An n-fold degenerate spectrum: ĝψ p α = ∑ k≤n ψk α D α kp (ĝ)
Matrices Dkp α (ĝ) form an n-dimensional representation of {G}
2 o Non-degenerate spectrum: ĝψ p α = D α (ĝ) ψ k q
Numbers D α (ĝ) form one-dimensional representations of {G}
In both cases the index α enumerating irreducible representations
plays a role of the label (quantum number) characterising symmetry
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Symmetry: Exact, Approximate, Spontaneously Broken
Consider an exact symmetry defined by transformations ĝ ∈ {G}:
ĝ H ĝ −1 = H ↔ Hψ = Eψ → ĝHĝ −1 (ĝψ) = E(ĝψ)
Conclusion: ψ and ĝψ are solutions. There are two possibilities:
1 o An n-fold degenerate spectrum: ĝψ p α = ∑ k≤n ψk α D α kp (ĝ)
Matrices Dkp α (ĝ) form an n-dimensional representation of {G}
2 o Non-degenerate spectrum: ĝψ p α = D α (ĝ) ψ k q
Numbers D α (ĝ) form one-dimensional representations of {G}
In both cases the index α enumerating irreducible representations
plays a role of the label (quantum number) characterising symmetry
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Symmetry: Exact, Approximate, Spontaneously Broken
Consider an exact symmetry defined by transformations ĝ ∈ {G}:
ĝ H ĝ −1 = H ↔ Hψ = Eψ → ĝHĝ −1 (ĝψ) = E(ĝψ)
Conclusion: ψ and ĝψ are solutions. There are two possibilities:
1 o An n-fold degenerate spectrum: ĝψ p α = ∑ k≤n ψk α D α kp (ĝ)
Matrices Dkp α (ĝ) form an n-dimensional representation of {G}
2 o Non-degenerate spectrum: ĝψ p α = D α (ĝ) ψ k q
Numbers D α (ĝ) form one-dimensional representations of {G}
In both cases the index α enumerating irreducible representations
plays a role of the label (quantum number) characterising symmetry
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Example: An Exactly Spherically-Symmetric Nucleus
Consider 132 Sn nucleus with 50 protons and 82 neutrons
Are the wave-functions obeying the spherical symmetry?
How the Pauli principle can be seen from the 3D perspective?
Hint:
To answer these questions we solve the corresponding realistsic problem
of motion and obtain the energies and wave functions
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Example: An Exactly Spherically-Symmetric Nucleus
Consider 132 Sn nucleus with 50 protons and 82 neutrons
Are the wave-functions obeying the spherical symmetry?
How the Pauli principle can be seen from the 3D perspective?
Hint:
To answer these questions we solve the corresponding realistsic problem
of motion and obtain the energies and wave functions
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Example: An Exactly Spherically-Symmetric Nucleus
Consider 132 Sn nucleus with 50 protons and 82 neutrons
Are the wave-functions obeying the spherical symmetry?
How the Pauli principle can be seen from the 3D perspective?
Hint:
To answer these questions we solve the corresponding realistsic problem
of motion and obtain the energies and wave functions
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Example: An Exactly Spherically-Symmetric Nucleus
Consider 132 Sn nucleus with 50 protons and 82 neutrons
Are the wave-functions obeying the spherical symmetry?
How the Pauli principle can be seen from the 3D perspective?
Hint:
To answer these questions we solve the corresponding realistsic problem
of motion and obtain the energies and wave functions
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Spatial Structure of Orbitals (Sperical 132 Sn) (|ψ(⃗r )| 2 )
Limit 80% Limit ??% Limit ??% Limit ??% Limit ??%
Density distribution |ψ π (⃗r )| 2 ≥ Limit, for π = [2, 0, 2]1/2 orbital
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Spatial Structure of Orbitals (Sperical 132 Sn) (|ψ(⃗r )| 2 )
Limit 80% Limit 50% Limit ??% Limit ??% Limit ??%
Density distribution |ψ π (⃗r )| 2 ≥ Limit, for π = [2, 0, 2]1/2 orbital
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Spatial Structure of Orbitals (Sperical 132 Sn) (|ψ(⃗r )| 2 )
Limit 80% Limit 50% Limit 10% Limit ??% Limit ??%
Density distribution |ψ π (⃗r )| 2 ≥ Limit, for π = [2, 0, 2]1/2 orbital
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Spatial Structure of Orbitals (Sperical 132 Sn) (|ψ(⃗r )| 2 )
Limit 80% Limit 50% Limit 10% Limit 3% Limit ??%
Density distribution |ψ π (⃗r )| 2 ≥ Limit, for π = [2, 0, 2]1/2 orbital
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Spatial Structure of Orbitals (Sperical 132 Sn) (|ψ(⃗r )| 2 )
Limit 80% Limit 50% Limit 10% Limit 3% Limit 1%
Density distribution |ψ π (⃗r )| 2 ≥ Limit, for π = [2, 0, 2]1/2 orbital
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Spatial Structure of Orbitals (Sperical 132 Sn) (|ψ(⃗r )| 2 )
Limit 80% Limit 50% Limit 10% Limit 3% Limit 1%
Limit 20% Limit ??% Limit ??% Limit ??% Limit ??%
Bottom: N=3 shell b-[303]7/2, w-[312]5/2, y-[321]3/2, p-[310]1/2
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Spatial Structure of Orbitals (Sperical 132 Sn) (|ψ(⃗r )| 2 )
Limit 80% Limit 50% Limit 10% Limit 3% Limit 1%
Limit 20% Limit 15% Limit ??% Limit ??% Limit ??%
Bottom: N=3 shell b-[303]7/2, w-[312]5/2, y-[321]3/2, p-[310]1/2
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Spatial Structure of Orbitals (Sperical 132 Sn) (|ψ(⃗r )| 2 )
Limit 80% Limit 50% Limit 10% Limit 3% Limit 1%
Limit 20% Limit 15% Limit 12% Limit ??% Limit ??%
Bottom: N=3 shell b-[303]7/2, w-[312]5/2, y-[321]3/2, p-[310]1/2
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Spatial Structure of Orbitals (Sperical 132 Sn) (|ψ(⃗r )| 2 )
Limit 80% Limit 50% Limit 10% Limit 3% Limit 1%
Limit 20% Limit 15% Limit 12% Limit 10% Limit ??%
Bottom: N=3 shell b-[303]7/2, w-[312]5/2, y-[321]3/2, p-[310]1/2
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Spatial Structure of Orbitals (Sperical 132 Sn) (|ψ(⃗r )| 2 )
Limit 80% Limit 50% Limit 10% Limit 3% Limit 1%
Limit 20% Limit 15% Limit 12% Limit 10% Limit 9%
Bottom: N=3 shell b-[303]7/2, w-[312]5/2, y-[321]3/2, p-[310]1/2
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Spatial Structure of N=3 Spherical Shell (|ψ ν (⃗r )| 2 )
132 Sn: Distributions |ψ ν (⃗r )| 2 for single proton orbitals. Top O xz ,
bottom O yz . Proton e ν ↔ [ν=30, 32, ... 38] for spherical shell
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Spatial Structure of N=3 Spherical Shell (|ψ ν (⃗r )| 2 )
132 Sn: Distributions |ψ ν (⃗r )| 2 for single proton orbitals. Top O xz ,
bottom O yz . Proton e ν ↔ [ν=40, 42, ... 48] for spherical shell
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Spatial Structure of N=3 Spherical Shell (|ψ ν (⃗r )| 2 )
132 Sn: distributions |ψ ν (⃗r )| 2 for consecutive pairs of orbitals. Top
O xz , bottom O yz . Proton e ν ↔ [n=30:32, ... 38:40], spherical shell
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Spatial Structure of N=3 Spherical Shell (|ψ ν (⃗r )| 2 )
132 Sn: distributions |ψ ν (⃗r )| 2 for consecutive pairs of orbitals. Top
O xz , bottom O yz . Proton e ν ↔ [n=40:42, ... 48:50], spherical shell
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
What Did We Learn?
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
In the 132 Sn nucleus with 50 protons and 82 neutrons:
... the wave functions do not obey the spherical symmetry ...
They transform under irreducible representations of the SO 3
Calculation shows that the energy eigen-values are degenerate:
(degeneracy equal to 2j+1)
Question: What happens if we continue adding nucleons to 132 Sn?
Experiments will show that the resulting systems are NOT spherically
symmetric! Strange! - since the Hamiltonian of the system obeys
the rotational symmetry exactly!! Let us have closer look...
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
What Did We Learn?
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
In the 132 Sn nucleus with 50 protons and 82 neutrons:
... the wave functions do not obey the spherical symmetry ...
They transform under irreducible representations of the SO 3
Calculation shows that the energy eigen-values are degenerate:
(degeneracy equal to 2j+1)
Question: What happens if we continue adding nucleons to 132 Sn?
Experiments will show that the resulting systems are NOT spherically
symmetric! Strange! - since the Hamiltonian of the system obeys
the rotational symmetry exactly!! Let us have closer look...
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
What Did We Learn?
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
In the 132 Sn nucleus with 50 protons and 82 neutrons:
... the wave functions do not obey the spherical symmetry ...
They transform under irreducible representations of the SO 3
Calculation shows that the energy eigen-values are degenerate:
(degeneracy equal to 2j+1)
Question: What happens if we continue adding nucleons to 132 Sn?
Experiments will show that the resulting systems are NOT spherically
symmetric! Strange! - since the Hamiltonian of the system obeys
the rotational symmetry exactly!! Let us have closer look...
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
What Did We Learn?
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
In the 132 Sn nucleus with 50 protons and 82 neutrons:
... the wave functions do not obey the spherical symmetry ...
They transform under irreducible representations of the SO 3
Calculation shows that the energy eigen-values are degenerate:
(degeneracy equal to 2j+1)
Question: What happens if we continue adding nucleons to 132 Sn?
Experiments will show that the resulting systems are NOT spherically
symmetric! Strange! - since the Hamiltonian of the system obeys
the rotational symmetry exactly!! Let us have closer look...
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
What Did We Learn?
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
In the 132 Sn nucleus with 50 protons and 82 neutrons:
... the wave functions do not obey the spherical symmetry ...
They transform under irreducible representations of the SO 3
Calculation shows that the energy eigen-values are degenerate:
(degeneracy equal to 2j+1)
Question: What happens if we continue adding nucleons to 132 Sn?
Experiments will show that the resulting systems are NOT spherically
symmetric! Strange! - since the Hamiltonian of the system obeys
the rotational symmetry exactly!! Let us have closer look...
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
What Did We Learn?
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
In the 132 Sn nucleus with 50 protons and 82 neutrons:
... the wave functions do not obey the spherical symmetry ...
They transform under irreducible representations of the SO 3
Calculation shows that the energy eigen-values are degenerate:
(degeneracy equal to 2j+1)
Question: What happens if we continue adding nucleons to 132 Sn?
Experiments will show that the resulting systems are NOT spherically
symmetric! Strange! - since the Hamiltonian of the system obeys
the rotational symmetry exactly!! Let us have closer look...
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
What Did We Learn?
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
In the 132 Sn nucleus with 50 protons and 82 neutrons:
... the wave functions do not obey the spherical symmetry ...
They transform under irreducible representations of the SO 3
Calculation shows that the energy eigen-values are degenerate:
(degeneracy equal to 2j+1)
Question: What happens if we continue adding nucleons to 132 Sn?
Experiments will show that the resulting systems are NOT spherically
symmetric! Strange! - since the Hamiltonian of the system obeys
the rotational symmetry exactly!! Let us have closer look...
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
What Did We Learn?
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
In the 132 Sn nucleus with 50 protons and 82 neutrons:
... the wave functions do not obey the spherical symmetry ...
They transform under irreducible representations of the SO 3
Calculation shows that the energy eigen-values are degenerate:
(degeneracy equal to 2j+1)
Question: What happens if we continue adding nucleons to 132 Sn?
Experiments will show that the resulting systems are NOT spherically
symmetric! Strange! - since the Hamiltonian of the system obeys
the rotational symmetry exactly!! Let us have closer look...
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
What Did We Learn?
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
In the 132 Sn nucleus with 50 protons and 82 neutrons:
... the wave functions do not obey the spherical symmetry ...
They transform under irreducible representations of the SO 3
Calculation shows that the energy eigen-values are degenerate:
(degeneracy equal to 2j+1)
Question: What happens if we continue adding nucleons to 132 Sn?
Experiments will show that the resulting systems are NOT spherically
symmetric! Strange! - since the Hamiltonian of the system obeys
the rotational symmetry exactly!! Let us have closer look...
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Fundamental Symmetries of Nuclear Forces [1]
Denote ˆx df . = {⃗r,⃗p,⃗s,⃗t}. Nuclear interactions have the form
̂V (ˆx 1 , ˆx 2 ) ≡ ̂V C (ˆx 1 , ˆx 2 ) + ̂V T (ˆx 1 , ˆx 2 ) + ̂V LS (ˆx 1 , ˆx 2 ) + ̂V LL 2(ˆx 1 , ˆx 2 )
where: C-central, T -tensor, LS-spin-orbit and LL 2 -quadratic LS
Central Interaction (r 12 ≡ |⃗r 1 −⃗r 2 |)
̂V C (ˆx 1 , ˆx 2 ) = V 0 (r 12 ) + V s (r 12 ) [⃗s (1) · ⃗s (2) ]
+ V t (r 12 ) [⃗t (1) ·⃗t (2) ]
+ V s−t (r 12 ) [⃗s (1) · ⃗s (2) ] [⃗t (1) ·⃗t (2) ]
Invariant under rotations, translations, inversion and time-reversal
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Fundamental Symmetries of Nuclear Forces [1]
Denote ˆx df . = {⃗r,⃗p,⃗s,⃗t}. Nuclear interactions have the form
̂V (ˆx 1 , ˆx 2 ) ≡ ̂V C (ˆx 1 , ˆx 2 ) + ̂V T (ˆx 1 , ˆx 2 ) + ̂V LS (ˆx 1 , ˆx 2 ) + ̂V LL 2(ˆx 1 , ˆx 2 )
where: C-central, T -tensor, LS-spin-orbit and LL 2 -quadratic LS
Central Interaction (r 12 ≡ |⃗r 1 −⃗r 2 |)
̂V C (ˆx 1 , ˆx 2 ) = V 0 (r 12 ) + V s (r 12 ) [⃗s (1) · ⃗s (2) ]
+ V t (r 12 ) [⃗t (1) ·⃗t (2) ]
+ V s−t (r 12 ) [⃗s (1) · ⃗s (2) ] [⃗t (1) ·⃗t (2) ]
Invariant under rotations, translations, inversion and time-reversal
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Fundamental Symmetries of Nuclear Forces [2]
Denote ˆx df . = {⃗r,⃗p,⃗s,⃗t}. Nuclear interactions have the form
̂V (ˆx 1 , ˆx 2 ) ≡ ̂V C (ˆx 1 , ˆx 2 ) + ̂V T (ˆx 1 , ˆx 2 ) + ̂V LS (ˆx 1 , ˆx 2 ) + ̂V LL 2(ˆx 1 , ˆx 2 )
where: C-central, T -tensor, LS-spin-orbit and LL 2 -quadratic LS
Tensor Interaction [Non-Central]
⃗ S
(12) df .
= 3 (⃗s 1 ·⃗r 12 )(⃗s 2 ·⃗r 12 ) − (⃗s 1 · ⃗s 2 ) r12
2
r12
2
and r 12
df .
= |⃗r 1 −⃗r 2 |
̂V T (ˆx 1 , ˆx 2 ) = [V t0 (r 12 ) + V t1 (r 12 )⃗t 1 ·⃗t 2 ] ⃗ S (12)
Invariant under rotations, translations, inversion and time-reversal
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Fundamental Symmetries of Nuclear Forces [3]
Denote ˆx df . = {⃗r,⃗p,⃗s,⃗t}. Nuclear interactions have the form
̂V (ˆx 1 , ˆx 2 ) ≡ ̂V C (ˆx 1 , ˆx 2 ) + ̂V T (ˆx 1 , ˆx 2 ) + ̂V LS (ˆx 1 , ˆx 2 ) + ̂V LL 2(ˆx 1 , ˆx 2 )
where: C-central, T -tensor, LS-spin-orbit and LL 2 -quadratic LS
Spin-Orbit Interaction [Non-Local]
⃗ L
df .
= 1 2 (⃗r 1 −⃗r 2 ) ∧ (⃗p 1 − ⃗p 2 ), r 12
df .
= |⃗r 1 −⃗r 2 | and ⃗ S df . = ⃗s 1 + ⃗s 2
̂V LS (ˆx 1 , ˆx 2 ) = V LS (r 12 ) ⃗ L · ⃗S
Invariant under rotations, translations, inversion and time-reversal
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Fundamental Symmetries of Nuclear Forces [4]
Denote ˆx df . = {⃗r,⃗p,⃗s,⃗t}. Nuclear interactions have the form
̂V (ˆx 1 , ˆx 2 ) ≡ ̂V C (ˆx 1 , ˆx 2 ) + ̂V T (ˆx 1 , ˆx 2 ) + ̂V LS (ˆx 1 , ˆx 2 ) + ̂V LL 2(ˆx 1 , ˆx 2 )
where: C-central, T -tensor, LS-spin-orbit and LL 2 -quadratic LS
Quadratic Spin-Orbit Interaction [Non-Local]
⃗ L
df .
= 1 2 (⃗r 1 −⃗r 2 ) ∧ (⃗p 1 − ⃗p 2 ) and r 12
df .
= |⃗r 1 −⃗r 2 |
̂V LL (ˆx 1 , ˆx 2 ) = V LL (r 12 ){(⃗s 1 · ⃗s 2 ) ⃗ L 2 − 1 2 [(⃗s 1 · ⃗L)(⃗s 2 · ⃗L) + (⃗s 2 · ⃗L)(⃗s 1 · ⃗L)]}
Invariant under rotations, translations, inversion and time-reversal
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Nuclear Forces and the Mean Field Theory
1 o From the above discussion it follows that the N-N interaction
̂V (ˆx 1 , ˆx 2 ) ≡ ̂V C (ˆx 1 , ˆx 2 ) + ̂V T (ˆx 1 , ˆx 2 ) + ̂V LS (ˆx 1 , ˆx 2 ) + ̂V LL 2(ˆx 1 , ˆx 2 )
is invariant under rotations, translations, inversion and time-reversal
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Nuclear Forces and the Mean Field Theory
1 o From the above discussion it follows that the N-N interaction
̂V (ˆx 1 , ˆx 2 ) ≡ ̂V C (ˆx 1 , ˆx 2 ) + ̂V T (ˆx 1 , ˆx 2 ) + ̂V LS (ˆx 1 , ˆx 2 ) + ̂V LL 2(ˆx 1 , ˆx 2 )
is invariant under rotations, translations, inversion and time-reversal
Spont. Sym. Breaking
Z=fixed
E
N N ’ N"
The Nuclear Mean Field Theory ...
... is usually very successful. It is based on
∫
̂V mf (ˆx) = ψ ∗ (x ′ ) ̂V (ˆx, ˆx ′ )ψ(x ′ ) dx ′
spherical
deformed
Def.
Some or all of the above symmetries will be
broken by the mean-field Hamiltonian
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Spontaneous Symmetry Breaking - An Example
Given a system with symmetry {G} i.e. [H(β), G] = 0. Here β is a
parameter. Often a critical value, β crit. , exists such that:
1 For β < β crit. symmetry of solution is compatible with {G}
2 For β > β crit. symmetry of solution is not compatible with {G}
A Classical (Macro) Example
Original system ↔ axial symmetry
For F > F crit. we find infinitely
many solutions at the same energy
Yet: The original axial symmetry
will be spontaneously broken and
only one among many directions -
privileged !!!
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Spontaneous Symmetry Breaking - An Example
Given a system with symmetry {G} i.e. [H(β), G] = 0. Here β is a
parameter. Often a critical value, β crit. , exists such that:
1 For β < β crit. symmetry of solution is compatible with {G}
2 For β > β crit. symmetry of solution is not compatible with {G}
A Classical (Macro) Example
Original system ↔ axial symmetry
For F > F crit. we find infinitely
many solutions at the same energy
Yet: The original axial symmetry
will be spontaneously broken and
only one among many directions -
privileged !!!
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Spontaneous Symmetry Breaking - An Example
Given a system with symmetry {G} i.e. [H(β), G] = 0. Here β is a
parameter. Often a critical value, β crit. , exists such that:
1 For β < β crit. symmetry of solution is compatible with {G}
2 For β > β crit. symmetry of solution is not compatible with {G}
A Classical (Macro) Example
Original system ↔ axial symmetry
For F > F crit. we find infinitely
many solutions at the same energy
Yet: The original axial symmetry
will be spontaneously broken and
only one among many directions -
privileged !!!
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Spontaneous Symmetry Breaking - An Example
Given a system with symmetry {G} i.e. [H(β), G] = 0. Here β is a
parameter. Often a critical value, β crit. , exists such that:
1 For β < β crit. symmetry of solution is compatible with {G}
2 For β > β crit. symmetry of solution is not compatible with {G}
A Classical (Macro) Example
Original system ↔ axial symmetry
For F > F crit. we find infinitely
many solutions at the same energy
Yet: The original axial symmetry
will be spontaneously broken and
only one among many directions -
privileged !!!
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Spontaneous Symmetry Breaking - An Example
Given a system with symmetry {G} i.e. [H(β), G] = 0. Here β is a
parameter. Often a critical value, β crit. , exists such that:
1 For β < β crit. symmetry of solution is compatible with {G}
2 For β > β crit. symmetry of solution is not compatible with {G}
A Classical (Macro) Example
Original system ↔ axial symmetry
For F > F crit. we find infinitely
many solutions at the same energy
Yet: The original axial symmetry
will be spontaneously broken and
only one among many directions -
privileged !!!
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Spontaneous Symmetry Breaking - Classical Physics
A small bead of mass m threaded on a rotating circle of radius r;
constant frequency ω. We use the Lagrangian formalism:
1 L = 1 2 m˙⃗r 2 −V (⃗r ) = 1 2 mr 2 ˙ϑ 2 + 1 2 mω2 r 2 sin 2 ϑ−mgr(1−cos ϑ)
df .
2 L = T ϑ − U ϑ ↔ U ϑ = mgr[ (1 − cos ϑ) − 1
} {{ }
2 (mω2 /g) sin 2 ϑ ]
} {{ }
Potential Centrifugal
Bead on a Circle
z
ω
π _ υ
y
r
m
x
f = −mg
U=mgr(1−cos υ)
The lowest energy solutions ↔ ˙ϑ = 0 ↔ dU
dϑ = 0
Result: sin ϑ s = 0 and cos ϑ o = 1/β; β df . = rω 2 /g
Conclusion. Two types of lowest-energy solutions:
1. ϑ s = 0 - with the axial symmetry of the problem
2. ϑ o ≠ 0 - breaking the original axial symmetry !!
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Spontaneous Symmetry Breaking - Classical Physics
A small bead of mass m threaded on a rotating circle of radius r;
constant frequency ω. We use the Lagrangian formalism:
1 L = 1 2 m˙⃗r 2 −V (⃗r ) = 1 2 mr 2 ˙ϑ 2 + 1 2 mω2 r 2 sin 2 ϑ−mgr(1−cos ϑ)
df .
2 L = T ϑ − U ϑ ↔ U ϑ = mgr[ (1 − cos ϑ) − 1
} {{ }
2 (mω2 /g) sin 2 ϑ ]
} {{ }
Potential Centrifugal
Bead on a Circle
z
ω
π _ υ
y
r
m
x
f = −mg
U=mgr(1−cos υ)
The lowest energy solutions ↔ ˙ϑ = 0 ↔ dU
dϑ = 0
Result: sin ϑ s = 0 and cos ϑ o = 1/β; β df . = rω 2 /g
Conclusion. Two types of lowest-energy solutions:
1. ϑ s = 0 - with the axial symmetry of the problem
2. ϑ o ≠ 0 - breaking the original axial symmetry !!
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Spontaneous Symmetry Breaking - Classical Physics
A small bead of mass m threaded on a rotating circle of radius r;
constant frequency ω. We use the Lagrangian formalism:
1 L = 1 2 m˙⃗r 2 −V (⃗r ) = 1 2 mr 2 ˙ϑ 2 + 1 2 mω2 r 2 sin 2 ϑ−mgr(1−cos ϑ)
df .
2 L = T ϑ − U ϑ ↔ U ϑ = mgr[ (1 − cos ϑ) − 1
} {{ }
2 (mω2 /g) sin 2 ϑ ]
} {{ }
Potential Centrifugal
Bead on a Circle
z
ω
π _ υ
y
r
m
x
f = −mg
U=mgr(1−cos υ)
The lowest energy solutions ↔ ˙ϑ = 0 ↔ dU
dϑ = 0
Result: sin ϑ s = 0 and cos ϑ o = 1/β; β df . = rω 2 /g
Conclusion. Two types of lowest-energy solutions:
1. ϑ s = 0 - with the axial symmetry of the problem
2. ϑ o ≠ 0 - breaking the original axial symmetry !!
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Spontaneous Symmetry Breaking - Classical Physics
A small bead of mass m threaded on a rotating circle of radius r;
constant frequency ω. We use the Lagrangian formalism:
1 L = 1 2 m˙⃗r 2 −V (⃗r ) = 1 2 mr 2 ˙ϑ 2 + 1 2 mω2 r 2 sin 2 ϑ−mgr(1−cos ϑ)
df .
2 L = T ϑ − U ϑ ↔ U ϑ = mgr[ (1 − cos ϑ) − 1
} {{ }
2 (mω2 /g) sin 2 ϑ ]
} {{ }
Potential Centrifugal
Bead on a Circle
z
ω
π _ υ
y
r
m
x
f = −mg
U=mgr(1−cos υ)
The lowest energy solutions ↔ ˙ϑ = 0 ↔ dU
dϑ = 0
Result: sin ϑ s = 0 and cos ϑ o = 1/β; β df . = rω 2 /g
Conclusion. Two types of lowest-energy solutions:
1. ϑ s = 0 - with the axial symmetry of the problem
2. ϑ o ≠ 0 - breaking the original axial symmetry !!
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Spontaneous Symmetry Breaking - Discussion
1 o Under which conditions are the so obtained solutions stable?
2 o Are the derivatives d2 U
dϑ 2 ≡ U ′′ = mgr(cos ϑ − β cos 2ϑ) > 0?
1 Solution ϑ s = 0 → U ′′ (ϑ s ) = 1 − β → ϑ s stable for β < 1
2 Solution ϑ o ≠ 0 → U ′′ (ϑ o ) = β − 1/β → ϑ o stable for β > 1
U is Symmetric when ϑ → −ϑ
U ( υ)
Define frequency: ω crit. ≡ √ g/r
ω < ω crit. : bead rests at ϑ s = 0
β1
ω > ω crit. : climbs up ϑ o : 0 → π 2
_ π /2
β crit. = 1
π /2
υ
ϑ s - sub-critical ϑ o - over-critical
... but why should there be any
spontaneous symmetry breaking?
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Spontaneous Symmetry Breaking - Discussion
1 o Under which conditions are the so obtained solutions stable?
2 o Are the derivatives d2 U
dϑ 2 ≡ U ′′ = mgr(cos ϑ − β cos 2ϑ) > 0?
1 Solution ϑ s = 0 → U ′′ (ϑ s ) = 1 − β → ϑ s stable for β < 1
2 Solution ϑ o ≠ 0 → U ′′ (ϑ o ) = β − 1/β → ϑ o stable for β > 1
U is Symmetric when ϑ → −ϑ
U ( υ)
Define frequency: ω crit. ≡ √ g/r
ω < ω crit. : bead rests at ϑ s = 0
β1
ω > ω crit. : climbs up ϑ o : 0 → π 2
_ π /2
β crit. = 1
π /2
υ
ϑ s - sub-critical ϑ o - over-critical
... but why should there be any
spontaneous symmetry breaking?
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Spontaneous Symmetry Breaking - Discussion
1 o Under which conditions are the so obtained solutions stable?
2 o Are the derivatives d2 U
dϑ 2 ≡ U ′′ = mgr(cos ϑ − β cos 2ϑ) > 0?
1 Solution ϑ s = 0 → U ′′ (ϑ s ) = 1 − β → ϑ s stable for β < 1
2 Solution ϑ o ≠ 0 → U ′′ (ϑ o ) = β − 1/β → ϑ o stable for β > 1
U is Symmetric when ϑ → −ϑ
U ( υ)
Define frequency: ω crit. ≡ √ g/r
ω < ω crit. : bead rests at ϑ s = 0
β1
ω > ω crit. : climbs up ϑ o : 0 → π 2
_ π /2
β crit. = 1
π /2
υ
ϑ s - sub-critical ϑ o - over-critical
... but why should there be any
spontaneous symmetry breaking?
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Spontaneous Symmetry Breaking - Discussion
1 o Under which conditions are the so obtained solutions stable?
2 o Are the derivatives d2 U
dϑ 2 ≡ U ′′ = mgr(cos ϑ − β cos 2ϑ) > 0?
1 Solution ϑ s = 0 → U ′′ (ϑ s ) = 1 − β → ϑ s stable for β < 1
2 Solution ϑ o ≠ 0 → U ′′ (ϑ o ) = β − 1/β → ϑ o stable for β > 1
U is Symmetric when ϑ → −ϑ
U ( υ)
Define frequency: ω crit. ≡ √ g/r
ω < ω crit. : bead rests at ϑ s = 0
β1
ω > ω crit. : climbs up ϑ o : 0 → π 2
_ π /2
β crit. = 1
π /2
υ
ϑ s - sub-critical ϑ o - over-critical
... but why should there be any
spontaneous symmetry breaking?
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Spontaneous Symmetry Breaking - Interpretation [1]
Essential Facts about our Physical System:
1 Considered system has 1 degree of freedom and axial symmetry
2 ... but in the generalised coordinates only 1 discrete symmetry:
U(ϑ) = 2mgr sin 2 (ϑ/2)[1 − β cos 2 (ϑ/2)] ↔ U(+ϑ) = U(−ϑ)
Essential Conclusions from Our Analysis:
1 This symmetry holds for β < β crit. and β > β crit. potentials
2 This symmetry holds only for one i.e. the β < β crit. solution
3 For the β > β crit. solutions we have either ϑ o > 0 or ϑ o < 0
4 For the β > β crit. solutions the original symmetry is violated
5 We have considered only the lowest energy (’ground’) states
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Spontaneous Symmetry Breaking - Interpretation [1]
Essential Facts about our Physical System:
1 Considered system has 1 degree of freedom and axial symmetry
2 ... but in the generalised coordinates only 1 discrete symmetry:
U(ϑ) = 2mgr sin 2 (ϑ/2)[1 − β cos 2 (ϑ/2)] ↔ U(+ϑ) = U(−ϑ)
Essential Conclusions from Our Analysis:
1 This symmetry holds for β < β crit. and β > β crit. potentials
2 This symmetry holds only for one i.e. the β < β crit. solution
3 For the β > β crit. solutions we have either ϑ o > 0 or ϑ o < 0
4 For the β > β crit. solutions the original symmetry is violated
5 We have considered only the lowest energy (’ground’) states
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Spontaneous Symmetry Breaking - Interpretation [1]
Essential Facts about our Physical System:
1 Considered system has 1 degree of freedom and axial symmetry
2 ... but in the generalised coordinates only 1 discrete symmetry:
U(ϑ) = 2mgr sin 2 (ϑ/2)[1 − β cos 2 (ϑ/2)] ↔ U(+ϑ) = U(−ϑ)
Essential Conclusions from Our Analysis:
1 This symmetry holds for β < β crit. and β > β crit. potentials
2 This symmetry holds only for one i.e. the β < β crit. solution
3 For the β > β crit. solutions we have either ϑ o > 0 or ϑ o < 0
4 For the β > β crit. solutions the original symmetry is violated
5 We have considered only the lowest energy (’ground’) states
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Spontaneous Symmetry Breaking - Interpretation [1]
Essential Facts about our Physical System:
1 Considered system has 1 degree of freedom and axial symmetry
2 ... but in the generalised coordinates only 1 discrete symmetry:
U(ϑ) = 2mgr sin 2 (ϑ/2)[1 − β cos 2 (ϑ/2)] ↔ U(+ϑ) = U(−ϑ)
Essential Conclusions from Our Analysis:
1 This symmetry holds for β < β crit. and β > β crit. potentials
2 This symmetry holds only for one i.e. the β < β crit. solution
3 For the β > β crit. solutions we have either ϑ o > 0 or ϑ o < 0
4 For the β > β crit. solutions the original symmetry is violated
5 We have considered only the lowest energy (’ground’) states
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Spontaneous Symmetry Breaking - Interpretation [1]
Essential Facts about our Physical System:
1 Considered system has 1 degree of freedom and axial symmetry
2 ... but in the generalised coordinates only 1 discrete symmetry:
U(ϑ) = 2mgr sin 2 (ϑ/2)[1 − β cos 2 (ϑ/2)] ↔ U(+ϑ) = U(−ϑ)
Essential Conclusions from Our Analysis:
1 This symmetry holds for β < β crit. and β > β crit. potentials
2 This symmetry holds only for one i.e. the β < β crit. solution
3 For the β > β crit. solutions we have either ϑ o > 0 or ϑ o < 0
4 For the β > β crit. solutions the original symmetry is violated
5 We have considered only the lowest energy (’ground’) states
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Spontaneous Symmetry Breaking - Interpretation [1]
Essential Facts about our Physical System:
1 Considered system has 1 degree of freedom and axial symmetry
2 ... but in the generalised coordinates only 1 discrete symmetry:
U(ϑ) = 2mgr sin 2 (ϑ/2)[1 − β cos 2 (ϑ/2)] ↔ U(+ϑ) = U(−ϑ)
Essential Conclusions from Our Analysis:
1 This symmetry holds for β < β crit. and β > β crit. potentials
2 This symmetry holds only for one i.e. the β < β crit. solution
3 For the β > β crit. solutions we have either ϑ o > 0 or ϑ o < 0
4 For the β > β crit. solutions the original symmetry is violated
5 We have considered only the lowest energy (’ground’) states
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Spontaneous Symmetry Breaking - Interpretation [1]
Essential Facts about our Physical System:
1 Considered system has 1 degree of freedom and axial symmetry
2 ... but in the generalised coordinates only 1 discrete symmetry:
U(ϑ) = 2mgr sin 2 (ϑ/2)[1 − β cos 2 (ϑ/2)] ↔ U(+ϑ) = U(−ϑ)
Essential Conclusions from Our Analysis:
1 This symmetry holds for β < β crit. and β > β crit. potentials
2 This symmetry holds only for one i.e. the β < β crit. solution
3 For the β > β crit. solutions we have either ϑ o > 0 or ϑ o < 0
4 For the β > β crit. solutions the original symmetry is violated
5 We have considered only the lowest energy (’ground’) states
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Spontaneous Symmetry Breaking - Interpretation [1]
This transition is called spontaneous since the ϑ o -sign chosen by the
system depends on ’irrelevant details’: fluctuations, non-uniformities
Aren’t there ”Some Miracles” Hidden Somewhere?
1 We used no information about those ’irrelevant details’ yet ...
2 We were able to predict the absolute value of ϑ o in experiment
3 Is spontaneous symmetry breaking something so very unusual?
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Spontaneous Symmetry Breaking - Interpretation [1]
This transition is called spontaneous since the ϑ o -sign chosen by the
system depends on ’irrelevant details’: fluctuations, non-uniformities
Aren’t there ”Some Miracles” Hidden Somewhere?
1 We used no information about those ’irrelevant details’ yet ...
2 We were able to predict the absolute value of ϑ o in experiment
3 Is spontaneous symmetry breaking something so very unusual?
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Spontaneous Symmetry Breaking - Interpretation [1]
This transition is called spontaneous since the ϑ o -sign chosen by the
system depends on ’irrelevant details’: fluctuations, non-uniformities
Aren’t there ”Some Miracles” Hidden Somewhere?
1 We used no information about those ’irrelevant details’ yet ...
2 We were able to predict the absolute value of ϑ o in experiment
3 Is spontaneous symmetry breaking something so very unusual?
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Symmetry Groups and Degeneracies of Levels
Hamiltonian and a Group of Transformations
Given Hamiltonian H and a point-group G = {O 1 , O 2 , . . . O f }
Assume that G is a symmetry group of H i.e.
[H, O k ] = 0 with k = 1, 2, . . . f
Spectra and Group’s Irreducible Representations
Let irreducible representations of G be {R 1 , R 2 , . . . R r }
Let their dimensions be {d 1 , d 2 , . . . d r }, respectively
Then the eigenvalues {ε ν } of the problem
Hψ ν = ε ν ψ ν
appear in multiplets d 1 -fold, d 2 -fold ... degenerate
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Symmetry Groups and Degeneracies of Levels
Hamiltonian and a Group of Transformations
Given Hamiltonian H and a point-group G = {O 1 , O 2 , . . . O f }
Assume that G is a symmetry group of H i.e.
[H, O k ] = 0 with k = 1, 2, . . . f
Spectra and Group’s Irreducible Representations
Let irreducible representations of G be {R 1 , R 2 , . . . R r }
Let their dimensions be {d 1 , d 2 , . . . d r }, respectively
Then the eigenvalues {ε ν } of the problem
Hψ ν = ε ν ψ ν
appear in multiplets d 1 -fold, d 2 -fold ... degenerate
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Symmetry Groups and Degeneracies of Levels
Hamiltonian and a Group of Transformations
Given Hamiltonian H and a point-group G = {O 1 , O 2 , . . . O f }
Assume that G is a symmetry group of H i.e.
[H, O k ] = 0 with k = 1, 2, . . . f
Spectra and Group’s Irreducible Representations
Let irreducible representations of G be {R 1 , R 2 , . . . R r }
Let their dimensions be {d 1 , d 2 , . . . d r }, respectively
Then the eigenvalues {ε ν } of the problem
Hψ ν = ε ν ψ ν
appear in multiplets d 1 -fold, d 2 -fold ... degenerate
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Symmetry Groups and Degeneracies of Levels
Hamiltonian and a Group of Transformations
Given Hamiltonian H and a point-group G = {O 1 , O 2 , . . . O f }
Assume that G is a symmetry group of H i.e.
[H, O k ] = 0 with k = 1, 2, . . . f
Spectra and Group’s Irreducible Representations
Let irreducible representations of G be {R 1 , R 2 , . . . R r }
Let their dimensions be {d 1 , d 2 , . . . d r }, respectively
Then the eigenvalues {ε ν } of the problem
Hψ ν = ε ν ψ ν
appear in multiplets d 1 -fold, d 2 -fold ... degenerate
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Symmetry Groups and Degeneracies of Levels
Hamiltonian and a Group of Transformations
Given Hamiltonian H and a point-group G = {O 1 , O 2 , . . . O f }
Assume that G is a symmetry group of H i.e.
[H, O k ] = 0 with k = 1, 2, . . . f
Spectra and Group’s Irreducible Representations
Let irreducible representations of G be {R 1 , R 2 , . . . R r }
Let their dimensions be {d 1 , d 2 , . . . d r }, respectively
Then the eigenvalues {ε ν } of the problem
Hψ ν = ε ν ψ ν
appear in multiplets d 1 -fold, d 2 -fold ... degenerate
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Symmetry Groups and Degeneracies of Levels
Hamiltonian and a Group of Transformations
Given Hamiltonian H and a point-group G = {O 1 , O 2 , . . . O f }
Assume that G is a symmetry group of H i.e.
[H, O k ] = 0 with k = 1, 2, . . . f
Spectra and Group’s Irreducible Representations
Let irreducible representations of G be {R 1 , R 2 , . . . R r }
Let their dimensions be {d 1 , d 2 , . . . d r }, respectively
Then the eigenvalues {ε ν } of the problem
Hψ ν = ε ν ψ ν
appear in multiplets d 1 -fold, d 2 -fold ... degenerate
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Point Groups - ... and What We Need to Remember
Consider a group of symmetry of an equilateral triangle (group C 3v )
Two Types of Transformations - Observe Existence of a Fixed Point
a
120 o
b
c
C^ 3
240 o
(C
^ 2
3)
b
a
c
Figure: This group contains rotations Ĉ3 and Ĉ 2 3 (120o and 240 o ), left,
and three vertical reflections ˆσ va , ˆσ vb and ˆσ vc , right, in addition to 1I.
Six Element Point Group: C 3v = {1I, Ĉ 3 , Ĉ 2 3 , ˆσ va , ˆσ vb , ˆσ vc }
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Point Groups - ... and What We Need to Remember
Consider a group of symmetry of an equilateral triangle (group C 3v )
Two Types of Transformations - Observe Existence of a Fixed Point
a
120 o
b
c
C^ 3
240 o
(C
^ 2
3)
b
a
c
Figure: This group contains rotations Ĉ3 and Ĉ 2 3 (120o and 240 o ), left,
and three vertical reflections ˆσ va , ˆσ vb and ˆσ vc , right, in addition to 1I.
Six Element Point Group: C 3v = {1I, Ĉ 3 , Ĉ 2 3 , ˆσ va , ˆσ vb , ˆσ vc }
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Discrete Symmetries in Nuclei
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Let us recall one of the magic forms introduced long time by Plato.
The implied symmetry leads to the tetrahedral group denoted T d
A tetrahedron has four equal walls.
Its shape is invariant with respect to
24 symmetry elements. Tetrahedron
is not invariant with respect to the
inversion. Of course nuclei cannot be
represented by a sharp-edge pyramid
... but rather in a form of a regular spherical harmonic expansion:
λ∑
max λ∑
R(ϑ, ϕ) = R 0 c({α})[1 + α λ,µ Y λ,µ (ϑ, ϕ)]
λ
µ=−λ
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Discrete Symmetries in Nuclei
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Let us recall one of the magic forms introduced long time by Plato.
The implied symmetry leads to the tetrahedral group denoted T d
A tetrahedron has four equal walls.
Its shape is invariant with respect to
24 symmetry elements. Tetrahedron
is not invariant with respect to the
inversion. Of course nuclei cannot be
represented by a sharp-edge pyramid
... but rather in a form of a regular spherical harmonic expansion:
λ∑
max λ∑
R(ϑ, ϕ) = R 0 c({α})[1 + α λ,µ Y λ,µ (ϑ, ϕ)]
λ
µ=−λ
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
A Basis for Tetrahedral Symmetry
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Only special combinations 1 of spherical harmonics may form a
basis for surfaces with tetrahedral symmetry:
Three Lowest Orders:
Rank ↔ Multipolarity λ
λ = 3 : α 3,±2 ≡ t 3
λ = 7 : α 7,±2 ≡ t 7 ; α 7,±6 ≡ − q 11
13 · t 7
λ = 9 : α 9,±2 ≡ t 9 ; α 9,±6 ≡ + q 28
198 · t 9
1 Collaboration with Andrzej GÓŹDŹ and Daniel ROS̷LY, UMCS Lublin, Poland
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
A Basis for Tetrahedral Symmetry
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Only special combinations 1 of spherical harmonics may form a
basis for surfaces with tetrahedral symmetry:
Three Lowest Orders:
Rank ↔ Multipolarity λ
λ = 3 : α 3,±2 ≡ t 3
λ = 7 : α 7,±2 ≡ t 7 ; α 7,±6 ≡ − q 11
13 · t 7
λ = 9 : α 9,±2 ≡ t 9 ; α 9,±6 ≡ + q 28
198 · t 9
1 Collaboration with Andrzej GÓŹDŹ and Daniel ROS̷LY, UMCS Lublin, Poland
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
A Basis for Tetrahedral Symmetry
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Only special combinations 1 of spherical harmonics may form a
basis for surfaces with tetrahedral symmetry:
Three Lowest Orders:
Rank ↔ Multipolarity λ
λ = 3 : α 3,±2 ≡ t 3
λ = 7 : α 7,±2 ≡ t 7 ; α 7,±6 ≡ − q 11
13 · t 7
λ = 9 : α 9,±2 ≡ t 9 ; α 9,±6 ≡ + q 28
198 · t 9
1 Collaboration with Andrzej GÓŹDŹ and Daniel ROS̷LY, UMCS Lublin, Poland
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
A Basis for Tetrahedral Symmetry
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Only special combinations 1 of spherical harmonics may form a
basis for surfaces with tetrahedral symmetry:
Three Lowest Orders:
Rank ↔ Multipolarity λ
λ = 3 : α 3,±2 ≡ t 3
λ = 7 : α 7,±2 ≡ t 7 ; α 7,±6 ≡ − q 11
13 · t 7
λ = 9 : α 9,±2 ≡ t 9 ; α 9,±6 ≡ + q 28
198 · t 9
1 Collaboration with Andrzej GÓŹDŹ and Daniel ROS̷LY, UMCS Lublin, Poland
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Nuclear Tetrahedral Shapes - 3D Examples (Part 1)
Illustrations below show the tetrahedral-symmetric surfaces at three
increasing values of rank λ = 3 deformations t 1 : 0.1, 0.2 and 0.3:
Figure: t 3 = 0.1 Figure: t 3 = 0.2 Figure: t 3 = 0.3
Observations:
There are infinitely many tetrahedral-symmetric surfaces
Nuclear ’pyramids’ do not resemble pyramids!
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Nuclear Tetrahedral Shapes - Proton Spectra
Double group Td
D has two 2-dimensional - and one 4-dimensional
irreducible representations → three distinct families of levels
Proton Energies [MeV]
-2
-3
-4
-5
-6
-7
-8
-9
-10
-11
-12
226
90 Th 136
{07}[5,4,1] 3/2
{19}[5,1,4] 7/2
{07}[5,0,5] 9/2
{20}[5,0,5] 9/2
{08}[5,2,3] 5/2
{06}[6,1,5] 11/2
{10}[6,2,4] 9/2
{08}[5,0,3] 7/2
{06}[6,3,3] 7/2
{04}[4,4,0] 1/2
{12}[5,1,4] 9/2
{09}[5,2,3] 7/2
{16}[5,0,5] 11/2
{08}[5,2,1] 1/2
{11}[5,0,5] 11/2
{09}[4,0,0] 1/2
{08}[4,4,0] 1/2
{16}[4,0,2] 3/2
{13}[4,1,3] 5/2
{10}[4,0,4] 7/2
{15}[4,0,4] 7/2
{09}[4,2,2] 3/2
{07}[4,3,1] 1/2
{09}[5,1,4] 9/2
100
64
82
58
100
64
90
70
94
-.2 -.1 .0 .1 .2 .3 .4
Tetrahedral Deformation
{04}[4,1,1] 3/2
{06}[4,2,0] 1/2
{06}[5,0,5] 9/2
{05}[6,1,5] 9/2
{07}[4,1,3] 7/2
{07}[4,2,2] 5/2
{11}[5,0,5] 11/2
{06}[5,4,1] 1/2
{05}[4,0,0] 1/2
{04}[3,0,1] 1/2
{04}[4,1,3] 7/2
{03}[4,1,3] 7/2
{04}[5,0,3] 7/2
{10}[3,1,2] 3/2
{05}[6,1,5] 11/2
{05}[4,3,1] 1/2
{04}[5,0,5] 11/2
{06}[4,0,4] 7/2
{06}[4,0,4] 7/2
{06}[3,0,1] 1/2
{04}[3,1,0] 1/2
{03}[3,1,0] 1/2
{10}[3,1,2] 5/2
{06}[3,1,2] 3/2
Strasbourg, August 2002 Woods-Saxon Universal Params.
α 32(min)=-.200, α32(max)=.400
Figure: Full lines ↔ 4-dimensional irreducible representations - marked
with double Nilsson labels. Observe huge gaps at N=64, 70, 90-94, 100.
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Nuclear Tetrahedral Shapes - Neutron Spectra
Double group Td
D has two 2-dimensional - and one 4-dimensional
irreducible representations → three distinct families of levels
Neutron Energies [MeV]
-4
-5
-6
-7
-8
-9
-10
-11
226
90 Th 136
{13}[6,2,4] 7/2
{05}[6,5,1] 3/2
{03}[6,6,0] 1/2
{07}[6,0,2] 5/2
{04}[6,4,0] 1/2
{11}[6,0,6] 11/2
{07}[6,0,6] 11/2
{06}[6,0,4] 9/2
{05}[7,2,5] 11/2
{06}[7,2,5] 11/2
{08}[6,0,4] 9/2
{04}[7,3,4] 9/2
{08}[6,1,5] 11/2
{09}[6,2,4] 9/2
{05}[6,1,5] 11/2
{04}[6,1,5] 11/2
{08}[6,3,1] 1/2
{19}[6,0,6] 13/2
{09}[6,0,6] 13/2
{08}[5,0,1] 1/2
{07}[5,5,0] 1/2
{13}[5,0,3] 5/2
{06}[5,1,2] 3/2
{07}[4,1,1] 1/2
{04}[5,5,0] 1/2
148
136 136
126
124
142
112
-.2 -.1 .0 .1 .2 .3 .4
Tetrahedral Deformation
{04}[6,0,6] 11/2
{02}[5,2,1] 3/2
{03}[5,2,1] 3/2
{04}[5,3,2] 5/2
{06}[6,5,1] 1/2
{10}[6,0,6] 13/2
{06}[5,2,3] 7/2
{04}[5,0,5] 9/2
{05}[5,3,2] 5/2
{08}[5,1,4] 9/2
{03}[5,0,1] 3/2
{03}[5,0,1] 3/2
{06}[6,2,4] 9/2
{04}[7,1,6] 13/2
{04}[4,1,3] 5/2
{04}[5,3,0] 1/2
{05}[7,2,5] 11/2
{04}[5,1,2] 5/2
{03}[5,4,1] 1/2
{02}[4,1,3] 5/2
{02}[3,0,1] 3/2
{06}[6,0,6] 13/2
{03}[5,0,3] 5/2
{03}[5,0,3] 5/2
{05}[4,1,1] 3/2
Strasbourg, August 2002 Woods-Saxon Universal Params.
α 32(min)=-.200, α32(max)=.400
Figure: Full lines ↔ 4-dimensional irreducible representations - marked
with double Nilsson labels. Observe huge gaps at N=112, 136.
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Introducing Nuclear Octahedral Symmetry
Let us recall one of the magic forms introduced long time by Plato.
The implied symmetry leads to the octahedral group denoted O h
An octahedron has 8 equal walls. Its
shape is invariant with respect to 48
symmetry elements that include inversion.
However, the nuclear surface
cannot be represented in the form of
¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦
§ § § § § § § § § § § § § § § § § § § § § § § § § § §
¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦
§ § § § § § § § § § § § § § § § § § § § § § § § § § §
¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦
§ § § § § § § § § § § § § § § § § § § § § § § § § § §
¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦
§ § § § § § § § § § § § § § § § § § § § § § § § § § §
¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦
§ § § § § § § § § § § § § § § § § § § § § § § § § § §
£ £ £ £ £ ¢ ¢ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢
¢ £ £ £ £ £ £
¤ ¤ ¤ ¤ ¤ ¢ ¢ ¢ £ £ £ ¢ ¤ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¤ ¢ ¢ ¢ ¢ £ £ £ ¤ ¤ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥
¡ ¡
¡ ¡ ¡ ¡ ¡ ¡
¡ ¡
¡ ¡ ¡ £ ¡ £ ¡ £ ¡ £ £ £ £ £ £ £ £ ¢ ¢ ¢ ¢ ¢ ¢ ¡ ¤ ¡ ¤ ¡ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¥ ¥
¡ ¡ ¥
¡ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥
¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¢ ¡ ¢ ¢ ¢ ¢ ¢ ¢
¢ ¢ ¢ ¢ ¢ ¢ £ £ £ £ £ £ ¡ ¡
¡ ¡ ¡ ¡ ¡ ¡ ¡
¡ ¡ ¡ ¡ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ £ £ £ £ £ £ £ £ £ ¥
¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ £ £ £ £ £ £ £ £ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¤
¤ ¡ ¡ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¡
¤ ¤ ¤ ¤ ¢ ¡ ¡ ¡ ¡ ¡
¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡
¨ ¨ ¨ ¨ ¡ ¨ ¨ © © © © © © © © © © © © © © © © © © © © © © © © © © © © © © © © © © © ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢
£ £ £ £ £ £ £ £ £ £ £ ¢ £ ¢ £ ¢ £ ¢ £ ¢ £ ¢ £ ¢ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡
¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢
¡
a diamond → → → ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡
¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡
→ → → → → ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢
£ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £
¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢
£ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £
¡
¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢
£ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £
¡
¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢
£ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £
¡
£ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £
... but rather in a form of a regular spherical harmonic expansion:
λ∑
max λ∑
R(ϑ, ϕ) = R 0 c({α})[1 + α λ,µ Y λ,µ (ϑ, ϕ)]
λ
µ=−λ
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
A Basis for Octahedral Symmetry
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Only special combinations 1 of spherical harmonics may form a
basis for surfaces with octahedral symmetry:
Three Lowest Orders:
Rank ↔ Multipolarity λ
λ = 4 : α 40 ≡ o 4 ; α 4,±4 ≡ ± q 5
14 · o 4
λ = 6 : α 60 ≡ o 6 ; α 6,±4 ≡ − q 7
2 · o 6
λ = 8 : α 80 ≡ o 8 ; α 8,±4 ≡ q 28
198 · o 8; α 8,±8 ≡ q 65
198 · o 8
1 Collaboration with Andrzej G ÓŹDŹ and Daniel ROS̷LY, UMCS Lublin, Poland
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
A Basis for Octahedral Symmetry
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Only special combinations 1 of spherical harmonics may form a
basis for surfaces with octahedral symmetry:
Three Lowest Orders:
Rank ↔ Multipolarity λ
λ = 4 : α 40 ≡ o 4 ; α 4,±4 ≡ ± q 5
14 · o 4
λ = 6 : α 60 ≡ o 6 ; α 6,±4 ≡ − q 7
2 · o 6
λ = 8 : α 80 ≡ o 8 ; α 8,±4 ≡ q 28
198 · o 8; α 8,±8 ≡ q 65
198 · o 8
1 Collaboration with Andrzej G ÓŹDŹ and Daniel ROS̷LY, UMCS Lublin, Poland
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
A Basis for Octahedral Symmetry
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Only special combinations 1 of spherical harmonics may form a
basis for surfaces with octahedral symmetry:
Three Lowest Orders:
Rank ↔ Multipolarity λ
λ = 4 : α 40 ≡ o 4 ; α 4,±4 ≡ ± q 5
14 · o 4
λ = 6 : α 60 ≡ o 6 ; α 6,±4 ≡ − q 7
2 · o 6
λ = 8 : α 80 ≡ o 8 ; α 8,±4 ≡ q 28
198 · o 8; α 8,±8 ≡ q 65
198 · o 8
1 Collaboration with Andrzej G ÓŹDŹ and Daniel ROS̷LY, UMCS Lublin, Poland
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
A Basis for Octahedral Symmetry
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Only special combinations 1 of spherical harmonics may form a
basis for surfaces with octahedral symmetry:
Three Lowest Orders:
Rank ↔ Multipolarity λ
λ = 4 : α 40 ≡ o 4 ; α 4,±4 ≡ ± q 5
14 · o 4
λ = 6 : α 60 ≡ o 6 ; α 6,±4 ≡ − q 7
2 · o 6
λ = 8 : α 80 ≡ o 8 ; α 8,±4 ≡ q 28
198 · o 8; α 8,±8 ≡ q 65
198 · o 8
1 Collaboration with Andrzej G ÓŹDŹ and Daniel ROS̷LY, UMCS Lublin, Poland
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Nuclear Octahedral Shapes - 3D Examples
Illustrations below show the octahedral-symmetric surfaces at three
increasing values of rank λ = 4 deformations o 4 : 0.1, 0.2 and 0.3:
Figure: o 4 = 0.1 Figure: o 4 = 0.2 Figure: o 4 = 0.2
Recall: α 40 ≡ o 4 ; α 4,±4 ≡ ± q 5
14 · o 4
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Nuclear Octahedral Shapes - Neutron Spectra
Double group Oh
D has four 2-dimensional and two 4-dimensional
irreducible representations → six distinct families of levels
Neutron Energies [MeV]
-2
-4
-6
-8
-10
-12
160
70 Yb 90
{17}[6,1,3] 7/2
{09}[6,4,2] 5/2
{08}[5,0,5] 9/2
{18}[6,1,5] 11/2
{21}[5,0,5] 9/2
{10}[5,4,1] 3/2
{19}[6,3,3] 7/2
{20}[6,1,5] 11/2
{07}[5,4,1] 3/2
{16}[6,2,4] 9/2
{11}[5,0,1] 3/2
{16}[5,4,1] 1/2
{09}[5,0,3] 7/2
{10}[5,3,0] 1/2
{07}[5,0,5] 11/2
{13}[5,2,1] 1/2
{12}[5,0,5] 11/2
{13}[5,0,5] 11/2
{21}[5,1,4] 7/2
{10}[5,2,3] 5/2
{12}[4,0,0] 1/2
{22}[4,4,0] 1/2
{21}[4,0,2] 3/2
118
114 116
94 94
88
100
-.35 -.25 -.15 -.05 .05 .15 .25 .35
Octahedral Deformation
82
126
110
86
88
{08}[6,0,6] 11/2
{07}[6,4,0] 1/2
{06}[8,0,2] 5/2
{10}[6,1,3] 7/2
{11}[6,5,1] 3/2
{17}[6,0,6] 13/2
{07}[5,1,4] 7/2
{12}[5,3,2] 3/2
{11}[5,2,1] 3/2
{11}[5,3,0] 1/2
{07}[8,8,0] 1/2
{08}[4,0,2] 5/2
{09}[6,0,4] 9/2
{08}[5,3,2] 5/2
{21}[5,2,3] 7/2
{14}[5,3,2] 5/2
{16}[5,1,4] 9/2
{08}[5,0,5] 9/2
{21}[4,1,3] 5/2
{13}[4,1,1] 1/2
{23}[4,2,2] 3/2
{08}[5,0,5] 11/2
{08}[5,4,1] 1/2
Strasbourg, August 2002 Dirac-Woods-Saxon
α 40(min)=-.350, α40(max)=.350
α 44(min)=-.209, α44(max)=.209
Figure: Full lines correspond to 4-dimensional irreducible representations -
they are marked with double Nilsson labels. Observe huge gap at N=114.
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Nuclear Octahedral Shapes - Proton Spectra
Double group Oh
D has four 2-dimensional and two 4-dimensional
irreducible representations → six distinct families of levels
Proton Energies [MeV]
0
2
-2
-4
-6
-8
-10
-12
160
70 Yb 90
{10}[5,0,5] 11/2
{08}[5,0,3] 7/2
{10}[5,0,3] 7/2
{10}[5,4,1] 3/2
{13}[5,0,5] 11/2
{19}[4,4,0] 1/2
{18}[4,0,2] 3/2
{11}[4,0,0] 1/2
{15}[5,1,4] 7/2
{07}[5,2,3] 5/2
{11}[5,1,2] 5/2
{10}[5,3,2] 5/2
{23}[5,1,4] 9/2
{13}[4,0,4] 7/2
{08}[4,2,2] 3/2
{08}[4,3,1] 1/2
{17}[5,2,3] 7/2
{15}[4,3,1] 1/2
{12}[4,0,2] 5/2
{11}[4,1,3] 5/2
{09}[4,2,0] 1/2
{13}[4,0,4] 9/2
{09}[4,3,1] 3/2
72
58
70
88
52
64
-.35 -.25 -.15 -.05 .05 .15 .25 .35
Octahedral Deformation
82
94
56
88
52
{09}[4,0,2] 5/2
{16}[5,3,2] 5/2
{18}[5,1,4] 9/2
{08}[5,0,5] 9/2
{21}[4,1,3] 5/2
{17}[4,1,3] 7/2
{24}[4,2,2] 3/2
{09}[5,1,2] 5/2
{10}[5,0,5] 9/2
{07}[5,1,0] 1/2
{11}[5,4,1] 3/2
{21}[4,2,0] 1/2
{10}[5,0,5] 11/2
{10}[4,3,1] 1/2
{09}[4,2,2] 3/2
{08}[3,3,0] 1/2
{13}[3,0,1] 3/2
{08}[5,0,3] 7/2
{25}[4,2,2] 5/2
{16}[4,1,3] 7/2
{09}[4,3,1] 3/2
{24}[3,1,2] 3/2
{10}[4,3,1] 1/2
Strasbourg, August 2002 Dirac-Woods-Saxon
α 40(min)=-.350, α40(max)=.350
α 44(min)=-.209, α44(max)=.209
Figure: Full lines correspond to 4-dimensional irreducible representations
- they are marked with double Nilsson labels. Observe huge gap at Z=70.
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Nuclear High-Rank Symmetries and Challenges
There are several new-physics aspects related to the nuclear highrank
symmetries - tetrahedral and octahedral ones!
Properties High Symmetries ’Usual’ symmetries
or features Tetrahedral Octahedral Ellipsoid
No. Sym. Elemts. 48 96 4 + . . .
Parity NO YES YES
New Degeneracies 4, 2, 2 4, 2, 2
} {{ }
π = +
4, 2, 2
} {{ }
π = −
}{{}
2
π = +
2
}{{}
π = −
New Q. Numbers 3 3 + 3 2: π = ±1
We call these new quantum numbers τρ ι − τιµυκoσ (tri-timeric)
’possessing three values’
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Point Groups: Simple and Double
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Vector Fields Ψ vs Spinor Fields Φ
If the objects Ψ do not change under the rotation ˆR(2π)
ˆR(2π) Ψ = +Ψ → bosons ↔ simple groups: Example T d
If the objects Φ change the sign under the rotation ˆR(2π)
ˆR(2π) Φ = −Φ → fermions ↔ double groups: Example T D d
Remarks - Comments
Objects in classical mechanics, integer rank tensors like boson
wave-functions transform under simple point-groups: D 2h , T d
Spinors - the non-collective wave functions of fermions - transform
under double point-groups: D D 2h , T D d , OD h ...
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Point Groups: Simple and Double
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Vector Fields Ψ vs Spinor Fields Φ
If the objects Ψ do not change under the rotation ˆR(2π)
ˆR(2π) Ψ = +Ψ → bosons ↔ simple groups: Example T d
If the objects Φ change the sign under the rotation ˆR(2π)
ˆR(2π) Φ = −Φ → fermions ↔ double groups: Example T D d
Remarks - Comments
Objects in classical mechanics, integer rank tensors like boson
wave-functions transform under simple point-groups: D 2h , T d
Spinors - the non-collective wave functions of fermions - transform
under double point-groups: D D 2h , T D d , OD h ...
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Point Groups: Simple and Double
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Vector Fields Ψ vs Spinor Fields Φ
If the objects Ψ do not change under the rotation ˆR(2π)
ˆR(2π) Ψ = +Ψ → bosons ↔ simple groups: Example T d
If the objects Φ change the sign under the rotation ˆR(2π)
ˆR(2π) Φ = −Φ → fermions ↔ double groups: Example T D d
Remarks - Comments
Objects in classical mechanics, integer rank tensors like boson
wave-functions transform under simple point-groups: D 2h , T d
Spinors - the non-collective wave functions of fermions - transform
under double point-groups: D D 2h , T D d , OD h ...
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Point Groups: Simple and Double
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Vector Fields Ψ vs Spinor Fields Φ
If the objects Ψ do not change under the rotation ˆR(2π)
ˆR(2π) Ψ = +Ψ → bosons ↔ simple groups: Example T d
If the objects Φ change the sign under the rotation ˆR(2π)
ˆR(2π) Φ = −Φ → fermions ↔ double groups: Example T D d
Remarks - Comments
Objects in classical mechanics, integer rank tensors like boson
wave-functions transform under simple point-groups: D 2h , T d
Spinors - the non-collective wave functions of fermions - transform
under double point-groups: D D 2h , T D d , OD h ...
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Point Groups: Simple and Double
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Vector Fields Ψ vs Spinor Fields Φ
If the objects Ψ do not change under the rotation ˆR(2π)
ˆR(2π) Ψ = +Ψ → bosons ↔ simple groups: Example T d
If the objects Φ change the sign under the rotation ˆR(2π)
ˆR(2π) Φ = −Φ → fermions ↔ double groups: Example T D d
Remarks - Comments
Objects in classical mechanics, integer rank tensors like boson
wave-functions transform under simple point-groups: D 2h , T d
Spinors - the non-collective wave functions of fermions - transform
under double point-groups: D D 2h , T D d , OD h ...
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Coexistence of T d - and O h -Symmetries in Nuclei
Octahedral Def. (Rank 1)
0.30
0.20
0.10
0.00
-0.10
-0.20
Tetrahedral Symmetry / Instability
1
1
1
-0.30
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
158 Tetrahedral Deformation (Rank 1)
70
Yb 88
1
1
1
E [MeV]
6.00
5.75
5.50
5.25
5.00
4.75
4.50
4.25
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
2.00
1.75
1.50
1.25
1.00
0.75
0.50
0.25
UNIVERS_COMPACT (D=3, 23)
UNIVERS_COMPACT (D=3, 23)
Emin=-2.34, Eo=-1.62
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Coexistence of T d - and O h -Symmetries in Nuclei
Octahedral Deformation
0.30
0.20
0.10
0.00
-0.10
-0.20
Tetrahedral Symmetry / Instability
1
2
2
3
3
-0.30
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
160 Tetrahedral Deformation
70
Yb 90
3
3
2
2
1
E [MeV]
6.00
5.75
5.50
5.25
5.00
4.75
4.50
4.25
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
2.00
1.75
1.50
1.25
1.00
0.75
0.50
0.25
UNIVERS_COMPACT (D=3, 23)
UNIVERS_COMPACT (D=3, 23)
Emin=-2.18, Eo= 0.40
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Coexistence of T d - and O h -Symmetries in Nuclei
Octahedral Def. (Rank 1)
0.30
0.20
0.10
0.00
-0.10
-0.20
Tetrahedral Symmetry / Instability
1
2
2
3
3
-0.30
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
162 Tetrahedral Deformation (Rank 1)
70
Yb 92
3
3
2
2
1
E [MeV]
6.00
5.75
5.50
5.25
5.00
4.75
4.50
4.25
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
2.00
1.75
1.50
1.25
1.00
0.75
0.50
0.25
UNIVERS_COMPACT (D=3, 23)
UNIVERS_COMPACT (D=3, 23)
Emin=-0.52, Eo= 2.37
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Coexistence of T d - and O h -Symmetries in Nuclei
Octahedral Def. (Rank 1)
0.30
0.20
0.10
0.00
-0.10
-0.20
1
Tetrahedral Symmetry / Instability
1
2
3
2
-0.30
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
164 Tetrahedral Deformation (Rank 1)
70
Yb 94
3
2
3
2
1
E [MeV]
6.00
5.75
5.50
5.25
5.00
4.75
4.50
4.25
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
2.00
1.75
1.50
1.25
1.00
0.75
0.50
0.25
UNIVERS_COMPACT (D=3, 23)
UNIVERS_COMPACT (D=3, 23)
Emin= 0.98, Eo= 4.20
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Coexistence of T d - and O h -Symmetries in Nuclei
Octahedral Def. (Rank 1)
0.30
0.20
0.10
0.00
-0.10
-0.20
1
Tetrahedral Symmetry / Instability
1
2
2
2
3
-0.30
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
166 Tetrahedral Deformation (Rank 1)
70
Yb 96
2
3
2
2
3
2
1
E [MeV]
6.00
5.75
5.50
5.25
5.00
4.75
4.50
4.25
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
2.00
1.75
1.50
1.25
1.00
0.75
0.50
0.25
UNIVERS_COMPACT (D=3, 23)
UNIVERS_COMPACT (D=3, 23)
Emin= 2.26, Eo= 5.81
Jerzy DUDEK
SYMMETRIES in PHYSICS

3
Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Coexistence of T d - and O h -Symmetries in Nuclei
Octahedral Def. (Rank 1)
0.30
0.20
0.10
0.00
-0.10
-0.20
1
Tetrahedral Symmetry / Instability
1
2
2
2
-0.30
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
168 Tetrahedral Deformation (Rank 1)
70
Yb 98
2
3
2
3
2
1
E [MeV]
6.00
5.75
5.50
5.25
5.00
4.75
4.50
4.25
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
2.00
1.75
1.50
1.25
1.00
0.75
0.50
0.25
UNIVERS_COMPACT (D=3, 23)
UNIVERS_COMPACT (D=3, 23)
Emin= 3.39, Eo= 7.18
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Coexistence of T d - and O h -Symmetries in Nuclei
Octahedral Def. (Rank 1)
0.30
0.20
0.10
0.00
-0.10
-0.20
1
Tetrahedral Symmetry / Instability
2
1
2
3
2
-0.30
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
170 Tetrahedral Deformation (Rank 1)
70
Yb 100
3
2
2
3
2
1
1
E [MeV]
6.00
5.75
5.50
5.25
5.00
4.75
4.50
4.25
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
2.00
1.75
1.50
1.25
1.00
0.75
0.50
0.25
UNIVERS_COMPACT (D=3, 23)
UNIVERS_COMPACT (D=3, 23)
Emin= 4.31, Eo= 8.25
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Coexistence of T d - and O h -Symmetries in Nuclei
Octahedral Def. (Rank 1)
0.30
0.20
0.10
0.00
-0.10
-0.20
1
Tetrahedral Symmetry / Instability
1
1
2
3
2
-0.30
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
172 Tetrahedral Deformation (Rank 1)
70
Yb 102
3
2
2
3
2
1
1
E [MeV]
6.00
5.75
5.50
5.25
5.00
4.75
4.50
4.25
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
2.00
1.75
1.50
1.25
1.00
0.75
0.50
0.25
UNIVERS_COMPACT (D=3, 23)
UNIVERS_COMPACT (D=3, 23)
Emin= 4.97, Eo= 9.00
Jerzy DUDEK
SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Coexistence of T d - and O h -Symmetries in Nuclei
Octahedral Def. (Rank 1)
0.30
0.20
0.10
0.00
-0.10
-0.20
1
Tetrahedral Symmetry / Instability
2
1
2
3
2
-0.30
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
174 Tetrahedral Deformation (Rank 1)
70
Yb 104
1
3
1
2
2
3
2
1
1
E [MeV]
6.00
5.75
5.50
5.25
5.00
4.75
4.50
4.25
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
2.00
1.75
1.50
1.25
1.00
0.75
0.50
0.25
UNIVERS_COMPACT (D=3, 23)
UNIVERS_COMPACT (D=3, 23)
Emin= 5.40, Eo= 9.45
Jerzy DUDEK
SYMMETRIES in PHYSICS

Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Nuclear Tetrahedral-Symmetry Effects
Symmetry and Spontaneous Symmetry Breaking
A32
0.30
0.20
0.10
0.00
-0.10
-0.20
1
Tetrahedral Symmetry / Instability
1
1
1
1
-0.30
0.0 0.1 0.2 0.3 0.4
156
A20
70
Yb 86
2
3
3
4
4
5
5
E [MeV]
6.00
5.75
5.50
5.25
5.00
4.75
4.50
4.25
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
2.00
1.75
1.50
1.25
1.00
0.75
0.50
0.25
UNIVERS_COMPACT (D=3, 23)
UNIVERS_COMPACT (D=3, 23)
Emin=-4.57, Eo=-3.94
Jerzy DUDEK
SYMMETRIES in PHYSICS

Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Nuclear Tetrahedral-Symmetry Effects
Symmetry and Spontaneous Symmetry Breaking
A32
0.30
0.20
0.10
0.00
-0.10
-0.20
1
1
Tetrahedral Symmetry / Instability
1
1
1
1
-0.30
0.0 0.1 0.2 0.3 0.4
158
A20
70
Yb 88
1
1
2
2
3
3
4
4
E [MeV]
6.00
5.75
5.50
5.25
5.00
4.75
4.50
4.25
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
2.00
1.75
1.50
1.25
1.00
0.75
0.50
0.25
UNIVERS_COMPACT (D=3, 23)
UNIVERS_COMPACT (D=3, 23)
Emin=-3.31, Eo=-1.71
Jerzy DUDEK
SYMMETRIES in PHYSICS

2
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Nuclear Tetrahedral-Symmetry Effects
Symmetry and Spontaneous Symmetry Breaking
Tetrahedral Deformation
0.30
0.20
0.10
0.00
-0.10
-0.20
1
2
2
1
Tetrahedral Symmetry / Instability
2
2
1
1
-0.30
0.0 0.1 0.2 0.3 0.4
160 Quadrupole Deformation
70
Yb 90
1
1
1
2
2
3
3
E [MeV]
6.00
5.75
5.50
5.25
5.00
4.75
4.50
4.25
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
2.00
1.75
1.50
1.25
1.00
0.75
0.50
0.25
UNIVERS_COMPACT (D=3, 23)
UNIVERS_COMPACT (D=3, 23)
Emin=-2.60, Eo= 0.34
Jerzy DUDEK
SYMMETRIES in PHYSICS

2
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Nuclear Tetrahedral-Symmetry Effects
Symmetry and Spontaneous Symmetry Breaking
A32
0.30
0.20
0.10
0.00
-0.10
-0.20
2
3
4
2
Tetrahedral Symmetry / Instability
4
3
3
3
3
2
1
-0.30
0.0 0.1 0.2 0.3 0.4
162
A20
70
Yb 92
1
1
1
1
2
2
3
E [MeV]
6.00
5.75
5.50
5.25
5.00
4.75
4.50
4.25
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
2.00
1.75
1.50
1.25
1.00
0.75
0.50
0.25
UNIVERS_COMPACT (D=3, 23)
UNIVERS_COMPACT (D=3, 23)
Emin=-2.55, Eo= 2.32
Jerzy DUDEK
SYMMETRIES in PHYSICS

Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Nuclear Tetrahedral-Symmetry Effects
Symmetry and Spontaneous Symmetry Breaking
A32
0.30
0.20
0.10
0.00
-0.10
-0.20
4
5
6
4
Tetrahedral Symmetry / Instability
6
5
5
5
5
4
4
3
-0.30
0.0 0.1 0.2 0.3 0.4
164
A20
70
Yb 94
3
2
2
1
1
1
1
2
3
E [MeV]
6.00
5.75
5.50
5.25
5.00
4.75
4.50
4.25
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
2.00
1.75
1.50
1.25
1.00
0.75
0.50
0.25
UNIVERS_COMPACT (D=3, 23)
UNIVERS_COMPACT (D=3, 23)
Emin=-2.94, Eo= 4.19
Jerzy DUDEK
SYMMETRIES in PHYSICS