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1. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
B EAUTY IN THE B REAKDOWN : S YMMETRY IN
P HYSICS
Zach McElrath
Covenant College
April 29, 2010
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
2. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
O UTLINE OF TALK
1 Introduction: Examples of Symmetries in Nature
2 Groups
3 Brief History of Use of Symmetry in Physics
4 Symmetry and Quantum Field Theory
5 Symmetry Breaking
6 Conclusion: Why Symmetry?
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
3. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
O UTLINE OF TALK
1 Introduction: Examples of Symmetries in Nature
2 Groups
3 Brief History of Use of Symmetry in Physics
4 Symmetry and Quantum Field Theory
5 Symmetry Breaking
6 Conclusion: Why Symmetry?
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
4. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
S YMMETRY ON THE SURFACE OF THE D EAD S EA
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
5. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
S YMMETRY ON THE SURFACE OF THE D EAD S EA
1 Translation
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
6. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
S YMMETRY ON THE SURFACE OF THE D EAD S EA
1 Translation
2 Rotation
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
7. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
M AKING R IPPLES – BREAKING SOME SYMMETRIES
AND CREATING OTHERS
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
8. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
M AKING R IPPLES – BREAKING SOME SYMMETRIES
AND CREATING OTHERS
1 Translational symmetry broken
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
9. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
M AKING R IPPLES – BREAKING SOME SYMMETRIES
AND CREATING OTHERS
1 Translational symmetry broken
2 Rotational symmetry about the center of the ripple
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
10. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
M AKING R IPPLES – BREAKING SOME SYMMETRIES
AND CREATING OTHERS
1 Translational symmetry broken
2 Rotational symmetry about the center of the ripple
3 How about 2 ripples?
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
11. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
M AKING R IPPLES – BREAKING SOME SYMMETRIES
AND CREATING OTHERS
1 Translational symmetry broken
2 Rotational symmetry about the center of the ripple
3 How about 2 ripples?
4 How about 3 ripples?
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
12. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
T HE S YMMETRIES OF E QUILATERAL T RIANGLES
Six Symmetry “Operations”
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
13. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
T HE S YMMETRIES OF E QUILATERAL T RIANGLES
Six Symmetry “Operations”
“Identity” (1)
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
14. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
T HE S YMMETRIES OF E QUILATERAL T RIANGLES
Six Symmetry “Operations”
“Identity” (1)
Rotation by 120◦ (R120 )
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
15. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
T HE S YMMETRIES OF E QUILATERAL T RIANGLES
Six Symmetry “Operations”
“Identity” (1)
Rotation by 120◦ (R120 )
Rotation by 240◦ (R240 )
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
16. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
T HE S YMMETRIES OF E QUILATERAL T RIANGLES
Six Symmetry “Operations”
“Identity” (1)
Rotation by 120◦ (R120 )
Rotation by 240◦ (R240 )
Reflection axis I (RI )
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
17. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
T HE S YMMETRIES OF E QUILATERAL T RIANGLES
Six Symmetry “Operations”
“Identity” (1)
Rotation by 120◦ (R120 )
Rotation by 240◦ (R240 )
Reflection axis I (RI )
Reflection axis II (RII )
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
18. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
T HE S YMMETRIES OF E QUILATERAL T RIANGLES
Six Symmetry “Operations”
“Identity” (1)
Rotation by 120◦ (R120 )
Rotation by 240◦ (R240 )
Reflection axis I (RI )
Reflection axis II (RII )
Reflection axis III (RIII )
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
19. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
R120 × RII = RI
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
20. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
R120 × RII = RI
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
21. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
RII × R120 = RIII
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
22. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
RII × R120 = RIII
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
23. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
T HE OPERATIONS DO NOT COMMUTE !
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
24. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
T HE OPERATIONS DO NOT COMMUTE !
R120 × RII = RI
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
25. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
T HE OPERATIONS DO NOT COMMUTE !
R120 × RII = RI
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
26. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
T HE OPERATIONS DO NOT COMMUTE !
R120 × RII = RI RII × R120 = RIII
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
27. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
M ULTIPLICATION TABLE FOR S3
1 R240 RI RIII
1 1 R120 R240 RI RII RIII
R120 R240 1 RIII RII
R240 R240 1 R120 RII RIII RI
RI RI RII RIII 1 R120 R240
RII RI R240 1 R120
RIII RIII RI RII R120 R240 1
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
28. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
M ULTIPLICATION TABLE FOR S3
1 R240 RI RIII
1 1 R120 R240 RI RII RIII
R120 R120 R240 1 RIII RII
R240 R240 1 R120 RII RIII RI
RI RI RII RIII 1 R120 R240
RII RI R240 1 R120
RIII RIII RI RII R120 R240 1
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
29. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
M ULTIPLICATION TABLE FOR S3
1 R240 RI RII RIII
1 1 R120 R240 RI RII RIII
R120 R120 R240 1 RIII RII
R240 R240 1 R120 RII RIII RI
RI RI RII RIII 1 R120 R240
RII RI R240 1 R120
RIII RIII RI RII R120 R240 1
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
30. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
M ULTIPLICATION TABLE FOR S3
1 R240 RI RII RIII
1 1 R120 R240 RI RII RIII
R120 R120 R240 1 RIII RI RII
R240 R240 1 R120 RII RIII RI
RI RI RII RIII 1 R120 R240
RII RI R240 1 R120
RIII RIII RI RII R120 R240 1
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
31. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
M ULTIPLICATION TABLE FOR S3
1 R240 RI RII RIII
1 1 R120 R240 RI RII RIII
R120 R120 R240 1 RIII RI RII
R240 R240 1 R120 RII RIII RI
RI RI RII RIII 1 R120 R240
RII RII RI R240 1 R120
RIII RIII RI RII R120 R240 1
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
32. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
M ULTIPLICATION TABLE FOR S3
1 R120 R240 RI RII RIII
1 1 R120 R240 RI RII RIII
R120 R120 R240 1 RIII RI RII
R240 R240 1 R120 RII RIII RI
RI RI RII RIII 1 R120 R240
RII RII RI R240 1 R120
RIII RIII RI RII R120 R240 1
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
33. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
M ULTIPLICATION TABLE FOR S3
1 R120 R240 RI RII RIII
1 1 R120 R240 RI RII RIII
R120 R120 R240 1 RIII RI RII
R240 R240 1 R120 RII RIII RI
RI RI RII RIII 1 R120 R240
RII RII RIII RI R240 1 R120
RIII RIII RI RII R120 R240 1
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
34. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
S YMMETRY OF THE C IRCLE
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
35. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
S YMMETRY OF THE C IRCLE
Rotate by any arbitrary angle θ ∈ R, and the circle still looks the
same
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
36. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
S YMMETRY O PERATIONS ON THE C IRCLE
Define this rotational symmetry operation as R(θ)
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
37. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
S YMMETRY O PERATIONS ON THE C IRCLE
Define this rotational symmetry operation as R(θ)
The set of possible symmetry operations on a circle forms
a continuous symmetry group
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
38. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
S YMMETRY O PERATIONS ON THE C IRCLE
Define this rotational symmetry operation as R(θ)
The set of possible symmetry operations on a circle forms
a continuous symmetry group
Continuous symmetry groups: infinite number of elements,
infinitesimal variation in parameters (i.e. θ).
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
39. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
S YMMETRY O PERATIONS ON THE C IRCLE
Define this rotational symmetry operation as R(θ)
The set of possible symmetry operations on a circle forms
a continuous symmetry group
Continuous symmetry groups: infinite number of elements,
infinitesimal variation in parameters (i.e. θ).
Impossible to construct a multiplication table as we did for
the discrete group S3 , which involves discrete steps and
has no infinitesimal operations.
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
40. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
S YMMETRY O PERATIONS ON THE C IRCLE
Define this rotational symmetry operation as R(θ)
The set of possible symmetry operations on a circle forms
a continuous symmetry group
Continuous symmetry groups: infinite number of elements,
infinitesimal variation in parameters (i.e. θ).
Impossible to construct a multiplication table as we did for
the discrete group S3 , which involves discrete steps and
has no infinitesimal operations.
Need a different way to describe all these operations
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
41. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
O UTLINE OF TALK
1 Introduction: Examples of Symmetries in Nature
2 Groups
3 Brief History of Use of Symmetry in Physics
4 Symmetry and Quantum Field Theory
5 Symmetry Breaking
6 Conclusion: Why Symmetry?
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
42. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
G ROUPS
A group G is essentially a set with a composition rule a · b, i.e.
if a, b ∈ G, then a · b ∈ G. There are three other defining
characteristics of groups:
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
43. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
G ROUPS
A group G is essentially a set with a composition rule a · b, i.e.
if a, b ∈ G, then a · b ∈ G. There are three other defining
characteristics of groups:
Associativity: a(bc) = (ab)c
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
44. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
G ROUPS
A group G is essentially a set with a composition rule a · b, i.e.
if a, b ∈ G, then a · b ∈ G. There are three other defining
characteristics of groups:
Associativity: a(bc) = (ab)c
The group has an identity element e s.t. ae = ea = a
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
45. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
G ROUPS
A group G is essentially a set with a composition rule a · b, i.e.
if a, b ∈ G, then a · b ∈ G. There are three other defining
characteristics of groups:
Associativity: a(bc) = (ab)c
The group has an identity element e s.t. ae = ea = a
The group has an inverse : ∀a ∈ G, ∃ a−1 ∈ G s.t.
aa−1 = a−1 a = e
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
46. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
D ISTINGUISHING F EATURE OF G ROUPS :
C OMMUTATIVITY
Key Distinguishing Feature: are the elements of the group
(in our case, the symmetry operations) commutative?
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
47. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
D ISTINGUISHING F EATURE OF G ROUPS :
C OMMUTATIVITY
Key Distinguishing Feature: are the elements of the group
(in our case, the symmetry operations) commutative?
i.e. is a · b = b · a?
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
48. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
D ISTINGUISHING F EATURE OF G ROUPS :
C OMMUTATIVITY
Key Distinguishing Feature: are the elements of the group
(in our case, the symmetry operations) commutative?
i.e. is a · b = b · a?
If the elements of the group are commutative, the group is
called abelian.
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
49. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
E XAMPLE OF AN A BELIAN G ROUP : T HE I NTEGERS
The set of integers form a group under the operation of
addition
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
50. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
E XAMPLE OF AN A BELIAN G ROUP : T HE I NTEGERS
The set of integers form a group under the operation of
addition
Take + as our composition rule
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
51. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
E XAMPLE OF AN A BELIAN G ROUP : T HE I NTEGERS
The set of integers form a group under the operation of
addition
Take + as our composition rule
Closure: For any elements a, b of the set of integers, a + b
will yield another integer
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
52. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
E XAMPLE OF AN A BELIAN G ROUP : T HE I NTEGERS
The set of integers form a group under the operation of
addition
Take + as our composition rule
Closure: For any elements a, b of the set of integers, a + b
will yield another integer (i.e. 2 + 3 = 5)
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
53. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
E XAMPLE OF AN A BELIAN G ROUP : T HE I NTEGERS
The set of integers form a group under the operation of
addition
Take + as our composition rule
Closure: For any elements a, b of the set of integers, a + b
will yield another integer (i.e. 2 + 3 = 5)
Identity element: 0 (because 19 + 0 = 19)
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
54. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
E XAMPLE OF AN A BELIAN G ROUP : T HE I NTEGERS
The set of integers form a group under the operation of
addition
Take + as our composition rule
Closure: For any elements a, b of the set of integers, a + b
will yield another integer (i.e. 2 + 3 = 5)
Identity element: 0 (because 19 + 0 = 19)
Inverse element: positive/negative (i.e. 23 + (−23) = 0)
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
55. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
E XAMPLE OF AN A BELIAN G ROUP : T HE I NTEGERS
The set of integers form a group under the operation of
addition
Take + as our composition rule
Closure: For any elements a, b of the set of integers, a + b
will yield another integer (i.e. 2 + 3 = 5)
Identity element: 0 (because 19 + 0 = 19)
Inverse element: positive/negative (i.e. 23 + (−23) = 0)
Associativity is satisfied, i.e. 3 + (5 + 7) = 15 = (3 + 5) + 7.
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
56. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
E XAMPLE OF AN A BELIAN G ROUP : T HE I NTEGERS
The set of integers form a group under the operation of
addition
Take + as our composition rule
Closure: For any elements a, b of the set of integers, a + b
will yield another integer (i.e. 2 + 3 = 5)
Identity element: 0 (because 19 + 0 = 19)
Inverse element: positive/negative (i.e. 23 + (−23) = 0)
Associativity is satisfied, i.e. 3 + (5 + 7) = 15 = (3 + 5) + 7.
Addition is commutative (2 + 3 = 3 + 2 = 5), so the set of
integers forms an abelian group under addition.
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
57. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
D ISCRETE VS . C ONTINUOUS G ROUPS
Discrete: Equilateral Triangle (S3 )–finite number of
distinguishable symmetry operations–6.
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
58. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
D ISCRETE VS . C ONTINUOUS G ROUPS
Discrete: Equilateral Triangle (S3 )–finite number of
distinguishable symmetry operations–6.
Continous: Circle (Rotations in the 2D plane)–infinite
number of possible operations
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
59. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
D ESCRIBING C ONTINUOUS G ROUPS : U(1)
U(1)
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
60. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
D ESCRIBING C ONTINUOUS G ROUPS : U(1)
U(1)–(extremely important in physics)
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
61. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
D ESCRIBING C ONTINUOUS G ROUPS : U(1)
U(1)–(extremely important in physics)
The group containing all complex numbers with length 1
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
62. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
D ESCRIBING C ONTINUOUS G ROUPS : U(1)
U(1)–(extremely important in physics)
The group containing all complex numbers with length 1
A subgroup of U(N), the group of N × N unitary matrices
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
63. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
D ESCRIBING C ONTINUOUS G ROUPS : U(1)
U(1)–(extremely important in physics)
The group containing all complex numbers with length 1
A subgroup of U(N), the group of N × N unitary matrices
Unitary Matrices: satisfy U † U = UU † = IN , where IN is the
N × N identity matrix
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
64. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
R EPRESENTATIONS
A representation of a group is a mapping which takes the
elements a, b ∈ G into linear operators F that preserve the
composition rule of the group
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
65. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
R EPRESENTATIONS
A representation of a group is a mapping which takes the
elements a, b ∈ G into linear operators F that preserve the
composition rule of the group
That is, F (a)F (b) = F (ab)
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
66. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
R EPRESENTATIONS
A representation of a group is a mapping which takes the
elements a, b ∈ G into linear operators F that preserve the
composition rule of the group
That is, F (a)F (b) = F (ab)
So, what’s the most basic representation of U(1)?
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
67. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
R EPRESENTATIONS
A representation of a group is a mapping which takes the
elements a, b ∈ G into linear operators F that preserve the
composition rule of the group
That is, F (a)F (b) = F (ab)
So, what’s the most basic representation of U(1)? eiθ
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
68. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
R(θ): A REPRESENTATION OF U(1)
cos(θ) sin(θ)
R(θ) =
− sin(θ) cos(θ)
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
69. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
R(θ): A REPRESENTATION OF U(1)
cos(θ) sin(θ)
R(θ) =
− sin(θ) cos(θ)
We can show that R(θ) is a group
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
70. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
R(θ): A REPRESENTATION OF U(1)
cos(θ) sin(θ)
R(θ) =
− sin(θ) cos(θ)
We can show that R(θ) is a group
i.e. Composition rule satisfied: R(θ1 )R(θ2 ) = R(θ1 + θ2 ).
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71. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
R(θ): A REPRESENTATION OF U(1)
cos(θ) sin(θ)
R(θ) =
− sin(θ) cos(θ)
We can show that R(θ) is a group
i.e. Composition rule satisfied: R(θ1 )R(θ2 ) = R(θ1 + θ2 ).
For R(θ) to be a representation of U(1), it too must satisfy
the properties of U(1)
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
72. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
R(θ): A REPRESENTATION OF U(1)
cos(θ) sin(θ)
R(θ) =
− sin(θ) cos(θ)
We can show that R(θ) is a group
i.e. Composition rule satisfied: R(θ1 )R(θ2 ) = R(θ1 + θ2 ).
For R(θ) to be a representation of U(1), it too must satisfy
the properties of U(1)
Unitarity: R(θ)† R(θ) = R(θ)R(−θ)
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
73. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
R(θ): A REPRESENTATION OF U(1)
cos(θ) sin(θ)
R(θ) =
− sin(θ) cos(θ)
We can show that R(θ) is a group
i.e. Composition rule satisfied: R(θ1 )R(θ2 ) = R(θ1 + θ2 ).
For R(θ) to be a representation of U(1), it too must satisfy
the properties of U(1)
Unitarity: R(θ)† R(θ) = R(θ)R(−θ) =
cos(θ) sin(θ) cos(θ) − sin(θ)
− sin(θ) cos(θ) sin(θ) cos(θ)
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
74. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
R(θ): A REPRESENTATION OF U(1)
cos(θ) sin(θ)
R(θ) =
− sin(θ) cos(θ)
We can show that R(θ) is a group
i.e. Composition rule satisfied: R(θ1 )R(θ2 ) = R(θ1 + θ2 ).
For R(θ) to be a representation of U(1), it too must satisfy
the properties of U(1)
Unitarity: R(θ)† R(θ) = R(θ)R(−θ) =
cos(θ) sin(θ) cos(θ) − sin(θ)
=
− sin(θ) cos(θ) sin(θ) cos(θ)
cos2 θ + sin2 θ sin θ cos θ − sin θ cos θ
sin θ cos θ − sin θ cos θ cos2 θ + sin2 θ
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
75. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
R(θ): A REPRESENTATION OF U(1)
cos(θ) sin(θ)
R(θ) =
− sin(θ) cos(θ)
We can show that R(θ) is a group
i.e. Composition rule satisfied: R(θ1 )R(θ2 ) = R(θ1 + θ2 ).
For R(θ) to be a representation of U(1), it too must satisfy
the properties of U(1)
Unitarity: R(θ)† R(θ) = R(θ)R(−θ) =
cos(θ) sin(θ) cos(θ) − sin(θ)
=
− sin(θ) cos(θ) sin(θ) cos(θ)
cos2 θ + sin2 θ sin θ cos θ − sin θ cos θ
=
sin θ cos θ − sin θ cos θ cos2 θ + sin2 θ
1 0
0 1
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76. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
L IE G ROUPS
Most of the continuous groups that turn up in studies of
symmetries in physics, including U(1), are Lie groups.
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77. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
L IE G ROUPS
Most of the continuous groups that turn up in studies of
symmetries in physics, including U(1), are Lie groups.
(1) Elements of the group depend on a finite set of
continuous parameters θ1 , θ2 , · · · , θn
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
78. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
L IE G ROUPS
Most of the continuous groups that turn up in studies of
symmetries in physics, including U(1), are Lie groups.
(1) Elements of the group depend on a finite set of
continuous parameters θ1 , θ2 , · · · , θn
(2) Derivatives of the group elements with respect to all of
the group parameters exist.
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79. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
L IE G ROUPS
Most of the continuous groups that turn up in studies of
symmetries in physics, including U(1), are Lie groups.
(1) Elements of the group depend on a finite set of
continuous parameters θ1 , θ2 , · · · , θn
(2) Derivatives of the group elements with respect to all of
the group parameters exist.
Finding the Generators of a group representation–Take
derivatives of the representation of the Lie group with
respect to each of its parameters, and evaluate the
derivative at zero
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
80. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
L IE G ROUPS
Most of the continuous groups that turn up in studies of
symmetries in physics, including U(1), are Lie groups.
(1) Elements of the group depend on a finite set of
continuous parameters θ1 , θ2 , · · · , θn
(2) Derivatives of the group elements with respect to all of
the group parameters exist.
Finding the Generators of a group representation–Take
derivatives of the representation of the Lie group with
respect to each of its parameters, and evaluate the
derivative at zero
If a group representation has n parameters, then there are
n generators Xi of that representation, given by
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
81. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
L IE G ROUPS
Most of the continuous groups that turn up in studies of
symmetries in physics, including U(1), are Lie groups.
(1) Elements of the group depend on a finite set of
continuous parameters θ1 , θ2 , · · · , θn
(2) Derivatives of the group elements with respect to all of
the group parameters exist.
Finding the Generators of a group representation–Take
derivatives of the representation of the Lie group with
respect to each of its parameters, and evaluate the
derivative at zero
If a group representation has n parameters, then there are
n generators Xi of that representation, given by
∂g
Xi =
∂θ θi =0
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82. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
U NITARY G ENERATORS
But we need a unitary group representation for our
purposes, because U(1) is unitary.
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83. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
U NITARY G ENERATORS
But we need a unitary group representation for our
purposes, because U(1) is unitary.
To do this, we must have Hermitian generators
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
84. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
U NITARY G ENERATORS
But we need a unitary group representation for our
purposes, because U(1) is unitary.
To do this, we must have Hermitian generators
How do you make a quantity involving a derivative
Hermitian?
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
85. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
U NITARY G ENERATORS
But we need a unitary group representation for our
purposes, because U(1) is unitary.
To do this, we must have Hermitian generators
How do you make a quantity involving a derivative
Hermitian? Multiply the generators by −i:
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
86. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
U NITARY G ENERATORS
But we need a unitary group representation for our
purposes, because U(1) is unitary.
To do this, we must have Hermitian generators
How do you make a quantity involving a derivative
Hermitian? Multiply the generators by −i:
∂g
Xi = −i
∂θ θi =0
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
87. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
T HE G ENERATORS OF THE 2D ROTATION G ROUP
Our 2-D rotation group representation of U(1) has only one
parameter, θ, so it has only one generator:
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88. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
T HE G ENERATORS OF THE 2D ROTATION G ROUP
Our 2-D rotation group representation of U(1) has only one
parameter, θ, so it has only one generator:
∂ cos(θ) sin(θ)
X (θ) = −i
∂θ − sin(θ) cos(θ) θ=0
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89. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
T HE G ENERATORS OF THE 2D ROTATION G ROUP
Our 2-D rotation group representation of U(1) has only one
parameter, θ, so it has only one generator:
∂ cos(θ) sin(θ) 0 −i
X (θ) = −i =
∂θ − sin(θ) cos(θ) θ=0
i 0
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
90. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
T HE G ENERATORS OF THE 2D ROTATION G ROUP
Our 2-D rotation group representation of U(1) has only one
parameter, θ, so it has only one generator:
∂ cos(θ) sin(θ) 0 −i
X (θ) = −i =
∂θ − sin(θ) cos(θ) θ=0
i 0
Lookie, lookie, its one of the Pauli Matrices!
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91. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
O UTLINE OF TALK
1 Introduction: Examples of Symmetries in Nature
2 Groups
3 Brief History of Use of Symmetry in Physics
4 Symmetry and Quantum Field Theory
5 Symmetry Breaking
6 Conclusion: Why Symmetry?
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92. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
H ISTORY OF S YMMETRY IN P HYSICS
Ancient Greeks: Regular polyhedra celebrated for their
defining proportionality relationships, regular solids
because they were recognized to be symmetric in the
sense of their invariance under geometric operations
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
93. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
H ISTORY OF S YMMETRY IN P HYSICS
Ancient Greeks: Regular polyhedra celebrated for their
defining proportionality relationships, regular solids
because they were recognized to be symmetric in the
sense of their invariance under geometric operations
Aristotle: Heavenly bodies perfect spheres, orbits circular
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
94. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
H ISTORY OF S YMMETRY IN P HYSICS
Ancient Greeks: Regular polyhedra celebrated for their
defining proportionality relationships, regular solids
because they were recognized to be symmetric in the
sense of their invariance under geometric operations
Aristotle: Heavenly bodies perfect spheres, orbits circular
Ptolemy: Theory of epicycles, dominant for 1500 years,
grounded in supposition that the circle must characterize
celestial orbits
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
95. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
H ISTORY OF S YMMETRY IN P HYSICS
Ancient Greeks: Regular polyhedra celebrated for their
defining proportionality relationships, regular solids
because they were recognized to be symmetric in the
sense of their invariance under geometric operations
Aristotle: Heavenly bodies perfect spheres, orbits circular
Ptolemy: Theory of epicycles, dominant for 1500 years,
grounded in supposition that the circle must characterize
celestial orbits
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
96. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
H ISTORY OF S YMMETRY IN P HYSICS
Copernicus: Heliocentric model shattered Ptolemaic
paradigm
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
97. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
H ISTORY OF S YMMETRY IN P HYSICS
Copernicus: Heliocentric model shattered Ptolemaic
paradigm
But it still preserved the cherished symmetry of the
circular celestial orbits
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
98. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
H ISTORY OF S YMMETRY IN P HYSICS
Copernicus: Heliocentric model shattered Ptolemaic
paradigm
But it still preserved the cherished symmetry of the
circular celestial orbits
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
99. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
H ISTORY OF S YMMETRY IN P HYSICS
Kepler: Believed that if he could find out where the
predictions of Copernicus’ theory diverged from Tycho
Brahe’s excellent data, he would “discover a magnificent
new symmetry.”
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
100. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
H ISTORY OF S YMMETRY IN P HYSICS
Kepler: Believed that if he could find out where the
predictions of Copernicus’ theory diverged from Tycho
Brahe’s excellent data, he would “discover a magnificent
new symmetry.”
By abandoning the cherished theory of circular orbits,
Kepler traded an approximate symmetry for a deeper
symmetry associated with the conservation of angular
momentum
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
101. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
H ISTORY OF S YMMETRY IN P HYSICS
Kepler: Believed that if he could find out where the
predictions of Copernicus’ theory diverged from Tycho
Brahe’s excellent data, he would “discover a magnificent
new symmetry.”
By abandoning the cherished theory of circular orbits,
Kepler traded an approximate symmetry for a deeper
symmetry associated with the conservation of angular
momentum
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
102. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
H ISTORY OF S YMMETRY IN P HYSICS
´
Studies of Crystal Structure by Rene Hauy in 1801: the
¨
symmetry of a geometric figure redefined as its
“invariance...when equal component parts are exchanged
according to one of the specified operations.”
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
103. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
H ISTORY OF S YMMETRY IN P HYSICS
´
Studies of Crystal Structure by Rene Hauy in 1801: the
¨
symmetry of a geometric figure redefined as its
“invariance...when equal component parts are exchanged
according to one of the specified operations.”
1st turning point in history of physics–19th century: The
idea of symmetry as invariance generalized to algebraic
structure of groups
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
104. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
H ISTORY OF S YMMETRY IN P HYSICS
´
Studies of Crystal Structure by Rene Hauy in 1801: the
¨
symmetry of a geometric figure redefined as its
“invariance...when equal component parts are exchanged
according to one of the specified operations.”
1st turning point in history of physics–19th century: The
idea of symmetry as invariance generalized to algebraic
structure of groups
The set of symmetry operations applicable to a given
geometric figure satisfies the conditions for a group
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
105. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
H ISTORY OF S YMMETRY IN P HYSICS
´
Studies of Crystal Structure by Rene Hauy in 1801: the
¨
symmetry of a geometric figure redefined as its
“invariance...when equal component parts are exchanged
according to one of the specified operations.”
1st turning point in history of physics–19th century: The
idea of symmetry as invariance generalized to algebraic
structure of groups
The set of symmetry operations applicable to a given
geometric figure satisfies the conditions for a group
In both its ancient and modern formulations, then,
symmetry is associated with a unity of parts which are
equal “with respect to the whole in the sense of their
interchangeability.”
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
106. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
H ISTORY OF S YMMETRY IN P HYSICS
Mid-19th century–Jacobi develops a method of solving
dynamical equations formulated using Hamilton’s
canonical variables by applying transformations of these
variables which leave the equations invariant
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
107. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
H ISTORY OF S YMMETRY IN P HYSICS
Mid-19th century–Jacobi develops a method of solving
dynamical equations formulated using Hamilton’s
canonical variables by applying transformations of these
variables which leave the equations invariant
This spawned a slew of research on how transformations
affect physical theories, culminating in the study of the
connection between the invariance of observable physical
quantities, such as momentum, with the algebraic and
geometric theory of invariants.
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
108. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
H ISTORY OF S YMMETRY IN P HYSICS
The idea of the classical “action” S
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
109. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
H ISTORY OF S YMMETRY IN P HYSICS
The idea of the classical “action” S = ˙
L(q, q, t)dt
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
110. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
H ISTORY OF S YMMETRY IN P HYSICS
The idea of the classical “action” S = ˙
L(q, q, t)dt
Hamilton’s Principle: The path that a particle actually takes
is the one that results in δS = 0 (at least to first-order)
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
111. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
H ISTORY OF S YMMETRY IN P HYSICS
The idea of the classical “action” S = ˙
L(q, q, t)dt
Hamilton’s Principle: The path that a particle actually takes
is the one that results in δS = 0 (at least to first-order)
This results in the Euler-Lagrange Equations:
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
112. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
H ISTORY OF S YMMETRY IN P HYSICS
The idea of the classical “action” S = ˙
L(q, q, t)dt
Hamilton’s Principle: The path that a particle actually takes
is the one that results in δS = 0 (at least to first-order)
This results in the Euler-Lagrange Equations:
d ∂L ∂L
= .
dt ˙
∂q ∂q
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
113. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
N OETHER ’ S T HEOREM
Using the Euler-Lagrange Equations, it can be shown that
IF a small transformation qk → qk + δqk leaves S invariant,
then there is an associated conserved quantity X , such
that
dX
=0
dt
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
114. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
N OETHER ’ S T HEOREM
Using the Euler-Lagrange Equations, it can be shown that
IF a small transformation qk → qk + δqk leaves S invariant,
then there is an associated conserved quantity X , such
that
dX
=0
dt
Emmy Noether rigorously generalized this idea–if a
transformation of one of the quantities in the action leaves
its form invariant, then there is a conserved quantity
associated with that symmetry in the physical system
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
115. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
N OETHER ’ S T HEOREM
Connection between global symmetries and conservation
principles
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
116. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
N OETHER ’ S T HEOREM
Connection between global symmetries and conservation
principles
Invariance Conserved Quantity
Translation in time Energy
Translation in space Linear Momentum
Rotation in space Angular Momentum
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117. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
H ISTORY OF S YMMETRY IN P HYSICS : T HE 20 TH
C ENTURY
1st 20th century turning point: Einstein’s paper on special
relativity in 1905
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118. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
H ISTORY OF S YMMETRY IN P HYSICS : T HE 20 TH
C ENTURY
1st 20th century turning point: Einstein’s paper on special
relativity in 1905
Up until Einstein, the global spatiotemporal symmetries
had been derived from the laws of classical
electrodynamics
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
119. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
H ISTORY OF S YMMETRY IN P HYSICS : T HE 20 TH
C ENTURY
1st 20th century turning point: Einstein’s paper on special
relativity in 1905
Up until Einstein, the global spatiotemporal symmetries
had been derived from the laws of classical
electrodynamics
He reversed this order: start with the universality of the
global symmetries, and then use as the test of the validity
of the laws of special relativistic dynamics
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
120. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
H ISTORY OF S YMMETRY IN P HYSICS : T HE 20 TH
C ENTURY
2nd turning point: Heisenberg’s description of the
indistinguishability of quantum particles in terms of a
“permutation symmetry”’ in 1926
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
121. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
H ISTORY OF S YMMETRY IN P HYSICS : T HE 20 TH
C ENTURY
2nd turning point: Heisenberg’s description of the
indistinguishability of quantum particles in terms of a
“permutation symmetry”’ in 1926
First time that a symmetry observed in quantum
mechanical phenomena dealt with using the techniques of
group theory
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
122. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
H ISTORY OF S YMMETRY IN P HYSICS : T HE 20 TH
C ENTURY
2nd turning point: Heisenberg’s description of the
indistinguishability of quantum particles in terms of a
“permutation symmetry”’ in 1926
First time that a symmetry observed in quantum
mechanical phenomena dealt with using the techniques of
group theory
Shout-out to fellow quantumers: the exchange operator!
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
123. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
H ISTORY OF S YMMETRY IN P HYSICS : T HE 20 TH
C ENTURY
2nd turning point: Heisenberg’s description of the
indistinguishability of quantum particles in terms of a
“permutation symmetry”’ in 1926
First time that a symmetry observed in quantum
mechanical phenomena dealt with using the techniques of
group theory
Shout-out to fellow quantumers: the exchange operator!
Hollah!
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
124. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
H ISTORY OF S YMMETRY IN P HYSICS : T HE 20 TH
C ENTURY
2nd turning point: Heisenberg’s description of the
indistinguishability of quantum particles in terms of a
“permutation symmetry”’ in 1926
First time that a symmetry observed in quantum
mechanical phenomena dealt with using the techniques of
group theory
Shout-out to fellow quantumers: the exchange operator!
Hollah!Pˆ |a, b >= |b, a >
12
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
125. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
R EST OF 20 TH C ENTURY H ISTORY OF S YMMETRY:
I T ’ S ALL SYMMETRY, BABY
“The history of the application of symmetry
principles in quantum mechanics and the quantum
field theory coincides with the history of the
developments of 20th century theoretical physics.”
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
126. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
O UTLINE OF TALK
1 Introduction: Examples of Symmetries in Nature
2 Groups
3 Brief History of Use of Symmetry in Physics
4 Symmetry and Quantum Field Theory
5 Symmetry Breaking
6 Conclusion: Why Symmetry?
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
127. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
T HE T HREE K EY D ISCRETE S YMMETRIES
Global symmetries–apply to all points in space.
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
128. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
T HE T HREE K EY D ISCRETE S YMMETRIES
Global symmetries–apply to all points in space.
Local symmetries apply only to individual space-time
locations
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
129. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
T HE T HREE K EY D ISCRETE S YMMETRIES
Global symmetries–apply to all points in space.
Local symmetries apply only to individual space-time
locations
Each quantum field theory possesses various global and
local symmetries
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
130. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
T HE T HREE K EY D ISCRETE S YMMETRIES
Global symmetries–apply to all points in space.
Local symmetries apply only to individual space-time
locations
Each quantum field theory possesses various global and
local symmetries
Each field theory has a global gauge symmetry associated
with its bosons
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
131. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
T HE T HREE K EY D ISCRETE S YMMETRIES
Invariance Conserved Quantity
Charge conjugation (C) Charge Parity
Coordinate inversion (P) Spatial Parity
Time reversal (T) Time Parity
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
132. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
T HE M AJOR I NTERNAL S YMMETRIES
Gauge Transformation Invariance Conserved Quantity
U(1) Electric Charge
U(1) Hypercharge
U(1)Y Weak Hypercharge
U(2) [U(1) × SU(2)] Electroweak force
SU(2) Isospin
SU(3) Quark Color
SU(3) (approximate) Quark Flavor
S(U(2) × U(3)) [U(1) × SU(2) × SU(3)] Standard Model
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
133. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
G AUGE B OSONS AND L OCAL S YMMETRIES
The QED Lagrangian–is invariant under a global U(1)
transformation
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
134. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
G AUGE B OSONS AND L OCAL S YMMETRIES
The QED Lagrangian–is invariant under a global U(1)
transformation
Associated gauge boson–the photon.
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
135. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
G AUGE B OSONS AND L OCAL S YMMETRIES
The QED Lagrangian–is invariant under a global U(1)
transformation
Associated gauge boson–the photon.
Weinberg-Salam Electroweak theory–invariant under an
SU(2) transformation
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
136. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
G AUGE B OSONS AND L OCAL S YMMETRIES
The QED Lagrangian–is invariant under a global U(1)
transformation
Associated gauge boson–the photon.
Weinberg-Salam Electroweak theory–invariant under an
SU(2) transformation
Associated gauge bosons–the W+, W-, and Z intermediate
vector bosons
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
137. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
G AUGE B OSONS AND L OCAL S YMMETRIES
The QED Lagrangian–is invariant under a global U(1)
transformation
Associated gauge boson–the photon.
Weinberg-Salam Electroweak theory–invariant under an
SU(2) transformation
Associated gauge bosons–the W+, W-, and Z intermediate
vector bosons
QCD Lagrangian–invariant under global SU(3)
transformation
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
138. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
G AUGE B OSONS AND L OCAL S YMMETRIES
The QED Lagrangian–is invariant under a global U(1)
transformation
Associated gauge boson–the photon.
Weinberg-Salam Electroweak theory–invariant under an
SU(2) transformation
Associated gauge bosons–the W+, W-, and Z intermediate
vector bosons
QCD Lagrangian–invariant under global SU(3)
transformation
(N 2 − 1 = 9 − 1 = 8) Generators, corresponding to 8
Gauge Bosons–the 8 gluons
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
139. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
W HAT U(1) INVARIANCE LOOKS LIKE
Invariance under U(1) transformation
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
140. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
W HAT U(1) INVARIANCE LOOKS LIKE
Invariance under U(1) transformation
Example: the Lagrangian for a complex scalar field:
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
141. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
W HAT U(1) INVARIANCE LOOKS LIKE
Invariance under U(1) transformation
Example: the Lagrangian for a complex scalar field:
L = ∂µ φ∗ ∂ µ φ − m 2 φ∗ φ
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
142. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
W HAT U(1) INVARIANCE LOOKS LIKE
Invariance under U(1) transformation
Example: the Lagrangian for a complex scalar field:
L = ∂µ φ∗ ∂ µ φ − m2 φ∗ φ is invariant under the U(1)
transformation φ → e−iθ φ:
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
143. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
W HAT U(1) INVARIANCE LOOKS LIKE
Invariance under U(1) transformation
Example: the Lagrangian for a complex scalar field:
L = ∂µ φ∗ ∂ µ φ − m2 φ∗ φ is invariant under the U(1)
transformation φ → e−iθ φ:
∗ ∗
L → ∂µ e−iθ φ ∂ µ e−iθ φ − m2 e−iθ φ e−iθ φ
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
144. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
W HAT U(1) INVARIANCE LOOKS LIKE
Invariance under U(1) transformation
Example: the Lagrangian for a complex scalar field:
L = ∂µ φ∗ ∂ µ φ − m2 φ∗ φ is invariant under the U(1)
transformation φ → e−iθ φ:
∗ ∗
L → ∂µ e−iθ φ ∂ µ e−iθ φ − m2 e−iθ φ e−iθ φ
= e+iθ e−iθ ∂µ φ∗ ∂ µ φ − e+iθ e−iθ m2 φ∗ φ
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
145. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
W HAT U(1) INVARIANCE LOOKS LIKE
Invariance under U(1) transformation
Example: the Lagrangian for a complex scalar field:
L = ∂µ φ∗ ∂ µ φ − m2 φ∗ φ is invariant under the U(1)
transformation φ → e−iθ φ:
∗ ∗
L → ∂µ e−iθ φ ∂ µ e−iθ φ − m2 e−iθ φ e−iθ φ
= e+iθ e−iθ ∂µ φ∗ ∂ µ φ − e+iθ e−iθ m2 φ∗ φ
= ∂µ φ∗ ∂ µ φ − m 2 φ∗ φ
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
146. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
I S PARITY CONSERVED ?
All of the fundamental forces are invariant under a local
coordinate inversion symmetry...except for the weak force
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
147. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
I S PARITY CONSERVED ?
All of the fundamental forces are invariant under a local
coordinate inversion symmetry...except for the weak force
The τ − θ problem
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
148. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
I S PARITY CONSERVED ?
All of the fundamental forces are invariant under a local
coordinate inversion symmetry...except for the weak force
The τ − θ problem
θ+ → π+ π0
τ + → π+π−π+
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
149. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
I S PARITY CONSERVED ?
All of the fundamental forces are invariant under a local
coordinate inversion symmetry...except for the weak force
The τ − θ problem
θ+ → π+ π0
τ + → π+π−π+
They appear to be the exact same particle
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
150. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
I S PARITY CONSERVED ?
All of the fundamental forces are invariant under a local
coordinate inversion symmetry...except for the weak force
The τ − θ problem
θ+ → π+ π0
τ + → π+π−π+
They appear to be the exact same particle
But...if the weak force is invariant under parity exchange,
the same particle can NOT decay in 2 different ways!
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
151. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
PARITY IS NOT CONSERVED
1956—Yang and Lee propose that parity might be violated
for weak interactions
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
152. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
PARITY IS NOT CONSERVED
1956—Yang and Lee propose that parity might be violated
for weak interactions
1957—Chien-Shung Wu’s group observes parity violation
in β-decay of cobalt-60
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
153. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
PARITY IS NOT CONSERVED
1956—Yang and Lee propose that parity might be violated
for weak interactions
1957—Chien-Shung Wu’s group observes parity violation
in β-decay of cobalt-60
1957—Garwin, Lederman, Weinrich and the decay of the
pion (π − ): π − → µ− + νµ
¯
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
154. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
PARITY IS NOT CONSERVED
1956—Yang and Lee propose that parity might be violated
for weak interactions
1957—Chien-Shung Wu’s group observes parity violation
in β-decay of cobalt-60
1957—Garwin, Lederman, Weinrich and the decay of the
pion (π − ): π − → µ− + νµ
¯
They expected to find approximately equal distribution of
positive and negative helicity muons
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
155. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
R ESULT OF G ARWIN , L EDERMAN , W EINRICH
EXPERIMENT
n (a), the helicity
is positive–both resultant particles’ velocities are aligned with
their spins; in (b), the helicity is negative. Only (b) is ever
actually observed.
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
156. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
R ESULT OF G ARWIN , L EDERMAN , W EINRICH
EXPERIMENT
In (a), the helicity
is positive–both resultant particles’ velocities are aligned with
their spins; in (b), the helicity is negative. Only (b) is ever
actually observed.
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
157. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
T IME R EVERSAL S YMMETRY
Laws of physics invariant under time reversal
transformation T
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
158. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
T IME R EVERSAL S YMMETRY
Laws of physics invariant under time reversal
transformation T
Doesn’t appear to hold at macroscopic level–poking hole in
a balloon example
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
159. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
T IME R EVERSAL S YMMETRY
Laws of physics invariant under time reversal
transformation T
Doesn’t appear to hold at macroscopic level–poking hole in
a balloon example
This is only a manifestation of the laws of statistical
mechanics
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
160. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
C HARGE C ONJUGATION
Dirac’s prediction and Carl Anderson’s 1932 discovery of
antimatter provides:
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
161. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
C HARGE C ONJUGATION
Dirac’s prediction and Carl Anderson’s 1932 discovery of
antimatter provides:
Another discrete symmetry: the invariance of the laws of
physics after replacing all particles with their antiparticles
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
162. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
C HARGE C ONJUGATION
Dirac’s prediction and Carl Anderson’s 1932 discovery of
antimatter provides:
Another discrete symmetry: the invariance of the laws of
physics after replacing all particles with their antiparticles
“Charge conjugation,” or C
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
163. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
C HARGE C ONJUGATION
Dirac’s prediction and Carl Anderson’s 1932 discovery of
antimatter provides:
Another discrete symmetry: the invariance of the laws of
physics after replacing all particles with their antiparticles
“Charge conjugation,” or C
If C is not violated, then atomic antimatter should be
exactly like atomic matter–antihydrogen
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
164. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
T HE TEST OF C INVARIANCE
The test of C invariance came shortly after the discovery
that P is violated
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
165. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
T HE TEST OF C INVARIANCE
The test of C invariance came shortly after the discovery
that P is violated
If we exchange particles for antiparticles in the π − decay
equation π − → µ− + νµ , we get the π + decay process,
¯
π + → µ+ + ν
µ
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
166. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
T HE TEST OF C INVARIANCE
The test of C invariance came shortly after the discovery
that P is violated
If we exchange particles for antiparticles in the π − decay
equation π − → µ− + νµ , we get the π + decay process,
¯
π + → µ+ + ν
µ
If C is a symmetry of this reaction, both µ+ and µ− should
have the same helicity
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
167. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
CP AND ITS VIOLATION
1957—The µ+ found to have positive helicity!
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
168. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
CP AND ITS VIOLATION
1957—The µ+ found to have positive helicity! C is violated!
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
169. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
CP AND ITS VIOLATION
1957—The µ+ found to have positive helicity! C is violated!
BUT...if C and P are considered as a joint CP operation,
then the reactions are invariant!
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
170. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
CP AND ITS VIOLATION
1957—The µ+ found to have positive helicity! C is violated!
BUT...if C and P are considered as a joint CP operation,
then the reactions are invariant!
Invert the parity and charge, and everything is alright
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
171. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
CP AND ITS VIOLATION
1957—The µ+ found to have positive helicity! C is violated!
BUT...if C and P are considered as a joint CP operation,
then the reactions are invariant!
Invert the parity and charge, and everything is alright
This IS what we observe
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
172. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
C RAP —CP IS VIOLATED TOO ! B UT CPT IS NOT
For 7 years, CP appeared to be inviolable
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
173. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
C RAP —CP IS VIOLATED TOO ! B UT CPT IS NOT
For 7 years, CP appeared to be inviolable
Cronon-Fitch experiment in 1964—CP is violated very
slightly
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
174. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
C RAP —CP IS VIOLATED TOO ! B UT CPT IS NOT
For 7 years, CP appeared to be inviolable
Cronon-Fitch experiment in 1964—CP is violated very
slightly
In every physical process we have yet observed, the
combination CPT IS conserved
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
175. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
C RAP —CP IS VIOLATED TOO ! B UT CPT IS NOT
For 7 years, CP appeared to be inviolable
Cronon-Fitch experiment in 1964—CP is violated very
slightly
In every physical process we have yet observed, the
combination CPT IS conserved
This combined CPT symmetry is a necessary condition for
probability to be conserved in quantum mechanics
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
176. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
O UTLINE OF TALK
1 Introduction: Examples of Symmetries in Nature
2 Groups
3 Brief History of Use of Symmetry in Physics
4 Symmetry and Quantum Field Theory
5 Symmetry Breaking
6 Conclusion: Why Symmetry?
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
177. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
S YMMETRY BREAKING
Pencil standing on its point
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
178. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
S YMMETRY BREAKING
Pencil standing on its point
Ferromagnetism—total rotational symmetry above Curie
point, but below it, the spins spontaneously line up in a
fixed direction
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
179. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
T HE 2008 N OBEL P RIZE : NAMBU AND THE B ROKEN
S YMMETRY OF BCS T HEORY
Yoichiro Nambu noticed that the BCS ground state violated
the gauge symmetry of QED
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
180. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
T HE 2008 N OBEL P RIZE : NAMBU AND THE B ROKEN
S YMMETRY OF BCS T HEORY
Yoichiro Nambu noticed that the BCS ground state violated
the gauge symmetry of QED
He recast BCS theory into perturbative quantum field
theory
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
181. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
T HE 2008 N OBEL P RIZE : NAMBU AND THE B ROKEN
S YMMETRY OF BCS T HEORY
Yoichiro Nambu noticed that the BCS ground state violated
the gauge symmetry of QED
He recast BCS theory into perturbative quantum field
theory
Showed that all of the phenomenon peculiar to
superconductivity are the result of the spontaneous
breaking of the underlying QED gauge symmetry
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
182. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
T HE 2008 N OBEL P RIZE : NAMBU AND THE B ROKEN
S YMMETRY OF BCS T HEORY
Yoichiro Nambu noticed that the BCS ground state violated
the gauge symmetry of QED
He recast BCS theory into perturbative quantum field
theory
Showed that all of the phenomenon peculiar to
superconductivity are the result of the spontaneous
breaking of the underlying QED gauge symmetry
Suggested that any theory in which a continuous
symmetry is spontaneously broken will give rise to
massless, spinless bosonic particles (Nambu-Goldstone
bosons) like those observed in BCS theory
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
183. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
T HE 2008 N OBEL P RIZE : NAMBU AND THE B ROKEN
S YMMETRY OF BCS T HEORY
Yoichiro Nambu noticed that the BCS ground state violated
the gauge symmetry of QED
He recast BCS theory into perturbative quantum field
theory
Showed that all of the phenomenon peculiar to
superconductivity are the result of the spontaneous
breaking of the underlying QED gauge symmetry
Suggested that any theory in which a continuous
symmetry is spontaneously broken will give rise to
massless, spinless bosonic particles (Nambu-Goldstone
bosons) like those observed in BCS theory
Showed that the same mechanism of spontaneous
symmetry breaking in BCS theory gives rise to the
mechanism required to support the existence of a nucleon
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
184. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
H IGGS F IELD
Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS