SIP Presentation

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


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?



S YMMETRY ON THE SURFACE OF THE D EAD S EA




1 Translation




1 Translation
2 Rotation



M AKING R IPPLES – BREAKING SOME SYMMETRIES
AND CREATING OTHERS



AND CREATING OTHERS

1 Translational symmetry broken



AND CREATING OTHERS

2 Rotational symmetry about the center of the ripple



AND CREATING OTHERS

3 How about 2 ripples?



AND CREATING OTHERS




T HE S YMMETRIES OF E QUILATERAL T RIANGLES

Six Symmetry “Operations”




“Identity” (1)




“Identity” (1)
Rotation by 120◦ (R120 )




“Identity” (1)




“Identity” (1)
Reﬂection axis I (RI )




“Identity” (1)
Reﬂection axis II (RII )




“Identity” (1)
Reﬂection axis II (RII )
Reﬂection axis III (RIII )



R120 × RII = RI



RII × R120 = RIII



T HE OPERATIONS DO NOT COMMUTE !




R120 × RII = RI




R120 × RII = RI RII × R120 = RIII



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




1 R240 RI RIII
R120 R120 R240 1 RIII RII
RII RI R240 1 R120




1 R240 RI RII RIII
R120 R120 R240 1 RIII RII
RII RI R240 1 R120




1 R240 RI RII RIII
R120 R120 R240 1 RIII RI RII
RII RI R240 1 R120




1 R240 RI RII RIII
RII RII RI R240 1 R120




1 R120 R240 RI RII RIII
RII RII RI R240 1 R120




1 R120 R240 RI RII RIII
RII RII RIII RI R240 1 R120



S YMMETRY OF THE C IRCLE



S YMMETRY OF THE C IRCLE

Rotate by any arbitrary angle θ ∈ R, and the circle still looks the
same



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.




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



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:



G ROUPS

Associativity: a(bc) = (ab)c



G ROUPS

The group has an identity element e s.t. ae = ea = a



G ROUPS

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



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?



C OMMUTATIVITY

i.e. is a · b = b · a?



C OMMUTATIVITY

i.e. is a · b = b · a?
If the elements of the group are commutative, the group is
called abelian.



E XAMPLE OF AN A BELIAN G ROUP : T HE I NTEGERS

The set of integers form a group under the operation of
addition




addition
Take + as our composition rule




addition
Closure: For any elements a, b of the set of integers, a + b
will yield another integer




addition
will yield another integer (i.e. 2 + 3 = 5)




addition
Identity element: 0 (because 19 + 0 = 19)




addition
Inverse element: positive/negative (i.e. 23 + (−23) = 0)




addition
Associativity is satisﬁed, i.e. 3 + (5 + 7) = 15 = (3 + 5) + 7.




addition
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.



D ISCRETE VS . C ONTINUOUS G ROUPS

Discrete: Equilateral Triangle (S3 )–ﬁnite number of
distinguishable symmetry operations–6.



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



D ESCRIBING C ONTINUOUS G ROUPS : U(1)

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




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



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



R EPRESENTATIONS

That is, F (a)F (b) = F (ab)



R EPRESENTATIONS

So, what’s the most basic representation of U(1)?



R EPRESENTATIONS

So, what’s the most basic representation of U(1)? eiθ



R(θ): A REPRESENTATION OF U(1)
cos(θ) sin(θ)
R(θ) =
− sin(θ) cos(θ)



cos(θ) sin(θ)
R(θ) =
− sin(θ) cos(θ)
We can show that R(θ) is a group



cos(θ) sin(θ)
R(θ) =
− sin(θ) cos(θ)
i.e. Composition rule satisﬁed: R(θ1 )R(θ2 ) = R(θ1 + θ2 ).



cos(θ) sin(θ)
R(θ) =
− sin(θ) cos(θ)
For R(θ) to be a representation of U(1), it too must satisfy
the properties of U(1)



cos(θ) sin(θ)
R(θ) =
− sin(θ) cos(θ)
Unitarity: R(θ)† R(θ) = R(θ)R(−θ)



cos(θ) sin(θ)
R(θ) =
− sin(θ) cos(θ)
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)



cos(θ) sin(θ)
R(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



cos(θ) sin(θ)
R(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



L IE G ROUPS
Most of the continuous groups that turn up in studies of
symmetries in physics, including U(1), are Lie groups.



L IE G ROUPS
(1) Elements of the group depend on a ﬁnite set of
continuous parameters θ1 , θ2 , · · · , θn



L IE G ROUPS
(2) Derivatives of the group elements with respect to all of
the group parameters exist.



L IE G ROUPS
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



L IE G ROUPS
derivative at zero
If a group representation has n parameters, then there are
n generators Xi of that representation, given by



L IE G ROUPS
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



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?




Hermitian? Multiply the generators by −i:




Hermitian? Multiply the generators by −i:

∂g
Xi = −i
∂θ θi =0



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





∂ cos(θ) sin(θ) 0 −i
X (θ) = −i =
i 0





∂ cos(θ) sin(θ) 0 −i
X (θ) = −i =
i 0

Lookie, lookie, its one of the Pauli Matrices!



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



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



Copernicus: Heliocentric model shattered Ptolemaic
paradigm



Copernicus: Heliocentric model shattered Ptolemaic
paradigm
But it still preserved the cherished symmetry of the
circular celestial orbits



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.”



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




´
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




´
¨
structure of groups
The set of symmetry operations applicable to a given
geometric ﬁgure satisﬁes the conditions for a group




´
¨
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.”




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




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.




The idea of the classical “action” S




The idea of the classical “action” S = ˙
L(q, q, t)dt




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)




L(q, q, t)dt
This results in the Euler-Lagrange Equations:




L(q, q, t)dt
This results in the Euler-Lagrange Equations:

d ∂L ∂L
= .
dt ˙
∂q ∂q



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




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




Connection between global symmetries and conservation
principles




Connection between global symmetries and conservation
principles
Invariance Conserved Quantity
Translation in time Energy
Translation in space Linear Momentum
Rotation in space Angular Momentum



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



C ENTURY

relativity in 1905
Up until Einstein, the global spatiotemporal symmetries
had been derived from the laws of classical
electrodynamics



C ENTURY

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



C ENTURY

2nd turning point: Heisenberg’s description of the
indistinguishability of quantum particles in terms of a
“permutation symmetry”’ in 1926



C ENTURY

First time that a symmetry observed in quantum
mechanical phenomena dealt with using the techniques of
group theory



C ENTURY

group theory
Shout-out to fellow quantumers: the exchange operator!



C ENTURY

group theory
Hollah!



C ENTURY

group theory
Hollah!Pˆ |a, b >= |b, a >
12



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.”



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




locations
Each quantum ﬁeld theory possesses various global and
local symmetries




locations
Each quantum ﬁeld theory possesses various global and
local symmetries
Each ﬁeld theory has a global gauge symmetry associated
with its bosons




Invariance Conserved Quantity
Charge conjugation (C) Charge Parity
Coordinate inversion (P) Spatial Parity
Time reversal (T) Time Parity



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



G AUGE B OSONS AND L OCAL S YMMETRIES

The QED Lagrangian–is invariant under a global U(1)
transformation




transformation
Associated gauge boson–the photon.




transformation
Weinberg-Salam Electroweak theory–invariant under an
SU(2) transformation




transformation
Associated gauge bosons–the W+, W-, and Z intermediate
vector bosons




transformation
vector bosons
QCD Lagrangian–invariant under global SU(3)
transformation




transformation
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



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




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





∗ ∗

= e+iθ e−iθ ∂µ φ∗ ∂ µ φ − e+iθ e−iθ m2 φ∗ φ

= ∂µ φ∗ ∂ µ φ − m 2 φ∗ φ



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+





θ+ → π+ π0
τ + → π+π−π+
They appear to be the exact same particle





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!



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
¯




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



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.



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.



T IME R EVERSAL S YMMETRY

Laws of physics invariant under time reversal
transformation T




transformation T
Doesn’t appear to hold at macroscopic level–poking hole in
a balloon example




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



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




“Charge conjugation,” or C
If C is not violated, then atomic antimatter should be
exactly like atomic matter–antihydrogen



T HE TEST OF C INVARIANCE

The test of C invariance came shortly after the discovery
that P is violated




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




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



CP AND ITS VIOLATION

1957—The µ+ found to have positive helicity!




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




Invert the parity and charge, and everything is alright
This IS what we observe



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




slightly
In every physical process we have yet observed, the
combination CPT IS conserved




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



S YMMETRY BREAKING

Pencil standing on its point



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



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



theory
Showed that all of the phenomenon peculiar to
superconductivity are the result of the spontaneous
breaking of the underlying QED gauge symmetry



theory
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



theory
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


H IGGS F IELD


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