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Symmetries and
conservation laws
Pierre-Hugues Beauchemin
PHY 006 –Talloire, May 2013
Symmetries in nature
 Many objects in nature presents a high level of
symmetry, indicating that the forces that produced
these objects feature the same symmetries
⇒ Learn structure of Nature by studying symmetries it features
Transformations and symmetries
 What is a symmetry?
It is a transformation that leaves an observable aspect
of a system unchanged
 The unchanged quantity is called an invariant
 E.g.: Any rotation of the needle of a clock change the
orientation of the needle, but doesn’t change is length
 The length of a clock needle is invariant under rotation
transformations
 Physics law can also be left invariant under transformation of
symmetries
 E.g. Coulomb’s law giving the electric force
of two charged particle on each other
Discrete symmetries
 A symmetry is discrete when there is a finite number of
transformations that leave an observable quantity
invariant

discrete
and
finite
symmetries
discrete
and
finite
symmetries
E.g. The set of transformation that leave a triangle invariant
 Rotation
!!! !!!by 120º and by 240º
! !respect to axis starting from a summit and
 Reflection with
•!!!symmetry!group!of!a!square!!
•!!!symmetry!group!of!a!triangle!!
bisecting
the opposite segment in two equal parts
This can be
used to describe
microscopic
systems
Continuous symmetries
 Characterized by an invariance following a continuous
change in the transformation of a system
 Infinite (uncountable) numbers of transformation leave the
system unchanged
 E.g.: continuous
The rotation of
a disk by groups
any angle q with respect to an axis
symmetry
 All these transformation are of a given kind and so can be easily
characterized by a small set of parameters
 E.g.: All the transformations that leaves the disk invariant can be
characterized by the axis of rotation and the angle of rotation
q
External symmetries
 These are the symmetries that leave a system invariant
under space-time transformations
 The external symmetries are:
 Spatial rotations
 Spatial translations
 Properties of a system unchanged under a continuous change of location
 Time translations
 Physics systems keep same properties over time
 Lorentz transformations
 Physics systems remain unchanged regardless of the speed at which they
moves with respect to some observer
 This is central to special relativity
Internal symmetries
 Symmetries internal to a system but which get
manifest through the various processes
 Parity transformation (P-Symmetry)
 Things look the same in a mirror image
 Same physics for left- and right-handed systems
 Time reversal (T-Symmetry)
 Laws of physics would be the same if they were
running backward in time
 Charge conjugation (C-Symmetry)
 Same laws of physics for particle and antiparticles
 Gauge transformation
 Laws of physics are invariant under changes of
redundant degrees of freedom
 E.g.: Rising the voltage uniformly through a circuit
 Internal symmetries can be global or local
depending on if the transformation that leaves
the system invariant is the same on each point
of space-time or varies with space-time
coordinates
Symmetries breaking
BROKEN symmetry
 Transformations or external effects
often break the symmetry of a system
 The system is still symmetric but the
symmetry is hidden to observation
 Can be inferred by looking at many
systems
 Residual symmetries can survive the breaking
 A symmetry can be:
 Explicitly broken:
the laws of physics don’t exhibit the symmetry
 The symmetry can be spotted when the breaking effect is
weak and the system is approximately symmetric
 Spontaneously broken:
the equations of motion are symmetric but the
state of lowest energy of the system is not
 The system prefer to break the symmetry to get
into a more stable energy state
Group theory
discrete
Consider the
setfinite
of permutation
of 1, 2, 3:
and
symmetries
!!!
(1,2,3), (2,3,1), (3,1,2), (2,1,3), (1,3,2), (3,2,1)
!
!!!symmetry!group!of!a!triangle!!
 •Now,
consider the symmetries of a triangle again :
We can see that:
• For the 120º rotation
A->B, B->C, C->A
• For the 240º rotation
A->C, C->B, B->A
Replace A, B, C for 1, 2, 3
and we have that the
symmetries of the triangle
are exactly the same as
the permutations of 1, 2, 3
There is a fundamental structure underlying symmetries: group theory
Invariance and conservation
 A quantity is conserved when it doesn’t vary
during a given process
 E.g. Money changes hands in a transaction, but
the total amount of money before and after the
transaction is the same
 In 1918, Emmy Noether published the proof of
two theorems now central to modern physics:
each continuous symmetry of a system is
equivalent to a measurable conserve quantity
 This is formulated in group theory and applies to
physics theory that are realizations of these groups
 By studying quantities that are conserved in
physics collision processes, we can learn what
are the fundamental symmetries determining
the underlying fundamental interactions
without knowing all the state of the system
Energy, momentum and
angular momentum (I)
 Energy is, in classical physics, the quantity needed
to perform mechanical work
 It can take various forms
 The total energy of an isolated system is conserved
 In HEP it describes the state of motion of a particle
 Momentum is another quantity describing the
state of motion of a particle
 It has a magnitude and a direction
 In Newton physics it corresponds to
 It each component of the momentum are
conserved in a particle collision process
Energy, momentum and
angular momentum (II)
 Angular momentum characterizes the state of rotating
motion of a system
 It also has a magnitude and a direction
 Particles have an extrinsic angular momentum when in
rotating motion, and an intrinsic angular momentum when
they have a spin
 In Newton physics, it is defined as
 The total angular momentum is conserved in
a particle collision
 These conservation rules proceed from
fundamental symmetries satisfied by the
physics laws governing the fundamental
interactions of nature:
 Invariance of the system with time 
conservation of energy
 Invariance with translation  conservation of momentum
 Invariance with rotation  conservation of angular momentum
Symmetries in HEP (I)
 The Standard Model Lagrangian is the equation
describing the dynamics of all known particles
 This equation corresponds to the most general equation
giving finite and stable observable predictions concerning all know
particles, and which satisfies a set of fundamental symmetries:
 It must be invariant under space translation, time translation, rotations and
Lorentz transformation
 The laws of physics are independent of the state of motion and the
position of observers
 This equation must satisfies three local gauge invariance on the internal
space of the particles (quantum fields):
 SUC(3): invariance of the theory under rotations in the 3-dimensional local
internal color space (b, r, g)
 SUL(2): invariance of the theory under rotations in the 2-dimensional local
internal weak charge space
 UY(1): invariance of the theory under rotations in the 1-dimensional local
internal hypercharge space
Requiring these invariances in the theory is sufficient to generate all
terms describing the fundamental interactions of Nature
Symmetries in HEP (II)
 While the SUC(3) part of the symmetry of the Standard
Model remains unbroken when particles acquires a
mass via the Higgs mechanism, the SUL(2)xUY(1) part
get broken to the Uem(1) weaker symmetry of the electromagnetism
 The SM is summarized by SUC(3) x SUL(2) x UY(1)  SUC(3) x Uem(1)
 The SM features some more symmetries:
 CPT: the mirror image of our universe filled of anti-particle rather than
particle and running backward in time would be exactly the same as our
universe
 This has been shown to be equivalent to Lorentz invariance
 Bring Feynman to interpret anti-particles as particles running backward in
time…
 Accidental symmetries such as Baryon (⅓ ×(nq-naq)) and lepton (nl-nal, l=e,
m, t) numbers conservation
 Approximate symmetries such as CP invariance
 The little CP-violating phase is crucial for matter domination over antimatter