1. Symmetries and conservation laws
Pierre-Hugues Beauchemin
PHY 006 –Talloire, May 2013

2. 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

3. 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

4. Discrete symmetries
A symmetry is discrete when there is a finite number of transformations that leave an observable quantity invariant
E.g. The set of transformation that leave a triangle invariant
Rotation by 120º and by 240º
Reflection with respect to axis starting from a summit and bisecting the opposite segment in two equal parts
This can be
used to describe
microscopic
systems

5. 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.: The rotation of a disk by any angle q with respect to an axis
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

6. 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

7. 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 anti- particles
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

8. Symmetries breaking
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

9. Group theory
Consider the set of permutation of 1, 2, 3:
(1,2,3), (2,3,1), (3,1,2), (2,1,3), (1,3,2), (3,2,1)
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

10. 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

11. 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

12. 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

13. terms describing the fundamental interactions of Nature
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

14. 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 anti- matter