PHY 042: Electricity and Magnetism Scalar Potential Prof. Hugo Beauchemin 1.
Symmetries and conservation laws Pierre-Hugues Beauchemin PHY 006 –Talloire, May 2013.
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Transcript of Symmetries and conservation laws Pierre-Hugues Beauchemin PHY 006 –Talloire, May 2013.
Symmetries in natureMany 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 symmetriesWhat 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 symmetriesA 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 beused to describe
microscopic systems
Continuous symmetriesCharacterized 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
External symmetriesThese 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 symmetriesSymmetries 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
Symmetries breakingTransformations 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
systemsResidual 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 theoryConsider 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 triangleare exactly the same asthe permutations of 1, 2, 3
There is a fundamental structure underlying symmetries: group theory
Invariance and conservationA 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
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
Energy, momentum and angular momentum (II)
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 spaceRequiring these invariances in the theory is sufficient to generate all
terms describing the fundamental interactions of Nature
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
Symmetries in HEP (II)