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)