Symmetries and conservation laws - PowerPoint Presentation

Slide1

Symmetries and conservation laws

Pierre-Hugues BeaucheminPHY 006 –Talloire, May 2013Slide2

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 featuresSlide3

Transformations and symmetries

What is a symmetry? It is a transformation that leaves an observable aspect of a system unchangedThe 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 otherSlide4

Discrete symmetries

A symmetry is discrete when there is a finite number of transformations that leave an observable quantity invariantE.g. The set of transformation that leave a triangle invariantRotation by 120º and by 240ºReflection with respect to axis starting from a summit and bisecting the opposite segment in two equal parts

This can be

u

sed to describe

m

icroscopic

systemsSlide5

Continuous symmetries

Characterized by an invariance following a continuous change in the transformation of a systemInfinite (uncountable) numbers of transformation leave the

system unchangedE.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

qSlide6

External symmetries

These are the symmetries that leave a system invariant under space-time transformations The external symmetries are:Spatial rotationsSpatial translati

onsProperties of a system unchanged under a continuous change of locationTime 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 relativitySlide7

Internal symmetries

Symmetries internal to a system but which get manifest through the various processesParity 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 coordinatesSlide8

Symmetries breaking

Transformations or external effects often break the symmetry of a systemThe system is still symmetric but the symmetry is hidden to observationCan 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 stateSlide9

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

a

nd we have that the

s

ymmetries of the triangle

a

re exactly the same as

t

he permutations of 1, 2, 3

There is a fundamental structure underlying symmetries: group theorySlide10

Invariance and conservation

A quantity is conserved when it doesn’t vary during a given processE.g. Money changes hands in a transaction, but the total amount of money before and after the transaction is the sameIn 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 systemSlide11

Energy, momentum and angular momentum (I)

Energy is, in classical physics, the quantity needed to perform mechanical work

It can take various forms

T

he 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 processSlide12

Angular momentum characterizes the state of rotating motion of a

systemIt also has a magnitude and a directionParticles have an extrinsic angular momentum when in rotating motion, and an intrinsic angular momentum when they have a spinIn 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 momentumSlide13

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):

SU

C

(3): invariance of the theory under rotations in the 3-dimensional local internal color space (b, r, g)

SU

L

(2): invariance of the theory under rotations in the 2-dimensional local internal weak charge space

U

Y

(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

t

erms describing the fundamental interactions of NatureSlide14

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 SU

C

(3) x SU

L

(2) x U

Y

(1)



SU

C

(3) x

U

em

(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 ×(

n

q

-n

aq

)) and lepton (

n

l

-n

al

, 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)