Symmetries and Conservation Laws

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Symmetries and Conservation Laws
1. Costituents of Matter 2. Fundamental Forces 3.
Particle Detectors (N. Neri) 4. Experimental
highlights (N. Neri) 5. Symmetries and
Conservation Laws 6. Relativistic Kinematics 7.
The Static Quark Model 8. The Weak Interaction 9.
Introduction to the Standard Model 10. CP
Violation in the Standard Model (N. Neri)
Five Deities Mandala Tibet, XVIIth Century
The word is used also to indicate a circular
diagram, basically made by the association of
different geometric figures (the most used being
the dot, the triangle, the circle and the
square). The drawing has spiritual and ritual
meaning in both Buddhism and Hinduism.
Mandala from it.wikipedia.org
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Class Invariance Conserved quantity
Proper orthochronousLorentz symmetry translation in time  (homogeneity) energy
translation in space  (homogeneity) linear momentum
rotation in space  (isotropy) angular momentum
Discrete symmetry P, coordinate inversion spatial parity
C, charge conjugation charge parity
T, time reversal time parity
CPT product of parities
Internal symmetry (independent ofspacetime coordinates) U(1) gauge transformation electric charge
U(1) gauge transformation lepton generation number
U(1) gauge transformation hypercharge
U(1)Y gauge transformation weak hypercharge
U(2) U(1) SU(2) electroweak force
SU(2) gauge transformation Isospin
SU(2)L gauge transformation weak isospin
P SU(2) G-parity
SU(3) "winding number" baryon number
SU(3) gauge transformation quark color
SU(3) (approximate) quark flavor
S(U(2) U(3)) U(1) SU(2) SU(3) Standard Model
A classification of symmetries in particle physics
Wikipedia
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Symmetires of a physical system
Quantum system
Classical system
Lagrangian Formalism
Lagrangian Formalism
Hamiltonian Formalism
Hamiltonian Formalism

• Invarianza Eq. Dinamica
• Invarianza relazioni di commutazione
• (Invarianza della probabilità)

Invariance of Equations of Motion
E. Noethers Theorem (valid for any lagrangian
theory, classical or quantum) relates symmetries
to conserved quantities of a physical system
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A classical example
Let us do a translation
The equations of motion are translation Invariant
!
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If one calculates the forces acting on 1 and 2
In the classical Lagrangian formalism
L invariant with respect to q
p conserved
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In the Hamiltonian formalism
Possible conservation of a dinamical quantity
Possible symmetry
This formalism can easily be used in Quantum
Mechanics
In Quantum Mechanics, starting from the
Schroedinger Equation
Time evolution (unitary)
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Schroedinger and Heisenberg Pictures
Heisenberg
Schroedinger
Operators in the Heisenberg picture
Taking the derivatives
Conserved quantities commute with H
In the case when there is an explicit time
dependence (non-isolated systems)
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Translational invariance a continuous spacetime
symmetry
The translation operator is naturally associated
to the linear momentum
For a finite translation
unitary
Self-adjoing the generator of space translations
If H does not depend on coordinates
The momentum is conserved
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Rotational invariance a continuous spacetime
symmetry
The rotation operator is naturally associated
with the angular momentum
Angular momentum (z-component) operator (angle
phi)
Self-adjoing rotation generator
A finite rotation
unitary
If H does not depend on the rotation angle f
around the z-axis
The angular momentum is conserved
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Time invariance (a continuous symmetry)
The generator of time translation is actually the
energy!
Using the equation of motion of the operators
If H does not dipend from t, the energy is
conserved
The continous spacetime symmetries Space
translation Space rotation Time translation
Linear momenum Angular momentum Energy
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Continuous symmetries and groups the case of
SU(2)
Combination of two transformations the result
depends on the commutation rules of the group
generators
Commutative (Abelian) algebra of translations
Translation operator along x
(two translations commute). Moreover
and clearly
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In the case of rotations
Commutation rules for the generators
A non-commutative algebra Rotations about
different axes do not commute
Why SU(2) ?
In the case of a two level quantum system, the
relevant internal symmetries are described by the
SU(2) group, having algebraic structure similar
to SO(3). SU(2) finds an application in the
Electroweak Theory SU(3) can be applied to QCD
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Isospin Symmetry
Let us consider a two-state quantum system (the
original idea of this came from the neutron and
the proton, considered to be degenerate states of
the nuclear force). Since they were considered
degenerate, they could be redefined
Degeneration
Ridefinition
Double degeneracy, similar to what happens in
s1/2 spin systems. The degeneration can be
removed by a magnetic field
One can introduce a two-components spinor
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The ridefinition now becomes
A symmetry for the Strong Interactions (broken by
electromagnetism)
SU(2) is a Lie group Properties can be deduced
from infinitesimal transformations
Which can be written in a general form
Pauli matrices
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Isospin
Infinitesimal rotation of the p-n doublet
A finite rotation in SU(2)

• Generalization of a global phase transformation
• Three phase angles
• Non-commuting operators (Non-abelian phase
  invariance)

example
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The two nucleon system
A one-nucleon state can be described with the
base of the nuclear spinors
A two nucleon state can be
In an Isospin rotation
Isospin singlet
The other three states transform into one another
in Isospin rotations, similar to a 3-d vector for
ordinary rotations
Isospin Invariance means there are two
amplitudes, I0 e I1 I1 states cannot be
distinguished by the Strong Interactions
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Nucleons and Quarks
I3 particle antiparticle particle antiparticle
1/2
-1/2
An Isospin triplet the Pion
I I3 Wave Function Q/e
1 1 1
1 -1 -1
1 0 o
0 o 0
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Building up strongly interacting particles
(hadrons) using Quarks as building blocks
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More on the two nucleon state
S
Isospin part
The total wave function
A
(non relativistic decomposition)

• The Deuton case
• As it is known that spin 1 ? a symmetric
• f has (-)l symm. It is known that the two
  nucleons are l0 or l2, f is symm.
• must be antisymmetric.
• This is because ? tot must be antisymmetric in
  nucleon exchange.

The Deuton is an Isospin singlet
Let us then consider the two reactions
(Deuton has I0, Pion has I1)
Isospin I 1 1
0,1 1
The reaction can proceed only via I1
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Gauge symmetries (global and local)
Gauge symmetries are continuous symmetries (a
continuous symmetry group). They can be global or
local. Global symmetries conserved quantities
(electric charge) Local symmetries new fields
and their transformation laws (Gauge theories)
Let us consider the Schoedinger equation
Let us consider a global phase transformation
the change in phase is the same everywhere
The Schroedinger equation is invariant for this
transformation. This invariance ia associated (E.
Noethers Theorem) to electric charge
conservation.
But then what happens if we consider a local
gauge tranformation ?
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How does one realize a local gauge invariance ?
Non invariant! And this is because
extra term !
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To solve the problem (and stick to the
invariance) we can introduce a new field. The
field would compensate for the extra term. The
new field would need to have an appropriate
transformation law.
Since the free Schroedinger Equation is not
invariant under
Let us modify the Equation
Compensating fields
Transformation laws
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This allows us to restore the invariance
To give the fields physical meaning
It is invariant, indeed

• The local gauge U(1) invariance of the free
  Schroedinger field
• Requires the presence of the Electromagnetic
  Field
• Dictates the field transformation law

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This gauge symmetry is the U(1) gauge symmetry
related to phase invariance There are, of
course, other possibilities.
In physics, a gauge principle specifies a
procedure for obtaining an interaction term from
a free Lagrangian which is symmetric with respect
to a continuous symmetry -- the results of
localizing (or gauging) the global symmetry group
must be accompanied by the inclusion of
additional fields (such as the electromagnetic
field), with appropriate kinetic and interaction
terms in the action, in such a way that the
extended Lagrangian is covariant with respect to
a new extended group of local transformations.
FREE
INTERACTING
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The U(1) Gauge Invariance and the Dirac field
Dirac Equation (1928) a quantum-mechanical
description of spin ½ elementary particles,
compatible with Special Relativity
Matrices 4x4 (gamma)
4 component spinor
Dirac probability current
Conjugate spinor
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The Dirac Lagrangian
Features a global gauge invariance
(trasformazioni di fase)
Let us require this global invariance to hold
locally. The invariance is now a dynamical
principle
Now the gauge transformation depends on the
spacetime points
Let us see how L behaves
Since
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This lagrangian is NOT gauge-invariant.
Se noi vogliamo una L gauge-invariante occorre
introdurre un campo compensante con una legge di
trasformazione opportuna
This new lagrangian is locally gauge-invariant.
This was made possible by the introduction of a
new field (the E.M. field).
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The gauge field A must however include a
free-field term. This will be the E.M. free
field term..
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Discrete symmetries P,C,T
Discrete symmetries describe non-continuous
changes in a system. They cannot be obtained by
integrating infinitesimal transformations. These
transformations are associated to discrete
symmetry groups
Parity P
Inversion of all space coordinates
The determinant of this transform is -1. In the
case of rotations, that would be 1
A unitary operator. Eigenvalues 1, -1 (if
definite-parity states) Eigenstates definite
parity states
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Parity is conserved in a system when
The case of the central potential
Bound states of a system with radial symmetry
have definite parity Example the hydrogen atom
Hydrogen atom wavefunction (no spin)
Radial part
Angular part
Electric Dipole Transition ?l 1
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The general parity of a quantum state
Let us consider a single particle a. The
intrinsic parity can be represented by a phase
Intrinsic
Spatial
Which is the physical meaning of the intrinsic
parity ? For instance, in a plane wave (momentum
eigenstates) representation
The intrinsic parity has the meaning of a parity
in the p0 system
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The parity of the photon from a classical analogy
A classical E field obeys
Let us take the P
To keep the Poisson Equation invariant, we need
to have the following law for E
On the other hand, in vacuum
And the parity operation would give
In order to make it consistent with the electric
field transformationt
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The action of parity on relevant physical
quantities
Position
Time
Momentum
Angular Momentum
Charge
Current
E field
B field
Spin
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Parity of a complex product of the parities of
the parts of the system
Spatial and intrinsic parity of particles the
pion
Exchange between the two neutrons
Angular momentum conservation J1 LS
Global nn symmetry
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Some intrinsic parities cannot be observed (p,
n). They are conventionally chosen to be 1.
Because of the Baryon Number conservation the
actual P value is not important as it cancels out
in any reaction.
Neutral pion parity. From the pair polarization
in
Transformation properties for rotations and space
reflections. Spin-parity
P(particle) - P (antiparticle) FERMIONS P(pa
rticle) P (antiparticle) BOSONS
Parity is violated in Weak Interactions
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Time Inversion T
T
It changes the time arrow
Classical dynamical equations are invariant
because of second order in time
Classical microscopic systems T invariance
Classical macroscopic systems time arrow
selected statistically (non decrease of entropy)
In the quantum case
Is not invariant for
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Let us now start from the Conjugate Schroedinger
Equation
Let us define the T-inversion operator
T
So, with this definition of T operator, we have
The operator representing T is an antilinear
operator. The square modulus of transition
amplitudes is conserved
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Wigner Theorem on Quantum Systems

• Any symmetry of a quantum system is given by
• either a unitary
• or an antiunitary operator

The Problem of Measurement in Quantum Mechanics
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An important consequence of T-invariance at the
microscopic level concerns the transition
amplitudes
(detailed balance)
Note the detailed balance DOES NOT imply the
equality of the reaction rates
A classical test, the study of the reaction
T is violated at the microscopic level il the
Weak Nuclear Interactions
Physical Review Letters 109 (2012) 211801. BaBar
experiment at SLAC Comparing the reactions
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Charge Conjugation C
An internal discrete symmetry
It changes the sign of the charges (and magnetic
moments)
In the case of a quantum state
The C eigenstates are the neutral states
For the photon case
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The C-parity of a state can be calculated for a
neutral state if we know the wave function of the
state
Since charge conjugation of two particles of
opposite charge, swaps the identify of the
particles, one has to account for the proper
quantum statistics
True also in general for spin zero particles
For a couple of femions, instead
In the pi-zero decay
This decay in 3 photons
Is forbidden if C is conserved in electromagnetic
interactions. In fact
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Action of C,P,T
C P
T CPT
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Positronium
Similar to the hydrogen atom. Actually, the
true atom.
The space part
The spin part
Triplet
Singlet
The C conjugation balance
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Positronium in the l0 (fundamental) state
Singlet Triplet
Antisymmetry by electron/positron exchange
C-parity conservation determines the Ps decay
modes
Singlet Triplet
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Photons, Spin, Helicity
Gauge - invariant
Coulomb Gauge
Free propagation
Plane wave solution
Transversality condition
Plane polarization Circular polarization
For instance
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Circular polarization
Which can be expressed by using the rotating
vectors
The polarization vectors can be associated to the
photon spin states
If the wave propagates along z
Jz only due to spin
Let us make a rotation around the z axis
Autostati di Jz
e
e
Per la trasversalità abbiamo solo
Fotoni con Jz0 sono i fotoni longitudinali.
Virtuali m?0
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Helicity
Proiezione dello spin nella direzione del momento
Un numero quantico approssimato per particelle
con massa Tanto più buono quanto la particella è
relativistica Rigorosamente buono per i fotoni
La legge di invarianza in azione. Le Interazioni
Elettromagnetiche conservano la Parità Ma
Nelle interazioni elettromagnetiche questa
quantità deve essere nulla. Nelle interazioni
elettromagnetiche i fotoni right e left handed
compaiono sempre in pari ampiezze, in modo da
compensarsi
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The Neutrino
C, P sono violate nelle Interazioni Nucleari
Deboli Il neutrino partecipa solo delle
Interazioni Nucleari Deboli Peraltro,
nellapprossimazione di neutrini senza massa,
abbiamo
Levidenza sperimentale indica che nelle
Interazioni Deboli I neutrini sono sempre
sinistrorsi. Gli antineutrini sono sempre
destrorsi !
P
C
CP
In buona approssimazione le Int. Deboli
conservano CP (non C e non P)
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The CPT Theorem
In a local, Lorentz-invariant quantum field
theory, the interaction (Hamiltonian) is
invariant with respect to the combined action of
C,P,T (Pauli, Luders, Villars, 1957)
A few consequences

• Mass of the particle Mass of the antiparticle
• (Magnetic moment of the particle) -- (Magnetic
  moment of antiparticle)
• 3) Lifetime of particle Lifetime of
  antiparticle

Proton Antiproton Electron Positron
Q e -e -e e
B o L(e) 1 -1 1 -1
µ
s
Protons, electrons
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Search for violations of C,P,T
A quantity is formed that would violate a
conservation law One checks if this quantity
exists for a pure state. Examle the Electric
Dipole Moment (EDM)
Cannot exist for a pure state is T, P invariant
situations
CPT Theorem (wikipedia) In quantum field theory
the CPT theorem states that any canonical (that
is, local and Lorentz-covariant) quantum field
theory is invariant under the CPT operation,
which is a combination of three discrete
transformations charge conjugation C, parity
transformation P, and time reversal T. It was
first proved by G.Lüders, W.Pauli and J.Bell in
the framework of Lagrangian field theory. At
present, CPT is the sole combination of C, P, T
observed as an exact symmetry of nature at the
fundamental level.
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Particle Numbers baryonic, flavor, and leptonic
Flavor
The flavor is the quark content of a hadron
Massa (MeV) Quark U D S C B
p uud 2 1 0 0 0
n udd 1 2 0 0 0
? uds 1 1 -1 0 0
?c udc 1 1 0 1 0
p u-dbar 1 -1 0 0 0
K- s-ubar -1 0 -1 0 0
D- d-cbar 0 1 0 -1 0
Ds c-sbar 0 0 1 1 0
B- b-ubar -1 0 0 0 -1
? b-bbar 0 0 0 0 0
938 940 1116 2285 140 494 1869 1970 5279 9460
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Favor quantum numbers refer to quark content of
hadrons They are conserved in Strong and
Electromagnetic Interactions They are violated in
Weak Interaction
Strangeness
Charm
Beauty
Top
In a Stong Nuclear (or E.M.) process, all flavors
are conserved
In Weak Interactions instead
Baryon Number
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Baryon Number
The Baryon Number is equivalent to
Baryons have B1 while Antibaryons have B
-1 Mesons have B 0 This law follows from the
conservation of the Quark Number. Quarks
transform into each other. They disappear (or
appear) in pairs.
Flavor quantum numbers refer to the identity of
quarks
(Isospin 1/2 o -1/2 in doublets) Strangeness
-1 for the s quark Charm 1 for the c
quark Bottom -1 for the b quark Top 1 for the
t quark
Violated in Weak Interactions
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The Leptonic Numbers
Electronic Lepton Number
Muonic Lepton Number
Numero leptonico tauonico
The Leptonic Numbers are conserved in any known
interaction WITH THE EXCEPTION OF Neutrino
Oscillations. In Neutrino Oscillations, they are
violated. However, one can define a total lepton
number
To the best of our knowledge the Total Lepton
Number (sum of the three leptonic numbers) is
conserved in every interaction. For instance,
the decay
Does not take place. .
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