Lecture 3: Invariance Principles, Conservation Laws - PowerPoint PPT Presentation

1 / 42
About This Presentation
Title:

Lecture 3: Invariance Principles, Conservation Laws

Description:

Symmetries of nature provide strong constraints to what nature can do. ... The formalism was the same as for spin-1/2, thus the name 'isospin' (self spin) ... – PowerPoint PPT presentation

Number of Views:585
Avg rating:3.0/5.0
Slides: 43
Provided by: hepPhy
Category:

less

Transcript and Presenter's Notes

Title: Lecture 3: Invariance Principles, Conservation Laws


1
Lecture 3Invariance Principles,Conservation
Laws
2
Symmetries
  • Symmetries of nature provide strong constraints
    to what nature can do.
  • They lead to conserved quantities
  • Parity, C-parity, CP, Charge, Isospin
  • Even simple ideas lead to non-trivial qualitative
    and quantitative predictions

3
Simple Symmetries
  • Time translation invariance
  • ? Conservation of energy
  • Space translation invariance
  • Conservation of linear momentum

4
Parity
  • One would expect physical laws to look the same
    in a mirror
  • That is, if we changed our coordinate system to
    X?X, Y?Y, Z?-Z
  • Parity is the most extreme case
  • X?-X, Y?-Y, Z?-Z
  • In quantum mechanics, it corresponds to
  • Discrete transformation ? not simply a rotation
  • PP1 so Unitary with eigenvalues of 1 or 1
  • Not all wave functions have well-defined parity

5
Hydrogen Wavefunctions
  • The wave functions you see in an intro QM course
    are still useful here
  • Represent the angular distribution of two bodies
    in motion around each other
  • Parity on these wave functions corresponds to

p-q
q
fp
6
Parity on angular functions
  • So working through the components of the
    spherical harmonics
  • In other words, the parity is only determined by
    the total angular momentum matters, not the
    particular state

7
Rules for Parity
  • Parity of a system is the product of the parity
    of its parts
  • Parity is conserved in strong and electromagnetic
    interactions only

8
Intrinsic Parity
  • We find that parity is conserved in strong and
    electromagnetic interactions
  • We assign an intrinsic parity to particles to
    make sure it balances in reactions
  • We will discuss the pion, but baryons are
    conserved (well also discuss this!) so their
    parity is a matter of convention
  • Parity of proton and neutron P1
  • Some reactions, like pp?pLK which involve
    strange particles always find particles produced
    in pairs. Can only measure pair parity

9
Example Parity of p
  • I must admit, these examples are tricky, but its
    worth working through slowly, just to appreciate
    the power of conservation laws
  • We will first establish the spin, and then the
    parity of the pion, by studying simple reactions
  • It is also worth noting that during the dawn of
    particle physics, nothing was assumed.
  • Today, we know pions have a quark and anti-quark,
    which have opposite parities, and so P-1
  • Back then, everything had to be learned from
    scratch!

10
Spin of p
  • Measured in study of reversible reaction
  • Rates should be the same if carried out with the
    same CMS energy by symmetry of matrix element
  • But sd1 and momenta are equal so

So we measurespin of pion!
11
Parity of p
  • Parity determined by capture in deuterium
  • Perkins tells us that capture proceeds through
    S-state of p relative to d
  • Since sd1 and sp0, then initial state has J1
  • Final state has two neutrons, JLS
  • NR QM lets us factorize space and spin of nn
    state

12
Parity of p , cont
  • Spin wave functions have well defined symmetry
    under interchange of spins
  • Spin function gets (-1)S1 under interchange of
    spin
  • Space function gets (-1)L under spatial
    interchange
  • So full interchange gives us (-1)LS1
  • But nn wave function must be fully antisymmetric
    under interchange ? LS is even!

13
Almost there!
  • So we have two conditions
  • J1 in initial state
  • LS even in final state
  • Satisfied by three possibilities
  • L0, S1 ? LS odd
  • L1, S0 or 1 ? LS even for L,S1
  • L2, S1 ? LS odd
  • So final neutron state have L1
  • Thus, parity of final state P1
  • P(n) and P(d)1, so P(p)-1 QED!

14
Parity of Anti-particles
  • Diracs theory predicts that particles and anti
    particles have opposite parities
  • Proven by experiment for e and e-
  • Good rule of thumb for parities of hadrons
  • Pions ? quark and antiquark in S state, so
    Pp(-1)L1 -1
  • Rho goes to two pions, but J1, so L1 in final
    state, so Pr-1 just from conservation of orbital
    angular momentum

15
Is Parity always conserved?
  • No
  • Weak interactions break the symmetry as much as
    they can, e.g. RH neutrinos do not exist!
  • All observation of parity violation in studies
    (nominally) of strong and EM can be attributed to
    small components of weak interactions in the
    hamiltonian
  • Not too surprising cant completely isolate
    yourself!

RH
LH
16
Charge Conjugation Invariance
  • C operator reverses charge (and thus magnetic
    moment) of a particle
  • For fermions in Dirac, theory, C exchanges
    particle and anti-particle
  • So the C operator results in many illegal
    transitions ? not a good symmetry for baryons and
    leptons
  • However, some hadronic and photonic states do
    have good C-parity

17
C-parity states
  • Particle-antiparticle annihilation
  • Strong interactions conserve charge
  • Neutral bosons
  • Sign determined by Cg
  • Photons determined by charges, which couple to
    fields as eA. C symmetry requires A to change
    sign Cg-1
  • So p0?gg ? Cp1
  • This implies that p0?ggg is forbidden, even if it
    does not violate charge conservation

18
Trick for Charge Conjugation
  • To think about C symmetry for multiparticle
    systems, just remember that a C-transformation
    corresponds to an exchange of particles
  • I.e. it looks a lot like a parity flip

19
CP Symmetry
  • Weak interactions also violate C for the same
    reason no LH anti-neutrinos!
  • However, CP corresponds to
  • Flip the handedness of the neutrino LH?RH
  • Flip the charge of the neutrino?anti-neutrino
  • Thus, you have a RH anti-neutrino, which does
    exist
  • CP looks like a good symmetry of the weak
    interaction.
  • Its not, but well get to this later!

20
Charge Conservation
  • Charge conservation is something we learn quite
    early
  • I1I2I3 in circuit theory
  • It is obeyed to an obscene level
  • Any net violation would give measureable net
    charge of the earth, which is not seen
  • How does it come about?
  • Typically, things which dont happen cant
    happen!
  • How does this conservation law come about?

I1
I3
I2
21
Gauge Invariance
  • We know that Maxwells equations are invariant
    under transformations
  • These gauge transformations leave the fields
    the same, so any observable quantities are
    unchanged
  • We never observe the absolute value of the scalar
    and vector potentials

22
Gauge Invariance in QM
  • Double slit experiment measures interference
    between two waves
  • Depends on relative phases
  • If we changed both phases by a constant over t
    x, no change ? no problem!
  • If the phase could change with x, then the
    physics at C would change

B
C
A
23
Gauge Symmetry
  • How do we deal with this?
  • Let phase be a smooth function
  • However, lets allow the particle to couple to a
    field (e.g. EM), which affects the phase
  • Then our problem looks like this
  • But EM is gauge invariant, so
    gives the same physics as A itself

And physics is unchanged!
24
Implications
  • The fact that physics should be invariant under
    local changes in phase implies two critical
    features of nature
  • Charges must be coupled to a long-range field
    which can change the phases of the electrons
  • Charge is conserved
  • Gauge invariance implies a very robust theory
  • Many of the problems encountered with field
    theories in general disappear when gauge symmetry
    is applied

25
Isospin
  • Heisenberg (1932) suggested that the similarity
    of the neutron and proton (M938 MeV, spin ½)
    indicated that they were two states of the same
    particle the nucleon
  • The formalism was the same as for spin-1/2, thus
    the name isospin (self spin)
  • Isospin was found to be conserved in strong
    interactions, not in weak EM
  • Only the magnitude I matters, not particular Iz
  • Evidence
  • Equivalence of nn, np, pp interactions
  • Equivalence of mirror nuclei

26
Isospin in the Quark Model
  • Isospin can be understood in the quark model of
    hadrons
  • The only difference betweenn and p is
    interchange d?u
  • Is d u had the same mass, this would be a true
    symmetry
  • No feature except charge would distinguish p/n
  • But its not perfect
  • We know that Mn-Mp 1.3 MeV
  • Coulomb effects should be stronger for p than n
  • So we conclude that DMqMd-Mu2-3 MeV
  • And DMq/MN.2 so a small effect!

27
Isospin in other hadrons
  • We now know the fundamental doublet for isospin
    is
  • The comparable doublet for anti-particles is
  • We can combine two spin-1/2 states just like spin

Bigger chargehas larger I3
28
Isospin in the N-N system
  • Consider a system of 2 nucleons
  • Nucleon, remember, is p or n
  • We again combine the isospins into triplet and
    singlet combinations

Symmetric
Anti-symmetric
29
Symmetry of NN wavefunction
  • Consider full wavefunction
  • For a deuteron
  • Spin function is symmetric (J1)
  • Space function is L0,2 so symmetric
  • Overall antisymmetry?I0!
  • Now consider and
  • Each final state has I1, so initial state must
    have that as well
  • Thus, only ½ initial states available

30
Pion-Nucleon scattering
  • If strong reactions only care about total
    isospin, then six reactionscan be described by
    two amplitudes!
  • We can classify them into groups
  • Processes are pure I3/2
    and so have the same cross section
  • The other processes will have I31/2 and thus
    I1/2 or I3/2
  • Now we need Clebsch-Gordon coefficients

31
Clebsch-Gordon Coefficients
  • Not going to make a formal introduction, but we
    can get a long way with 4 simple rules for
    angular momentum in QM
  • If we act on the I3/2, I33/2 state, we get

32
CGC Table
33
Comparing Reaction Rates
  • Consider three similar processes of pion-nucleon
    scattering
  • What can we say about their reaction rates just
    from isospin considerations?

34
Matrix elements
  • Cross section goes as the matrix element squared
  • H is a Hamiltonian which has pieces connecting
    different I states

35
Elastic Scattering (Pure)
  • The reactionis pure I3/2, I33/2 so
  • There is a leading factor (phase space, spins,
    etc.) and then a matrix element squared,
    incorporating the force
  • We dont have a fundamental model for this, but
    you will see how even presenting it as an unknown
    variable is useful sometimes!

36
Elastic Scattering (Mixed)
  • However, is a
    combination of I3/2 and I1/2 states

37
Charge Exchange
  • Now we have to consider two different wave
    functions, one for initial and one for final state

38
Cross-Section Ratios
  • We can now look at ratios of the cross sections
  • Consider limiting cases, when one or other
    isospin exchange process dominates

39
Hypercharge
  • There is a compact description of the charges of
    pions and nucleons
  • Adding strangeness to the mix is easy

Hypercharge
40
G-parity
  • There is another operator used to characterize
    hadron states
  • For nucleon-antinucleon system
  • The neutral pion has G-parity 1
  • All of the pions are chosen to have the same
    G-parity as the neutral one

41
Selection Rule for G
  • So ultimately we have a nice rule for the
    multi-pion states
  • Constrains the number of pions a particular
    isospin state of nucleons can decay into!

42
Conservation laws
  • All forces we know of conserve
  • Energy-momentum
  • Charge
  • Baryon Number
  • Lepton Number
  • CPT
  • Weak force violates P,C,CP(or T)
  • Strong and EM Conserve them
  • Only strong force obeys isospin
  • Weak and EM violate them
Write a Comment
User Comments (0)
About PowerShow.com