PHYS 3446Lecture #23PHYS 3446 –Lecture #23Monday, Nov. 17 2008y,Dr. Andrew Brandt1. Symmetry1. Symmetry• Local gauge symmetryEC for those2. Parity• Properties of ParityEC for those who couldn’t go to seminar: 2 pars abouta particle physics Nobel• Determination of Parity• Conservation and violationa particle physics Nobel prize winnerMonday, Nov. 17, 2006 PHYS 3446, Fall 2008 Andrew Brandt1U(1) Local Gauge Invariance Dirac Lagrangian for free particle of spin ½ and mass m;()()2μ∂()()2ic mcμμψγ ψ ψψ=∂−hLii i t d lblh t f ti (lblis invariant under a global phase transformation (global gauge transformation) since .ieθψψ→ieθψψ−→However, if the phase, θ, varies as a function of space-time coordinate xμisLstill invariant underspacetime coordinate, xμ, is L still invariant under the local gauge transformation, ?()ixeθψψ→No, because it adds an extra term from derivative of θ.Monday, Nov. 17, 2006 PHYS 3446, Fall 2008 Andrew Brandt2U(1) Local Gauge InvarianceRequiring the complete Lagrangian to be invariant under λ(x) local gauge transformation will require additional terms in the free Dirac Lagrangian to cancel the extra term()⎡⎤()Wh Aifildhf()()2ic mcμμψγ ψ ψψ⎡⎤=∂−⎣⎦hL()qAμμψγ ψ−Where Aμis a new vector gauge field that transforms under local gauge transformation as follows:ggAμ→ffAμμλ+∂Addition of this vector field to L keeps L invariant under local gauge transformation Monday, Nov. 17, 2006 PHYS 3446, Fall 2008 Andrew Brandt3ggGauge Fields and Local Symmetries• To maintain a local symmetry, additional fields must be introduced– This is in general true even for more complicated symmetries– Crucial information for modern physics theories• A distinct fundamental forces in nature arises from local invariance of physical theoriespy• The associated gauge fields generate these forces–These gauge fields are the mediators of the given force–These gauge fields are the mediators of the given force• This is referred as gauge principle, and such theories are gauge theoriesgauge theories– Fundamental interactions are understood through this theoretical frameworkMonday, Nov. 17, 2006 PHYS 3446, Fall 2008 Andrew Brandt4frameworkGauge Fields and Mediators• To keep local gauge invariance, new particles had to be introduced in gauge theories– U(1) gauge introduced a new field (particle) that mediates the electromagnetic force: Photon–SU(2) gauge introduces three new fields that mediates weak force• Charged current mediator: W+and W-Ntl tZ0•Neutral current: Z0– SU(3) gauge introduces 8 mediators (gluons) for the strong forceU ifi ti f l t ti d k f SU(2) U(1)•Unification of electromagnetic and weak force SU(2)xU(1) gauge introduces a total of four mediators–Neutral current: Photon, Z0– Charged current: W+and W-Monday, Nov. 17, 2006 PHYS 3446, Fall 2008 Andrew Brandt5Th i i f i ( i i )ÎSihParity•The space inversion transformation (mirror image)ÎSwitch right- handed coordinate system to left-handedctx⎛⎞⎜⎟⎜⎟P arityctx⎛⎞⎜⎟−⎜⎟yz⎜⎟⎜⎟⎜⎟⎝⎠P arityuuuuuuryz⎜⎟⎜⎟−⎜⎟−⎝⎠• How is this different from a normal spatial rotation?–Rotation is continuous in a given coordinate systemgy• Quantum numbers related to rotational transformation are continuous – Space inversion cannot be obtained through any set of rotational tf titransformation• Quantum numbers related to space inversion are discrete•Parity is an example of a discrete transformationMonday, Nov. 17, 2006 PHYS 3446, Fall 2008 Andrew Brandt6•Parity is an example of a discrete transformationProperties of Parity• Position and momentum vectors change sign under space inversionrrrr& Puuurr−rPrr&• Whereas their magnitudes do not change signspmr= Puuurmr p−=−rrr=⋅rr Puuur()()rr rrr−⋅− = ⋅=rrrrVectors (particles w/ JP1) change sign underppp=⋅rr Puuur()()pp ppp−⋅− = ⋅=rrrr•Vectors (particles w/ JP=1-) change sign under space-inversion while the scalars (particles w/ JP=0+) dtMonday, Nov. 17, 2006 PHYS 3446, Fall 2008 Andrew Brandt7do not.Some vectors however behave like a scalar under Parity opProperties of Parity•Some vectors, however, behave like a scalar under Parity op.– Angular momentumLrp×rrrP()()×rrrpL×=rrr– These are called pseudo-vectors or axial vectors (particles w/ JP=1+)Lik i l b h lik tLrp=× Puuur()()rp−×−=rpL×=•Likewise some scalars behave like vectorsP()abc⋅×rrr Puuur()()()()abc−⋅−×− =rrr()abc−⋅ ×rrr–These are called pseudo-scalars(particles w/ JP=0-)• Two successive application of parity operations must turn the di t b k t i i l()()()coordinates back to original–Th ibll(i l)fitP +1( )12Pψ=()PPψ=()Pψ−=ψ–The possible values (eigen values) of parity, P, are +1 (even) or -1 (odd).•Parity is amultiplicative quantum numberMonday, Nov. 17, 2006 PHYS 3446, Fall 2008 Andrew Brandt8•Parity is a multiplicative quantum numberParity• Two parity quantum numbers– Intrinsic parity: Bosons have the same intrinsic parities as their anti-particles while fermions have opposite parity of their anti-particle P it d ti l t f ti f ll th l P (1)l–Parity under spatial transformation follows the rule: P=(-1)l•lis the orbital angular momentum quantum numberAlt tid ittilf i itd•Are electromagnetic and gravitational forces invariant under parity operation or space inversion?2drr–Newton’s equation of motion for a point-like particle2drmFdt=r–For electromagnetic and gravitational forces we can write the forces , and thus are invariant under parity.222ˆdr CmFrdt r==rrMonday, Nov. 17, 2006 PHYS 3446, Fall 2008 Andrew Brandt9dt rDetermination of Parity Quantum Numbers• How do we find out the intrinsic parity of particles?Determination of Parity Quantum Numbers– Use observation of decays and production processes–Absolute determination of parity is not possible, just like py p ,jelectrical charge or other quantum numbers.–Thus the accepted convention is to assign +1 intrinsic pgparity to proton, neutron and the Λ hyperon.• The parities of other particles are determined relative to these assignments through the analysis of parity conserving interactions involving these particles.•Λhyperon is always produced with a K in pair So one can•Λhyperon is always produced with a K in pair. So one can determine parity
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