P780.02 Spring 2003 L4 Richard KassConservation LawsWhen something doesn’t happen there is usually a reason!

Read:M&S Chapters 2, 4, and 5.1,

n\ pe- or p\ ne+v or \ e That something is a conservation law !

A conserved quantity is related to a symmetry in the Lagrangian thatdescribes the interaction. (“Noether’s Theorem”)A symmetry is associated with a transformation that leaves the Lagrangian invariant.

time invariance leads to energy conservationtranslation invariance leads to linear momentum conservationrotational invariance leads to angular momentum conservation

Familiar Conserved QuantitiesQuantity Strong EM Weak

Commentsenergy Y Y Y

sacredlinear momentumY Y Y sacredang. momentum Y Y Y sacredelectric charge Y Y Y

sacred

P780.02 Spring 2003 L4 Richard KassOther Conserved Quantities

Quantity Strong EM Weak Comments Baryon number Y Y Y no p+0

Lepton number(s) Y Y Y no -e-Nobel 88, 94, 02 top Y Y N discovered 1995 strangeness Y Y N discovered 1947 charm Y Y N discovered 1974, Nobel 1976 bottom Y Y N discovered 1977 Isospin Y N N proton = neutron (mumd) Charge conjugation (C) Y Y N particle anti-particle Parity (P) Y Y N Nobel prize 1957 CP or Time (T) Y Y y/n small No, Nobel prize 1980 CPT Y Y Y sacred G Parity Y N N works for pions only

Classic example of strangeness violation in a decay: p- (S=-1 S=0)Very subtle example of CP violation:

expect: Kolong+0-

BUT Kolong+-(1 part in 103)

Neutrino oscillations give first evidence of lepton # violation! These experiments were designed to look for baryon # violation!!

P780.02 Spring 2003 L4 Richard KassSome Reaction ExamplesProblem 2.1 (M&S page 43):a) Consider the reaction vu+p++nWhat force is involved here? Since neutrinos are involved, it must be WEAK interaction.Is this reaction allowed or forbidden?Consider quantities conserved by weak interaction: lepton #, baryon #, q, E, p, L, etc.

muon lepton number of vu=1, +=-1 (particle Vs. anti-particle) Reaction not allowed!

b) Consider the reaction ve+pe-+++pMust be weak interaction since neutrino is involved.

conserves all weak interaction quantitiesReaction is allowed

c) Consider the reaction e-+++ (anti-ve)Must be weak interaction since neutrino is involved.

conserves electron lepton #, but not baryon # (1 0)Reaction is not allowed

d) Consider the reaction K+-+0+ (anti-v)Must be weak interaction since neutrino is involved.

conserves all weak interaction (e.g. muon lepton #) quantitiesReaction is allowed

P780.02 Spring 2003 L4 Richard KassMore Reaction ExamplesLet’s consider the following reactions to see if they are allowed:

a) K+p ++ b) K-p n c) K0+-

First we should figure out which forces are involved in the reaction.All three reactions involve only strongly interacting particles (no leptons)so it is natural to consider the strong interaction first. a) Not possible via strong interaction since strangeness is violated (1-1)b) Ok via strong interaction (e.g. strangeness –1-1) c) Not possible via strong interaction since strangeness is violated (1 0)If a reaction is possible through the strong force then it will happen that way!Next, consider if reactions a) and c) could occur through the electromagnetic interaction.Since there are no photons involved in this reaction (initial or final state) we can neglect EM.Also, EM conserves strangeness.Next, consider if reactions a) and c) could occur through the weak interaction.Here we must distinguish between interactions (collisions) as in a) and decays as in c).The probability of an interaction (e.g. a) involving only baryons and mesons occurring through the weak interactions is so small that we neglect it.Reaction c) is a decay. Many particles decay via the weak interaction through strangeness changing decays, so this can (and does) occur via the weak interaction.To summarize:a) Not possible via weak interactionc) OK via weak interactionDon’t even bother to consider Gravity!

dsK

usK

usK

0

P780.02 Spring 2003 L4 Richard Kass

Conserved Quantities and SymmetriesEvery conservation law corresponds to an invariance of the Hamiltonian (or Lagrangian) of the system under some transformation.We call these invariances symmetries. There are 2 types of transformations: continuous and discontinuousContinuous give additive conservation laws

x x+dx or +d examples of conserved quantities:

electric chargemomentumbaryon #

Discontinuous give multiplicative conservation lawsparity transformation: x, y, z (-x), (-y), (-z)charge conjugation (particleantiparticle): e- e+

examples of conserved quantities: parity (in strong and EM)charge conjugation (in strong and EM)parity and charge conjugation (strong, EM, almost always in weak)

P780.02 Spring 2003 L4 Richard Kass

Conserved Quantities and SymmetriesExample of classical mechanics and momentum conservation.In general a system can be described by the following Hamiltonian: H=H(pi,qi,t) with pi=momentum coordinate, qi=space coordinate, t=timeConsider the variation of H due to a translation qi only.

dt

dppwith

q

Hp

p

Hq i

ii

ii

i

dtt

Hdp

p

Hdq

q

HdH

ii

iii

i

3

1

3

1

For our example dpi=dt=0 so we have:

3

1ii

i

dqq

HdH

Using Hamilton’s canonical equations:

We can rewrite dH as:

3

1

3

1 iii

ii

i

dqpdqq

HdH

If H is invariant under a translation (dq) then by definition we must have:

03

1

3

1

iii

ii

i

dqpdqq

HdH

This can only be true if:00

3

1

3

1

ii

ii p

dt

dp or

Thus each p component is constant in time and momentum is conserved.

P780.02 Spring 2003 L4 Richard Kass

Conserved Quantities and Quantum MechanicsIn quantum mechanics quantities whose operators commute with theHamiltonian are conserved.Recall: the expectation value of an operator Q is:

),,(and),(with* txxQQtxxdQQ How does <Q> change with time?

xdt

Qxdt

QxdQ

txdQ

dt

dQ

dt

d ***

*

Recall Schrodinger’s equation:

Ht

iHt

i **

and

Substituting the Schrodinger equation into the time derivative of Q gives:

xdQHi

xdt

QxdQH

ixdQ

dt

dQ

dt

d **** 11

H+= H*T= hermitian conjugate of H

Since H is hermitian (H+= H) we can rewrite the above as:

xdHQit

QQ

dt

d)],[

1(*

So if Q/t=0 and [Q,H]=0 then <Q> is conserved.

P780.02 Spring 2003 L4 Richard Kass

Conservation of electric charge and gauge invarianceConservation of electric charge: Qi=Qf

Evidence for conservation of electric charge: Consider reaction e-ve whichviolates charge conservation but not lepton number or any other quantum number.If the above transition occurs in nature then we should see x-rays from atomictransitions. The absence of such x-rays leads to the limit:e > 2x1022 years

There is a connection between charge conservation, gauge invariance,and quantum field theory.Recall Maxwell’s Equations are invariant under a gauge transformation:

tc

AA

1:potentialscalar

:potentialvector

A Lagrangian that is invariant under a transformation U=ei is saidto be gauge invariant.

There are two types of gauge transformations:local: =(x,t)global: =constant, independent of (x,t)

Conservation of electric charge is the result of global gauge invariancePhoton is massless due to local gauge invariance

Maxwell’s EQs are locally gauge invariant

P780.02 Spring 2003 L4 Richard Kass

Gauge invariance, Group Theory, and StuffConsider a transformation (U) that acts on a wavefunction ():ULet U be a continuous transformation then U is of the form:

U=ei is an operator.If is a hermitian operator (=*T) then U is a unitary transformation:

U=eiU+=(ei)*T= e-i*T = e-iUU+= eie-i=1Note: U is not a hermitian operator since UU+

In the language of group theory is said to be the generator of UThere are 4 properties that define a group:1) closure: if A and B are members of the group then so is AB2) identity: for all members of the set I exists such that IA=A3) Inverse: the set must contain an inverse for every element in the set AA-1=I4) Associativity: if A,B,C are members of the group then A(BC)=(AB)C

If = (1, 2, 3,..) then the transformation is “Abelian” if:U(1)U(2) = U(2)U(1) i.e. the operators commute

If the operators do not commute then the group is non-Abelian.The transformation with only one forms the unitary abelian group U(1)The Pauli (spin) matrices generate the non-Abelian group SU(2)

10

01

0

0

01

10zyx i

i S= “special”= unit determinantU=unitaryn=dimension (e.g.2)

P780.02 Spring 2003 L4 Richard Kass

Global Gauge Invariance and Charge ConservationThe relativistic Lagrangian for a free electron is:

ii ee

is the electron field (a 4 component spinor)m is the electrons massu= “gamma” matrices, four (u=0,1,2,3) 4x4 matrices that satisfy uv+ vu =2guv

u= (0, 1, 2, 3)= (/t, /x, /y, /z)

Let’s apply a global gauge transformation to L

1 cmiL uu

LmiL

eemeeiL

eemeeiL

miL

uu

iiuu

ii

iiiuu

i

uu

constantaisλsince

By Noether’s Theorem there must be a conserved quantity associatedwith this symmetry!

This Lagrangian gives the Dirac equation:

0 mci uu

P780.02 Spring 2003 L4 Richard KassThe Dirac Equation on One Page0 mci u

u

0][ 3210

mczyxt

i

0

0

0

0

1000

0100

0010

0001

]

0010

0001

1000

0100

000

000

000

000

0001

0010

0100

1000

1000

0100

0010

0001

[ mczy

i

i

i

i

xti

0

0

10

010

i

ii

])(/(exp[])(/(exp[

)(

0

1

/)|(|

2

2

2 rpEtiUrpEti

McE

ippcMcE

cpcmcE

yx

z

A solution (one of 4, two with +E, two with -E) to the Dirac equation is:

0)( Umcpuu

The function U is a (two-component) SPINOR and satisfies the following equation:

The Dirac equation:

Spinors are most commonly used in physics to describe spin 1/2 objects. For example:

)2/(downspinrepresents1

0while/2)(upspinrepresents

0

1

Spinors also have the property that they change sign under a 3600 rotation!

P780.02 Spring 2003 L4 Richard Kass

E-L equation in 1D

Global Gauge Invariance and Charge ConservationWe need to find the quantity that is conserved by our symmetry.In general if a Lagrangian density, L=L(, xu) with a field, is invariantunder a transformation we have:

u

u

xx

LLL

0

For our global gauge transformation we have:

uuu xi

xxii

and)1(

Plugging this result into the equation above we get (after some algebra…)

u

u

u

uu

u x

L

xx

L

x

L

xx

LLL )(0

The first term is zero by the Euler-Lagrange equation.The second term gives us a continuity equation. 0)(

x

L

dt

d

x

L

Result fromfield theory

)1( iei

P780.02 Spring 2003 L4 Richard Kass

Global Gauge Invariance and Charge ConservationThe continuity equation is:

u

u

u

u

u

u

u

u

x

LiJ

x

Ji

x

L

xx

L

xwith0

Recall that in classical E&M the (charge/current) continuity equation has the form:

0

Jt

Also, recall that the Schrodinger equation give a conserved (probability) current:

])([and2

***22

ciJcVmt

i

If we use the Dirac Lagrangian in the above equation for L we find:

uuJThis is just the relativistic electromagnetic current density for an electron.The electric charge is just the zeroth component of the 4-vector:

xdJQ

0

Therefore, if there are no current sources or sinks (J=0) charge is conserved as:

000

t

J

x

J

u

u

(J0, J1, J2, J3) =(, Jx, Jy, Jz)

Result fromquantum field theory

Conserved quantity

P780.02 Spring 2003 L4 Richard Kass

Local Gauge Invariance and PhysicsSome consequences of local gauge invariance:a) For QED local gauge invariance implies that the photon is massless.

b) In theories with local gauge invariance a conserved quantum number impliesa long range field.

e.g. electric and magnetic fieldHowever, there are other quantum numbers that are similar to electric charge(e.g. lepton number, baryon number) that don’t seem to have a long range forceassociated with them!

Perhaps these are not exact symmetries! evidence for neutrino oscillation implies lepton number violation.c) Theories with local gauge invariance can be renormalizable, i.e. can useperturbation theory to calculate decay rates, cross sections, etc.

Strong, Weak and EM theories are described by local gauge theories.U(1) local gauge invariance first discussed by Weyl in 1919SU(2) local gauge invariance discussed by Yang&Mills in 1954 (electro-weak)e i(x,t) is represented by the 2x2 Pauli matrices (non-Abelian)SU(3) local gauge invariance used to describe strong interaction (QCD) in 1970’s

e i(x,t) is represented by the 3x3 matrices of SU(3) (non-Abelian)

P780.02 Spring 2003 L4 Richard Kass

Local Gauge Invariance and QEDConsider the case of local gauge invariance, =(x,t) with transformation:

),(),( txitxi ee

The relativistic Lagrangian for a free electron is NOT invariant under this transformation.

1 cmiL uu

The derivative in the Lagrangian introduces an extra term:

)],([),(),( txiee utxiutxi

We can MAKE a Lagrangian that is locally gauge invariant by addingan extra piece to the free electron Lagrangian that will cancel the derivative term.

We need to add a vector field Au which transforms under a gauge transformation as: AuAu+u(x,t) with (x,t)=-q(x,t) (for electron q=-|e|)

The new, locally gauge invariant Lagrangian is:u

uuvuvu

u AqFFmiL 16

1

P780.02 Spring 2003 L4 Richard Kass

The Locally Gauge Invariance QED Lagrangianu

uuvuvu

u AqFFmiL 16

1

Several important things to note about the above Lagrangian:1) Au is the field associated with the photon.

2) The mass of the photon must be zero or else there would be a term in the Lagrangian of the form:

mAuAu

However, AuAu is not gauge invariant!

3) Fuv=uAv-vAu and represents the kinetic energy term of the photon.

4) The photon and electron interact via the last term in the Lagrangian. This is sometimes called a current interaction since:u

uu

u AJAq In order to do QED calculations we apply perturbation theory (via Feynman diagrams) to JuAu term.

5) The symmetry group involved here is unitary and has one parameter U(1)

e-e-

Au

Ju