fundamental principles of particle physics our description of the fundamental interactions and...
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Fundamental principles of particle physics
Our description of the fundamental interactions and particles rests on two fundamental structures :
Quantum Mech s anic
Symme tries
Symmetries
Central to our description of the fundamental forces :
Relativity - translations and Lorentz transformations
Lie symmetries - (3) (2) (1)SU SU U
Copernican principle : “Your system of co-ordinates and units is nothing special”
Physics independent of system choice
Quantum Mechanics
Relativity+ } Quantum Field theory
Fundamental principles of particle physics
Special relativity
( , , , )a ct x y z
( ) ( , , , )a a a a c t x y z
Space time point not invariant under translations
Space-time vector
Invariant under translations …but not invariant under rotations or boosts
Einstein postulate : the real invariant distance is
32 2 2 2 20 1 2 3
, 0
a a a a g a a a a a
( 1, 1, 1, 1)g diag
Physics invariant under all transformations that leave all such distances invariant :
Translations and Lorentz transformations
Lorentz transformations :
3
0
x x x x
g x x g x x g g
Solutions :
3 rotations R
(Summation assumed)
1 0 0 0
0 cos sin 0
0 sin cos 0
0 0 0 1
( )zR ct
x
y
z
ct
x
y
z
cos sin
cos sin
ct
x y
y x
z
Lorentz transformations :
3
0
x x x x
g x x g x x g g
Solutions :
3 rotations R
1 0 0 0
0 cos sin 0
0 sin cos 0
0 0 0 1
3 boosts B
cosh sinh 0 0
sinh cosh 0 0
0 0 1 0
0 0 0 1
Space reflection – parity P
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
Time reflection, time reversal T
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
(Summation assumed)
The Lorentz transformations form a group, G 1 2 1 2( , )g g G if g g G
. /( ) ,iR e J
( . . )c f J r p
Angular momentumoperator
Rotations
( )..z y xJ i x y
/ziJ
0 0 0 0
0 0 1 0
0 1 0 0
0 0 0 0
/ziJe ( )zR 1 /ziJ 1/ . / ...
2 z ziJ iJ
1 0 0 0
0 cos sin 0
0 sin cos 0
0 0 0 1
( )zR
The Lorentz transformations form a group, G 1 2 1 2( , )g g G if g g G
. /( ) ,iR e J
( . . )c f J r p
Angular momentumoperator
3
1
[ , ]i j ijk kk
J J i J
123 231 312 213 132 321
totally antisymmetric Levi-Civita symbol,
1; 1
ijk
Rotations
( )..z y xJ i x y
iJThe are the “generators” of the group.
Their commutation relations define a “Lie algebra”.
Demonstration that ( ) ziJZR e
cos sin '( ) ( , ) ( ) ( ', ')
sin cos 'Z Z
x x xR x y R x y
y y y
'For small = ,
'
x x y
y y x
( ) ( , ) ( , ) ( , ) ( )ZR x y x y y x x y y xx y
(1 ( )) ( , )y xi xp yp x y
. . ( ) (1 ( )) 1Z y x zi e R i xp yp i J
Hence ( ) (1 ) ziJnZ Z nR n i J e
( ) ( ) ( ) ( )x y x yR R R R
1 0 0 1 0 1 0 0 1 0
0 1 0 1 0 0 1 0 1 0
0 1 0 1 0 1 0 1
1 0
1 0 ( ) (1 )
0 0 1z zR i J
( ) ( ) ( ) ( )x y x yR R R R
(1 )(1 )(1 )(1 ) ( )x y x y x y y xi J i J i J i J J J J J
Equating the two equations implies
[ , ]x y zJ J iJ QED
Derivation of the commutation relations of SO(3) (SU(2))
, (infinitesimal)small
Quantum Mechanics
Relativity+ } Quantum Field theory
Fundamental principles of particle physics
Relativistic quantum field theory
Fundamental division of physicist’s world :
slow fast
large
small
ClassicalNewton
Classicalrelativity
ClassicalQuantum mechanics
Quantum Fieldtheory
speed
Ac
t
ion
c
( )S
( . . . .)B
A
t
classical
t
S K E P E dt
Action
“Principle of Least Action” Feynman Lectures in Physics Vol II Chapter 19
Relativistic quantum field theory
Fundamental division of physicist’s world :
slow fast
large
small
ClassicalNewton
Classicalrelativity
ClassicalQuantum mechanics
Quantum Fieldtheory
speed
Ac
t
ion
c
( )
( amplitude )S
iQM e
( )S
1Sie
2Sie
3Sie
Quantum Mechanics : Quantization of dynamical system of particles
Quantum Field Theory : Application of QM to dynamical system of fields
Why fields?
No right to assume that any relativistic process can be explained by single particle since E=mc2 allows pair creation
(Relativistic) QM has physical problems. For example it violates causality
2 2
20
20
( / 2 ) 3 ( / 2 )0 0
( )3 ( / 2 )3
3/ 2( ) / 2
( ) | | | | |
1
2
... f nonzero or all , 2
i p m t i p m t
ip x xi p m t
im x x t
U t x e x d p x e p p x
d p e e
me x t
it
2 2-m xRelativistic case : U(t) e t ... nonzero for all , x t
Amplitude for free propagation from x0 to x
Quantum Mechanics
22 0
2i
t m
2
02
pE
m Classical – non relativistic
Quantum Mechanical : Schrodinger eq
Quantum Mechanics
22 0
2i
t m
2
02
pE
m Classical – non relativistic
Quantum Mechanical : Schrodinger eq
2 2E m 2p
2
2
2
tm
2
Classical – relativistic
Quantum Mechanical - relativistic
Relativistic QM - The Klein Gordon equation (1926)
Scalar particle (field) (J=0) : (x)
2
2
2 2 2
tE m m
2 2p
Energy eigenvalues 2 1/ 2( ) ???E m 2p
1934 Pauli and Weisskopf revived KG equation with E<0 solutions as E>0solutions for particles of opposite charge (antiparticles). Unlike Dirac’s hole theory this interpretation is applicable to bosons (integer spin) as well as tofermions (half integer spin).
1927 Dirac tried to eliminate negative solutions by writing a relativistic equation linear in E (a theory of fermions)
As we shall see the antiparticle states make the field theory causal
(natural units)
