Quantum Mechanics

Relativity

+ } Quantum Field theory

Relativistic quantum mechanics

Special relativity

( , , , )a ct x y zµ=

( ) ( , , , )a a a a c t x y zµ µ µ

+ ! " = ! = ! ! ! !

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

( ) ( ) ( ) ( ) ( )3

2 2 2 2 20 1 2 3

, 0

a a a a g a a a a aµ ! µ

µ! µ

µ ! =

" # " # " # " = " " = " " = "$

( 1, 1, 1, 1)g diag g µ!

µ! = + " " " =

• Physics invariant under all transformations that leave all such distances invariant :

Translations and Lorentz transformations

Lorentz transformations :

!3

0

x x x xµµ µ ! µ !

! !

! =

= "#$ $ =%! !g x x g x x g gµ ! µ ! µ !

µ! µ! µ! " # "#$ $%= =

Solutions :

• 3 rotations R

1 0 0 0

0 cos 0 sin

0 0 1 0

0 sin 0 cos

! !

! !

" #$ %$ %$ %$ %$ %&' (

• 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)

0 0( , ), ( , )A A A B B Bµ µ= =

4 vector notation

0 0( , ), ( , )A A A B B Bµ µ= ! = !

contravariant

covariant

0 0. .A B A B A B A B A B

µ µ

µ µ= = = !

A g A A g A! µ µ!

µ µ! != =

( , )ct x xµ

!

4 vectors

( , ) ( , )c t c t

µ

µ

! !! = "# ! = #

! !

( , )Ep p

c

µ!

p iµ µ! "!

22 2p p E p

µ µ

µ µ! = ! " ! # $ $!

The Klein Gordon equation (1926)

Scalar field (J=0) : (x)!

2

2

2 2 2

tE m m

! ! !"

"= + # $ +% =

2 2p

Energy eigenvalues 2 1/ 2( ) ???E m= ± +2p

2 2m

µ

µ ! ! !"# # = " =!

(Natural units)

Physical interpretation of Quantum Mechanics

Schrödinger equation (S.E.)21

20

t mi

! !"

"+ # =

2

=! "

* *( . .) ( . .)i S E i S E! !" continuity 0 eq. .t

!"

"+# =j

“probability current”

* *

2( )i

m! ! ! != " # " #j

“probability density”

Klein Gordon equation

**( )

t ti

! !" ! !# #

# #= $

2., 2

ip xNe E N! "#

= =

32

V

dV d x E! != =" "

* *( )

( , )

i

jµ

! ! ! !

"

= # $ # $

=

j

j

0

. 1

2

ip x

pp V

f e±=

!

Normalised free particle solutions

2

2

2

tm

! ! !"

"# +$ =

2

Physical interpretation of Quantum Mechanics

Schrödinger equation (S.E.)21

20

t mi

! !"

"+ # =

2

=! "

* *( . .) ( . .)i S E i S E! !" continuity 0 eq. .t

!"

"+# =j

“probability current”

* *

2( )i

m! ! ! != " # " #j

“probability density”

Klein Gordon equation

**( )

t ti

! !" ! !# #

# #= $

2., 2

ip xNe E N! "#

= =

* *( )

( , )

i

jµ

! ! ! !

"

= # $ # $

=

j

j

0

. 1

2

ip x

pp V

f e±=

!

Normalised free particle solutions

2

2

2

tm

! ! !"

"# +$ =

2

Negative probability?

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

•

Feynman – Stuckelberg interpretation

( 0)E!+

> ! ( 0)E!"

<

iEte! ( )( )i E t

e! ! !

time

space

Two different time orderings giving same observable event :

1 2t t

! ! !+ + +""# ""# 1 2

t t

! ! ! ! !+ + " + +##$ ##$

But energy eigenvalues 2 1/ 2) ?( ??E m= ± +2p

1t

2t