Download Pt-Symmetric Scarf-II Potential :an Update

XVII European
Workshop on
String Theory
2011
de
Italy, Padua
By:Farrin Payandeh
University of Payame Noor,
Tabriz,Iran
September
2011







Dirac’s equation and the sea of negative
energy.
Standard quantum field theory leading to the
infinites.
Renormalization : Sweeping the infinites
under the rug.
EPR Paradox and Antiparticles.
Quantum Mechanics in complex spacetime.
The Miracle of creation.
Conservation of Angular momentum(a.k.a.
“The photon”).
2
 Neglecting
negative
Energies:
“At what cost?”
3
4
Case Study
P. A. M. Dirac: Bakerian Lecture ,
Proceedings of the Royal Society A (1941)
4

Quantization of the minimally coupled scalar field and calculating
the gravitaton two point function in de Sitter spacetime:

Breaking of the covariance.
(a)
Appearance of the infrared divergence.
(b)
Covariant Quantization adding Negative Frequency States and
Free of divergence graviton field operator in krein Space
Field operator in Krein space:
( x ) 
1
(p ( x )  n ( x ))
2
Krein QFT calculations in Minkowski spacetime
Two point function in Krein space
iGT ( x, x)  0 | T ( x) ( x) | 0 
J1 ( 2m 2 0 )
1
m2
GT ( x, x)  Re G ( x, x)    ( 0 ) 
 ( 0 )
8
8
2m 2 0
P
F
2 0   ( x   x )( x  x )  0
Feynman propagator
G ( x, x)  
P
F
 J ( 2m 2 )  iN ( 2m 2 )  im 2
K1 ( 2m 2 ( 0 ) )
d 4 k e ik.( x  x)
1
m2
1
0
1
0
  2  ( 0 )
   ( 0 ) 
 ( 0 ) 
(2 ) 4 k 2  m 2  i
8
8

 4
2m 2 0
2m 2 ( 0 )
The divergence appears in the imaginary part of this eq. and the real part is convergent :
m2
1
P

lim Re GF ( x, x ) 
  ( 0 )
 0 0
16 8
lim
z 0
,
J1 ( z ) 1
N ( z)
2 1

, lim 1

z 0
z
2
z
 z2
lim Re GFP ( x, x)  0
 0 
, lim
z 0
K1 ( z ) 1
 2
Z
z
z  (1) s  z 
J1 ( z )  
2 s 0 s!( s  1)!  2 
z 2
N1 ( z )  2 J1 ( z ) log 
2 z
K1 ( z )  

2
2s
J1 (iz )  iN1 (iz )
10
Divergent Integrals of QED in
Krein Space
General form of divergent integrals of QED is:
and using integral representation of the causal propagator in momentum space:

i
is ( k 2  m2  i )

ds
e
k 2  m 2  i 0
we can see that the real part is convergent:
(1)
Convergent expressions of vacuum polarization and vertex graphs in krein space
will be:
†
  ( p, q, p)   p ( p, q, p)    0  p ( p, q, p)   0
13
14
15
16
17
18
Negative-energy modes as a candidate solution to some
problems in quantum physics?
Quantum Field Theory with the N.F.S. ,
an answer to Feynman reply , “ A
Nobel prize for sweeping the rushes
(infinties) under the rug” ?
20
21
22
23
GHOSTBUSTING:
Case Study
Reviving
quantum
P. A. M. Dirac: Bakerian Lecture ,
Proceedings of the Royal Society
A (1941)
theories
that were thought
to be dead
Thank you for your
kind attention
24
24