Preliminary: some notation from SSP 06: the fluctuation/dissipation theorem, which motivates us to pursue Green functions, R A( , ) 2 Im G ( , ); iG ( , ) A( , ) 1 n F ( ) ; iG (, ) A( , ) n F ( ); (1.1) † 1 1 n n c c iG ( , 0) i G d A d ( , ) ( , ) ; 2 2
First: recall the solution to SSP 17 - 098 - pr 04 - localized field position operator for electrons is,
(x)
(x) (x) ( x) † 3 (x) ( x) ( x x) ( x) d x vac
Pr. 5.4
... ( x x) † ( x)
vac
vac
(1.2)
Time: introduce † (xt ) (xt ) , and consider expectation values,
expectation-value (xt ) † (xt )
(xt ) † (xt ) Tr (xt ) † (xt );
;
(1.3)
† † † Example: unperturbed ground state of Fermi gas. Then, e h h e , and,
(1) † (1) (xt ) † ( xt ) e (1) †e (1) †h (1) h (1) e (1) h (1) †h (1) †e (1)
(1.4)
Time ordering: you have either the Wick or Dyson D yson time-ordering operator,
(1) (1) (1) (1)(t t ) (1)(1)(t t ); T (1) (1) (1) (1)(t t ) (1) (1)(t t );
P
†
†
†
†
†
†
(1.5)
Time ordered/causal Green functions: The Wick time ordering operator T is used to define the one particle Green functions, G
G(1;1) i
T
(1) (1) †
i (1) † (1) (t t ) i † (1) (1) (t t )
(1.6)
Two important elements: (1) an extension to 2 particles, and (2) special case of Galilean invariance, K
K (1; 1; 1; 1)
T
(1)(1)
†
(1) † (1) ???;
Galilei invar. G G (1) i Ga
T
(1)
†
( 0) ; (1.7)
Example: consider one-dimensional system of non-interacting fermions of mass m , contained in length L , in the true vacuum state. Compute the one-particle Green function. Show, as an a n intermediary, that it is causal.
Expand the field operator for a particle pa rticle of mass
(1)
k
ck (t )e
ik x
k
ck e
m
in a free-particle basis,
i ( k x k t )
(1)
† i (k x k t )
†
k
c ke
1
; k
2
k
2m
;
(1.8)
2
Then for the vacuum state the (Galilei-invariant ) one-particle Green's function , beginning with (1.6) and losing no generality by regarding t 0 , is,
G(1) i
(1) (0) †
1
Meaning: we lose no generality in saying G
2
The Green function’s containment of causality-physics causality -physics can be, here, noted; send t 0 , and see how G ( x, 0
.
time to act, is nonzero only at the “infinitely“infinitely-local” x G ( x, 0
T
)i
vac | T
†
k
ck ck e
ik x
| vac
i
0: vac | T
k
( kk ck† ck )eik x | vac i ( x)
) , having zero having zero
G
G (1; 1) G (1; 0) i
(1) vac | T
vac | T
i (1) † (0) (t 0) i
†
( 0) | vac i † (0) (1) (0 t ) i (1) † (0 ) (t 0) †
kk
ckc k e
i ( k x k t )
In (1.9), you have T 1 since time-ordering has already been effected, and appended, due to orthogonality of the temporary intermediary), and write, G
G (1; 0) i
vac | T
k -states.
ck ck kk e
iL exp[i(k x 2 m )]dk 2
k t
†
vac | ck ck | vac
can have a kk
Let us introduce a bounded-interval of length L (as a
†
kk
(1.9)
| vac
i ( k xk t )
2
2 mL / te
| vac 2
i [ 1 mx 2
/ t 34 ]
i
k
1e
i ( k x k t )
i e
i ( k x k t )
L dk
(1.10)
;
Show that this Green function (1.10) is a solution to the time-dependent Schrödinger equation, HG i t G . HG
2
2m
2
d G dx
2
2
e
2 mL2
i 34
2m
t
2
d
dx
2
e
2 i 1 mx 2
t
(1.11)
We also have,
2 iG0 2 iG0 ( x,0 ) 2 i
k F
e
ikx
dk
k F
e
ikx
vac |
dk
†
kk
ck ck e
2 ( x)
i ( k x kx )
k F
k F
i kx
e
dk
(t 0 )
k F
2 ( k
F
†
kk
ck cke
i ( k x kx )
x ) 2sinc k F x
(0 t ) | vac
t 0
(1.12)
;
Then,
2 iG0 ( x, 0 )
k F
k F
e
ik x
dk
2k F sinc kF x ;
(1.13)
