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Z e t a F u n c t i o n R e g u l a r i z a t i o n
N i c o l a s M R o b l e s
D e p a r t m e n t o f T h e o r e t i c a l P h y s i c s
I m p e r i a l C o l l e g e L o n d o n
F e b r u a r y 2 0 t h 2 0 0 9
S u p e r v i o r : D r A r t t u R a j a n t i e
S u b m i t t e d i n p a r t i a l f u l l l m e n t
o f t h e r e q u i r e m e n t s f o r t h e d e g r e e o f M a s t e r o f S c i e n c e i n
Q u a n t u m F i e l d s a n d F u n d a m e n t a l F o r c e s o f I m p e r i a l C o l l e g e L o n d o n
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A b s t r a c t
n o z l z o n d n n m m n n d ( l d o F
e n o d n z n o n o o l o m o n n o n o o o n
n d m o n m o n o l l o n n m m n n d d n n n E
o n l o n m ( l d n n o o J o n d n d m n n o
o o n d n d ' n l d o o I E l o o o d F
n n o o n l n v n n o 4 o o l o n o l l o n o J o d n d l n d n d o n n o n
o m o n o l l o n ( l d o o m d n d l n d n l m o
m F
l n k n z n o n D k n l n d w l l n n o m l n d
n d l n n z n d d m n o n l l z o n o n n d l l
d d o
4 o F
p m o D n o m o n o n I E l o o ' v n n l o l l d F
p n l l D g m ' n o d d F
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I
o m l
1 + 1 + 1 + 1 +
=
1
2 o o m n s o m e n F
e n o n m o
@ F F F A t h e u s e o f t h e p r o c e d u r e o f a n a l y t i c c o n t i n u a t i o n t h r o u g h t h e z e t a f u n c t i o n r e q u i r e s a
g o o d d e a l o f m a t h e m a t i c a l w o r k . I t i s n o s u r p r i s e t h a t [ i t ] h a s b e e n o f t e n a s s o c i a t e d w i t h
m i s t a k e s a n d e r r o r s F
i F i l z l d n d e F o m o
W e m a y - p a r a p h r a s i n g t h e f a m o u s s e n t e n c e o f G e o r g e O r w e l l - s a y t h a t ' a l l m a t h e m a t i c s
i s b e a u t i f u l , y e t s o m e i s m o r e b e a u t i f u l t h a n t h e o t h e r ' . B u t t h e m o s t b e a u t i f u l i n a l l m a t h -
e m a t i c s i s t h e z e t a f u n c t i o n . T h e r e i s n o d o u b t a b o u t i t F
u z z o w l n k
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C O N T E N T S P
C o n t e n t s
1 I n t r o d u c t i o n 3
2 I n t r o d u c t i o n t o t h e R i e m a n n Z e t a F u n c t i o n 9
P F I q m m p n o n (s) F F F F F F F F F F F F F F F F F F F F F F F F F F F W P F P r z p n o n (s, a) F F F F F F F F F F F F F F F F F F F F F F F F F F I T P F Q e n l o n n o n n d n o n l o n o (s, a) F F F F F F F F F P H P F R f n o l l n m n d l o (0) F F F F F F F F F F F F F F F F F F F F P R P F S l o (0) F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F P T P F T o l m m n o n (m)(s) F F F F F F F F F F F F F F F F F F F F F F F F P V P F U v n o
(s) n o n F F F F F F F F F F F F F F F F F F F F F F F P W P F V g o n l d n m k o n d o n o m n m n d z o o
(s) Q H
3 Z e t a R e g u l a r i z a t i o n i n Q u a n t u m M e c h a n i c s 3 7
Q F I n l o m o n o l l o F F F F F F F F F F F F F F F F F F F F Q U
Q F P o l o n o l F F F F F F F F F F F F F F F F F F F F F F F F F F F Q W
Q F Q o o n o n n o n F F F F F F F F F F F F F F F F F F F F F F F F F R I
Q F R l z o n o l o n o o o n o n n o n F F F F F F F F R Q
Q F S e l n o l o n F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F R S
Q F T p m o n o n n o n F F F F F F F F F F F F F F F F F F F F F F F F R S
Q F U l z o n o l o n o m o n o n n o n F F F F F F F R V
4 D i m e n s i o n a l R e g u l a r i z a t i o n 5 1
R F I q n n n o n l n d o l m l d n n o o J S I
R F P p n o n l n D o n n d o n l n d l l ( l d
cl(x) F F F F S R R F Q h o n o
4 o n l
cl(x) F F F F F F F F F F F F F F F F F F F F F F F S W R F R E l o n o d m n o n l l o o n l F F F F F F F F F F F F F F F F F F F T S R F S i n o n o Z n i l n m F F F F F F F F F F F F F F F F F F F F F T V
5 Z e t a R e g u l a r i z a t i o n i n F i e l d T h e o r y 7 1
S F I r k n l n d w l l n n o m F F F F F F F F F F F F F F F F F F F F F F F U I
S F P h o n o
4 o n l
cl n l z o n F F F F F F F F F F F F U Q S F Q g o l n o n n F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F U T
S F R o n n o n n ( l d o F F F F F F F F F F F F F F F F F F F F F F F F U V
S F S r m l m F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F V Q
S F T i l n o n d d m n o n l l z o n n 4 o F F F F F F F F V V
6 C a s i m i r E e c t 9 4
T F I i m n l F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F W R
T F P
l o F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F W R
T F Q i m n l d n F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F W T
7 C o n c l u s i o n 9 7
A A p p e n d i x 1 0 1
e F I q n l z d q n n l F F F F F F F F F F F F F F F F F F F F F F F F F I H I
e F P q m n x m F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F I H Q
e F Q w l l n n o m n d n o n o log (s + 1) F F F F F F F F I H U
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C O N T E N T S Q
B R e f e r e n c e s 1 1 2
f F I f o o k F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F I I P
f F P F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F I I P
f F Q s n n e l F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F I I P
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1 I N T R O D U C T I O N R
1 I n t r o d u c t i o n
l n o n n o n n l n d m l l n m m l n d o l
o o s n d n F
s n d d D d o o l l d l n o n X h D v n d D
f l D r n k l D r m n o n E o n m E n d n n n d o m o
m o d n o n F m o o m n n o n n E n m m
n o n n d o o l o n o d ' n l o n m o d l l m F
m n n z n o n d ( n d o o m l l s V
(s) :=n=1
1
ns=p
1
1 ps
o d ( n o n l d o (s) > 1 n d n n l l l o n n d o o l o m l l n s = 1 m l o l d I F s l l o n o d n o l l m p F
s n o o l o n o n l l m o d d ' n l o n I
o m o l n o n m o n n l m n n F
d o n l l D m n n n o n d l o n n n l n m o n d l l n d o n o m n m F e n d d m o l
n o n l l o m l l n l m o m m n d m o l
l d d F
l l n d n n d n o n F l k o o o n l n o n s m l g o l l l o m o o n o d n o o l F
o o n l l o o d ( n o n E o m E o o l n d o n l n l E
o o o m l n l X o n n D n l o n n o n D d l l
n d p o n d w l l n n o m F
l m d d o n D o D o n l o n o n l l n d o o d o n n d o m m n d l o n n o n
n n m o n d
n o n o n l ) F
n o n l o n o l m o n
(s) = 2(2)s1(1 s)sins
2
(1 s)
l l o m l
(0) = 12
, (0) = 12
log(2)
l l d m n d n F y o D n d n o o l ( o n F n o n n n
(s) =1
1 21sn=1
(1)n+1ns
.
o m n o n o n F r o D n (s) < 1 r o n l o n n o n n d o o 12 s 0 n (0) = 12 n n F
h n l o n o m F r k n R D F i l z l d D F
y d n o n d e F o m o @ i y A Q l n d o l z o n n ( n l
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1 I N T R O D U C T I O N T
m n m = iTFe l l m o o n o n n d d o m o n l l d r z n o n d ( n d V
(s, a) =n=0
1
(n + a)s
a = 0, 1, 2, F n m o n o d m n d n E o m m o n l o n l l n d q m n n n m n d o l l l n n d l o n l n o
m n d d o o m o n n o n o m o n F
m n l d n o n o o n n o n
TreH =
dd
|eH| e. n d o n j q m n n n m l d S
Z() = 2n=1
1 +
(2n 1)
2
= 2 cosh 2
.
s n n m m n l n l n l l k n o n o m E o n o
A l l o m d o ( l d o l F k n o l
A n o n o o o o n d d n o n D o n n n o o n
boson(s) =n=1
n
2s=
2s(2s).
n m o n
fermion(s) =
k=1
2(k 1/2)
s
=
2s
(s, 1/2),
l l m k o o m l o
(0) n d (0) o n n o n d d
n o n l n m o m o n l l o o n n n n E
l z o n n n m o n o o n o o m l n l D n m l
n l o n n o n F
f d m n n o d ' n l d o o n ( l d o o
n l n n o o J d o d o l o l n
o n o n F o o m l m n o p G e p o o m w F
o n l ( l d o l l o n d 4 o m o n o n o E l z o n n l d o m o o m l ( l d o n d n o n o Q F
l l k o n n d o d o p n m n d m n o l F
i d l l o o l d l o d n d n o n l l o
n o n n d n ( l d o F l l l m n o n o o m l o d n o n P D n d l o d n o n Q n d o l n d
n o n R F p m o D l l n l o n F
n m ( l d o o l ( l d n n o n n l o J l l
d d F s l l o n o n n n o n l o o l m l d T
ZE [J] = eiE[J] =
|eiHT|
J=
D exp
i
d4x(L[] + J)
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1 I N T R O D U C T I O N U
n n n i l d n m Q D U
ZE [J] = NE exp {SEuc[(x0), J]}
det
( + m2 + V[(x0)])(x1 x2)
s l n l l d m n n n d n o m n o l d o n n d o
n D n m l d ' n l d o o F f k n o n l o
o
4 o D o o o m
AF T
= 2 + m2 + 2
20(x),
s n o n o n m m n l D l n k o
A n o n m m o o m l n d m k o k n l F o o n d o n d o n D
o l o n o o n
AxGA(x, y, t) = t
GA(x, y, t)
n l o d
GA(x, y, t) =
x|etA|y = n
etnn(x)n(y).
n n l o o o n d n o o o n l n o F o o l o n
d4xGA(x,x,t) =n
etn = TrGA(t)
n d l n k A n o n o m o m w l l n n o m R
A(s) =n
sn =1
(s)
0
dtts1TrGA(t).
l l o k n n o o n ( o d n m o o n D o n l n
v n n o ( l d o n o m l z d
V(c l
) =
4!4
c l
+24
c l
2562
log
2c l
M2 25
6
c l
l l ( l d d ( n d n m o o J n d o n d
cl(x) = |(x)|J =
J(x)E[J] = i
J(x)log Z.
l l n d o o l n o n n o l n m o l d n d n n d
l o n n l o n ( l d o n d l m n F p m o D l l
o U
log Z =1
2A(0) =
1
2 log(1 e).
o n n o n o p m o n o l l o F n m o d m o
o n
log Z =1
2B(0) = V
M3
12+
2
903 M
2
24+
M4
322
+ log
4
+
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1 I N T R O D U C T I O N V
B = 2
t2 + 22 + m2 + 2 20
2
n l m o m
0 F n l l l o o o ' v n n ' d F
e m n o k o m k n l d m l o E
l z o n n l l d m l n m o n d d m o d
o n E l o o n o n o F r o D l l d o n o n
o n l n n d l l o l l o o 4 o F
i y d o n l m o n o o o k o n g m ' F g o n n l
n o d o n o l l o n d o n g m n l I W R H F l l o n E
m o 9 l d 9 o F
o n l o n o n n l ( o n o n o m o j d d n n d
l l d ( n n o n o n l l o o n l o o o n o n n n d o n o z o o n o n D l z o n n d d m n o n l l z E o n F
n d o n n o m o m l d d n o ( n n o n n d m k
o l d o n l m o l E o n n d F n n o d d n n d d l
D n l d n m n d n l o d F y l l D l
o n
l z o n F m n o o d o n n q o n d
n D u l n D m o n d n d i l z l d D y d n o D n d o m o F
s n m o k n o l d o D o n l E o m o m ( l d o
o n o o p G e p o m w n p p p F
s n n o j o o l o m d ' n o n n l D
l n o o n d n o n o m l n d l o n l d
o o n d o n n o m l F
p n l l D m k s n o n l o ( n d n l n o m d
o n n n o n l l o n n n o n n d m l o l m n E l F h ' n n o n o m n o l n n m m n o m n o n n o n F e d o D m n n (s) n o n d n o o n E o n n o n D r z (s, a) n o n n d d o m o n o n n o n F
m n n n o n o o n r z n o n m o n
s o l d n n n j o o n o n o n o o o n d n l D n d k n d o k n o l d n o
l n l n o n F p o n n D o o n n o d n o n D n d o n n o n n o m d n k n o n o
n o n D o n o
n d D k n o n o o o n m n l ( o n o o o
n o n F
l d m n n o l m l l n o n l z o o m n n
n o n o o m ( 12 + it) t l F e n o o o n s n o n l
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1 I N T R O D U C T I O N W
o ( n d o l d o o n d n o o n n o n o o m ( 12 + it) n d n o o n o n o o o F o l d m n 9 n l o n 9 o m n n o n d d o l n o n F
i p i x g i
E I t F f n g o n T h e R i e m a n n H y p o t h e s i s
E P r F f F q F g m n d h F o l d T h e I n u e n c e o f R e t a r d a t i o n o n t h e L o n d o n - v a n d e r
W a a l s F o r c e s
E Q i l z l d D y d n o D o m o D f n k n d n Z e t a R e g u l a r i z a t i o n T e c h n i q u e s w i t h
A p p l i c a t i o n s
E R n r k n Z e t a F u n c t i o n R e g u l a r i z a t i o n o f P a t h I n t e g r a l s i n C u r v e d S p a c e t i m e
E S r n u l n I n t e g r a l s i n Q u a n t u m M e c h a n i c s , S t a t i s t i c s , P o l y m e r P h y s i c s a n d
F i n a n c i a l M a r k e t s
E T w l i F k n n d h n l F o d A n I n t r o d u c t i o n t o Q u a n t u m F i e l d T h e o r y
E U m o n d D F i e l d T h e o r y : A M o d e r n P r e m i e r
E V i d d m T h e T h e o r y o f t h e R i e m a n n Z e t a F u n c t i o n
E W n n T h e Q u a n t u m T h e o r y o f F i e l d s
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2 I N T R O D U C T I O N T O T H E R I E M A N N Z E T A F U N C T I O N I H
2 I n t r o d u c t i o n t o t h e R i e m a n n Z e t a F u n c t i o n
2 . 1 T h e G a m m a F u n c t i o n
(s)y n n m m D m o n l o d ( n n o n n m o n n l d E
n d n o n m n o o F n o n o n F d o n l l D n o d o d o m E d n d n
n l F o o n o n o d n o n o l l o o m o n o m l n l d l d p n d m P D l l m T n d
k n d o n U F
d o n o o n s = + it n o d d m n n n I V S W n d o m n d d n l o n o n F v ( d ( n i l n o n F
D e n i t i o n 1 n l
(s) :=
0
dtts1et@ P F I A
l l d ( n d n d d ( n o l o m o n o n n l o m l l n D
Re(s) = > 0F
( l m m n l o l n o n o l l o
L e m m a 1 p o n n N (n) = (n 1)!FP r o o f F x o (1) :=
0
dtet n d n o n
(s + 1) =
0
dttset = tset|0 + s
0
dtts1et = s(s)
o n s n l E l n F x o D o n o n n D
(n) = (n 1)(n 1) = (n 1)!(1) = (n 1)! @ P F P A o l o d F
s n o d o o m l o n o n n d o n d o m o m o E n o n n o l o m l l n F
L e m m a 2 v cn n Z+ n o o m l n m m n=0 |cn| o n F p m o D l S = {n|n Z+ n d cn = 0} F n
f(s) =
n=0
cns + n
o n o l l o s CS n d n o m l o n o n d d o CSF n o n f m o m o n o n o n C m l o l o n n S n d d
n s=nf(s) = cn o n n SFP r o o f F v ( n d n o n d F s |s| < RD n |s + n| |n R| o l l n RF o D |s + n|1 (n R)1 o |s| < R n d n RF p o m n d d o
n0 > R
n=n0
cns + n
n=n0
|cn||s + n|
n=n0
|cn|n R
1
n R
n=n0
|cn|.
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2 I N T R O D U C T I O N T O T H E R I E M A N N Z E T A F U N C T I O N I I
e
n>R cn/(s + n) o n o l l n d n o m l o n d k |s| < R
n d d ( n o l o m o n o n F s o l l o n=0 cn/(s + n) m o m o n o n o n d k m l o l o n o S o n d k |s| < R F D
n=0 cn/(s + n) m o m o n o n m l o l o n n S n d o n n S n
f(s) =cn
s + n+
kS{n}
cks + k
=cn
s + n+ g(s),
g o l o m o nF p o m d n d d s=nf(s) = cn F o n l d o o F
i d l m m n o o n o n d n o n n d F
T h e o r e m 1 n o n n d o m o m o n o n o n o m l l n F s m l o l 0, 1, 2, 3, F d o n
ress=k
(s) =(1)k
k!, @ P F Q A
o n k Z+ FP r o o f . v l n o n
(s) =
10
dtts1et +
1
dtts1et,
o n d n l o n o n o m l s n d n n n o n F v n d
o n n l n o n n ( n l
10
dtts1et =
10
dtts1k=0
(1)kk!
tk =k=0
(1)kk!
10
dttk+s1 =k=0
(1)kk!
1
s + k,
o o n l d o
s C o n n l n o n n n d o n n o m l o n o m o o m l l n F n o n n n o n n o m v m m P n d D F F
(s) =
1
dtts1et +k=0
(1)kk!
1
s + k, @ P F R A
o n s n l E l n F f v m m P D r d ( n m o m o n o n
o n o m l l n m l o l 0,
1,
2,
3,
F d n
d l o n o l m m F
T h e o r e m 2 p o s C n
(s + 1) = s(s). @ P F S A
P r o o f . o l l o d l o m v m m I n d o m I F
E d m n o n l n o n o n o n l o o k l k e n o m o n n o n l d o n o n D n d l o d o d i l D
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2 I N T R O D U C T I O N T O T H E R I E M A N N Z E T A F U N C T I O N I P
p P F I X |(x + iy)| o 5 x 3 n d 1 y 1
f n o n o d o d l o o l l o F v (p), (q) > 0 n d n n l d ( n n o n D m k n o l t = u2 o o n
(p) =
0
dttp1et = 2
0
duu2p1eu2
.
s n n n l o o o m
(q) = 2
0
dvv2q1ev2
w l l n o o
(p)(q) = 4
0
0
dudve(u2+v2)u2p1v2q1,
n d n o o l o o d n u = r cos , v = r sin , dudv = rdrd
(p)(q) = 4
0
/20
drd er2
r2(p+q)1 cos2p1 sin2q1
=
2
0
dr er2
r2(p+q)1
2
/20
d cos2p1 sin2q1
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2 I N T R O D U C T I O N T O T H E R I E M A N N Z E T A F U N C T I O N I Q
= 2(p + q)
/2
0
d cos2p1 sin2q1
n l n m l ( d n z = sin2
2
/20
d cos2p1 sin2q1 =
10
dz zq1(1 z)p1.
x d ( n
B(p,q) :=
10
dz zp1(1 z)q1
o Re(p), Re(q) > 0, n d d n
B(p,q) = B(q, p) =(p)(q)
(p + q).
d n o B f n o n F w o o D 0 < x < 1
(x)(1 x) = (x)(1 x)(1)
= B(x, 1 x) =1
0
dz zx1(1 z)x.
l l l n l n o o n o D ( n d o m k o n
l n z = u/(u + 1) l d
1
0
dz zx1(1
z)x =
0
du
(u + 1)2
ux1
(u + 1)x1 1
u
u + 1x
=
0
duux1
1 + u
.
L e m m a 3 p o 0 < y < 1 @ I F I H A
0
duuy
1 + u=
sin y.
@ P F T A
P r o o f . v k o l o n o D o m l d n o m l l n
l o n o l F y n o n d ( n n o n
f(s) =sy
1 + s
m n o sy l o H o n d o F p m o D n o n f ( o d o l s = 1 d eiy F p P F P l o F n o o n n o n l o n d d n p P F P X o
l o n d o o m > 0 o R D n l o n l CR o d R n d o n D n l o n d o o m R o
n d n d o n d o n
l
C o d l o n d o n F e n l o n o g d o m
2ieiy =
R
duuy
1 + u+
CR
dzzy
1 + z e2iy
R
duuy
1 + uC
dzzy
1 + z.
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2 I N T R O D U C T I O N T O T H E R I E M A N N Z E T A F U N C T I O N I R
p P F P X C = CR C [, R] [R, ]
n d o n l o n d o m F x o o
o l l o n
|zy| = |ey log z| = eyRe(log z) = ey log |z| = |z|y, zy1 + z |z|y|1 + z| |z|
y
|1 |z|| ,
n d n l n m d
CR
dzzy
1 + z
2R1y
R 1 R0,C
dzzy
1 + z
21y
1 0 0;
o l
(1 e2iy)0
duuy
1 + u= 2ieiy .
m E o o n ( n l l
(eiy
eiy)
0
duuy
1 + u
= 2i
0
duuy
1 + u
=
sin y
.
o l m o l m m F
v n o m k o n l d n m k F
T h e o r e m 3 @ i l ) o n p o m l A p o l l
s C o n
(s)(1 s) = sin s
. @ P F U A
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2 I N T R O D U C T I O N T O T H E R I E M A N N Z E T A F U N C T I O N I S
P r o o f . l m m j o d n n
(x)(1 x) = sin (1 x) = sin x ,
o 0 < x < 1 F r o D o d o o n o m o m o D n o d o m F
C o r o l l a r y 1 y n
1
2
=
0
dtt1/2et =
dtet2
=
.@ P F V A
P r o o f F o l l o n s = 1/2 n i l ) o n o m l @ P F U A F
T h e o r e m 4 n o n n o z o F P r o o f . n s sin(s) n n n o n D r o o m Q n o z o D o (s) = 0 o n l n s (1 s) o l F r o D d o D o l o 0, 1, 2, 3, o o l l o (1 s) m o l 1, 2, 3, F f o l o m l D (n + 1) = n! = 0 n d o n o z o F
s n n l l o n n d o n o n i l o n n F v ( d ( n n d o n F
L e m m a 4 s sn := 1 +12 + + 1n log n, n limn sn F l m l l d
i l o n n F P r o o f . g o n d tn = 1 +
12 + + 1n1 log n o m l l D n o
n 1 o n n m n n m n d l o n1 dx x1 F o tn n n F n tn =
n1k=1
1
k log k + 1
k
, limntn =
k=1
1
k log
1 +
1
k
.
o n o n o o o n n n
0 1 o n n n ( n o d n z o o n l o n o o n z o F v 1, 2, z o o (s) F s o l l o o m n o n l o n (s) n o z o o < 0 o l o n F n n o n l o n @ P F P P A (1 s)
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2 I N T R O D U C T I O N T O T H E R I E M A N N Z E T A F U N C T I O N Q R
n o z o o < 0 D n d sin(s/2) m l z o s = 2, 4, n d (1 s) n o z o F
l D l z o o (s) d o n o o o n d o z o o (s) n n l l d o l o (s/2)F o D o l l o (s) n o z o o < 0 n d o > 1 F z o 1, 2, l n 0 1 F r o D z o o (s) l o n s(s 1)(s/2) n o z o n o n s = 1 n l l d m l o l o (s) F o
(s) n n ( n n m o z o 1, 2, n 0 1 n d n
(1 21s)(s) =n=1
(1)n+1ns
> 0
o 0 < s < 1 n d (0) = 1/2 = 0 n (s) n o z o o n l n H n d I D F F z o 1, 2, l l o m l F r o m l o o n F z o o m n o n j X n (s) l o n l n d z o o 1 n o n l o n n o 1 F s = + i n 1 = 1 + iF g o n n l z o l o n = 1/2 o o n m m l o l n F
p P F W X |(x + iy)|1 o 2 x 2 n d 0 y 50X z o o m o l
m n n k o o o o o m o n o n m N(T) o z o o 0 < t < T
N(T) =T
2log
T
2 T
2+ O(log T). @ P F S I A
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2 I N T R O D U C T I O N T O T H E R I E M A N N Z E T A F U N C T I O N Q S
n l o d o n w n o l d F o n m o m o
o o ' v n n I E l o o o d @ o n S F T i o n @ S F I V W A A F
m n n n n n d o n j d l l z o o n l n Re(s) = 12 Fs n I W I R r d o d n n ( n n m o z o o n l l n Re(s) = 12 d o n o m n a l l z o l o d F
m n n 9 m n o m l m n o I V S W n l o n n o n o n o n n d o o @ o A o n o n l o n F p m o D o
n n n o p o n o m o d
log (s) = s
2
du(u)
u(us 1)
n l d m o o o l l o n m k l l o d n l o m l o (x)
(x) = R(x) 12
R(x1/2) 13
R(x1/3) 15
R(x1/5) + 16
R(x1/6) + =n=1
(n)n
R(x1/n)
(n) w o n o n d ( n d H n d l m D I n o d o n n n m o d n m n d 1 n o d o n o d d n m o d n m n d
R(x) = Li(x)
:()=0
Li(x) log 2 +x
dt
t(t2 1) log t ,
m d 9 n 9 D F F 1 o
Li(x
) =
Im>0 Li(x
) + Li(x
1
).
s n I V W T r d m d n d d l l l o n o d D n d n d n l n d l m o m l n E
o l D m n m o m o n l n o
(1 + it) = 0 D F F n o z o o n Re(s) = 1 F m n n 9 r o l n o
(x) x
2
dt
log t= O(x1/2 log x)
x Fp n l l D r d m d l o o d o d n o n d @ o l d o o A
m n n
(s) =eHs
2(s 1) s2 + 1
:()=0
1 s
es/
@ P F S P A
H = log 2 1 /2 F (s) n o n l o d m o d n o n
(s) =1
2
1 s
.
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2 I N T R O D U C T I O N T O T H E R I E M A N N Z E T A F U N C T I O N Q T
v E d ( n o n o (s)
(s) = s
2
(s 1)s/2(s) m n n 9 n o o n o d n o n
(s 1) := (s). l o m d o (s) o n o n n d
d
dslog
1 s
=
1
s
n d o n o n d
d
dslog
s
2 1
2log +
1
s
1
+(s)
(s).
i l n o o n s = 0 l d
1
=
1
2+
1
2log + 1 log2
o Im>0
1
+
1
1
=1
2[2 + log4] @ P F S Q A
n (0) = (1) = F x o log4 m m l o n n l l o n l n n o m l z o n D n d n n o l n k n n m o n d
z o o
n o n F o m l n d o o m z o
F o m o
( z o = 12 + iti
p P F I H X |( 12 + it)| o 0 t 50
D
t1 = 14.134725t2 = 21.022040t3 = 25.010858t4 = 30.424878t5 = 32.935057
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2 I N T R O D U C T I O N T O T H E R I E M A N N Z E T A F U N C T I O N Q U
i p i i x g i
r o l l o d o o m
E I o m e o o l 9 I n t r o d u c t i o n t o A n a l y t i c N u m b e r T h e o r y o n o n o n o n n d n l o n n o n F e o o l n n o l l o
E Q D i s t r i b u t i o n o f P r i m e N u m b e r s s n m F
o n o n o n o l l o o m o @ n d A o n E P C o m p l e x A n a l y s i s p n d f m D m 9 T h e o r y o f F u n c t i o n s T n d
k n d o n 9 M o d e r n A n a l y s i s U F
m n o o d l o m n o m n n
n o n F m o o m E
n o n
E R s 9 R i e m a n n ' s Z e t a F u n c t i o n D n d E S i F g F m @ d h F F r E
f o n A T h e T h e o r y o f t h e R i e m a n n Z e t a F u n c t i o n F
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3 Z E T A R E G U L A R I Z A T I O N I N Q U A N T U M M E C H A N I C S Q V
3 Z e t a R e g u l a r i z a t i o n i n Q u a n t u m M e c h a n i c s
3 . 1 P a t h i n t e g r a l o f t h e h a r m o n i c o s c i l l a t o r
y n o ( n n m n n n o n o n n m n n l d l o m n o m o n o l l o Y ( l l k l n
o m n o n n o n o m o m o n o l l o F
l l o l l o i W F P Q I P o m k n n d o d P n d n l
d l o m n o m g P o u l n I F o n o o n E d m n o n l m o n
o l l o n
S =
tfti
dtL, @ Q F I A
v n n
L =1
2mx2 1
2m2x2. @ Q F P A
e k n o o m n o n l o o n m m n D n o n m l d
n o n l n l
xf, tf|xi, ti =
DxeiS[x(t)]. @ Q F Q A
m m o S D xc(t) D (
S[x]
x x=xc(t)
= 0. @ Q F R A
n o o d o n d o n o n d xc(t)F n d xc(t) l l j o o n n n o E m o n o m l d n d o (
i l E v n o n
xc + 2xc = 0. @ Q F S A
o l o n o o n o o n d o n xc(ti) = xi n d xc(tf) = xf
xc(t) = (sin T)1[xf sin (t ti) + xi sin (tf t)], @ Q F T A
T = tf ti F n l o l o n n o o n S
Sc := S[xc] =m
2sin T [(x2f + x2i )cos T 2xfxi]. @ Q F U A
e n n d d o n l l D n o n d
S[x] o n d x = xc o o n
S[xc + z] = S[xc] +
dtz(t)
S[x]
x(t)
x=xc
+1
2!
dt1dt2z(t1)z(t2)
2S[x]
x(t1)x(t2)
x=xc
@ Q F V A
z(t) ( o n d o n d o n
z(ti) = z(tf) = 0. @ Q F W A
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3 Z E T A R E G U L A R I Z A T I O N I N Q U A N T U M M E C H A N I C S R H
o ( @ Q F W A F n l n o n n l
T0
dt(z2 2z2) = T2
n=1
a2n
nT
2 2
. @ Q F I V A
s n o d o l l E d ( n d n o m o n m k n m o l
m o n d n o m o n F s n d d D p o n o m o n o m
y(t) o an m o o n o l n n o n F o k D k n m o m l o
N + 1 D n l d n o t = 0 n d t = TD o
N 1 n d n d n zk l F o D m an = 0 o l l n > N 1 Fx D o m o o n d n t o n F h n o
tk k m l n n l [0, T] l n o N n ( n m l D n
JN = detzk
an= detsin ntk
T . @ Q F I W A
l t o n o o l D o l F
o D l m k d o n o l o m n o n d o l m l d F
3 . 2 S o l u t i o n o f t h e f r e e p a r t i c l e
p o l D v n n L = 12 mx2
F g l l d d o l o n o
l n o n d n g P o u l n I l l n g Q o q o n d
n R F m l d o m d ( n o n r m l o n n n
H = px L = p2
2m, @ Q F P H A
o
xf, tf | xi, ti =
xf|eiHT| xi
=
dp
xf| exp(iHT)|p
p| xi
=
dp
2eip(xfxi)eiTp
2/(2m) =
m
2iTexp
im(xf xi)2
2T
@ Q F P I A
T = tf ti n o d o n d d n o d z o n o m = T /NF m l l o o o n o n l n o n m l d
xf, tf | xi, ti = limn
m2i
n/2 dx1 dxn1 exp
i
nk=1
m
2
xk xk1
2. @ Q F P P A
n o n o o d n o zk
= (m/(2))1/2xk
o m l d o m
xf, tf | xi, ti = limn
m2i
n/2 2m
(n1)/2 dz1 dzn1 exp
i
nk=1
(zk zk1)2
.
@ Q F P Q A
s n n d D o n d o n
dz1 dzn1 exp
i
nk=1
(zk zk1)2
=
(i)n1
n
1/2ei(znz0)
2/n. @ Q F P R A
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3 Z E T A R E G U L A R I Z A T I O N I N Q U A N T U M M E C H A N I C S R I
o n d o
xf, tf | xi, ti = limn
m
2i
n/2
2im
(n1)/21n
eim(xfx0)2/(2n)
=
m
2iTexp
im
2T(xf xi)2
.
@ Q F P S A
k n @ Q F P S A n o o n
xf, T | xi, 0 =
1
2iT
1/2exp
im
2T(xf xi)2
=
1
2iT
1/2eiS[xc]. @ Q F P T A
n n m o n l o n
eiS[xc] z(0)=z(T)=0
Dz expim
2
tf
ti
dtz2. @ Q F P U A x o D o m @ Q F I V A
m
2
T0
dtz2 mN
n=1
a2nn22
4T@ Q F P V A
n d n o m o n l o n o n l
1
2iT
1/2=
z(0)=z(T)=0
Dz exp
i m
2
tfti
dtz2
= limN
JN
12i1/2
da1 . . . d aN1 exp
im
N1n=1
a
2
nn
2
2
4T
. @ Q F P W A
x o o m o o l n q n n l 1
2iT
1/2= lim
NJN
1
2i
N/2 N1n=1
1
n
4iT
2
1/2
= limN
JN
1
2i
N/21
(N 1)!
4iT
2
(N1)/2@ Q F Q H A
n d o m o n o m l o t o n
JN = 2(N1)/2NN/2N/1(N 1)!
N @ Q F Q I A
t o n d n D d n n o l n JN o m n d o d n o F
v n o n o o n l o l m o o l m l d o m o n
o l l o F m l d
xf, T | xi, 0 = limN
JN
1
2iT
N/2eiS[xc]
da1 . . . d aN1 exp
i
mT
4
N1n=1
a2n
nT
2 2
. @ Q F Q P A
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3 Z E T A R E G U L A R I Z A T I O N I N Q U A N T U M M E C H A N I C S R P
e d d l D o o m o n o q n n l
n o m l
dan exp
imT
4a2n
nT
2 2
=
4iT
n2
1/2 1
T
n
21/2. @ Q F Q Q A
k m l d @ Q F Q P A n o m l l
xf, tf | xi, ti = limN
JN
N
2iT
N/2eiS[xc]
N1k=1
1
k
4iT
1/2N1n=1
1
T
n
21/2
= limN
1
2iT
1/2eiS[xc]
N1n=1
1
T
n
21/2. @ Q F Q R A
s n o n o d l o
limN
Nn=1
1
T
n
2=
sin T
T. @ Q F Q S A
d n o Jn n l d n o o m n d o l ( n l o n ( m n d o o n n n l n o J F n n l o o d ( n l l @ P F Q T A
xf, tf | xi, ti =
2i sin T
1/2eiS[xc]
=
2i sin T
1/2
expi
2sin T(x2f + x2i )cos T
2xixf . @ Q F Q T A 3 . 3 T h e b o s o n i c p a r t i t i o n f u n c t i o n
s r m l o n n H o m o n d d o m l o n D d d n o o n n o r m l o n n D n m k H o d ( n D F F
spec(H) = {0 < E0 E1 En } . @ Q F Q U A
e l o m o n d n o d n F l d o m o o n o
eiHt
eiHt =
neiEnt |n n| @ Q F Q V A
n d d o m o o n n l n l o l E l n o t D H|n = En |n Fe d o n n l n q n n l n o d k o o n
t = i l n d o D x = idx/d n d eiHt = eH o
i
tfti
dt
1
2mx2 V(x)
= i(i)
fi
dt
1
2m
dx
d
2 V(x)
=
fi
dt
1
2m
dx
d
2+ V(x)
@ Q F Q W A
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3 Z E T A R E G U L A R I Z A T I O N I N Q U A N T U M M E C H A N I C S R Q
g o n n l D n l o m
xf, tf | xi, ti = xf, tf |e H(fi)| xi, ti=
Dx exp
fi
d
1
2m
dx
d
2+ V(x)
, @ Q F R H A Dx n o n m n m n m F i o n @ Q F R H A o o n n o n n n o n l o n d l m n F
v n o d ( n o n n o n I D P D R D S o r m l o n n H
Z() = TreH, @ Q F R I A
o o n n n d o r l o d HF o n n n n m o n o n {|En}
H|En = En |En , Em| En = mn. @ Q F R P A s n
Z() =n
En|eH| En
=n
En|eEn | En
=n
eEn , @ Q F R Q A
o n m o n o |x o o o n o o xD
Z() =
dx
x|eH| x
.@ Q F R R A
s n l l d n o = iT ( n d
xf|eiHT| xi
=
xf|eH| xi
, @ Q F R S A
n d o m n l o n o o n n o n
Z() =
dz
x(0)=x()=z
Dx exp
0
d
1
2mx2 + V(x)
=
periodic
Dx exp
0
d
1
2mx2 + V(x)
, @ Q F R T A o d n l n d n l o l l o d
n n l [0, ] F n l o m o n o l l o D o n n o n m l
TreH =
n=0
e(n+1/2). @ Q F R U A
e l o n m o o l n o n n o n D o o
o n o l z o n l l d F o d o l l o F m n m = iT o o n n l z(0)=z(T)=0
Dz exp
i
2
dtz
d
2
dt2 2
z
z(0)=z()=0
Dz exp
1
2
d z
d
2
dt2+ 2
z
,
@ Q F R V A
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3 Z E T A R E G U L A R I Z A T I O N I N Q U A N T U M M E C H A N I C S R R
n Dz n d n o n m m n m F o n n
n r m n m M o E d ( n n l k 1
k
n
n o n n d @ j n Q A
nk=1
dxk
exp
1
2
p,q
xpMpqxq
= n/2
nk=1
1k
=n/2
detM.
@ Q F R W A
m n l z o n o l q n n l
dx exp
1
2x2
=
2
, > 0.
n k o d ( n d m n n o n o o O n ( n o d o
n l
kF o m l d n
DetO =
k kF x o h l
d d n o d m n n o n o o D m l l d d n o d m n n
o m D m l o n d F
3 . 4 Z e t a r e g u l a r i z a t i o n s o l u t i o n o f t h e b o s o n i c p a r t i t i o n f u n c t i o n
n n n n l o m n m
z(0)=z()=0
Dz exp
1
2
d z
d
2
dt2+ 2
z
=
DetD
d
2
d2+ 2
1/2@ Q F S H A
D d n o h l o n d o n d o n z(0) = z() = 0F m l l o d d @ Q F I V A n l o l o n
z() =1
n=1
zn sinn
. @ Q F S I A
d o n o 0 n zn l z l n o n F n o n o o n o o m l o n F u n o n n l o n o n
sin(n/) n = (n/)2 + 2 m d m n n o o o
DetD
d
2
d2+ 2
=
n=1
n =n=1
n
2+ 2
=
n=1
n
2 m=1
1 +
m
2.
@ Q F S P A
s n o m o d n ( n ( n o d n o n l d m n n D F F
DetD
d
2
d2
n=1
n
2. @ Q F S Q A
r n o n o m n o l F o O n o o o E d ( n n l n F p o l l o n I D R D S n D k l o
log DetO = logn
n = Tr log O =n
log n. @ Q F S R A
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3 Z E T A R E G U L A R I Z A T I O N I N Q U A N T U M M E C H A N I C S R S
x o d ( n l n o n
O(s) :=n
sn . @ Q F S S A
m o n o 0 n l l @ s A n d
O(s) n l n s n o n F e d d o n l l D n n l l l o n n d o o l s l n o l
( n n m o o n F d o l n o n l n k d o n o n l d m n n
dO(s)
ds
s=0
= n
log n. @ Q F S T A
e n d o o n o DetO
DetO = exp dO(s)ds s=0 . @ Q F S U A
o o n d n O = d2/d2 o l d
d2/d2(s) =n=1
n
2s=
2s(2s). @ Q F S V A
e o d n g I D n o n n l o o l o m l s l n m l o l s = 1 F l @ P F Q H A n d @ P F Q Q A
(0) = 12
(0) = 12
log(2)
l o l l d D n d n m n o o o n
d2/d2(0) = 2 log
(0) + 2(0) = log(2). @ Q F S W A
n n o o n o d m n n h l o n d o n
DetD
d
2
d2
= elog(2) = 2 @ Q F T H A
n d ( n l l
DetD d2
d2+ 2 = 2
p=11 +
p 2
. @ Q F T I A x o o n ( n o d n o o m ( n d o (0) n d (0)Fv o k o o n n o n
TreH =n=0
e(n+1/2) =
2
p=1
1 +
p
21/2
tanh(/2)
1/2
=
2
sinh
1/2
tanh(/2)
1/2=
1
2 sinh(/2).
@ Q F T P A
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3 Z E T A R E G U L A R I Z A T I O N I N Q U A N T U M M E C H A N I C S R V
g o n n l
Z() =n
dd(1 )
0|eH| n n | 0 1|eH| n n | 1 + 0|eH| n n | 1 + 1|eH| n n | 0 .@ Q F U R A
x o n o l m o n n d d o n o o n o n l n d o
m o m l
Z() =n
dd(1 )
0|eH| n n | 0 1|eH| n n | 1 + 0|eH| n n | 1 1|eH| n n | 0=
dde
|eH|
. @ Q F U S A
n l k o o n D o m o n n E o d o n d o [0, ] n n q m n n l
n
= 0 n d n = Ff n o k n o n
eH = limN
(1 H/N)N @ Q F U T A n d n n o m l n l o n @ e F T H A o n o l l o n o n
o o n n o n
Z() = limN
dde
|(1 H/N)N|
= limN
dd
N1
k=1dkdk exp
N1
n=1nn
|(1 H)| N1 N1| | 1 1|(1 H)|
= limN
Nk=1
dkdk exp
Nn=1
nn
N|(1 H)| N1 N1| | 1 1|(1 H)| N , @ Q F U U A l o n n o n n d n n l l l o n
= /N and = N = 0, =
N =
0 . @ Q F U V A
w l m n l d @ o ( o d A
k|(1 H)| k1 = k| k1
1 k|H| k1k| k1
= k| k1 exp( k|H| k1 / k|k1)= e
kk1e(
kk11/2) = e/2e(1)
kk1 .
@ Q F U W A
s n m o n l o n n o n o m @ P F I I Q A
Z() = limN
e/2N
k=1
dkdk exp
Nn=1
nn
exp
(1 )
Nn=1
nn
= e/2 limN
Nk=1
dkdk exp
Nn=1
{n(n n1) + nn1}
,
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3 Z E T A R E G U L A R I Z A T I O N I N Q U A N T U M M E C H A N I C S R W
n d o n m l ( o n o o n n l I
Z() = e/2 limN
Nk=1
dkdk exp(
B ), @ Q F V H A
o l l o n o n d m l m n
=
1F
F
F
N
, = 1 N , BN =
1 0 0 yy 1 0 00 y 1 0
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
0 0 y 1
, @ Q F V I A
y = 1 + F o m o n n d d l l n o m d ( n o n o q m n n q n
n l o n d
Z() = e/2 limN
det BN = e/2 lim
N
1 + (1 /N)N = e/2(1 + e) = 2 cosh
2.
@ Q F V P A
e o o n o n n o n D n o m l n
n o n
@ n l z o n o A D n d l l o l l F
l l o d
Z() = e/2 limN
Nk=1
dkdk exp(
B )
= e/2 Dk
Dk
exp
0
d (1 )d
d+
= e/2Det()=(0)
(1 ) d
d+
. @ Q F V Q A
() = (0) n d n l o l d l d o o l o n o n E o d o n d o n d o n
() = (0)Fp D n d o
() n p o m o d F n m o d n d o o n d n n l
exp
i(2n + 1)
, (1 ) i(2n + 1)
+ , @ Q F V R A
n n n = 0, 1, 2, F n m o d o d o m N(= /) o o n @ o A o m l F n
n o o l l o n o
n ( n o n o n o n l ( n F
3 . 7 Z e t a r e g u l a r i z a t i o n s o l u t i o n o f t h e f e r m i o n i c p a r t i t i o n f u n c t i o n
n o n o m l l o l d o d o m D n d o n o d
N/4 k N/4 F p o l l o n o n D o n
Z() = e/2 limN
N/4k=N/4
i(1 ) (2n 1)
+
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3 Z E T A R E G U L A R I Z A T I O N I N Q U A N T U M M E C H A N I C S S H
= e/2e/2
k=1
2(n 1/2)
2
+ 2=
k=1
(2k 1)
2 n=1
1 +
(2n 1)2
@ Q F V S A
o l o m o m ( n ( n o d D D d n n d n n d o l o o m ( n F n o m l d o l l o
log = 2k=1
log
2(k 1/2)
,
@ Q F V T A
n d d ( n o o n d n n o n @ r z n o n A
fermion(s) =
k=1
2(k 1/2)
s=
2s
(s, 1/2), @ Q F V U A
@ g P D i P F I I A
(s, a) =k=0
(k + a)s,
0 < a < 1F
= exp(2fermion(0)). @ Q F V V A
o n o l o d ' n n fermion n o n s = 0 d o n o l l o I
fermion(0) = log
2
(0, 1/2) + (0, 1/2) = 1
2log2, @ Q F V W A
n o d n g P @ o m I H A
(0, 12 ) = 0, (0, 12 ) = 12 log2.
n l l o o D o n n l
= exp(2fermion(0)) = elog 2 = 2. @ Q F W H A
l n d n d n d n o o n l z o n o m d F p n l l D o n n o n l d n l l
Z() = 2n=1
1 +
(2n 1)2
= 2 cosh
2, @ Q F W I A
o o m l
n=1
1 +
x
(2n 1)2
= coshx
2. @ Q F W P A
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3 Z E T A R E G U L A R I Z A T I O N I N Q U A N T U M M E C H A N I C S S I
x o m l n @ Q F Q S A n d @ Q F W P A l l @ Q F T P A n d @ Q F W I A o o o n
n d m o n l F
d p n m n n d o o n o l o n o k n o n o l m n n k n o n o E
l m D o n m 9 T h e s a m e e q u a t i o n s h a v e t h e s a m e s o l u t i o n s 9 F o n l n d
m n o n o l m m l o l m D n E o l o n n n o
l o n F p n m n k l l d n n o m n o l m n o o n o l d
o l F l d o n n F n o @ Q F W P A n
n l F
i p i i x g i
d o n o o l o n o n n n m m n n d z n o n E l z o n l n d n o n d k o
E I r n u l n 9 P a t h I n t e g r a l s i n Q u a n t u m M e c h a n i c s V I o V Q D I T I o I T Q n d
T H H o T I R
E P k n n d o d P W W o Q H I n d
E Q A d v a n c e d Q u a n t u m F i e l d T h e o r y o j n o d l o m n o q m n n
n m F
s l o o m l m n d o m
E R g F q o n d p F n H a n d b o o k o f F e y n m a n P a t h I n t e g r a l s Q U o R R D S S o
S W o m o n n d
E S i y 9 T S o T U o m o n F
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4 D I M E N S I O N A L R E G U L A R I Z A T I O N S P
4 D i m e n s i o n a l R e g u l a r i z a t i o n
4 . 1 G e n e r a t i n g f u n c t i o n a l a n d p r o b a b i l i t y a m p l i t u d e s i n t h e p r e s -
e n c e o f a s o u r c e J
p o m n m m n l n l d n l z o n l d o
d o m X ( l d o F l l l l d l 4, (x) l l ( l d F v m m z n d d l o m ( l d o o m k n n d o d P l l
j n Q F o n l o m v n n
S =
dxL((x), (x)), @ R F I A
n d o o d L v n n d n F o n o m o o n @ i y w A
n i l E v n o n
x
L((x))
= L(x)
. @ R F P A
p o m l ( l d v n n
L0((x), (x)) = 12
( + m22) @ R F Q A
n d u l n E q o d o n o n
( m2) = 0. @ R F R A
n o J n m m l d n o n l n o n
0, |0, J = Z[J] = ND expidxL0 + J + i2 2, @ R F S A ( l
i d d d o m k n l o n F n n k o J
d n o D F F n m l l o d o d m n n n d
m o n F s n n o n
Z[J] =
D exp
i
dx
L0 + J +
i
22
=
D exp
i
dx
1
2
(
m2) + i2+ J. @ R F T A s n D u l n E q o d o n o n o m l l m o n l z d o n
(
m2
+ i)c = J. @ R F U A o k n n d d m n o n n d d ( n n p n m n o o
(x y) = 1(2)d
ddk
eik(xy)
k2 + m2 i @ R F V A
o l o n o n l z d u l n E q o d o n o n o m
c(x) =
dy(x y)J(y). @ R F W A
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4 D I M E N S I O N A L R E G U L A R I Z A T I O N S Q
p n m n o o o
( m2 + i)(x y) = d(x y). @ R F I H A r n m m l d n n n m o o J
0, |0, J = Nexp i
2
dxdyJ(x)(x y)J(y)
@ R F I I A
o
Z0[J] = Z0[0] exp
i
2
dxdyJ(x)(x y)J(y)
. @ R F I P A
n 0, |0, J := Z0[J]F p n m n o o l o o m d n o n l d o Z0[J]
(x y) = iZ0[0]
2Z0[J]J(x)J(y)
J=0
. @ R F I Q A
s n o d o l
Z0[0] @ m o m m l d n n o o A n d o n o d m n m
x4 = t = ix0 n d o o = 2 + 2 o
Z0[0] =
D exp
1
2
dx(
m2)
=1
Det( m2), @ R F I R A
l d D d m n n o d o n l o o n d n o n d
o n d o n F m o D v n n o o m l l ( l d k o m
L0 = m2 ||2 + J + J, @ R F I S A n d o n n l n n n o n l o m
Z0[J, J] =
DD exp
i
dx(L0 i ||2)
=
DD exp
i
dx(
m2 i) + J + J
,@ R F I T A
d ' n n o n o o
(x y) = iZ0[0, 0]
2Z0[J, J]
J(x)J(y)
J=J=0
. @ R F I U A
m l n o n o u l n E q o d o n o n ( m2) = J n d (
m2) = J
Z0[J, J] = Z0[0, 0] exp
i
dxdyJ(x)(x y)J(y)
@ R F I V A
n d n n o k o o n
Z0[0, 0] =
DD exp
i
dx( m2 i)
=
1
Det( m2). @ R F I W A
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4 D I M E N S I O N A L R E G U L A R I Z A T I O N S R
n o o n l n v n n
L(, ) = L0(, ) V() @ R F P H A
o m d o l X o m o o n l l m d m m n d n o m l E
z o n o o n d o n d o n d l d l F o n l
l l o o m V() = n! n
l n m n o n o n n d n > 2 n n F e o D n n n o n l P D R
Z[J] =
D exp
i
dx
1
2(
m2) V() + J
=
D exp
i
dxV()
exp
i
dx (L0(, ) + J)
= expi dxVi J(x)D expidx (L0(, ) + J)
=k=0
dx1
dxk
(i)kk!
V
i
J(x1)
V
i
J(xk)
Z0[J]. @ R F P I A
q n n o n @ m o n o o d m o d o ( l d
o o A
Gn(x1, , xn) := 0|T[(x1) (xn)]|0 = (i)nn
J(x1) J(xn) Z[J]J=0
@ R F P P A
n d n n n o n l
Z[J]Fr o D n n n o n l d o Z[J] o n d J = 0 n d o m l n o l o n o n o o n n l o n d o n
Z[J] =k=1
1
k!
n
i=1
dxiJ(xi)
0|T[(x1) (xn)]|0 =
0
Texp
dxJ(x)(x)
0
.
@ R F P Q A
g o n n d n E o n n o n n d
Z[J] = exp(W[J]), @ R F P R A
n d ' o n d ( n d v n d n o m o n o l l o
[cl] := W[J] ddxiJcl @ R F P S A
cl := J =W[J]
J. @ R F P T A
l l l o [cl] n I E l d l d m @ f l n n d v o I n d m o n d R A F
s o n n n n o o d o d o n n o m l m n n n d l o
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4 D I M E N S I O N A L R E G U L A R I Z A T I O N S S
n l o o l m n F n n n o n l o o l o n n o n o
( l d o v n n L n @ R F S A
Z[J] =
D exp
i
d4x(L + J)
, @ R F P U A
m l o n n d n E T n d T D T (1 i) F p m o o l l o n
0|T (x1)(x2)|0 = limT(1i)
D(x1)(x2)exp
iTT
d4xL
D exp
iTT
d4xL
= Z[J]1
i J(x1)
i
J(x2)
Z[J]
J=0
. @ R F P V A
v d o o m m n l o n o n m l Y n d d n l
o m l o n o n m m n @ e p Q A o n m n o n
l d n o o m l l n n d o n o l d l l o o n o o n o n
o o d l o k o n o m n F m d o n l n ( n m l
o o n o m n n ( n m l o o d p n m n o o F
x o D k o o n o m o o d n t ix0 l d i l d n R E o o d
x2 = t2 |x|2 (x0)2 |x|2 = |xE |2 , @ R F P W A
n d m l l m n l o n n o n o m l n n q n 9
n o n o n m ( l d o o d o l o n n o n i n v a r i a n t n d
o o n l m m o o E d m n o n l i l d n F
4 . 2 F u n c t i o n a l e n e r g y , a c t i o n a n d p o t e n t i a l a n d t h e c l a s s i c a l e l d
cl(x)
v n o l o 4 o F e k n o o n n
S =
d4x(L + J) =
d4x
1
2()
2 12
m22 4!
4 + J
, @ R F Q H A
n d o m n k o o n
i
d4xE(LE J) = i
d4xE
1
2(E)
2 +1
2m22
4!4 J
, @ R F Q I A
n n k E o d n n n o n l
Z[J] =
D exp
d4xE(LE J)
. @ R F Q P A
n o n l LE[] o n d d o m l o n d n ( l d l m l d o l d n n o n l o m l F o o l d m l LE [] o m o n n n d o n n l o l n d d o l o
) o n o F n l D k o d n o n l Z[J] o n n o n d n l m n o m o o m n o m n ) n l
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4 D I M E N S I O N A L R E G U L A R I Z A T I O N S T
( l d F
p n l l D l n l o n ( l d o n d l m n
n n q n 9 n o n o (xE) P
(xE1)(xE2) =
d4kE(2)4
eikE (xE1xE2)
k2E + m2
, @ R F Q Q A
n p n m n o o l d n l k o n n d l l o '
exp(m|xE1 xE2|) F o o n d n n n m ( l d o n d l m n l n m o n n n d n d n l o l d n F
l l n n n n o n l o o l o n n o n D d ( n n n n o n l
E[J]
Z[J] = exp(iE[J]) =D exp
i
d4x(L + J)
=
|eiHT|
, @ R F Q R A
o n n o n m l n d o F x o i o n o n n l n o n o @ R F P R A F n o n l
E[J] D d o D m n n o n o n l o J F v o m n o n l d o
E[J] o J(x)
J(x)E[J] = i
J(x)log Z
=
D exp
i
d4x(L + J)
1 D(x)exp
i
d4x(L + J)
@ R F Q S A
n d
J(x) E[J] = |(x)|J @ R F Q T A
m o n l n n o o J F
x D d ( n
E l l ( l d
cl(x) = |(x)|J , @ R F Q U A d o l l o l ) o n D n d d n d n o n o J F
E ' o n v n d n o m o n o
E[J] F F n @ R F P S A
[cl] := E[J]
d4xJ(x)cl(x). @ R F Q V A
f o @ R F Q T A o l l o n
cl(x)[cl] =
cl(x)E[J]
d4x
J(x)
cl(x)cl(x
) J(x)
=
d4xJ(x)
cl(x)
E[J]
J(x)
d4xJ(x)
cl(x)cl(x
) J(x) = J(x) @ R F Q W A
m n n o o z o D o n
cl(x)[cl] = 0 @ R F R H A
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4 D I M E N S I O N A L R E G U L A R I Z A T I O N S U
( d ' o n F o n o l o n l o
(x)
n l o o F s l l m d o l m
n n n d n l o n n d v o n z n o m o n F m l n l
m l ( o n o @ R F R H A o o l m o o n d n o l o n cl l l n d n d n o x D n d n j o l n n y h i o o n l F
m o d n m l l D o o o n l o o l m o m o n o n o n l n l k n D n d o D n l n F g o n n l D n
m o o l m V n d o d T o o n m
[cl] = (V T)Veff(cl), @ R F R I A Veff ' p o t e n t i a l F s n o d [cl] n m m n d o l l o n o o l d
clVeff(cl) = 0. @ R F R P A
i o l o n o @ R F R P A n l o n n n o o D F J = 0F o D ' o n @ R F Q V A E n ( = E) n d o n n l Veff(cl) l d o l o n o @ R F R P A n d n o o o n d n F
' o n l d ( n d @ R F R I A n d @ R F R P A l d n o n o m n m z o n
d ( n m o ( l d o n l d n l l ' o n m o o n F
l o n o
Veff(cl) l l o l l o o m n l o m l o n F s n o d o o m l l l o l l o k n n d o d 9 m o d n n o l l o o m
F t k S n d d k o I W U R F d o o m ' o n d l o m n l d ( n o n n d n o n Veff o n o n o n n l o cl Ff n n o m l z d o n o D v n n
L =1
2
()2
1
2
m202
0
4!
4@ R F R Q A
o o l
L = L1 + L, @ R F R R A
n l o o o l o v n n n n o m l z d
4 o d o n n e p D
j n Q n d f l n n d v o g U F R I D F F l n ( l d = Z1/2r Z d n v d o n o m l
d4x | T (x)(0)| eipx = iZp2 m2 + (terms regular at p
2 = m2), @ R F R S A
m n l m F l n v n n n o
L =1
2
(r)2
1
2
m202r
0
4!
4r +1
2
Z(r)2
1
2
m2r
4!
4r, @ R F R T A
o Z = Z 1D m = m20Z m2 n d = 2Z m n d l l m d F l m k n o n o n m n d k
n o o n n ( n n d n o l n m n d
l m F
e l o o d n o n o l o n n o n d
l l ( l d
L
=cl
+ J(x) = 0. @ R F R U A
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4 D I M E N S I O N A L R E G U L A R I Z A T I O N S V
f n o n l Z[J] d n d o n cl o d n d n o n J o o l o o m n o n o cl F l l n o n F x D d ( n J1 o n o n l ( l l ( l d o n o o o d D F F n L = L1
L1
=cl
+ J1(x) = 0, @ R F R V A
n d d ' n n o o J n d J1 l l n @ k n n d o d I I F R A
J(x) = J1(x) + J(x), @ R F R W A
J o d m n d D o d o d n o n o o @ R F Q U A D n o n (x)J = cl(x)F m n o @ R F Q R A
eiE[J] =
D exp
i
d4x(L1[] + J1)
exp
i
d4x(L[] + J )
, @ R F S H A
l l o n m n o n d o n n l F v o n n o n (
o n n l ( F i n d n o n n l o (x) = cl(x) + (x) l d d4x(L1[] + J1) =
d4x(L1[cl] + J1cl) +
d4x(x)
L1
+ J1
+1
2!
d4xd4y(x)(y)
2L1
(x)(y)
+1
3! d4xd4yd4z(x)(y)(z)
3L1(x)(y)(z)+ @ R F S I A
n d n d o o d n o n l d o L1 l d cl F o n d n l o n r n @ R F R V A n d o n l o q n n l D o o n n n d
m F
v m o 0 n o @ R F S I A @ F F n o n l d o
L1 A l l E d ( n d o o F s k o n l m o d o d n n d o n l o o ( n l o @ R F S H A ( n d q n n l
n l d n m o n o n l d m n n o m d n e n d
D exp
i
d4x(L1[cl] + J1cl
+1
2 d4xd4y
2L1(x)(y)
= exp
i
d4x(L1[cl] + J1cl)
det
2L1
(x)(y)
1/2, @ R F S P A
l o E o d n m o o n o ' o n n d m n n F s
n o o n d o n d n l o @ R F S H A o n o o n m o
v n n n d n d n d o n o
(L[cl] + J cl) + (L[cl + ] L[cl] + J ). @ R F S Q A
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4 D I M E N S I O N A L R E G U L A R I Z A T I O N T H
4 . 3 D e r i v a t i o n o f 4 p o t e n t i a l a t cl(x)
s n n ( o n o D o n o n n o m m k n
d l o m n o m o n l @ i A o m l ( l d F p m o D
i d m n d m n m o n o ' o n l D
dV
dcl= 0. @ R F S V A
' o n l n l l o n l V o v n n n o
n m ' k n n o o n F o n o l l o o l n m
m D o o l d l n o n E n o o n n o m m
k n F l n m l o o n o n n o d F
v E o o n d @ R F P R A o l l o F n n n o n l X o
o n n d q n n o n n l ( l d o D
Z[J] = exp(i1X[J]) = ND exp
i1
d4x(L + J)
@ R F S W A
n o m l z o n o n n N o n o
Z[0] = 1 X[0] = 0. @ R F T H A
y n E l E d l q n n o n (n) n d ' o n [cl] o I D P D Q D R
[cl] =n=1
in
n!
d4x1
d4xn
(n)(x1, , xn)cl(x1) cl(xn), @ R F T I A
n d n o o n
G(N)(x1, x2, , xN) = 1iN
J1
J2
JNZ[J]
J=0
.
s n @ R F S W A o 1 m l l o l v n n @ n o j n o n A
o V n n d m l l 1 o n d o I n n l l n l l
o F i o E n l l n n q n n o n G(E) o o F n o m o n D o l l o
V+I+E = L+1E , @ R F T P A
o L = I V + 1 n d n o n E n m o n l l n F p m o D o o LE n n d m n n o n o 1
XF o n E l E d l
q n n o n (E) n o o o D n m l l n o o n l L F m n o o n d n m o l o o p s F n n o l o o
(L = 0) o n l n o n E n n (E)
(2)(p, p) = p2 2 @ R F T Q A
n d
(4)(p1, p2, p3, p4) = , @ R F T R A
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4 D I M E N S I O N A L R E G U L A R I Z A T I O N T I
o m o n o o n l @ f l n n d v o R F R A
V(cl) = n=1
inn!
(n)(0, , 0)ncl V0(cl) = 12 22cl +
14!
4cl, @ R F T S A
l l l d o n m o o n F x o D o D o l o
n o n l o d o n n o m o n o z n m d D o n l n
n o n m F f l n n d v o P H I m ' o n [cl]
[cl] = i log N 18
dx[iF(0)]
2 12
dxcl(x)(
+ 2)cl(x)
14
iF(0)
dx2cl(x)
1
24
dx4cl(x) + O(
2). @ R F T T A
m o n n p n m n o o F(0) o m o m d n l o o n l
n d n d o n o o n n z o o d D o n n l m
0[cl] = 12
dxcl(x)(
+ 2)cl(x) 1
4!
dx4cl(x) @ R F T U A
n k n n o o n N = 1 m l n [0] = 0 F ( o d o D l o o n o n D o ' o n l V n d
' o n n o m d n
(x) = 0(x) + (x), @ R F T V A
0 n z o o d o m o n o cl F r n n n o n l n o n l m o l l o n i y w
(
+ 2
)0(x) +
6 30(x) = J(x).
@ R F T W A
n l n n o v n n d n
L() =1
2()() 1
222 1
4!4 @ R F U H A
o l l o n n l d4x(L + J) =
d4x(L(0(x)) + J0)
+
d4x
()(0) 20 1
630 + J
+
d4
x
L2(, 0) 1
6 3
0 1
4 4
@ R F U I A
L2 o n n o l l d l o n n o m d D F F
L2 =1
2()() 1
222 1
420
2.@ R F U P A
x o D n
m n m l l o n n l n m n
o @ R F U I A d F
y n o E l n o n l o l l o
= 1/2 @ R F U Q A
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4 D I M E N S I O N A L R E G U L A R I Z A T I O N T Q
m @ R F T U A F d d o n l m n n d d ' o n
1[cl] = X0[J] 0[cl]
d4xJcl + X1[J]
=
d4x[L(0) + J0]
d4x[L(cl) + Jcl] +
i
2trlog
A(x, x , 0)
A(x, x, 0). @ R F V R A
f 0 o l o n o @ R F T W A d ' n o o n l n @ R F V R A o o d (0 cl)2 = O(2), n d n n n 0 n d cl l l o F o
1[cl] =i
2trlog
A(x, x , cl)
A(x, x, 0). @ R F V S A
x D ' o n l V(cl) n d d o m [cl] n cl o n n D n
[cl] =
d4xV(cl). @ R F V T A
h l n o n l l o o d o n l A(x, x , cl) o o l d ( n l o m o @ R F V S A F d o n o l l o
A(x, x , cl) =
xx + 2 +
1
22cl
(x x)
=
d4k
(2)4
xx + 2 +
1
22cl
eik(x
x)
=
d4k
(2)4 k2 + 2 + 1
22cl
eik(xx)
=
d4k
(2)2d4k
(2)2eix
k
k2 + 2 + 12
2cl
(k k)eikx @ R F V U A
r n o m n l o n d o o n
log A(x, x , cl) =
d4kd4k
eixk
(2)2log
k2 + 2 + 1
22cl
(k k) e
ixk
(2)2@ R F V V A
o
trlog A =
d4xd4x(x x)log A(x, x , cl) =
d4x
d4k
(2)4log
k2 + 2 + 1
22cl
.
@ R F V W A
o m l o d m n n o A n
Tr log A = log det A. @ R F W H A
m m z n n n o o l l o n X o n E l o o o d o n o n o '
o n l
V1(cl) =i2
d4k
(2)4log
k2 + 2 + 12 2clk2 + 2
@ R F W I A
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4 D I M E N S I O N A L R E G U L A R I Z A T I O N T R
n d l o d D ' o n l o m d
V(cl) V0(cl) + V1(cl) = 12
22cl + 14!4cl i2
d
4
k(2)4
log
1 12
2cl
k2 2
, @ R F W P A
D q k m m D F F o n n n v n n F
n l m n o n l l o l d n n d d m n o n l E
l z o n o d l F F @ o n n f l n n d v o D F V H A
I(, B) =
d2k
(2)2(k2 2B + i)1 =
1
i
d2 k(2)2
(k2 + 2B)1
=i2B
162(M2)2
1
2 + (1) + 1 log
2B
4M2+ O( 2)
. @ R F W Q A
f o o l l o n n o m o n o m
B(x) = Z1/2(x) Z2B = 2 + 2 Z2B = + , @ R F W R A
n n o m o o n l @ R F W P A o
V(cl) =1
222cl +
1
4!4cl +
1
222cl +
1
4!4cl
i
2
d2k
(2)2log
1 1
2
2clk2 2
,
@ R F W S A
n q k m n o n n o m l z d m n
F s m o n o n o D o D m o n n m o d
MS m F s n m o q n n o n D d l o n l n 2 n d n l z d o n m I
= M42
a0(,M/,) +k=1
ak(,M/)(2 )k
@ R F W T A
2 = 2
b0(,M/,) +
k=1
bk(,M/)
(2 )k
@ R F W U A
Z = c0(,M/,) +k=1
ck(,M/)
(2 )k @ R F W V A
a0, b0 n d c0 l 2 n d = M24 F f o n l m l o l D k E n l o n l m l o l 2
F m n
k > 1
ak = bk = 0 F
o n E o n E d l q n n o n o n o m l z d o
2(p, p) = p2(1 + Z1) 2 21 +2
322
1
2 + 1 log2
4M+ O( 2)
.
@ R F W W A
n o n n
I4(B) =
d4k
(2)4(k2 2B + i)1 @ R F I H H A
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4 D I M E N S I O N A L R E G U L A R I Z A T I O N T S
n d @ R F W Q A o n p n m n o o
F(0) =
d4k(2)4
(k2 2B + i)1 = M24 i2162
12 + 1 log
24M
+ O( 2)
.
@ R F I H I A
o o D n n D n n q n n o n
2(p, p) = p2(1 + Z1) +
2 +1
2iF(0) +
21
+ O(2), @ R F I H P A
n n n p n m n d m o l d n o d F
o m o n o 4 l n o l l o m l n D n l l D n 4 ( n n n n o m l o n m
2 o l l o n o l d
3322 12 2 const, @ R F I H Q A
=k=2
k. @ R F I H R A
( n o 2 F q o n k o o d o n o @ R F W T A o @ R F W V A n o o l n 2 n l z d n
a1 =3
3222, b1 =
1
322. @ R F I H S A
o n k o n o n l o n
V(cl) =1
222cl
1 + b0
322
3
2 + log 4
+1
4!4cl
+ a0 3
2
322
3
2 + log 4
+1
642
2 +
1
2cl
2log
2 + 12 2cl
M2 4 log
2
M2
. @ R F I H T A
s n n o n MS m m o o @ j n o f l n n d v o A
MS = 3
322
12 + log 4
+ O(3) @ R F I H U A
aMS0 =32
322[+ log 4] + O(3), aMS1 =
32
322+ O(3), @ R F I H V A
l l d o + log 4 o F r o D n d m n o j o o m o n n o n l E z o n D o l d l o n o m l z V(cl) n n o n o l m n d
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