zeta function regularization

8/14/2019 Zeta Function Regularization

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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


4/115

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


6/115

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


15/115

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.


16/115

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


17/115

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)


36/115

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


37/115

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

.


38/115

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


39/115

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


40/115

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


41/115


42/115

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


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


43/115

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


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


44/115

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


45/115

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


46/115

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


48/115


49/115


50/115

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


54/115

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


55/115

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


56/115

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


57/115

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


58/115

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


60/115

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


68/11

zeta function regularization

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