Field Quantization Without Divergences

John R. KlauderUniversity of Florida

Gainesville, FL

Dirac on Divergences Most physicists are very satisfied with the

situation. They say: 'Quantum electrodynamics is a good theory and we do not have to worry about it any more.' I must say that I am very dissatisfied with the situation, because this so-called 'good theory' does involve neglecting infinities which appear in its equations, neglecting them in an arbitrary way. This is just not sensible mathematics. Sensible mathematics involves neglecting a quantity when it is small - not neglecting it just because it is infinitely great and you do not want it!

.

Frequently Asked Questions

• No divergences. Is that possible?

YES• What is the ‘‘cost’’?

AN UNUSUAL LOCAL COUNTER TERM• Basic strategy?

SOLVE NONRENORMALIZABLE CASES• Which fields?

SCALARS (HIGGS), [GRAVITY, FERMIONS]

Outline

• Background (Scalar Fields); Basic Proposal• Free/Pseudofree Models• Why Divergences Arise• How Divergences Appear• Relevance for Scalar Fields• The Cure: ‘‘Measure Mashing’’• Lattice Hamiltonian• Lattice Action• Monte Carlo Evidence• Other Fields• Origin of Measure Mashing

Background (Scalar Fields)

,6,5 ; 4 ; 3,2 :tionregulariza Lattice

,6,5 ; 4 ; 3,2 :analysison Perturbati

)(

)()()(

)(

)(

0202

1000

}])[({)/1(

)()()()2/1(

]})[({)/1(00

4

0

22

0

221

00

22

0

221

nnn

nnn

rmscounter te

hSghSghS

DeMhS

e

DeMhS

xdgmhgg

ydxdyhyxxh

xdmh

n

nn

n

values allfor limit )0( classicalproper

the toleads andexpansion free-divergence a

has integral that thisso ),( rmcounter te

dependent- an choose tois goal desired The

)(

:ydifferentl issues theseexamine topropose We

0

)},(])[({)/1( 4

0

22

0

221

00

g

C

DeMhS xdCgmhgg

n

Basic Proposal

Free/Pseudofree Models

)()(

lim 0,,

lim

})(])()([{

]})()([{

)2/1(

0

}][{00

00

42221

2221

0

42221

Tn

nnn

dtgxxxg

gg

g

exhxh

DxeNxTx

AA

dttgxtxtxA

dttxtxA

F CON.

g

Free/Pseudofree Models

])1(1)[()()(

lim 0,,

lim

})(])()([{

]})()([{

)2/1(

0

}][{00

000

42221

2221

0

42221

Tnn

nnn

dtgxxxg

gg

g

exhxhxx

DxeNxTx

AAA

dttgxtxtxA

dttxtxA

F CON.

g

DISCON.PF

AN “AMPUTATED” ACTION FUNCTIONAL

Scalar Fields

F REN.

g

NONREN.PF

o

THEORY. A TO CONNECTED

LYCONTINUOUS ARE THEY INSTEAD

.THEORY FREE OWN THEIR TO CONNECTED

ARE THEORIESG INTERACTIN SOME

5 ; 3/44

}){(}{

}])[({

222/14

40

220

221

0

PSEUDOFREE

NOT

CnCn

xdCxd

xdgmA

nn

ng

AN “AMPUTATED” ACTION FUNCTIONAL

B.I.

Why Divergences Arise

supportdisjoint

)( ; )(

)1( ; )(

measuressingular Mutually

support equal

)( ; )(

/ ; /

measures Equivalent

43

3443

43

21

1221

)1(21

22

mm

dmxVdmdmxUdm

dxxdmdxxdm

mm

dmxvdmdmxudm

dxeAdmdxeAdm xAAx

B.I.

Many Variables

; ; support

)(

/)( ;

; ; support

)(

/)( ;

2

111

2

2

111

2

)4/1(

1

)4/1(

1

R

R

llllllll

l

l

N

lll

N

lll

N

l

l

fxfixfi

l lx

NNN

fN

xfixfi

Nl l

xN

exdee

dxexdN

exdee

dxexdN

Support when N = Infinity

if :Hence

lim :order toExpanding

lim :ly Consequent

0||lim

]1[

||||

;

21

1212

4/1

1

2

2/12/)(,1,

1

222

4/1

1

2

22

2

2

Nj jNN

cicxNjNN

NNN

cN

cxxicNNkjN

NNNN

cNN

icxNjNN

xc

ee

YY

eee

YYYY

edYYeY

j

kj

j

How Divergences Appear

Nverge asmoments diAll such

x

ms!unting terJust by co

NOdxeM

dxexM

pl

Nl

pl

Nl

x

lNl

xpl

Nl

l

N

l

l

N

l

etc. , ][for Also

)()(

][

41

1

12

12

1

2

1

Relevance for Scalar Fields

' )(

)(12

21'2

21

21

20

2

0

2

2

),,,( :nHamiltonia Lattice

; ; 0 :limit Continuum

;

: withlattice spatial ldimensiona-

cubic-hyper periodic,by Regulate

ks

ks

s

kk

aa

kkkk

aLLa

acinglattice spaach edgesites on eL

s

H

Ground State Distribution

'

)( )'(

'][][

')(

/'2'2'

/'

2

,

2

,

s

p

kkaAps

kkps

kk

kkaAf

LNsDiverges a

ms!No. of terNO

deaMa

deMds

llkklk

s

llkklk

WILL THE GUILTY VARIABLES THAT LEAD TO DIVERGENCES PLEASE RAISE THEIR HANDS

Ground State Distribution

dinates rical coorhyper-sphe

LNsDiverges a

ms!No. of terNO

deaMa

deMd

k

kkkkkk

s

p

kkaAps

kkps

kk

kkaAf

s

llkklk

s

llkklk

11 ; 0

1 ; ;

SCOORDINATE OF CHANGEA MAKE

'

)( )'(

'][][

')(

2'2'2

/'2'2'

/'

2

,

2

,

Moments in the New Coordinates

? HANDS THEIR RAISE PLEASE

SDIVERGENCE TO LEAD THAT

VARIABLES GUILTY THE ILL W

)'(][ toleads which )'(

:integral ofdescent steepest by integral Estimate

)1(2

][

2'2

'2')1'(

22' 2'

,

2

ppskk

kkkkN

aAspppskk

NOaNO

dd

eaMas

llkklk

Moments in the New Coordinates

! been has every strongly;

variables theseconstrains 1hat relation t The

sdivergence toscontribute that variableONLY theis

)'(][ toleads which )'(

:integral ofdescent steepest by integral Estimate

)1(2

][

2'

2'2

'2')1'(

22' 2'

,

2

dneutralize

NOaNO

dd

eaMa

k

kk

ppskk

kkkkN

aAspppskk

s

llkklk

Moments in the New Coordinates

!!measures! EQUIVALENT tomeasures SINGULAR

MUTUALLY from measures changes ; DISAPPEAR

sdivergence then , , to Change

.][ : variableONE from arise sDivergence

)'(][ toleads which )'(

:integral ofdescent steepest by integral Estimate

)1(2

][

)1(1)'(

2/12'

2'2

'2')1'(

22' 2'

,

2

R

NOaNO

dd

eaMa

RN

kk

ppskk

kkkkN

aAspppskk

s

llkklk

The Cure: ‘‘Measure Mashing’’

0'2

21' 22

021

22',2

1'221

),,('2/)21(2,

),,(')'(),,(

)1()1'(

)1'(

)(

)(

}]'['{

)'2( , ''''

remedy! achieve toGSD model pseudofreeDesign

, :identifiedRemedy

:identified sdivergence of Source

* *2

2

pfs

k ks

k k

skkk kk

spf

aUballklk

saURNaU

RN

N

Eaam

aa

eJ

NbaRee

R

k

s

F

H

B.I.

Counter Term

/)()()(

then][ )(Let 2212

4/)21(2,

''

ks

k

ballklk

TTa

JTs

F

limit! continuum in the potential localA

; ''/1'')(

][][2)21(

][)21()(

}{,121

,2

2,

,2

22,

22,

2

21

2

2,

,241

}{

)(

nnkklkslkk

lltl

kts

lltl

kkts

ts

lltl

kkts

ts

k

J

J

Ja

J

Jaba

J

Jaba

F

F

s=2

Lattice Hamiltonian

83

43

0'2

21' 4

0

' 2202

12',

221'2

21

' )()(

1221'2

21

' ;

4 TO EXTENDED ; 5 BY INSPIRED

)(

)(

* *2

2

20

2

0

2

2

ggive e nonnegatSquares ar

E aced by OPering replNormal ord

nn

Eaag

amaa

aa

sk k

sk k

sk k

skk kkk

s

ks

ks

k

kk

F

H

H

B.I.

Spacetime and Space Averages

kkpp

ppk

pk

pkk

kkp

kkp

k

kkaIp

k

pk

skk

skk

dFF

FFa

FFaaF

deaFM

aF

agamF

pp

pp

20

/1

/),,(

40

220

)()()(

|)()(|

|)()(||])([|

])([

])([

} , ,{)(

010010

0100100

0

0

Lattice Action

gqaNgag

mqaNmam

qaNZahZ

deMe

aag

amaaI

spnk k

pnk k

spnkk k

p

kkaIahZahZ

nk kk

nk

nk k

nkkkk

n

kkk

n

kkk

)2(30

40

2120

220

)1(22

/),,(//

2214

0

2202

122,2

1

)( ][

)(' ][

)( ][

)(

)(),,(

2/12/1

**

F

Monte Carlo Evidence

ed)(unpublish ,Stankowicz J. Deumens, E. 2.

(1982) 486-481 113B,

n, WeingarteD. Smolensky, P. Freedman, B. 1.

1)( ; 1 ; 63.3 : parametersChosen

)0(~/])0(~3)0(~[)(

: constant coupling edRenormaliz

|)(~|/]|)(~|)0(~[

: mass edRenormaliz

)/2,,0,0( ; )(~

:onansformatiFourier tr Discrete

22224

22222

.Phys. Lett

Lam

Lamg

g

pppm

m

Lapep

R

nRR

R

R

R

kkkaip

g_R vs. g_0 for n=3 and n=4

Phi^4_4 With Counter Term

Other Fields

bosons gauge NO

only) sindicationfar (So :Fermions

0)( , )( ; )()( )(

Gravity) Quantum (Affine :Gravity

][ ; ][ ; )(

},,2,1{ ; :fields like-Higgs2,,

',

'2,

2,,,,

,

**

bab

aabcb

acab

llklkp

k kkkkk

kk

uxguxgxgxx

J

A

Origin of Measure Mashing

ytion theorr perturba cutoffs oed without Both solv

xddtMgmiA

xddtgmA

s

s

'mashing' measure'' of) (analog toleadmay

})({

! 1973in Solved

2008in 'mashing' measure'' toled

}][{

! 1970in Solved

2200

440

220

221

Ultralocal Scalar Fields

NbaR

ambam

deM

defba

dexfxdbfC

edeMfC

and

xddtxtgxtmxtA

sRN

ss

kkba

kkamafi

babmk

sk

bmspf

xdxfmkk

amafif

sg

ss

kk

s

kkk

s

ss

kk

s

kkk

2 ; :y Effectivel

; : Note

][

}||/)]cos(1[)(1{

}||/)])(cos(1[exp{)(

)(

! Trivial lizableNonrenorma

}),(]),(),([{

)1()1(

0

2/)21(2

)21(

)()4/1(

40

220

221

2

0

2

2

2

0

2

0

Summary

• Lattice ground state wave function ‘‘=’’ Lattice Hamiltonian ‘‘=’’ Lattice action

• Origin of divergences traced to power of the hyper-spherical radius

• Measure mashing changes mutually singular measures into equivalent measures

• Finite spatial moments implies finite spacetime moments

• Monte Carlo supports non-triviality

2/12]'[ kk

Feynman on Divergences .

The shell game that we play ... is technically called 'renormalization'. But no matter how clever the word, it is still what I would call a dippy process! Having to resort to such hocus-pocus has prevented us from proving that the theory of quantum electrodynamics is mathematically self-consistent. It's surprising that the theory still hasn't been proved self-consistent one way or the other by now; I suspect that renormalization is not mathematically legitimate.

Thank You

References

• ``Scalar Field Quantization Without Divergences In All Spacetime Dimensions'' J. Phys. A: Math. Theor. 44, 273001 (2011); arXiv:1101.1706

• ``Divergences in Scalar Quantum Field Theory: The Cause and the Cure'', Mod. Phys. Lett. A 27, 1250117 (9pp) (2012); arXiv:1112.0803

• Ultralocal model scalar quantum fields: ‘‘Beyond Conventional Quantization’’ (Cambridge, 2000 & 2005)

• ‘‘Recent Results Regarding Affine Quantum Gravity’’, J. Math. Phys. 53, 082501 (19pp) (2012); arXiv:1203.0691