Field Quantization Without Divergences
-
Upload
colette-collins -
Category
Documents
-
view
56 -
download
2
description
Transcript of Field Quantization Without Divergences
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