The Many Faces of QFT, Leiden 2007 Some remarks on an old problem Gerard ‘t Hooft Utrecht...
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![Page 1: The Many Faces of QFT, Leiden 2007 Some remarks on an old problem Gerard ‘t Hooft Utrecht University.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d4b5503460f94a28b64/html5/thumbnails/1.jpg)
The Many Faces of QFT, Leiden 2007The Many Faces of QFT, Leiden 2007
Some remarks on an old problemGerard ‘t Hooft
Utrecht University
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The Many Faces of QFT, Leiden 2007
Lattice regularization of gauge theories without loss of chiral symmetry.Gerard 't Hooft (Utrecht U.) . THU-94-18, Nov 1994. 11pp. Published in Phys.Lett.B349:491-498,1995. e-Print: hep-th/9411228
SPIRES:
P. van Baal, Twisted Boundary Conditions: A Non-perturbative Probe for Pure Non-Abelian Gauge Theories thesis: 4 July, 1984.
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The Many Faces of QFT, Leiden 2007
Gauge theory on the lattice:
1 2
4 3
Plaquette1234
Site x=1
Link 23
2
1(12)
ig A dxU e
P
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The Many Faces of QFT, Leiden 2007
1234(12) (23) (34) (41)
2 2 4 2112 122
Tr ( ) Tr
Tr ( )
( )ig A dx
U U U U e
C iga F g a F
P
1 14 2Tr ( ) Tr ( )F F UUUU
2
1(12)
ig A dxU e
P
After symmetrization :
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The Many Faces of QFT, Leiden 2007
The Fermionic Action (first without gauge fields) :
Dirac Action
2 11 1
links
( ) ( )( ) ( )( )x x
L x m xa
Species doubling
(and same for 2, 3 )
12 1( )U x12 1( )U x
However, in the limit , the equation
has several solutions besides the vacuum solution :
since
( ) ( )( ) 0x e x stC
11 4 1 1 4 1 5 1'( , , ) ( 1) ( , , ) ;xx x C x x C
1 1 1 1C C
0m
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The Many Faces of QFT, Leiden 2007
Wilson Action
1 1links
( ) ( )( ) ( ) (1 ) (1 )
x e x eL x m x
a a
This forces us to treat the two eigenvalues of separately,and species doubling is then found to disappear.
Effectively, one has added a “mass renormalization term”
However, now chiral symmetry has been lost !
Nielsen-Ninomiya theorem
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The Many Faces of QFT, Leiden 2007
ABJ anomaly
Lattice
( ) (1) ( ) (1)
( ) ( ) (1)
( ) (1)
F L L F R R
F L F R V
F V V
SU N U SU N U
SU N SU N U
SU N U
It could not have been otherwise: even in the continuum limit invariance is broken by the Adler-Bell-Jackiw anomaly.5
However, in the chiral limit, , the symmetry pattern is0M
Can one modify lattice theory in such a way thatsymmetry is kept?( ) ( ) (1)F L F R VSU N SU N U
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The Many Faces of QFT, Leiden 2007
(2)SU
3
The BPST instanton(A.A. Belavin, A.M. Polyakov, A.S. Schwartz and Y.S. Tyupkin)
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The Many Faces of QFT, Leiden 2007
Instanton
Fermi level
timetime
LEFT
RIGHT
The massless fermions
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The Many Faces of QFT, Leiden 2007
The fermionic zero-mode:
In Euclidean time:
e e
In Minkowski time: i te i te
negativeenergy
positiveenergy
Thus, the number zero modes determines how many fermions are lifted from the Dirac sea intoreal space.Left – right: a left-handed fermion transmutes into a right-handedone, breaking chirality conservation / chiral symmetry
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The Many Faces of QFT, Leiden 2007
The instanton breaks chiral symmetry explicitly:
2 2 implies
(2) (2) (2) (2) (1)
m m
U U SU SU U
2 2' implies
(3) (3) (3) (3) (1)
Km m
U U SU SU U
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The Many Faces of QFT, Leiden 2007
Each quark species makes oneleft - right transition at the instanton.
Leftu
Leftd
Lefts
Rightu
Rightd
Rights
LeftcRightc
charmm
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The Many Faces of QFT, Leiden 2007
The interior is mapped
onto 4
(2)SU
3
The number of left-minus-right zero modes of the fermions = the number of instantons there.
Atiyah-Singer index
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The Many Faces of QFT, Leiden 2007
How many “small” instantons or anti-instantons are there inside any 4-simplex between the lattice sites? These numbers are ill-defined !
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The Many Faces of QFT, Leiden 2007
The number of instantons is ill-defined on the lattice!
If one does keep this number fixed, one will neveravoid the species-doubling problem.
Therefore, the number of fermionic modes cannot dependsmoothly on the gauge-field variables on the links!( )U x
Domain-wall fermions are an example of a solution to theproblem: there is an extra dimension, allowing for anunspecified number of fermions in the Kaluza-Klein tower!
Is there a more direct way ?
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The Many Faces of QFT, Leiden 2007
We must specify # ( instantons) inside every 4-simplex.
This can be done easily !
Construct the gauge vector potential at all , starting
from the lattice link variables (defined only on the links)
( )A x x
( )U x
Step #1: on the 1-simplicesdef
( )iagA
U x e
Note: this merely fixes a gauge choice in between neighboringlattice sites, and does not yet have any physical meaning.
Next: Step #2: on the 2-simplices
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The Many Faces of QFT, Leiden 2007
This is unambiguous only in the elementary, faithful representation,which means that we have to exclude invariant U(1) subgroups – the space of U variables must be simply connected
– we should not allow for a clash of the fluxes !
First choose local gauge :
12F a A
Then subsequently, if so desired, gauge-transform back
This procedure is local, as well asgauge- and rotation-invariant
( The subset of gauge- transformations needed
to rotate is Abelian )
U I iagAU e
U I
U I
1
2
Here, we may now choose the minimal flux F , which means that
all eigenvalues must obey:
2 1 2 1( , ) ( / )A x x A x a
Aa g
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The Many Faces of QFT, Leiden 2007
Step #3: on the 3-simplices
Step #4: on the 4-simplices
We have on the entire
boundary. Extend the field in the 3-d bulk by choosing it to obey sourceless 3-d field eqn’s
(extremize the 3-d action , and in
Euclidean space, take its absolute minimum ! )
( )A x
3 ( ) ( )( )ij ijd x F x F x
Exactly as in step #3, but then for the 4-simplices. Taking theabsolute minimum of the action here fixes the instantonwinding number !
This prescription is gauge-invariant and it is local !!
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The Many Faces of QFT, Leiden 2007
Thus, there is a unique, gauge-independent and local way todefine as a smooth function of starting from the
link variables ( )A x x
( )U x
In principle, we can now leave the fermionic part of theaction continuous:
fermion ( )( )( )xm igA L
Our theory then is a mix of a discrete lattice sum(describing gauge fields and scalars) and acontinuous fermionic functional integral.
The fermionic integral needs no discretization because it ismerely a determinant (corresponding to a single-loop diagramthat can be computed very precisely)
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The Many Faces of QFT, Leiden 2007
0 0 1log det ( ) log det log 1iiD ig A D C A
The first four diagrams can be regularized in the standard way – giving only the standard U(1) anomaly
1 + + + + + ···
The sum over the higher order diagrams can be boundedrigorously in terms of bounds on the A fields.
(Ball and Osborn, 1985, and others)
- one might choose to put the fermions on a very dense lattice: , to do practical lattice calculations, but this is not necessary for the theory to be finite !
fermion gaugea a
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The Many Faces of QFT, Leiden 2007
The procedure proposed here is claimed to be non localin the literature. This is not true.
The extended gauge field inside a d -dimensional simplexis uniquely determined by its (d – 1) -dimensional boundary
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The Many Faces of QFT, Leiden 2007
The prescription is: solve the classical equations, and of all solutions, take the one that minimizes the total action.
However, imagine squeezing an instanton ina 4-simplex, using a continuous process (such as gradually reducing its size).
As soon as a major fraction of the instanton fits inside the 4-simplex, a solution with different winding number will show up, whose action is smaller.
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The Many Faces of QFT, Leiden 2007
→ the gauge field extrapolation procedure itself is discontinuous ! Depending on the configuration of the link variables U, the number of instantons within given 4-simplices may vary discontinuously.
This is as it should be!
The most essential part of the gauge field extrapolation procedureconsists of determining the flux quanta on the 2-simplices, andthe instanton winding numbers of the 4-simplices. We demandthem to be minimal, which usually means that the Atiyah-Singerindex on one simplex 2
42
1232
gF F d x
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The Many Faces of QFT, Leiden 2007
Lattice regularization of gauge theories without loss of chiral symmetry. Gerard 't Hooft (Utrecht U.) . THU-94-18, Nov 1994. 11pp. Published in Phys.Lett.B349:491-498,1995. e-Print: hep-th/9411228
We claim that this procedure is important for resolvingconceptual difficulties in lattice theories.
The END