Transcript talk_pacific - University of Kentucky

Lattice QCD and the QCD
Vacuum Structure
Ivan Horváth
University of Kentucky
Outline
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3 Why’s (What’s)
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QCD = Quantum Chromodynamics
Why Quantum QCD?
Why Lattice QCD?
Why Vacuum?
Vacuum & Path Integral
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Summation over the Paths
Configurations and Vacuum
Structure
Degree of Space-Time Order
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Topological Vacuum
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What is Topological Vacuum?
Lattice Topological Field
Surprising Structure of
Topological Vacuum
 Fundamental Structure
 Global Nature
 Low-Dimensionality
 Space-Filling Feature
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Ivan Horváth@University of the Pacific, Apr 2006
3 Why’s: Why Quantum Chromodynamics?
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Goal of physics is to explain and predict natural phenomena
Historically this proceeded via discovering/understanding
forces driving them
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Gravity
Long-range
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Electromagnetism
Long-range
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Weak Force
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Strong Force
1018 meter
1015 meter
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Ivan Horváth@University of the Pacific, Apr 2006
Why Quantum Chromodynamics continued…
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Weak and strong force require quantum description
Quest for unified description of all fundamental forces (reductionism)
At present this means gauge invariant quantum field theory
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Ivan Horváth@University of the Pacific, Apr 2006
Why QCD continued…
Standard Model
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Ivan Horváth@University of the Pacific, Apr 2006
3 Why’s: Why Lattice QCD?
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Strange behavior of QCD relative to QED
Elementary fields of QED:
Elementary fields of QCD:
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A ( x),  ( x)
photon
electron
Aa ( x), a=1,2,...,8
gluons
 b ( x), b  1, 2,3
quarks
Elementary fields/particles of QCD are never observed!
Elementary particles of QCD are influenced by interaction
strongly and approximate methods involving them do not
work!
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Ivan Horváth@University of the Pacific, Apr 2006
Why Lattice QCD continued…
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Defining fields and interaction on
space-time lattice
allows to define the theory and
treat it numerically
Kenneth Wilson (1974)
Michael Creutz (1979)
A ( x)  U n , 
 ( x)   n
S
QCD
S
LQCD
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Ivan Horváth@University of the Pacific, Apr 2006
3 Why’s: Why Vacuum?
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Vacuum in Quantum Field Theory (QFT) – state in the
Hilbert space with lowest energy
Pays the role of the medium where everything happens
Medium can be very important – in QFT medium is pretty
much everything!
Look back at the non-observability of elementary particles in
QCD: this is usually referred to as the confinement
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Ivan Horváth@University of the Pacific, Apr 2006
Why Vacuum continued…
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Ivan Horváth@University of the Pacific, Apr 2006
Why Vacuum continued…
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Ivan Horváth@University of the Pacific, Apr 2006
Why vacuum continued…
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Ivan Horváth@University of the Pacific, Apr 2006
Why Vacuum continued…
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Understanding of QCD Vacuum is crucial for understanding
of strong interactions!
Calculation of all observables in QFT involves calculating
vacuum expectation values
Origin of all observables can be traced to vacuum
structure!
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Ivan Horváth@University of the Pacific, Apr 2006
Why Vacuum continued…
(masses)
Hadron propagator
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Ivan Horváth@University of the Pacific, Apr 2006
Why Vacuum continued…
(masses)
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Ivan Horváth@University of the Pacific, Apr 2006
Vacuum and the Path Integral (Paths)
How does one grasp the task of understanding
QCD vacuum?
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In Quantum Theory vacuum is not a “uniform medium”.
Rather it is a fluctuating medium.
This fluctuating nature is most naturally expressed in
Feynman’s path integral formulation of quantum theory.
Consider a Quantum-Mechanical particle described by
Hamiltonian H and corresponding classical action S.
 x f , t f | xi , ti    x f | e
 iH ( t f ti )
| xi  
xf

iS [ x ( t )]
Dx
e

x(ti )  xi
x (t f )  x f
xi
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Ivan Horváth@University of the Pacific, Apr 2006
Summation over the paths continued…
Every path x(t) can be thought
of as a configuration of this
one-dimensional system.
Path integration is a summation
over the configurations!!!
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Ivan Horváth@University of the Pacific, Apr 2006
Summation over the paths continued…
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What is a generalization to Quantum field Theory?
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For a QM particle the configuration/path is one possible history for
the dynamical variable involved (its coordinate)
For quantum field it is the same:
the history of field values in 3-d space
x(t )  (x,t)
Configuration/Path is a function of space-time variables!
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Ivan Horváth@University of the Pacific, Apr 2006
Summation over the paths continued…
But how do we sum these paths up?
There is a representation of QFT (Euclidean field theory)
where this is particularly transparent!
ensemble
QFT
 (, P() ) 
All content is stored in the probability distribution!
 =  P() ()

In lattice field theory such
statistical sum is meaningfully
defined
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Ivan Horváth@University of the Pacific, Apr 2006
Configurations & the Vacuum Structure
VACUUM
STATISTICAL ENSEMBLE OF
CONFIGURATIONS
Isn’t this too much fluctuation? Can we learn anything?
BASIC ASSUMPTION of path-integral approach to vacuum structure:
The statistical sum is dominated by a specific kind of configurations with
high degree of space-time order (typical configurations)!
VACUUM STRUCTURE is associated with SPACE-TIME STRUCTURE
in typical configurations.
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Ivan Horváth@University of the Pacific, Apr 2006
Degree of space-time order
How do we quantify degree of space-time order in a configuration?
( x)
01011001011010101110…
binary string S
Kolmogorov complexity of S is a measure of order in
P(S)
Universal
Turing
machine
( x)
S
Minimal length of P(S) in bits is the Kolmogorov complexity of
( x)
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Ivan Horváth@University of the Pacific, Apr 2006
Topological Vacuum
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(What is…)
In QCD it is important to understand behavior of
various composite fields
Aa ( x), a=1,2,...,8
fundamental fields
 b ( x), b  1, 2,3
Fa ( x)    Aa ( x)   Aa ( x)  g f abc Ab Ac
composite
field
 Important composite field is topological charge density
q( x)  64 2   Fa ( x) Fa ( x)
g2
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Ivan Horváth@University of the Pacific, Apr 2006
What is topological vacuum?
continued…
Topological charge density is a topological field
(stable under deformations)

 A ( z )
a
d
4
x q ( x) 

 A ( z )
a
Q  0
Topological vacuum is the vacuum defined by the
ensemble of q(x) induced by the QCD ensemble
configuration of A(x)
configuration of q(x)
Understanding topological vacuum is considered
an important key to understanding QCD vacuum
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Ivan Horváth@University of the Pacific, Apr 2006
Lattice Topological Field
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Topological properties are frequently thought to be tied
to continuity of the underlying space-time. Can the lattice
analog of topological field be strictly topological?
Yes it can!
(Hasenfratz, Laliena, Niedermayer, 1998)
It behaves in a continuum-like manner (integer global
charge, index theorem)
Related to defining lattice theory with exact chiral
symmetry (Ginsparg-Wilson fermions)
q( x) 
1
tr  5 D( x, x)
2
SF 
  ( x ) D ( x, y )  ( y )
x,y
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Ivan Horváth@University of the Pacific, Apr 2006
Lattice topological field continued…

U  ( z)
a ,b
 q ( x) 
x

U  ( z)
a ,b
Q  0
Strictly topological on the space-time lattice!
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Ivan Horváth@University of the Pacific, Apr 2006
Surprising Structure of Topological
Vacuum
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How do we examine the structure of topological vacuum?
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Define gauge theory on a finite lattice
Generate the ensemble via Monte-Carlo simulation
 (U , P(U ) ) 
ensemble
...,U
( i 1)
, U (i ) , U ( i 1) ,...
probabilistic chain
Elements of probabilistic chain are “typical configurations”
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Calculate the probabilistic chain of topological density
..., q
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( i 1)
, q ( i ) , q ( i 1) ,...
Examine the space-time behavior in typical configurations
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Ivan Horváth@University of the Pacific, Apr 2006
Fundamental Structure
I.H. et al, 2003
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Ivan Horváth@University of the Pacific, Apr 2006
Global Nature of the Structure
Characteristics of global behavior saturate faster than physical observables
Structure has to be viewed as global!
I.H. et. al. 2005
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Ivan Horváth@University of the Pacific, Apr 2006
Low-Dimensional Nature
Claim: It is impossible to embed 4-d manifold in sign-coherent
regions of QCD topological structure (I.H. et.al. 2003)
Topological structure has low-dimensional character
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Ivan Horváth@University of the Pacific, Apr 2006
Space-Filling Feature
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Two seemingly contradictory facts:
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Coherent topological structure is low-dimensional
Occupies finite fraction of space-time
Finite line occupies zero fraction of a surface
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In geometry there are intriguing objects defying this
space-filling curves (Peano, 1890)
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Space-Filling Feature continued…
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Ivan Horváth@University of the Pacific, Apr 2006
Space-Filling Feature continued…
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Ivan Horváth@University of the Pacific, Apr 2006
Space-Filling Feature continued…
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Peano curve: continuous surjection
[0,1]  [0,1]2
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QCD structure: continuous surjection
[0,1]  [0,1]d
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d is the embedding dimension of the structure
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1 d  4
QCD topological structure is a quantum analog of spacefilling object!
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Ivan Horváth@University of the Pacific, Apr 2006
Thanks to my collaborators
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Andrei Alexandru
Jianbo Zhang
Ying Chen
Shao-Jing Dong
Terry Draper
Frank Lee
Keh-Fei Liu
Nilmani Mathur
Sonali Tamhankar
Hank Thacker
University of Kentucky
University of Adelaide
Academia Sinica
University of Kentucky
University of Kentucky
George Washington Univ.
University of Kerntucky
Jefferson Laboratory
Hamline University
University of Virginia
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Ivan Horváth@University of the Pacific, Apr 2006