Lattice Gauge Theory
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N.D. Hari Dass |
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Institute of Mathematical Sciences |
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Chennai |
Quark Confinement,
Hadronic Strings and the KABRU Supercomputer
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N.D. Hari Dass |
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Institute of Mathematical Sciences |
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Chennai |
The Periodic Table
The Scale of Things
Artists view of an atom
The Atom
Nuclear forces mediated
pions
Neutron spallation
Are neutrons and protons
elementary?
Hitting particles with
particles
A particle accelerator
High energy electron
proton collision
Birth of the hadronic
string
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Remarkably, if one plotted the angular
momentum of the state vs mass(or square of the mass) they all fell on
families of parallel straight lines. |
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Nielsen, Nambu, Goto, Susskind in 1968
made the astonishing suggestion that a relativistically rotating string had
states of excitation this spectrum! |
Slide 14
The quark model
Pion in the quark model
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The relativity theory says that there
is an antiparticle for each particle. |
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The antiparticle of u-quark with
electric charge 2/3 is u-bar antiquark with
charge -2/3. |
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The pion in this model is made up of u
and d-bar . |
What holds the quarks
together?
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If there were no forces between quarks
(anti – quarks) they would just fly apart without forming hadrons like
protons, pions, neutrons etc… |
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The next important question is : what
is the nature of these forces? |
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Are they attractive, repulsive … |
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How do they vary with distance and what
is their dependence on the quark type? |
The way electric charges
interact: Quantum Electrodynamics
Quantum Chromodynamics
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It turns out that fully consistency
with the hadrons requires that each quark type comes in 3 species which will
symbolically be denoted as red , blue and green . |
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Observed hadrons only correspond to
combinations that are white. |
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For example : proton is u(red), u(blue)
and d(green). |
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The analog of photons are gluons and
they come in 8 colours. |
Quarks and Gluons of
SU(3) QCD
What is QCD? – More
technically…
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QCD is a non-Abelian Gauge Theory. |
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The gauge group is SU(3). |
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The Quarks carry the fundamental
representation 3. |
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The Gluons, which transmit the forces
between quarks, carry the adjoint representation 8. |
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3 has triality, 8 has no triality. |
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The theory has defied analytical
results despite the best efforts for more than 25 years. |
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It is an outstanding problem of
theoretical physics. |
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Mathematical Formulation
of QCD
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First let us look at electrodynamics
described by Maxwell’s equations . |
"Now we have 8
vector..."
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Now we have 8 vector potentials and 8
scalar potentials: |
Discretising QCD
More on updating
The Wilson Loop
The Great Quark Puzzle
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It should be possible to release quarks by supplying sufficient
energy to protons and neutrons. |
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But no quarks were liberated even in
very high energy collisions! |
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Making A Virtue Out Of An
Embarassment
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“Mama Mia, if you don’t see something
you should be seeing, there must be a very deep reason”---- N. Testa |
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The theory of Quarks must be such that
they can never be free! |
Is Such A Thing Possible
At All?
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‘t Hooft, Mandelstam, Nambu: A thin flux
tube connects the quarks and keeps them from being liberated. |
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This can happen to magnetic monopoles
submerged in a superconductor. |
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Dual Superconductor Picture. |
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Can this be proven in Quantum
Chromodynamics (QCD), the best candidate theory of Quarks and Gluons? |
Slide 30
The dual picture
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Instead of charges consider magnetic
monopoles: |
Numerical Simulations
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Discretise the space-time manifold. |
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Continue from Minkowski signature to
Euclidean signature. |
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The Quantum Field Theory is mapped
exactly into Classical Statistical Mechanics in Four spatial dimensions. |
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Simulate that numerically. |
Numerical Evidence For Flux Tubes
The latest simulation
results on flux tubes
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Due to a path breaking development in algorithms
it is now possible to measure extremely tiny quantities on the lattice. |
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The flux tube is actually a string ! It
is not rigid but can actually fluctuate like a string. |
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Even its states of excitations can be
probed numerically and they indeed look like hadronic states. |
The Polyakov Loop
The string picture of
protons, neutrons and pions…
Coming Back To Flux Tubes
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If quarks are not too heavy, the flux
tube can snap. |
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If quarks carry adjoint charge no flux
tube should be formed. |
Unresolved Issues
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What causes the flux tube formation? |
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Colour Monopoles? |
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Colour Vortices(Z3)? |
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Instantons and topological charges? |