Docstoc

An introduction to Lattice Gauge theories

Document Sample
An introduction to Lattice Gauge theories Powered By Docstoc
					          An introduction to Lattice Gauge theories

                                   Antonio Rago

                                 University of Wuppertal


                         CP3 -Origins Odense-17 August 2010




A. Rago (U. Wuppertal)               Lattice Introduction     17 August 2010   1 / 20
Overview of the Lattice Approach


    LGT calculations are a non-perturbative implementation of field theory using the Feynman
    path integral approach.




   A. Rago (U. Wuppertal)            Lattice Introduction               17 August 2010   2 / 20
Overview of the Lattice Approach


    LGT calculations are a non-perturbative implementation of field theory using the Feynman
    path integral approach.
    The calculations proceed exactly as if the field theory was being solved analytically (had
    we the ability to do the calculations).




   A. Rago (U. Wuppertal)             Lattice Introduction                 17 August 2010   2 / 20
Overview of the Lattice Approach


    LGT calculations are a non-perturbative implementation of field theory using the Feynman
    path integral approach.
    The calculations proceed exactly as if the field theory was being solved analytically (had
    we the ability to do the calculations).
    The starting point is the partition function in Euclidean space-time


                                      Z=              ¯
                                                DAµDψDψeS


    where S is the gauge action, for example:

                                                1           ¯
                               S=      d4 x       Fµν Fµν − ψM ψ .
                                                4

    and M is the Dirac operator. The fermions are represented by Grassmann variables ψ and
    ¯
    ψ.




   A. Rago (U. Wuppertal)              Lattice Introduction                17 August 2010   2 / 20
The ambitious plan

   Defining a finite dimensional system
   The Yang-Mills action for gauge fields have to be transcribed onto a discrete space-time
   lattice in such a way as to preserve
       1   gauge invariance
       2   chiral symmetry
       3   topology,
       4   and a one-to-one relation between continuum and lattice fields.
   Discretisation Errors & Continuum limit
   it is necessary to quantify and remove the discretisation errors by a reliable extrapolation
   to a = 0.
   Generating background gauge fields
   the path integral is analogous to the Boltzmann factor in the partition function for
   statistical mechanics systems. It can be regarded as a probability weight for generating
   configurations.
   Expectation values & Correlation functions




  A. Rago (U. Wuppertal)              Lattice Introduction                  17 August 2010    3 / 20
Discretisation


There are a number of possible ways to discretized space-time in 4 Euclidean dimensions:
hypercubic, body-centered cubic, hexagonal, random lattices.

Of these the simplest is the isotropic hypercubic grid with spacing:
                           a = aS = aT and size NS × NS × NS × NT                               .




                                       aT

                                      aS
                             NT




                                              NS



    A. Rago (U. Wuppertal)              Lattice Introduction               17 August 2010   4 / 20
Discretisation
    The lattice transcription of the matter fields ψ(x) → ψi is straightforward.




   A. Rago (U. Wuppertal)             Lattice Introduction                 17 August 2010   5 / 20
Discretisation
    The lattice transcription of the matter fields ψ(x) → ψi is straightforward.
    The construction of gauge fields is less intuitive.
    In the continuum: Aµ (x) carry 4-vector Lorentz indices
    Wilson noticed that                        y
                                 ψ(y) = Pe x igAµ (x)dxµ ψ(x)
    This suggested that gauge fields be associated with links that connect sites on the lattice.

                                                                  ˆ
                                                                  µ
                               U (i, i + µ) ≡ Uµ (i) = eiagAµ (i+ 2 )




   A. Rago (U. Wuppertal)             Lattice Introduction                 17 August 2010   5 / 20
Discretisation
    The lattice transcription of the matter fields ψ(x) → ψi is straightforward.
    The construction of gauge fields is less intuitive.
    In the continuum: Aµ (x) carry 4-vector Lorentz indices
    Wilson noticed that                        y
                                 ψ(y) = Pe x igAµ (x)dxµ ψ(x)
    This suggested that gauge fields be associated with links that connect sites on the lattice.

                                                                   ˆ
                                                                   µ
                               U (i, i + µ) ≡ Uµ (i) = eiagAµ (i+ 2 )




                                              aT
                                                              ψi
                                             aS
                                  NT


                                                    Aµ



                                                     NS

   A. Rago (U. Wuppertal)              Lattice Introduction                17 August 2010   5 / 20
Symmetries
      The SO(4) symmetric group is broken by the discretisation and replaced with the dihedral
      group D3 .
      Translations have to be by at least one lattice unit, so the allowed momenta are discrete
                                                   2πn
                                           k=            n = 0, 1, ..., N
                                                   Na
      A local gauge transformation Vi acts
                                     ψi → Vi ψi                     ψi → ψi Vi†
                                                                    ¯    ¯
                                                                     †
                                          Uµ (i)    →     Vi Uµ (i)Vi+µ




The only gauge invariant operators must be written like:

                         ¯
                         ψi U (Ci,j )ψj
                             U (Ci,i )




    A. Rago (U. Wuppertal)                   Lattice Introduction                 17 August 2010   6 / 20
Quenched Approximation
In the rest of this lecture we will work in the “quenched approximation”.
All the dynamics of the fermions and the related problems will be presented in next lecture (A. Patella ... good luck)




      A. Rago (U. Wuppertal)                          Lattice Introduction                             17 August 2010    7 / 20
Quenched Approximation
In the rest of this lecture we will work in the “quenched approximation”.
All the dynamics of the fermions and the related problems will be presented in next lecture (A. Patella ... good luck)

                                  Quenched approximation:
Once integrated integrated out exactly the fermionic fields
                                                               4   1
                                       Z=         DAµ det M e d x( 4 Fµν Fµν )




      A. Rago (U. Wuppertal)                          Lattice Introduction                             17 August 2010    7 / 20
Quenched Approximation
In the rest of this lecture we will work in the “quenched approximation”.
All the dynamics of the fermions and the related problems will be presented in next lecture (A. Patella ... good luck)

                                  Quenched approximation:
Once integrated integrated out exactly the fermionic fields
                                                               4   1
                                       Z=         DAµ det M e d x( 4 Fµν Fµν )


        det M is an highly non-local term




      A. Rago (U. Wuppertal)                          Lattice Introduction                             17 August 2010    7 / 20
Quenched Approximation
In the rest of this lecture we will work in the “quenched approximation”.
All the dynamics of the fermions and the related problems will be presented in next lecture (A. Patella ... good luck)

                                  Quenched approximation:
Once integrated integrated out exactly the fermionic fields
                                                               4   1
                                       Z=         DAµ det M e d x( 4 Fµν Fµν )


        det M is an highly non-local term
        the partition function is an integral over only background gauge configurations




      A. Rago (U. Wuppertal)                          Lattice Introduction                             17 August 2010    7 / 20
Quenched Approximation
In the rest of this lecture we will work in the “quenched approximation”.
All the dynamics of the fermions and the related problems will be presented in next lecture (A. Patella ... good luck)

                                  Quenched approximation:
Once integrated integrated out exactly the fermionic fields
                                                               4   1
                                       Z=         DAµ det M e d x( 4 Fµν Fµν )


     det M is an highly non-local term
     the partition function is an integral over only background gauge configurations
One can write the action, by re-exponentiating

                                                                   1
                   S = Sgauge + Squarks =                 d4 x       Fµν Fµν − i           log (det Mi )
                                                                   4                   i

where the sum is over the quark flavors.

                           We define the quenched approximation setting

                                                    det M = constant




      A. Rago (U. Wuppertal)                          Lattice Introduction                             17 August 2010    7 / 20
A pure gauge action


The simplest loop you can draw on a lattice is a square composed by the product of four links.

                                                     ˆ †       ˆ †
                                 Uµν = Uµ (i)Uν (i + µ)Uµ (i + ν )Uν (i)

If you make an a expansion of such a term what you will find is:

                                 2                   a4
                                  2
                                    ReTr(1 − Uµν ) =    Fµν F µν + O(a6 )
                                g                    4

with
                                   Fµν = ∂µ Aν − ∂ν Aµ + g [Aµ , Aν ]
with the derivative term obtained from

                                                1
                                   ∂x f (x) → ∆x (f (x + a) − f (x))
                                                a




       A. Rago (U. Wuppertal)               Lattice Introduction            17 August 2010   8 / 20
The Haar Measure


The last ingredient needed to complete the formulation of LGT as a quantum field theory via
the path integral is to define the measure of integration over the gauge degrees of freedom.
Unlike the continuum fields Aµ , lattice fields are SU(3) matrices with elements that are
bounded in the range [0, 1].
Wilson proposed an invariant group measure, the Haar measure.

                             f (U )dU =      f (V U )dU =        f (U V )dU

As an example for SU(2) we have:

                                   U   =      x0 Id + ix · σ
                              detU     =      x2 = (x0 )2 + x2 = 1
                                              1
                                dU     =        δ(x2 − 1)d4 x
                                              π




    A. Rago (U. Wuppertal)                Lattice Introduction                17 August 2010   9 / 20
Continuum limit and critical behaviour

We just mentioned the possibility of taking the limit for a → 0.
It’s clear that at large distance correlator of two fields goes like

                                    φ(x1 )φ(x2 )   c   ∝ e−m|x1 −x2 |

The correlation length which determines the rate of the exponential decay is therefore given by:

                                                        1
                                              ξ=
                                                       am


If a continuum limit with a finite physical mass exists
By a suitable choice of bare parameters we can approach a limit where a → 0 while m stays
finite
                                            ξ→∞
We are in presence of a critical point (second order phase transition)
Our concern is SU(N) gauge theories. Such theories are characterised by having an UV fixed
point at g = 0.




    A. Rago (U. Wuppertal)                Lattice Introduction              17 August 2010   10 / 20
The Wilson loop
Elitzur’s theorem:
      In gauge theories it is impossible to find a local quantity acting as an order parameter.




    A. Rago (U. Wuppertal)               Lattice Introduction                17 August 2010   11 / 20
The Wilson loop
Elitzur’s theorem:
      In gauge theories it is impossible to find a local quantity acting as an order parameter.
      Average values cannot distinguishes the Coulomb phase, in which charge states can be
      produced, from the so called confined phase.




    A. Rago (U. Wuppertal)               Lattice Introduction                17 August 2010   11 / 20
The Wilson loop
Elitzur’s theorem:
      In gauge theories it is impossible to find a local quantity acting as an order parameter.
      Average values cannot distinguishes the Coulomb phase, in which charge states can be
      produced, from the so called confined phase.
      To study confinement in pure gauge theories we need a gauge invariant non local quantity




    A. Rago (U. Wuppertal)               Lattice Introduction                17 August 2010   11 / 20
The Wilson loop
Elitzur’s theorem:
      In gauge theories it is impossible to find a local quantity acting as an order parameter.
      Average values cannot distinguishes the Coulomb phase, in which charge states can be
      produced, from the so called confined phase.
      To study confinement in pure gauge theories we need a gauge invariant non local quantity

                     Consider a rectangular loop CR,T of side lengths R and T


                                      W (CR,T ) =               U
                                                      CR,T


     The static quark potential V (R) is defined by means of the large T behaviour of the
                corresponding Wilson loop in the fundamental representation.


                                                    1
                                 V (R) ≡ − lim        log W (CR,T )
                                           T →∞     T

                                               so that

                                   W (CR,T )    ∼     Ce−T V (R)
                                               T →∞


    A. Rago (U. Wuppertal)               Lattice Introduction                17 August 2010   11 / 20
Strong coupling

    One well-established nonperturbative method is strong coupling.




   A. Rago (U. Wuppertal)            Lattice Introduction             17 August 2010   12 / 20
Strong coupling

    One well-established nonperturbative method is strong coupling.
    Strong coupling is tied to the lattice and cannot be derived directly for a continuum
    theory.




   A. Rago (U. Wuppertal)             Lattice Introduction                17 August 2010    12 / 20
Strong coupling

    One well-established nonperturbative method is strong coupling.
    Strong coupling is tied to the lattice and cannot be derived directly for a continuum
    theory.
    Once expanded each lattice integral in power of 1/g 2




   A. Rago (U. Wuppertal)             Lattice Introduction                17 August 2010    12 / 20
Strong coupling

    One well-established nonperturbative method is strong coupling.
    Strong coupling is tied to the lattice and cannot be derived directly for a continuum
    theory.
    Once expanded each lattice integral in power of 1/g 2
    It’s possible to perform the integrations that turns out to be only factorised group
    integration




   A. Rago (U. Wuppertal)              Lattice Introduction                17 August 2010   12 / 20
Strong coupling

      One well-established nonperturbative method is strong coupling.
      Strong coupling is tied to the lattice and cannot be derived directly for a continuum
      theory.
      Once expanded each lattice integral in power of 1/g 2
      It’s possible to perform the integrations that turns out to be only factorised group
      integration

     The largest contribution correspond to the minimal area bounded to the Wilson loop

                                                                2
                                     W (CR,T ) ∼ e−RT ln g

If no phase transition occurs when g 2 varies from infinity to zero the gauge theory underlying is
                                 always in a confinement phase
                                     Ex. SU(2) SU(3) in 4d

On the other side is a phase transition exists, the phase in which we can demonstrate the
existence of a confining behaviour is not connected to the continuum limit.
                                   Ex. SU(N) with N> 3 in 4d


    A. Rago (U. Wuppertal)               Lattice Introduction                17 August 2010   12 / 20
Thermodynamics of Quantum Field Theory and the Lattice


In order to describe a finite temperature theory we require that one dimension (the Time) is
compactified with periodic boundary conditions.



                                                       1
              The compactification length LT a =        T
                                                           , where T is the temperature.
      Periodic boundary conditions implies a new global symmetry of the action.
      The symmetry group is the center C of the gauge group (SU(N ) → ZN ).


This symmetry can be realised as follows.

                     UT (i) → W0 UT (i)
       ∀ i belonging to an hyperplane a fixed time




    A. Rago (U. Wuppertal)                Lattice Introduction                   17 August 2010   13 / 20
Thermodynamics of Quantum Field Theory and the Lattice


     The space-like plaquettes are not affected by the transformation
                                                            −1
     In timelike plaquette two contribution appear: W0 and W0 . Since they belong to the
     center they cancel each other.
     The Wilson action is invariant under such transformation.
     It is impossible to re-absorb this global twist by means of local gauge transformations.

  It is possible to define gauge invariant observables which are topologically non-trivial: The
                                         Polyakov loop

                                                  LT
                                     P (x) ≡ Tr         UT (i).
                                                  i=1




   A. Rago (U. Wuppertal)               Lattice Introduction                17 August 2010   14 / 20
Thermodynamics of Quantum Field Theory and the Lattice


The relevant feature of the Polyakov loop is that it transforms under the above discussed
symmetry as follows:


                     P (x) → W0 P (x);
thus it is a natural order parameter for this symmetry.


      It will acquire a non zero expectation value if the center symmetry is spontaneously
      broken.
      Its expectation value is related to the free energy of a single isolated quark.
      Signature of deconfinement: phase transition which separates the regime in which the
      center symmetry is unbroken from the broken symmetric phase.




    A. Rago (U. Wuppertal)                Lattice Introduction                 17 August 2010   15 / 20
A partial summary



What we have learned (or at least mentioned) so far
     Discretized the theory on a lattice




    A. Rago (U. Wuppertal)                 Lattice Introduction   17 August 2010   16 / 20
A partial summary



What we have learned (or at least mentioned) so far
     Discretized the theory on a lattice
     Define the Wilson action




    A. Rago (U. Wuppertal)                 Lattice Introduction   17 August 2010   16 / 20
A partial summary



What we have learned (or at least mentioned) so far
     Discretized the theory on a lattice
     Define the Wilson action
     Search for a continuum limit




    A. Rago (U. Wuppertal)                 Lattice Introduction   17 August 2010   16 / 20
A partial summary



What we have learned (or at least mentioned) so far
     Discretized the theory on a lattice
     Define the Wilson action
     Search for a continuum limit
     Order parameter for confinement at zero Temperature (Wilson Loop)




    A. Rago (U. Wuppertal)                 Lattice Introduction         17 August 2010   16 / 20
A partial summary



What we have learned (or at least mentioned) so far
     Discretized the theory on a lattice
     Define the Wilson action
     Search for a continuum limit
     Order parameter for confinement at zero Temperature (Wilson Loop)
     Order parameter for confinement at non-zero Temperature (Polyakov Loop)




    A. Rago (U. Wuppertal)                 Lattice Introduction         17 August 2010   16 / 20
A partial summary



What we have learned (or at least mentioned) so far
     Discretized the theory on a lattice
     Define the Wilson action
     Search for a continuum limit
     Order parameter for confinement at zero Temperature (Wilson Loop)
     Order parameter for confinement at non-zero Temperature (Polyakov Loop)
     An analytical method to justify confinement of the continuum theory if the theory does
     not have any bulk PT




    A. Rago (U. Wuppertal)                 Lattice Introduction          17 August 2010   16 / 20
A partial summary



What we have learned (or at least mentioned) so far
     Discretized the theory on a lattice
     Define the Wilson action
     Search for a continuum limit
     Order parameter for confinement at zero Temperature (Wilson Loop)
     Order parameter for confinement at non-zero Temperature (Polyakov Loop)
     An analytical method to justify confinement of the continuum theory if the theory does
     not have any bulk PT

What we miss now is to understand how to obtain results from our discretized model.




    A. Rago (U. Wuppertal)                 Lattice Introduction          17 August 2010   16 / 20
Dynamics



Ergodic Hypotheses:
average on a statistical ensemble           time average

Problem: give a dynamic to the system

Fundamental request: at the equilibrium you must generate configuration
according to the Boltzmann distribution




   A. Rago (U. Wuppertal)     Lattice Introduction         17 August 2010   17 / 20
Markov Chain


                            n           t                     t−1
Series of configurations Cm for which Cm depends only on Cn according
to a probability distribution Pnm (upper index time, lower index
configuration index)
    Irreducible if for each Cj it’s possible to reach Cl l > j, or
    equivalently it exists a time k for which
      k
    Pjl = i1 ...in Pji1 Pi1 i2 ...Pin l = 0 for every j, k
                   k
    Aperiodic if Pii = 0 for each i
    Ci state is said to be positive if it reappear in a finite time




   A. Rago (U. Wuppertal)       Lattice Introduction         17 August 2010   18 / 20
Equilibrium distribution
Given a {Ci } Markov chain irreducible, aperiodic and with only positive
states
    The equilibrium distribution exists and is unique
    hence it doesn’t depend on the start state
                                        N
                                  lim Pij = Pi
                                N →∞




   A. Rago (U. Wuppertal)      Lattice Introduction        17 August 2010   19 / 20
Equilibrium distribution
Given a {Ci } Markov chain irreducible, aperiodic and with only positive
states
    The equilibrium distribution exists and is unique
    hence it doesn’t depend on the start state
                                        N
                                  lim Pij = Pi
                                N →∞

    The equilibrium distribution is stationary
                                                  1
                                Pj =            Pij Pi
                                            i




   A. Rago (U. Wuppertal)      Lattice Introduction        17 August 2010   19 / 20
Equilibrium distribution
Given a {Ci } Markov chain irreducible, aperiodic and with only positive
states
    The equilibrium distribution exists and is unique
    hence it doesn’t depend on the start state
                                                N
                                          lim Pij = Pi
                                        N →∞

    The equilibrium distribution is stationary
                                                          1
                                        Pj =            Pij Pi
                                                    i

    if the variance of the reappear time is finite
                                                                 N
                                                             1
                                Pi O(Ci ) = O = lim                    O(Cj )
                                                        N →∞ N
                            i                                    j=1


   A. Rago (U. Wuppertal)              Lattice Introduction                17 August 2010   19 / 20
Detailed balance



    Montecarlo Dynamics: any Markov dynamics




   A. Rago (U. Wuppertal)   Lattice Introduction   17 August 2010   20 / 20
Detailed balance



    Montecarlo Dynamics: any Markov dynamics
    Problem: Given an action, build a Markov dynamic




   A. Rago (U. Wuppertal)    Lattice Introduction      17 August 2010   20 / 20
Detailed balance



    Montecarlo Dynamics: any Markov dynamics
    Problem: Given an action, build a Markov dynamic
    Necessary condition: not known




   A. Rago (U. Wuppertal)    Lattice Introduction      17 August 2010   20 / 20
Detailed balance



    Montecarlo Dynamics: any Markov dynamics
    Problem: Given an action, build a Markov dynamic
    Necessary condition: not known
    Sufficient condition: Detailed balance

                            e−βH(Ci ) Pij = e−βH(Cj ) Pji




   A. Rago (U. Wuppertal)        Lattice Introduction       17 August 2010   20 / 20
Detailed balance



    Montecarlo Dynamics: any Markov dynamics
    Problem: Given an action, build a Markov dynamic
    Necessary condition: not known
    Sufficient condition: Detailed balance

                            e−βH(Ci ) Pij = e−βH(Cj ) Pji

    and you are still free to choose Pij , decide you algorithm




   A. Rago (U. Wuppertal)        Lattice Introduction       17 August 2010   20 / 20

				
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views:5
posted:8/13/2011
language:English
pages:47