16.1 Introduction

• Is p r ecisely th e reg im e in the form of hadrons.
• The vast amount of data at ex ists in low-energy hadronic physics should be explained by the true theory of the in teractio n between quarks.

• It is not likely that the answers to these questions will be found by performing calculations.
• One integral encounter has to be evaluated numerically.
• The standard way to approximate the integral is by some kind of sum.
• It would make sense to formulate the theory on a mesh of points in the first place.

• There is a more fundamental point involved.
• The success of renormalization techniques, first in 'taming' the ultraviolet divergences of quantum field theories and then in providing quantitative predictions for short-distance phenomena, has been one of the triumphs of theoretical physics over the past 50 years.
• We were able to regulate the diverg ences in a way that didn't rely on the appearance of the graphs.
• Wilson was the first to propose a way to approximate continuous sp acetime by a d iscrete lattice.
• There are points between them.
• The lattice version of the theory should be a good approximation if the lattice spacing is not too small.
• The following one will show how a gauge theory is discretized.

• The quantum field theory should be used for computation.
• It would be the same for ordinary quantum mechanics.

• Section 16.4 introduces the method of tial star tin g p o in t f o r th e lattice a p p r o ach to q.
• The sum over paths approach doesn't work, bu t fermio is still h ave to b e accommo d ated.

• Section 16.4 describes the way this is done.

• This relationship has allowed physical insights and numerical techniques to be used from one subject to another in a way that has been very beneficial to both.
• The physics of renormalizatio n an d o f th e RGE is a lattice/statistical perspective.
• The chapter ends with me sample r e su lts o b tain e d f r o m lattice sim.

• All of spacetime is represented by a finite-volume 'hypercube'.

• In the same way as in previous work, we can consider a formula in momentum space, which will be discretized.

• O f vo lu me 1, f U n c tio n r elatio n g iven in.

• We are in a one-dimensional space.
• Readers are familiar with the theory of lattice vibrations and phonons.

• The procedure for fermion field leads to difficulties.

• Thefermion doubling problem is a phenomenon in which there are 16 corners of the hyper cube.

• The Dirac Lagrangian is linear in the d erivatives.

• In M o n tva y and M"unster (1994), Fu r th e r d iscu ssio.

• We will return to this question in section 16.4.

• It is time to think about gauge invariance after exploring th e iscretizatio.
• It is very important to see how the same id arises in the lattice case.

• We have a case of Ab elian U( 1 )th e o r y, QED.

• The discussion in chapter 13 should have prepared us for this, as we are trying to compare twovectors at two different points.
• Before they can be compared, we need to parallel transport one field to the other.
• The solution shows us how to do it.

• I re 16.

• The appropriate matrices are in the continuum form of the covariant d erivative.

• The gauge-invariant discretized derivatives can be constructed.

• The gauge is verified by the accumulated phase factors and the exponentials have no matrices.
• We will see how to recover the action.

• The exponential may be expanded.

• A classical field theory is defined on a lattice with a suitable continuum limit.

• We don't know how we are going to turn the classical lattice theory into a quantum one.
• The fact that the calculations are mostly going to have to be done numerically seems to require a formula that avoids non-commuting operators.
• This formalism is given a brief introduction in this section.

• In ordinary quantum mechanics, observables and state vectors are related to each other.

• The two pictures are at the same time.

• We can evaluate (16.47) directly.

• If we are in teg r atio n var iab le.

• As it stan ds, th e in teg r al in.
• It is at this point that 'Euclid ean' sp acetime arises.

• The result can be represented in a form.

• Since the integral of a Gaussian is again a Gaussian, we may perform all the integrations analytically.
• The method of Feynman and Hibbs was followed by us.

• The free-particle case is established.

• There are more general cases discussed in Peskin and Schroeder.

• We are going to discuss further aspects of the path-integral formula.

• The ground state gives the dominant contribution.

• The results of the field theory case are generalizations.

• Four-dimensional generalizations of expressions such as (16.7).

• This formalism can be used to develop perturbation theory.

• One finds the same 'Feynman rules' as in the canonical approach to analyse this theory.

• We already know from chapter 7 that we can't construct a well-defin ed perturbatio n theory in this way.
• The route followed by Faddeev and Popov was to obtain the Feynman rules for non-Ab elian g auge th eories.

• The 'Hermitean matrix' can be parametrized in such a way that it is convenient for me.
• The variables in th e p arametrizatio n o f U vary over some domain, as in the simple U(1) case.
• If the action is gauge invariant, the integration measure for the link variables can be chosen so as to be gauge invariant.

• This is similar to the neglect of closed fermion loops in a graph approach.

• A highly suggestive connection to be set in quantum field theory is not the least advantage.
• There is a preliminary discussion of renormalization in the following section.

• Each configuration was evaluated by IOP Publishing.

• Statistical mechanics deals in three spatial dimensions, not the four of our spacetime.
• It is remarkable that quantum field theory in three spatial dimensions appears to have a close relationship to equilibrium statistical mechanics in four spatial dimensions.

• In the case of pure gauge actions (16.39 or 16.40), the gaugecoupling is seen to be analogous to an inverse temperature, by comparison with (16.84).

• The second point is related to this.