Lattice
Field Theories| Lattice
Field Theories |
In General > s.a. Cellular
Automaton; lattice [math
notion]; number theory [geometric]; {& N
Cabibbo, SU lecture 10.1986}.
* Motivation: They allow
some approximate calculations to be done, and, in the case of lattice spacetimes,
they are a trick to introduce a cutoff for
non-perturbative
regularization; But one can consider the approach as more
fundamental than
the continuum one, and take more seriously its results, hoping
to get
them in some approximation from a full quantum gravity theory (> see discrete
spacetime); It
is natural in the context of (Euclidean) path integral quantization.
* Remark: It emphasizes
the close connection with statistical mechanics; The strong coupling limit
is equivalent to a high-T expansion.
@ General references: Kadanoff RMP(77);
Drouffe & Itzykson PRP(78);
Dashen & Gross PRD(81);
Rebbi ed-83; Rebbi SA(83)feb; 't Hooft et al ed-84;
Friedberg et al JMP(94).
@ Texts, reviews: Kogut RMP(79);
Creutz 83; Montvay & Münster 94; DeGrand ht/96-ln
[non-perturbative
quantum field theory], comment Neuberger hp/04;
Smit 02; Di Pierro IJMPA(06)hl/05
[quantum field theory, especially gauge theory and QCD].
Scalar Fields > s.a. klein-gordon.
* 
4 theory:
It becomes the free scalar field theory when the spacing goes to zero; Possible
reactions are (1) The theory cannot be considered seriously;
(2) The lattice approximation is ok but there just isn't any 4 continuum
theory (no self-interacting field, except maybe coupled to other fields); (3)
Fix the problem [Klauder].
@ 4 theory:
Caiani et al JPA(98)
[2D, Hamiltonian]; Klauder LMP(03)ht/02 [D > 3,
non-trivial limit]; Albeverio et al JMP(04)
[regularization and continuum lim]; Butera
& Comi PRB(05)hl [high-T expansion].
@ Related topics: Felder & Tkachev
hp/00 [C++
program for expanding universe]; Adib & Almeida PRE(01)ht [kink
dynamics]; Aoki & Kusnezov AP(02)
[non-equilibrium statistical mechanics]; Borasoy & Krebs NPB(06)
[2-loop renormalization].
Fermions, Spinor Fields > s.a. dirac
fields; ising model; spinors and
2-spinors;
spin models; {random lattices below}.
* In lattice
gauge theory: There
is a problem, the absence of a chiral anomaly (no-go Nielsen-Ninomiya theorem); > s.a. spinors.
@ General references: Anthony et al PLB(82)
[Monte Carlo, proposal]; Bullinaria AP(85);
Friedberg et al JMP(94);
Foster & Jacobson ht/03 [massless,
path integral on tetrahedral lattice]; Del Debbio et al JHEP(08) [higher representations].
@ Chiral fermions: Quinn & Weinstein PRD(86);
Bornyakov PTP(98)hl-in,
NPPS(99)hl/98-in;
Zenkin PRD(98)hl [no-go];
Chiu hl/99-in;
Jahn & Pawlowski NPB(02)hl;
Kerler IJMPA(03)
[rev]; Poppitz & Shang a0706 [decoupling of mirror fermions].
@ Supersymmetric models: Fendley et al JPA(03)
[N = 2 susy];
Giedt & Poppitz JHEP(04)
[superfields and renormalization]; Santachiara & Schoutens JPA(05);
Giedt IJMPA(06)
[rev and deconstruction approach]; Takimi a0705 [relationships].
@ Related topics: Jourjine PRD(85)
[Dirac-Kähler fermions]; Bietenholz et al NPB(97)
[staggered, perfect lattice actions]; Polley qp/01;
Master et al qp/02 [energy
spectrum quantum algorithm]; Araki & Moriya RVMP(03)mp/02 [statistical
mechanics]; Inagaki & Suzuki
JHEP(04)hl [Majorana,
Majorana-Weyl, various dimensions]; > s.a. correlations.
Other Theories > s.a. composite and constrained
systems; integrable
systems; lattice gravity and gauge
theory; topological theories.
* Quantum mechanics:
We want to calculate GE(x2, t;
x1, 0) = 
0t 
x(t)
exp[–SE/
],
and we substitute in the action dx/dt by
[x(ti) – x(ti–1)]
/ 
t;
One then uses the Monte Carlo method to integrate over random paths.
@ General references: Ueltschi mp/01-in
[bosonic particles]; Kuzemsky RNC(02) [many-particle].
@ State entanglement: Narnhofer PRA(05)qp/04 [high-T]; Cramer
et al PRA(06)
[area scaling law].
> Related topics: see crystals; wave
function collapse.
Random Lattices > s.a. dirac fields;
ising model;
spin models;
lattice
gauge theory; regge
calculus.
* Idea: Build a lattice
using the Voronoi construction on a uniformly random set of points in space; Can
be done in Riemannian geometry, but not in Lorentzian geometry.
* And localization: Simulation of diffusion processes shows that geometrical
defects, sites with abnormally low or large connectivities, produce localization
of eigenmodes.
* And phase transitions: Some phase transitions observed in simulations
with regular lattices are softened in random lattices (e.g., 10-state Potts
model on 2D lattices), in other systems the critical behavior persists.
@ References: Ren NPB(88)
[massless fermions]; Eynard & Kristjansen NPB(98)
[three-color problem]; Biroli & Monasson
JPA(99)
[localized state description]; Zinn-Justin EPL(00)cm/99,
Kostov NPB(00)ht/99 [6-vertex
model]; Orlandini et al JPA(02)
[thermodynamic self-averaging]; Sanpera et
al PRL(04)
[atomic Fermi-Bose mixtures]; Feng & Siegel PRD(06)ht [superstrings];
Pérez Vicente & Coolen JPA(08) [Ising model].
Other Types of Lattices > s.a. cell
complex; fractal;
graph; network;
regge calculus; Tangle; tiling.
* Non-Abelian lattices:
Ones with a non-Abelian symmetry group; For example, C60.
@ Asymmetric lattices: Csikor & Fodor PLB(96) [SU(2)-Higgs].
@ Cell complexes: Jourjine PRD(85)
[dimensional phase transitions], PRD(86)
[spinors and gauge fields]; Vanderseypen pr(93) [Langevin equation].
@ Supersymmetric lattice: Grosse CMP(97)
[field theory].
@ Non-commutative: Bimonte et al JGP(96)ht/95 [continuum
limit]; Ercolessi
et al RVMP(98)qa/96 [posets
and representations of non-commutative algebras], qa/96 [K-theory];
Balachandran et al MPLA(00)ht/99 [and
fermion doubling]; Häußling AP(02)ht/01.
@ Related topics: Chandrasekharan & Wiese NPB(97)
[quantum links].
Special Concepts and Techniques > s.a. algebraic
quantum field theory;
field theory and types;
green functions in quantum field theory; holography; Many-Body
Systems; Percolation; phase
transition; types of manifolds [discretized].
* Techniques: One uses
methods of statistical and many-body physics, and discrete approximations to
differential equations.
* Coordination sequence: The sequence {S(n)} gives
the number of nodes n links away from a given node.
@ Renormalization group: Balaban CMP(87);
Yamamoto LMP(01); Bertini et al CMP(05) [cluster expansion]; > s.a. specific
theories.
@ Small vs large lattice: Patrascioiu & Seiler PLB(96); Lin et
al PRL(99) [1D Fermi-Pasta-Ulam chain].
@ Continuum limits: de Lyra et al PRD(91);
[differentiability and continuity]; Bimonte et al JGP(96)ht/95;
Nielsen & Rugh2 cd/96,
ht/96-in
[gauge theory].
@ Coordination sequences: Conway & Sloane PRS(97).
@ Monte Carlo: Creutz et al PRP(83);
NS(91)jan5, 41-44; Easther et al
hl/03-in
[modified schemes].
@ Other techniques: Ben-Av & Solomon MPLA(89)
= IJMPA(90);
García et
al PLB(94)
[non-probabilistic]; Markopoulou ht/00 [coarse-graining];
Capitani PRP(03)
[perturbation theory]; Schmidt & Stamatescu MPLA(03)
[matter determinants]; Succi JPA(07)
[Boltzmann discretization, 1+1 Klein-Gordon and Dirac
fields].
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15 jul 2008