QFT on a Lattice

1. QFT on a Lattice Todd Kempel--Phys 624

2. Why a lattice? • Most symmetries can still be preserved. • A non-perturbative approach is possible • Computing algorithms can be written naturally Todd Kempel--Phys 624

3. L x a Discretizing in General Simple! yn take a 0, L   Problem: No guarantee that symmetries of the action will be maintained—even in the limit a->0. Also, we will be helped greatly if we require periodicity (next slide). Todd Kempel--Phys 624

4. Why Require Periodicity? Parallelization! Implementation becomes much more reasonable if we can run on multiple processors are the same time and combine results at the end. Todd Kempel--Phys 624

5. Give it a try with QED • First Step: Find a gauge invariant action that satisfies periodic boundary conditions. • Second Step: Discretize correlation functions. • If we can do these two things without sacrificing all symmetries we can, in principle at least, calculate expectation values of operators (observables) • If we get QED right, QCD is ‘simply’ a matter of using the same prescription in the non-Abelian case (i.e. with extra D.O.F.) Main Reference will be: Phys. Rev. D 10, 2445 (1974): Wilson - Confinement of quarks Todd Kempel--Phys 624

6. Give it a try with QED We work in the Euclidean metric (complex time) for convenience so that there is no factor of ‘i’ on the derivative term of the action . No Gauge fixing yet… we need to retry it in the discretized version Todd Kempel--Phys 624

7. Gauge Fixing Continuous with } g(x) arbitrary Todd Kempel--Phys 624

8. Gauge Fixing Discrete } g, yn arbitrary But how do we construct a covariant derivative? Todd Kempel--Phys 624

9. Gauge Fixing Discrete } g, yn arbitrary We need to work a bit harder to make the derivative gauge invariant and require periodicity: n is a dummy index--Shift n→n+μ in second term of derivative Todd Kempel--Phys 624

10. FμνFμν term Continuous Discrete Periodicity can be preserved with i.e. when a2 g Fμν= 2 π the action doesn’t change (a2 g Fμν acts like an angular variable) We also have Todd Kempel--Phys 624

11. FμνFμν term Constant term: Irrelevant in the action First Order Term: 0 since Fμν is odd Third and Higher Order Terms: 0 in continuum limit since integral only divides out a4 and we have a→0 as we want Todd Kempel--Phys 624

12. A Gauge Invariant Discretized Action with Compare to: Todd Kempel--Phys 624

13. Short Review of Path Integral Formalism Time evolution of the wave equation can be written as Good Ole’ Fashioned Quantum Mechanics where G is the Green’s Function describing the evolution of Ψ(x,t) with We define an ‘n-point’ Green’s Function to determine the time evolution of a state with a ‘partition function’ Todd Kempel--Phys 624

14. Discretized Version For a lattice theory, we only care about correlation functions between fixed lattice sites (i.e. correlation functions involving our gauge bosons) ~Fμν ~Aμ Integrations only happen over one site—No need for more than this because of periodicity Todd Kempel--Phys 624

15. A Pictorial Summary ψn Use a given Action (I’ve presented a simple one here) and carry out path integrations over all paths on the lattice Anμ Todd Kempel--Phys 624

16. A Pictorial Summary ψn L Anμ a Hopefully take a ~ 0, L  ~  Todd Kempel--Phys 624

17. A Pictorial Summary ψn Anμ Connect lattices from different processors (i.e. L→~∞) Todd Kempel--Phys 624

18. On to non-Abelian Gauge Theories Lattice Visualization of QCD vacuum from http://hermes.physics.adelaide.edu.au/theory/staff/leinweber/VisualQCD/Nobel/index.html Todd Kempel--Phys 624

19. Problems with Lattice Gauge Theory • For realistic systems, enormous computing time is necessary. • Gauge invariance (which we have shown here) is not always necessary—but other symmetries can be very hard to maintain. • Lorentz invariance is often impossible. Todd Kempel--Phys 624

20. Transition Temperature at RHIC hep-lat/0609040 Recent Result from the Lattice Todd Kempel--Phys 624

21. Thank You! Todd Kempel--Phys 624