1. Štefan Olejník Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Simple (approximate) YM vacuum wave-functional in 2+1 dimensions from gauge invariance J. Greensite, ŠO: Confinement from gauge invariance in 2+1 dimensions, archive/0609nnn (to appear probably this week). [Topic announced in the program can be found in: J. Greensite, ŠO: PR D74 (2006) 014502, hep-lat/0603024.]

2. ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia QCD: Facts and Prospects, Oberwölz, Austria, September 12, 2005 2 Introduction Confinement is the property of the vacuum of quantized non-abelian gauge theories. In the hamiltonian formulation in D=d+1 dimensions and temporal gauge: Strong-coupling lattice-gauge theory – systematic expansion: Greensite (1980)

3. ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia QCD: Facts and Prospects, Oberwölz, Austria, September 12, 2005 3 At large distance scales one expects: Greensite (1979) Greensite, Iwasaki (1989) Karabali, Kim, Nair (1998) Property of dimensional reduction: Computation of a spacelike loop in d+1 dimensions reduces to the calculation of a Wilson loop in Yang-Mills theory in d Euclidean dimensions.

4. ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia QCD: Facts and Prospects, Oberwölz, Austria, September 12, 2005 4 At weak couplings, one would like to similarly expand: For g!0 one has simply: Wheeler (1962)

5. ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia QCD: Facts and Prospects, Oberwölz, Austria, September 12, 2005 5 A possibility to enforce gauge invariance: No handle on how to choose f’s.

6. ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia QCD: Facts and Prospects, Oberwölz, Austria, September 12, 2005 6 Suggestion Building blocks of a systematic expansion: covariant derivatives and covariant Green’s functions of local operators. Then the choice of Q 0 is essentially unique: If the kernel is of finite range, one would have far enough:

7. ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia QCD: Facts and Prospects, Oberwölz, Austria, September 12, 2005 7 Self-consistency check in 2+1 dimensions In 2 dimensions there is no Bianchi identity; the field-strength correlation length can only arise from a non-zero range of the vacuum kernel. The range of the vacuum kernel determines the correlation length of vacuum fluctuations, but vacuum fluctuations determine the range of the kernel. Take a vacuum fluctuation typical for massless free field theory:

8. ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia QCD: Facts and Prospects, Oberwölz, Austria, September 12, 2005 8 Compute:

9. ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia QCD: Facts and Prospects, Oberwölz, Austria, September 12, 2005 9 Result: The kernel is finite range, in any stochastic background. The vacuum functional has the dimensional-reduction form, therefore it is confining!

10. ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia QCD: Facts and Prospects, Oberwölz, Austria, September 12, 2005 10 Possible explanation The result – at first sight rather surprising – may have a natural explanation. Weak (Anderson) localization: Consider a non-relativistic particle moving in a stochastic potential in one and two dimensions. Eigenstates of the corresponding hamiltonian are all localized, no matter how the weak the stochastic potential may be. Then the inverse operator to the hamiltonian has a finite range. Our covariant laplacian is fully analogous to the Anderson hamiltonian.

11. ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia QCD: Facts and Prospects, Oberwölz, Austria, September 12, 2005 11 How far is the simple vacuum from the true one? Estimate of the glueball mass: compute the range of the kernel in true vacuum configurations from MC simulations of lattice YM theory in D=3 – estimate glueball mass as inverse of the kernel range. Meyer, Teper (2003)

12. ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia QCD: Facts and Prospects, Oberwölz, Austria, September 12, 2005 12 Consistent with

13. ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia QCD: Facts and Prospects, Oberwölz, Austria, September 12, 2005 13 Conclusions and open questions We have proposed a simple approximate form of the confining YM vacuum wave-functional in 2+1 dimensions. Essential ingredients: known g=0 limit, gauge invariance, and (probably) the phenomenon of weak localization. This cannot be the whole story – with the simple wave-functional one would expect Casimir scaling of higher-representation asymptotic string tensions, while asymptotic string tensions should in fact depend only on the N-ality of representations. The simple zeroth-order state is clearly missing the center-vortex or domain structure which has to dominate the vacuum at sufficiently large scales. Hope for 3+1 dimensions: Adjoint covariant laplacian has an interval of localized states also in D=4 Euclidean dimensions, so the relevant kernel in the vacuum wave-functional may again be of finite range. We should have some answer in the near future, though the task is numerically much more challenging. Greensite, Kovalenko, Polikarpov, Syritsyn, Zakharov, ŠO, hep-lat/0606008 No obvious method for systematic improvement.

14. ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia QCD: Facts and Prospects, Oberwölz, Austria, September 12, 2005 14 Dimensions