20.-21.11.2009

Erwin Schrödinger Symposium 2009, Prague 1

Ground-state solution of the Yang–Mills Schrödinger equation in 2+1

dimensions

(in collaboration with Jeff Greensite, SFSU)

J. Greensite, ŠO, Phys. Rev. D 77 (2008) 065003, arXiv:0707.2860

[hep-lat]

20.-21.11.2009

Erwin Schrödinger Symposium 2009, Prague 2

A few quotes plus a bit of philately

“The mathematical framework of quantum theory has passed countless successful tests and is now universally accepted as a consistent and accurate description of all atomic phenomena.” [Erwin Schrödinger (?)]

20.-21.11.2009

Erwin Schrödinger Symposium 2009, Prague 3

“QCD field theory with six flavors of quarks with three colors, each represented by a Dirac spinor of four components, and with eight four-vector gluons, is a quantum theory of amplitudes for configurations each of which is 104 numbers at each point in space and time. To visualize all this qualitatively is too difficult. The thing to do is to take some qualitative feature to try to explain, and then to simplify the real situation as much as possible by replacing it by a model which is likely to have the same qualitative feature for analogous physical reasons.

The feature we try to understand is confinement of quarks. We simplify the model in a number of ways.

First, we change from three to two colors as the number of colors does not seem to be essential.

Next we suppose there are no quarks. Our problem of the confinement of quarks when there are no dynamic quarks can be converted, as Wilson has argued, to a question of the expectation of a loop integral. Or again even with no quarks, there is a confinement problem, namely the confinement of gluons. […]

The next simplification may be more serious. We go from the 3+1 dimensions of the real world to 2+1. There is no good reason to think understanding what goes on in 2+1 can immediately be carried by analogy to 3+1, nor even that the two cases behave similarly at all. There is a serious risk that in working in 2+1 dimensions you are wasting your time, or even that you are getting false impressions of how things work in 3+1. Nevertheless, the ease of visualization is so much greater that I think it worth the risk. So, unfortunately, we describe the situation in 2+1 dimensions, and we shall have to leave it to future work to see what can be carried over to 3+1.” [Richard P. Feynman (1981)]

20.-21.11.2009

Erwin Schrödinger Symposium 2009, Prague 4

SLOVENSKO1 €

fyzik

20.-21.11.2009

Erwin Schrödinger Symposium 2009, Prague 5

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:

20.-21.11.2009

Erwin Schrödinger Symposium 2009, Prague 6

At large distance scales one expects:

Halpern (1979), Greensite (1979) Greensite, Iwasaki (1989)

Kawamura, Maeda, Sakamoto (1997) Karabali, Kim, Nair (1998)

Property of dimensional reduction: Computation of a spacelike Wilson (-Wegner) loop in d+1 dimensions reduces to the calculation of a loop in Yang-Mills theory in d Euclidean dimensions.

20.-21.11.2009

Erwin Schrödinger Symposium 2009, Prague 7

Suggestion for an approximate vacuum wavefunctional

“It is normal for the true physicist not to worry too much about mathematical rigor. […] This goes with a certain attitude of physicists towards mathematics: loosely speaking, they treat mathematics as a kind of prostitute. They use it in an absolutely free and shameless manner, taking any subject or part of a subject, without having the attitude of the mathematician who will only use something after some real understanding.”

(Alain Connes, interview with C. Goldstein and G. Skandalis, EMS Newsletter, 03/2008)

20.-21.11.2009

Erwin Schrödinger Symposium 2009, Prague 8

Warm-up example: Abelian ED

20.-21.11.2009

Erwin Schrödinger Symposium 2009, Prague 9

20.-21.11.2009

Erwin Schrödinger Symposium 2009, Prague 10

Free-field limit (g!0)

20.-21.11.2009

Erwin Schrödinger Symposium 2009, Prague 11

Zero-mode, strong-field limit

D. Diakonov (private communication)

Let’s assume we keep only the zero-mode of the A-field, i.e. fields constant in space, varying in time. The lagrangian is

and the hamiltonian operator

Solution (up to 1/V corrections):

20.-21.11.2009

Erwin Schrödinger Symposium 2009, Prague 12

Now the proposed vacuum state coincides with this solution in the strong-field limit, assuming

The covariant laplacian is then

In the above limit:

20.-21.11.2009

Erwin Schrödinger Symposium 2009, Prague 13

Dimensional reduction and confinement

What about confinement with such a vacuum state? Define “slow” and “fast” components using a mode-number cutoff:

Then:

20.-21.11.2009

Erwin Schrödinger Symposium 2009, Prague 14

Effectively for “slow” components

we then get the probability distribution of a 2D YM theory and can compute the string tension analytically (in lattice units):

Non-zero value of m implies non-zero string tension and confinement! Let’s revert the logic: to get with the right scaling behavior ~ 1/ 2, we need to

choose

20.-21.11.2009

Erwin Schrödinger Symposium 2009, Prague 15

Non-zero m is energetically preferred

Take m as a variational parameter and minimize <H > with respect to m:

Assuming the variation of K with A in the neighborhood of thermalized configurations is small, and neglecting therefore functional derivatives of K w.r.t. A one gets:

20.-21.11.2009

Erwin Schrödinger Symposium 2009, Prague 16

Abelian free-field limit: minimum at m2 = 0 → 0.

20.-21.11.2009

Erwin Schrödinger Symposium 2009, Prague 17

Non-abelian case: Minimum at non-zero m2 (~ 0.3), though a higher value (~ 0.5) would be required to get the right string tension.

Could (and should) be improved!

20.-21.11.2009

Erwin Schrödinger Symposium 2009, Prague 18

Calculation of the 0++ glueball mass (mass gap)

To extract the mass gap, one would like to compute

in the probability distribution:

Looks hopeless, K[A] is highly non-local, not even known for arbitrary fields.

But if - after choosing a gauge - K[A] does not vary a lot among thermalized configurations … then something can be done.

20.-21.11.2009

Erwin Schrödinger Symposium 2009, Prague 19

20.-21.11.2009

Erwin Schrödinger Symposium 2009, Prague 20

Summary (of apparent pros)

Our simple approximate form of the confining YM vacuum wavefunctional in 2+1 dimensions has the following properties: It is a solution of the YM Schrödinger equation in the weak-

coupling limit … … and also in the zero-mode, strong-field limit. Dimensional reduction works: There is confinement (non-zero

string tension) if the free mass parameter m is larger than 0. m > 0 seems energetically preferred. If the free parameter m is adjusted to give the correct string

tension at the given coupling, then the correct value of the mass gap is also obtained.

Coulomb-gauge ghost propagator and color-Coulomb potential come out in agreement with MC simulations of the full theory (not covered in this talk).

20.-21.11.2009

Erwin Schrödinger Symposium 2009, Prague 21

Open questions (or contras?)

Can one improve (systematically) our vacuum wavefunctional Ansatz?

Can one make a more reliable variational estimate of m?

How to go to 3+1 dimensions? Much more challenging (Bianchi identity, numerical treatment

very CPU time consuming). The zero-mode, strong-field limit argument valid (in certain

approxima-tion) also in D=3+1.

I acknowledge support by the Slovak Grant Agency for Science, Project VEGA No. 2/0070/09, by ERDF OP R&D, Project CE QUTE ITMS 26240120009, and via QUTE – Center of Excellence of the Slovak Academy of Sciences.