Transcript Document

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]
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A few quotes plus a bit of philately
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“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 (?)]
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“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.
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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)]
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SLOVENSKO
1€
fyzik
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Introduction
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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:
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At large distance scales one expects:
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Halpern (1979), Greensite
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Greensite, Iwasaki
Kawamura, Maeda, Sakamoto
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Karabali, Kim, Nair
(1979)
(1989)
(1997)
(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.
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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)
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Warm-up example: Abelian ED
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Free-field limit (g!0)
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Zero-mode, strong-field limit
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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
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Solution (up to 1/V corrections):
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Now the proposed vacuum state coincides with this solution in the strong-field limit,
assuming
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The covariant laplacian is then
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In the above limit:
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Dimensional reduction and confinement
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What about confinement with such a vacuum state?
Define “slow” and “fast” components using a mode-number cutoff:
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Then:
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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):
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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
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Non-zero m is energetically preferred
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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:
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Abelian free-field limit: minimum at m2 = 0 → 0.
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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!
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Calculation of the 0++ glueball mass (mass gap)
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To extract the mass gap, one would like to compute
in the probability distribution:
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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.
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Summary (of apparent pros)
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Our simple approximate form of the confining YM vacuum
wavefunctional in 2+1 dimensions has the following properties:
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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).
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Open questions (or contras?)
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Can one improve (systematically) our vacuum wavefunctional
Ansatz?
Can one make a more reliable variational estimate of m?
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How to go to 3+1 dimensions?
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Much more challenging (Bianchi identity, numerical treatment very CPU
time consuming).
The zero-mode, strong-field limit argument valid (in certain approximation) also in D=3+1.
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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.