A bit of Physics
The first Landscape in physics appeared after the work of Vladimir Gribov. It represents Gribov copies in non-Abelian gauge theories.
Non-Abelian gauge theories are defined on configuration spaces with a redundancy, the gauge orbit, consisting of all
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The first Landscape in physics appeared after the work of Vladimir Gribov. It represents Gribov copies in non-Abelian gauge theories.
Non-Abelian gauge theories are defined on configuration spaces with a redundancy, the gauge orbit, consisting of all
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configurations related by local gauge transformations. In the continuum, Gribov showed in 1977 that standard gauge-fixing conditions are satisfied by multiple inequivalent field configurations (the so-called Gribov copies). The Fundamental Modular Region (FMR), defined
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as the set of absolute minima of the gauge-fixing functional, is bounded by the Gribov horizon where the Faddeev-Popov operator develops zero modes (Singer, 1978, Zwanziger, 1989).
I have imagined the FMR as if it were a geographical zone with many different valleys,
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I have imagined the FMR as if it were a geographical zone with many different valleys,
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but all of them on the same large basin, such that the Gribov horixon is a sort of equipotential rounding that valley.
More concrete, one has a landscape, being the space of all possible gauge configurations as a vast terrain. Because the "terrain" is non-linear and
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More concrete, one has a landscape, being the space of all possible gauge configurations as a vast terrain. Because the "terrain" is non-linear and
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complicated, the functional often has multiple local minima. Each local minimum is a "valley," and each represents a Gribov copy (different mathematical descriptions of the exact same physical state.)
The First Gribov Region is defined as the set of points where the
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The First Gribov Region is defined as the set of points where the
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Faddeev-Popov operator is positive definite. In my analogy the Gribov Horizon is like "ridge". As long as one is inside the "valley," the ground is curving upward in every direction (technically, the Hessian is positive). The Gribov Horizon is the boundary where the "ground"
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becomes flat in at least one direction (the first zero eigenvalue). Beyond that horizon, the terrain starts to slope downward into another valley. The Gribov Region: Contains all the "valleys" (local minima).
While the Gribov Horizon defines the limit of the first valley,
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While the Gribov Horizon defines the limit of the first valley,
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the FMR is the "deepest part of the deepest valley." It is the set of absolute minima of the gauge functional; it is the FMR is the unique "lowest basin" in the center of the terrain.
In my analogy, the "equipotential" aspect is related to the horizon that is defined by
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In my analogy, the "equipotential" aspect is related to the horizon that is defined by
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the point where the lowest eigenvalue of the Faddeev-Popov operator M(A)=−∂⋅D(A) hits zero: det(M(A))=0
Following with the analogy, at high energies (short distances), the terrain is simple. But at low energies (long distances), the "terrain" becomes incredibly rugged.
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Following with the analogy, at high energies (short distances), the terrain is simple. But at low energies (long distances), the "terrain" becomes incredibly rugged.
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The Gribov horizon effectively acts as a "wall" that modifies the behavior of gluons, potentially explaining why we never see them in isolation.
10/10
(I love analogies)
10/10
(I love analogies)
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