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Authors: Scientific American Editors
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coordinate system may be needed. The lines of longitude and latitude employed on the earth constitute such a system, since they follow the curvature of the earth.
In such a system a local coordinate transformation can readily be imagined. Suppose height is defined as vertical distance from the ground rather than from mean sea level. The digging of a pit would then alter the coordinate system, but only at those points directly over the pit. The digging itself represents the local coordinate transformation. It would appear that the laws of physics (or the rules of navigation) do not remain invariant after such a transformation, and in a universe without gravitational forces that would be the case. An airplane set to fly at a constant height would dip suddenly when it flew over the excavation, and the accelerations needed to follow the new profile of the terrain could readily be detected.
As in electrodynamics, local symmetry ean be restored only by adding a new field to the theory; in general relativity the field is of course that of gravitation.
The presence of this field offers an alternative explanation of the accelerations detected in the airplane: they could result not from a local change in the coordinate grid but from an anomaly in the gravitational field. The source of the anomaly is of no concern: it could be a concentration of mass in the earth or a distant object in space. The point is that any local transformation of the coordinate system could be reproduced by an appropriate set of gravitational fields.
The pilot of the airplane could not distinguish one effect from the other.
Both Maxwell's theory of electromagnetism and Einstein's theory of gravitation owe much of their beauty to a local gauge symmetry; their success has long been an inspiration to theoretical physicists. Until recently theoretical acco unts of the other two forces in nature have been less satisfactory. A theory of the weak force formulated in the 1930's by Enrico Fermi accounted for some basic features of the weak interaction,
but the theory lacked local symmetry.
The strong interactions seemed to be a jungle of mysterious fields and resonating particles. It is now clear why it took so long to make sense of these forces: the necessary local gauge theories were not understood.
The first step was taken in 1954 in a theory devised by C. N. Yang and Robert L. Mills, who were then at the Brookhaven National Laboratory. A similar idea was proposed independently at about the same time by R. Shaw of the University of Cambridge. Inspired by the success of the other gauge theories,
these theories begin with an established global symmetry and ask what the consequences would be if it were made a local symmetry.
The symmetry at issue in the Yang-Mills theory is isotopic-spin symmetry,
the rule stating that the strong interactions of matter remain invariant (or nearly so) when the identities of protons and neutrons are interchanged. In the global symmetry any rotation of the internal arrows that indicate the isotopic-spin state must be made simultaneously everywhere. Postulating a local symmetry allows the orientation of the arrows to vary independently from place to place and from moment to moment. Rotations of the arrows can depend on any arbitrary function of position and time.
This freedom to choose different conventions for the identity of a nuclear particle in different places constitutes a local gauge symmetry.
As in other instances where a global symmetry is converted into a local one,
the invariance can be maintained only if something more is added to the theory.
Because the Yang-Mills theory is more complicated than earlier gauge theories it turns out that quite a lot more must be added. When isotopic-spin rotations are made arbitrarily from place to place, the laws of physics remain invariant only if six new fields are introduced. They are all vector fields, and they all have infinite range.
The Yang-Mills fields are constructed on the model
In such a system a local coordinate transformation can readily be imagined. Suppose height is defined as vertical distance from the ground rather than from mean sea level. The digging of a pit would then alter the coordinate system, but only at those points directly over the pit. The digging itself represents the local coordinate transformation. It would appear that the laws of physics (or the rules of navigation) do not remain invariant after such a transformation, and in a universe without gravitational forces that would be the case. An airplane set to fly at a constant height would dip suddenly when it flew over the excavation, and the accelerations needed to follow the new profile of the terrain could readily be detected.
As in electrodynamics, local symmetry ean be restored only by adding a new field to the theory; in general relativity the field is of course that of gravitation.
The presence of this field offers an alternative explanation of the accelerations detected in the airplane: they could result not from a local change in the coordinate grid but from an anomaly in the gravitational field. The source of the anomaly is of no concern: it could be a concentration of mass in the earth or a distant object in space. The point is that any local transformation of the coordinate system could be reproduced by an appropriate set of gravitational fields.
The pilot of the airplane could not distinguish one effect from the other.
Both Maxwell's theory of electromagnetism and Einstein's theory of gravitation owe much of their beauty to a local gauge symmetry; their success has long been an inspiration to theoretical physicists. Until recently theoretical acco unts of the other two forces in nature have been less satisfactory. A theory of the weak force formulated in the 1930's by Enrico Fermi accounted for some basic features of the weak interaction,
but the theory lacked local symmetry.
The strong interactions seemed to be a jungle of mysterious fields and resonating particles. It is now clear why it took so long to make sense of these forces: the necessary local gauge theories were not understood.
The first step was taken in 1954 in a theory devised by C. N. Yang and Robert L. Mills, who were then at the Brookhaven National Laboratory. A similar idea was proposed independently at about the same time by R. Shaw of the University of Cambridge. Inspired by the success of the other gauge theories,
these theories begin with an established global symmetry and ask what the consequences would be if it were made a local symmetry.
The symmetry at issue in the Yang-Mills theory is isotopic-spin symmetry,
the rule stating that the strong interactions of matter remain invariant (or nearly so) when the identities of protons and neutrons are interchanged. In the global symmetry any rotation of the internal arrows that indicate the isotopic-spin state must be made simultaneously everywhere. Postulating a local symmetry allows the orientation of the arrows to vary independently from place to place and from moment to moment. Rotations of the arrows can depend on any arbitrary function of position and time.
This freedom to choose different conventions for the identity of a nuclear particle in different places constitutes a local gauge symmetry.
As in other instances where a global symmetry is converted into a local one,
the invariance can be maintained only if something more is added to the theory.
Because the Yang-Mills theory is more complicated than earlier gauge theories it turns out that quite a lot more must be added. When isotopic-spin rotations are made arbitrarily from place to place, the laws of physics remain invariant only if six new fields are introduced. They are all vector fields, and they all have infinite range.
The Yang-Mills fields are constructed on the model
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