In QFT, the initial and final state is the vacuum. We can envision an instanton as a sort of path that correlates or links initial and final states (that have different topological winding numbers) and since those winding numbers can be infinite in extent, the vacuum not only becomes a state of lowest energy but also an aggregation of an infinite number of apparently homogeneous yet topologically different vacua. The lawn-mowing analogy is helpful as the power-lead leading over the tops of trees and shrubs act as barriers to movement of the electric lawn-mower. In field theory, this is equivalent to an energy barrier, instantons surpass this barrier via quantum tunnelling (linking one distinct topological state to another, measured by the θ-parameter. But how do instantons solve the U(1) problem? We could just invoke a respectable particle to account for the symmetry breaking like the η meson; but it's a Goldstone boson and the particle with the next mass up is too heavy. Instantons, like goldilocks, give just the right symmetry disturbance. A massless spiral of gluons and inverts right-handed quarks to left-handed ones. Such an inversion of handedness breaks the chiral symmetry and deals with the additional U(1) symmetry without the need for particles.
Thursday, 5 December 2013
Instantons- Topology of the Vacuum
In QFT, the initial and final state is the vacuum. We can envision an instanton as a sort of path that correlates or links initial and final states (that have different topological winding numbers) and since those winding numbers can be infinite in extent, the vacuum not only becomes a state of lowest energy but also an aggregation of an infinite number of apparently homogeneous yet topologically different vacua. The lawn-mowing analogy is helpful as the power-lead leading over the tops of trees and shrubs act as barriers to movement of the electric lawn-mower. In field theory, this is equivalent to an energy barrier, instantons surpass this barrier via quantum tunnelling (linking one distinct topological state to another, measured by the θ-parameter. But how do instantons solve the U(1) problem? We could just invoke a respectable particle to account for the symmetry breaking like the η meson; but it's a Goldstone boson and the particle with the next mass up is too heavy. Instantons, like goldilocks, give just the right symmetry disturbance. A massless spiral of gluons and inverts right-handed quarks to left-handed ones. Such an inversion of handedness breaks the chiral symmetry and deals with the additional U(1) symmetry without the need for particles.
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