The operators that multiply the zero modes in the expansion play a special role. Each term in the sum has a $c$-number valued function of the "space" or "spacetime" and it is multiplied by an operator corresponding to the mode. But its real meaning is that the whole quantum field should be expanded into modes - zero modes and non-zero modes. The zero mode is a $c$-number-valued solutions of the right differential equation with the right operator. Any fermionic particles in 3 1 dimensions that has a zero mode will actually appear as a factor in a complicated product of many fermionic fields - and this multi-product interaction term is actually induced by the instanton. Then this total Dirac operator will again have some zero modes on $R^$ the zero mode will be nonzero and nontrivial especially in the region where the instanton is located. And include the gauge field term to the Dirac operator. But consider a different background, e.g. We may continue with solutions to Dirac and similar equations. In heterotic string theory, it is interpreted as the number of generations (minus the number of antigenerations) of leptons and quarks we obtain in 3 1 dimensions. It is invariant under all continuous transformations. So the number of zero modes of the Dirac equation (more precisely the number of solutions minus the number of "anti-solutions" with some opposite charge or chirality etc.) is a special number known as the "index". When one discusses supersymmetry, the non-zero modes typically come in pairs and it can be proved, but the zero modes - massless particles in 3 1 dimensions, as I mentioned - may come non-paired. All the spinors $\psi$ on the 6-dimensional manifold may be written as a combination of the "modes", some of which are the zero modes but most of them are non-zero modes. These non-zero modes are eigenstates of an operator, so if you include all possible eigenvalues and the right degeneracy, you may reconstruct any function - by Fourier expansions. They are eigenvalues of the same operators with different eigenvalues.įor example, the operator $-d^2/d\sigma^2$ on the string may act on functions such as $\exp(in\sigma)$ and it will produce $n^2$ - the same with sines and cosines. In this context, it's useful to mention what are the "non-zero modes" or "normal modes". In both cases above, there was an operator - either $-d^2/d\sigma^2$, or the Dirac operator $D^\mu \gamma_\mu$, that annihilated the solution and that's why the solution was ultimately called a "zero mode". Such a field will then behave as a massless particle in 4 dimensions - it will solve the 3 1-dimensional equation with a vanishing mass. But if you consider both 6 compact and the 3 1 non-compact coordinates, one may consider spinors $\psi$ whose dependence on the 6 compact coordinates is given by the "zero mode" - the solution of the equation above. The equation above only included functions of the compact coordinates. The solutions $\psi$ to this massless Dirac equations are just zero modes. For example, we may have a Dirac equation If one solves differential equations on compact manifolds, such as the Calabi-Yau manifolds, one may find zero modes, too. And indeed, in some approximation, the string may be considered to be a particle, and the zero modes are the usual properties of the particle. These two zero modes, the center-of-mass position and the total momentum, are the quantities we would normally associate with a point-like particle. The dual quantity is the "zero mode" of the momentum - the integral of $P_\mu(\sigma)$: that's nothing else than the total momentum. The "zero mode" of $X$ is then nothing else than the average over the string - the center-of-mass coordinate. When one discusses the motion of an extended object, such as the string, the position of a point along the string - the point is parameterized by $\sigma$ - is $X^\mu(\sigma)$. Moshe is of course completely right, but let me be a bit less abstract for a while and list a few examples that cover most of the contexts where the term "zero mode" is used.
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