I am studying instantons in quantum mechanics. My question is regarding the the zero mode of the fluctuation determinant that we get because the solution for the instanton breaks time translation invariance. The solution is given by $${x(\tau)=-x_0\tanh(\tau-\tau_0)}$$ for the potential ${V(x)=\frac{m}{2x_0 ^2}(x^2-x_0 ^2)^2}$. I can see that it breaks time translation invariance because there is an arbitrary integration constant ${\tau_0}$ in the solution but what I don't get is what the zero mode actually means. Do different values of the constant give different instanton solutions? If so, how exactly do these solutions differ physically?
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