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I am studying one of the paper of Sidney Coleman, "Fate of the False Vacuum. II. First quantum corrections". Just before eq. (2.18) he says

"Because of time translation invariance, this equation necessarily possessed an eigenfunction of eigenvalue zero".

Could you help me in understanding this statement?

Edit: the paper is here. In particular, he says that the equation $$\frac{d^2x_n}{d {t}^2}+V''(\bar{x})x_n=\lambda_n x_n \tag{2.9}$$ must have a zero eigenvalue due to translational invariance of the theory. He then says which is this eigenfunction and I have checked that its eigenvalue is zero, after a brief computation. However, I still do not understand why expected this zero eigenvalue.

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  1. Ref. 1 is interested in the limit where the characteristic duration of an instanton/bounce is much smaller than the full temporal integration region $[t_i,t_f]$, cf. the first column of p. 1764. (This is known as the dilute instanton gas model.) In other words, we can effectively consider an unbounded integration region $\mathbb{R}$ from $t_i=-\infty$ to $t_f=\infty$. This is time translation invariant, unlike its bounded counterparts.

  2. Combined with the time translation invariance of the Lagrangian $L$, this implies that the instant of a instanton/bounce profile is arbitrary, i.e. the instanton/bounce has a moduli parameter along the time-axis, i.e. the classical path (satisfying the pertinent Dirichlet boundary conditions) is not unique, i.e. the EL equation has a zero-mode.

  3. See also this related Phys.SE post, which deals with the very same Ref. 1.

References:

  1. C.G. Callan, Jr. & S. Coleman, Fate of the false vacuum. II. First quantum corrections, Phys. Rev. D 16 (1977) 1762.
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