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How Effective is the Effective Potential?

Motivated by bubble nucleation in first order phase transitions, we question the validity of the effective potential for inhomogeneous configurations. In an attempt to get some insight into the importance of derivative terms, we analyze a simple model, a kink in (1+1) dimensions and zero temperature. We evaluate the energy shift from the quantum fluctuations about the non-uniform background ({\it i.e.}, the effective action) and compare it to the energy from the effective potential. Our results clearly show that for inhomogeneous configurations it may be inadequate to omit derivative terms and confine oneself to the effective potential. We then couple the kink field to an additional scalar field and perform the same comparison. The addition of the second field allows us to vary the mass of the fluctuations and their coupling to the underlying kink. If the mass of the second field is large, it does not feel the inhomogeneities in the kink field and consequently does not give rise to important derivative corrections in the effective action. In contrast, if the mass is small, derivative terms are significant and the effective potential fails. In the latter regime we can, however, rely on the Born approximation to calculate the effective action.