Nonstandard expansion techniques for the finite-temperature effective potential in λ${\phi }^4$ quantum field theory

The finite-temperature effective potential in scalar quantum field theory in n-dimensional space-time is calculated using different expansions of a path-integral representation for the partition function. First-order results in an optimized and mean-field expansion are compared with a conventional one-loop result. In one dimension the comparison with the finite-temperature effective potential calculated numerically shows that only the mean-field method gives qualitative agreement in the entire range of Lagrangian parameters. In four dimensions the mean-field method seems also to be most reliable. The thermal properties of the renormalized theory indicate that a scalar theory is noninteracting.