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The Finite Temperature Effective Potential for Local Composite Operators

The method of the effective action for the composite operators $Φ^2(x)$ and $Φ^4(x)$ is applied to the termodynamics of the scalar quantum field with $λΦ^4$ interaction. An expansion of the finite temperature effective potential in powers of $\hbar$ provides successive approximations to the free energy with an effective mass and an effective coupling determined by the gap equations. The numerical results are studied in the space-time of one dimension, when the theory is equivalent to the quantum mechanics of an anharmonic oscillator. The approximations to the free energy show quick convergence to the exact result.