Convexity of the effective potential
Technical Report
·
OSTI ID:6557202
The effective potential V(phi) in field theories is a convex function of phi. V(lambda phi/sub 1/ + (1 - lambda)phi/sub 2/) less than or equal to lambdaV(phi/sub 1/) + (1 - lambda)V(phi/sub 2/), 0 less than or equal to lambda less than or equal to 1, all phi/sub 1/, phi/sub 2/. A linear interpolation of V(phi) is always larger than or equal to V(phi). There are numerous examples in the tree approximation and in perturbation theory for which this is not the case, the most notorious example being the double dip potential. More complete solutions may or may not show this property automatically. However, a non-convex V(phi) simply indicates that an unstable vacuum state was used in implementing the definition of V(phi). A strict definition will instruct one to replace V(phi) with its linear interpolation in such a way as to make it convex. (Alternatively one can just as well take the view that V(phi) is undefined in these domains.) In this note, attention is called to a very simple argument for convexity based on a construction described by H. Callen in his classic book Thermodynamics.
- Research Organization:
- Louisiana State Univ., Baton Rouge (USA). Dept. of Physics and Astronomy
- DOE Contract Number:
- AS05-77ER05490
- OSTI ID:
- 6557202
- Report Number(s):
- DOE/ER/05490-32; ON: DE83005265
- Country of Publication:
- United States
- Language:
- English
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Related Subjects
645400* -- High Energy Physics-- Field Theory
658000 -- Mathematical Physics-- (-1987)
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
CONVEX MANIFOLDS
FIELD THEORIES
INTERPOLATION
MATHEMATICAL MANIFOLDS
NUMERICAL SOLUTION
PERTURBATION THEORY
VACUUM STATES
658000 -- Mathematical Physics-- (-1987)
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
CONVEX MANIFOLDS
FIELD THEORIES
INTERPOLATION
MATHEMATICAL MANIFOLDS
NUMERICAL SOLUTION
PERTURBATION THEORY
VACUUM STATES