ps6 - Physics 509 Relativistic Quantum Field Theory Problem...

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Physics 509: Relativistic Quantum Field Theory Problem Set 5 Due Tuesday, 22 November 2011 23. Spontaneous symmetry breaking: scalars. A theory with a higher amount of symmetry in the action than in the a particular ground-state is said to undergo spontaneous symmetry breaking: the dynamics of the theory break the symmetry group down to a smaller sub-group. Spontaneous symmetry breaking (SSB) is of crucial importance in situations where field theories encapsulate the dynamical degrees of freedom of a system, notably in second-order phase transitions where order parameters are described by fluctuating quantum fields. We will work out some basic and important aspects of SSB below. a) Consider the following Lagrangian for a real scalar field, φ : L = 1 2 ( μ φ )( μ φ ) + | m 2 | φ 2 - λ 4 φ 4 . (1) Note the wrong sign for the mass term. Is this theory stable about h φ i = 0? Where are the stable classical minima? This system is invariant under a discrete symmetry.
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