d 3 x\u03b2 x h \u03a9 \u03c6 x t \u03a9 i \u03c6 x 523 53 with respect to \u03b1 \u03b2 x and minimizing it with

D 3 xβ x h ω φ x t ω i φ x 523 53 with respect

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d 3 ~xβ ( ~x ) ( h Ω | φ ( ~x, t ) | Ω i - φ 0 ( ~x )) (5.23) 53
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with respect to α , β ( ~x ), and minimizing it with respect to unconstrained | Ω i . We obtain in particular H | Ω i = α | Ω i + Z d 3 ~xβ ( ~x ) φ ( ~x, t ) | Ω i . (5.24) This equation can be read as one that determines | Ω i given α and β ( ~x ). α and β ( ~x ) are then adjusted so that (5.22) is satisfied (at a given time t ). We can now interpret β ( ~x ) as an external current, and H - Z d 3 ~xβ ( ~x ) φ ( ~x, t ) (5.25) as the Hamiltonian in the presence of the current β ( ~x ), derived from the modified action Z d 4 x L + β ( ~x ) φ ( ~x, t ) . (5.26) α is the ground state energy in the presence of the current β ( ~x ). This follows from the assumption that h Ω | H | Ω i is minimized by our | Ω i . This ground state energy α can also be computed from the transition amplitude h 0 , out | 0 , in i β = e - iαT = e iW [ β ] , (5.27) where the current β ( ~x ) is turned on adiabatically, and turned off adiabatically after a long time T . In this adiabatic process the vacuum changes into the new ground state with respect to the current modified Hamiltonian. The result is nothing but our generating function e iW [ β ] . Now we can derive the minimal energy h Ω | H | Ω i = α h Ω | Ω i + Z d 3 ~xβ ( ~x ) h Ω | φ ( ~x, t ) | Ω i = α + Z d 3 ~xβ ( ~x ) φ 0 ( ~x ) = 1 T - W [ β ] + Z d 4 ( ~x ) φ 0 ( ~x ) = - 1 T Γ[ φ 0 ] = V 3 V ( φ 0 ) . (5.28) This result shows that the effective potential V ( φ 0 ) has the interpretation of the min- imal energy subject to the constraint h φ ( x ) i = φ 0 . Note that the second order derivative of V ( φ 0 ) in φ 0 is the inverse of the two-point function of φ in the matrix sense, evaluated at zero momentum. The latter is positive definite in the matrix sense. This implies that V ( φ 0 ) is a convex function. Suppose we have a scalar field theory with a quartic coupling, and negative m 2 , in its classical action. The potential is still bounded from below, but is not convex. What we just 54
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saw is that the quantum effective potential must be convex, seemingly contradicting (5.20). In fact, in deriving (5.20) we have assumed a stable vacuum with vanishing expectation value of φ ( x ), which is not the case when m 2 < 0. In this case, there are two vacuum states | Ω 1 i and | Ω 2 i , in which the expectation value of φ ( x ) (to leading order in perturbation theory) is at the one of the two minima of the classical potential, h φ ( x ) i = φ ± ≡ ± s 6 | m 2 | g . (5.29) If we restrict h φ ( x ) i to lie between φ - and φ + , we can minimize the energy with a state of the form | Ω i = c 1 | Ω 1 i + c 2 | Ω 2 i , h φ i Ω = | c 1 | 2 φ - + | c 2 | 2 φ + ( φ - , φ + ) . (5.30) Thus the true effective potential V ( φ 0 ) is a constant in between φ - and φ + ! We see that perturbation theory fails to capture this effective potential (in fact, the one-loop calculation would give a complex effective potential in between φ - and φ + ). 5.3 Renormalization of nonabelian gauge theory We will now deal with the renormalization of nonabelian gauge theory by studying the 1PI effective action Γ[ A μ , ψ, η, ¯ η ] (5.31) where ψ denotes matter fields (transforming in some representation R of the gauge group G ), and η, ¯ η are the Fadeev-Popov ghosts. Γ is computed by shifting A μ A 0 μ + A μ , etc., and summing up 1PI diagrams of the quantum fluctuations
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