d 3 ~xβ ( ~x ) ( h Ω | φ ( ~x, t ) | Ω i - φ 0 ( ~x )) (5.23) 53
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 xβ ( ~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
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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