periodic case or if the double well potential has a re\ufb02ection symmetry then the

# Periodic case or if the double well potential has a

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periodic case, or if the double-well potential has a reﬂection symmetry, then the termsin this series are independent of the minimum we expand around. The degeneracy isnot lifted to any order in perturbation theory. We will choose the classical energy atthe minimum to be zero.Time-dependent saddle points of the Euclidean action, which we will dubinstantons,lead to contributions to the ground-state energy that are of order eS0/g2. Normally,we would neglect these in comparison with the terms in the expansion, but they willgive the leading contribution to the energysplittingsbetween the classically degeneratelevels. In order to have a finite actionS0, in the limitβ→ ∞, the instanton solutionmust have the asymptoticsx(t)x±minast→ ±∞. It must interpolate between twodifferent minima. Clearly this means that|dx/dt| →0 for large|t|.Saddle points of the Euclidean action are given by solutions ofd2xdt2=V(x),which are Newton’s equations in a potentialV, for a particle of unit mass. TheEuclidean energy,12˙x2V, is conserved in this motion. On evaluating it att= ±∞,we find˙x2=2V,which can be solved by quadratures. Notice that, for any pair of minima, there aretwo solutions, related by time reversalt→ −t. We (arbitrarily) call one of them theinstanton, and the other the anti-instanton. Note that, in any relativistic quantum fieldtheory, we always have a kind of time-reversal symmetry, namely TCP. Thus, in thiscontext, every instanton will always have an anti-instanton.Given the form of the solution, we can now understand the term instanton. Expandthe equation of motion around a minimumx(t)=xmin+δ. Then¨δ=V(xmin.SinceVis positive the solutions are rising and falling exponentials. We have to choosethe falling exponential at botht→ ±∞, in order to obey the finite action boundaryconditions. Thus, the instantonx(t)differs from the classical ground state only in1We will not deal with the sub-leading terms inβ. From them, one can extract information about excitedstates. See the book by Zinn-Justin . 20910.2 Instantons in quantum mechanicsthe local vicinity of some pointt1(which may be chosen arbitrarily because of time-translation invariance). It thus has a particle-like aspect, which explains the suffixon,but localized in time rather than space, which is indicated by the prefixinstant. Inhigher-dimensional field theories instantons will be localized in both time and space.We can also reinterpretd-dimensional instantons as localized (d+1)-dimensional staticsolutions. The latter are known assolitons, and actually do define particle states in thequantum theory.Time-translation invariance implies that, in theβ→ ∞limit, we have a continuousset of solutions, labeled byt1. We can write them asx(t)=X(t+t1), whereX(t)is the solution whose maximum deviation fromx±minoccurs att=0. Obviously, wehave to integrate over this whole set of saddle points, because they have the same  • • • 