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Notes on Instantons Notes on Instantons Leonardo Almeida Lessa November 2019 Contents 1 Instantons in Quantum Mechanics 2 1.1 Saddle Point Approximation . .2 1.2 Quadratic Potential . .3 1.3 Double Well . .4 1.4 Periodic Potential . .5 2 Instantons in Gauge Field Theories 5 2.1 A Minimum of Homotopy Groups . .5 2.2 Topological Solutions and Winding Number . .6 2.3 Many Vacua of Gauge Theories . .6 2.4 General Features of Instantons . .7 2.5 Constructing the Instanton . .8 3 Applications of Instantons 9 3.1 Abelian Higgs Model in 1+1 Dimensions . .9 3.2 ’t Hooft Solution of the U(1) Problem . 10 4 Conclusion 10 We introduce the concept of instantons first in Quantum Mechanics (Sec. 1) and then in Quantum Field Theory (Sec. 2). Similarly to the WKB approximation, the technique behind instantons involves a small-~ nonperturbative approximation. It also connects the topology of spacetime to the the vacuum structure of quantum theories (Sec. 2.3). Of the many applications of instantons, we discuss the confinement effect in the Abelian Higgs Model in 1+1 dimensions (Sec. 3.1) and the ’t Hooft solution of the U(1) problem (Sec. 3.2). These notes are based mostly on Chapter 3 of Sidney Coleman’s “Aspects of Symmetry” [1]. Other references include Chapter 3 of “Condensed Matter Field Theory”, by A. Altland and B. Simon [2], 1 INSTANTONS IN QUANTUM MECHANICS Notes on Instantons Warning: Euclidean Spaces As we will work almost solely within Euclidean Space, we shall remind ourselves of the necessary changes: x4 = ix0; t = iτ; SE = −iS: If, for example, the relation between the action S of a massive particle in Minkowski space " # Z m dx2 S = dτ − V ; R 2 dτ and the corresponding Euclidean action SE is the following " # Z m dx2 S = −i dt − − V ; iR 2 dt " # Z m dx2 = i dt + V ; iR 2 dt = iSE: Note that the kinetic and potential terms of SE have the same plus sign. Since the following discussion uses only Euclidean spaces, we will drop the subscript E from Euclidean quantities. 1 Instantons in Quantum Mechanics We will first introduce the instantons in the context of Nonrelativistic Quantum Mechanics (NRQM), which will form the basis for the calculations in QFT. In all cases, the definition of instantons is Instantons are (nontrivial) solutions φ of the classical equations of motion of an action S, i.e. δS [φ] = 0: δφ More that that, the main idea behind instantons is that we can have a nonperturbative results arising from a small ~ (semiclassical) approximation. As we shall see, this amounts to nontrivial solutions of the equation of motion, around which we will calculate quantum perturbations. Nonperturbative results in a small ~ approximation may seem strange, but there are known examples of this behavior in ordinary NRQM, such as the WKB approximation. It is a small ~ approximation that predicts effects that don’t appear in any order in perturbation theory, such as tunneling probabilities. 1.1 Saddle Point Approximation Suppose a field action S has a stationary point φ, which, by definition, satisfies the equations of motion of S. In this case, we can do a Taylor expansion around φ to calculate the partition function: Z Z = N Dφ exp{−S[φ]=~g; Z Z 2 1 4 4 1 δ S[φ] ≈ N Dφ exp −S[φ]=~ − d x d y ∆φ(x) ∆φ(y) ; 2 ~ δφ(x)δφ(y) 2 −1=2 −S[φ]= 1 δ S[φ] = Ne ~ det [1 + O(~)]; 2π~ δφ(x)δφ(y) 2 1 INSTANTONS IN QUANTUM MECHANICS Notes on Instantons where ∆φ = φ − φ and N is a normalization factor. This formula also works for NRQM action 0 1 R h m dx 2 i with substitutions x ! t; y ! t ; φ ! x . In the case of S = dt 2 dt + V (x(t)) , the path integral becomes −1=2 ~ −S[x]=~ 2 2 Z = Ne det m(−@t + ! ) [1 + O(~)]; 2 00 2 2 where m! = V (x(t)) and the operator m(−@t + ! ) is to be viewed as a diagonal infinite matrix in t and t0. If S has many stationary points, we sum over them. The stationary point condition for the NRQM action is mx¨ = V 0(x), Newton’s Law with inverted potential −V . −HT= The partition function will be useful for analyzing the states of lowest energies, since hxf j e ~ jxii = Z with boundary conditions x(0) = xi and x(T ) = xf . If H jni = En jni, with E0 being the energy of the ground state, −HT= X −EnT= G(xf ; xi; T ) := hxf j e ~ jxii = e ~ hxf jni hnjxii ; n T !1 −E0T= = e ~ hxf j0i h0jxii ; (1.1) so the ground state mode dominates the behavior of Z in the limit T ! +1. 1.2 Quadratic Potential To test our saddle point approximation, we will calculate the propagator ±V −HT= G(0; 0; T ) = hxf = 0j e ~ jxi = 0i 1 2 2 for the locally quadratic potential V (x) ∼ 2 ! x (see Figure 1). x The stationary points of the action are classical solutions for the inverted potential −V (dashed curve) with x(0) = x(T ) = 0. For a quadratic potential, only x = 0 satisfy these requirements, for which S[x] = 0. Now, it remains to calculate the operator determinant. The 2 2 eigenvectors of the −@t + ! with boundary conditions x(0) = Figure 1: Quadratic potential x(T ) = 0 are rn(t) = sin(πnt=T ), n = 1; 2;:::, with eigenvalues 2 2 n = (πn=T ) + ! . Thus, 1 −1=2 −1=2 Y πn2 det−@2 + !2 = + !2 ; t T n=1 2 2−1=2 which appears in the propagator as G(0; 0; T ) = J det −@t + ! . To calculate the prefactor J, we use that G(0; 0; T ) should reduce to the free propagator 2 −1=2 (xf − xi) Gfree(xf ; xi; T ) = (2π~T ) exp − ; 2~T in the limit ! ! 0. Thus, G(0; 0; T ) G(0; 0; T ) = Gfree(0; 0; T ); G(0; 0; T )j!=0 1 −1=2 Y T! −1=2 = 1 + (2π T ) ; πn ~ n=1 r ! r ! = T−!!1 e−!T=2; (1.2) 2π~ sinh !T π~ 1NRQM is just a 0 space + 1 time = 1 dimensional field theory. 3 1 INSTANTONS IN QUANTUM MECHANICS Notes on Instantons Q −1 where we used that n≥1[1 + (x/nπ)] = x= sinh x. In the limit T ! 1, we can use (1.1) and 1 2 1=2 (1.2) to conclude that E0 = 2 !~[1 + O(~)] and jhx = 0jn = 0ij = (!/π~) [1 + O(~)], which reproduces the semiclassical results. From now on, we will drop the O(~) when it is clear from the context that we are approximating at first order in ~. 1.3 Double Well Now, let’s consider a double well potential (see Fig- ure 2) with minima in ±a. We can investigate the prop- ±V agators G(±a; ∓a; T ), between −a and a, and between themselves G(±a; ±a; T ). Looking at the inverted po- tential (dashed curve), we can see that there are classical solutions that start at x = −a and go to x = a. Their en- p x ergy is zero, so they satisfy dx=dx = 2V and the action p R h 1 dx 2 i R a −a a for them is S0 = dt 2 dt + V = −a dx 2V . Moreover, these solutions are non trivial stationary points of the action and are called instantons (see Fig- ure 3). The inverse solution is called an anti-instanton and we can combine any number of them centered at different times to construct a multi-instanton solution. Figure 2: Double well potential The dynamical part of an instanton is localized within a time ∼ !−1. Thus, taking T ! 1, we can assume the transitions between minima are instantaneous and infinitely far apart for a multi-instanton solution (see Figure 4). Therefore, we can treat them as independent instantons. According to the instanton gas approximation, we need to sum over all of configurations that take a certain minimum point to a x another one, and then integrate over all their centers. The result of this for the propagator is r n t ! X T G(−a; −a; T ) = e−!T=2 (Ke−S0=~)n; π n! ~ even n p ! −!T=2 where the external factor π e comes from the quadratic ~ 2 n −a potential propagator; K is a correction term ; T =n! comes from the integration over all instantons’ centers; and the sum over even Figure 3: Instanton solution n guarantees that the system comes back to the initial state. Anal- ogously, 1r ! h i G(±a; −a; T ) = e−!T=2 exp Ke−S0=~T ∓ exp −Ke−S0=~T : (1.3) 2 π~ Comparing equations (1.1) and (1.3), we see not only the ground a x state contributing to the propagator, but two low energy states, with 1 −S0=~ energies E± = 2 ~! ±~Ke , separated by an exponentially small factor in ~, the barrier penetration factor, characteristic of an instan- t ton solution. These states are exactly the symmetric and antisym- metric combinations of the ground states of the harmonic oscillators. 4 1 µ Observation: The λφ model with negative mass L = 2 @ φ∂µφ+ 1 2 2 1 4 −a 2 m φ − 4 λφ exhibits spontaneous symmetry breaking (SSB) of the φ ! −φ symmetry. If we discretize spacetime and ignore the spa- Figure 4: Multi-instanton tial derivative term, there would be a double well potential for each spacetime point. According to our analysis, the energy difference be- tween the lowest energy states would be ∼ e−VS0=~, where V is the volume of the space.
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