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Introduction to Instantons Introduction to Instantons T. Daniel Brennan Quantum Mechanics Quantum Introduction to Instantons Field Theory Effects of Instanton- Matter T. Daniel Brennan Interactions February 18, 2015 Introduction to Instantons T. Daniel Brennan Quantum Mechanics 1 Quantum Mechanics Quantum Field Theory Effects of Instanton- Matter 2 Quantum Field Theory Interactions 3 Effects of Instanton-Matter Interactions Instantons in Quantum Mechanics Path Integral Formulation of Quantum Mechanics Introduction to Instantons T. Daniel Brennan Quantum mechanics is based around the propagator: Quantum Mechanics −iHT =~ Quantum hxf je jxi i Field Theory Effects of Instanton- In path integral formulation of quantum mechanics we Matter Interactions relate the propagator to a sum over all possible paths with a phase: Z −iHt=~ iS[x]=~ hxf je jxi i = N Dx[t] e Instantons in Quantum Mechanics Wick Rotation Introduction to Instantons In order to make the integral well defined we perform a T. Daniel Brennan Wick Rotation: Quantum Mechanics t ! T = −it Quantum −iHt=~ −HT =~ Field Theory e ! e Effects of Z 2 Instanton- 1 dx Matter S = i dt − V (x) Interactions 2 dt " # Z 1 dx 2 ! S = dT − − V (x) 2 dT " # Z 1 dx 2 = − dT + V (x) 2 dt This is equivalent working with an inverted potential. Instantons in Quantum Mechanics Semi-Classical Solution Introduction to Instantons If we are to solve the Classical equations of motion to an T. Daniel Brennan inverted potential we arrive at the usual equation: Quantum 2 Mechanics d x 0 2 − V (x) = 0 Quantum dt Field Theory Effects of we also have the conserved \energy" quantity: Instanton- Matter 2 Interactions 1 dx E = − V (x) 2 dt Now assume that x(t) has quantum corrections, we can expand it in ~. To first order we have the equation: d2x(1) − V 00(x(0))x(1) = 0 dt2 Instantons in Quantum Mechanics Introducing Quantumness Introduction to Instantons More formally we can perturb around the classical solution T. Daniel x¯(t): Brennan X x(t) =x ¯(t) + c x (t) Quantum n n Mechanics n Quantum Field Theory But now let the xn(t) be eigenvalues of the equation: 2 Effects of d xn 00 Instanton- − V (¯x)xn = λnxn Matter dt2 Interactions This introduces "quantumness" to the equations of motion. Integrating over all of these perturbations is equivalent to integrating over all paths. Now we want them to be orthogonal so there is no over counting of perturbations in our integral: Z T =2 0 0 0 xn(T )xm(T )dT = δnm −T =2 Instantons in Quantum Mechanics Formal Integration over Modes Introduction to Instantons Now if we expand the potential around the classical T. Daniel solution we get: Brennan 0 1 00 2 Quantum V (x) = V (¯x) − (x − x¯)V (¯x) + V (¯x)(x − x¯) + ::: Mechanics 2 Quantum Field Theory Now integrating, using the perturbative expansion of x, −S0= Effects of the first term will give us a factor of e ~ and the last Instanton- Matter term will give us: Interactions Y 1 2 00 −1=2 p = det(−@t + V (¯x)) n λn This is a problem if there exist any λn = 0. But since these satisfy the semi-classical equation, these are actually just a deformation of the classical solution so integrating over them corresponds to integrating over the free parameters of the classical solution. Instantons in Quantum Mechanics Quick Summary Introduction to Instantons T. Daniel Brennan Quantum Consider the Path Integral formulation of Quantum Mechanics Mechanics Quantum Field Theory Wick Rotate (make time imaginary t ! −it), causes us to Effects of Instanton- invert the potential Matter Interactions Solve the classical equations of motion there, plug into action Integrate over parameter space (zero modes) and multiply by Q p1 (quantum fluctuations). λn Instantons in Quantum Mechanics What are Instantons? Introduction to Instantons Instantons are classical solutions to the Wick rotated T. Daniel equations of motion which have non-trivial topology... Brennan To illustrate this, we will work out the canonical double Quantum Mechanics well potential. Quantum Field Theory Effects of Instanton- Matter Interactions Instantons in Quantum Mechanics Double Well Introduction 2 2 2 to Instantons Double well has potential: V (x) = λ(x − a ) T. Daniel Brennan Quantum Mechanics Quantum Field Theory Effects of Instanton- Matter Interactions Interested in the classical solutions which are \topologically nontrivial". In this example that is when a particle will tunnel from one minima to another. Each tunneling event goes as ±a tanh(c(t − t0)) We can glue solutions together in alternating order to have more general paths. Limit of large separation (quick tunneling): dilute gas approximation. Instantons in Quantum Mechanics Double Well (cont.) Introduction to Instantons T. Daniel Now when we do the path integral we will have an Brennan expression that looks like: Quantum Mechanics haje−HT =~j − ai Quantum Field Theory N Z T =2 Z tn−1 X n nS0=~ Effects of = dt1::: dtnK e p 2 Instanton- det0(−@ + V 00) −T =2 −T =2 Matter t n odd Interactions Since the particle spends most of its time at the bottom of one of the wells which is approximately harmonic, we can approximate the normalization by the harmonic oscillator: N ! 1=2 ≈ e−!T =2 p 0 2 00 det (−@t + V ) π~ Instantons in Quantum Mechanics Double Well (cont. cont.) Introduction n to Instantons The integration time integration gives us a factor of T . T. Daniel n! Brennan Now we get the expression: Quantum Mechanics ! 1=2 X T n haje−HT =~j − ai = e−!T =2 K nenS0=~ Quantum π n! Field Theory ~ n odd Effects of 1=2 −S = −S = Instanton- ! −!T =2 1 h KTe 0 ~ −KTe 0 ~ i Matter = e e − e Interactions π~ 2 And similarly: ! 1=2 X T n haje−HT =~jai = e−!T =2 K nenS0=~ π n! ~ n even ! 1=2 1 h −S = −S = i = e−!T =2 eKTe 0 ~ + e−KTe 0 ~ π~ 2 Instantons in Quantum Mechanics Double Well (cont.cont.cont.) Introduction to Instantons Examining the exponentials we find: T. Daniel ~! −S = Brennan E = ± Ke 0 ~ 2 ~ Quantum Mechanics It is now important to determine K. As it turns out, these Quantum are generally very annoying to calculate so I will just tell Field Theory you: Effects of Instanton- 1=2 0 2 2 1=2 Matter S0 det (−@t + ! ) Interactions K = 0 2 00 2π~ det (−@t + V (¯x)) This is determined by comparing the exact calculation for the one instanton with the single instanton term in the sum. Furthermore there is a formula for computing the fraction of coefficients: det0(−@2 + !2) (T =2) t = SHO (1) 0 2 00 det (−@t + V (¯x)) 0(T =2) Instantons in Quantum Mechanics Uneven Double Well Decay Introduction to Instantons We can also consider the uneven well. These are pictures T. Daniel Brennan of the classical solution and zero mode: Quantum Mechanics Quantum Field Theory Effects of Instanton- Matter Since there is a node there will actually be a single mode Interactions which has a lower (negative) eigenvalue. This makes K imaginary. We now have that the energy has an imaginary part which is exactly the decay width: S0 Im[E] = jKje−S0=~ = Γ=2 2π~ Instantons in Quantum Field Theory Derrick's Theorem Introduction to Instantons There are no non-trivial topological solutions to the double T. Daniel well's qft equivalent in dimension other than 2. There are Brennan no non-trivial matter solutions due to Derrick's Theorem. Quantum Mechanics Take a scalar field theory: Quantum 1 2 Field Theory L = (@µφ) − V (φ) Effects of 2 Instanton- Z d−1 1 2 Matter E = d x (@µφ) + V (φ) = I1 + I2 Interactions 2 Assume there is time dependent solution of finite energy φ¯(x). Now define: φ¯λ(x) = φ¯(λx). Then the energy changes as: 2−d −d Eλ = λ I1 + λ I2 Want λ = 1 minimize the energy otherwise the solution is unstable Instantons in Quantum Field Theory Derricks' Theorem (cont.) Introduction to Instantons T. Daniel Brennan Varying the Energy with respect to λ we find: Quantum @Eλ Mechanics = (2 − d)I1 − d I2 = 0 Quantum @λ λ=1 Field Theory @2E λ = (d − 2)(d − 1)I + d(d − 1) I Effects of 2 1 2 Instanton- @λ λ=1 Matter Interactions Solving these we find: 2 − d I = I 2 d 1 2 @ Eλ = −2(d − 2)I1 < 0 for d > 2 @λ2 λ=1 Instantons in Quantum Field Theory Gauge Instantons Introduction to Instantons We can however have topological solutions to gauge fields. T. Daniel Take SU(N) gauge theory: Brennan Z 4 1 a 2 Quantum S = − d x 2 (Fµν) Mechanics 2g Quantum Require finite energy and finite action. This enforces: Field Theory Effects of −1 1 Instanton- Aµ −! U @µU + O 2 Matter r!1 r Interactions These can be classified by their winding number at r = 1: 1 Z k = − d4x trF ∗ F µν 16π2 µν Z 1 3 µ 2 = − 2 d xn µνρσ tr Aν@ρAσ + AνAρAσ 8π r=1 3 1 Z 2 = − 2 tr A ^ dA + A ^ A ^ A 8π r=1 3 Instantons in Quantum Field Theory Winding Number Classification Introduction to Instantons This classification holds because: T. Daniel Brennan 1 Z S = − d4x trF 2 Quantum 2g 2 Mechanics 1 Z 1 Z Quantum = − d4x tr(F ∓ ∗F )2 ∓ d4 trF ∗ F Field Theory 4g 2 2g 2 Effects of Z 2 Instanton- 1 4 8π Matter ≥ ∓ 2 d x tr(F ∗ F ) = 2 (±k) Interactions 2g g This bound is saturated when F is (anti-)self dual.
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