Introduction to Instantons

Home , Instanton, Wick rotation

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 |e |xi 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 |e |xi 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 √ = 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 √1 (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 ha|e−HT /~| − 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 ω 1/2 ≈ e−ωT /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 ω 1/2 X T n ha|e−HT /~| − ai = e−ωT /2 K nenS0/~ Quantum π n! Field Theory ~ n odd Eﬀects of 1/2 −S / −S / Instanton- ω −ωT /2 1 h KTe 0 ~ −KTe 0 ~ i Matter = e e − e Interactions π~ 2 And similarly:

ω 1/2 X T n ha|e−HT /~|ai = e−ωT /2 K nenS0/~ π n! ~ n even ω 1/2 1 h −S / −S / i = e−ωT /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 ~ω −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 + ω ) 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 + ω2) ψ (T /2) t = SHO (1) 0 2 00 det (−∂t + V (¯x)) ψ0(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] = |K|e−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→∞ r Interactions These can be classified by their winding number at r = ∞: 1 Z k = − d4x trF ∗ F µν 16π2 µν Z 1 3 µ 2 = − 2 d xn µνρσ tr Aν∂ρAσ + AνAρAσ 8π r=∞ 3 1 Z 2 = − 2 tr A ∧ dA + A ∧ A ∧ A 8π r=∞ 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. These winding numbers are determined by the types of gauges we can have, that is the class of functions U(x) which maps spatial inﬁnity S3 for Euclidean space to the gauge group. This tells us that the k are exactly determined by π3(G). Instantons in Quantum Field Theory Pure Gauge Theory

Introduction to Instantons

T. Daniel Brennan For a pure gauge theory, everything else is the same as

Quantum before Mechanics Determinant Quantum Field Theory Integrate over Zero modes Classical action Eﬀects of Instanton- Matter Now we can use Index Theorems to determine how many Interactions zero modes Can show exact correspondence and orthogonality of zero modes from free parameters Can use Fadeev-Popov method to convert integration over zero modes to integration over parameters Eﬀects of Instanton-Matter Interactions Fermions???

Introduction to Instantons When add fermions to theory, everything is the same T. Daniel Brennan except for integration over the fermionic zero modes... There are grassmanian degrees of freedom: N=2 C(R) k Quantum Mechanics Integration over grassmanian variable picks out zero modes Quantum from operators in the expectation value of operators Field Theory

Eﬀects of Instanton- i X Matter ψ(yi ) = ψ ψ = ψcl + ψn ψzero ∼ f (x)K Interactions n 1 N hO(A, ψ, ψ¯)ψ ...ψ )ik Z N ¯ Y ¯ 1 N − S[A,ψ] = N DADψDψ DKi O(A, ψ, ψ)ψ ...ψ e ~ i=1 8π2 − 2 |k| ∼ hO(A, ψ, ψ¯i0e g Eﬀects of Instanton-Matter Interactions Breaking of U(1) Symmetry

Introduction to Instantons Now that we have operators with uneven number of T. Daniel fermions and their conjugates, the symmetry of the Brennan Lagrangian: Quantum iϕ Mechanics ψ → e ψ Quantum Field Theory is broken. Eﬀects of If we have a term in Lagrangian: Instanton- Matter θ Interactions L = trF ∗ F µν 16π2 µν then this anomalous phase can be compensated by: θ → θ + 2C(R)kϕ θ ∼ θ + 2π Now we have a discrete symmetry: 2π ψ → eiϕψ ϕ = 2C(R)k Important when up in N=2 SUSY Moduli space Eﬀects of Instanton-Matter Interactions Baryon Decay

Introduction to Instantons

T. Daniel Consider the Strong Force: SU(3) Brennan The k=1 instanton background leads to an integration Quantum over 6 zero modes. (it removes 6 quark operators from Mechanics expectation values) Quantum Field Theory This leads to a 6 (anti-)fermion vertex in our eﬀective Eﬀects of action Instanton- Matter 1 2 3 1 2 3 Interactions V ∼ uLuLuLdL dL dL

This clearly violates baryon conservation and can lead to proton, neutron decay:

+ p + n −→ e +ν ¯µ

2 − 8π |k| Suppressed by e g2 ∼ 10−34. Conclusion

Introduction to Instantons

T. Daniel Brennan In Quantum Mechanics, instantons describe Quantum Mechanics tunneling/decay phenomena by using a semi-classical

Quantum approximation to equations of motion with imaginary time. Field Theory In Quantum Field Theory instantons are described by Effects of Instanton- gauge fields with non-trivial winding at infinity. They also Matter Interactions lead to a description of tunneling and decay. In QFT when instantons interact with matter they give rise to fermionic zero modes which allow for chiral asymmetric operators which lead to U(1) symmetry breaking and baryon decay. The End

Introduction to Instantons

T. Daniel Brennan THE END

Quantum Mechanics

Quantum Field Theory

Eﬀects of Instanton- Matter Interactions