Instantons and Monopoles

Transcription

1 An Introduction to Instantons and Monopoles Zainab Nazari Advisors: K. Narain, G. Thompson The Thesis Submitted in fulfilment of the requirements To The Abdus Salam International Centre for Theoretical Physics in the Postgraduate Diploma Programme High Energy Physics

2 I think nature s imagination is so much greater than man s, she s never going to let us relax Richard Feynman ( ) 1

3 ACKNOWLEDGEMENTS It is my pleasure to express my gratitude to Prof. Kumar Narain who introduced this interesting topic to me and for his enlightening advises, encouragemnts and explanations, I am immensely thankful of Prof. George Thompson who made alot of efforts to clear up my messy mind with his great and clear explanations and for his advices to make things nicer. I would like to thank our lecturers who brought us to the point with better and deeper insights in to the era of High Energy Phyiscs as well as indispensable and complementary role of our tutorails, I am also indepted to my classmates with their kindness and useful discussions. 2

4 Dedicated to Prof. Abdus SALAM In the past Particle Physics was driven by troika which consisted of (1) Theory, (2) Experiment, and (3) Accelerator and Detection-Devices technology. To this troika have been added two more horses. Particle Physics is now synonymous with (4) Early cosmology (from 10 ( 43) sec. up to the end of the first three minutes of the Universe s life) and (5) it is strongly interacting with Pure Mathematics. One may recall Res Jost who made the statement (towards the end of the 1950 s) that all the mathematics which a particle physicists needed to know was a rudimentary knowledge of Latin and Greek alphabets so that one can populate ones equations with indices. this is no longer true today. The situation in this regard has changed so drastically that a theoretical particle physicists must know algebraic geometry, topology, Riemann surface theory, index theorems and the like. More mathematics that one knows, the deeper the insights one may aspire. Abdus Salam-1988, Particle Physics today 1 1 Ann. Inst. Henri Poincare, Vol. 49, n 3, 1988, p

5 Contents 1 Introduction 5 2 Yang-Mills Instantons Instantons as Yang-Mills Solutions Boundary Conditions and Winding Numbers An n = 1 Instanton Homotopic classes Definition S 1 S S 3 S Monopoles in Maxwell s Theory Classical electromagnetism and duality transformation Monopoles in the Georgi-Glashow model Bounds on the Action Exact Classical Solution, Prasad-Sommerfield Limit A Conventions 17 B Vacuum tunelling 17 References 19 Abstract This thesis is concerned with the study of instantons and monopoles as nontrivial solution for the Yang-Mills theory in 4 and 3 dimensions, respectively. Instantons arise as special classes of solutions to the pure Yang-Mills equations. I showed that the BPST instanton with topological number n = 1 satisfies the equations of motion and I studied the connection between winding number and boundary conditions. I introduce the magnetic monopole firstly in the context of electromagnetism. I then showed that in SO(3) Yang-Mills theory coupled with an adjoint valued Higgs field there is a monopole solution in the Prasad-Sommerfield limit. 4

6 1 Introduction An Instanton is a configuration that is an absolute minima of the Yang-Mills action. More precisely, it is a non-trivial solution to the equations of motion of the classical field theory with nontrivial topological properties on a (4-dimentional) Euclidean space (indeed, in general, there are such solutions on compact 4-dimensional manifolds). As we are working in Euclidean space, there is no time. The 4-th direction is simply another space direction. We can think of it as an imaginary time direction, however. Instantons depend on all four space directions and they are called pseudo particles solutions in the literature because of their dependence on imaginary time. The theoretical construction of Instanton is popular among the interest of many physicists and mathematicians, Instantons are essential as they play an important role in the path integral formalism of quantum field theory and they can be used to study the tunneling behavior in various systems such as a Yang-Mills theory B. Instantons as the topological structure of gauge group gives a real deep reason for the phenomena of quark confinement[9]. Instantons also occur in theories like general relativity and string theory. In this thesis I will consider the Belavin-Polyakov-Schwarz-Tyupkin (BPST) self-dual gauge-field solution with topological number n = 1 [2]. To illustarte the topological properties of the Instanton we need to have a look at to the homotopy classification of continuous functions. As analogy Monopole is also a minima of the equation of motion of Yang-Mills theory coupled to an adjoint valued scalar field in 4-dimensional space-time, we can also consider instanton solutions as monopole solutions, in the static limit where all time derivatives are zero. An interesting aspect of the non-abelian gauge theory is that it has objects with the properties of magnetic monopole. Historically, in 1864 J.C. Maxwell banished isolated magnetic charges from his four equations because no isolated magnetic pole had ever been observed. His brilliant simplification of electromagnetism theory however, led to asymmetric equations, which called for the aesthetically more attractive symmetric theory that would result if a magnetic charge did exist. Paul Dirac, in a paper published 1948 [4] proved that the existence of the magnetic monopole was consistent with quantum theory. In this paper he showed that the existence of the magnetic monopole not only symmetrized Maxwell s equations, but also explained the quantization of electric charge [6]. The discovery of the magnetic fundamentally changes our understanding of electromagnetism; and Provides powerful insights into theories of the nature of the Universe such as Grand Unified Theories, String Theory and M-theory, it has revolutionary implications for astrophysics and cosmology since if monopoles are discovered, they would also have been produced in the early Universe.[8] It is expected that monopoles associated with the spantaneous symmetry breakdown of grand unified gauge theories and should be superheavy and 5

7 such object may well have escaped detection [3]. MoEDAL 1 experiment is the latest accelerator experiment designed to search for direct production of magnetic monopoles or dyons (particles with electric and magnetic charge)[8]. The solutions that I look for are time-independent so that they may also be considered as solutions to the Yang-Mills plus matter action in 3-dimensional Euclidean space, we seek solutions such that the square of the Higgs field goes to a constant as r, the Prasad-Sommerfield limit [10]. The conventions that I use within the thesis are explained in Appendix A 2 Yang-Mills Instantons In this section we wish to find solutions to the Yang-Mills equations in 4-dimensional Euclidean space. In particular we wish to obtain absolute minima of the action. In order to specify the type of solution one also needs to specify the boundary conditions. Those conditions, as we will see later, are given by topological considerations. The Yang-Mills Lagrangian of a non-abelian gauge theory with group G on 4-dimansional Euclidean space is L = 1 4 Tr F µν F µν (2.1) where the trace is in the fundamental representation for G = SU(n) and we raise and lower indices with the Kronecker delta, see (A.4). The action is S = 1 d 4 x Tr F µν 2 (2.2) 4 R 4 where is the Euclidean norm so that we are integrating the norm of some function. We also presume that the integral makes sense, so that as x the field strength decays at least as fast as for some ɛ > 0. We note that F µν 1 x 2+ɛ (2.3) αβ d 4 x Tr F µν 1 2 ɛ µν F αβ 2 0 (2.4) as this is a convergent integral of a norm and where my choice of ɛ αβγδ is spelt out in (A.5). Now (2.4) can be rewritten as d 4 x Tr F µν F µν 1 2 d 4 x Tr F µν ɛ µν ρσf ρσ (2.5) The right hand side of this expression is a topological number as I will explain in the next section. In any case, for any gauge configuration we have that S 1 d 4 x Tr F µν ɛ µν ρσ F ρσ (2.6) 8 1 the Monopole and Exotics Detector At the LHC 6

8 and clearly the absolute minimum of the action is when the inequality is actually an equality. Looking back at (2.4) we see then that the absolute minimum of the action occurs when the integrand is zero and this corresponds to having αβ F µν = ± 1 2 ɛ µν F αβ (2.7) A gauge field that satisfies either of these equations (and which one is satisfied is determined by the boundary data) is an absolute minimum configuration of the action. The equation with the plus sign is known as the instanton equation and the negative sign is the anti-instanton equation. I should mention that the argument that leads to (2.6) is correct as long as we are in Euclidean space otherwise we could not use positivity of the norm (as it is not true in Minkowski space-time). Notice that (2.7) are first order differential equations and so are easier to solve than the usual Yang-Mills equations of motion. In the original paper where instantons were introduced [2] only a particular SU(2) instanton (with winding number 1- see the following section) was constructed. It was Atiyah, Drinfeld, Hitchin and Manin [1], however, who found a trick to construct instantons for gauge theories with structure group SU(n). They discovered a set of quadratic algebraic equations, called the ADHM equations, whose solutions are in 1-1 correspondence with the instanton solutions. 2.1 Instantons as Yang-Mills Solutions Since the solutions A µ of the first order equations (2.7) are absolute minima of the action, they are automatically solutions of the Yang-Mills equations. The equation of motion that follows from the action is µ F µν = 0 (2.8) Clearly it should be easier to solve the first order equations rather than the second-order Yang-Mills equations of motion (2.8). In this section I show explicitly that the (anti)-self-dual equations imply the Yang-Mills equation of motion by making use of the Bianchi identity µ F αβ + α F βµ + β F µα = 0 (2.9) which, in 4-dimensions can equivalently be written as ɛ αβµν ν F αβ = 0 (2.10) Now acting (and contracting) with µ on both sides of the (anti)-self-duality equation αβ F µν = ± 1 2 ɛ µν F αβ (2.11) gives αβ µ F µν = ± 1 2 µ ɛµν F αβ (2.12) 7

9 The right hand side of this equation vanishes thanks to the Bianchi identity (2.10) and we are left with the Yang-Mills equation of motion µ F µν = 0 (2.13) I should emphasis that strictly we do not need this proof, because it is a principle that absolute minima of the action exists in the set of all the minima (providing the boundary conditions agree)that is in the set of solutions to the equations of motion. 2.2 Boundary Conditions and Winding Numbers The gauge fields that we are interested in have the boundary condition, that x F µν 0 (2.14) fast enough so as to ensure that the action is a convergent integral. This means that at infinity A µ g 1 (x) µ g(x) (2.15) The expresion for A µ implies that it s a pure gauge, and if we consider the boundary of R 4 to be very large sphere S 3 then every field A µ produce a certain mapping of S 3 onto our gauge group G, namely g : S 3 G. In the particular case that G = SU(2), then the mapping would be from S 3 S 3. The discussion on homotopy classes in section 3 implies that maps of the type that we are interested in g : S 3 G fall into different classes labelled by integers and maps in the different classes cannot be smoothly deformed into each other. We say that these maps are not homotopic to each other. For the SU(2) gauge group the classes count the number of times that the S 3 at infinity wraps around the group SU(2) which is thought of as the topological space S 3. For other gauge groups the homotopy classes arise from the fact that π 3 (G) = Z. Given that at infinity the gauge field is constructed from the maps (2.15) it is true that two gauge fields A 1 µ and A 2 µ constructed from two non-homotopically equivalent maps can not be continuously deformed into each other. It is possible to assign an integer n corresponding to the class that the gauge field belongs to. It can be shown that n = 1 8π 2 ɛµν αβ d 4 x Tr F µν F αβ (2.16) We note that ɛ µν αβ d 4 x Tr F µν F αβ = d 4 x α J α (2.17) where J α is J α = ɛ µνλα Tr (A µ ν A λ + 23 ) A µa ν A λ (2.18) 8

10 Then after a straightforward calculation we have that 1 8π 2 ɛµν αβ d 4 x Tr F µν F αβ = 1 6π 6 ɛµν αβ d 3 σ µ Tr (A ν A α A β ) (2.19) where, as the integral on the right hand side is over the 3-sphere at infinity, A µ (x) = g 1 (x) µ g(x) (2.20) Therefore we can show that according to the inequality that we have already discussed in (2.4) we can show S(A) 2π 2 n (2.21) This formula(2.21) gives the lowst bound for the energy of the quasiparticles in each homotopy class.it impies that from (2.4) that the action is minimized (i.e. equality achieved) when (2.11) is satisfied. We remark that the usual solution A µ = 0 which has trivial quantum number (n = 0) clearly satisfies the condition (2.21) 2.3 An n = 1 Instanton The instanton solution for SU(2) discovered by [2] corresponds to g(x)s with a nontrivial winding number n = 1. To find non-trivial self-dual gauge field solutions, they employed the strategy of considering F µν of O(4) gauge theory, which isomorphic to SU(2) SU(2). The gauge fields for O(4) are A αβ µ Where A µ is antisymmetric on αβ. Here, we will content ourselves in presenting an SU(2) gauge field that satisfies the instanton equation. Consider the following configuration ( ) r 2 A µ (x) = r 2 + ρ 2 U 1 µ U (2.22) where r 2 = x x2 and ρ is some arbitrary scale parameter (often refered to as the instanton size) and with U(x) = x 4 + ix.σ r (2.23) a map from SU(2) to S 3. The gauge field (2.22) satisfies the instanton equation as I will now show. Firstly, as far as the boundary conditions are concerned as r, we have A µ (x) U 1 µ U (2.24) as required. For convenience we re-write the group valued field U(x) as U = i xµ r σ µ (2.25) 9

11 where we have defined σ µ = (σ i, ii), σ µ = (σ i, ii) (2.26) and the σ i are Pauli matrices. A short calculation shows that A 4 = ixi.σ i r 2 + ρ 2, A j = ix 4σ j + ix k ɛ kji σ i r 2 + ρ 2 (2.27) As the instanton equation (2.7) is self dual it is the same as F 4i = ɛ jk 4i F jk (2.28) and a short calculation verifies that our configurations satisfies this equation. Actually, by acting with translations we do not need to consider that the instanton is centred at x µ = (0, 0, 0, 0), rather we may take it to be centred at x µ = a µ. If we understand r 2 = (x a) 2 in the above formulae the proof that the configuration satisfies the instanton equation is not invalidated. Hence our most general solution for the instanton has 5 free parameters (a µ, ρ), its centre and scale. With this change we have (x a) λ A µ (x) = i ((x a) 2 + ρ 2 ). ηa λµ σ a (2.29) where I have introduced an t Hooft symbol 0 µ = λ = 4 η λµ a = δj a λ = 4 µ = j λ = i µ = j ɛ a ij (2.30) 3 Homotopic classes 3.1 Definition To study the topologocal properties of Instanton solution let s have a look at topologocal properties of continuous function,to illustrate the concept, I will divide the discussion in 2 steps to get reach to the point that we are more interested in. First let s understand what is homotopic classes; It s a class that we can divide continuous function by that; each class is made up functions that can be deformed continuously into each other. Let X and Y be two topologocal spaces and f 0 (x), f 0 (x),two continuous functions from X to Y, let I be the unit interval on the real line: 0 t I; f 0 andf 1 are said to be homotopic if and only if there is a continuous function F(x,t) which maps direct product of X and I to Y such that F (x, 0) = f 0 (x) and F (x, 1) = f 1 (x). The continuous function F(x,t) which does this difformation is called homotopy. So that we can divide all functions from X to Y in the same homotopy class if they are homotopic.to illustrate let s go to 10

12 Figure 1: Winding numbers S 1 S S 1 S 1. Let X be the points on a unite circle labelled by θ, with θ and θ + 2π identified, and let Y be a set of complex numbers u 1 = e iσ, which is topological equivalent to a unite circle, one-dimansional sphere. We consider mapping from θ e iσ. The continuous functions given by f (θ) = exp[i(nθ + a)] (3.1) which form a homotopic class for different values of a and a fixed integer n. This is because we can construct a homotopy F (θ, t) = exp{i[nθ + (1 t)θ 0 + tθ 1 ]} (3.2) such that: and f 0 (θ) = exp[i(nθ + θ 0 )] (3.3) f 1 (θ) = exp[i(nθ + θ 1 )] (3.4) are homotopic. We can visualize f (θ) as mapping of a circle on to another, a point of the first circle is mapped on n points on another circle (3.2) and we can think of it as winding number n. The winding number n for a given mapping f (θ) can be written n = 2π 0 dθ 2π [ i f (θ) df (θ) ]. (3.5) dθ A particular interest can be the mapping with the lowest nontrivial winding number, n = 1. f (1 ) (θ) = e iθ. (3.6) We can get mappings of higher winding number by taking power of this mapping. For instance, the mapping [f (1 ) (θ)] m will have winding number m. We can write the previous mapping in cartesian coordinates system as: f (x, y) = x + iy with x 2 + y 2 = 1 (3.7) 11

13 Now, we can generalize the doamin X of this mapping from the unite circle to the whole real line x by identifying the end-points x = and x = to be the same point, i.e. the mapping are required to satisfy the property f (x = ) = f (x = ). This has the same topology as the unite circle. Example of this type of mapping with winding number n = 1 are f 1 (x) = exp{iπx/(x 2 + λ 2 ) 1 2 } (3.8) f 1 (x) = exp{i2πsin 1 [x/(x 2 + λ 2 ) 1 2 ]} = (λ + ix) 2 λ 2 + x 2 (3.9) Where λ is an arbitrary number. In this case the topological winding number for a general mapping can be expressed as n = 1 2π + dx[ i f (x) As we can see n = 1 yeilds for the functions f 1 (x) and f 1 (x). df (x) ]. (3.10) dx 3.3 S 3 S 3. We now consider mapping from a three-sphere to SU(2) space, i.e mapping from the points on S 3, the sphere in 4-dimensional Euclidean space laballed by three angles, to the elements of SU(2) group, which are also charachterized by three parameters. As we know the manifold of the SU(2) group is the same as three-sphere. Mapping in this case also charachterized by the topological winding number n. and it can be shown that number can be expressed as where n = 1 24π 2 dθ 1 dθ 2 dθ 3 Tr (ε ijk A i A j A k ) (3.11) A i = f 1 (x 0, x) i f (x 0, x) (3.12) and θ 1,θ 2 and θ 3 are the three angles that parameterize S 3. Now, we want to generalize the case where the domain X is the whole three-dimensional space with all points at infinity identified. Example of this mapping with n = 1 are f 1 (x) = exp{iπx.τ/(x 2 + λ 2 ) 1 2 } (3.13) f 1 (x) = (λτ + ix) 2 /(x 2 + λ 2 ) (3.14) Which is the generalizations of the previous mappings. As for winding number, we now have n = 1 24π 2 d 3 x Tr (ε ijk A i A j A k ) (3.15) v A i = f 1 (x) i f (x) (3.16) We can see explicitly this nice analogy with our previous discussion. 12

14 4 Monopoles in Maxwell s Theory Maxwell s equations propose that there is electric charge, but no magnetic charge (magnetic monopoles) in the Universe. Magnetic charge has never been observed and may not exist. If they did exist, both Gauss s law for magnetism and Faraday s law would need to be modified, and the resulting four equations would be fully symmetric under the interchange of electric and magnetic fields. 4.1 Classical electromagnetism and duality transformation Classical electromagnetism is described by Maxwell s equations E = ρ B 0 E = j (4.1) B = 0 E 0 B = 0 (4.2) We can write these four equations simply by replacing them with field strength F µν and we have combination of two terms ν F µν = j µ (4.3) ν F µν = 0 (4.4) Where and In vacuum, where j µ transformation j µ = (x, j), F 0i = E i, F ij = ɛ ijk B k, (4.5) F µν = 1 2 ɛµνρσ F ρσ (4.6) = 0, Maxwell s equations are symmetric under the duality F µν F µν, F µν F µν (4.7) it means that under exchange of E B, B E, theory is invariant. This symmetry will be broken by the presence of the electric current j µ in (4.4). Dirac proposed that we can introduce the magnetic current k µ = (σ, k) on the right hand side of the (4.4) so that ν F µν = j µ ν F µν = k µ (4.8) therefore they would be symmetric under duality transformation j µ k µ, k µ j µ (4.9) 13

15 In the sence that we have introduced magnetic current in our theory it automatically requires the existence of magnetic monopoles, So that we use the analogy that we had before to the electric current produced by point particles at x i with charge q i j µ (x) = q i dx µ i δ4 (x x i ), (4.10) i we can introduce magnetic charges with strength g i as following k µ (x) = g i dx µ i δ4 (x x i ), (4.11) i 5 Monopoles in the Georgi-Glashow model The SU(2) Georgi-Glashow theory, in 3-dimesions, of electromagnetic and weak interactions, is given by S = d 3 1 x Tr [ 4e 2 F µν ( µφ) λ(φ2 + η 2 ) 2 ] (5.1) where φ is an anti-hermitian adjoint valued field (which explains the relative sign between φ 2 and η 2 ). With my conventions F a µν = µ A a ν ν Aµ a ɛ abc Aµ b Aν c µ φ a = µ φ a ɛ abc A b µ φ c (5.2) Two equations of motion can be derived on varying the gauge field and Higgs field respectively they are 1 e 2 µf µν + [φ, ν φ] = 0 (5.3) µ µ φ λφ(φ 2 + η 2 ) = 0 (5.4) These coupled second order differential equations are difficult to solve. However, solutions can be found at least numerically. Indeed one can find monopole solutions to these equations. That is solutions which carry both an electric and a magnetic charge. I will not show that here, rather in the Section 5.2 I will consider a limit where the equations boil down to a coupled set of first order differential equations which allow for an exact solution. 5.1 Bounds on the Action We now simplify the Georgi-Glashow action by dropping the scalar potential. In this case the action is S = d 3 1 x Tr [ 4e 2 F µν ( µφ) 2 ] (5.5) 14

16 This is our previous action (Georgi-Glashow) but in the limit when λ 0, though, we also insist that at spatial infinity that φ 2 η 2 (to simplify matters further I set η = 1) which is a limit that Prasad and Sommerfield studied. The following inequality is true d 3 x Tr ( 1 e F µν ± ɛµν α α φ) 2 0 (5.6) as the right hand side is the integral of a positive integrand. This squaring trick allows us to bound the action, just as we did for the instanton. By expanding the square in (5.6) we obtain S 1 2e α d 3 x α Tr (F µν ɛµν φ) (5.7) We might ask ourselves what would the quantity on the right hand side of (5.7) represent? Write it as 1 2e S 2 α dσ α Tr (F µν ɛµν φ) (5.8) At infinity φ picks a U(1) direction in the Lie algebra of SO(3). Consequently we can think of the combination φ a Fµν a as a choice of Abelian field strength. Furthermore, as the only components of F µν that appear are space like we have α B α = 1 2e Tr (F µν ɛµν φ) (5.9) is rightly thought of as the magnetic field. Consequently we define the magnetic charge to be g = 1 2e α d 3 x α Tr (F µν ɛµν φ) (5.10) 5.2 Exact Classical Solution, Prasad-Sommerfield Limit Prasad and Sommerfield presented an exact solution to the nonlinear field equations by considering the limit where the φ 4 coupling goes to zero λ 0. In this limit one considers that the Higgs field at spatial infinity, nevertheless, satisfies φ 2 = η 2. The solution they found describes a classical object which is both magnetically and electrically charged. The first order equation that we want to solve is We look for a solution with the ansatz 1 2e ɛ µν α F a µν = ± α φ a (5.11) A a µ = ɛ a µνx ν g(r), φ a = x a f(r) (5.12) 15

17 where space indices and group indices are liberally confused and r 2 = x 2 + y 2 + z 2. A straightforward calculation leads to α φ a = δα a ( f(r) + r 2 g(r)f(r) ) ( f + x a x α ) (r) g(r)f(r) r (5.13) here a prime indicates a derivative with respect to r. We also have ɛ α µν Fµν a = 2δα a ( 2g(r) + rg (r) ) ( g + 2x a x α ) (r) g(r) 2 r Comparing terms proportional to δ a α and x a x α, respectively, in (5.11) we obtain (2g(r) + rg (r)) = ( f(r) + r 2 g(r)f(r) ) ( g ) ( (r) f g(r) 2 ) (r) = ± g(r)f(r) r r (5.14) (5.15) To make contact with the equationss as they are usually presented, and which depend on dimensionless variables, we let f(r) = H(er)/er 2 = eh(ξ)/ξ 2, ξ = er g(r) = [K(er) 1]/r 2 = e 2 [K(ξ) 1]/ξ 2 (5.16) In this way we arrive at ξ dk(ξ) dξ ±ξ dh(ξ) dξ = ±K(ξ).H(ξ) = ±H(ξ) [K(ξ) 2 1] (5.17) As one can see one can change the sign of ± by changing the sign of H. These equations agree with those derived by Goddard and Olive [7] (eq (4.48) there). Therefore, an exact solution is given by K(ξ) = ξ ; H(ξ) = ξcoth(ξ) 1 (5.18) Sinh(ξ) 16

18 Appendices A Conventions We use the following convention for the covariant derivative D µ = µ + A µ (A.1) As the covariant derivative should be anti-hermitian, I take the connection, or gauge field to also be anti-hermitian A µ = A µ (A.2) which is a condition on the generators of the gauge group G. The definition of the field strength is F µν = [D µ, D ν ] = µ A ν ν A µ + [A µ, A ν ] (A.3) Throughout the text I will take the SU(n) generators T a in the fundamental representation to be normalised so that ( Tr T a T b) = δ ab (A.4) The ɛ αβγδ tensor is such that ɛ 1234 = +1 and satisfies +1 even permutation ɛ αβγδ = 1 odd permutation 0 otherwise (A.5) The Pauli matrices are ( 0 1 σ 1 = 1 0 ), σ 2 = ( 0 i i 0 ), σ 3 = ( ) (A.6) and are clearly Hermitian. B Vacuum tunelling The physical interpretation of the instanton as quantum mechanical events correspond to tunnelling between vacuum states of differenttopological numbers. Let s have a review on tunneling in quantum mechanics. In feynman path-integral formalism the basic vacuum-to-vacuum transition amplitude is expected as sum over all possible paths between the initial and final states (an infinite 17

19 number of convolutions to get from t i to t f ), weighted by the exponential of i times the action for the particular path. Z[A µ ] = N (DA µ )e is (B.1) In the previous discussion we have included in our sum of path-integral only the condition A µ 0 on boundary. We will see that non-abelian gauge field theory has vacuum which corresponds to a superposition of vacuum states with different topological winding numbers. The instanton field configurations correspond to paths that connect initial and final vacuum states with different winding numbers. Here, I will breifly discuss the tunneling behavour of Instanton by way of a simple example Consider the model of scalar field with interaction λφ 4 L = ( t φ) 2 ( x φ) 2 + µ 2 λ 4 φ4, (µ 2, λ > 0) (B.2) The equation of motion can be found: 2 t 2 x = µ 2 φ λφ 3 (B.3) The solution for that would be The minima of potential φ = 0, φ = ±µ/ λ (B.4) V (φ) = (µ 2 /2)φ 2 + (λ/4)φ 4 (B.5) are the points φ = ±µ/ λ, vacuum solutions. Equation (B.3) has constant solution, so called kink, or domain wall, with the vacuum boundary conditions φ kink = φ(x x 0 ) = ±(µ/ λ)tanh[(µ/ λ)(x x 0 )], φ( ) = µ/ λ; φ( ) = µ/ λ (B.6) (B.7) (x 0 is an arbitrary point on the straight line R 1, called the center of solution. This can be easily seen to be independent of the choice of x 0 ). The solution is local minimum of potential and since we fixed a boundary condition we fixed topological number and therefore, it can be called Instanton. As we can see from the solution (B.6) it is a tunnelling through the minimas of the action. the discussion about tunneling behavour can be traced on textbooks on Quantum Mechanics [5]. 18

20 References [1] M. Atiyah, Drinfeld, N. Hitchin and Y. Manin, Construction of instantons, Phys. Lett. A65 (1978) [2] A. Belavin, A. Polyakov, A. Schwarz and Y. Tyupkin, Pseudoparticle Solutions of the Yang-Mills Equations, Phys. Lett. B59 (1975) [3] T.P. Cheng, L.F. Li Gauge Theory of elementary particle physics, (1988)Magnetic monopoles [4] P. A. M. Dirac The Theory of Magnetic Poles (1948) Phys. Rev [5] S. Gasiorowicz, Quantum Physics, John Willey and Sons 3-rd ed. (2003) [6] F. Moret, G. Senjanovic Magnetic Monopoles, ICTP, Diploma thesis HEP, (2011), [7] P. Goddard and D. Olive, Magnetic Monopoles in Gauge Theories, Rep. Prog. Phys. 41 (1978) [8] J.L. Pinfold Dirac s Dream the Search for the Magnetic Monopole, AIP Conf. Proc. (2010) [9] A.M.Polyakov,Quark Confinement and Topology of Gauge Theories, Nuclear-Phys. B120 (1977) [10] M. K. Prasad, C. M. Sommerfield Exact Classical Solution for the t Hooft and Julia-Zee Dyon Phys. Lett. 35 (1975)