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Majorana Fermions QFFF MSc Dissertation Aran Sivaguru September 2012 Advisor: Dr. A. Rajantie Contents 1 Introduction 1 2 Particles and Anti-Particles 2 2.1 The Dirac Equation . 2 2.1.1 Starting from Schr¨odinger . 2 2.1.2 Enter Dirac . 3 2.2 Field Theory and Anti-Matter . 5 2.2.1 The Dirac Field . 5 2.2.2 Neutral Particles . 7 3 Neutrinos 10 3.1 Lepton Number Conservation . 10 3.2 Massive Neutrinos and the Standard Model . 12 3.2.1 Neutrino Oscillations . 12 3.2.2 The Seesaw Mechanism . 14 3.3 Neutrinoless Double Beta Decay . 17 3.3.1 In Search of 0νββ ............................. 17 3.3.2 Theory vs. Experiment . 18 4 Condensed Matter Physics 19 4.1 Electrons and Holes . 19 4.2 Superconductors . 20 4.2.1 Cooper Pairs . 20 4.2.2 Vortices . 21 4.3 Non-Abelian Symmetries . 22 4.3.1 Discovering the Majorana Fermion? . 23 4.3.2 Quantum Computing . 24 5 Discussion 25 5.0.3 Supersymmetry . 25 5.0.4 Dark Matter . 25 5.1 Conclusion . 26 1 INTRODUCTION 1 Introduction In 1937 Ettore Majorana suggested that chargeless fermions could be described by a real wave equation, leading to the possibility that this particle would be identical to its anti- particle[1]. These Majorana fermions seemed a mathematical quirk without any physical basis. If fermions and anti-fermions were indeed indistinguishable, they could co-exist with- out annihilating one another - not something that we seem to see in nature. However, recent experiments have indicated the existence of these particles within condensed matter systems. If further data bears this out, it could lead to myriad applications in fields as diverse as quantum computing and cosmology[2]. With its grounding in quantum field theory, Majorana fermions provide a useful insight into the utility of field theories, and the interplay between theoretical physics and its exper- imental cousin - with a 75 year-old theoretical prediction only recently gaining experimental credence. This dissertation will discuss some of the physics and applications of Majorana fermions, looking at the implications of their discovery. Firstly, this report will look at some of the basic concepts in field theory that underpin the theoretical aspects of these particles. As Majorana fermions could be integral to the description of massive neutrinos - an exten- sion of the Standard Model, the next section will be devoted to this. Then, as recent results in condensed matter physics have shown glimpses of these particles, a qualitative look at Majorana fermions in this branch of research will be sought. Finally, the last section will briefly discuss some further applications of Majorana fermions and its utility in theoretical and experimental physics. 1 2 PARTICLES AND ANTI-PARTICLES 2 Particles and Anti-Particles 2.1 The Dirac Equation Quantum field theory provides a convenient mathematical framework to view particles and their anti-particles. So, as with all discussions surrounding quantum field theory, it is prob- ably best to start with looking at the Dirac equation[3]. This equation was Dirac's way 1 of formulating a description of elementary spin- 2 particles that was consistent with both quantum mechanics and special relativity. 2.1.1 Starting from Schr¨odinger Dirac's famous equation was motivated by Schr¨odinger'sequation for a free particle h2 @ − r2 = i (2.1) 2m ~@t The left hand side of this equation gives the non-relativistic kinetic energy. However, rel- ativity treats space and time on the same footing. Therefore a relativistic treatment of Schr¨odinger'sequation must have differential equations of the same order with respect to space and time. We can try to do this by starting with the relativistic invariance of 4- momentum E2 − p2 = m2c2 (2.2) c2 ^ @ If we express the energy and momentum as operators i.e. E = i~ @t andp ^ = −i~r, we can get the Klein-Gordon equation1 @2 − + r2 = m2 (2.3) @t2 1From here on c = ~ = 1 2 2 PARTICLES AND ANTI-PARTICLES The wavefunction is now a relativistic scalar and the space and time derivatives are both second order. However, although equation 2.3 is a valid wave equation, it is not a good relativistic generalisation of Schr¨odinger'sequation. This is because the initial values of and @ @t can be chosen freely, and as a result the probability density is no longer positive definite. This leaves open the possibility of negative probabilities - which is somewhat worrisome. 2.1.2 Enter Dirac So making the connection between Schr¨odinger'sequation and relativity was not so straight- forward. Equation 2.3 gives a relativistic wave equation, but with the possibility of negative probabilities. Dirac wondered if instead of getting a second order equation in space and time (as in the Klein-Gordon equation), he could find a first order equation that would be positive definite. He started by trying to take the square root of the wave equation @2 @ @ @ @ 2 r2 − = A + B + C + iD (2.4) @t2 @x @y @z @t @2 @2 when the right hand side of this equation is multiplied out, the cross terms ( @x@y , @x@z etc) cancel if the anti-commutators2 of the coefficients vanish fA; Bg = fC; Dg = fA; Dg = fB; Cg = ::: = 0 (2.5) and they each square to give the identity 2 2 2 2 A = B = C = D = 14×4 (2.6) From this Dirac realised that to get a generalisation of quantum mechanics that was com- patible with relativity you need to construct matrices that are at least 4 × 4 - leading to a 2with fx; yg = xy + yx being the usual anti-commutation relation. 3 2 PARTICLES AND ANTI-PARTICLES wavefunction that has 4 components, known as a spinor. Using equations 2.3 and 2.4 we can write @ @ @ @ A + B + C + iD = X (2.7) @x @y @z @t with X still to be determined. Applying X to both sides we have @2 r2 − = X2 (2.8) @t2 by setting X = m we have a first order equation in space and time @ @ @ @ A + B + C + iD − m = 0 (2.9) @x @y @z @t setting A = iβα1, B = iβα2, C = iβα3 and D = β @ @ @ @ iβ α + α + α + − m = m (2.10) 1 @x 2 @y 3 @z @t inserting r @ iβ α · r + = m (2.11) @t From 2.6 it follows that β2 = 1 and β = β−1 and therefore we get @ iα · r + i = mβ (2.12) @t Then, if we insert the momentum operatorp ^ = −ir, we have the original form of the Dirac equation @ i = (mβ + α · p^) (2.13) @t 4 2 PARTICLES AND ANTI-PARTICLES By defining the gamma matrices γµ γ0 = β (2.14) i γ = βαi (2.15) we get the more usual form of Dirac's equation µ (iγ @µ − m) = 0 (2.16) with being the four component spinor. 2.2 Field Theory and Anti-Matter 1 So equation 2.16 allows us to describe relativistic spin- 2 particles, but what does this tell us about particles and their anti-particle partners? And what does this have to do with the Majorana fermion? 2.2.1 The Dirac Field When building a quantum filed theory you can start with the Lagrangian for your fields then canonically quantise the Lagrangian by promoting the fields to operators. The Lagrangian of the free Dirac field is given by µ L = (iγ @µ − m) (2.17) 5 2 PARTICLES AND ANTI-PARTICLES with ≡ yγ0. It is easy to see that when this Lagrangian is varied we get back the Dirac equation, which is just what we want. The gamma matrices γµ follow the Clifford Algebra fγµ; γνg = γµγν + γνγµ (2.18) = 2ηµν (2.19) Dirac found gamma matrices that were complex. From the form of equation 2.17 this means that the spinor field must also be complex. This makes sense from the point of view of field theory as a complex field would create particles and annihilate anti-particles while its complex conjugate would create anti-particles and annihilate particles. s s y s More explicitly we can expand in terms of creation and annihilation operators ap, ap ,bp, s y bp Z d3p 1 X (x) = (as us(p)e−ip·x + bs yvs(p)eip·x) (2.20) (2π)3 p p p 2Ep s Z d3p 1 X (x) = (bs vs(p)e−ip·x + as yus(p)eip·x) (2.21) (2π)3 p p p 2Ep s With the creation and annihilation operators obeying the normal anti-commutation relations r s y r s y 3 (3) rs fap; ap g = fbp; bp g = (2π) δ (p − q)δ (2.22) s y s y The explanation for these two types of operators is that ap creates a fermion and bp creates an anti-fermion, both with energy Ep and momentum p. Therefore, one-particle states are 6 2 PARTICLES AND ANTI-PARTICLES created by operating on the vacuum state j0i − p s y j e ; p; si ≡ 2Epap j0i (2.23) + p s y j e ; p; si ≡ 2Epbp j0i (2.24) Here, e− is the electron and e+ is its anti-particle partner, the positron. These anti-particles have the same mass as their particle partners but with opposite electric charge. When pairs meet they annihilate each other to produce photons, thus conserving total charge. Field theory gives a neat way to describe particles and their anti-particles, each being created by its own operator. However, as anti-particles are usually defined as having the same mass but opposite charge from their partners, what does this mean for neutral particles? 2.2.2 Neutral Particles It is possible for neutrally charged particles to have a distinct anti-particle.
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  • The Bound States of Dirac Equation with a Scalar Potential

    The Bound States of Dirac Equation with a Scalar Potential

    THE BOUND STATES OF DIRAC EQUATION WITH A SCALAR POTENTIAL BY VATSAL DWIVEDI THESIS Submitted in partial fulfillment of the requirements for the degree of Master of Science in Applied Mathematics in the Graduate College of the University of Illinois at Urbana-Champaign, 2015 Urbana, Illinois Adviser: Professor Jared Bronski Abstract We study the bound states of the 1 + 1 dimensional Dirac equation with a scalar potential, which can also be interpreted as a position dependent \mass", analytically as well as numerically. We derive a Pr¨ufer-like representation for the Dirac equation, which can be used to derive a condition for the existence of bound states in terms of the fixed point of the nonlinear Pr¨uferequation for the angle variable. Another condition was derived by interpreting the Dirac equation as a Hamiltonian flow on R4 and a shooting argument for the induced flow on the space of Lagrangian planes of R4, following a similar calculation by Jones (Ergodic Theor Dyn Syst, 8 (1988) 119-138). The two conditions are shown to be equivalent, and used to compute the bound states analytically and numerically, as well as to derive a Calogero-like upper bound on the number of bound states. The analytic computations are also compared to the bound states computed using techniques from supersymmetric quantum mechanics. ii Acknowledgments In the eternity that is the grad school, this project has been what one might call an impulsive endeavor. In the 6 months from its inception to its conclusion, it has, without doubt, greatly benefited from quite a few individuals, as well as entities, around me, to whom I owe my sincere regards and gratitude.