Majorana and Majorana Fermions. From Romance to Reality

Transcription

1 Majorana and Majorana Fermions From Romance to Reality

2 The Romance of Ettore Majorana ( ?)

3 There are many categories of scientists: People of second and third rank, who do their best, but do not go very far. There are also people of first-class rank, who make great discoveries, fundamental to the development of science. But then there are the geniuses, like Galileo and Newton. Well Ettore Majorana was one of them. - Enrico Fermi

4 Majorana s most famous work is a 1937 paper containing results he derived several years before. I ll be discussing the legacy of that paper.

5 On March Ettore Majorana boarded a steamer from Palermo to Naples. He was never seen again.

6 Majorana Fermions

7 In 1928 Dirac proposed his relativistic wave equation for electrons (the Dirac equation). This was a watershed in theoretical physics. It led to a new understanding of spin, predicted antimatter, and impelled quantum field theory.

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10 It also inaugurated a new method in theoretical physics, emphasizing mathematical esthetics.

11 Majorana out-diraced Dirac.

12 The photon is a neutral spin-1 particle that is its own antiparticle. These facts lead to Maxwell s equations. Majorana asked if there could be a spin-1/2 version. More specifically, Majorana asked if there could be a modified version of the Dirac equation, suitable for describing electron-like particles that are their own antiparticles.

13 [technical interlude 1]

14 Dirac s equation connects the 4 components of a field ψ. In modern (covariant) notation, it reads: (i γ μ μ - m)ψ = 0.

15 The γ matrices obey the rules of Clifford algebra, i.e. {γ μ,γ ν } = 2η μν or more explicitly: a) (γ 0 ) 2 = - (γ 1 ) 2 = - (γ 2 ) 2 = - (γ 3 ) 2 = 1 b) γ i γ j = - γ j γ i if i j c) γ 0 is Hermitean, the others are anti- Hermitean

16 Dirac postulated a representation of the γ matrices that involves both real and imaginary numbers. Then for the Dirac equation to make sense, the ψ field must be complex. Dirac (and others) regarded this as a good feature. Electrons are charged, and charged particles require complex fields, even at the level of the Schrödinger equation.

17 Another perspective: In quantum field theory, ψ destroys electrons and creates positrons, while its complex conjugate ψ* does the opposite. So ψ and ψ* must be distinct.

18 Even more specifically: Can there be an equation of Dirac s type that works for a field with ψ* = ψ, i.e. a realnumber field? For this, we require γ matrices that are pure imaginary (and satisfy the Clifford algebra). Majorana found such matrices. Here they come:

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21 [end of technical interlude 1]

22 Are Neutrinos Majorana Fermions?

23 Majorana speculated that his equation might apply to neutrinos, which in 1937 were still very hypothetical particles with many unknown properties.

24 Neutrinos were discovered in 1956, but their properties appeared to disfavor Majorana s idea. Specifically, there seemed to be a strict distinction between neutrinos and antineutrinos. The distinction is connected with the laws of lepton number conservation.

25 The law of electron number conservation takes the form: N(e) - N(e)+ N(νe)- N(νe ) Le = constant There are similar laws for muon number and tau lepton number.

26 These laws lead to many successful selection rules. For example: Neutrinos from π + decay (π + μ + + νμ ) will induce νμ + n μ - + p but not νμ + p μ + + n; while (anti)neutrinos from π - μ - + νμ obey the opposite pattern.

27 Of course, if neutrinos really differ from antineutrinos, then they are not Majorana particles. In recent years, however, the situation has come to seem less clear-cut. It has been discovered that neutrinos oscillate. In some sense this is a small effect, but neutrinos can travel a long way, and then they have time to do rare things. The oscillations indicate that at best only the sum Le + Lμ + Lτ is strictly conserved.

28 Awakened from our dogmatic slumber, we re-open Majorana s question: Could the apparent distinction between neutrino and antineutrino be superficial? Consider the apparent difference between the morning star and the evening star: Yet they re both Venus! But how can ν = ν be reconciled with the many observations, which seemed to indicate they are different?

29 The point is that the ν particles produced in (for example) π + μ + + νμ are in a very different state of motion from the ν particles produced in π - μ - + νμ. The former are left-handed while the latter are right-handed:

30 So, logically, ν and ν might be the same particle seen in different states of motion! If you could bring neutrinos and antineutrinos to rest, and do experiments with them, you could test whether they behave the same way. This is impractical, unfortunately. (Theoretically, the cosmos bathes us in slow neutrinos, but they are too hard to detect.)

31 Many experiments to test conservation of Le + Lμ + Lτ are underway. The most sensitive are searches for neutrinoless double beta decay, e.g. Ge 76 Se 76 + e + e So far there is no clear positive result*. *(But google Klapdor.)

32 Meanwhile, the leading theory of neutrino masses, rooted in grand unification, predicts that neutrinos are Majorana particles.

33 Supersymmetry, Dark Matter, and Majorana Fermions

34 There are good reasons to expect that (broken) supersymmetry is a feature of our world, with some superpartners weighing in at 1 TeV. (I ll convince you momentarily!) If so, they should be discovered at the LHC, in the next few years.

35 The Standard Model, As It Comes six fundamental materials three fundamental forces (plus gravity) (plus 2 repeats) (plus Higgs particle)

36 Unification SO(10) One material One force 03/01/2003 Frank Wilczek, MIT light Majorana neutrino Coincidence? - I think not heavy Majorana neutrino

37 electric inverse coupling strength weak strong large energy, short distance

38 electron quarks photon gluons

39 Now Add SUSY electron quarks photon gluons

40 Why I SUSY electric inverse coupling strength weak Gravity fits too! (roughly) strong large energy, short distance

41 The spin-1/2 superpartners of spin-1 or spin-0 particles that are their own antiparticles will be Majorana fermions. This includes photinos, zinos, gluinos, Higgsinos, and axinos. The lightest superpartner (LSP) is a prime candidate to provide the astronomers dark matter. In any case, if low-energy symmetry is correct, the LHC will provide many realizations of Majorana s idea.

42 Majorana Modes Today s Toys -Tomorrow s Qubits?

43 Majorana modes are 0+1-dimensional massless Majorana fermions, that attach to special points (e.g. ends or junctions of wires) or small objects (e.g. Josephson junctions). I will call them Majorinos.

44 Ordinarily, electrons can t behave as Majorinos, simply because the sign of electric charge distinguishes whether they re present (particle) or absent (antiparticle).

45 In superconductors, however, it s a different story, because there is an omnipresent condensate of electron pairs. A hole can tap into the condensate, grab a pair, and look like a particle.

46 Majorana Modes and Their Algebra [Technical Interlude 2]

47 We can t count the number of electrons (since 2 = 0), but we can keep track of even and odd:

48 P = ( 1) N e {P, b j } = 0 [P, H eff. ] = 0 P 2 = 1

49 Most of this workshop has been devoted to the question of how you make Majorinos, in practice. I will assume that is solved, and consider assembling 5 such particles together.

50 Consider, at a junction of 5 wires, the total product operator Γ b 1 b 2 b 3 b 4 b 5 It satisfies Γ 2 = 1 [H eff., Γ ] = 0 {P, Γ } = 0

51 Γ creates a new composite Majorino: It is hermitean and squares to 1. It commutes with the Hamiltonian. *It is not a function of the Hamiltonian, since it anticommutes with P.*

52 A similar construction works for a junction of any odd number of wire ends, to produce a Γ with the same properties. Their Γ matrices, plus P, satisfy a euclidean Clifford algebra; the states they create make a spinor in Hilbert space. One can also include any finite number of interacting electron modes, simply by augmenting Γ.

53 If we diagonalize both H and P, then Γ connects degenerate states with P = ±1. The junction spectrum is doubled, in a way reminiscent of Kramers doubling. Majorana doubling occurs not only for the ground state, but for all states, through the action of Γ.

54 The Γ s and their algebra have many unusual properties: they re associated with the product wave function (extreme Heitler-London). their symmetry algebra involves anticommutators, not commutators (supersymmetry). they change sign under 2π rotations (fermionic)*. *This changes states, not only phases. Q: In what sense do they define observables?

55 [End of technical interlude 2]

56 When we have several Majorinos -- e.g., a circuit with several junctions -- their wave function lives in a huge Hilbert space. Their wave function keeps track not only of where they are, but also of where they ve been. Moving them around gives us a powerful way to play with knots (or actually braids), and then read out the result.

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60 These observations inspire the vision of topological quantum computing.

61 Conclusion Ettore Majorana s esoteric question and exotic mathematics have become central to several promising frontiers of physics: neutrinos supersymmetry and dark matter topological quantum computing

62 Will one or more of them break through? I think the question is not if, but when.

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64 What I Didn t Talk About Understanding (generalized) Majorana modes and their statistics as JR (GW) charges: walls, vortices tangles of vortices anyon statistics monopoles and their clouds? ribbon statistics