The Dirac Equation. Topic 3 Spinors, Fermion Fields, Dirac Fields Lecture 13

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1 The Dirac Equation Dirac s discovery of a relativistic wave equation for the electron was published in 1928 soon after the concept of intrisic spin angular momentum was proposed by Goudsmit and Uhlenbeck to explain the double degeneracy of atomic energy levels. His article, P.A.M. Dirac, The Quantum Theory of the Electron, Proc. R. Soc. Lond. A 117, is one of the most important publications of the twentieth century. In this article, Dirac introduced all of the concepts and notation needed for the fermion part of the QED Lagrangian density [Peskin-Schroeder Eq. 4.3] L QED = L Dirac + L int + L Maxwell = ψx [iγ µ µ m] ψx e ψxγ µ A µ xψx 1 4 F µνf µν, including the minimal coupling to the electromagnetic field. Eq. 14 in Dirac s article gives the Hamiltonian wave equation [p 0 + e c A 0 + ρ 1 σ, p + e c A ] + ρ 3 mc ψ = 0, where p 0 = i c t, p = i, σ, p + e c A σ p + e c A. PHY /25/2013

2 The Dirac matrices are defined by ρ 1 = , ρ 3 = , σ = σpauli 0 0 σ Pauli. These are essentially the 4 4 Dirac γ-matrices which obey the Clifford algebra anticommutators in modern notation {γ µ, γ ν } = 2g µν. The essential new feature that was introduced into quantum field theory by Pauli and Dirac is the use of anticommutators instead of commutators to describe the Fock space of fermions. PHY /25/2013

3 Fermionic Harmonic Oscillator The ladder operators âk, λ and â k, λ are used to construct the Fock space for the photon fields. This is simply the direct product of simple harmonic oscillator Hilbert spaces, one for each mode with momentum k and polarization λ. The Fock space of a simple harmonic oscillator is defined by the Hamiltonian and commutation relations Ĥ = â â + ââ ω 2, [â, â ] = 1, [â, â ] = 0, [â, â] = 0, and the energy eigenstates and eigenvalues are â n â 0 = 0, n = 0, Ĥ n = E n n, E n = n! n + 1 ω. 2 The operator â increases the number of quanta in an eigenstate by one and its energy by ω. The quanta represent non-interacting bosonic particles with mass m = ω in Minkowski spacetime M 1,0. Ladder Operators, Hamiltonian, and Fock Space Let ˆb be the annihilation operator and ˆb the creation operator for fermionic particles, and try replacing the commutation relations of âˆ,a with anti-commutators {ˆb, ˆb } = ˆbˆb + ˆb ˆb = 1, {ˆb, ˆb } = 2 ˆb 2 = 0, {ˆb, ˆb} = ˆb2 = 0. The + sign in the bosonic Hamiltonian must be changed to Ĥ = ˆb ˆb ˆbˆb ω 2 = 1 ˆb ˆb 2 PHY /25/2013 ω

4 so that Ĥ 0. To define the vaccuum state annihilated by ˆb, let ψ be any state. Then ˆb either annihilates ψ or it does not. In either case a normalized vacuum state can be defined: 0 = { ψ / ψ ψ if ˆb ψ = 0 χ / χ χ if ˆb ψ = χ 0 { b 0 = 0 b 0 = b 2 ψ/ χ χ = 0 because b 2 = 0. Apply the creation operator to the vacuum to obtain the one-fermion state which is turns out to be properly normalized: 1 = ˆb 0, 1 1 = ˆb 0 1 = 0 ˆb 1 = 0 ˆbˆb 0 = 0 Applying the creation operator to the one-fermion state gives ˆb 1 = ˆb 2 0 = 0, because ˆb 2 = 0. 1 ˆb ˆb 0 = 1. The fermionic oscillator Hilbert space represents a two-state system Ĥ n = E n n, E n = n 1 ω, n = 0, 1, 2 The zero-point energy of the fermionic oscillator is negative, and the quanta obey the Pauli exclusion principle in agreement with Fermi-Dirac statistics. PHY /25/2013

5 The Supersymmetric Harmonic Oscillator The fact that the vacuum energies of the bosonic B and fermionic F oscillators have opposite signs raises the interesting possibility that the divergent vacuum energy due to quantum fluctuations, and possibly other divergences in quantum field theories, can be cancelled if the theories were somehow made symmetric between bosons and fermions. Define a supersymmetric harmonic oscillator to be independent bosonic and fermionic oscillators with the same mass ω = m. The ladder operators mutually commute [ â, ˆb ] = 0, [ â, ˆb ] = 0, [ â, ˆb ] = 0, [ â, ˆb ] = 0, the Hamiltonian is the sum Ĥ = ĤB + ĤF = NB + N F ω = â â + ˆb ˆb ω, and the Fock space is spanned be the direct product states n B, n F = n B n F, Ĥ n B, n F = E n n B, n F, E n = n B + n F ω. The vacuum state is non-degenerate with E 0 = 0. Every excited state is doubly degenerate. The state with energy E n and n > 0 has either n bosons and no fermion, or one fermion and n 1 bosons. An interesting question is whether linear superpositions of states with n F = 0, 1 are allowed. If there is a superselection rule that forbids such superpositions, then the supersymmetry is accidental and PHY /25/2013

6 rather trivial. Fermion number does appear to be strictly conserved in all observed elementary particle interactions. Supersymmetry has been proposed as a solution to the cosmological constant problem. If Nature is supersymmetic the vaccum energy density from quantum fluctuations is zero. Supersymmetry must be broken at some level to explain the Casimir effect. Supercharges and Superalgebra To construct superpositions of states with different numbers of fermions and to study the properties of such states, it is important to be able to transform the two types of ladder operators. This can be done with two hermitian supercharge operators Q α, α = 1, 2, Q1 = ωˆbâ + ˆb â, Q2 = i ωˆbâ ˆb â, Q α = Q α. The operators Q α change the fermion and boson numbers of an energy eigenstate by ±1 without changing the energy, for example Q 1 n B, n F = { nb + 1, n F 1 if n F = 1 ω n B 1, n F + 1 if n B > 0 and n F = 0 0 otherwise The supercharges and the Hamiltonian obey the operator algebra { Qα, Q } [ Ĥ] β = 2δ αβ Ĥ, Qα, = 0. PHY /25/2013

7 Supersymmetric Quantum Mechanics Supersymmetry has been used to study many interesting problems in non-relativistic quantum mechanics. Fore a review, see F. Cooper, A. Khare, and U. Sukhatme, Supersymmetry and Quantum Mechanics. Consider a one-dimensional potential V 1 x and measure energy relative to the ground state E 0 = 0. The Schrödinger equation for the ground state wave function can be solved for H 1 ψ 0 x = 2 d 2 dx 2 ψ 0x + V 1 xψ 0 x = E 0 ψ 0 x = 0 V 1 x = 2 ψ 0 x ψ 0 x, which is well defined because the ground state wave function has no nodes. Define a superpotential wx and ladder operators A, A wx = ψ 0x ψ 0 x, A = i d dx + iwx, A = i d dx iwx. Then the potential and Hamiltonian can be derived from the superpotential and ladder operators V 1 x = w 2 x w x, H 1 = A A. PHY /25/2013

8 The operator A annihilates the ground state The Hamiltonian H 1 has a superpartner Aψ 0 x = 0. H 2 = AA = 2 d 2 dx 2 + V 2x, where V 2 x = w 2 x + w x. The energy levels of the supersymmetric partner Hamiltonians are shown in the figure. PHY /25/2013

9 The partner Hamiltonians can be written in terms of the Pauli matrix σ 3 = σ z H = A A 0 0 AA = H1 0 0 H 2 = p2 + w2 x w xσ 3, and a pair of supercharges can be defined using σ 1 = σ x and σ 2 = σ y Q 1 = Q 2 = p σ 1 + wxσ 2 = p σ 1 wxσ 2 = 0 A, A 0 0 ia ia 0. The supersymmetry algebra for this system is {Q 1, Q 1 } = {Q 2, Q 2 } = 2H, {Q 1, Q 2 } = 0, [Q 1, H] = [Q 2, H] = 0, which is the same as that defined for the supersymmetric harmonic oscillator. PHY /25/2013

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