Lecture 4 - Relativistic wave equations Postulates Relativistic wave equations must satisfy several general postulates. These are;. The equation is developed for a field amplitude function, ψ 2. The normal statistical interpretation of quantum mechanics is satisfied 3. Superposition is valid (the equation must be linear) 4. The equation must be Lorentz invarient 5. Because of the above, the probability density, ρ, must be positive definite and; ρ dx 3 ρ = invarient d dt dx 3 ρ = 6. The theory must be consistent with the correspondance principle, i.e. it reduces to the non-relativistic Schrodinger equation in the limit when β. 7. The equation must describe the time evolution of the amplitude ψ. Thus; Hψ = i ψ ψ(t) = e t iht/ ψ() with H/i the time evolution operator. 2 Klein-Gordon equation The relativistic energy for a free particle is E 2 = p 2 c 2 + m 2 c 4. Applying the QM time and momentum operators, the Schrodinger picture takes the form; 2 /c 2 ( 2 t 2 )ψ = 2 2 ψ + m 2 c 4 ψ This is the Klein-Gordon equation. The wave function is a Lorentz scalar and the equation of continuity for the probability density, ρ is; ρ t + J =
Here a probability current is defined by the Hermitian form; J = i 2m [ψ ψ ψ Ψ ] The definitions preserve the conservation of probability. So in the case of the Klein-Gordon equation; ρ = i 2m [ψ ( t )ψ ( t ψ) Ψ] J = i 2m [ψ ψ ( ψ) ψ] This conserves the probability density; dρ dt = ρ t + v ρ Thus the Klein-Gordon wave equation has the form; [ 2 m/c 2 2 t 2 k 2 ]ψ = Then ψ is a Lorentz scalar, but while the probability density ρ = E/mv 2 ψ ψ has the correct non-relativisitc limit, ρ is not positive definite, (or E can be < as both positive and negative values of ω are allowed in the solution). This can t happen unless the Klein-Gordon equation represents spin= particles and the probability is defined as a charged density, which would allow both positive and negative charged particles. 3 Dirac equation Dirac recognized that the problem with the Klien Gordon equation was the second order time operator, so he sought to linearize the equation. To be covarient, an equation in first order in t must also be first order in x. Thus Dirac factored the the Klein Gordon equation into 2 components. In what follows take c= (natural units). E 2 = p 2 + m 2 = (p m)(p + m) Then assume p is a linear combination of momentum operators, which have the Lorentz covarient form where p l is the momentum 4-vector; p l = k g lk (γ k p k ) In the above g kl is the space-time metric and γ k are operator coefficients. Identify p l with 2
the time and momentum operators in quantum mechanics. p = i t p i = i x i Require that p 2 = p i p i = m 2 (E 2 p p) and that (p m)(p + m) =. This gives the anti-commutation equation; γ n γ m + γ m γ n = 2g mn To satisfy the equation the γ n are four 4-component matricies. The Dirac equations are; [i n [i n γ n x n + m]ψ + = γ n x n m]ψ = The standard representation of the γ matrix operators is; ( ) ( ) σ I γ = γ σ = I Here the σ are the Pauli spin matricies. ( ) ( i σ x = σ y = i ) σ z = ( and I is the unit matrix. The wave function,ψ, is a vector with 4 components which written in vector-matrix form and separated is; ( ) ( ) ( ) ψ u ψ = with ψ u ψ = ψ u ψ3 = ψ d ψ 2 Using the standard representation for the γ operators, the upper ψ u and lower, ψ l, wave function components result in the coupled equations; ( σ p)ψ u = (E + m)ψ l ( σ p)ψ l = (E m)ψ u These equations may be separated to obtain; ( σ p) 2 ψ u = (E 2 m 2 )ψ u ( σ p) 2 ψ l = (E 2 m 2 )ψ l ψ 4 ) 3
Then ( σ A)( σ B) = A B + i σ ( aa B) or ( σ p) 2 = p 2 which gives the eigenvalue problem; p 2 + m 2 E 2 p 2 + m 2 E 2 p 2 + m 2 E 2 p 2 + m 2 E 2 with eigenvalues; E = (p 2 + m 2 ) /2 two eigenvalues E = (p 2 + m 2 ) /2 two eigenvalues ψ ψ 2 ψ 3 ψ 4 = This means there are 4 eigenstates of H, two with positive energy and 2 with negative energy. The negative energy states are assumed to be anti-particles, for example positrons. If the positive energy states are electrons. Because the time evolution operator has the form, e ( ih/ )t ψ, the negative energy states can be assigned as positive energy particles moving backward in time. The Dirac equation thus involves spin ( the two values of ψ u and ψ l ) and also anti-particles. The transformation properties of the wave functions are; ψ ψ ψ γ n ψ ψ γ 5 ψ ψ γ 5 γ n ψ scalar vector pseudo-scalar pseudo-vector here γ5 = γ n γ n and the adjoint wave function ψ is; ( ) I ψ = ψ I 4 Helicity Helicity is defined as the spin direction relative to the momentum. The helicity operator is; Σ = σ p/ p The σ operators are 4 dimensional; ( ) σ σ 4
Thus the eigenvectors of helicity take the form given in the Table. Table : State helicities in Dirac equation State Energy Helicity Chiralty > + + 2 > - - 3 < + - 4 < - + Chirality is obtained by operating on the Dirac wave function by w ± as defined below. w ± = /2( ± iγ 5 ) Obviously the chirality can be the same as helicity for positive energies and the negative of the helicity for negaive energies if the particle is massless. If a system is not identical to its mirror image it will posess chirality. Chirality and helicity are the same for massless particles, but helicity is frame dependent for massive particles since it is possible to transform into a frame where the momentum of the particle changes sign. ψ + is a right handed particle (positive helicity - use the right hand rule) and ψ is a left handed particle (positive helicity). Figure. P P + Helicity Helicity Figure : Spin, momentum, and helicity Thus one obtains; w + = w + = 5
w + = w + = 5 Plane wave solution ( ) u The plane wave solution for ψ = takes the form; l ( ) ( χ /2 ψ + = a + σ p ψ = a + E + m χ/2 The χ vectors are the spin /2 projections; ( ) ( ) χ /2 = χ /2 = χ /2 σ p E + m χ /2 ) 6 The Weyl equation When m = the Dirac equation takes the form; (E γ γ )ψ = m ψ which has 3 anti-communicating matricies. Choose to identify these with the 3 Pauli matricies, σ. In this case the solutions are 2-component wave functions (spinors for spin ±/2). The solutions are identical to those of the Dirac 4-component equation after applying the chiral projection operators. This equation is not partiy invarient, and the plane wave has only one possible spin state. Then a massless neutrino would have left handed helicity and an anti-neutrino right handed helicity. The reduction from the Dirac 4 state solution to a 2 state solution occurs because only one spin state is allowed. 7 Majorana fermion A Majorana fermion is its own anti-particle, thus it cannot be charged. ψ = ψ. Majorana fermions might exist in super-symmetric theories when the real and imaginary wave functions are the same. However, right-handed neutrinos with majorana mass can result from the seesaw mechanism which has 6 neutrino fields. Three of these have a masses approximately ev and 3 have masses comparable to the GUT scale after spontaneous symmetry breaking. In the standard model the Weyl spinor, χ, could be the neutrino component of a lefthanded weak-isospin doublet (see the next lecture). The right-handed spinor is an uncharged singlet, and represented as a sterile neutrino. If right-handed neutrinos exist but do not 6
have a Majorana mass, the neutrinos behave as three Dirac fermions with anti-particles with masses created by the Higgs interaction. Thus Majorana fermions are connected with supersymmetric theories and spontaneous symmetry breaking. 7