Fermi Fields without Tears

Transcription

1 Fermi Fields without Tears Peter Cahill and Kevin Cahill Abstract One can construct Majorana and Dirac fields from fields that are only slightly more complicated than scalar fields.

2 Introduction Some problems in quantum field theory are intrinsically difficult or insoluble; We intend to shed no light on them. Others are solved and pose few difficulties for students. There is, however, a class of solved problems that consistently confuse students year after year. Among the more prickly and more important of these is the construction of fields of spin one-half. Weinberg has written the clearest and most complete discussions of this subject in his papers on massive [Weinberg, 964a] and massless [Weinberg, 964b] fields of any spin and in his magnificent books [Weinberg, 995, Weinberg, 996, Weinberg, ] on quantum field theory. We follow his notation and show how to make Majorana and Dirac fields out of simple scalar-like fields. Fermi Fields without Tears

3 A Scalar Field First let's recall the usual formula for a spin-zero field ϕ(x ϕ(x = d 3 p (π3 p [ a(pe ipx + a (pe ipx]. ( The annihilation and creation operators a(p and a (p satisfy the commutation relations [a(p, a (p ] = δ(p p [a(p, a(p ] = [a (p, a (p ] =. ( They destroy and create spin-zero particles of mass m, momentum p, and energy p = m + p. These particles are their own anti-particles. In our units, = c =. Fermi Fields without Tears 3

4 Gamma Matrices Weinberg's [Weinberg, 995] choice of γ-matrices is γ k = i ( σk σ k k =,, 3, and γ = i They satisfy the anti-commutation relations in which the flat space-time metric is η ab = ( I. (3 I [γ a, γ b ] + = η ab, (4. (5 Under hermitian conjugation, they transform as (γ k = γ k and (γ = γ. Fermi Fields without Tears 4

5 A Scalar-like Field For this choice of γ-matrices, we may define Majorana and Dirac fields in terms of the scalar-like field d 3 [( ( ] p I φ(x = A(pe ipx σ + i A (pe ipx (6 (π 3 p (p + m I σ where A(p and A (p are the -vectors ( a(p, A(p = a(p, and A (p = ( a (p, a (p, (7 and the operators a(p, ± and a (p, ± satisfy the anti-commutation relations [a(p, σ, a (p, σ ] + = δ σ σ δ(p p [a(p, σ, a(p, σ ] + = [a (p, σ, a (p, σ ] + =. (8 They destroy and create spin-one-half particles of mass m, momentum p, spin one-half in the ±ẑ direction, and energy p = m + p. Fermi Fields without Tears 5

6 These particles are their own anti-particles. The matrices I and σ I = ( are and σ = ( i i. (9 The scalar-like field φ(x is φ(x = d 3 p (π 3 p (p + m a(p, a(p, a(p, a(p, eipx + a (p, a (p, a (p, a (p, e ipx. ( Fermi Fields without Tears 6

7 An equivalent formula for the scalar-like field φ(x is φ(x = d 3 p (π3 p (p + m [ u(σ a(p, σe ipx + v(σ a (p, σe ipx] σ= in which the spinors u(σ and v(σ are ( and u( = v( =, u( =, v( =, (. (3 They are the spinors of zero momentum of the Majorana and Dirac fields. Fermi Fields without Tears 7

8 The Klein-Gordon Equation In the definition (6 of the scalar-like field φ(x, the energy p m + p, and so is m + p = m + p (p =. (4 Thus the scalar-like field φ(x satisfies the Klein-Gordon equation (m + φ(x = (m η ab a b φ(x = (m + p φ(x =. (5 Fermi Fields without Tears 8

9 The Majorana Field The Majorana field χ(x is obtained from derivatives of the scalar-like field φ(x: χ(x = (m γ a a φ(x. (6 Because the scalar-like field φ(x satisfies the Klein-Gordon equation (5 and because the γ-matrices satisfy the anti-commutation relations (4, the Majorana field χ(x satisfies the Dirac equation: (γ a a + m χ(x = (γ a a + m (m γ a a φ(x = ( m γ a γ b a b φ(x = ( m [γa, γ b ] + a b φ(x = ( m η ab a b φ(x =. (7 Fermi Fields without Tears 9

10 Spinors It follows from Eqs.( & 6 that the explicit form of the Majorana field χ(x is χ(x = (m γ a a φ(x d 3 p = (π3 p (p + m σ [ (m iγ a p a u(σ a(p, σe ipx + (m + iγ a p a v(σ a (p, σe ipx] = d 3 p (π 3/ σ [ u(p, σ a(p, σe ipx + v(p, σ a (p, σe ipx] (8 where the spinors u(p, σ and v(p, σ are u(p, σ = (m iγa p a u(σ p (p + m and v(p, σ = (m + iγa p a v(σ p (p + m. (9 Fermi Fields without Tears

11 The Dirac Field Suppose there are two spin-one-half particles of the same mass m described by the two operators a (p, σ and a (p, σ which satisfy the anti-commutation relations [a i (p, σ, a j (p, σ ] + = δ ij δ σσ δ 3 (p p. ( Then by following Eqs.(6--7 and defining two -vectors A i (p, σ as in (7, we may construct two scalar-like fields d 3 [( ( ] p I φ i (x = A (π 3 p (p + m I i (pe ipx σ + i A i (pe ipx ( σ and from them two Majorana fields that satisfy the Dirac equation χ i (x = (m γ a a φ i (x ( (γ a a + m χ i (x =. (3 Fermi Fields without Tears

12 Anti-Particles Because the two fields φ i (x are of the same mass, we may combine them into a complex, scalar-like field Φ(x = [φ (x + iφ (x]. (4 The complex operators a(p, σ = [a (p, σ + ia (p, σ] (5 and a c (p, σ = [a (p, σ ia (p, σ], (6 destroy particles that are each other's anti-particles. Fermi Fields without Tears

13 From the complex -vectors A(p = [A (p + ia (p] = ( a(p, a(p, (7 and with A c (p = [A (p ia (p] = ( a c (p, a c (p, A c (p = [A (p ia (p] = [A (p + ia (p] = we can make a complex, scalar-like field Φ(x Φ(x = d 3 p (π 3 p (p + m [( I I (8 ( a c (p, a c (p,, (9 ( ] A(pe ipx σ + i A c (pe ipx. (3 σ Fermi Fields without Tears 3

14 The Dirac field is then The Dirac Field ψ(x = (m γ a a Φ(x = (m γ a a [φ (x + iφ (x] = [χ (x + iχ (x]. (3 It satisfies the Dirac equation (γ a a + m ψ(x = (3 because the Majorana fields χ and χ do. It follows from Eqs.(8, 3, & 3 that the explicit form of the Dirac field is ψ(x = d 3 p (π 3/ σ [ u(p, σ a(p, σe ipx + v(p, σ a c (p, σe ipx] (33 where the spinors are the same as for the Majorana field, Eq.(9. Fermi Fields without Tears 4

15 Other Conventions We have defined Majorana and Dirac fields in terms of Weinberg's choice of γ-matrices. If one uses a different set of γ-matrices γ a = Sγ a S, (34 then the fields should be multiplied from the left by the matrix S: Φ (x = S Φ(x, ψ (x = S ψ(x, etc. (35 Fermi Fields without Tears 5

16 Very Light Fermions In the m limit, it follows from their formulas (9 that the spinors u(p, σ and v(p, σ for p = p ẑ are u(p ẑ, = i ( γ γ 3 u(, = u(p ẑ, = =, v(p ẑ, = = u(,, (36, & v(p ẑ, =. (37 So the upper (lower two components of a Majorana or Dirac field (the (, ((, part can only destroy particles of helicity (+. Fermi Fields without Tears 6

17 Neutrinos The upper two components of a Majorana field (the (, part can only create particles of helicity +. The lower two components of a Majorana field (the (, part can only create particles of helicity. The upper two components of a Dirac field (the (, part can only create antiparticles of helicity +. The lower two components of a Dirac field (the (, part can only create antiparticles of helicity. The Standard Muddle represents neutrinos by fields with only the upper two components (the (, part. Nobody knows whether these fields are Majorana or Dirac. If Majorana, then the neutrino field uses u(p ẑ, a(p ẑ, to destroy neutrinos of helicity and uses v(p ẑ, a (p ẑ, to create neutrinos of helicity. The hermitian adjoint of that field creates neutrinos of helicity and destroys neutrinos of helicity. If Dirac, then the neutrino field uses u(p ẑ, a(p ẑ, to destroy neutrinos of helicity and uses v(p ẑ, ac (p ẑ, to create antineutrinos of helicity. The hermitian adjoint of that field creates neutrinos of helicity and destroys antineutrinos of helicity. Fermi Fields without Tears 7

18 Acknowledgments Thanks to Michael Gold for a helpful conversation. Fermi Fields without Tears 8

19 References [Weinberg, 964a] Weinberg, S. (964a. 33. Phys. Rev. 33 (5B, B38-- [Weinberg, 964b] Weinberg, S. (964b. Phys. Rev. 34 (4B, B [Weinberg, 995] Weinberg, S. (995. The Quantum Theory of Fields, volume I Foundations. Cambridge, UK: Cambridge University Press. [Weinberg, 996] Weinberg, S. (996. The Quantum Theory of Fields, volume II Modern Aplications. Cambridge, UK: Cambridge University Press. [Weinberg, ] Weinberg, S. (. The Quantum Theory of Fields, volume III Supersymmetry. Cambridge, UK: Cambridge University Press. Fermi Fields without Tears 9