Types of Fields

• Scalar Fields: Φ(x) describe spin-0 particles. They are scalar bosons.

  • A complex scalar field has two real degrees of freedom which correspond to a particle and an anti-particle. They carry opposite charges.
  • A real-valued scalar field has one real degree of freedom, corresponding to a particle = its anti-particle. This is a neutral particle.

• Dirac Spinor Fields: ψ(x) describe spin-1/2 particles.

  • Complex-valued-4-component spinor. In the chiral basis for γ matrices, ψ(x) can be decomposed into two 2-component spinors ψL(x) and ψR(x), called the left-handed and right-handed Weyl Spinors. $\psi (x)=\left(\begin{array}{c}{\psi }_1\\ {\psi }_2\\ {\psi }_3\\ {\psi }_4\end{array}\right)=\left(\begin{array}{c}{\psi }_L\\ {\psi }_R\end{array}\right)$
  • The left-handed Weyl spinor describes (creation and annihilation of) particles ofhelicity = -1/2 and its Hermitian-conjugate corresponds to its anti-particle. Thus, 2 degrees of freedom for left-handed Weyl spinors.
  • The right-handed Weyl spinor describes particles ofhelicity = +1/2 and its Hermitian-conjugate corresponds to its anti-particle. Thus, 2 degrees of freedom for right-handed Weyl spinors.
  • So, in total, 4 degrees of freedom are contained in a Dirac Spinor field.
  • helicity is the projection of spin on 3-momentum of a particle. It is a good quantum number for mass-less particles. If a particle is massive, then its helicity can be changed by performing a Lorentz boost.
    • Left-handed helicity ⇒ h=-1/2
    • Right-handed helicity ⇒ h=+1/2

• Majorana Fermion Fields: describe spin-1/2 particles that are their own anti-particles.

  • Essentially, we identify ψR=ψL†, ${\psi }_M(x)=\left(\begin{array}{c}{\psi }_L\\ {\psi }_L^{†}\end{array}\right)=\left(\begin{array}{c}\psi \\ {\psi }^{†}\end{array}\right)$

• Vector Fields: A(x) describe spin-1 particles (known as vector bosons)

  • In QFT, vector fields originate from gauge fields (real-valued).
  • But when gauge symmetry is exact, all gauge fields must be strictly massless ⇒ massless vector bosons (photons in QED and gluons in QCD - theory of strong interactions).
  • Gauge symmetry can also be spontaneously broken and Higgs mechanism becomes operational and the gauge field becomes massive ⇒ massive vector boson.
  • We technically have 4 real-valued degrees of freedom for A(x) since it is a Lorentz 4-vector. But, since it has to satisfy the Euler-Lagrange equations, the number reduces to 3.
  • Furthermore, the requirement of exact gauge invariance dictates that another degree of freedom corresponds to a choice of gauge (unphysical) and can be removed by gauge fixing.
  • Thus, for a massless vector boson, we are left with 2 real-valued degrees of freedom ⇒ 2 transverse spin polarisations. They are often combined into the left-handed and right-handed polarisations: $A^{\pm }=\frac{1}{\sqrt{2}(A_x\pm i{A}_y)}$
  • On the other hand, massive vector bosons have 3 independent real-valued degrees of freedom. This is because gauge invariance is broken spontaneously and the longitudinal component of the vector field no longer decouples from the system. (W(+/-) and Z0 bosons of weak interactions have spin projections 0,+1,-1)