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14 Chapter 3. Relativistic Bound State Wave Functions
A Dirac formalism based on these scalar and vector potentials, makes it possible to
maintain a distinction between them.
Various relativistic models with varying degrees of accuracy have been developed
up until now [29, 30, 31, 32, 33, 34]. One such an approach is based on a relativistic
quantum field theory, where nucleons interact with one another by exchanging
mesons, and was originally introduced by Walecka in Ref. [35].
3.1 Formalism
A relativistic quantum field theory for nucleons (ψ) interacting with scalar mesons
(φ) through a Yukawa coupling ¯ψψφ and with neutral vector mesons (V µ ) that couple
to the conserved baryon current ¯ψγ µ ψ, can be described through a lagrangian density
of the type [16, 29, 36]
L 0 = ¯ψ(ı̸ ∂ − M)ψ + 1 2 (∂ µφ∂ µ φ − m 2 sφ 2 ) − 1 4 G µνG µν
+ 1 2 m2 vV µ V µ − g v ¯ψγµ ψV µ + g s ¯ψψφ , (3.1)
with M, m s and m v the nucleon, scalar meson and vector meson masses, respectively,
and G µν ≡ ∂ µ V ν − ∂ ν V µ the vector meson field strength. The scalar and vector
fields may be associated with the σ and ω mesons. The model can be extended to
include also π and ρ mesons, as well as the coupling to the photon field. Then, the
corresponding lagrangian has the form
L = L 0 + 1 2 (∂ µ⃗π · ∂ µ ⃗π − m 2 π⃗π · ⃗π) − ıg π ¯ψγ5 ⃗τ · ⃗πψ − 1 4 ⃗ B µν ·
⃗B
µν
+ 1 2 m2 ρ ⃗ b µ ·⃗b µ − 1 2 g ρ ¯ψγ µ ⃗τ ·⃗b µ ψ − 1 4 F µνF µν − eA µ [ ¯ψγ µ 1 2 (1 + τ 3)ψ
+( ⃗ b ν × ⃗ B νµ ) 3 + (⃗π × (∂ µ ⃗π + g ρ (⃗π × ⃗ b µ ))) 3 ] . (3.2)
Here ⃗π, ⃗ b µ and A µ are the pion, rho and Maxwell fields, respectively, while B ⃗ µν ≡
∂ µ ⃗ b ν − ∂ ν ⃗ b µ − g ρ ( ⃗ b µ × ⃗ b ν ) is the ρ meson field strength and F µν is the electromagnetic
field strength. In the lagrangian of Eq. (3.2), higher order electromagnetic
corrections and non-linear meson terms are omitted.
As discussed by Walecka in Ref. [35], the complete quantum field theory, which is
a non-trivial problem, can be approximated by replacing the meson field operators
with their expectation values at high densities. In infinite matter one then takes
< φ >≡ φ 0 and < V µ >≡ δ µ0 V 0 . This model can be extended to include the
interactions of pions and ρ mesons [37]. The resulting mean-field theory may then
be solved exactly.
In the relativistic Hartree approximation under study here, use is made of a
coordinate-space Green’s function approach to derive the relevant equations of motion
(for details, see Refs. [16, 29]). Eventually one arrives at a theory similar in