102
A. Lattanzi
is the fact that making suitable simplifications and approximations it stems from
the Bethe–Salpeter equation [6], which is the most orthodox tool for discussing
the relativistic two-body problem in quantum field theory [9]. Firstly proposed by
Nambu [1] without derivation, the equation was then derived by Bethe and Salpeter
[2] using Feynman-graphs and by Gell-Mann and Low [3] on the basis of the
rigorous quantum field theory considerations. After some initial difficulties, the
Bethe–Salpeter equation has been the object of intense theoretical studies [7, 9, 19–
22]. As a general quantum field theoretical tool, the Bethe–Salpeter equation finds
applications in several areas of quantum field theory, as, for instance, in connection
with electron–positron pairs and excitons (i.e. bound states of electron–hole pairs).
A more detailed discussion on the approximations needed for obtaining the
spinless Salpeter equation from the Bethe–Salpeter equation can be found in [12, 13]
and a summary is delineated here. By eliminating the dependence on time-like
variables through the assumption of static or instantaneous interactions, the Bethe–
Salpeter equation [2] reduces to the Salpeter equation [4]. Neglecting furthermore
all spin degrees of freedom and restricting it only to positive-energy solutions, one
obtains the spinless Salpeter equation [6].
The spinless Salpeter equation is frequently employed in the phenomenological
description of hadrons. Moreover, the agreement of the predictions of the spinless
Salpeter equation with the experimental spectrum of mesonic atoms is as good as
those of the Klein–Gordon equation.
In (1+1)D, the spinless Salpeter equation incorporates the relativistic expression
for the energy of the particle, which in the presence of a potential V (x, t) is
E =
m 2 c 4 + p 2 c 2 + V (x, t),
(1)
where m and p denote, respectively, the rest mass and the momentum of the particle,
while c is the speed of light in vacuum and x denotes the position.
In accord with the standard quantization rules
E → i ¯
h∂ t ,
r → →
r,
p → −i ¯
h∇,
the relativistic (1 + 1)D Hamiltonian reads
ˆ
H =
m 2 c 4 − c 2 ¯
h 2 ∂ 2
∂x 2 + V (x, t),
(2)
whose main and most remarkable feature is the presence of the square-root operator.
In this work, we consider the free-particle Salpeter equation (V (x, t) = 0):
i ¯
h∂ t ψ(x, t) =
m 2 c 4 − c 2 ¯
h 2 ∂ 2
∂x 2 ψ(x, t),
ψ(x, 0) = ψ 0 (x).
(3)
A. Lattanzi
is the fact that making suitable simplifications and approximations it stems from
the Bethe–Salpeter equation [6], which is the most orthodox tool for discussing
the relativistic two-body problem in quantum field theory [9]. Firstly proposed by
Nambu [1] without derivation, the equation was then derived by Bethe and Salpeter
[2] using Feynman-graphs and by Gell-Mann and Low [3] on the basis of the
rigorous quantum field theory considerations. After some initial difficulties, the
Bethe–Salpeter equation has been the object of intense theoretical studies [7, 9, 19–
22]. As a general quantum field theoretical tool, the Bethe–Salpeter equation finds
applications in several areas of quantum field theory, as, for instance, in connection
with electron–positron pairs and excitons (i.e. bound states of electron–hole pairs).
A more detailed discussion on the approximations needed for obtaining the
spinless Salpeter equation from the Bethe–Salpeter equation can be found in [12, 13]
and a summary is delineated here. By eliminating the dependence on time-like
variables through the assumption of static or instantaneous interactions, the Bethe–
Salpeter equation [2] reduces to the Salpeter equation [4]. Neglecting furthermore
all spin degrees of freedom and restricting it only to positive-energy solutions, one
obtains the spinless Salpeter equation [6].
The spinless Salpeter equation is frequently employed in the phenomenological
description of hadrons. Moreover, the agreement of the predictions of the spinless
Salpeter equation with the experimental spectrum of mesonic atoms is as good as
those of the Klein–Gordon equation.
In (1+1)D, the spinless Salpeter equation incorporates the relativistic expression
for the energy of the particle, which in the presence of a potential V (x, t) is
E =
m 2 c 4 + p 2 c 2 + V (x, t),
(1)
where m and p denote, respectively, the rest mass and the momentum of the particle,
while c is the speed of light in vacuum and x denotes the position.
In accord with the standard quantization rules
E → i ¯
h∂ t ,
r → →
r,
p → −i ¯
h∇,
the relativistic (1 + 1)D Hamiltonian reads
ˆ
H =
m 2 c 4 − c 2 ¯
h 2 ∂ 2
∂x 2 + V (x, t),
(2)
whose main and most remarkable feature is the presence of the square-root operator.
In this work, we consider the free-particle Salpeter equation (V (x, t) = 0):
i ¯
h∂ t ψ(x, t) =
m 2 c 4 − c 2 ¯
h 2 ∂ 2
∂x 2 ψ(x, t),
ψ(x, 0) = ψ 0 (x).
(3)