Variational two-particle wave equation in scalar quantum field theory
We study two-particle systems in a model quantum field theory, in which scalar particles of different mass interact via a mediating scalar field. The Lagrangian of the model is reformulated using covariant Green's functions to solve for the mediating field in terms of the particle fields. This r...
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2000
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| Цитувати: | Variational two-particle wave equation in scalar quantum field theory / J. Darewych // Condensed Matter Physics. — 2000. — Т. 3, № 3(23). — С. 633-639. — Бібліогр.: 6 назв. — англ. |
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irk-123456789-1210022017-06-14T03:04:02Z Variational two-particle wave equation in scalar quantum field theory Darewych, J. We study two-particle systems in a model quantum field theory, in which scalar particles of different mass interact via a mediating scalar field. The Lagrangian of the model is reformulated using covariant Green's functions to solve for the mediating field in terms of the particle fields. This results in a Hamiltonian in which the mediating-field propagator appears directly in the interaction term. The variational method, with a simple Fock-state trial state, is used to derive a relativistic momentum-space two-particle wave equation. Non-relativistic and one-particle limits of the equation are determined and discussed briefly Ми вивчаємо двочастинкові системи в модельній квантовій теорії поля, в якій скалярні частинки різної маси взаємодіють через посередкове скалярне поле. Використовуючи коваріантні функції Ґріна для розв’язку посередкового поля в термінах частинкових полів, переформульовано ляґранжіян моделі. В результаті в гамільтоніяні виникає пропагатор посередкового поля прямо в члені, який описує взаємодію. Варіяційна метода з простим пробним фоковим станом використовується для того, щоб вивести релятивістичне двочастинкове хвильове рівняння в імпульс-просторі. Одержуються і обговорюються нерелятивістичні і одночастинкові границі цього рівняння. 2000 Article Variational two-particle wave equation in scalar quantum field theory / J. Darewych // Condensed Matter Physics. — 2000. — Т. 3, № 3(23). — С. 633-639. — Бібліогр.: 6 назв. — англ. 1607-324X DOI:10.5488/CMP.3.3.633 PACS: 11.10.Qr, 11.10.St http://dspace.nbuv.gov.ua/handle/123456789/121002 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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We study two-particle systems in a model quantum field theory, in which
scalar particles of different mass interact via a mediating scalar field. The
Lagrangian of the model is reformulated using covariant Green's functions
to solve for the mediating field in terms of the particle fields. This results in
a Hamiltonian in which the mediating-field propagator appears directly in
the interaction term. The variational method, with a simple Fock-state trial
state, is used to derive a relativistic momentum-space two-particle wave
equation. Non-relativistic and one-particle limits of the equation are determined and discussed briefly |
| format |
Article |
| author |
Darewych, J. |
| spellingShingle |
Darewych, J. Variational two-particle wave equation in scalar quantum field theory Condensed Matter Physics |
| author_facet |
Darewych, J. |
| author_sort |
Darewych, J. |
| title |
Variational two-particle wave equation in scalar quantum field theory |
| title_short |
Variational two-particle wave equation in scalar quantum field theory |
| title_full |
Variational two-particle wave equation in scalar quantum field theory |
| title_fullStr |
Variational two-particle wave equation in scalar quantum field theory |
| title_full_unstemmed |
Variational two-particle wave equation in scalar quantum field theory |
| title_sort |
variational two-particle wave equation in scalar quantum field theory |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2000 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/121002 |
| citation_txt |
Variational two-particle wave equation in scalar quantum field theory / J. Darewych // Condensed Matter Physics. — 2000. — Т. 3, № 3(23). — С. 633-639. — Бібліогр.: 6 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT darewychj variationaltwoparticlewaveequationinscalarquantumfieldtheory |
| first_indexed |
2025-07-08T19:00:11Z |
| last_indexed |
2025-07-08T19:00:11Z |
| _version_ |
1837106424464801792 |
| fulltext |
Condensed Matter Physics, 2000, Vol. 3, No. 3(23), pp. 633–639
Variational two-particle wave equation
in scalar quantum field theory
J.Darewych
Department of Physics and Astronomy, York University
Toronto, Ontario M3J 1P3, Canada
Received April 29, 2000
We study two-particle systems in a model quantum field theory, in which
scalar particles of different mass interact via a mediating scalar field. The
Lagrangian of the model is reformulated using covariant Green’s functions
to solve for the mediating field in terms of the particle fields. This results in
a Hamiltonian in which the mediating-field propagator appears directly in
the interaction term. The variational method, with a simple Fock-state trial
state, is used to derive a relativistic momentum-space two-particle wave
equation. Non-relativistic and one-particle limits of the equation are deter-
mined and discussed briefly
Key words: quantum field theory, two-particle wave equation
PACS: 11.10.Qr, 11.10.St
This paper is dedicated to Prof. Ihor Yukhnovsky on the occasion of his 75th
birthday.
1. Introduction
In earlier papers [1, 2], a relativistic wave equation for a scalar particle-antipar-
ticle system, interacting via a mediating scalar field, was derived variationally for
the scalar Yukawa model (which is also called the Wick-Cutkosky model [3-6]). We
consider the complementary problem of two scalar particles with different masses in
the present paper. The model quantum field theory being studied is defined by the
Lagrangian density (h̄ = c = 1)
L =
2
∑
k=1
[
∂νφ†
k(x) ∂νφk(x)−m2
kφ
†
k(x)φk(x)− gkφ
†
k(x)φk(x)χ(x)
− λk(φ
†
k(x)φk(x))
2
]
+
1
2
∂νχ(x) ∂νχ(x)−
1
2
µ2χ2(x), (1)
where φ1(x) and φ2(x) are the scalar fields corresponding to the particles of masses
m1 and m2 respectively, while g1, g2, λ1, λ2 are positive coupling constants. The me-
c© J.Darewych 633
J.Darewych
diating “chion” field can be massive (µ 6= 0) or massless (µ = 0).
The fields φk and χ satisfy the Euler-Lagrange equations
∂ν∂νχ(x) + µ2χ(x) = ρ(x), (2)
where ρ(x) = −g1φ†
1(x)φ1(x)− g2φ
†
2(x)φ2(x),
∂ν∂νφk(x) +m2
kφk(x) + 2λk(φ
†
k(x)φk(x))φk(x) = −gkφk(x)χ(x), (3)
and the conjugates of (3). Equation (2) has the formal solution
χ(x) = χ0(x) +
∫
dx′D(x− x′) ρ(x′), (4)
where dx = dNx dt in N + 1 dimensions, and χ0(x) satisfies the homogeneous (or
free field) equation (equation (2) with ρ = 0), while D(x− x′) is a covariant Green
function (or chion propagator, in the terminology of QFT ), such that
(
∂ν∂ν + µ2
)
D(x− x′) = δN+1(x− x′). (5)
Substitution of the formal solution (4) into equation (3) yields the equations
∂ν∂νφk(x) +m2
kφk(x) + 2λk(φ
†
k(x)φk(x))φk(x) =
= −gkφk(x)χ0(x)− gkφk(x)
∫
dx′D(x− x′)ρ(x′). (6)
Equations (6) are derivable from the action principle δ
∫
dxL = 0, corresponding
to the modified Lagrangian density
L =
2
∑
k=1
[
∂νφ†
k(x) ∂νφk(x)−m2
kφ
†
k(x)φk(x)− gkφ
†
k(x)φk(x)χ0(x)
− λk(φ
†
k(x)φk(x))
2
]
+
1
2
∫
dx′ρ(x)D(x− x′)ρ(x′), (7)
provided that D(x − x′) = D(x′ − x). (We suppress the Lagrangian density of the
free chion field.)
The QFTs based on (1) and (7) are equivalent in the sense that they lead to the
same invariant matrix elements in various order of covariant perturbation theory.
The difference is that, in the formulation based on (7), the interaction term, which
contains the propagator leads to Feynman diagrams involving virtual chions, while
the term that contains χ0 corresponds to diagrams that cannot be generated using
the term with D(x− x′), such as those with external (physical) chion lines.
The Hamiltonian density corresponding to the Lagrangian (7) is given by
H(x) = Hφ1
(x) +Hφ2
(x) +Hχ(x) +HI(x) +HII(x), (8)
634
Variational two-particle wave equation in scalar quantum field theory
where
Hφk
(x) = φ̇†
k(x)φ̇k(x) +∇φ†
k(x) · ∇φk(x) +m2
k φ
†
k(x)φk(x), (9)
Hχ(x) =
1
2
χ̇2
0 +
1
2
(∇χ0)
2 +
1
2
µ2χ2
0, (10)
HI(x) =
2
∑
k=1
gk φ
†
k(x)φk(x)χ0(x), (11)
HII(x) = −1
2
∫
dx′ ρ(x)D(x− x′)ρ(x′) +
2
∑
k=1
λk(φ
†
k(x)φk(x))
2, (12)
and
D(x− x′) =
∫
dk
(2π)N+1
e−ik·(x−x′) 1
µ2 − k · k , (13)
where dk = dN+1k and k · k = k2 = kνkν .
To specify our notation, we quote the Fourier decomposition of the fields in N+1
dimension:
φk(x) =
∫
dNq [(2π)N2ω(q, mk)]
− 1
2 [Ak(q)e
−iq·x +B†
k(q)e
iq·x], (14)
where ω(p, m) =
√
p2 +m2, q · x = qνxν and qν = (q0 = ω(q, mk),q), that is
q2 = m2
k,
χ0(x) =
∫
dNk [(2π)N2ω(k, µ)]−
1
2 [d(k)e−ik·x + d†(k)eik·x] (15)
with k2 = µ2. The momentum-space operators obey the usual commutation rela-
tions. The nonvanishing operators are
[Ai(p), A
†
j(q)] = [Bi(p), B
†
j (q)] = δij δ
N(p− q), (16)
[d(p), d†(q)] = δN (p− q). (17)
These operators have the usual interpretation, namely that A†
k are creation operators
of the (free) scalar particles of mass mk (k = 1, 2), B†
k are the corresponding an-
tiparticle creation operators, while d† is the creation operator of the mediating-field
quantum (which may be massive, µ > 0, or massless, µ = 0).
The Hamiltonian operator, Ĥ =
∫
dNx Ĥ(x), of the QFTheory is expressed in
terms of the creation and the annihilation operators A†
k, Ak, B
†
k, Bk, d
†, d in the
usual way. Since we are not interested in vacuum-energy questions in this work, we
commute these operators so that they stand in normal order in the Hamiltonian.
2. Two-particle trial state and variational equations
We seek approximate two-particle states variationally by evaluating the expec-
tation value of the Hamiltonian operator of the QFT given in equation (8). The
simplest possible two-particle trial state is
| ψ2〉 =
∫
dNp1 d
Np2 F (p1,p2)A
†
1(p1)A
†
2(p2) | 0〉, (18)
635
J.Darewych
where |0〉 is the vacuum state annihilated by all the annihilation operators, Ak, Bk, d,
of the theory, and F (p1,p2) is an adjustable function to be determined variationally.
Note that the commutation properties of the operators, together with the definition
of |ψ2〉, imply that F (p1,p2) = F (p2,p1)
We shall consider the simplified case with λk = 0 in this paper. The relevant
matrix elements needed to implement the variational principle are
〈ψ2 |: Ĥφ1
+ Ĥφ2
+ Ĥχ :| ψ2〉 =
=
∫
dNp1 d
Np2 F
∗(p1,p2)F (p1,p2)
[
ω(p1, m1) + ω(p2, m2)
]
, (19)
〈ψ2 |: ĤI :| ψ2〉 = 0, (20)
and
〈ψ2 |: ĤII :| ψ2〉 =
= − g1g2
8(2π)N
∫
dNp dNq dNp′ dNq′ F ∗(p′,q′)F (p,q) δN(p+ q− p′ − q′)
× e−i(ω(p′,m1)−ω(p,m1)+ω(q′,m2)−ω(q,m2))t
1
√
ω(p′, m1)ω(p, m1)ω(q′, m2)ω(q, m2)
×
[
1
µ2 − (p′(1) − p(1))2
+
1
µ2 − (q′(2) − q(2))2
]
, (21)
where p(k) = (ω(p, mk),p) .
We have normal-ordered the entire Hamiltonian, since this circumvents the need
for mass renormalization which would otherwise arise. Not that there is a difficulty
with handling mass renormalization in the present formalism (as shown in various
earlier papers; see, for example, [6] and citations therein). It is simply that mass
renormalization has no effect on the two-body states that we obtain in this paper.
Furthermore, the approximate trial state (18), which we use in this work, is incapable
of sampling loop effects.
We now specialize to the rest frame of the two-particle system. The momentum
operator of this quantum field theory is given by
: P̂ : =
∫
dNq q
[
d†(q)d(q) +
2
∑
k=1
(
A†
k(q)Ak(q) +B†
k(q)Bk(q)
)
]
. (22)
The requirement that : P̂ : |ψ2〉 = 0 implies that
F (p1,p2) = f(p1)δ
N(p1 + p2) (23)
in the rest-frame of the two-particle system. Then, the matrix elements (19) and
(21) reduce to
〈ψ2 |: Ĥφ1
+ Ĥφ2
+ Ĥχ :| ψ2〉 = δN(0)
∫
dNpf ∗(p)f(p)
[
ω(p, m1) + ω(p, m2)
]
, (24)
636
Variational two-particle wave equation in scalar quantum field theory
and
〈ψ2 |: ĤII :| ψ2〉 =
= − g1g2
8(2π)N
δN(0)
∫
dNp dNp′ f ∗(p′)f(p)
× e−i(ω(p′,m1)−ω(p,m1)+ω(p′,m2)−ω(p,m2))t
1
√
ω(p′, m1)ω(p, m1)ω(p′, m2)ω(p, m2)
×
[
1
µ2 − (p′(1) − p(1))2
+
1
µ2 − (p′(2) − p(2))2
]
, (25)
where p2(k) = m2
k .
We evaluate the matrix elements at t = 0, and choose f(p) in accordance with
the variational principle
δ
〈ψ2| : Ĥ : |ψ2〉
〈ψ2|ψ2〉
= 0, (26)
whereupon we find that f(p) must be a solution of the momentum-space wave
equation
[
ω(p, m1) + ω(p, m2)− E
]
f(p) =
=
g1g2
8(2π)N
∫
dNp′
f(p′)
√
ω(p, m1)ω(p′, m1)ω(p, m2)ω(p′, m2)
×
[
1
µ2 + (p′ − p)2 − (ω(p, m1)− ω(p′, m1))2
+
1
µ2 + (p′ − p)2 − (ω(p, m2)− ω(p′, m2))2
]
, (27)
where the Lagrange multiplier E represents the total rest-frame energy of the two-
particle system, that is the total mass of a bound two-particle system. Note that the
kernel (momentum-space potential) in this equation contains terms corresponding
to one-chion exchange (this is perhaps more obvious from the manifestly covariant
terms in equation (25)).
3. Nonrelativistic and one particle limit
In the nonrelativistic limit, p2/m2 ≪ 1, equation (27) reduces to
[
p2
2mr
− ǫ
]
f(p) =
g1g2
4(2π)Nm1m2
∫
dNp′ f(p′)
1
µ2 + (p′ − p)2
, (28)
where mr = m1m2/(m1 +m2) and ǫ = E − 2m. In coordinate space, equation (28)
is the usual time-independent Schrödinger equation for the relative motion of the
two-particle system:
− 1
2mr
∇2ψ(r) + V (r)ψ(r) = ǫψ(r). (29)
637
J.Darewych
The potential V (r) is an attractive Yukawa potential (due to one-chion exchange).
In 3+1 dimensions it is, explicitly,
V (r) = −αe
−µr
r
, (30)
where α =
g1g2
16πm1m2
is the effective dimensionless coupling constant.
In the limit when one of the particles becomes very heavy, say m1 → ∞, equation
(27) becomes, in 3 + 1 dimensions,
[
ω(p, m2)− ε
]
f(p) =
α
2π2
∫
d3p′f(p′)
√
m2
ω(p, m2)
m2
ω(p′, m2)
[
1
µ2 + (p′ − p)2
− 1
2
(ω(p, m2)− ω(p′, m2))
2
[µ2 + (p′ − p)2] [µ2 + (p′ − p)2 − (ω(p, m2)− ω(p′, m2))2]
]
,(31)
where ε = E−m1. This is a Salpeter-like equation, with a Yukawa-like potential and
retardation terms in the kernel. In the non-relativistic limit, this equation reduces
to the usual one-particle momentum-space Schrödinger equation with a Yukawa
(µ > 0) or Coulombic (µ = 0) potential.
4. Concluding remarks
We have used the variational method to derive a relativistic two-particle wave
equation (27) from the underlying scalar quantum field theory (the scalar Yukawa, or
Wick-Cutkosky, model). The momentum-space potential describing the interaction
between the two scalar particles corresponds to one-chion exchange (the mediating,
or “chion”, field is also a scalar field). The equation has only positive-energy solu-
tions, that is, it is free of any negative-energy pathologies. It has the Schrödinger
equation for the relative motion of the two particles (with a Yukawa interparticle
potential) as its non-relativistic limit. It has a Salpeter-like equation, rather than a
Klein-Gordon equation, as its one-body limit. That is not surprising, since the Klein-
Gordon equation has negative-energy solutions, which do not (and should not) arise
in the present formalism, since we use the standard Dirac (“filled-negative-energy-
sea”) vacuum.
The two-particle equation (27) cannot be solved analytically, even for a massless
mediating field (µ = 0), for either bound or scattering states. Nevertheless, approx-
imate numerical or variational solutions can be readily obtained, as was done in
the case of particle-antiparticle equations [1,2]. However, approximate solutions of
equation (27) will be presented in a separate work.
The support of the Natural Sciences and Engineering Research Council of Canada
for this work is gratefully acknowledged.
638
Variational two-particle wave equation in scalar quantum field theory
References
1. Di Leo L., Darewych J.W. // Can. J. Phys., 1993, vol. 71, p. 365.
2. Ding B., Darewych J. // J. Phys. G, 2000, vol. 26, p. 907.
3. Wick G.C. // Phys. Rev., 1954, vol. 96, p. 1124.
4. Cutkosky R.E. // Phys. Rev., 1954, vol. 96, p. 1135.
5. Nakanishi N. // Prog. Theor. Phys. Suppl., 1988, No. 95.
6. Darewych J.W. // 1996, Ukr. J. Phys., vol. 41, p. 41.
Варіяційне двочастинкове хвильове рівняння у
скалярній квантовій теорії поля
Ю.Даревич
Відділ фізики і астрономії, Йоркський Унiверситет,
Канада, Торонто
Отримано 29 квітня 2000 р.
Ми вивчаємо двочастинкові системи в модельній квантовій теорії по-
ля, в якій скалярні частинки різної маси взаємодіють через посеред-
кове скалярне поле. Використовуючи коваріантні функції Ґріна для
розв’язку посередкового поля в термінах частинкових полів, пере-
формульовано ляґранжіян моделі. В результаті в гамільтоніяні ви-
никає пропагатор посередкового поля прямо в члені, який описує
взаємодію. Варіяційна метода з простим пробним фоковим станом
використовується для того, щоб вивести релятивістичне двочастин-
кове хвильове рівняння в імпульс-просторі. Одержуються і обгово-
рюються нерелятивістичні і одночастинкові границі цього рівняння.
Ключові слова: квантова теорія поля, двочастинкове хвильове
рівняння
PACS: 11.10.Qr, 11.10.St
639
640
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