An Axiomatic Approach to Semiclassical Field Perturbation Theory

0.26 MB·English

by  O. Yu. Shvedov

An Axiomatic Approach to Semiclassical Field Perturbation
Theory
O.Yu.Shvedov
5
0 Sub-Dept. of Quantum Statistics and Field Theory,
0 Dept. of Physics, Moscow State University,
2
119992, Moscow, Vorobievy Gory, Russia

n
a
J hep-th/0412302

9

2 Abstract
v
2
0
Semiclassical perturbation theory is investigated within the framework of axiomatic field
3
2 theory. Axiomsofperturbationsemiclassical theoryareformulated. Theircorrespondence
1 with LSZ approach and Schwinger source theory is studied. Semiclassical S-matrix, as
4
0 well as examples of decay processes, are considered in this framework.
/ Keywords: Maslovsemiclassical theory, axiomatic quantumfield theory, BogoliubovS-
h
t matrix, Lehmann-Symanzik-Zimmermann approach, Schwinger sources, Peierls brackets.
-
p
e
h
:
v
i
X
r
a
0e-mail: [email protected]
0This work was supported by the Russian Foundation for Basic Research, project 02-01-01062
0
1 Introduction
The main difficulty of quantum field theory (QFT) is that there is no nontrivial model
satisfying all the axioms. In fact, QFT is constructed within the perturbation theory
framework.
However, the perturbation theory is a partial case of the semiclassical theory. There-
fore, it is useful to generalize an axiomatic approach to perturbation QFT to the semi-
classical theory. This is the problem to be considered in this paper. The ideas of [1] are
developed here.
Generalstructureofsemicalssical perturbationtheoryinQFTisinvestigated insection
2. The main object of semiclassical theory is a semiclassical bundle [2]. Points on the
space of the bundle areinterpreted as possible semiclassical states. The base of the bundle
is the classical state space, fibres are spaces of quantum states in a given external classical
background.
In addition to the ”point-type” states, one can also consider their superpositions.
Important objects are introduced in QFT. These are Poincare transformation unitary
operators h and Heisenberg field operators. There analogs should arise in the semiclas-
Ug
sical theory as well. These semiclassical structures are also discussed in section 2.
Section 3 deals with the specific features of the covariant approach to the semiclassical
field theory. Its relationship with the axiomatic field theory [3], Schwinger source theory
[4], LSZ approach [5], S-matrix Bogoliubov theory [6, 7].
Section 4 is devoted to the leading order of the semiclassical theory. All axioms of the
semiclassical field theory are checked.
The semiclassical perturbation theory is discussed in section 5. The calculations can
be simplified, provided that the asymptotic condition of the S-matrix approach [3, 6, 7, 8]
is satisfied.
It is well-known that there are difficulties of the S-matrix approach due to unstable
particles and bound states [3]. In section 6 we show how one can develop the semiclassical
perturbation theory with unstable particles. An example of particle decay is considered.
Section 7 contains the concluding remarks.
2 General structure of semiclassical perturbation
theory in QFT
2.1. We consider the quantum field system with the Lagrangian that depends on the
1
small parameter h as follows:
1 1
= ∂ ϕ∂µϕ V(√hϕ). (2.1)
L 2 µ − h
Different methods to develop the semiclassical approximation theory for this model are
known. One can use both Hamiltonian and manifestly covariant approaches. One consid-
ers ”semiclassical” states that depend on h as h 0 as follows (see [9], [10] and references
→
therein). For the Hamiltonian approach,
Ψ ehiSe√ih dx[Π(x)ϕˆ(x)−Φ(x)πˆ(x)]f Kh f, (2.2)
≃ ≡ S,Π,Φ
R
Here ϕˆ and πˆ are field and momenta operators, f is a state vector that expands into a
series in √h. For the manifestly covariant approach,
Ψ ehiSTe√ih dxJ(x)ϕˆ(x)f ehiSThf Kh f. (2.3)
≃ ≡ J ≡ S,J
R
Here ϕˆ(x) is a Heisenberg field, J(x) is a classical source with a compact source, f is
expanded in √h. A semiclassical state of the form (2.2) (or (2.3)) can be viewed as
a point on the space of the semiclassical bundle [2]. Base of the bundle is set X =
{
(S,Π(x),Φ(x)) (or X = (S,J(x)) ) of classical states, fibers are spaces f ( f ) of
} { } { } { }
quantum states in the external field. States (2.2) and (2.3) are written as Khf; the
X
operator Kh is called as a canonical operator.
X
The Maslov theory of Lagrangian manifolds with complex germs [11, 12] is a general-
ization of the Maslov complex germ theory. One considers states of the more complicated
form:
dαKh f(α), α = (α ,...,α ), (2.4)
X(α) 1 k
Z
They can be interpreted as k-dimensional surface on the semiclassical bundle.
Within the semiclassical theory, one can formulate the following problems:
- let h be a Poincare transformation corresponding to an element g of the Poincare
Ug
group G, ϕˆ(x) be a Heisenberg field; one should investigate these operators as h 0;
→
- one should find norm of the state (2.4) as h 0;
→
- one should investigate whether the expressions (2.3) corresponding to different
sources J may approximately coincide as h 0.
→
2.2. It happens that the following commutation rules are satisfied:
hKhf = Kh U (u X X)f;
Ug X ugX g g ← (2.5)
√hϕˆ(x)Khf = KhΦ(x X)f,
X X |
2
Here U (u X X) is an unitary operator which is expanded into an asymptotic series
g g ←
in √h, u X is a Poincare transformation of classical state, Φ(x X) is an operator-valued
g
|
distribution. It is also presented as an asymptotic series in √h. Its leading order is a
c-number quantity Φ(x X):
|
Φ(x X) = Φ(x X)+√hΦ(1)(x X)+... (2.6)
| | |
Investigate the properties of the introduced objects. Since the operators h should
Ug
obey the group property
h = ,
Ug1g2 Ug1Ug2
it follows from eq.(2.5) that
u = u u ;
g1g2 g1 g2 (2.7)
U (u X X) = U (u X u X)U (u X X).
g1g2 g1g2 ← g1 g1g2 ← g2 g2 g2 ←
Moreover, the Poincare covariance of the fields implies that
h ϕˆ(x) h = ϕˆ(w x), w x = Λ 1(x a) (2.8)
Ug−1 Ug g g − −
Therefore, one finds from eq.(2.5) that
Φ(x u X)U (u X X) = U (u X X)Φ(w x X). (2.9)
| g g g ← g g ← g |
2.3. Let us estimate the square of the norm of state (2.4) as h 0. The plan is as
→
follows (cf. [13]). The integral
dαdα(Kh f(α),Kh f(α)) (2.10)
′ X(α) X(α) ′
′
Z
is calculated with the help of the substitution α = α + √hβ. Then one performs an
′
expansion in √h. To do this, it is necessary to use the formula of expansion of the vector
Kh f(α+β√h) in √h.
X(α+β√h)
One can obtain this formula from the commutation rule:
∂ ∂X
ih Kh = Kh ω [ ]. (2.11)
∂α X(α) X(α) X(α) ∂α
a a
Here ω [δX] is an operator-valued 1-form. It assins an operator in f to each tangent
X
{ }
vector δX to the base. In the leading order, the 1-form is a c-number:
ω [δX] = ω [δX]+√hω(1)[δX]+...
X X X
3
Property [ih ∂ ,ih ∂ ] = 0 implies the commutation relation for 1-forms:
∂αa ∂αb
∂X ∂X ∂ ∂X ∂ ∂X
ω [ ];ω [ ] = ih ω [ ] ω [ ] , (2.12)
" X(α) ∂αa X(α) ∂αb # − "∂αa X(α) ∂αb − ∂αb X(α) ∂αa #
or
[ω [δX ],ω [δX ]] = ihdω (δX ,δX ).
X 1 X 2 − X 1 2
The operator Kh can be expanded then in √h as follows. Set
X(α+√hβ)
Kh = Kh V (α,β). (2.13)
X(α+√hβ) X(α) h
Differentiate (2.13) with respect to β . Making use of eq.(2.11), we obtain
a
∂ i ∂X
V (α,β) = V (α,β)ω [ (α+β√h)]. (2.14)
∂β h −√h h X(α+β√h) ∂α
a a
Therefore, the leading order in h gives us a relation
Vh(α,β) e−√ihωX(α)[∂∂αXa]βa.
∼
The inner prooduct (2.10) is taken to the form
hk/2 dαdβ(f(α),V (α,β)f(α+β√h)); (2.15)
h
Z
The integrand is a rapidly oscillating quantity. Therefore, the integral will be exponen-
tially small, except for the special case when the Maslov isotropic condition is satisfied:
∂X
ω [ ] = 0. (2.16)
X(α) ∂α
a
For the case (2.16), the system (2.14) can be solved within the perturbation framework,
iff the self-consistent condition (2.12) is satisfied. For the leading order, one has
Vh(α,β) e−iωX(1()α)[∂∂αXa]βa,
≃
The inner product (2.15) takes the form
∂X
hk/2 dαdβ(f(α), 2πδ(ω(1)[ ]) f(α)).
{ X ∂α }
Z a a
Y
4
Notice that eqs.(2.5) and (2.11) imply the relations
U (u X X)ω [∂X ] = ω [∂(ugX)]U (u X X)+ih ∂ U (u X X);
g g ← X ∂αa ugX ∂αa g g ← ∂αa g g ← (2.17)
ih ∂ Φ(x X) = Φ(x X);ω [∂X ] .
∂αa | | X ∂αa
h i
2.4. Inthecovariant framework, someofstates (2.3)approximately coincideash 0.
→
Thismeansthatonesouldintroduceanequivalencerelationofthebaseofthesemiclassical
bundle (on the classical state space). Some of classical states are equivalent. Moreover,
if X X , then
1 2
∼
Kh f Kh f (2.18)
X1 1 ≃ X2 2
iff
f = V(X X )f .
2 2 ← 1 1
Investigate the properties of the operator V(X X ). First of all, the following
2 1
←
relation
V(X X ) = V(X X )V(X X ) (2.19)
3 1 3 2 2 1
← ← ←
should be satisfied. Moreover, eq. (2.18) implies that hKh f hKh f ; therefore
Ug X1 1 ≃ Ug X2 2
V(u X u X )U (u X X ) = U (u X X )V(X X ). (2.20)
g 2 ← g 1 g g 1 ← 1 g g 2 ← 2 2 ← 1
It follows from the relation √hϕˆ(x)Kh f √hϕˆ(x)Kh f that
X1 1 ≃ X2 2
Φ(x X )V(X X ) = V(X X )Φ(x X ). (2.21)
2 2 1 2 1 1
| ← ← |
Finally, let (X ,f ) depend on α. Differentiate (2.18) with respect to α : ih ∂ Kh f
i i a ∂αa X1 1 ≃
ih ∂ Kh f ; therefore,
∂αa X2 2
∂X ∂X ∂
V(X X )ω [ 1] = ω [ 2]V(X X )+ih V(X X ) (2.22)
2 ← 1 X1 ∂α X2 ∂α 2 ← 1 ∂α 2 ← 1
a a a
provided that X (α) X (α).
1 2
∼
2.5. Thus, all the problems of semiclassical theory can be solved within the pertur-
bation framework iff one specifies:
- the Poincare transformations u (classical) U (u X X) (unitary operator ex-
g g g ←
panded into a formal series in √h);
- semiclassical series in √h for Φ(x X) ω [δX] (these opeators are c-numbers in the
| X
leading order);
5
- semiclassical series in √h for the operators V(X X ) as X X (if the equiva-
2 1 1 2
← ∼
lence relation is introduced on the classical state space);
These objects should satisfy the properties (2.7), (2.9), (2.12), (2.17), (2.19), (2.20),
(2.21), (2.22).
Therefore, let us say that a model of semiclassical field theory is given iff the objects
u , U (u X X), Φ(x X), ω [δX], V(X X ) are specified and they obey the
g g g ← | X 2 ← 1
required properties. We suppose the semiclassical model to be well-defined, even if the
corresponding exact QFT model is ill-defined.
3 Specific features of the covariant approach to semi-
classical perturbation theory
3.1. Let us investigate the objects arising in the covariant approach to semiclassical field
theory. First, notice that eq. (2.8) implies that
hThf = Th hf,
Ug J ugJUg
with u J(x) = J(w x), w being of the form (2.8). Therefore, an explicit form of
g g g
transformation u is known, the property u = u u is satisifed, while the opera-
g g1g2 g1 g2
tor U (u X X) U = h is X-independent and satisfies the group property and
g g ← ≡ g Ug
property of invariance of the fields.
U = U U , U ϕˆ(x)U = ϕˆ(w x). (3.1)
g1g2 g1 g2 g−1 g g
The 1-form ω can be expressed via the LSZ R-functions [5]:
δTh
R(x J) ih(Th)+ J . (3.2)
| ≡ − J δJ(x)
Namely,
ihδKh = Kh [ δS dxR(x J)δJ(x)],
S,J S,J − − |
Z
therefore,
ω [δX] = δS dxR(x J)δJ(x), (3.3)
X − − |
Z
Moreover, R(x J) should be expanded into a formal series in √h
|
R(x J) = Φ(x J)+√hR(1)(x J)+...
| | |
6
The c-number function Φ(x J) is called as a classical field generated by the source J.
|
Eq.(3.2) implies the following properties of the Hermitian R-function:
- Poincare invariance
U R(x u J)U = R(w x J); (3.4)
g−1 | g g g |
- Bogoliubov causality property:
δR(x J) >
| = 0, y x. (3.5)
δJ(y)
∼
- commutation relation
δR(x J) δR(y J)
[R(x J);R(y J)] = ih | | ; (3.6)
| | − δJ(y) − δJ(x) !
- boundary condition
<
R(x J) = ϕˆ(x)√h, x suppJ. (3.7)
|
∼
The operator Φ(x J) can be expressed via the R-function at x>suppJ:
|
∼
>
Φ(x X) = R(x J), x suppJ. (3.8)
| |
∼
3.2. Investigate the equivalence property. Say that J 0 iff
∼
TJhf ≃ ehiIJWJf (3.9)
for some c-number phase I and operator W presented as a formal perturbation series
J J
in √h. The following properties are satisfied:
- relativistic invariance:
U W U = W , I = I ; (3.10)
g J g−1 ugJ ugJ J
- unitarity
W+ = W 1; (3.11)
J −J
- Bogoliubov causality: if J +∆J 0, J +∆J +∆J 0 and supp∆J >supp∆J
2 1 2 2 1
∼ ∼
then the operator (W )+W and c-number I + I ∼ do not
J+∆J2 J+∆J1+∆J2 − J+∆J2 J+∆J1+∆J2
depend on ∆J ;
2
7
- variational property:
δI ihW+δW = dxR(x J)δJ(x); (3.12)
J − J J |
Z
- boundary condition
>
R(x J) = W+ϕˆ(x)√hW , x suppJ. (3.13)
| J J
∼
It follows from eq.(3.13) that Φ(x J) = 0 as x>suppJ. Therefore, the classical field
|
generated by the source J 0 has a compact supp∼ort. The following requirement allows
∼
us to construct the covariant semiclassical field theory without additional postulating
equations of motion and commutation relations. Namely, suppose that for any field
configuration Φ(x) with compact support there exists a source J 0 (denoted as J =
∼
J = J(x Φ)) that generates the configuration Φ: Φ(x) = Φ(x J). It satisfies the locality
Φ
| |
property: δJ(x|Φ) = 0 for x = y.
δΦ(y) 6
Eq.(3.12) implies in the leading order in h that the functional
I[Φ] = I dxJ (x)Φ(x) (3.14)
JΦ − Φ
Z
obeys the relation
δI[Φ]
J(x) = . (3.15)
−δΦ(x)
It is a classical equation of motion. The functional I[Φ] satisfying the locality property
δ2I
= 0, x = y (3.16)
δΦ(x)δΦ(y) 6
will be called as a classical action of the theory.
Denote W[Φ] W . The obtained relations can be formulated as follows:
≡ JΦ
- relativistic invariance
U W[Φ]U = W[u Φ]; (3.17)
g g 1 g
−
- unitarity
W+[Φ] = (W[Φ]) 1; (3.18)
−
- Bogoliubov causality
δ δW[Φ] >
W+[Φ] = 0, y x; (3.19)
δΦ(y) δΦ(x) !
∼
8
- the Yang-Feldman relation:
δ2I δW[Φ]
dy [R(y J) Φ(y J)] = ihW+[Φ] ; (3.20)
δΦ(x)δΦ(y) | − | δΦ(x)
Z
- the boundary condition
>
W+[Φ]ϕˆ(x)√hW[Φ] = R(x J), x suppΦ. (3.21)
|
∼
Differential form of the Bogoliubov causality property (3.19) is obtained by a standard
procedure. The Yang-Feldman relation (3.20) is derived from the variational property
(3.12) with the help of the substitution
δ2I
δJ(x) = dy δΦ(y).
− δΦ(x)δΦ(y)
Z
3.3. It happens that all objects of the semiclassical field theory considered in the
previous subsection can be reconstructed if one specifies:
- action I[Φ] satisfying the locality and Poincare invariance property;
- operators ϕˆ(x) and U expanded in √h and satisfying the properties (3.1);
g
-HermitianoperatorsR(x J)expandedin√handsatisfyingtheproperties(3.4),(3.5),
|
(3.6), (3.7); in the leading order, the operators R(x J) should be equal to the solution
|
Φ(x J) of eq.(3.15) under condition Φ = 0;
| |x<suppJ
- operator W[Φ] expanded in √h an∼d satisfying the relations (3.17), (3.18), (3.19),
(3.20), (3.21).
These properties are not independent. In section 5, we will show that they are related
with each other. Let us show now how one can reconstruct all the structures of the
semiclassicalfieldtheoryandcheckthepropertiesofsection2. Letusconsiderthesimplest
example - the theory with classical action
1
I[Φ] = dx[ ∂ Φ∂µΦ V(Φ)], (3.22)
µ
2 −
Z
where V(Φ) m2Φ2, Φ 0.
∼ 2 →
Notice that the classical Poincare transformation is already constructed. Namely,
u J(x) = J(w x); U is X-independent. The 1-form ω is of the form (3.3).
g g g
Say that J J iff for some source J with the support suppJ > suppJ, suppJ >
′ + + +
∼
suppJ the properties J +J 0, J +J 0 are satisfied. This definition is equivalent
′ + ′ +
∼ ∼
to the following:
>
Φ(x J) = Φ(x J ), x suppJ,J J J (3.23)
′ ′ ′
| | ⇔ ∼
∼
9