Spec. Matrices 2015; 3:169–174
Communication
Open Access
Igor V. Beloussov
Another formulation of the Wick’s theorem.
Farewell, pairing?
DOI 10.1515/spma-2015-0015
Received February 19, 2015; accepted July 2, 2015
Abstract: The algebraic formulation of Wick’s theorem that allows one to present the vacuum or thermal
averages of the chronological product of an arbitrary number of field operators as a determinant (permanent)
of the matrix is proposed. Each element of the matrix is the average of the chronological product of only
two operators. This formulation is extremely convenient for practical calculations in quantum field theory,
statistical physics, and quantum chemistry by the standard packages of the well known computer algebra
systems.
Keywords: vacuum expectation value; chronological product; contractions
AMS: 15-04, 15A15, 65Z05, 81U20
1 Introduction
Wick’s theorems are used extensively in quantum field theory [1–4], statistical physics [5–7], and quantum
chemistry [8]. They allow one to use the Green’s functions method, and consequently to apply the Feynman’s
diagrams for investigations [1–3]. The first of these, which can be called Wick’s Theorem for Ordinary Products,
gives us the opportunity to reduce in almost automatic mode the usual product of operators into a unique sum
of normal products multiplied by c–numbers. It can be formulated as follows [4]. Let A i (x i ) (i = 1, 2, . . . , n )
are “linear operators”, i.e., some linear combinations of creation and annihilation operators. Then the ordinary product of linear operators is equal to the sum of all the corresponding normal products with all possible
contractions, including the normal product without contractions, i.e.,
A1 . . . A n
=
: A1 . . . A n : + : A1 A2 . . . A n : + . . . + : A1 . . . A n−1 A n :
+ : A1 . . . A n : + : A1 A2 A3 A4 . . . A n : + . . . ,
where A i A j = A i A j − : A i A j : (i, j = 1, 2, . . . , n) is the contraction between the factors A i and A j . Since the
vacuum expectation value of the normal ordered product is equal zero, this theorem provides us a way of
expressing the vacuum expectation values of n linear operators in terms of vacuum expectation values of
two operators.
Wick’s Theorem for Chronological Products [4] asserts that the T–product of a system of n linear operators
is equal to the sum of their normal products with all possible chronological contractions, including the term
without contractions. It follows directly from the previous theorem and gives the opportunity to calculate the
vacuum expectation values of the chronological products of linear operators.
Finally, from Wick’s theorem for chronological products the Generalized Wick’s Theorem [4] can be obtained. It asserts that the vacuum expectation value of the chronological product of n + 1 linear operators
A, B1 , . . . , B n can be decomposed into the sum of n vacuum expectation values of the same chronological prod-
Igor V. Beloussov: Institute of Applied Physics, Academy of Sciences of Moldova, 5 Academy Str., Kishinev, 2028, Republic of
Moldova, e-mail: [email protected]
© 2015 Igor V. Beloussov, licensee De Gruyter Open.
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License.
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170 | Igor V. Beloussov
ucts with all possible contractions of one of these operators (for example A ) with all others, i.e.,
)︁⟩
∑︁ ⟨ (︁
⟨T (AB1 . . . B n )⟩0 =
T AB1 . . . B i . . . B n
.
0
1≤i≤n
(1)
(︀
)︀
⟨︀ (︀
)︀⟩︀
Here A i A i = T A i A j − : A i A j := T A i A j 0 (i, j = 1, 2, . . . , n ) i is the chronological contraction between
the factors A i and A j . It should be noted that, in contrast to the usual Wick’s theorem for chronological products, there are no expressions involving the number of contractions greater than one on the right-hand side
of (1).
Wick’s theorem for chronological products or its generalized version are used for the calculation of matrix elements of the scattering matrix in each order of perturbation theory [1–4]. The procedure is reduced
to calculation of the vacuum expectation of chronological products of the field operators in the interaction
representation. As factors in these products a number of operators ψ i of the Fermi fields and the same num(−)
ber of their “conjugate” operators ψ̄ i , as well as operators of the Bose fields φ s = φ(+)
s + φ s may be used.
Here all continuous and discrete variables are included in the index. In the interaction representation the
operators ψ i ,ψ̄ i , and φ(±)
correspond to free
the commutation
relationships
of the form
[︁ fields and
]︁ satisfy
[︁
]︁
[︁
]︁
[︀
[︀
]︀ s [︀
]︀
]︀
(−)
(+)
(+)
(+)
(−)
ψ i , ψ j + = ψ̄ i , ψ̄ j + = 0, ψ i , ψ̄ j + = D ij , φ(−)
,
φ
=
φ
,
φ
=
0,
φ
,
φ
=
D̄
.
rs Therefore,
r
s
r
s
r
s
−
−
the averaging of the Fermi and Bose fields can be performed independently.
−
2 Basic theorem
Since we may rearrange the order of the operators inside T–products taking into account the change of the
sign, which arises when the order of the Fermi operators is changed, we present our vacuum expectation
value of the chronological product of the Fermi operators in the form
⟨︀ [︀(︀
)︀ (︀
)︀
(︀
)︀]︀⟩︀
± T ψ i1 ψ̄ j1 ψ i2 ψ̄ j2 . . . ψ i n ψ̄ j n 0 .
(2)
To calculate (2), we can use Wick’s theorem for chronological products. However, while considering the
higher-order perturbation theory, the number of pairs ψ i ψ̄ j of operators ψ i and ψ̄ j becomes so large that
the direct application of this theorem begins to represent certain problems because it is very difficult to sort
through all the possible contractions between ψ i and ψ̄ j .
A consistent use of generalized Wick’s theorem would introduce a greater accuracy in our actions. However, in this case we expect very cumbersome and tedious calculations. Hereinafter we show that the computation of (2) can be easily performed using a simple formula
(︁
)︁
⟨︀ [︀(︀
)︀
(︀
)︀]︀⟩︀
T ψ i1 ψ̄ j1 . . . ψ i n ψ̄ j n 0 = det ∆ i α j β ,
(3)
where
⟨ (︁
)︁⟩
∆ i α j β = ψ i α ψ̄ j β = T ψ i α ψ̄ j β
0
(α, β = 1, 2, . . . , n) .
(4)
The proof of this theorem is by induction. Let us assume now that (3) is true for n pairs ψ i α ψ̄ j β , and consider
it for the case n + 1. Using generalized Wick’s theorem we have
⟨︀ [︀(︀
)︀ (︀
)︀
(︀
)︀]︀⟩︀
⟨︀ [︀(︀
)︀ (︀
)︀
(︀
)︀]︀⟩︀
T ψ i1 ψ̄ j1 ψ i2 ψ̄ j2 . . . ψ i n*1 ψ̄ j n*1 0 = −∆ i1 j n+1 T ψ i n*1 ψ̄ j1 ψ i2 ψ̄ j2 . . . ψ i n ψ̄ j n 0
−
n−1
∑︁
γ=2
⟨︀ [︀(︀
)︀
(︀
)︀ (︀
)︀ (︀
)︀
(︀
)︀]︀⟩︀
∆ iγ j n+1 T ψ i1 ψ̄ j1 . . . ψ iγ−1 ψ̄ jγ−1 ψ i n+1 ψ̄ jγ ψ iγ+1 ψ̄ jγ+1 . . . ψ i n ψ̄ j n 0
⟨︀ [︀(︀
)︀
(︀
)︀]︀⟩︀
⟨︀ [︀(︀
)︀
(︀
)︀ (︀
)︀]︀⟩︀
− ∆ i n j n+1 T ψ i1 ψ̄ j1 . . . ψ i n−1 ψ̄ j n−1 ψ i n+1 ψ̄ j n 0 + ∆ i n+1 j n+1 T ψ i1 ψ̄ j1 . . . ψ i n ψ̄ j n 0 .
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Another formulation of the Wick’s theorem |
171
Taking into account (3) we obtain
⃒
⃒ ∆
⃒ i n+1 j1 ∆ i n+1 j2 · · ·
⃒
∆ i2 j2
···
⃒ ∆ i2 j1
⟨︀ [︀(︀
)︀ (︀
)︀
(︀
)︀]︀⟩︀
T ψ i1 ψ̄ j1 ψ i2 ψ̄ j2 . . . ψ i n*1 ψ̄ j n*1 0 = −∆ i1 j n+1 ⃒⃒ .
..
..
.
.
⃒ .
.
⃒
⃒ ∆i j
∆ i n j2
···
n 1
⃒
⃒
⃒
⃒ ∆
⃒
⃒
∆ i1 j2
· · · ∆ i1 j n ⃒
∆ i1 j2
⃒ i1 j1
⃒ ∆ i1 j1
⃒
⃒
⃒ .
..
⃒ ∆ i n+1 j1 ∆ i n+1 j2 · · · ∆ i n+1 j n ⃒
⃒ ..
.
⃒ − . . . − ∆i j ⃒
−∆ i2 j n+1 ⃒⃒ .
.
.
n n+1 ⃒
.
⃒
.
.
.
.
.
⃒ .
⃒
⃒ ∆ i n−1 j1 ∆ i n−1 j2
.
.
⃒
⃒
⃒
⃒ ∆i j
⃒
⃒ ∆i j
∆
·
·
·
∆
∆ i n+1 j2
i
j
i
j
n 1
n 2
n n
n+1 1
⃒
⃒
⃒ ∆
⃒
⃒ i1 j1 ∆ i1 j2 · · · ∆ i1 j n ⃒
⃒
⃒
⃒ ∆ i2 j1 ∆ i2 j2 · · · ∆ i2 j n ⃒
⃒
⃒ .
+∆ i n+1 j n+1 ⃒ .
..
.
..
⃒
. ..
⃒ ..
⃒
.
⃒
⃒
⃒ ∆i j
∆i j · · · ∆i j ⃒
n 1
n 2
∆ i n+1 j n
∆ i2 j n
..
.
∆ in jn
···
..
.
···
···
⃒
⃒
⃒
⃒
⃒
⃒
⃒
⃒
⃒
⃒
∆ i1 j n
...
∆ i n−1 j n
∆ i n+1 j n
⃒
⃒
⃒
⃒
⃒
⃒
⃒
⃒
⃒
⃒
(5)
n n
Rearranging the rows in the determinants in (5) it is easy to see that the right hand side is the expansion of
the
⃒
⃒
⃒ ∆
∆ i1 j2
· · · ∆ i1 j n+1 ⃒⃒
⃒ i1 j1
⃒
(︁
)︁ ⃒⃒ ∆ i2 j1
∆ i2 j2
· · · ∆ i2 j n+1 ⃒
⃒ (α, β = 1, 2, . . . , n + 1)
det ∆ i α j β = ⃒⃒ .
.
.
..
⃒
..
. ..
⃒ ..
⃒
⃒
⃒
⃒ ∆i j
∆ i n+1 j2 · · · ∆ i n+1 j n+1 ⃒
n+1 1
along the last column [9]. The validity of (3) for n = 1 follows from the definition ∆ i α j β in (4).
Note that this result does not depend on the way how we divide the operators on the left hand side of
(3) into pairs ψ i α ψ̄ j β . Indeed, if on the left hand side of (3) we permute, for example, ψ̄ j β and ψ̄ jγ (β ≠ γ), it
changes its sign. The same happens on the right hand side of (3) since this change leads to the permutation
of two columns in the determinant, and it also changes its sign. Similarly, in the case of a permutation of ψ i α
and ψ i δ (α ≠ δ). Obviously, when the whole pair ψ i α ψ̄ j β is transposed, the left and right hand sides of (3) do
not change.
The above theorem has an important consequence. In fact, it establishes a perfect coincidence between
the vacuum expectation values of the chronological products of n pairs of field operators and the n–order
determinant. If to present this determinant as the sum of the elements and cofactors of one any row or column,
and thereafter to use again the indicated coincidence for the (n − 1)–order determinants included in each
summand, we will return to the generalized Wick’s theorem. Alternatively, we can select in the our n–order
determinant arbitrary m rows or columns (1 < m < n) and use the Generalized Laplace’s Expansion [9] for
its presentation as the sum of the products of all m–rowed minors using these rows (or columns) and their
algebraic complements. Then, taking into account our theorem, we obtain a representation of the vacuum
expectation values of the chronological products of n pairs of field operators as the sum of the products of
vacuum expectation values of the chronological products of m pairs of operators and vacuum expectation
values of the chronological products of n −m pairs. The number of terms in this sum is equal to n!/m! (n − m)!.
This decomposition can be useful for the summation of blocks of diagrams.
Obviously, the formula similar to (3) can be obtained and in the case of Bose fields
)︁
(︁
)︁]︁⟩
(︁
)︁
)︁ (︁
⟨ [︁(︁
(−)
(−)
(−)
= perm ∆ i α j β ,
(6)
φ(+)
. . . φ(+)
T φ(+)
in φ jn
i2 φ j2
i1 φ j1
0
where
⟨ (︁
)︁⟩
¯ i α j = φ(+) φ(−) = T φ(+) φ(−)
∆
β
iα
jβ
iα
jβ
0
(α, β = 1, 2, . . . , n) .
(7)
In quantum statistics the n–body thermal, or imaginary-time, Green’s functions in the Grand Canonical
Ensemble are defined as the thermal trace of a time-ordered product of the fields operators in the imaginarytime Heisenberg representation [5–7]. To calculate them in each order of perturbation theory, Wick’s theorem
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172 | Igor V. Beloussov
is also used. Obviously, in this case, the theorem also may be formulated in the form (3) and (6) convenient
for practical calculation.
3 Simple examples
In order to demonstrate the usability of the proposed formulation of Wick’s theorem, we find the first-order
correction to the one- and two-particle thermal Green’s functions for the Fermi system described in the interaction representation by the Hamiltonian
∫︁
1
H int (τ) =
dr1 dr2 ψ̄ α (r1 , τ)ψ̄ β (r2 , τ) U (r1 − r2 ) ψ β (r2 , τ) ψ α (r1 , τ)
2
that contains the product of the field operators ψ α (r1 , τ) and ψ̄ α (r1 , τ) in this representation (parameter α
indicates the spin projections, τ is the imaginary-time). The one-particle Green’s function can be represented
as [5, 6]
⟨︀ [︀
]︀⟩︀
T τ ψ (x1 ) ψ̄ (x2 ) S 0
GI ( x 1 , x 2 ) = −
,
⟨S⟩0
where ⟨. . .⟩0 is the symbol for the Gibbs average over the states of a system of noninteracting particles, x ≡
(r, τ, α) and
⎧ τ
⎫
⎨ ∫︁
(︁ )︁ ⎬
′
′
S (τ) = T τ exp − dτ H int τ
.
⎩
⎭
0
We obtain
∫︁
1
GI (x1 , x2 ) = − T τ ψ (x1 ) ψ̄ (x2 ) 0 +
dz1 dz2 V (z1 − z2 )
2⟨S⟩0
⟨︀ [︀(︀
)︀ (︀
)︀ (︀
)︀]︀⟩︀
× T τ ψ (x1 ) ψ̄ (x2 ) ψ (z1 ) ψ̄ (z1 ) ψ (z2 ) ψ̄ (z2 ) 0
∫︁
1
dz1 dz2 V (z1 − z2 )
= − ∆ (x1 , x2 ) +
2⟨S⟩0
⃒
⃒
⃒ ∆ ( x , x ) ∆ ( x , z ) ∆ ( x , z )⃒
1 2
1 1
1 2 ⃒
⃒
⃒
⃒
× ⃒ ∆ (z1 , x2 ) ∆ (z1 , z1 ) ∆ (z1 , z2 )⃒ ,
⃒
⃒
⃒ ∆ (z2 , x2 ) ∆ (z2 , z1 ) ∆ (z2 , z2 )⃒
⟨︀
[︀
]︀⟩︀
(8)
where V (x1 − x2 ) = U (r1 − r2 ) δ (τ1 − τ2 ). We can immediately take into account the reduction of the disconnected diagrams, if we assume in (8) ⟨S⟩0 = 1 and ∆ (x1 , x2 ) = 0 [6]. Then,
∫︁
1
GI (x1 , x2 ) = −∆ (x1 , x2 ) +
dz1 dz2 V (z1 − z2 )
2⟨S⟩0
⃒]︃
⃒
⃒
⃒
[︃
⃒∆ z , x
⃒
⃒∆ z , x
⃒
⃒ ( 1 2 ) ∆ (z1 , z2 )⃒
⃒ ( 1 2 ) ∆ ( z 1 , z 1 )⃒
× ∆ (x1 , z2 ) ⃒
⃒ − ∆ (x1 , z1 ) ⃒
⃒
⃒ ∆ ( z 2 , x 2 ) ∆ ( z 2 , z 1 )⃒
⃒∆ (z2 , x2 ) ∆ (z2 , z2 )⃒
∫︁
= −∆ (x1 , x2 ) + dz1 dz2 ∆ (x1 , z1 ) V (z1 − z2 ) ∆ (z1 , z2 ) ∆ (z2 , x2 )
∫︁
− dz1 dz2 ∆ (x1 , z1 ) V (z1 − z2 ) ∆ (z2 , z2 ) ∆ (z1 , x2 ) .
Taking into account G(0) (x1 , x2 ) = −∆ (x1 , x2 ) we obtain finally
∫︁
GI (x1 , x2 ) = G(0) (x1 , x2 ) + dz1 dz2 G(0) (x1 , z1 ) Σ (1) (z1 , z2 )2 G(0) (z2 , x2 ) ,
Σ (1) (z1 , z2 ) = −V (z1 − z2 ) G(0) (z1 , z2 ) + δ (z1 − z2 )
∫︁
dzV (z1 − z) G(0) (z, z) .
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Similarly, for the two-particle Green’s function
⟨︀ [︀
]︀⟩︀
T τ ψ (x1 ) ψ (x2 ) ψ̄ (x3 ) ψ̄ (x4 ) S 0
GII (x1 , x2 , x3 , x4 ) = −
,
⟨S⟩0
we have
⟨︀ [︀(︀
)︀ (︀
)︀]︀⟩︀
GII (x1 , x2 , x3 , x4 ) = T ψ (x1 ) ψ̄ (x3 ) ψ (x2 ) ψ̄ (x4 ) 0 −
⃒
⃒ ∆ x ,x
⃒ ( 1 3)
=⃒
⃒ ∆ (x2 , x3 )
−
∫︁
1
2⟨S⟩0
∫︁
dz1 dz2 V (z1 − z2 )
⟨︀ [︀(︀
)︀ (︀
)︀ (︀
)︀ (︀
)︀]︀⟩︀
× T ψ (x1 ) ψ̄ (x3 ) ψ (x2 ) ψ̄ (x4 ) ψ (z1 ) ψ̄ (z1 ) ψ (z2 ) ψ̄ (z2 ) 0
⃒
⃒ 0
0
∆ (x1 , z1 )
⃒
⃒
∫︁
⃒ 0
⃒
0
∆ (x2 , z1 )
∆ (x1 , x4 ) ⃒ 1
⃒
dz1 dz2 V (z1 − z2 ) ⃒
⃒−
⃒ ∆ (z1 , x3 ) ∆ (z1 , x4 ) ∆ (z1 , z1 )
∆ (x2 , x4 ) ⃒ 2
⃒
⃒ ∆ (z2 , z3 ) ∆ (z2 , x4 ) ∆ (z2 , z1 )
∆ (x1 , z2 )
∆ (x2 , z2 )
∆ (z1 , z2 )
∆ (z2 , z2 )
⃒
⃒
⃒
⃒
⃒
⃒
⃒
⃒
⃒
= G(0) (x1 , x3 ) G(0) (x2 , x4 ) − G(0) (x1 , x4 ) G(0) (x2 , x3 )
[︁
]︁
dz1 dz2 G(0) (x1 , z1 ) G(0) (x2 , z2 ) V (z1 − z2 ) G(0) (z1 , x3 ) G(0) (z2 , x4 ) − G(0) (z1 , x4 ) G(0) (z2 , x3 ) .
4 Conclusions
Representations (3) and (6) not only greatly simplify all calculations, but also allow one to perform them
using a computer with programs of symbolic mathematics. Development of the methods of symbolic algebra for computing in higher-orders of many-body perturbation theory has been realized for several decades.
The SCHOONSCHIP program [10, 11] and other symbolic packages (FEYNCALC [12], HIP [13], REDUCE [14],
FORM [15], XLOOPS [16], SNEG [17], etc.) are employed for evaluating Feynman diagrams in quantum electrodynamics and high-energy physics. Similar investigations are under way in quantum chemistry [18] (see
also [19] and references therein).
The starting point of symbolic computations is the Wick’s theorem (see, e.g., discussion in [20]). However, there is an alternative approach based on the symbolic replacement rules that are suitable to be used
in symbolic algebra [21, 22]. Some of these programs are designed for a narrow range of specific tasks; however, they provide algorithms appropriate for their possible use in other areas. Moreover, they require a certain specific software with mastering add-on packages. Calculations using representations (3) and (6) and
subsequent simplifications are possible using standard packages of well-known computing systems such as
Mathematica [23], Maple [24], Mathcad [25] and so on.
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[2]
[3]
[4]
[5]
[6]
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174 | Igor V. Beloussov
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