Table Of ContentAnother Formulation of the Wick’s Theorem. Farewell, Pairing?
Igor V. Beloussov∗
Institute of Applied Physics, Academy of Sciences of Moldova,
5 Academy Str., Kishinev, 2028, Republic of Moldova
(Dated: January 19, 2015)
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)ofthematrixisproposed. Eachelementofthematrixistheaverageofthechronological
product of only two operators. This formulation is extremely convenient for practical calculations
in quantum field theory and statistical physics by the methods of symbolic mathematics using
computers.
5
1
PACSnumbers: 02.10.Ud,02.70.Wz,05.30.Fk,11.10.Wx
0
2
n INTRODUCTION A,B ,...,B can be decomposed into the sum of n vac-
1 n
a uum expectation values of the same chronological prod-
J
Wick’s theoremsareusedextensivelyinquantumfield ucts with all possible contractions of one of these opera-
6
tors (for example A ) with all others, i.e.,
1 theory[1–4]andstatisticalphysics[5–7]. Theyallowone
to use the Green’s functions method, and consequently
] toapplytheFeynman’sdiagramsforinvestigations[1–3]. hT (AB ...B )i = T AB ...B ...B .
h 1 n 0 1 i n
p The first of these, which can be called Wick’s Theorem 1≤Xi≤nD (cid:16) (cid:17)E0
- forOrdinaryProducts,givesustheopportunitytoreduce (1)
h
inalmostautomaticmodetheusualproductofoperators Here A A = T(A A )− : A A := hT (A A )i (i,j =
t i i i j i j i j 0
a into a unique sum of normal products multiplied by c– 1,2,...,n ) is the chronologicalcontraction between the
m
numbers. It canbe formulatedas follows [4]. Let Ai(xi) factors Ai and Aj. It should be noted that, in contrast
[ (i = 1,2,...,n ) are “linear operators”, i.e., some lin- to the usual Wick’s theorem for chronological products,
ear combinations of creation and annihilation operators. there are no expressions involving a number of contrac-
1
v Then the ordinary product of linear operators is equal to tions greater than one on the right-hand side of (1).
5 the sum of all the corresponding normal products with all Wick’s theorem for chronological products or its ge-
8 possible contractions, including the normal product with- neralized version are used for the calculation of matrix
9 out contractions, i.e., elementsofthescatteringmatrixineachorderofpertur-
3
bationtheory[1–4]. Theprocedureisreducedtocalcula-
0
A ...A = : A ...A : +: A A ...A : +...
. 1 n 1 n 1 2 n tionofthevacuumexpectationofchronologicalproducts
1
0 +: A1...An−1An: +: A1...An: of the field operators in the interaction representation.
5 As factors in these products a number of operators ψi
+: A A A A ...A : +... ,
1 1 2 3 4 n of the Fermi fields and the same number of their con-
: jugate operators ψ¯ , as well as operators of the Bose
v i
Xi where AiAj = AiAj−: AiAj: (i,j = 1,2,...,n) is the fields ϕs = ϕ(s+) +ϕs(−) may be used. Here all continu-
contraction between the factors Ai and Aj. Since the ous and discrete variables are included in the index. In
r vacuumexpectationvalueofthenormalorderedproduct the interaction representation the operators ψ ,ψ¯, and
a i i
isequalzero,thistheoremprovidesusawayofexpressing ϕ(±) correspond to free fields and satisfy the commuta-
s
the vacuum expectation values of n linear operators in tion relationships of the form [ψ , ψ ] = ψ¯, ψ¯ =0,
terms of vacuum expectation values of two operators. i j + i j +
ψ , ψ¯ = D , ϕ(−), ϕ(−) = ϕ(+)(cid:2), ϕ(+) (cid:3) = 0,
Wick’s Theorem for Chronological Products [4] asserts (cid:2) i j(cid:3)+ ij h r s i− h r s i−
that the T–product of a system of n linear operators is ϕ(+), ϕ(−) = D¯ . Therefore, the averaging of the
r s rs
equal to the sum of their normal products with all possi- h i−
Fermi and Bose fields can be performed independently.
blechronological contractions, includingthetermwithout
contractions. It follows directly from the previous theo-
rem and gives the opportunity to calculate the vacuum
BASIC THEOREM
expectationvaluesofthechronologicalproductsoflinear
operators.
Finally, from Wick’s theorem for chronological prod- Since we may rearrange the order of the operators in-
ucts the Generalized Wick’s Theorem [4] can be ob- side T–products taking into account the change of the
tained. It asserts that the vacuum expectation value sign, which arises when the order of the Fermi operators
of the chronological product of n + 1 linear operators is changed, we present our vacuum expectation value of
2
the chronological product of the Fermi operators in the lations. Hereinafterweshowthatthecomputationof(2)
form can be easily performed using a simple formula
± T ψi1ψ¯j1 ψi2ψ¯j2 ... ψinψ¯jn 0. (2) T ψi1ψ¯j1 ... ψinψ¯jn 0 =det ∆iαjβ , (3)
(cid:10) (cid:2)(cid:0) (cid:1)(cid:0) (cid:1) (cid:0) (cid:1)(cid:3)(cid:11)
(cid:10) (cid:2)(cid:0) (cid:1) (cid:0) (cid:1)(cid:3)(cid:11) (cid:0) (cid:1)
To calculate (2), we can use Wick’s theorem for chrono-
where
logical products. However, while considering the higher-
order perturbation theory, the number of pairs ψ ψ¯ of
operators ψi and ψ¯j becomes so large that the idirject ∆iαjβ =ψiαψ¯jβ = T ψiαψ¯jβ 0
application of this theorem begins to represent certain (α,β =1,(cid:10)2,.(cid:0)..,n) .(cid:1)(cid:11) (4)
problems because it is very difficult to sort through all
the possible contractions between ψ and ψ¯ . Theproofofthistheoremisbyinduction. Letusassume
i j
A consistent use of generalized Wick’s theorem would now that (3) is true for n pairs ψ ψ¯ , and consider it
iα jβ
introduce a greater accuracy in our actions. However,in for the case n+1. Using generalizedWick’s theorem we
this case we expect very cumbersome and tedious calcu- have
T ψ ψ¯ ψ ψ¯ ... ψ ψ¯ =−∆ T ψ ψ¯ ψ ψ¯ ... ψ ψ¯
i1 j1 i2 j2 in∗1 jn∗1 0 i1jn+1 in∗1 j1 i2 j2 in jn 0
(cid:10) (cid:2)(cid:0) n−1 (cid:1)(cid:0) (cid:1) (cid:0) (cid:1)(cid:3)(cid:11) (cid:10) (cid:2)(cid:0) (cid:1)(cid:0) (cid:1) (cid:0) (cid:1)(cid:3)(cid:11)
− ∆ T ψ ψ¯ ... ψ ψ¯ ψ ψ¯ ψ ψ¯ ... ψ ψ¯
iγjn+1 i1 j1 iγ−1 jγ−1 in+1 jγ iγ+1 jγ+1 in jn 0
γX=2 (cid:10) (cid:2)(cid:0) (cid:1) (cid:0) (cid:1)(cid:0) (cid:1)(cid:0) (cid:1) (cid:0) (cid:1)(cid:3)(cid:11)
−∆ T ψ ψ¯ ... ψ ψ¯ ψ ψ¯ +∆ T ψ ψ¯ ... ψ ψ¯ .
injn+1 i1 j1 in−1 jn−1 in+1 jn 0 in+1jn+1 i1 j1 in jn 0
(cid:10) (cid:2)(cid:0) (cid:1) (cid:0) (cid:1)(cid:0) (cid:1)(cid:3)(cid:11) (cid:10) (cid:2)(cid:0) (cid:1) (cid:0) (cid:1)(cid:3)(cid:11)
Taking into account (3) we obtain
T ψ ψ¯ ψ ψ¯ ... ψ ψ¯ =
i1 j1 i2 j2 in∗1 jn∗1 0
(cid:10) (cid:2)(cid:0) (cid:1)(cid:0) (cid:1) (cid:0) (cid:1)(cid:3)(cid:11)
∆ ∆ ··· ∆ ∆ ∆ ··· ∆
in+1j1 in+1j2 in+1jn i1j1 i1j2 i1jn
−∆i1jn+1(cid:12)(cid:12)(cid:12)(cid:12)∆...i2j1 ∆...i2j2 ·.·..· ∆...i2jn (cid:12)(cid:12)(cid:12)(cid:12)−∆i2jn+1(cid:12)(cid:12)(cid:12)(cid:12) ∆...in+1j1 ∆...in+1j2 ·.·..· ∆...in+1jn (cid:12)(cid:12)(cid:12)(cid:12)−... (5)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12)∆ ∆ ··· ∆ (cid:12) (cid:12) ∆ ∆ ··· ∆ (cid:12)
(cid:12) inj1 inj2 injn (cid:12) (cid:12) inj1 inj2 injn (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) ∆i1j1 ∆i1j2 ··· ∆i1jn(cid:12) (cid:12) ∆i1j1 ∆i1j2 ··· ∆i1jn (cid:12)
−∆injn+1(cid:12)(cid:12)(cid:12)(cid:12) ∆... ∆... ·.·..· ∆... (cid:12)(cid:12)(cid:12)(cid:12)+∆in+1jn+1(cid:12)(cid:12)(cid:12)(cid:12)∆...i2j1 ∆...i2j2 ·.·..· ∆...i2jn (cid:12)(cid:12)(cid:12)(cid:12) .
(cid:12) in−1j1 in−1j2 in−1jn (cid:12) (cid:12) (cid:12)
(cid:12) ∆ ∆ ··· ∆ (cid:12) (cid:12)∆ ∆ ··· ∆ (cid:12)
(cid:12) in+1j1 in+1j2 in+1jn (cid:12) (cid:12) inj1 inj2 injn (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
Rearrangingtherowsinthedeterminantsin(5)itiseasy Note that this result does not depend on the way
to see that the right hand side is the expansion of the how we divide the operators on the left hand side of (3)
into pairs ψ ψ¯ . Indeed, if on the left side of (3) we
iα jβ
∆i1j1 ∆i1j2 ··· ∆i1jn+1 permute, for example, ψ¯jβ and ψ¯jγ (β 6= γ), it changes
det ∆iαjβ =(cid:12)(cid:12)(cid:12)(cid:12) ∆...i2j1 ∆...i2j2 ·.·..· ∆...i2jn+1 (cid:12)(cid:12)(cid:12)(cid:12) i(t3s)ssiignnc.e Tthhise csahmanegehalpeapdesnstoonthteheperrigmhuttahtainodn soifdetwoof
(cid:0) (cid:1) (cid:12) (cid:12) columns in the determinant, and it also changes its
(cid:12) ∆ ∆ ··· ∆ (cid:12)
(cid:12) in+1j1 in+1j2 in+1jn+1 (cid:12) sign. Similarly, in the case of a permutation of ψ and
(α(cid:12)(cid:12),β =1,2,...,n+1) (cid:12)(cid:12) ψ (α 6= δ). Obviously, when the whole pair ψ iψα¯ is
iδ iα jβ
transposed, the left and right hand sides of (3) do not
along the last column [8]. The validity of (3) for n = 1
change.
follows from the definition ∆ in (4).
iαjβ
3
where V(x −x ) = U(r −r )δ(τ −τ ). We can im-
1 2 1 2 1 2
Obviously, the formula similar to (3) can be obtained mediately take into account the reduction of the dis-
and in the case of Bose fields connected diagrams, if we assume in (8) hSi = 1 and
0
T ϕ(+)ϕ(−) ϕ(+)ϕ(−) ... ϕ(+)ϕ(−) ∆(x1,x2)=0 [6]. Then,
D h(cid:16) i1 j1 (cid:17)(cid:16) i2 j2 (cid:17) (cid:16) in jn (cid:17)iE0
=perm ∆ , (6)
iαjβ
(cid:0) (cid:1)
where 1
G (x ,x )=−∆(x ,x )+ dz dz V(z −z )
I 1 2 1 2 1 2 1 2
2hSi Z
∆¯ =ϕ(+)ϕ(−) = T ϕ(+)ϕ(−) 0
iαjβ iα jβ D (cid:16) iα jβ (cid:17)E0 × ∆(x ,z ) ∆(z1,x2) ∆(z1,z1)
(α,β =1,2,...,n) . (7) (cid:20) 1 2 (cid:12)(cid:12)∆(z2,x2) ∆(z2,z1)(cid:12)(cid:12)
(cid:12) (cid:12)
In quantum statistics the n–body thermal, or −∆(x ,z ) ∆(cid:12) (z1,x2) ∆(z1,z2) (cid:12)
imaginary-time,Green’s functions in the Grand Canoni- 1 1 (cid:12)∆(z2,x2) ∆(z2,z2)(cid:12)(cid:21)
(cid:12) (cid:12)
cal Ensemble are defined as the thermal trace of a time- =(cid:12)(cid:12)−∆(x1,x2) (cid:12)(cid:12)
orderedproduct ofthe fields operatorsin the imaginary-
time Heisenberg representation [5–7]. To calculate them + dz dz ∆(x ,z )V(z −z )∆(z ,z )∆(z ,x )
1 2 1 1 1 2 1 2 2 2
Z
in each order of perturbation theory, Wick’s theorem is
also used. Obviously, in this case, the theorem also may − dz dz ∆(x ,z )V(z −z )∆(z ,z )∆(z ,x ) .
1 2 1 1 1 2 2 2 1 2
beformulatedintheform(3)and(6)convenientforprac- Z
tical calculation.
Taking into account G(0)(x ,x ) = −∆(x ,x ) we ob-
SIMPLE EXAMPLES 1 2 1 2
tain finally
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 Greens GI(x1,x2)=G(0)(x1,x2)
functions for the Fermi system described in the interac-
+ dz dz G(0)(x ,z )Σ(1)(z ,z ) G(0)(z ,x ) ,
tion representation by the Hamiltonian Z 1 2 1 1 1 2 2 2 2
1
H (τ)= dr dr ψ¯ (r ,τ)ψ¯ (r ,τ)
int 1 2 α 1 β 2
2Z
×U(r −r )ψ (r ,τ)ψ (r ,τ)
1 2 β 2 α 1
thatcontainstheproductofthefieldoperatorsψ (r ,τ)
α 1
and ψ¯ (r ,τ) in this representation (parameter α indi-
α 1 Σ(1)(z ,z )=−V(z −z )G(0)(z ,z )
catesthe spinprojections,τ is the imaginary-time). The 1 2 1 2 1 2
one-particleGreen’sfunctioncanbe representedas[5,6] +δ(z −z ) dzV(z −z)G(0)(z,z) .
1 2 1
T ψ(x )ψ¯(x )S Z
G (x ,x )=− τ 1 2 0 ,
I 1 2 (cid:10) (cid:2) hSi (cid:3)(cid:11)
0
where h...i is the symbol for the Gibbs average over
0
the states of a system of noninteracting particles, x ≡ Similarly, for the two-particle Green’s function
(r,τ,α) and
τ
S(τ)=Tτexp−Z dτ′Hint(τ′) . GII(x1,x2,x3,x4)=
0
We obtain
G (x ,x )=− T ψ(x )ψ¯(x )
I 1 2 τ 1 2 0
1 (cid:10) (cid:2) (cid:3)(cid:11)
+ dz dz V(z −z )
1 2 1 2
2hSi0 Z T ψ(x )ψ(x )ψ¯(x )ψ¯(x )S
× T ψ(x )ψ¯(x ) ψ(z )ψ¯(z ) ψ(z )ψ¯(z ) − τ 1 2 3 4 0 ,
τ 1 2 1 1 2 2 0 (cid:10) (cid:2) hSi (cid:3)(cid:11)
(cid:10) (cid:2)(cid:0) (cid:1)(cid:0) 1 (cid:1)(cid:0) (cid:1)(cid:3)(cid:11) 0
=−∆(x ,x )+ dz dz V(z −z )
1 2 1 2 1 2
2hSi Z
0
∆(x1,x2) ∆(x1,z1) ∆(x1,z2) we have
×(cid:12)∆(z1,x2) ∆(z1,z1) ∆(z1,z2)(cid:12) , (8)
(cid:12) (cid:12)
(cid:12)(cid:12)∆(z2,x2) ∆(z2,z1) ∆(z2,z2)(cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
4
G (x ,x ,x ,x )= T ψ(x )ψ¯(x ) ψ(x )ψ¯(x )
II 1 2 3 4 1 3 2 4 0
1 (cid:10) (cid:2)(cid:0) (cid:1)(cid:0) (cid:1)(cid:3)(cid:11)
− dz dz V(z −z ) T ψ(x )ψ¯(x ) ψ(x )ψ¯(x ) ψ(z )ψ¯(z ) ψ(z )ψ¯(z )
2hSi Z 1 2 1 2 1 3 2 4 1 1 2 2 0
0 (cid:10) (cid:2)(cid:0) (cid:1)(cid:0) (cid:1)(cid:0) (cid:1)(cid:0) (cid:1)(cid:3)(cid:11)
0 0 ∆(x ,z ) ∆(x ,z )
1 1 1 2
= ∆(x1,x3) ∆(x1,x4) − 1 dz dz V(z −z )(cid:12)(cid:12) 0 0 ∆(x2,z1) ∆(x2,z2)(cid:12)(cid:12)
(cid:12) ∆(x2,x3) ∆(x2,x4)(cid:12) 2Z 1 2 1 2 (cid:12) ∆(z1,x3) ∆(z1,x4) ∆(z1,z1) ∆(z1,z2) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) ∆(z ,z ) ∆(z ,x ) ∆(z ,z ) ∆(z ,z ) (cid:12)
(cid:12) (cid:12) (cid:12) 2 3 2 4 2 1 2 2 (cid:12)
=G(0)(x1,x3)G(0)(x2,x4)−(cid:12)(cid:12)G(0)(x1,x4)G(0)(x2,x3) (cid:12)(cid:12)
− dz dz G(0)(x ,z )G(0)(x ,z )V(z −z ) G(0)(z ,x )G(0)(z ,x )−G(0)(z ,x )G(0)(z ,x ) .
1 2 1 1 2 2 1 2 1 3 2 4 1 4 2 3
Z h i
CONCLUSIONS Representations (3) and (6) not only greatly simplify
allcalculations,butalsoallowonetoperformthemusing
a computer with programs of symbolic mathematics [9].
The above theorem has an important consequence. In
fact,itestablishesaperfectcoincidencebetweenthevac-
uum expectation values of the chronological products of
n pairs of field operators and the n–order determinant. ∗ [email protected]
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