A Representation for Fermionic Correlation Functions Joel Feldman∗ Department of Mathematics University of British Columbia Vancouver, B.C. CANADA V6T 1Z2
Horst Kn¨ orrer, Eugene Trubowitz Mathematik ETH-Zentrum CH-8092 Z¨ urich SWITZERLAND
∗
Research supported in part by the Natural Sciences and Engineering Research Council of Canada, the Schweizerischer Nationalfonds zur F¨ orderung der wissenschaftlichen Forschung and the Forschungsinstitut f¨ ur Mathematik, ETH Z¨ urich
§I. Introduction Let A(a1 , · · · , an ) be the finite dimensional, complex Grassmann algebra freely n o generated by a1 , · · · , an . Let Mr = (i1 ,···,ir ) 1 ≤ i1 ,···,ir ≤ n be the set of all multi indices of degree r ≥ 0 . For each multi index
I
= (i1 ,···,ir ) , set aI = ai1 · · · air . By
convention, a∅ = 1 . Let I =
n
[
o (i1 ,···,ir ) 1 ≤ i1 |I| |J|
The number Λ is morally the supremum kSk∞ =
|Sij | of the covariance S . The number Φ (|I|) is intuitively the degree of can-
i,j∈{1,···,n}
cellation between the at most |I|! nonzero terms contributing to the Pfaffian Z SI SI,J Pf = aI : aJ :S dµS (a) SJ,I 0 At one extreme (see, Example III.4), we can always choose Φ (|I|) = I
|I| !
for all multi indices
and Λ = kSk∞ . At the other extreme (see, Example III.5), suppose that 0 Σ S = −Σ T 0
for some matrix Σ = Σij
of order
n 2
, and further that there is a complex Hilbert space
H , elements vi , wi ∈ H , i = 1, · · · , n2 , and a constant Λ > 0 with hvi , wj iH
= Σij 1
kvi kH , kwj kH ≤ L 2
for all i, j = 1, · · · , n2 . Then, by a variant of Gram’s inequality, we can choose Φ (|I|) = 1 for all multi indices
I
and Λ = 2L . 6
For example, the Grassmann algebra associated to a many fermion system has an equal number of “annihilation” a1 , · · · , am and “creation” a ¯1 , · · · , a ¯m generators. Furthermore, the physical covariance C cannot pair two annihilation or two creation generators. 0 Cı ¯ That is, C = . It is also often possible to write Cı ¯ as an inner product C¯ı 0 between vectors in an appropriate Hilbert space with “naturally” bounded norms so that Φ (|I|) = 1 , I ∈ M , can be achieved in models of physical interest. See, [FMRT, p.682]. Now, let
P
f (a) =
f (m) (a)
m≥0
be a Grassmann polynomial in A(a1 , · · · , an ) where, for each m ≥ 0 , f (m)(a) =
P j1 ,···,jm
fm (j1 ,···,jm ) aj1 · · · ajm
and the kernel fm (j1 ,···,jm ) is an antisymmetric function of its arguments. Definition I.7 For all α ≥ 2 , the “external” and “internal” naive power counting norms kf kα and |||f ||| α of the Grassmann polynomial f (a) are kf kα =
P
kf (m) kα =
m≥0
and |||f ||| α =
P
P
|||f (m) ||| α =
kfm k1 = kfm k1,∞ =
1
αm Λ 2
m
kfm k1
m≥0
m≥0
where
P
αm
1
kSk1,∞
Λ2
(m −2)
kfm k1,∞
m≥0
P j1 ,···,jm
fm (j1 ,···,jm ) P
sup j1 ∈{1,···,n}
fm (j1 ,j2 ,···,jm )
j2 ,···,jm
are the L1 and “ mixed L1 , L∞ ” norms of the antisymmetric kernels fm (j1 ,···,jm ) .
In Section III we prove Theorem I.8 Suppose 2 |||W|||α+1 ≤ 1 . Then, for all polynomials f in the Grassmann algebra A(a1 , · · · , an ) , kR (f )kα ≤ 2 Φ(n) |||W|||α+1 kf kα 7
In particular, the spectrum of R is bounded away from 1 uniformly in the number of degrees of freedom n when Φ is constant and |||W|||α+1 is small enough.
A simple consequence of Theorem I.8 is the archetypical bound on correlation functions
Theorem I.9 Suppose indices
2 (1+Φ(n)) |||W|||α+1
< 1 . Then, for each m ≥ 0 and all sequences of
1 ≤ j1 ,···,jm ≤ n ,
|Sm (j1 ,···,jm )| ≤
Φ(n) 1−2 Φ(n) |||W|||α+1
1
αm Λ 2
m
In particular, the correlation functions are bounded uniformly in the number of degrees of freedom n when Φ is constant and kWkα+1 is small enough.
Decomposition of Feynman Graphs into Annuli In the rest of the introduction we motivate and interpret Definition I.4 and Theorem I.3 “graphically”. However, we emphasize that the purely algebraic proof of Theorem I.3 given in the next section is completely independent of this discussion and, in particular, does not refer to graphs. Recall that, for
I
= (i1 ,···,ir ) in Mr with r even,
Z aI dµS (a) = Pf SI
r X
=
k1 ,···,kr =1 k2i−1 0 . Integration by parts with respect to a]j1 gives Z Z aI : aJ :S dµS = aI a]J dµS ] Z |I| = (−1) a]j1 aI a]J\{j1 } dµS ] Z |I| P `−1 |I| aI\{i` } a]J\{j1 } dµS ] (−1) = (−1) S j1 i ` `=1
=
|I|
(−1)
|I| P
Z (−1)
`−1
`=1
24
S j1 i `
aI\{i` } : aJ\{j1 } :S dµS
Our induction hypothesis implies that Z 1 2 (|I|+|J|−2) aI\{i` } : aJ\{j1 } :S dµS ≤ kSk∞
(|I|−1)!
for each ` = 1, · · · , |I| . Now, Z Z |I| P aI : aJ :S dµS ≤ |Sj1 i` | aI\{i` } : aJ\{j1 } :S dµS `=1
2 (|I|+|J|−2) ≤ kSk∞ 1
1
2 (|I|+|J|) ≤ kSk∞
|I| P
(|I|−1)!
`=1
(|I|−1)!
|Sj1 i` |
|I|
This “perturbative bound” is obtained by ignoring all potential cancellations between the at Z aI a]J dµS ] . most |I| ! nonzero terms appearing in Pfaffian equal to
Example III.5 (Gram’s Inequality) Suppose that S = Sij metric matrix of the form
S =
where Σ = Σij
is a matrix of order
n 2
0 −Σ t
Σ 0
is a complex, skew sym-
. Suppose, in addition, that there is a complex
Hilbert space H , elements vi , wi ∈ H , i = 1, · · · , n2 , and a constant Λ > 0 with Σij = hvi , wj iH and kvi kH , kwj kH ≤
Λ 2
12
for all i, j = 1, · · · , n2 . Then, the “nonperturbative bound” 1 (|I|+|J|) Z Λ2 , aI : aJ :S dµS ≤ 0, holds for all multi indices
I
and
J.
≤ |I| |J| > |I| |J|
The proof is presented in the Appendix.
Now, let f (a) =
P m≥0
25
f (m) (a)
be a Grassmann polynomial in A(a1 , · · · , an ) where, for each m ≥ 0 , P
f (m)(a) =
j1 ,···,jm
fm (j1 ,···,jm ) aj1 · · · ajm
and the kernel fm (j1 ,···,jm ) is an antisymmetric function of its arguments. Fix a complex, skew symmetric matrix S = Sij of order n satisfying Hypothesis III.3. We recall Definition I.4 For all α ≥ 2 , the “external” and “internal” naive power counting norms kf kα and |||f ||| α of the Grassmann polynomial f (a) are P
kf kα =
m≥0
and
P
|||f ||| α =
P
kf (m) kα =
1
αm Λ 2
m
kfm k1
m≥0
P
|||f (m) ||| α =
m≥0
αm
1
kSk1,∞
Λ2
(m −2)
kfm k1,∞
m≥0
By the triangle inequality, P
P
`≥1
r,s ∈ IN`
kR (f )kα ≤
P
kR r s (f )kα ≤
m≥0
and consequently, kR (f )kα ≤
P
m P
P
m≥1
`=1
r,s ∈ IN`
P
P
`≥1
r,s ∈ IN`
kR r s (f (m) )kα
kR r s (f (m) )kα
since, R r s (f (m)) = 0 for all r, s ∈ IN` when ` > m . Furthermore, `
kR
rs
(f
(m)
)kα = α
since, R r s (f (m) ) =
Σ (ri −si ) i=1
P j1 ,···,j M
with
M
Λ
1 2
`
Σ (ri −si ) i=1
kAlt R r s (f (m) )k1
Alt R r s (f (m) ) (j1 ,···,j M ) aj1 · · · aj M
= (r1 −s1 )+···+(r` −s` ) . Altogether, kR (f )kα ≤
P m≥1
m P
P
`=1
r,s ∈ IN`
`
α
Σ (ri −si ) i=1
Λ
1 2
`
Σ (ri −si ) i=1
kR r s (f (m))k1
Proposition III.2 will now be used to obtain a bound on the norm kR r s (f (m) )k1 of the kernel R r s (f (m)) (K1 ,···, K` ) . 26
Lemma III.6 Let H , J1 , ···, J` be multi indices with |H| = m ≥ ` . Then, Z ` ` Q Q P : a Ji :S a H dµS ≤ M(H , J1 , ···, J` ) |Shµi ji1 | 1 ≤ µ1 ,···,µ` ≤ m pairwise different
i=1
i=1
where M(H , J1 , ···, J` ) =
Proof:
Z ` Q : a : dµ a H \{hµ ,···,hµ } Ji \{ji1 } S S 1 `
sup 1 ≤ µ1 ,···,µ` ≤ m pairwise different
i=1
For convenience, set ki = ji1 , i = 1, · · · , ` . By antisymmetry, the integrand can be
rewritten so that Z aH :
` Q i=1
Z a Ji :S dµS = ±
a]k` · · · a]k1 a H
` Q i=1
a]Ji \{ki } dµS ]
Now, integrate by parts successively with respect to a]k` , · · · , a]k1 , and then apply Leibniz’s rule to obtain Z ` n Z ` Q Q P a Ji :S dµS = ± Ski m aH : i=1
m=1
i=1
P
=
1 ≤ µ1 ,···,µ` ≤ m pairwise different
since
` P n Q i=1
m=1
∂ Ski m ∂a m
±
` Q i=1
aH
P 1 ≤ µ1 ,···,µ` ≤ m pairwise different
a]Ji \{ki } dµS ]
a H \{hµ1 ,···,hµ` }
i=1
±
` Q i=1
Z
Q `
1 ≤ µ1 ,···,µ` ≤ m pairwise different
=
Ski hµi
P
aH =
∂ ∂am
Ski hµi
` Q i=1
∂ ∂ahµ
i
` Q i=1
a]Ji \{ki } dµS ]
aH
Ski hµi a H \{hµ1 ,···,hµ` }
It follows immediately that Z ` Q a Ji :S dµS aH : i=1
≤
P
` Q
1 ≤ µ1 ,···,µ` ≤ m pairwise different
i=1
≤ M(H , J1 , ···, J` )
Z ` Q a H \{hµ1 ,···,hµ` } : |Shµi ki | a Ji \{ki } :S dµS i=1
P
` Q
1 ≤ µ1 ,···,µ` ≤ m pairwise different
i=1
27
|Shµi ki |
Proposition III.7 Let P
f (m) (a) =
h1 ,···,hm
fm (h1 ,···,hm ) ah1 · · · ahm
be a homogeneous Grassmann polynomial of degree m , where fm (h1 ,···,hm ) is an antisymmetric function of its arguments. Let Z ` Q (m) : b Ji :S f (b) dµS ≤ `! i=1
J1 , ···, J`
m `
be multi indices with m ≥ ` . Then, P
M(m, J1 , ···, J` )
` Q
|fm (h1 ,···,hm )|
i=1
h1 ,···,hm
|Shi ji1 |
where M(m, J1 , ···, J` ) = sup M(H , J1 , ···, J` ) . |H|=m
Proof: For convenience, set ki = ji1 , i = 1, · · · , ` . By the preceding lemma, Z ` ` P Q Q P b Ji :S f (m) (b) dµS ≤ |fm (H)| M(H , J1 , ···, J` ) |Shµi ki | : |H|=m
i=1
P
≤ M(m, J1 , ···, J` )
|H|=m
|fm (H)|
1 ≤ µ1 ,···,µ` ≤ m pairwise different
i=1
P
` Q
1 ≤ µ1 ,···,µ` ≤ m pairwise different
i=1
|Shµi ki |
Observe that, by the antisymmetry of fm , P |H|=m
|fm (H)|
P
` Q
1 ≤ µ1 ,···,µ` ≤ m pairwise different
i=1
|Shµi ki | =
P
P
1 ≤ µ1 ,···,µ` ≤ m pairwise different
h1 ,···,hm
P
P
1 ≤ µ1 ,···,µ` ≤ m pairwise different
h1 ,···,hm
=
= `!
m `
P
|fm (h1 ,···,hm )|
` Q i=1
|fm (h1 ,···,hm )|
|fm (h1 ,···,hm )|
h1 ,···,hm
` Q i=1
` Q i=1
|Shµi ki | |Shi ki |
|Shi ki |
Proposition III.8 Let f (m) (a) =
P h1 ,···,hm
fm (h1 ,···,hm ) ah1 · · · ahm
be as above. Let r, s ∈ IN` with m ≥ ` . Then, the L1 norm kR r s (f (m))k1 of the kernel R r s (f (m) ) (K1 ,···, K` ) is bounded by 28
(a)
kR r s (f (m) )k1 ≤ where M (m, s) =
sup |Ji |=si i=1,···,`
M (m, s) kfm k1
m `
Q `
(rsii ) kSk1,∞ kwri k1,∞
i=1
M(m, J1 , ···, J` ) .
(b) kR r s (f
(m)
)k1 ≤ Φ(m−`)
m `
Λ
1 2
`
m − Σ (ri −si )
Q `
kfm k1
i=1
(rsii )
i=1
1
kSk1,∞
Λ2
(ri −2)
kwri k1,∞
when, in addition, Hypothesis III.3 is satisfied.
Proof:
To verify (a), set ui (j, Ki ) =
P |J 0 |=si −1 i i=1,···,`
|wri (j , Ji0 , Ki )|
for each i = 1, · · · , ` . By construction, kui k1,∞ = kwri k1,∞ , i = 1, · · · , ` . Also, set P
T (K1 ,···, K` ) =
` P n Q
|fm (h1 ,···,hm )|
i=1
h1 ,···,hm
j=1
|Shi j | |ui (j, Ki )|
By Proposition III.7, R r s (f ) (K1 ,···, K` ) ≤
m `
` Q
M (m, s)
i=1
(rsii )
T (K1 ,···, K` )
since, P
M(m, J1 , ···, J` )
|Ji |=si i=1,···,`
≤ M (m, s)
= M (m, s)
P
|fm (h1 ,···,hm )|
i=1
h1 ,···,hm
P
P
|J 0 |=si −1 i i=1,···,`
h1 ,···,hm
P
` Q
|fm (h1 ,···,hm )|
` P n Q i=1
` P n Q
|fm (h1 ,···,hm )|
i=1
h1 ,···,hm
j=1
|Shi j i1 | |wri (Ji , Ki )|
j=1
|Shi j | |wri (j , Ji0 , Ki )|
|Shi j | |ui (j, Ki )|
It follows from Proposition III.2 that
kR r s (f )k1 ≤
m `
≤
m `
M (m, s)
` Q i=1
M (m, s)
` Q i=1
(rsii ) (rsii ) 29
kTk1 kfm k1
` Q i=1
kSk1,∞ kwri k1,∞
For (b), simply observe that, by Hypothesis III.3, Z 1 ` Q a H \{hµ1 ,···,hµ` } : a Ji \{ji1 } :S dµS ≤ Φ(m−` ) Λ 2
`
m+Σ
i=1
(|Ji |−2)
i=1
for any multi indices
H , J1 , ···, J`
with
|H|
= m ≥ ` and any pairwise different sequence of
indices 1 ≤ µ1 , · · · , µ` ≤ m . Consequently, M (m, s) =
sup
sup
|Ji |=si i=1,···,`
|Hi |=m
M(H , J1 , ···, J` ) ≤ Φ(m−`) Λ
1 2
`
m + Σ (si −2)
i=1
We have developed all the material required for a useful bound on the operator R . For the rest of this section we assume Hypothesis III.3.
Lemma III.9 Let
P
f (m) (a) =
fm (h1 ,···,hm ) ah1 · · · ahm
h1 ,···,hm
be as above and let m ≥ ` . Then, for all α ≥ 2 , P r,s ∈ IN`
Proof:
By Proposition III.8 (b), `
kR
rs
(f
kR r s (f (m))kα ≤ Φ(m−`) kf (m) kα |||W|||`α+1
(m)
)kα ≤ α
Σ (ri −si ) i=1
Λ
1 2
` Q
where, for convenience, P r s =
i=1
m `
1
`
Σ (ri −si ) i=1
kR r s (f (m) )k1 ≤ Φ(m−`)
(rsii ) αri −si
Λ 2 m kfm k1 =
1 αm
m `
1
kSk1,∞
Λ2
(ri −2)
m `
kwri k1,∞ . However,
1
αm Λ 2 m kfm k1 ≤ kf (m) kα
when α ≥ 2 , and consequently, P r,s ∈ IN`
kR r s (f (m) )kα ≤ Φ(m−`) kf (m) kα
30
1
Λ 2 m kfm k1 P r s
P ri ≥si ≥1 i=1,···,`
Prs
Observe that P ri ≥si ≥1 i=1,···,`
P
Prs =
r s
r−s
()α
kSk1,∞
Λ
`
1 2 (r −2)
kwr k1,∞
≤ |||W|||`α+1
r≥s≥1
since P
(rs) αr−s
1
kSk1,∞
Λ2
(r −2)
P
kwr k1,∞ ≤
r≥s≥1
(rs) αr−s
1
kSk1,∞
Λ2
(r −2)
kwr k1,∞
r≥s≥0
=
P
(α+1)
r
1
kSk1,∞
Λ2
(r −2)
kwr k1,∞
r≥0
= |||W|||α+1 Therefore,
P r,s ∈ IN`
kR r s (f (m) )kα ≤ Φ(m−`) kf (m) kα |||W|||`α+1
We can now prove
Theorem I.8 Suppose 2 |||W|||α+1 ≤ 1 . Then, for all polynomials f in the Grassmann algebra A(a1 , · · · , an ) , kR (f )kα ≤ 2 Φ(n) |||W|||α+1 kf kα
Proof:
By Lemma III.9, kR (f )kα ≤
P m≥1
≤ Φ(n)
m P
P
`=1
r,s ∈ IN`
P
kf (m) kα
m≥1
≤ Φ(n)
kR r s (f (m) )kα m P `=1
1 kf kα 1−|||W||| α+1
≤ 2 Φ(n) kf kα |||W|||α+1
31
|||W|||`α+1 |||W|||α+1
Corollary III.10 Suppose
2 (1+Φ(n)) |||W|||α+1
< 1 . Then, for all polynomials f in the
Grassmann algebra A(a1 , · · · , an ) , k(1l − R)−1 (f )kα ≤
1 1−2 Φ(n) |||W|||α+1
kf kα
Lemma III.11 For all Grassmann polynomials f in A(a1 , · · · , an ) , Z f (a) dµS ≤ Φ(n) kf kα
Proof:
As usual, write
P
f (a) = where, for each m ≥ 0 , f
(m)
m≥0
P
(a) =
f (m) (a)
fm (j1 ,···,jm ) aj1 · · · ajm and the kernel fm (j1 ,···,jm )
j1 ,···,jm
is an antisymmetric function of its arguments. Then, by Hypothesis III.3, Z Z f (a) dµS ≤ P f (m)(a) dµS m≥0
≤
P
|fm (j1 ,···,jm )|
P
m≥0 j1 ,···,jm
≤
P
P
P
aj1 · · · ajm dµS 1
m≥0 j1 ,···,jm
≤
Z
|fm (j1 ,···,jm )| Φ (m) Λ 2 1
kfm k1 Φ (m) αm Λ 2
m
m
m≥0
≤ Φ (n) kf kα
Recall that the correlation functions Sm (j1 ,···,jm ) , m ≥ 0 , corresponding to the interaction W(a) and the propagator S = Sij are given by Z 1 Sm (j1 ,···,jm ) = Z aj1 · · · ajm eW(a) dµS (a) Theorem I.9 Suppose indices
2 (1+Φ(n)) |||W|||α+1
< 1 . Then, for each m ≥ 0 and all sequences of
1 ≤ j1 ,···,jm ≤ n ,
|Sm (j1 ,···,jm )| ≤
Φ(n) 1−2 Φ(n) |||W|||α+1
32
1
αm Λ 2
m
Proof:
Fix
1 ≤ j1 ,···,jm ≤ n
aJ =
and rewrite the monomial aJ = aj1 · · · ajm as P
k1 ,···,km
Alt(δk1 ,j1 · · · δkm ,jm ) (k1 ,···,km ) ak1 · · · akm
Then, 1
kaJ kα = αm Λ 2
m
1
kAlt(δ ·,j1 · · · δ ·,jm )k1 ≤ αm Λ 2
By Theorem I.3, Lemma III.11 and Corollary III.10, |Sm (j1 ,···,jm )| =
Z
(1l − R)−1 (aJ ) dµS
≤ Φ(n) k(1l − R)−1 (aJ )kα ≤
Φ(n) 1−2 Φ(n) |||W|||α+1
kaJ kα
so that |Sm (j1 ,···,jm )| ≤
Φ(n) 1−2 Φ(n) |||W|||α+1
33
1
αm Λ 2
m
m
Appendix: Gram’s Inequality for Pfaffians
Proposition. Suppose that S = Sij
is a complex, skew symmetric matrix of the form
S = where Σ = Σij
is a matrix of order
n 2
0 −Σ t
Σ 0
. Suppose, in addition, that there is a complex
Hilbert space H , elements vi , wi ∈ H , i = 1, · · · , n2 , and a constant Λ > 0 with Σij = hvi , wj iH and kvi kH , kwj kH ≤ for all i, j = 1, · · · , n2 . Then, for all multi indices
I
Z aI dµS ≤ and
Proof:
Λ 2
12
and Λ 2
J,
12 |I|
1 (|I|+|J|) Z , Λ2 aI : aJ :S dµS ≤ 0,
To prove the first inequality, suppose Z
i1