Fermionic Functional Integrals and the Renormalization Group Joel Feldman Department of Mathematics University of British Columbia Vancouver B.C. CANADA V6T 1Z2
[email protected] http://www.math.ubc.ca/∼feldman Horst Kn¨ orrer, Eugene Trubowitz Mathematik ETH–Zentrum CH-8092 Z¨ urich SWITZERLAND
Abstract
The Renormalization Group is the name given to a technique for analyzing
the qualitative behaviour of a class of physical systems by iterating a map on the vector space of interactions for the class. In a typical non-rigorous application of this technique one assumes, based on one’s physical intuition, that only a certain finite dimensional subspace (usually of dimension three or less) is important. These notes concern a technique for justifying this approximation in a broad class of Fermionic models used in condensed matter and high energy physics.
These notes expand upon the Aisenstadt Lectures given by J. F. at the Centre de Recherches Math´ematiques, Universit´e de Montr´eal in August, 1999.
Table of Contents §I Fermionic Functional Integrals p 1 §I.1 Grassmann Algebras p 1 §I.2 Grassmann Integrals p 6 §I.3 Differentiation and Integration by Parts p 11 §I.4 Grassmann Gaussian Integrals p 14 §I.5 Grassmann Integrals and Fermionic Quantum Field Theories p 18 §I.6 Wick Ordering p 24 §I.7 Bounds on Grassmann Gaussian Integrals p 28 §II Fermionic Expansions §II.1 Notation and Definitions §II.2 Expansion – Algebra §II.3 Expansion – Bounds §II.4 Sample Applications Gross–Neveu2 Naive Many-fermion2 Many-fermion2 – with sectorization
p 34 p p p p p p p
Appendices §A Infinite Dimensional Grassmann Algebras
p 65
§B Pfaffians
p 76
§C Propagator Bounds
p 82
References
p 88
34 36 37 47 49 53 57
I. Fermionic Functional Integrals
This chapter just provides some mathematical background. Most of it is easy algebra – primarily the definition of Grassmann algebra and the definition and basic properties of a class of linear functionals on Grassmann algebras known as Grassmann Gaussian integrals. There is also one piece of analysis – the Gram bound on Grassmann Gaussian integrals – and a brief discussion of how Grassmann integrals arise in quantum field theories. To make this chapter really trivial, we consider only finite dimensional Grassmann algebras. A simple–minded method for handling the infinite dimensional case is presented in Appendix A.
I.1 Grassmann Algebras Definition I.1 (Grassmann algebra with coefficients in C) Let V be a finite dimensional vector space over C. The Grassmann algebra generated by V is
V0
V = C and
V
Vn
V=
∞ M Vn n=0
V
V is the n–fold antisymmetric tensor product of V with itself. V Thus, if {a1 , . . . , aD } is a basis for V, then V is a vector space with elements of the form
where
f (a) =
D X
X
n=0 1≤i1 0 m,p
ℓ DkW k(α+1)F
=
DkW k 2 kf kαF 1−DkW(α+1)F α k(α+1)F
≤
3 α kf kαF DkW k(α+1)F
The proof for the other norm is similar.
Lemma II.12 Assume Hypothesis (HG). If α ≥ 1 then, for all g(a, c) ∈ AC Z g(a, c) dµS (a) ≤ |||g(a, c)|||αF αF
Z
[g(a, c) − g(a, 0)] dµS (a) ≤ kg(a, c)kαF αF
Proof:
Let P
g(a, c) =
P
gl,r (L, J) cL aJ
L∈Ml J∈Mr
l,r≥0
with gl,r (L, J) antisymmetric under separate permutations of its L and J arguments. Then Z P Z P gl,r (L, J) cL aJ dµS (a) g(a, c) dµS (a) = αF
P
=
l
l
αF
≤
P P
L∈Ml
l≥0
P
αF
L∈Ml J∈Mr
l,r≥0
l
l+r
αF
l,r≥0
Similarly,
Z
[g(a, c) − g(a, 0) dµS (a)
αF
=
P
l≥1
≤
P
l≥1 r≥0
l
P
=
l≥1 r≥0
l
αF
P
sup
1≤k≤n L∈M ˜ l−1
αl Fl+r sup 1≤k≤n
≤ kg(a, c)kαF
44
|gl,r (L, J)|
gl,r (L, J) cL
L∈Ml J∈Mr
P
gl,r (L, J)
r≥0 J∈Mr
P
L∈Ml J∈Mr
≤ |||g(a, c)|||αF
P
P
P
P
Z
aJ dµS (a)
aJ dµS (a)
˜ J) gl,r (k, L,
r≥0 J∈Mr
˜ L∈M l−1 J∈Mr
Z
˜ J)| |gl,r (k, L,
αF
Z
aJ dµS (a)
Set g = (1l − R)−1 f . Then
Z
= g(a, c) − g(a, 0) dµS (a)
Proof of Corollary II.7: kS(f )(c) − S(f )(0)kαF
(Theorem II.2)
αF
≤ k(1l − R)−1 (f )kαF ≤
1 1−3DkW k(α+1)F /α
kf kαF ≤
(Lemma II.12)
1 1−1/α
kf kαF
(Theorem II.6)
The argument for |||S(f )|||αF is identical.
we have
With the more detailed notation R f (a, c) eW (a,c) dµS (a) R S(f, W ) = eW (a,c) dµS (a) kΩ(W )kαF
Z 1
= S(W, εW )(c) − S(W, εW )(0) dε αF 0 Z 1 α α ≤ α−1 kW kαF dε = α−1 kW kαF 0
We end this section with a proof of the continuity of the Schwinger functional Z Z W (c,a) 1 S(f ; W, S) = Z(c;W,S) f (c, a) e dµS (a) where Z(c; W, S) = eW (c,a) dµS (a) and the renormalization group map Z 1 Ω(W, S)(c) = log ZW,S eW (c,a) dµS (a) where
ZW,S =
Z
eW (0,a) dµS (a)
with respect to the interaction W and covariance S.
Theorem II.13 Let F, D > 0, 0 < t, v ≤ Z bH :bJ :S dµS (b) ≤ F|H|+|J| kSk ≤ F2 D
DkW k(α+2)F ≤ then
1 6
1 2
S and α ≥ 4. If, for all H, J ∈ r≥0 Mr Z √ bH :bJ :T dµT (b) ≤ ( t F)|H|+|J| kT k ≤ tF2 D
DkV k(α+2)F ≤
v 6
|||S(f ; W + V, S + T ) − S(f ; W, S)|||αF ≤ 8 (t + v) |||f |||αF kΩ(W + V, S + T ) − Ω(W, S)kαF ≤ 45
3 D
(t + v)
Proof:
First observe that, for all |z| ≤ 1t Z bH :bJ :zT dµzT (b) ≤ F|H|+|J| Z bH :bJ :S+zT dµS+zT (b) ≤ (2F)|H|+|J|
by Problem II.1.a by Problem II.1.b
kS + zT k ≤ 2F2 D ≤ (2F)2 D
Also, for all |z ′ | ≤ v1 ,
DkW + z ′ V k(α+2)F ≤
1 3
Hence, by Corollary II.7, with F replaced by 2F, α replaced by and S replaced by S + zT |||S(f ; W + z ′ V, S + zT )|||αF ≤ for all |z| ≤
1 t
kΩ(W + z ′ V, S + zT )kαF ≤ Z
f (c, a) eW (c,a)+z
′
W replaced by W + z ′ V
α α−2
|||f |||αF
α α−2
kW + z ′ V kαF ≤
and |z ′ | ≤ v1 .
The integral
α 2,
V (c,a)
α 1 α−2 3D
(II.2)
dµS+zT (a)
is a polynomial in z and z ′ . Hence both S(f ; W + z ′ V, S + zT ) and Ω(W + z ′ V, S + zT )
are meromorphic functions of z and z ′ (even with the log appearing in the definition of
Ω – the definition of log(1 + s1 ) in Problem I.2 is a polynomial in s1 ). By (II.2), both S(f ; W + z ′ V, S + zT ) and Ω(W + z ′ V, S + zT ) must be analytic functions of z and z ′ on
|z| ≤ 1t , |z ′ | ≤ v1 .
By the Cauchy integral formula, if f (z) is analytic and bounded in absolute value by M on |z| ≤ r, then
′
f (z) = and, for all |z| ≤
r 2,
Hence, for all |z| ≤
1 , 2t
′ f (z) ≤
1 2πı
Z
|ζ|=r
f (ζ) dζ (ζ−z)2
1 M 2πr 2π (r/2)2
= 4M r1
1 |z ′ | ≤ 2v d α S(f ; W + z ′ V, S + zT ) ≤ 4t α−2 |||f |||αF dz αF d α ′ S(f ; W + z ′ V, S + zT ) ≤ 4v α−2 |||f |||αF dz αF
d
α 1
Ω(W + z ′ V, S + zT ) ≤ 4t α−2 dz 3D αF
d α 1
′ Ω(W + z ′ V, S + zT ) ≤ 4v α−2 dz 3D αF 46
By the chain rule, for all |z| ≤ 1, d α S(f ; W + zV, S + zT ) ≤ 4(t + v) α−2 |||f |||αF dz αF
d α 1
Ω(W + zV, S + zT ) ≤ 4(t + v) α−2 dz 3D αF
Integrating z from 0 to 1 and α S(f ; W + V, S + T ) − S(f ; W, S) |||f |||αF ≤ 4(t + v) α−2 αF
α 1
Ω(W + V, S + T ) − Ω(W, S) ≤ 4(t + v) α−2 3D αF As α ≥ 4,
α α−2
≤ 2 and the Theorem follows.
II.4 Sample Applications We now apply the expansion to a few examples. In these examples, the set of Grassmann algebra generators {a1 , · · · , an } is replaced by ψx,σ , ψ¯x,σ x ∈ IRd+1 , σ ∈ S
with S being a finite set (of spin/colour values). See §I.5. To save writing, we introduce the notation
ξ = (x, σ, b) ∈ IRd+1 × S × {0, 1} Z X XZ dξ . . . = dd+1 x . . . b∈{0,1} σ∈S
ψξ =
ψx,σ ψ¯x,σ
if b = 0 if b = 1
Now elements of the Grassmann algebra have the form ∞ Z X f (ψ) = dξ1 · · · dξr fr (ξ1 , · · · , ξr ) ψξ1 · · · ψξr r=0
the norms kfr k and kf (ψ)kα become kfr k = max sup 1≤i≤r ξi
kf (ψ)kα =
∞ X r=0
Z
r Q
j=1 j6=i
αr kfr k
dξj fr (ξ1 , · · · , ξr )
When we need a second copy of the generators, we use
Ψξ ξ ∈ IRd+1 × S × {0, 1} (in
place of {c1 , · · · , cn }). Then elements of the Grassmann algebra have the form f (Ψ, ψ) =
∞ Z X
l,r=0
dξ1′ · · · dξl′ dξ1 · · · dξr fl,r (ξ1′ , · · · , ξl′ ; ξ1 , · · · , ξr ) Ψξ1′ · · · Ψξl′ ψξ1 · · · ψξr 47
and the norms kfl,r k and kf (Ψ, ψ)kα are kfl,r k = max sup 1≤i≤l+r ξi
kf (Ψ, ψ)kα =
∞ X
l,r=0
Z
l+r Q j=1 j6=i
αl+r kfl,r k
dξj fl,r (ξ1 , · · · , ξl ; ξl+1 , · · · , ξl+r )
All of our Grassmann Gaussian integrals will obey Z
Z
ψx,σ ψ dµS (ψ) = 0 ψ¯x,σ ψ¯x′ ,σ′ dµS (ψ) = 0 Z Z ψx,σ ψ¯x′ ,σ′ dµS (ψ) = − ψ¯x′ ,σ′ ψx,σ dµS (ψ) x′ ,σ ′
Hence, if ξ = (x, σ, b), ξ ′ = (x′ , σ ′ , b′ )
where
0 ′ ′ S(ξ, ξ ′ ) = Cσ,σ (x, x′ ) −Cσ′ ,σ (x , x) 0 ′
Cσ,σ′ (x, x ) =
Z
if if if if
b = b′ = 0 b = 0, b′ = 1 b = 1, b′ = 0 b = b′ = 1
ψx,σ ψ¯x,σ′ dµS (ψ)
That our Grassmann algebra is no longer finite dimensional is, in itself, not a big deal. The Grassmann algebras are not the ultimate objects of interest. The ultimate objects of interest are various expectation values. These expectation values are complex numbers that we have chosen to express as the values of Grassmann Gaussian integrals. See (I.3). If the covariances of interest were to satisfy the hypotheses (HG) and (HS), we would be able to easily express the expectation values as limits of integrals over finite dimensional Grassmann algebras using Corollary II.7 and Theorem II.13. The real difficulty is that for many, perhaps most, models of interest, the covariances (called propagators by Physicists) do not satisfy (HG) and (HS). So, as explained in §I.5, the covariance is expressed as a sum of terms, each of which does satisfy the hypotheses. These terms, called single scale covariances, will, in each example, be constructed by substituting a partition of unity of IRd+1 (momentum space) into the full covariance. The partition of unity will be constructed using a fixed “scale parameter” M > 1 and a 48
function ν ∈ C0∞ ([M −2 , M 2 ]) that takes values in [0, 1], is identically 1 on [M −1/2 , M 1/2 ] and obeys
∞ X j=0
for 0 < x < 1.
ν M 2j x = 1
Example (Gross–Neveu2 ) The propagator for the Gross-Neveu model in two space-time dimensions has Z Z d2 p ip·(x′ −x) 6 pσ,σ′ + mδσ,σ′ ′ ¯ e Cσ,σ′ (x, x ) = ψx,σ ψx,σ′ dµS (ψ) = (2π)2 p2 + m2 where 6p =
ip0 −p1
p1 −ip0
is a 2 × 2 matrix whose rows and columns are indexed by σ ∈ {↑, ↓}. This propagator does not satisfy Hypothesis (HG) for any finite F. If it did satisfy (HG) for some finite F, R Cσ,σ′ (x, x′ ) = ψx,σ ψ¯x′ ,σ′ dµS (ψ) would be bounded by F2 for all x and x′ . This is not
the case – it blows up as x′ − x → 0. Set
ν νj (p) = ν 1
Then
M 2j p2 2j
M p2
S(x, y) =
∞ X
if j > 0 if j = 0, |p| ≥ 1
if j = 0, |p| < 1 S (j) (x, y)
j=0
with
and
0 C (j) (x, x′ ) σ,σ ′ S (j) (ξ, ξ ′ ) = (j) −Cσ′ ,σ (x′ , x) 0 C
(j)
′
(x, x ) =
Z
if b = b′ = 0 if b = 0, b′ = 1 if b = 1, b′ = 0 if b = b′ = 1
d2 p ip·(x′ −x) 6 p + m e νj (p) (2π)2 p2 + m
We now check that, for each 0 ≤ j < ∞, S (j) does satisfy Hypotheses (HG) and
(HS). The integrand of C (j) is supported on M j−1 ≤ |p| ≤ M j+1 for j > 0 and |p| ≤ M 49
for j = 0. This is a region of volume at most const M 2j and on this region, the integrand is bounded by const M1 j . By Corollary I.28, the value of F for this propagator is bounded by
Z 1/2
d2 p
Fj = 2 p6 p+m ν (p) ≤ CF j 2 +m (2π)2
1/2 1 M 2j Mj
is the matrix norm of for some constant CF . Here p6 p+m 2 +m
6 p+m p2 +m .
= CF M j/2
By the following Lemma,
the value of D for this propagator is bounded by Dj =
1 M 2j
We have increasing the value of CF in Fj in order to avoid having a const in Dj .
Lemma II.14 sup x,σ
XZ σ′
(j)
d2 y |Cσ,σ′ (x, y)| ≤ const M1j
Proof: We have already observed that the integrand of C (j) is bounded by const M1 j and is supported on a region of volume at most const M 2j . Hence (j) sup |Cσ,σ′ (x, y)| x,y σ,σ ′
≤
Z
d2 p 6 +m p p2 +m νj (p) (2π)2
≤ const M1 j M 2j ≤ const M j
(II.3)
(j)
To show that Cσ,σ′ (x, y) decays sufficiently quickly in x − y, we play the usual integration by parts game. (y − x)
The support of
4
(j) Cσ,σ′ (x, y)
∂2 ∂p21
+
Z
d2 p 6 p + m ∂2 ∂ 2 2 ip·(y−x) ν (p) = + e 2 2 j ∂p1 ∂p2 (2π)2 p2 + m Z d2 p ip·(y−x) ∂ 2 ∂2 2 6 p + m + e ν (p) = 2 2 j ∂p1 ∂p2 (2π)2 p2 + m
∂ 2 2 6p+m p2 +m ∂p22
νj (p) is contained in the support of νj (p). So the
integrand is still supported in a region of volume const M 2j . Each
∂ ∂pi
acting on
6p+m p2 +m
increases the difference between the degree of the denominator and the degree of the numerator by one. So does each
∂ ∂pi
acting on νj (p), provided you count both M j and p 50
as having degree one. Hence, recalling that |p| is bounded above and below by const M j (of course with different constants),
∂2 ∂p21
+
∂2 2 2 ∂p2
6p + m
p2 + m
on the support of the integrand and
νj (p) ≤ const M14j
1 Mj
(j) sup M 4j (y − x)4 Cσ,σ′ (x, y) ≤ const M1j M 2j ≤ const M j
(II.4)
x,y σ,σ ′
Multiplying the
1 4
power of (II.3) by the
3 4
power of (II.4) gives
(j) sup M 3j |y − x|3 Cσ,σ′ (x, y) ≤ const M j
(II.5)
x,y σ,σ ′
Adding (II.3) to (II.5) gives
(j) sup [1 + M 3j |y − x|3 ]Cσ,σ′ (x, y) ≤ const M j x,y σ,σ ′
Dividing across
j
(j)
M |Cσ,σ′ (x, y)| ≤ const 1+M 3j |x−y|3
Integrating Z
2
d y
(j) |Cσ,σ′ (x, y)|
≤
Z
2
d y const
Mj 1+M 3j |x−y|3
= const
1 Mj
Z
d2 z
1 1+|z|3
≤ const M1j
We made the change of variables z = M j (y − x). To apply Corollary II.7 to this model, we fix some α ≥ 2 and define the norm R P kW kj of W (ψ) = r>0 dξ1 · · · dξr wr (ξ1 , · · · , ξr ) ψξ1 · · · ψξr to be kW kj = Dj kW kαFj =
X
(αCF )r M j
r
r−4 2
kwr k
Let J > 0 be a cutoff parameter (meaning that in the end, the model is defined by taking the limit J → ∞) and define, as in §I.5, Z 1 eW (Ψ+ψ) dµS (≤J ) (ψ) GJ (Ψ) = log ZJ 51
where
ZJ =
Z
eW (ψ) dµS (≤J ) (ψ)
and Ωj (W )(Ψ) = log Z
1
W,S (j)
Z
W (Ψ+ψ)
e
dµS (j) (ψ)
where
ZW,S (j) =
Z
eW (ψ) dµS (j) (ψ)
Then, by Problem I.11, GJ = ΩS (1) ◦ ΩS (2) ◦ · · · ◦ ΩS (J ) (W ) Set Wj = ΩS (j) ◦ ΩS (j+1) ◦ · · · ◦ ΩS (J ) (W ) Suppose that we have integrated out all scales from the ultraviolet cutoff J down to j and have shown that kW kj ≤ 13 . To integrate out scale j − 1 we use Theorem II.15GN Suppose α ≥ 2 and M ≥
α α−1
2 α+1 α
for r ≤ 4, then kΩj−1 (W )kj−1 ≤ kW kj . Proof:
6
. If kW kj ≤
1 3
and wr vanishes
We first have to relate kW (Ψ + ψ)kα to kW (ψ)kα , because we wish to apply
Corollary II.7 with W (c, a) replaced by W (Ψ + ψ). To do so, we temporarily revert to the old notation with c and a generators, rather than Ψ and ψ generators. Observe that W (c + a) =
X X m
=
I∈Mm
X X X I∈Mm J⊂I
m
=
wm (I)(c + a)I wm (J, I \ J)cJ aI\J
X X X l,r
J∈Ml K∈Mr
l+r l
wl+r (J, K)cJ aK
We have renamed I \ J = K. The l+r arises because, given two ordered sets J, K, there l K| are |J|+| ordered sets I with J ⊂ I, K = I \ J. Hence |J| kW (c + a)kα =
X l,r
αl+r
l+r l
X kwl+r k = αm 2m kwm k = kW (a)k2α m
Similarly, kW (Ψ + ψ)kα = kW (ψ)k2α 52
To apply Corollary II.7 at scale j − 1, we need 1 3
Dj−1 kW (Ψ + ψ)k(α+1)Fj−1 = Dj−1 kW (ψ)k2(α+1)Fj−1 ≤ But Dj−1 kW k2(α+1)Fj−1 = =
X r
r r−4 2(α + 1)CF M (j−1) 2 kwr k
X
1
r≥6
≤
X
2
2 α+1 M −( 2 − m ) α 1
M−6 2 α+1 α
r≥6
≤ 2 α+1 α
6
r
1 kW kj M
6 as M > 2 α+1 and kW kj ≤ 31 . By Corollary II.7, α
r
(αCF )r M j
(αCF )r M j
≤
r−4 2
r−4 2
kwr k
kwr k
1 3
α Dj−1 kW (Ψ + ψ)kαFj−1 kΩj−1 (W )kj−1 = Dj−1 kΩj−1 (W )kαFj−1 ≤ α−1 6 1 α 2 α+1 kW kj ≤ kW kj ≤ α−1 α M
Theorem II.15GN is just one ingredient used in the construction of the Gross– Neveu2 model. It basically reduces the problem to the study of the projection X Z P W (ψ) = dξ1 · · · dξr wr (ξ1 , · · · , ξr ) ψξ1 · · · ψξr r=2,4
of W onto the part of the Grassmann algebra of degree at most four. A souped up version of Theorem II.15GN can be used to reduce the problem to the study of the projection P ′ W of W onto a three dimensional subspace of the range of P .
Example (Naive Many-fermion2 ) The propagator, or covariance, for many–fermion models is Z 1 d3 k ik·(x′ −x) ′ e Cσ,σ′ (x, x ) = δσ,σ′ 3 (2π) ik0 − e(k) where k = (k0 , k) and e(k) is the one particle dispersion relation (a generalisation of
k2 2m )
minus the chemical potential (which controls the density of the gas). The subscript on 53
many-fermion2 signifies that the number of space dimensions is two (i.e. k ∈ IR2 , k ∈ IR3 ). For pedagogical reasons, I am not using the standard many–body Fourier transform
conventions. We assume that e(k) is a reasonably smooth function (for example, C 4 ) that has a nonempty, compact, strictly convex zero set, called the Fermi curve and denoted F . We further assume that ∇e(k) does not vanish for k ∈ F , so that F is itself a reasonably smooth curve. At low temperatures only those momenta with k0 ≈ 0 and k near F are important, so we replace the above propagator with Z d3 k ik·(x′ −x) U (k) ′ e Cσ,σ′ (x, x ) = δσ,σ′ (2π)3 ik0 − e(k)
The precise ultraviolet cutoff, U (k), shall be chosen shortly. It is a C0∞ function which takes values in [0, 1], is identically 1 for k02 + e(k)2 ≤ 1 and vanishes for k02 + e(k)2 larger than some constant. This covariance does not satisfy Hypotheses (HS) for any finite D. If it did, Cσ,σ′ (0, x′ ) would be L1 in x′ and consequently the Fourier transform would be uniformly bounded. But
U(k) ik0 −e(k)
U(k) ik0 −e(k)
blows up at k0 = 0, e(k) = 0. So we write the
covariance as sum of infinitely many “single scale” covariances, each of which does satisfy (HG) and (HS). This decomposition is implemented through a partition of unity of the set of all k’s with k02 + e(k)2 ≤ 1.
We slice momentum space into shells around the Fermi curve. The j th shell is
defined to be the support of ν (j) (k) = ν M 2j (k02 + e(k)2 ) By construction, the j th shell is a subset of
k
1 M j+1
≤ |ik0 − e(k)| ≤
1 M j−1
As the scale parameter M > 1, the shells near the Fermi curve have j near +∞. Setting Z d3 k ik·(x′ −x) ν (j) (k) (j) ′ Cσ,σ′ (x, x ) = δσ,σ′ e (2π)3 ik0 − e(k) and U (k) =
∞ P
ν (j) (k) we have
j=0 ′
Cσ,σ′ (x, x ) =
∞ X j=0
54
(j)
Cσ,σ′ (x, x′ )
The integrand of the propagator C (j) is supported on a region of volume at most const M −2j (k0 is restricted to an interval of length const M −j and k must remain within a distance const M −j of F ) and is bounded by const M j . By Corollary I.28, the value of F for this propagator is bounded by 1/2 Z 1/2 ν (j) (k) d3 k ≤ CF M j M12j = CF M1j/2 Fj = 2 |ik0 −e(k)| (2π)3
(II.6)
for some constant CF . Also
(j) sup |Cσ,σ′ (x, y)| x,y σ,σ ′
Each derivative
∂ ∂ki
≤
ν (j) (k) ik0 −e(k)
acting on
Z
ν (j) (k) d3 k |ik0 −e(k)| (2π)3
≤ const M1j
increases the supremum of its magnitude by a factor
of order M j . So the naive argument of Lemma II.14 gives XZ (j) (j) 1/M j |Cσ,σ′ (x, y)| ≤ const [1+M −j |x−y|]4 ⇒ sup d3 y |Cσ,σ′ (x, y)| ≤ const M 2j x,σ
σ′
In fact, using Corollary C.3, with lj = M1j/2 , yields the better bound XZ (j) d3 y |Cσ,σ′ (x, y)| ≤ const l1j M j ≤ const M 3j/2 sup x,σ
Here, the factor
1 lj
(II.7)
σ′
is the number of terms in the partition of unity used to write C (j) as
(j)
a sum of Cχ ’s, each term of which is bounded using Corollary C.3. So the value of D for this propagator is bounded by Dj = M 5j/2 This time we define the norm kW kj = Dj kW kαFj =
X
(αCF )r M −j
r
r−5 2
kwr k
Again, let J > 0 be a cutoff parameter and define, as in §I.5, Z Z W (Ψ+ψ) 1 GJ (c) = log ZJ e dµS (≤J ) (a) where ZJ = eW (ψ) dµS (≤J ) (a) and Ωj (W )(c) = log
1 ZW,S (j)
Z
W (Ψ+ψ)
e
dµS (j) (a) 55
where
ZW,S (j) =
Z
eW (ψ) dµS (j) (a)
Then, by Problem I.11, GJ = ΩS (J ) ◦ · · · ◦ ΩS (1) ◦ ΩS (0) (W ) Also call Gj = Wj . If we have integrated out all scales from the ultraviolet cutoff, which in this (infrared) problem is fixed at scale 0, to j and we have ended up with some interaction that obeys kW kj ≤
1 , 3
then we integrate out scale j + 1 using the following analog of
Theorem II.15GN. α Theorem II.15MB1 Suppose α ≥ 2 and M ≥ 2 α−1
vanishes for r < 6, then kΩj+1 (W )kj+1 ≤ kW kj . Proof:
2
α+1 12 . α
If kW kj ≤
1 3
and wr
To apply Corollary II.7 at scale j + 1, we need Dj+1 kW (Ψ + ψ)k(α+1)Fj+1 = Dj+1 kW k2(α+1)Fj+1 ≤
But Dj+1 kW k2(α+1)Fj+1 = =
X r
r−5 2(α + 1 CF )r M −(j+1) 2 kwr k
X r≥6
≤
X
1
− 12 2 α+1 α M
r≥6
≤ 2 α+1 α By Corollary II.7
5
1
− 2 (1− r ) 2 α+1 α M
6
r
r
(αCF )r M −j
(αCF )r M −j
1 kW kj M 1/2
≤ kW kj ≤
r−5 2
1 3
r−5 2
kwr k
kwr k
1 3
α Dj+1 kW (Ψ + ψ)kαFj+1 kΩj+1 (W )kj+1 = Dj+1 kΩj+1 (W )kαFj+1 ≤ α−1 6 1 α kW kj ≤ kW kj 2 α+1 ≤ α−1 α M 1/2
It looks, in Theorem II.15MB1 , like five-legged vertices w5 are marginal and all vertices wr with r < 5 have to be renormalized. Of course, by evenness, there are no five–legged vertices so only vertices wr with r = 2, 4 have to be renormalized. But it still 56
looks, contrary to the behaviour of perturbation theory, like four–legged vertices are worse than marginal. Fortunately, this is not the case. Our bounds can be tightened still further. In the bounds (II.6) and (II.7) the momentum k runs over a shell around the Fermi curve. Effectively, the estimates we have used to count powers of M j assume that all momenta entering an r–legged vertex run independently over the shell. Thus the estimates fail to take into account conservation of momentum. As a simple illustration of R (j) (j) this, observe that for the two–legged diagram B(x, y) = d3 z Cσ,σ (x, z)Cσ,σ (z, y), (II.7)
yields the bound
sup x
Z
3
d y |B(x, y)| ≤ sup x
Z
(j) (x, z) d z Cσ,σ 3
Z
(j) (z, y) d3 y Cσ,σ
≤ const M 3j/2 M 3j/2 = const M 3j
But B(x, y) is the Fourier transform of W (k) =
ν (j) (k)2 [ik0 −e(k)]2
= C (j) (k)C (j) (p) p=k. Conser-
vation of momentum forces the momenta in the two lines to be the same. Plugging this W (k) and lj =
1 M j/2
into Corollary C.2 yields Z sup d3 y |B(x, y)| ≤ const l1j M 2j ≤ const M 5j/2 x
We exploit conservation of momentum by partitioning the Fermi curve into “sectors”.
Example (Many-fermion2 – with sectorization) We start by describing precisely what sectors are, as subsets of momentum space. Let, for k = (k0 , k), k′ (k) be any reasonable “projection” of k onto the Fermi curve F=
k ∈ IR2 e(k) = 0
In the event that F is a circle of radius kF centered on the origin, it is natural to choose
k′ (k) =
kF k. |k|
For general F , one can always construct, in a tubular neighbourhood of F ,
a C ∞ vector field that is transverse to F , and then define k′ (k) to be the unique point of F that is on the same integral curve of the vector field as k is. Let j > 0 and set if k ∈ F 1 (≥j) P ν (k) = ν (i) (k) otherwise i≥j
57
Let I be an interval on the Fermi surface F . Then s=
k k′ (k) ∈ I, k ∈ supp ν (≥j−1)
is called a sector of length length(I) at scale j. Two different sectors s and s′ are called neighbours if s′ ∩ s 6= ∅. A sectorization of length lj at scale j is a set Σj of sectors of length lj at scale j that obeys - the set Σj of sectors covers the Fermi surface - each sector in Σj has precisely two neighbours in Σj , one to its left and one to its right - if s, s′ ∈ Σj are neighbours then
1 16 lj
≤ length(s ∩ s′ ∩ F ) ≤ 18 lj
Observe that there are at most 2 length(F )/lj sectors in Σj . In these notes, we fix lj =
1 M j/2
and a sectorization Σj at scale j. s1
s2 s3 s4
F
Next we describe how we “sectorize” an interaction X Z r Q dxi Wr = wr (x1 , σ1 , κ1 ), · · · , (xr , σr , κr ) ψσ1 (x1 , κ1 ) · · · ψσr (xr , κr ) i=1
σi ∈{↑,↓} κi ∈{0,1}
where
ψσi (xi ) = ψσi (xi , κi ) κ
ψ¯σi (xi ) = ψσi (xi , κi ) κ
i =0
i =1
Let F (r, Σj ) denote the space of all translation invariant functions
r fr (x1 , σ1 , κ1 , s1 ), · · · , (xr , σr , κr , sr ) : IR3 × {↑, ↓} × {0, 1} × Σj → C
whose Fourier transform, fˆr (k1 , σ1 , κ1 , s1 ), · · · , (kr , σr , κr , sr ) , vanishes unless ki ∈ si .
An fr ∈ F (r, Σj ) is said to be a sectorized representative for wr if
P ˆ w ˆr (k1 , σ1 , κ1 ), · · · , (kr , σr , κr ) = fr (k1 , σ1 , κ1 , s1 ), · · · , (kr , σr , κr , sr ) si ∈Σj 1≤i≤r
58
for all k1 , · · · , kr ∈ supp ν (≥j) . It is easy to construct a sectorized representative for wr
by introducing (in momentum space) a partition of unity of supp ν (≥j) subordinate to Σj .
Furthermore, if fr is a sectorized representative for wr , then Z r Q dxi wr (x1 , σ1 , κ1 ), · · · , (xr , σr , κr ) ψσ1 (x1 , κ1 ) · · · ψσr (xr , κr ) i=1 Z r Q P dxi fr (x1 , σ1 , κ1 , s1 ), · · · , (xr , σr , κr , sr ) ψσ1 (x1 , κ1 ) · · · ψσr (xr , κr ) = i=1
si ∈Σj 1≤i≤r
for all ψσi (xi , κi ) “in the support of” dµC (≥j) , i.e. provided ψ is integrated out using a Gaussian Grassmann measure whose propagator is supported in supp ν (≥j) (k). Furthermore, by the momentum space support property of fr , Z r Q fr (x1 , σ1 , κ1 , s1 ), · · · , (xr , σr , κr , sr ) ψσ1 (x1 , κ1 ) · · · ψσr (xr , κr ) dxi i=1 Z r Q = fr (x1 , σ1 , κ1 , s1 ), · · · , (xr , σr , κr , sr ) ψσ1 (x1 , κ1 , s1 ) · · · ψσr (xr , κr , sr ) dxi i=1
where
ψσ (x, b, s) =
Z
d3 y ψσ (y, b, s)χ ˆs(j)(x − y)
(j)
and χ ˆs is the Fourier transform of a function that is identically one on the sector s. This function is chosen shortly before Proposition C.1. We have expressed the interaction Z r r Q P Q Wr = fr (x1 , σ1 , κ1 , s1 ), · · · , (xr , σr , κr , sr ) ψσi (xi , κi , si ) dxi i=1
si ∈Σj σi ∈{↑,↓} κi ∈{0,1}
i=1
in terms of a sectorized kernel fr and new “sectorized” fields, ψσ (x, κ, s), that have propagator (j) Cσ,σ′
′
(x, s), (y, s ) =
Z
=δ
ψσ (x, 0, s)ψσ′ (y, 1, s′) dµC (j) (ψ)
σ,σ ′
The momentum space propagator (j)
Z
(j)
(j)
d3 k ik·(y−x) ν (j) (k)χs (k)χs′ (k) e (2π)3 ik0 − e(k) (j)
Cσ,σ′ (k, s, s′) = δσ,σ′
(j)
ν (j) (k)χs (k)χs′ (k) ik0 − e(k)
59
vanishes unless s and s′ are equal or neighbours, is supported in a region of volume const lj M12j and has supremum bounded by const M j . By an easy variant of Corollary I.28, the value of F for this propagator is bounded by Fj ≤ CF
1 M 2j
M j lj
1/2
= CF
q
lj Mj
for some constant CF . By Corollary C.3, XZ (j) sup d3 y |Cσ,σ′ (x, s), (y, s′) | ≤ const M j x,σ,s
σ ′ ,s′
so the value of D for this propagator is bounded by Dj =
2j 1 lj M
We are now almost ready to define the norm on interactions that replaces the unsectorized norm kW kj = Dj kW kαFj of the last example. We define a norm on F (r, Σj ) by kf k = max
X
max
1≤i≤r xi ,σi ,κi ,si
σk ,κk ,sk k6=i
Z
Q
ℓ6=i
dxℓ f (x1 , σ1 , κ1 , s1 ), · · · , (xr , σr , κr , sr )
and for any translation invariant function
we define
r wr (x1 , σ1 , κ1 ), · · · , (xr , σr , κr ) : IR3 × {↑, ↓} × {0, 1} → C kwr kΣj = inf
n
o kf k f ∈ F (r, Σj ) a representative for W
The sectorized norm on interactions is kW kα,j = Dj
X X r−2 r−4 (αFj )r kwr kΣj = (αCF )r lj 2 M −j 2 kwr kΣj r
r
Proposition II.16 (Change of Sectorization) Let j ′ > j ≥ 0. There is a constant CS , independent of M, j and j ′ , such that for all r ≥ 4
l r−3 kwr kΣj kwr kΣj′ ≤ CS l j′ j
60
Proof:
The spin indices σi and bar/unbar indices κi play no role, so we suppress them.
Let ǫ > 0 and choose fr ∈ F (r, Σj ) such that wr (k1 , · · · , kr ) =
P
si ∈Σj 1≤i≤r
fr (k1 , s1 ), · · · , (kr , sr )
for all k1 , · · · , kr in the support of supp ν (≥j) and kwr kΣj ≥ kfr k − ǫ Let 1=
X
χs′ (k′ )
s′ ∈Σj ′
be a partition of unity of the Fermi curve F subordinate to the set intervals that obeys
sup ∂km′ χs′ ≤ k′
Fix a function ϕ ∈ C0∞
constm lm j′
s′ ∩ F s′ ∈ Σj of
[0, 2) , independent of j, j ′ and M , which takes values in [0, 1]
and which is identically 1 for 0 ≤ x ≤ 1. Set
ϕj ′ (k) = ϕ M 2(j
′
−1)
[k02 + e(k)2 ] ′
Observe that ϕj ′ is identically one on the support of ν (≥j ) and is supported in the support of ν (≥j
′
−1)
. Define gr ∈ F (r, Σj ′ ) by
r Q P gr (k1 , s′1 ), · · · , (kr , s′r ) = fr (k1 , s1 ), · · · , (kr , sr ) χs′m (km )ϕj ′ (km ) m=1
sℓ ∈Σ 1≤ℓ≤r
=
P
sℓ ∩s′ 6=∅ ℓ 1≤ℓ≤r
fr (k1 , s1 ), · · · , (kr , sr )
r Q m=1
χs′m (km )ϕj ′ (km )
Clearly wr (k1 , · · · , kr ) =
P
s′ ∈Σ ′ j ℓ 1≤ℓ≤r
gr (k1 , s′1 ), · · · , (kr , s′r )
′
for all kℓ in the support of supp ν (≥j ) . Define Momi (s′ ) = (s′1 , · · · , s′r ) ∈ Σrj′ s′i = s′ and there exist kℓ ∈ s′ℓ , 1 ≤ ℓ ≤ r P such that (−1)ℓ kℓ = 0 ℓ
61
Here, I am assuming, without loss of generality, that the even (respectively, odd) numbered ¯ legs of wr are hooked to ψ’s (respectively ψ’s). Then kgr k = max sup 1≤i≤r
xi ∈IR3 s′ ∈Σ ′ j
X
Momi (s′ )
Z
dxℓ gr (x1 , s′1 ), · · · , (xr , s′r )
Q
ℓ6=i
Fix any 1 ≤ i ≤ r, s′ ∈ Σ′j and xi ∈ IR3 . Then X
Momi (s′ )
≤
X
Z
Momi (s′ )
Q
ℓ6=i
dxℓ gr (x1 , s′1 ), · · · , (xr , s′r )
P
s1 ,···,sr sℓ ∩s′ 6=∅ ℓ
Z
Q
ℓ6=i
kχ ˆs′′ ∗ ϕˆj ′ kr dxℓ fr (x1 , s1 ), · · · , (xr , sr ) max ′′ s ∈Σj ′
By Proposition C.1, with j = j ′ and φ(j) = ϕˆj ′ , maxs′′ ∈Σj′ kχ ˆs′′ ∗ ϕˆj ′ kr is bounded by a
constant independent of M, j ′ and lj ′ . Observe that Z X Q P dxℓ fr (x1 , s1 ), · · · , (xr , sr ) Momi (s′ )
s1 ,···,sr sℓ ∩s′ 6=∅ ℓ
≤
ℓ6=i
X
X
s1 ,···,sr Mom (s′ ) i si ∩s′ 6=∅ s ∩s′ 6=∅ ℓ ℓ 1≤ℓ≤r
Z
Q
ℓ6=i
dxℓ fr (x1 , s1 ), · · · , (xr , sr )
I will not prove the fact that, for any fixed s1 , · · · , sr ∈ Σj , there are at most r−3 l elements of Momi (s′ ) obeying sℓ ∩ s′ℓ 6= ∅ for all 1 ≤ ℓ ≤ r, but I will try to CS′ l j′ j
motivate it below. As there are at most two sectors s ∈ Σj that intersect s′ , X X Z Q dxℓ fr (x1 , s1 ), · · · , (xr , sr ) s1 ,···,sr Mom (s′ ) i si ∩s′ 6=∅ s ∩s′ = 6 ∅ ℓ ℓ 1≤ℓ≤r
ℓ6=i
l r−3 sup ≤ 2 CS′ l j′ j
s∈Σj
l r−3 ≤ 2 CS′ l j′ kfr k
X
s1 ,···,sr si =s
Z
Q
ℓ6=i
dxℓ fr (x1 , s1 ), · · · , (xr , sr )
j
and
′ lj r−3 r ′′ ∗ ϕ ′k kfr k kwr kΣj′ ≤ kgr k ≤ 2 max k χ ˆ ˆ CS l ′ s j ′′ s ∈Σj ′
j
l r−3 k kwl,r kΣj + ǫ ≤ CS l j′ j
62
with CS = 2 maxs′′ ∈Σj′ kχ ˆs′′ ∗ ϕˆj ′ k4 CS′ . Now, I will try to motivate the fact that, for any fixed s1 , · · · sr ∈ Σj , there are l r−3 elements of Momi (s′ ) obeying sℓ ∩ s′ℓ 6= ∅ for all 1 ≤ ℓ ≤ r. We may at most CS′ l j′ j
assume that i = 1. Then s′1 must be s′ . Denote by Iℓ the interval on the Fermi curve F
that has length lj + 2lj ′ and is centered on sℓ ∩ F . If s′ ∈ Σj ′ intersects sℓ , then s′ ∩ F is contained in Iℓ . Every sector in Σj ′ contains an interval of F of length 34 lj ′ that does not intersect any other sector in Σj ′ . At most [ 34
lj +2lj ′ lj ′
] of these “hard core” intervals can be
l
contained in Iℓ . Thus there are at most [ 34 l j′ + 3]r−3 choices for s′2 , · · · , s′r−2 . j
Fix
s′1 , s′2 , · · · , s′r−2 .
Once
s′r−1
is chosen, s′r is essentially uniquely determined by
conservation of momentum. But the desired bound on Momi (s′ ) demands more. It says, roughly speaking, that both s′r−1 and s′r are essentially uniquely determined. As kℓ runs Pr−2 over s′ℓ for 1 ≤ ℓ ≤ r − 2, the sum ℓ=1 (−1)ℓ kℓ runs over a small set centered on some
point p. In order for (s′1 , · · · , s′r ) to be in Mom1 (s′ ), there must exist kr−1 ∈ s′r−1 ∩ F and
kr ∈ s′r ∩ F with kr − kr−1 very close to p. But kr − kr−1 is a secant joining two points of the Fermi curve F . We have assumed that F is convex. Consequently, for any given p 6= 0 in IR2 there exist at most two pairs (k′ , q′ ) ∈ F 2 with k′ − q′ = p. So, if p is not
near the origin, s′r−1 and s′r are almost uniquely determined. If p is close to zero, then Pr−2 ℓ ′ ′ ′ ℓ=1 (−1) kℓ must be close to zero and the number of allowed s1 , s2 , · · · , sr−2 is reduced.
Theorem II.15MB2 Suppose α ≥ 2 and M ≥
2 α α−1
2CS α+1 α
wr vanishes for r ≤ 4, then kΩj+1 (W )kα,j+1 ≤ kW kα,j . Proof:
≤
X r≥6
X r≥6
. If kW kα,j ≤
1 3
and
We first verify that kW (Ψ + ψ)kα+1,j+1 ≤ 31 .
kW (Ψ + ψ)kα+1,j+1 = kW k2(α+1),j+1 = ≤
12
2 α+1 α
r
2CS α+1 α
lj+1 lj
r
r−2 2
M−
M−
r−4 2
r−4 2
lj lj+1
l
j CS lj+1
r−4 2
X
2(α + 1)CF
r
r−3
r
(r−2)/2
lj+1
(r−2)/2
(αCF )r lj (r−2)/2
(αCF )r lj 63
M −j
r−4 2
M −j
M −(j+1)
r−4 2
kwr kΣj
r−4 2
kwr kΣj
kwr kΣj+1
=
X
1
4
− 4 (1− r ) 2CS α+1 α M
r≥6
≤ 2CS α+1 α
6
r
1 kW kα,j M 1/2
≤
(r−2)/2
(αCF )r lj
M −j
r−4 2
kwr kΣj
1 3
By Corollary II.7, kΩj+1 (W )kα,j+1 ≤
α α−1 kW (Ψ
+ ψ)kα,j+1 ≤
64
α α−1
2CS α+1 α
6
1 kW kα,j M 1/2
≤ kW kα,j
Appendix A: Infinite Dimensional Grassmann Algebras
To generalize the discussion of §I to the infinite dimensional case we need to
add topology. We start with a vector space V that is an ℓ1 space. This is not the only possibility. See, for example [Be]. Let I be any countable set. We now generate a Grassmann algebra from the vector space V = ℓ1 (I) = Equipping V with the norm kαk =
n
P o |αi | < ∞ α:I→C i∈I
P
i∈I
|αi | turns it into a Banach space. The algebra
will again be an ℓ1 space. The index set will be I, the (again countable) set of all finite subsets of I, including the empty set. The Grassmann algebra generated by V is 1
A(I) = ℓ (I) =
n
P o |αI | < ∞ α:I→C I∈I
Clearly A = A(I) is a Banach space with norm kαk =
It is also an algebra under the multiplication (αβ)I =
X J⊂I
P
I∈I
|αI |.
sgn(J, I\J) αJ βI\J
The sign is defined as follows. Fix any ordering of I and view every I ∋ I ⊂ I as being listed in that order. Then sgn(J, I\J) is the sign of the permutation that reorders (J, I\J) to I. The choice of ordering of I is arbitrary because the map {αI } 7→ {sgn(I)αI }, with sgn(I) being the sign of the permutation that reorders I according to the reordering of I, is an isometric isomorphism. The following bound shows that multiplication is everywhere defined and continuous. kαβk =
X I∈I
X X XX sgn(J, I\J) αJ βI\J ≤ |(αβ)I | = |αJ | |βI\J | ≤ kαk kβk (A.1) I∈I
J⊂I
I∈I J⊂I
Hence A(I) is a Banach algebra with identity 1lI = δI,∅ . In other words, 1l is the function on I that takes the value one on I = ∅ and the value zero on every I 6= ∅. 65
Define, for each i ∈ I, ai to be the element of A(I) that takes the value 1 on I = {i} and zero otherwise. Also define, for each I ∈ I, aI to be the element of A(I) that takes the value 1 on I and zero otherwise. Then aI =
Y
ai
i∈I
where the product is in the order of the ordering of I and α=
X
αI aI
I⊂I
If f : C → C is any function that is defined and analytic in a neighbourhood of 0, then the power series f (α) converges for all α ∈ A(I) with kαk strictly smaller than the radius of convergence of f since, by (A.1), ∞
X
kf (α)k =
n=0
1 (n) (0)αn n! f
≤
∞ X
n=0
(n) 1 (0)| kαkn n! |f
If f is entire, like the exponential of any polynomial, then f (α) is defined on all of A(I).
Problem A.1 Here is an easy generalization of the above construction. Let I be any countable set and I the set of all finite subsets of I (including the empty set). Let
w : I → IR>0 be any strictly positive function on I. Define V = ℓ1 (I, w) =
n
P o wi |αi | < ∞ α:I→C i∈I
The algebra will again be a weighted ℓ1 space. The weighting function is WI =
Y
wi
i∈I
By convention W∅ = 1. The Grassmann algebra generated by V is 1
A(I, w) = ℓ (I, W ) = The multiplication is (αβ)I = P WI |αI |. I∈I
P
J⊂I
n
P o α:I→C WI |αI | < ∞ I∈I
sgn(J, I \J) αJ βI\J again and the norm is kαk =
66
a) Show that kαβk ≤ kαk kβk b) Show that if f : C → C is any function that is defined and analytic in a neighbourhood of 0, then the power series f (α) converges for all α ∈ A with kαk smaller than the radius of convergence of f .
Problem A.2 Here are two more generalizations of the above construction. Let I be any countable set and I the set of all finite subsets of I. Let S=
α:I→C
be the set of all sequences indexed by I. Observe that our standard product (αβ)I = P P sgn(J, I\J) αJ βI\J is well–defined on S – for each I ∈ I, J⊂I is a finite sum. We now J⊂I define, for each integer n, a norm on (a subset of) S by kαkn =
P
I∈I
2n|I| |αI |
It is defined for all α ∈ S for which the series converges. Observe that this is precisely
the norm of Problem A.1 with wi = 2n for all i ∈ I. Also observe that, if m < n, then kαkm ≤ kαkn . Define
α ∈ S kαkn < ∞ for all n ∈ ZZ A− = α ∈ S kαkn < ∞ for some n ∈ ZZ A+ =
a) Prove that if α, β ∈ A+ then αβ ∈ A+ .
b) Prove that if α, β ∈ A− then αβ ∈ A− . c) Prove that if f (z) is an entire function and α ∈ A+ , then f (α) ∈ A+ . d) Prove that if f (z) is analytic at the origin and α ∈ A− has |α∅ | strictly smaller than the radius of convergence of f , then f (α) ∈ A− . Before moving on to integration, we look at some examples of Grassmann algebras generated by ordinary functions or distributions on IRd . In these algebras we can have 67
elements like ¯ =− P A(ψ, ψ)
σ∈S
−
λ 2
Z
P
dd+1 k (2π)d+1
σ,σ ′ ∈S
Z
4 Q
i=1
ik0 −
k2 2m
− µ ψ¯k,σ ψk,σ
dd+1 ki (2π)d+1 δ(k1 +k2 −k3 −k4 )ψ¯k1 ,σ ψk3 ,σ u ˆ(k1 (2π)d+1
− k3 )ψ¯k2 ,σ′ ψk4 ,σ′
from §I.5, that look like the they are in an algebra with uncountable dimension. But they really aren’t. Example A.1 (Functions and distributions on IRd /LZZd .) There is a natural way to express the space of smooth functions on the torus IRd /LZZd (i.e. smooth periodic functions) as a space of sequences: Fourier series. The same is true for the space of distributions on IRd /LZZd . By definition, a distribution on IRd /LZZd is a map d d IR /LZZ → C f :C ∞
h 7→< f, h >
that is linear in h and is continuous in the sense that there exist constants Cf ∈ IR, νf ∈ IN such that | < f, h > | ≤
sup x∈IRd /LZZd
Q d 1− Cf j=1
νf d2 h(x) dx2j
(A.2)
for all h ∈ C ∞ . Each L1 function f (x) is identified with the distribution < f, h >=
Z
dx f (x)h(x)
which has Cf = kf kL1 , νf = 0. The delta “function” supported at x is really the distribution < δx , h >= h(x) and has Cδx = 1, νδx = 0 . Let γ ∈ ZZ. We now define a Grassmann algebra Aγ (IRd /LZZd ), γ ∈ ZZ. It is the Grassmann algebra of Problem A.1 with index set I=
2π d Z L Z
68
and weight function wpγ =
d Q
1 + p2j
j=1
γ
To each distribution f , we associate the sequence
f˜p = f, e−i
indexed by I. In the event that f is given by integration against an L1 function, then Z ˜ fp = dx f (x)e−i ZZd and f (x) = is the usual Fourier coefficient. For example, let q ∈ 2π L 1 if p = q f˜p = 0 if p = 6 q and
X
p∈ 2π Zd L Z
1 i e . Ld
Then
wpγ f˜p = wqγ
is finite for all γ. Call this sequence (the sequence whose pth entry is one when p = q and zero otherwise) ψq . We have just shown that ψq ∈ Vγ (IRd /LZZd ) ⊂ Aγ (IRd /LZZd ) for all
γ ∈ ZZ. Define, for each x ∈ IRd /LZZd ,
ψ(x) =
X
e−i ψq
Zd q∈ 2π L Z
The pth entry in the sequence for ψ(x) is X
p∈ 2π Zd L Z
P
q
e−i δp,q = e−i . As
wpγ e−i =
X
wpγ
Zd p∈ 2π L Z
converges for all γ < − 12 , ψ(x) ∈ Vγ (IRd /LZZd ) ⊂ Aγ (IRd /LZZd ) for all γ < − 12 . By (A.2), for each distribution f , there are constants Cf and νf such that
|f˜p | = f, e−i ≤
Because
sup x∈IRd /LZZd
Q d 1− Cf
X
Zd p∈ 2π L Z
j=1
d Q
j=1
(1 + p2j )γ
69
νf −i d2 e 2 dxj
= Cf
d Q
j=1
(1 + p2j )νf
converges for all γ < −1/2 , the sequence {f˜p | p ∈
2π d Z L Z
} is in Vγ (IRd /LZZd ) =
ℓ1 (I, wγ ) ⊂ Aγ (IRd /LZZd ), for all γ < −νf − 1/2 . Thus, every distribution is in Vγ for
some γ (often negative). On the other hand, by Problem A.3 below, a distribution is given by integration against a C ∞ function if and only if f˜p decays faster than the inverse of any polynomial in p and then it (or rather f˜) is in Vγ for every γ.
Problem A.3 a) Let f (x) be a C ∞ function. Define f˜p =
Z
dx f (x)e−i
Prove that for every γ ∈ ZZ, there is a constant Cf,γ such that d Q f˜p ≤ Cf,γ (1 + p2j )−γ j=1
for all p ∈
2π d Z L Z
b) Let f be a distribution on IRd /LZZd . Suppose that, for each γ ∈ ZZ , there is a constant Cf,γ such that d −i f, e ≤ Cf,γ Q (1 + p2 )−γ j j=1
for all p ∈
2π d ZZ L
Prove that there is a C ∞ function F (x) on IRd /LZZd such that hf, hi =
Z
dx F (x)h(x)
Example A.2 (Functions and distributions on IRd ). Example A.1 exploited the Qd d2 basis of L2 IRd /LZZd given by the eigenfunctions of j=1 1 − dx . If instead we use 2 j Qd 2 the basis of L2 (IRd ) given by the eigenfunctions of j=1 x2j − ddx2 we get a very useful j
identification between tempered distributions and functions on the index set o n I = i = (i1 , · · · , id ) ij ∈ ZZ≥0 70
that we now briefly describe. For more details see [RS, Appendix to V.3]. Define the differential operators A=
√1 2
x+
d dx
and the Hermite functions
hℓ (x) =
A† =
(
1 2 1 e− 2 x π 1/4 √1 (A† )i h0 (x) ℓ!
√1 2
x−
d dx
ℓ=0 ℓ>0
for ℓ ∈ ZZ≥0 . Problem A.4 Prove that a) AA† f = A† Af + f b) Ah0 = 0 c) A† Ahℓ = ℓhℓ
R
d) hhℓ , h′ℓ i = 0. Here hf, gi = e) x2 −
d2 dx2
= 2A† A + 1
f¯(x)g(x)dx.
Define the multidimensional Hermite functions hi (x) =
d Y
hij (xj )
j=1
for i ∈ I. The latter form an orthonormal basis for L2 (IRd ) and obey d Q
j=1
x2j −
γ d2 hi 2 dxj
=
d Q
(1 + 2ij )γ hi
j=1
By definition, Schwartz space S(IRd ) is the set of C ∞ functions on IRd all of
whose derivatives decay faster than any polynomial. That is, a function f (x) is in S(IRd ),
if it is C ∞ and
Q d x2j − j=1
γ d2 f (x) 2 dxj
71
that is linear in h and is continuous in the sense that there exist constants C ∈ IR, γ ∈ IN such that
Q d | < f, h > | ≤ sup C x2j − x∈IRd
j=1
γ d2 h(x) 2 dxj
To each tempered distribution we associate the function f˘i =< f, hi >
on I. In the event that the distribution f is given by integration against an L2 function, that is, < f, h >=
Z
F (x)h(x) dd x
for some L2 function, F (x), then f˘i =
Z
F (x)hi (x) dd x
The distribution f is Schwartz class if and only if f˘i decays faster than any polynomial in i. For any distribution f˘i is bounded by some polynomial in i. So, if we define wiγ
=
d Y
(1 + 2ij )γ
j=1
then every Schwartz class function (or rather their f˘’s) are in Vγ = ℓ1 (I, wγ ) for every γ and every tempered distribution is in Vγ for some γ. The Grassmann algebra generated
by Vγ is called Aγ (IRd ). In particular, since the Hermite functions are continuous and
uniformly bounded [AS, p.787], every delta function δ(x − x0 ) and indeed every finite P measure has |f˘i | ≤ const . Thus, since i∈ZZ≥0 (1 + i)γ converges for all γ < −1, all finite measures are in Vγ for all γ < −1. The sequence representation of Schwartz class functions and of tempered distributions is discussed in more detail in [RS, Appendix to V.3].
Problem A.5 Show that the constant function f = 1 is in Vγ for all γ < −1. Hint: the
Fourier transform of hℓ is (−i)ℓ hℓ .
72
Infinite Dimensional Grassmann Integrals Once again, let I be any countable set, I the set of all finite subsets of I, V = ℓ1 (I)
and A = ℓ1 (I). We now define some linear functionals on A. Of course the infinite R dimensional analogue of · da1 · · · daD will not make sense. But we can define the R analogue of the Grassmann Gaussian integral, · dµS , at least for suitable S. Let S : I × I → C be an infinite matrix. Recall that, for each i ∈ I, ai is the
element of A that takes the value 1 on I = {i} and zero otherwise, and, for each I ∈ I, aI is the element of A that takes the value 1 on I and zero otherwise. By analogy with the finite dimensional case, we wish to define a linear functional on A by Z
p Q
j=1
aij dµS = Pf Sij ,ik 1≤j,k≤p
We now find conditions under which this defines a bounded linear functional. Any A ∈ A can be written in the form A=
∞ X X p=0
αI aI
I⊂I |I|=p
We want the integral of this A to be Z
A dµS =
αI
∞ X X
αI Pf SIj ,Ik 1≤j,k≤p
p=0
=
Z
∞ X X
p=0
I⊂I |I|=p
I⊂I |I|=p
aI dµS (A.3)
where I = {I1 , · · · , Ip } with I1 < · · · < Ip in the ordering of I. We now hypothesise that there is a number CS such that sup Pf SIj ,Ik 1≤j,k≤p ≤ CSp
(A.4)
I⊂I |I|=p
Such a bound is proven in the following simple Lemma. The bound proven in this Lemma is not tight enough to be of much use in quantum field theory applications. A much better bound for those applications is proven in Corollary I.28. 73
Lemma A.3 Let S be a bounded linear operator on ℓ2 (I). Then sup Pf SIj ,Ik 1≤j,k≤p ≤ kSkp/2
I⊂I |I|=p
Proof:
Recall that Hadamard’s inequality (Problem I.19) bounds a determinant by the product of the lengths of its columns. The k th column of the matrix SIj ,Ik 1≤j,k≤p has
length
v uX sX u p t |Sj,Ik |2 |SIj ,Ik |2 ≤ j=1
j∈I
But the vector Sj,Ik j∈I is precisely the image under S of the Ith k standard unit basis vector, δj,Ik j∈I , in ℓ2 (I) and hence has length at most kSk. Hence
det SIj ,Ik
1≤j,k≤p
≤
p s X Y
k=1
j∈I
|Sj,Ik |2 ≤ kSkp
By Proposition I.18.d, 1/2 ≤ kSkp/2 Pf SIj ,Ik 1≤j,k≤p ≤ det SIj ,Ik 1≤j,k≤p
Under the hypothesis (A.4), the integral (A.3) is bounded by Z ∞ X X AdµC ≤ |αI | CSp ≤ kAk p=0
(A.5)
I∈I |I|=p
provided CS ≤ 1. Thus, if CS ≤ 1, the map A ∈ A 7→ with norm at most one.
R
A dµS is a bounded linear map
The requirement that CS be less than one is not crucial. Define, for each n ∈ ZZ, the Grassmann algebra An =
P n|I| 2 |αI | < ∞ α : I → C kαkn = I∈I
74
R If 2n ≥ CS , then (A.5) shows that A ∈ A 7→ A dµS is a bounded linear functional on An . R Alternatively, we can view · dµS as an unbounded linear functional on A with domain
of definition the subalgebra An of A. Observe that ◦ An is dense in A ◦ V ⊂ An
◦ if A ∈ An and f is an entire function (or even a function whose radius of convergence about the origin is strictly larger than kAkn ), then f (A) ∈ An .
75
Appendix B: Pfaffians
Definition B.1
Let S = (sij ) be a complex, skew symmetric matrix of even order
n = 2m . By definition, the Pfaffian Pf(S) of S is Pf(S) =
1 2m m!
n X
i1 ,···,in =1
where εi1 ···in =
(
εi1 ···in si1 i2 · · · sin−1 in
(B.1)
1 if i1 , · · · , in is an even permutation of 1, · · · , n −1 if i1 , · · · , in is an odd permutation of 1, · · · , n 0 if i1 , · · · , in are not all distinct
By convention, the Pfaffian of a skew symmetric matrix of odd order is zero.
Example B.2 For all s12 , s21 ∈ C with s21 = −s12 , P 0 s12 εkℓ skℓ = Pf = 12 s21 0 1≤k,ℓ≤2
1 2
s12 − s21
= s12
Let Pm =
n
o {(k1 , ℓ1 ), · · · , (km , ℓm )} {k1 , ℓ1 , · · · , km , ℓm } = {1, · · · , 2m}
be the set of of all partitions of {1, · · · , 2m} into m disjoint ordered pairs. Observe that, the expression εk1 ℓ1 ···km ℓm sk1 ℓ1 · · · skm ℓm is invariant under permutations of the pairs in
the partition {(k1 , ℓ1 ), · · · , (km , ℓm )} , since εkπ(1) ℓπ(1) ···kπ(m) ℓπ(m) = εk1 ℓ1 ···km ℓm for any permutation π . Thus, Pf(S) =
1 2m
X
P∈Pm
Let < Pm =
n
εk1 ℓ1 ···km ℓm sk1 ℓ1 · · · skm ℓm
o {(k1 , ℓ1 ), · · · , (km , ℓm )} ∈ Pm ki < ℓi for all 1 ≤ i ≤ m 76
(B.1′ )
Because S is skew symmetric εk1 ℓ1 ···ki ℓi ···km ℓm ski ℓi = εk1 ℓ1 ···ℓi ki ···km ℓm sℓi ki so that Pf(S) =
X
< P∈Pm
εk1 ℓ1 ···km ℓm sk1 ℓ1 · · · skm ℓm
(B.1′′ )
Problem B.1 Let α1 , · · · , αr be complex numbers and let S be the 2r × 2r antisymmetric matrix
r M 0 S= −αm m=1
αm 0
All matrix elements of S are zero, except for r 2 × 2 blocks running down the diagonal. For example, if r = 2,
0 −α1 S= 0 0
α1 0 0 0
0 0 0 −α2
Prove that Pf(S) = α1 α2 · · · αr .
0 0 α2 0
Proposition B.3 Let S = (sij ) be a skew symmetric matrix of even order n = 2m . a) Let π be a permutation and set S π = (sπ(i)π(j) ) . Then, Pf(S π ) = sgn(π) Pf(S) b) For all 1 ≤ k 6= ℓ ≤ n , let Skℓ be the matrix obtained from S by deleting rows k and ℓ and columns k and ℓ . Then, Pf(S) =
n P
ℓ=1
sgn(k − ℓ) (−1)k+ℓ skℓ Pf (Skℓ )
In particular, Pf(S) =
n P
ℓ=2
77
(−1)ℓ s1ℓ Pf (S1ℓ )
Proof:
To verify (a), observe that Pf(S π ) =
X
1
2m m!
i1 ,···,in
=
X
1 2m m!
εi1 ···in sπ(i1 )π(i2 ) · · · sπ(in−1 )π(in ) επ
−1
(j1 )···π −1 (jn )
j1 ,···,jn
sj1 j2 · · · sjn−1 jn
= sgn(π −1 ) Pf(S) To verify (b), fix 1 ≤ k ≤ n . Let P = {(k1 , ℓ1 ), · · · , (km , ℓm )} be a partition in Pm . We may assume, by reindexing the pairs, that k = k1 or k = ℓ1 . Then, Pf (S) =
1 2m
X
P∈Pm k1 =k
εkℓ1 ···km ℓm skℓ1 · · · skm ℓm +
1 2m
X
P∈Pm ℓ1 =k
εk1 k···km ℓm sk1 k · · · skm ℓm
By antisymmetry and a change of summation variable, X
P∈Pm ℓ1 =k
εk1 k···km ℓm sk1 k · · · skm ℓm =
Thus, Pf(S) =
2 2m
X
P∈Pm k1 =k
=
2 2m
n P
ℓ=1
X
P∈Pm k1 =k
εkℓ1 ···km ℓm skℓ1 · · · skm ℓm
εkℓ1 ···km ℓm skℓ1 · · · skm ℓm X
P∈Pm k1 =k ℓ1 =ℓ
εk ℓ···km ℓm skℓ · · · skm ℓm
Extracting skℓ from the inner sum, Pf(S) =
k−1 P
skℓ
ℓ=1
+
n P
ℓ=k+1
2 2m
X
P∈Pm k1 =k ℓ1 =ℓ
skℓ
2 2m
εk ℓ k2 ℓ2 ···km ℓm sk2 ℓ2 · · · skm ℓm
X
P∈Pm k1 =k ℓ1 =ℓ
εk ℓ k2 ℓ2 ···km ℓm sk2 ℓ2 · · · skm ℓm
The following Lemma implies that, Pf(S) =
k−1 P
skℓ (−1)k+ℓ Pf (Skℓ ) +
ℓ=1
n P
ℓ=k+1
78
skℓ (−1)k+ℓ+1 Pf (Skℓ )
Lemma B.4 For all 1 ≤ k 6= ℓ ≤ n , let Skℓ be the matrix obtained from S by deleting rows k and ℓ and columns k and ℓ . Then,
Proof:
X
(−1)k+ℓ 2m−1
Pf (Skℓ ) = sgn(k − ℓ)
εk ℓ k2 ℓ2 ···km ℓm sk2 ℓ2 · · · skm ℓm
P∈Pm k1 =k ℓ1 =ℓ
For all 1 ≤ k < ℓ ≤ n , Skℓ =
where, for each 1 ≤ i ≤ n − 2 , i′ =
si′ j ′ ; 1 ≤ i, j ≤ n − 2
(
i if 1 ≤ i ≤ k − 1 i + 1 if k ≤ i ≤ ℓ − 1 i + 2 if ℓ ≤ i ≤ n − 2
By definition, Pf (Skℓ ) =
X
1
2m−1 (m−1)!
1≤i1 ,···,in−2 ≤n−2
εi1 ···in−2 si′1 i′2 · · · si′n−3 i′n−2
We have ′
′
εi1 ···in−2 = (−1)k+ℓ−1 εkℓi1 ···in−2 for all 1 ≤ i1 , · · · , in−2 ≤ n − 2 . It follows that X k+ℓ−1 Pf (Skℓ ) = 2(−1) m−1 (m−1)! = =
k+ℓ−1
1≤i1 ,···,in−2 ≤n−2
(−1) 2m−1 (m−1)! k+ℓ−1
(−1) 2m−1 (m−1)!
X
′
1≤k2 ,ℓ2 ···,km ,ℓm ≤n
X
P∈Pm k1 =k ℓ1 =ℓ
′
εkℓi1 ···in−2 si′1 i′2 · · · si′n−3 i′n−2 εk ℓ k2 ℓ2 ···km ℓm sk2 ℓ2 · · · skm ℓm
εk ℓ k2 ℓ2 ···km ℓm sk2 ℓ2 · · · skm ℓm
If, on the other hand, k > ℓ , Pf (Skℓ ) = Pf (Sℓk ) =
X
(−1)k+ℓ+1 2m−1
P∈Pm k1 =ℓ ℓ1 =k
=
(−1)k+ℓ 2m−1
X
P∈Pm k1 =k ℓ1 =ℓ
79
εℓ k k2 ℓ2 ···km ℓm sk2 ℓ2 · · · skm ℓm
εk ℓ k2 ℓ2 ···km ℓm sk2 ℓ2 · · · skm ℓm
Proposition B.5 Let S =
0 −C t
C 0
where C = (cij ) is a complex m × m matrix. Then, 1
Pf(S) = (−1) 2 m(m−1) det(C)
Proof:
The proof is by induction on m ≥ 1 . If m = 1 , then, by Example B.2, 0 s12 Pf = s12 s21 0
Suppose m > 1 . The matrix elements sij , i, j = 1, · · · , n = 2m , of S are 0 if 1 ≤ i, j ≤ m ci j−m if 1 ≤ i ≤ m and m + 1 ≤ j ≤ n sij = −cj i−m if m + 1 ≤ i ≤ n and 1 ≤ j ≤ m 0 if m + 1 ≤ i, j ≤ n For each k = 1, · · · , m , we have
S1k+m =
0 t −C 1,k
C 1,k 0
where C 1,k is the matrix of order m − 1 obtained from the matrix C by deleting row 1 and column k . It now follows from Proposition B.3 b) and our induction hypothesis that n P
Pf(S) = = = =
ℓ=m+1 n P
(−1)ℓ s1ℓ Pf (S1ℓ ) (−1)ℓ c1 ℓ−m Pf (S1ℓ )
ℓ=m+1 m P
(−1)k+m c1 k Pf (S1k+m )
k=1 m P
k=1
1 (−1)k+m c1 k (−1) 2 (m−1)(m−2) det C 1,k
Collecting factors of minus one
1
Pf(S) = (−1)m+1 (−1) 2 (m−1)(m−2)
m P
(−1)k+1 c1 k det C 1,k
k=1
= (−1)
1 2 m(m−1)
m P
k=1
= (−1)
1 2 m(m−1)
(−1)k+1 c1 k det C 1,k
det(C)
80
Proposition B.6 Let S = (si,j ) be a skew symmetric matrix of even order n = 2m . a) For any matrix B of even order n = 2m , Pf B t S B = det(B) Pf(S)
b) Pf(S)2 = det(S)
Proof: a) The matrix element of B t S B with indices i1 , i2 is Pf B t S B
=
1 2m m!
X
εi1 ···in
i1 ,···,in
=
1 2m m!
X
j1 ,···,jn
X
j1 ,···,jn i1 ,···,in
The expression
X
i1 ,···,in
X
P
j1 ,j2
bj1 i1 sj1 j2 bj2 i2 . Hence
bj1 i1 · · · bjn in sj1 j2 · · · sjn−1 jn
εi1 ···in bj1 i1 · · · bjn in sj1 j2 · · · sjn−1 jn
εi1 ···in bj1 i1 · · · bjn in
is the determinant of the matrix whose ℓth row is the jℓth row of B. If any two of j1 , · · · , jn are equal, this determinant is zero. Otherwise it is εj1 ···jn det(B). Thus X εi1 ···in bj1 i1 · · · bjn in = εj1 ···jn det(B) i1 ,···,in
and
Pf B t S B
= det(B)
1
2m m!
X
j1 ,···,jn
εj1 ···jn sj1 j2 · · · sjn−1 jn = det(B) Pf(S)
b) It is enough to prove the identity for real matrices, since Pf(S)2 and det(S) are polynomials in the matrix elements sij , i, j = 1, · · · , n , of S . So, let S be real and, as in Lemma I.10 (or Problem I.5), let R be an orthogonal matrix such that Rt SR = T with m M 0 αi T = −αi 0 i=1
for some real numbers α1 , · · · , αm . Then, by part a and Problem B.1, Pf(S) = det(R) Pf(S) = Pf Rt SR = Pf(T ) = α1 · · · αm and
det(S) = det Rt SR
= α21 · · · α2m = Pf(S)2
81
Appendix C: Propagator Bounds
The propagator, or covariance, for many–fermion models is the Fourier transform of Cσ,σ′ (k) =
δσ,σ′ ik0 − e(k)
where k = (k0 , k) and e(k) is the one particle dispersion relation minus the chemical potential. For this appendix, the spins σ, σ ′ play no role, so we suppress them completely. We also restrict our attention to two space dimensions (i.e. k ∈ IR2 , k ∈ IR3 ) though it is trivial to extend the results of this appendix to any number of space dimensions. We assume that e(k) is a reasonably smooth function (for example, C 4 ) that has a nonempty, compact zero set F , called the Fermi curve. We further assume that ∇e(k) does not vanish for k ∈ F , so that F is itself a reasonably smooth curve. At low temperatures only those momenta with k0 ≈ 0 and k near F are important, so we replace the above propagator with C(k) =
U (k) ik0 − e(k)
The precise ultraviolet cutoff, U (k), shall be chosen shortly. It is a C0∞ function which takes values in [0, 1], is identically 1 for k02 + e(k)2 ≤ 1 and vanishes for k02 + e(k)2 larger than some constant. We slice momentum space into shells around the Fermi surface. To do this, we fix M > 1 and choose a function ν ∈ C0∞ ([M −2 , M 2 ]) that takes values in [0, 1], is identically
1 on [M −1/2 , M 1/2 ] and obeys
∞ X j=0
ν M 2j x = 1
for 0 < x < 1. The j th shell is defined to be the support of ν (j) (k) = ν M 2j k02 + e(k)2 By construction, the j th shell is a subset of
k
1
M j+1
≤ |ik0 − e(k)| ≤ 82
1
M j−1
As the scale parameter M > 1, the shells near the Fermi curve have j near +∞. Setting C (j) (k) = C(k)ν (j) (k) and U (k) =
∞ P
ν (j) (k) we have
j=0
C(k) =
∞ X
C (j) (k)
j=0
To analyze the Fourier transform of C (j) (k), we further decompose the j th shell into more or less rectangular “sectors”. To do so, we fix lj ∈ M1 j , M1j/2 and choose a
partition of unity
1=
X
χs(j) (k′ )
s∈Σ(j) (j)
of the Fermi curve F with each χs
supported on an interval of length lj and obeying
sup ∂km′ χs(j) ≤ k′
constm lj m
Given any function χ(k′ ) on the Fermi curve F , we define Cχ(j) (k) = C (j) (k)χ(k′ (k)) where, for k = (k0 , k), k′ (k) is any reasonable “projection” of k onto the Fermi curve. In the event that F is a circle of radius kF centered on the origin, it is natural to choose
k′ (k) =
kF |k| k.
For general F , one can always construct, in a tubular neighbourhood of F ,
a C ∞ vector field that is transverse to F , and then define k′ (k) to be the unique point of F that is on the same integral curve of the vector field as k is. Proposition C.1 Let χ(k′ ) be a C0∞ function on the Fermi curve F which takes values in [0, 1], which is supported on an interval of length lj ∈ M1j , M1j/2 and whose derivatives
obey
sup ∂kn′ χ(k′ ) ≤ k′
83
constn ln j
ˆ be unit tangent and normal vectors to Fix any point k′c in the support of χ. Let ˆt and n the Fermi curve at k′c and set ρ(x, y) = 1 + M −j |x0 − y0 | + M −j |x⊥ − y⊥ | + lj |xk − yk | ˆ. where xk is the component of x parallel to ˆt and x⊥ is the component parallel to n Let φ be a C0∞ function which takes values in [0, 1] and set φ(j) = φ M 2j [k02 +
e(k)2 ] . For any function W (k) define (j) Wχ,φ (x, y)
=
Z
d3 k ik·(x−y) W (k)φ(j) (k) χ (2π)3 e
k′ (k)
Let γ ∈ IN. If e(k) has bounded max{2, γ}th derivatives, then there is a constant, const , depending on γ, const0 , · · · , constγ , φ and e(k), but independent of j, x and y such that (j)
l
|Wχ,φ (x, y)| ≤ const Mj2j ρ(x, y)−γ max3 α∈IN |α|≤γ
α1 α2 α0 ˆ ˆ ∂ n ·∇ t·∇ W (k) k k M j(α0 +α1 ) k0 α
sup k∈supp χφ(j)
lj 2
ˆ n O(1/M j ) k′c
ˆt
O(lj )
Proof:
Use S to denote the support of φ(j) (k)χ (k′ (k) . Observe that S has volume at
most const M −2j lj , since k0 is supported in an interval of length const M −j , the component
of k normal to F is supported in an interval of length const M −j and the component of k tangential to F runs over an interval of length const lj . Hence (j)
l
sup |Wχ,φ (x, y)| ≤ vol(S) sup |W (k)| ≤ const Mj2j sup |W (k)| x,y
k∈S
k∈S
which is the desired bound for γ = 0. 84
(j)
To bound supx,y ρ(x, y)γ |Wχ,φ (x, y)| by 4γ C, it suffices to bound x0 −y0 β0 x⊥ −y⊥ β1 j j M M Z =
lj (xk − yk )
β2
(j) Wχ,φ (x, y)
β0 1 d3 k ik·(x−y) 1 ˆ (2π)3 e M j ∂k0 Mj n
· ∇k
β1
lj ˆt · ∇k
β2 W (k)φ(j) (k) χ k′ (k)
by C for all x, y ∈ IR3 and β ∈ IN3 with |β| ≤ γ. The volume of the domain of integration l
is still bounded by const Mj2j , so by the product rule, to prove the desired bound it suffices to prove that max sup
|β|≤γ
k
β0 1 1 ˆ ∂ n M j k0 Mj
Since lj ≥
1 Mj
max sup
β0 1 β1 1 ˆ ∂ n ·∇ j k j k 0 M M
|β|≤γ
k
· ∇k
β1
lj ˆt · ∇k
and all derivatives of k′ (k) to order γ are bounded, lj ˆt·∇k
β2
so, by the product rule, it suffices to prove max sup
|β|≤γ k∈S
β0 1 1 ˆ ∂ n j k 0 M Mj
χ k′ (k) ≤ const max
1 M j ∂k0 ˆ · ∇k di = M1 j n lj ˆt · ∇k
′
and, for each I ′ ⊂ I, dI = d φ
(k) =
|β| X
Q
i∈I ′
X
β
lj 2 1 jβ β +β β1 +β2 ≤γ M 1 lj 1 2
· ∇k
Set I = {1, · · · , |β|},
I (j)
β2 (j) φ (k) χ k′ (k) ≤ const
β1
lj ˆt · ∇k
β2
(j)
φ
≤ const
(k) ≤ const
if 1 ≤ i ≤ β0
if β0 + 1 ≤ i ≤ β0 + β1
if β0 + β1 + 1 ≤ i ≤ |β|
di . By the product and chain rules dm φ dxm
M 2j k02 + e(k)2
m=1 (I1 ,···,Im )∈Pm
m Q
i=1
M 2j dIi k02 + e(k)2
where Pm is the set of all partitions of I into m nonempty subsets I1 , · · · , Im with, for
all i < i′ , the smallest element of Ii smaller than the smallest element of Ii′ . For all m m, ddxmφ M 2j k02 + e(k)2 is bounded by a constant independent of j, so to prove the Proposition, it suffices to prove that max sup M 2j
|β|≤γ k∈S
β0 1 1 ˆ ∂ n j k 0 M Mj
· ∇k
β1
85
lj ˆt · ∇k
β2
k02
+ e(k) ≤ const 2
If β0 6= 0
(
2k0 M j if β0 = 1, β1 = β2 = 0 lj ˆt · ∇k + e(k) = 2 M if β0 = 2, β1 = β2 = 0 0 otherwise is bounded, independent of j since |k0 | ≤ const M1 j on S. Thus it suffices to consider 2j
1 M j ∂k0
β0
1 ˆ M j n · ∇k
β1
β2
2
k02
β0 = 0. Applying the product rule once again, this time to the derivatives acting on M 2j e(k)2 = [M j e(k)] [M j e(k)], it suffices to prove β1 β2 j 1 ˆ ˆ · ∇k lj t · ∇k e(k) ≤ const max sup M M j n |β|≤γ k∈S
If β1 = β2 = 0, this follows from the fact that |e(k)| ≤ const M1j on S. If β1 ≥ 1 or β2 ≥ 2, it follows from
Mj
β M β1 j lj 2
1 .) M j/2
≤ 1. (Recall that lj ≥
This leaves only β1 = 0, β2 = 1.
ˆ. If ˆt · ∇k e(k) is evaluated at k = k′c , it vanishes, since ∇k e(k′c ) is parallel to n The second derivative of e is bounded so that, M j lj sup ˆt · ∇k e(k) ≤ const M j lj sup |k − k′c | ≤ const M j l2j ≤ const k∈S
k∈S
since lj ≤
1 . M j/2
Corollary C.2 Under the hypotheses of Proposition C.1, (j) l sup Wχ,φ (x, y) ≤ const Mj2j sup
k∈supp χφ(j)
|W (k)|
and, if e(k) has bounded fourth derivatives, Z Z (j) (j) sup dy Wχ,φ (x, y) , sup dx Wχ,φ (x, y) y
x
α1 α2 α0 ˆ ˆ · ∇k t · ∇k W (k) ∂ n M j(α0 +α1 ) k0 α
≤ const max3 α∈IN |α|≤4
Proof:
sup
k∈supp χφ(j)
lj 2
The first claim is simply a restatement of Proposition C.1 with γ = 0, For the
second statement just use Z Z Z 2j 1 1 1 1 sup dy ρ(x,y)4 , sup dx ρ(x,y)4 = dx ρ(x,0) 4 ≤ const M lj x
y
For the last inequality, just make the change of variables x0 = M j z0 , x⊥ = M j z1 , xk = 1 lj z2 .
86
Corollary C.3 Under the hypotheses of Proposition C.1, sup Cχ(j) (x, y) ≤ const
lj Mj
and, if e(k) has bounded fourth derivatives, sup x
Proof:
Z
dy Cχ(j) (x, y) , sup y
Z
Apply Corollary C.2 with W (k) =
dx Cχ(j) (x, y) ≤ const M j 1 ik0 −e(k)
and φ = ν. To achieve the desired
bounds, we need max
sup
|α|≤4 k∈supp χν (j)
α0 1 1 ˆ ∂ n j k 0 M Mj
· ∇k
α1
lj ˆt · ∇k
α2
1 ik0 −e(k)
≤ const M j
In the notation of the proof of Proposition C.1, with β replaced by α, I (j)
d ν
(k) = M
j
|α| X
(−1)m m!
m=1
X
(I1 ,···,Im )∈Pm
1/M j ik0 −e(k)
m+1 Q m
On the support of χν (j) , |ik0 − e(k)| ≥ const M1j so that
i=1
M j dIi (ik0 − e(k))
1/M j ik0 −e(k)
m+1
is bounded uni-
formly in j. That M j dIi (ik0 − e(k)) is bounded uniformly in j was proven during the
course of the proof of Proposition C.1.
87
References
[AS] M. Abramowitz and I. A. Stegun eds., Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover, 1964. [Be] F. A. Berezin, The method of second quantization, Academic Press, 1966. [BS] F. Berezin, M. Shubin, The Schr¨ odinger Equation, Kluwer 1991, Supplement 3: D. Le˘ites, Quantization and supermanifolds. [RS] M. Reed and B. Simon, Methods of Modern Mathematical Physics, I: Functional Analysis, Academic Press, 1972.
The “abstract” fermionic expansion discussed in §II is similar to that in • J. Feldman, H. Kn¨orrer, E. Trubowitz, A Nonperturbative Representation for Fermionic Correlation Functions, Communications in Mathematical Physics, 195, 465-493 (1998). and is a simplified version of the expansion in • J. Feldman, J. Magnen, V. Rivasseau and E.Trubowitz, An Infinite Volume Expansion for Many Fermion Green’s Functions, Helvetica Physica Acta, 65 (1992) 679-721. The latter also contains a discussion of sectors. Both are available over the web at http://www.math.ubc.ca/e feldman/research.html
88