Fermionic Functional Integrals and the Renormalization Group

150 Pages·2002·0.742 MB·English

by  Horst Knorrer, and Eugene Trubowitz Joel Feldman

Fermionic Functional Integrals and
the Renormalization Group
Joel Feldman
Department of Mathematics
University of British Columbia
Vancouver B.C.
CANADA V6T 1Z2
[e-mail]
http://www.math.ubc.ca/ feldman
∼
Horst Kno¨rrer, Eugene Trubowitz
Mathematik
ETH–Zentrum
CH-8092 Zu¨rich
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 (usuallyof dimensionthree orless)isimportant. These notes concern atechnique
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
Preface p i
I Fermionic Functional Integrals p 1
§
I.1 Grassmann Algebras p 1
§
I.2 Grassmann Integrals p 7
§
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 The Renormalization Group p 24
§
I.7 Wick Ordering p 30
§
I.8 Bounds on Grassmann Gaussian Integrals p 34
§
II Fermionic Expansions p 40
§
II.1 Notation and Definitions p 40
§
II.2 Expansion – Algebra p 42
§
II.3 Expansion – Bounds p 44
§
II.4 Sample Applications p 56
§
Gross–Neveu p 57
2
Naive Many-fermion p 63
2
Many-fermion – with sectorization p 67
2
Appendices
A Infinite Dimensional Grassmann Algebras p 75
§
B Pfaffians p 86
§
C Propagator Bounds p 93
§
D Problem Solutions p 100
§
References p 147
Preface
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
justifyingthisapproximationinabroadclassoffermionicmodelsused incondensed matter
and high energy physics.
The first chapter provides the necessary mathematical background. Most of it is
easy algebra – primarily the definition of Grassmann algebra and the definition and basic
properties of a family of linear functionals on Grassmann algebras known as Grassmann
Gaussian integrals. To make I really trivial, we consider only finite dimensional Grass-
§
mann algebras. A simple–minded method for handling the infinite dimensional case is
presented in Appendix A. There is also one piece of analysis in I – the Gram bound on
§
Grassmann Gaussian integrals – and a brief discussion of how Grassmann integrals arise
in quantum field theories.
The second chapter introduces an expansion that can be used to establish ana-
lytic control over the Grassmann integrals used in fermionic quantum field theory models,
when the covariance (propagator) is “really nice”. It is also used as one ingredient in a
renormalization group procedure that controls the Grassmann integrals when the covari-
ance is not so nice. To illustrate the latter, we look at the Gross–Neveu model and at
2
many-fermion models in two space dimensions.
i
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 prop-
erties of aclass of linear functionals on Grassmann algebras known asGrassmann 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 the-
ories. 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 be a finite di-
V
mensional vector space over C. The Grassmann algebra generated by is
V
∞
n
=
V V
n=0
(cid:86) (cid:77) (cid:86)
0 n
where = C and is the n–fold antisymmetric tensor product of with itself.
V V V
Thus, i(cid:86)f a1,...,aD is(cid:86)a basis for , then is a vector space with elements of the form
{ } V V
(cid:86)
D
f(a) = β a a
i1,···,in i1 ··· in
n(cid:88)=0 1≤i1<(cid:88)···<in≤D
with the coefficients β C. (In differential geometry, the traditional notation uses
i1,···,in ∈
a a in place of a a .) Addition and scalar multiplication is done compo-
i1 ∧ ···∧ in i1 ··· in
nentwise. That is,
D D
α β a a +γ δ a a
i1,···,in i1 ··· in i1,···,in i1 ··· in
(cid:16)n(cid:88)=0 1≤i1<(cid:88)···<in≤D (cid:17) (cid:16)n(cid:88)=0 1≤i1<(cid:88)···<in≤D (cid:17)
D
= αβ +γδ a a
i1,···,in i1,···,in i1 ··· in
n(cid:88)=0 1≤i1<(cid:88)···<in≤D(cid:0) (cid:1)
1
The multiplication in is determined by distributivity and
V
(cid:86)
0 i , ,i j , ,j =
1 m 1 n
(a a )(a a ) = { ··· }∩{ ··· } (cid:54) ∅
i1 ··· im j1 ··· jn (cid:26)sgn(k) ak1 ···akm+n otherwise
where (k , ,k ) is the permutation of (i , ,i ,j ,j ) with k < k < <
1 m+n 1 m 1 n 1 2
··· ··· ··· ···
k and sgn(k) is the sign of that permutation. In particular
m+n
a a = a a
i j j i
−
and a a = 0 for all i. For example
i i
a +3a a +2a a = a a +3a a +2a a a +6a a a
2 3 1 1 2 2 1 3 1 2 1 2 3 1 2
(cid:2) (cid:3)(cid:2) (cid:3) = a a 3a a 2a a a 6a a a
1 2 1 3 1 2 2 1 3 2
− − − −
= a a 3a a +6a a a
1 2 1 3 1 2 3
− −
Remark I.2 If a ,...,a is a basis for , then has basis
1 D
{ } V V
(cid:86)
a a n 0, 1 i < < i D
i1 ··· in ≥ ≤ 1 ··· n ≤
(cid:8) (cid:12) (cid:9)
(cid:12)
The dimension of n is D and the dimension of is D D = 2D. Let
V n V n=0 n
(cid:86) (cid:0) (cid:1) (cid:86) (cid:80) (cid:0) (cid:1)
= (i , ,i ) 1 i , ,i D
n 1 n 1 n
M ··· ≤ ··· ≤
(cid:8) (cid:12) (cid:9)
(cid:12)
be the set of all multi–indices of degree n 0. Note that i ,i does not have to be in
1 2
≥ ···
increasing order. For each I set a = a a . Also set a = 1. Every element
∈ Mn I i1 ··· in ∅
f(a) has a unique representation
∈ V
(cid:86)
D
f(a) = β a
I I
n(cid:88)=0 I∈(cid:88)Mn
with the coefficients β C antisymmetric under permutations of the elements of I. For
I ∈
example, a a = 1 a a a a = β a +β a with β = β = 1.
1 2 2 1 2 − 2 1 (1,2) (1,2) (2,1) (2,1) (1,2) − (2,1) 2
(cid:2) (cid:3)
2
Problem I.1 Let be a complex vector space of dimension D. Let s . Then s
V ∈ V
D n
has a unique decomposition s = s0 +s1 with s0 ∈ C and s1 ∈ n=1 V. P(cid:86)rove that, if
s0 = 0, then there is a unique s(cid:48) with ss(cid:48) = 1 and a uni(cid:76)que s(cid:48)(cid:48) (cid:86) with s(cid:48)(cid:48)s = 1
(cid:54) ∈ V ∈ V
and furthermore (cid:86) (cid:86)
D
s(cid:48) = s(cid:48)(cid:48) = s10 + (−1)nsnsn1+1
0
n=1
(cid:88)
It will be convenient to generalise the concept of Grassmann algebra to allow for
coefficients in more general algebras than C. We shall allow the space of coefficients to be
any superalgebra [BS]. Here are the definitions that do it.
Definition I.3 (Superalgebra)
(i) A superalgebra is an associative algebra S with unit, denoted 1, together with a de-
composition S = S S such that 1 S ,
+ +
⊕ − ∈
S S S S S S
+ + + +
· ⊂ − · − ⊂
S S S S S S
+ +
· − ⊂ − − · ⊂ −
and
ss = s s if s S or s S
(cid:48) (cid:48) + (cid:48) +
∈ ∈
ss = s s if s,s S
(cid:48) (cid:48) (cid:48)
− ∈ −
The elements of S are called even, the elements of S odd.
+
−
(ii) A graded superalgebra is an associative algebra S with unit, together with a decompo-
sition S = ∞m=0Sm such that 1 ∈ S0, Sm ·Sn ⊂ Sm+n for all m,n ≥ 0, and such that the
decomposit(cid:76)ion S = S+ S with
⊕ −
S = S S = S
+ m m
−
m even m odd
(cid:77) (cid:77)
gives S the structure of a superalgebra.
Example I.4 Let be a complex vector space. The Grassmann algebra S = =
V V
m m m
over isagradedsuperalgebra. Inthiscase,Sm = ,S+ = (cid:86) and
V V V V
m 0 m even
S(cid:76)≥=(cid:86) m . In these notes, all of the superalgebras tha(cid:86)t we use will (cid:76)be G(cid:86)rassmann
− V
m odd
algebras(cid:76). (cid:86)
3
Definition I.5 (Tensor product of two superalgebras)
(i) Recall that the tensor product of two finite dimensional vector spaces S and T is the
vector space S T constructed as follows. Consider the set of all formal sums n s t
⊗ i=1 i⊗ i
with all of the si’s in S and all of the ti’s in T. Let be the smallest equivale(cid:80)nce relation
≡
on this set with
s t+s t s t +s t
(cid:48) (cid:48) (cid:48) (cid:48)
⊗ ⊗ ≡ ⊗ ⊗
(s+s ) t s t+s t
(cid:48) (cid:48)
⊗ ≡ ⊗ ⊗
s (t+t ) s t+s t
(cid:48) (cid:48)
⊗ ≡ ⊗ ⊗
(zs) t s (zt)
⊗ ≡ ⊗
for all s,s S, t,t T and z C. Then S T is the set of all equivalence classes under
(cid:48) (cid:48)
∈ ∈ ∈ ⊗
this relation. Addition and scalar multiplication are defined in the obvious way.
(ii) Let S and T be superalgebras. We define multiplication in the tensor product S T by
⊗
s (t +t ) (s +s ) t = ss t t+ss t t+ss t t ss t t
+ + + + + +
⊗ − − ⊗ ⊗ ⊗ − − ⊗ − − ⊗ −
(cid:2) (cid:3)(cid:2) (cid:3)
for s S, t T, s S , t T . This multiplication defines an algebra structure on
∈ ∈ ± ∈ ± ± ∈ ±
S T. Setting (S T) = (S T ) (S T ), (S T) = (S T ) (S T ) we
+ + + + +
⊗ ⊗ ⊗ ⊕ −⊗ − ⊗ − ⊗ − ⊕ −⊗
get a superalgebra. If S and T are graded superalgebras then the decomposition S T =
⊗
∞m=0(S⊗T)m with
(cid:76) (S T) = S T
⊗ m m1 ⊗ m2
m1+(cid:77)m2=m
gives S T the structure of a graded superalgebra.
⊗
Definition I.6 (Grassmann algebra with coefficients in a superalgebra) Let
V
be a complex vector space and S be a superalgebra, the Grassmann algebra over with
V
coefficients in S is the superalgebra
= S
SV ⊗ V
(cid:94) (cid:94)
If S is a graded superalgebra, so is .
SV
(cid:86)
4
Remark I.7 Itis naturaltoidentify s S withthe element s 1 of = S and to
∈ ⊗ SV ⊗ V
identify ai1 ···ain ∈ V with the element 1⊗ai1 ···ain of SV. Un(cid:86)der this ide(cid:86)ntification
(cid:86) (cid:86)
s a a = s 1 1 a a = s a a
i1 ··· in ⊗ ⊗ i1 ··· in ⊗ i1 ··· in
(cid:0) (cid:1)(cid:0) (cid:1)
Every element of has a unique representation
SV
(cid:86)
D
s a a
i1,···,in i1 ··· in
n(cid:88)=0 1≤i1<(cid:88)···<in≤D
with the coefficients s S. Every element of also has a unique representation
i1,···,in ∈ SV
(cid:86)
D
s a
I I
n(cid:88)=0 I∈(cid:88)Mn
with the coefficients s S antisymmetric under permutation of the elements of I. If
I ∈
S = , with the complex vector space with basis b , ,b , then every element of
(cid:48) (cid:48) 1 D
V V { ··· }
(cid:86)has a unique representation
SV
(cid:86)
D
β b a
I,J J I
n,(cid:88)m=0 JI∈∈(cid:88)MMmn
with the coefficients β C separately antisymmetric under permutation of the elements
I,J ∈
of I and under permutation of the elements of J.
Problem I.2 Let be a complex vector space of dimension D. Every element s of
V V
D n
has a unique decomposition s = s0 +s1 with s0 ∈ C and s1 ∈ n=1 V. Define (cid:86)
(cid:76) (cid:86)
D
es = es0 1 sn
n! 1
n=0
(cid:110) (cid:111)
(cid:80)
Prove that if s,t with st = ts, then, for all n IN,
∈ V ∈
(cid:86) n
(s+t)n = n smtn m
−
m
m=0
(cid:88) (cid:0) (cid:1)
and
eset = etes = es+t
5
Problem I.3 Use the notation of Problem I.2. Let, for each α IR, s(α) .
∈ ∈ V
Assume that s(α) is differentiable with respect to α (meaning that if we write s(α(cid:86)) =
D
s (α)a a , every coefficient s (α) is differentiable with
n=0 1≤i1<···<in≤D i1,···,in i1 ··· in i1,···,in
r(cid:80)espect to(cid:80)α) and that s(α)s(β) = s(β)s(α) for all α and β. Prove that
d s(α)n = ns(α)n 1s (α)
− (cid:48)
dα
and
d es(α) = es(α)s (α)
(cid:48)
dα
Problem I.4 Use the notation of Problem I.2. If s > 0, define
0
D
lns = lns0 + (−1n)n−1 ss10 n
n=1
(cid:88) (cid:0) (cid:1)
with lns IR.
0
∈
a) Let, for each α IR, s(α) . Assume that s(α) is differentiable with respect to α,
∈ ∈ V
that s(α)s(β) = s(β)s(α) for all(cid:86)α and β and that s0(α) > 0 for all α. Prove that
d lns(α) = s(cid:48)(α)
dα s(α)
b) Prove that if s with s IR, then
0
∈ V ∈
(cid:86)
lnes = s
Prove that if s with s > 0, then
0
∈ V
(cid:86)
elns = s
Problem I.5 Use the notation of Problems I.2 and I.4. Prove that if s,t with
∈ V
st = ts and s0,t0 > 0, then (cid:86)
ln(st) = lns+lnt
6
Problem I.6 Generalise Problems I.1–I.5 to with S a finite dimensional graded
SV
superalgebra having S0 = C. (cid:86)
Problem I.7 Let be a complex vector space of dimension D. Let s = s +s
0 1
V ∈ V
D n
with s0 ∈ C and s1 ∈ n=1 V. Let f(z) be a complex valued function that is anal(cid:86)ytic
in |z| < r. Prove that(cid:76)if |s0| <(cid:86)r, then ∞n=0 n1!f(n)(0)sn converges and
(cid:80) D
∞
1 f(n)(0)sn = 1 f(n)(s )sn
n! n! 0 1
n=0 n=0
(cid:88) (cid:88)
I.2 Grassmann Integrals
Let be a finite dimensional complex vector space and S a superalgebra. Given
V
any ordered basis a ,...,a for , the Grassmann integral da da is defined to
1 D D 1
{ } V · ···
be the unique linear map from SV to S which is zero on ⊕Dn(cid:82)=−01 nV and obeys
(cid:86) (cid:86)
a a da da = 1
1 D D 1
··· ···
(cid:90)
D
Problem I.8 Let a , ,a be an ordered basis for . Let b = M a , 1 i D,
1 ··· D V i j=1 i,j j ≤ ≤
be another ordered basis for . Prove that (cid:80)
V
da da = det M db db
D 1 D 1
· ··· · ···
(cid:90) (cid:90)
In particular, if b = a for some permutation σ S
i σ(i) D
∈
da da = sgnσ db db
D 1 D 1
· ··· · ···
(cid:90) (cid:90)
Example I.8 Let be a two dimensional vector space with basis a ,a and be a
1 2 (cid:48)
V { } V
second two dimensional vector space with basis b ,b . Set S = . Let λ C 0
1 2 (cid:48)
{ } V ∈ \ { }
and let S be the 2 2 skew symmetric matrix (cid:86)
×
0 1
S = −λ
1 0
(cid:20) λ (cid:21)
7