JOURNAL OF MATHEMATICAL PHYSICS 51, 053305 共2010兲
Power series representations for complex bosonic
effective actions. I. A small field renormalization group step
Tadeusz Balaban,1,a兲 Joel Feldman,2,b兲 Horst Knörrer,3,c兲 and
Eugene Trubowitz3,d兲
1
Department of Mathematics, Rutgers, The State University of New Jersey, 110
Frelinghuysen Rd., Piscataway, New Jersey 08854-8019, USA
2
Department of Mathematics, University of British Columbia, Vancouver, British Columbia
V6T 1Z2, Canada
3
Mathematik, ETH-Zentrum, CH-8092 Zürich, Switzerland
共Received 27 October 2009; accepted 27 January 2010; published online 21 May 2010兲
We develop a power series representation and estimates for an effective action of
the form: ln关兰e f共␣1,. . .,␣s;z*,z兲d␮共z* , z兲 / 兰e f共0,. . .,0;z*,z兲d␮共z* , z兲兴. Here, f共␣1 , . . . , ␣s ;
z* , z兲 is an analytic function of the complex fields ␣1共x兲 , . . . , ␣s共x兲, z*共x兲, z共x兲
indexed by x in a finite set X, and d␮共z* , z兲 is a compactly supported product
measure. Such effective actions occur in the small field region for a renormalization
group analysis. Using methods similar to a polymer expansion, we estimate the
power series of the effective action. © 2010 American Institute of Physics.
关doi:10.1063/1.3329425兴
I. INTRODUCTION
Let X be a finite set and CX the space of complex valued bosonic fields 共i.e., functions兲 on X.
Furthermore, let d␮共z* , z兲 be a product measure on CX of the form
d␮共z*,z兲 =
兿 d␮0共z*共x兲,z共x兲兲,
共1.1兲
x苸X
where d␮0共␨* , ␨兲 is a normalized measure on C that is supported in 兩␨兩 艋 r for some constant r.
That is,
冕
兩 ␨ 兩 kd ␮ 0共 ␨ *, ␨ 兲 艋 r k
for all k 苸 N.
共1.2兲
For an analytic function f共␣1 , . . . , ␣s ; z* , z兲 of fields ␣1 , . . . , ␣s , z* , z, we develop criteria under
which
g共␣1, . . . , ␣s兲 = ln
兰e f共␣1,. . .,␣s;z*,z兲d␮共z*,z兲
兰e f共0,. . .,0;z*,z兲d␮共z*,z兲
共1.3兲
exists. This is done using norms for f which are defined in terms of the expansion of f in powers
of the fields. The construction also gives estimates on the corresponding norms of g.
In Ref 2, we described the analogous construction for real valued fields, which is technically
simpler. As pointed out there, expression like 共1.3兲 occur during the course of each iteration step
in a Wilson style renormalization group flow. Here 共z* , z兲 are the fluctuation fields integrated out
a兲
Electronic mail: tbalaban@math.rutgers.edu.
Electronic mail: feldman@math.ubc.ca. URL: http://www.math.ubc.ca/~feldman/.
c兲
Electronic mail: knoerrer@math.ethz.ch. URL: http://www.math.ethz.ch/~knoerrer/.
d兲
Electronic mail: trub@math.ethz.ch.
b兲
0022-2488/2010/51共5兲/053305/30/$30.00
51, 053305-1
© 2010 American Institute of Physics
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp
053305-2
J. Math. Phys. 51, 053305 共2010兲
Balaban et al.
at the current scale, while ␣1 , . . . , ␣s can be fields that are to be integrated out in future scales or
can be source fields that are used to generate and control correlation functions and are never
integrated out. We shall use the methods developed here to control the ultraviolet limit in the
coherent state functional integral representation for many boson systems described in Ref. 1
共Theorems 2.2 and 3.7兲.
To give an example of the norms used to control, 共1.3兲, assume for simplicity that s = 1 and
write ␣1 = ␣. Then, f has a power series expansion
f共␣ ;z*,z兲 =
兺
兺
n1,n2,n3艌0 xជ 苸Xn1
a共xជ ;yជ *,yជ 兲␣共x1兲 ¯ ␣共xn1兲z*共y*1兲 ¯ z*共y*n2兲z共y1兲 ¯ z共yn3兲
yជ 苸Xn2
*
yជ 苸Xn3
with coefficients a共xជ ; yជ * , yជ 兲 that are symmetric under permutations of the components of the
vectors xជ , yជ *, and yជ , respectively.
Assume that X is a metric space with metric d. Fix a parameter ␬. An example of a norm that
we use is
储f储 = 兩f共0;0,0兲兩 +
兺
sup
兺
max
n1+n2+n3艌1 x苸X 1艋ᐉ艋n1+n2+n3 共xជ ,yជ ,yជ 兲苸Xn1⫻Xn2⫻Xn3
w共xជ ;yជ *,yជ 兲兩a共xជ ;yជ *,yជ 兲兩,
*
共xជ ,yជ ,yជ 兲ᐉ=x
*
where the weight system w is defined as
w共xជ ;yជ *,yជ 兲 = ␬n1共4r兲n2+n3e␶d共xជ ,yជ *,yជ 兲
for 共xជ ,yជ *,yជ 兲 苸 Xn1 ⫻ Xn2 ⫻ Xn3
共1.4兲
and ␶d共xជ , yជ * , yជ 兲 is the minimal length of a tree whose set of vertices contains the set
兵x1 , . . . , xn1 , y*1 , . . . , y*n2 , y1 , . . . , yn3其. For this norm, our main result, Theorem 3.4, states that
g共␣兲 = ln
兰e f共␣;z*,z兲d␮共z*,z兲
兰e f共0;z*,z兲d␮共z*,z兲
1
exists provided that 储f储 ⬍ 16
, and that, in this case,
储g储 艋
储f储
.
1 − 16储f储
共1.5兲
Theorem 3.4 applies to more general norms than those described above 共see Definitions 2.3,
2.6, and 3.1.兲 To reflect the geometry and scale structure of a “large field/small field” decomposition of X, one can replace the constant ␬ by a “weight factor” ␬ : X → 共0 , ⬁兴 and the factor ␬n1 in
共1.4兲 by ␬共x1兲 ¯ ␬共xn1兲 关see Example 2.4, part 共i兲兴. Another variation in the norms comes from the
fact that one is often led to bound source fields by their sup norm rather than by their L1 norm 共see
Definition 2.6兲.
If the measure d␮0共␨* , ␨兲 is rotation invariant and there are no nontrivial monomials of the
form a共xជ ; yជ , yជ 兲␣共x1兲 ¯ ␣共xn1兲共z*共y1兲z共y1兲兲 ¯ 共z*共yn2兲z共yn2兲兲 in the power series of f, then 共1.5兲 can
be improved to a quadratic bound 共see Corollary 3.5兲. This situation occurs in our analysis of
many boson systems. There we also need information on how the g of 共1.3兲 varies with f. This is
provided by Corollary 3.6.
In the analysis of an infrared limit, one often has an increasing sequence
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp
053305-3
J. Math. Phys. 51, 053305 共2010兲
Complex bosonic effective actions. I
X1 傺 X2 傺 ¯ 傺 Xn 傺 ¯
of subsets of a lattice in Rd that exhausts the lattice. For each index n, there will be a function f n
that is 共close to兲 the restriction of a function on the whole lattice to Xn. Theorem 3.4 can possibly
be used to construct
gn = log
兰e f n共␣1,. . .,␣s;z*,z兲d␮共z*,z兲
兰e f n共0,. . .,0;z*,z兲d␮共z*,z兲
.
To take the limit n → ⬁ of the functions gn, we need to compare gn and the “restriction” of gm to
Xn when n ⬍ m. In the context of this paper, think of X = Xm and of Xn as a subset X⬘ of X.
The problem described above amounts to the following: with the notation of 共1.3兲, set
f ⬘共␣1, . . . , ␣s ;z*,z兲 = 兩f共␣1, . . . , ␣s ;z*,z兲兩 ␣k共x兲=z*共x兲=z共x兲=0 .
for x苸X\X⬘,k=1,. . .,s
Then, we want to compare
g = log
兰e f共␣1,. . .,␣s;z*,z兲d␮共z*,z兲
and
兰e f共0,. . .,0;z*,z兲d␮共z*,z兲
g⬘ = log
兰e f ⬘共␣1,. . .,␣s;z*,z兲d␮共z*,z兲
兰e f ⬘共0,. . .,0;z*,z兲d␮共z*,z兲
.
Actually, in the multiscale analysis even more complicated comparisons arise.
To facilitate such comparisons, we introduce an auxiliary real valued “history field” h on X. It
will only be evaluated with h共x兲 苸 兵0 , 1其 so that h2 = h. Set
f̃共␣1, . . . , ␣s ;z*,z;h兲 = f共␣1h, . . . , ␣sh;z*h,zh兲.
That is, we replace ␣i共x兲 by ␣i共x兲h共x兲 everywhere in the power series expansion for f, and the
same for z*共x兲 and z共x兲. Clearly,
f = 兩f̃兩h共x兲=1
for all x苸X,
f ⬘ = 兩f̃兩
h共x兲=1 for all x苸X⬘
h共x兲=0 for all x苸X\X⬘
.
Theorem 3.4 can be applied to construct
˜
g̃ = log
兰e f 共␣1,. . .,␣s;z*,z;h兲d␮共z*,z兲
˜
兰e f 共0,. . .,0;z*,z;h兲d␮共z*,z兲
.
Clearly,
g = 兩g̃兩h共x兲=1
for all x苸X,
g⬘ = 兩g̃兩
h共x兲=1 for all x苸X⬘
h共x兲=0 for all x苸X\X⬘
.
The measures that typically arise in renormalization group steps are rarely product measures.
To apply the results of this paper, one must first perform a change in variables so as to diagonalize
the 共essential part兲 of the covariance of the measure. Linear changes in variables, as well as
substitutions that typically occur in renormalization group steps, are controlled in Sec. IV and
Appendix A.
II. NORMS
To get a general setup for the norms that we shall use, we need a number of definitions.
Definition 2.1: (n-tuples)
共i兲
Let n 苸 Z with n 艌 0 and xជ = 共x1 , . . . , xn兲 苸 Xn be an ordered n-tuple of points of X. We
denote by n共xជ 兲 = n the number of components of xជ . Set
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp
053305-4
J. Math. Phys. 51, 053305 共2010兲
Balaban et al.
␾共xជ 兲 = ␾共x1兲 ¯ ␾共xn兲.
If n共xជ 兲 = 0, then ␾共xជ 兲 = 1. The support of xជ is defined to be
supp xជ = 兵x1, . . . ,xn其 傺 X.
共ii兲
For each s 苸 N, we denote
X共s兲 =
艛
n1,. . .,ns艌0
X n1 ⫻ ¯ ⫻ X ns .
关We distinguish between Xn1 ⫻ ¯ ⫻ Xns and Xn1+¯+ns. We use Xn1 ⫻ ¯ ⫻ Xns as the set of
possible arguments for ␺1共xជ 1兲 ¯ ␺s共xជ s兲, while Xn1+¯+ns is the set of possible arguments for
␺1共xជ 1 ⴰ ¯ ⴰ xជ s兲, where ⴰ is the concatenation operator of part 共iii兲.兴
The support of 共xជ 1 , . . . , xជ s兲 苸 X共s兲 is
s
supp共xជ 1, . . . ,xជ s兲 = 艛 supp共xជ j兲.
j=1
共iii兲
If 共xជ 1 , . . . , xជ s−1兲 苸 X共s−1兲, then 共xជ 1 , . . . , xជ s−1 , −兲 denotes the element of X共s兲 having n共xជ s兲 = 0.
In particular, X0 = 兵−其 and ␾共−兲 = 1.
We define the concatenation of xជ = 共x1 , . . . , xn兲 苸 Xn and yជ = 共y1 , . . . , ym兲 苸 Xm to be
xជ ⴰ yជ = 共x1, . . . ,xn,y1, . . . ,ym兲 苸 Xn+m .
For 共xជ 1 , . . . , xជ s兲, 共yជ 1 , . . . , yជ s兲 苸 X共s兲
共xជ 1, . . . ,xជ s兲 ⴰ 共yជ 1, . . . ,yជ s兲 = 共xជ 1 ⴰ yជ 1, . . . ,xជ s ⴰ yជ s兲
Definition 2.2: (Coefficient systems)
共i兲
共ii兲
A coefficient system of length s is a function a共xជ 1 , . . . , xជ s兲 that assigns a complex number
to each 共xជ 1 , . . . , xជ s兲 苸 X共s兲. It is called symmetric if, for each 1 艋 j 艋 s, a共xជ 1 , . . . , xជ s兲 is
invariant under permutations of the components of xជ j.
Let f共␣1 , . . . , ␣s兲 be a function, which is defined and analytic on a neighborhood of the
origin in Cs兩X兩. Then, f has a unique expansion of the form
f共␣1, . . . , ␣s兲 =
兺
共xជ 1,. . .,xជ s兲苸X共s兲
a共xជ 1, . . . ,xជ s兲␣1共xជ 1兲 ¯ ␣s共xជ s兲
with a共xជ 1 , . . . , xជ s兲 as a symmetric coefficient system. This coefficient system is called the
symmetric coefficient system of f.
Definition 2.3: (Weight systems) A weight system of length s is a function that assigns a
positive extended number w共xជ 1 , . . . , xជ s兲 苸 共0 , ⬁兴 to each 共xជ 1 , . . . , xជ s兲 苸 X共s兲 and satisfies the following conditions.
共a兲
共b兲
For each 1 艋 j 艋 s, w共xជ 1 , . . . , xជ s兲 is invariant under permutations of the components of xជ j.
w共共xជ 1, . . . ,xជ s兲 ⴰ 共yជ 1, . . . ,yជ s兲兲 艋 w共xជ 1, . . . ,xជ s兲w共yជ 1, . . . ,yជ s兲
for all 共xជ 1 , . . . , xជ s兲, 共yជ 1 , . . . , yជ s兲 苸 X共s兲 with supp共xជ 1 , . . . , xជ s兲 艚 supp共yជ 1 , . . . , yជ s兲 ⫽ 쏗.
Example 2.4: (Weight systems)
共i兲
If ␬1 , . . . , ␬s are functions from X to 共0, ⬁兴 共called weight factors兲, then
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp
053305-5
J. Math. Phys. 51, 053305 共2010兲
Complex bosonic effective actions. I
s n共xជ j兲
w共xជ 1, . . . ,xជ s兲 = 兿
兿 ␬ j共x j,ᐉ兲
j=1 ᐉ=1
共ii兲
is a weight system of length s.
Let d be a metric on X. The length of a tree T with vertices in X is simply the sum of the
lengths of all edges of T 共where the length of an edge is the distance between its vertices兲.
For a subset S 傺 X, denote by ␶d共S兲 the length of the shortest tree in X whose set of vertices
contains S. Then,
w共xជ 1, . . . ,xជ s兲 = e␶d共supp共xជ 1,. . .,xជ s兲兲
共iii兲
is a weight system of length s.
Assume again that d is a metric on X. Let ⍀ 傺 X. The “maximum distance” of any subset
S of X to ⍀c is D共S , ⍀c兲 = supx苸S d共x , ⍀c兲. Then,
c
w共xជ 1, . . . ,xជ s兲 = eD共supp共xជ 1,. . .,xជ s兲,⍀ 兲
共iv兲
is a weight system of length s.
Let ␬ 艌 1, N 苸 N, 1 艋 s0 艋 s, and Y 傺 X. Denote by ␯Y 共xជ 1 , . . . , xជ s0兲 the number of components of xជ 1 , . . . , xជ s0 that are in Y. Then,
w共xជ 1, . . . ,xជ s兲 = ␬max兵N−␯Y 共xជ 1,. . .,xជ s0兲,0其
is a weight system of length s. The verification of condition 共b兲 of Definition 2.3 follows
from
␬max兵N−m−n,0其 艋 ␬max兵N−m,0其␬max兵N−n,0其
共v兲
for all m,n 艌 0.
If w1共xជ 1 , . . . , xជ s兲 and w2共xជ 1 , . . . , xជ s兲 are two weight systems of length s, then
w3共xជ 1, . . . ,xជ s兲 = w1共xជ 1, . . . ,xជ s兲w2共xជ 1, . . . ,xជ s兲
is also a weight systems of length s.
Definition 2.5: Let d be a metric on X. Given weight factors ␬ j : X → 共0 , ⬁兴 for j = 1 , . . . , s, we
call
s n共xជ j兲
w共xជ 1, . . . ,xជ s兲 = e␶d共supp共xជ 1,. . .,xជ s兲兲 兿
兿 ␬ j共x j,ᐉ兲
j=1 ᐉ=1
the weight system with metric d that associates the weight factor ␬ j with the field ␣ j. It follows
from parts 共i兲, 共ii兲, and 共v兲 of Example 2.4 that this is indeed a weight system.
Using weight systems as defined above, we define norms for functions that depend analytically on the complex fields ␣1 , . . . , ␣s and the additional “history” field h.
Definition 2.6: (Norms) Let w be a weight system of length s + 1. Let 0 艋 s⬘ 艋 s. We think of
the fields ␣ j with 1 艋 j 艋 s⬘ as being sources 共that is, we differentiate with respect to these fields to
generate correlation functions兲, the fields ␣ j with s⬘ ⬍ j 艋 s as being internal fields 共that is, they
will be integrated out兲 and the field ␣s+1 as the history field.
共i兲
For any n1 , . . . , ns+1 艌 0 and any function b共xជ 1 , . . . , xជ s , xជ s+1兲 on Xn1 ⫻ ¯ ⫻ Xns+1, we define
the norm 储b储n1,. . .,ns+1 as follows:
⬘ n ⫽ 0, then
• If there are external fields, that is, if 兺sj=1
j
储b储n1,. . .,ns+1 = max
xជ ᐉ苸Xnᐉ
兺
兩b共xជ 1, . . . ,xជ s,xជ s+1兲兩.
xជ ᐉ苸Xnᐉ
1艋ᐉ艋s⬘ s⬘⬍ᐉ艋s+1
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp
053305-6
J. Math. Phys. 51, 053305 共2010兲
Balaban et al.
⬘ n = 0, but
• If there are no external fields but there are internal fields, that is, if 兺sj=1
j
兺sj=1n j ⫽ 0, then
储b储n1,. . .,ns+1 = max max max
x苸X s⬘⬍j艋s 1艋i艋n j
n j⫽0
兺
兩b共xជ 1, . . . ,xជ s,xជ s+1兲兩.
xជ ᐉ苸Xnᐉ
s⬘⬍ᐉ艋s+1
共xជ j兲i=x
Here 共xជ j兲i is the ith component of the n j-tuple xជ j.
• If there are no external or internal fields but there are history fields, that is, if 兺sj=1n j = 0,
but ns+1 ⫽ 0, then we take the pure L1 norm
储b储n1,. . .,ns+1 =
兺
兩b共− , . . . ,− ,xជ s+1兲兩.
xជ s+1苸Xns+1
• Finally, for the constant term, that is, if 兺s+1
j=1 n j = 0,
储b储n1,. . .,ns+1 = 兩b共− , . . . ,− 兲兩.
共ii兲
We define the norm, with weight w, of a coefficient system a of length s + 1 to be
兩a兩w =
共iii兲
兺
n1,. . .,ns+1艌0
储w共xជ 1, . . . ,xជ s,xជ s+1兲a共xជ 1, . . . ,xជ s,xជ s+1兲储n1,. . .,ns+1 .
In some applications, it will be more convenient to turn this norm into a seminorm by
ignoring the constant term a共−兲. The results of this paper apply equally well to such
seminorms.
Let f共␣1 , . . . , ␣s , ␣s+1兲 be a function that is defined and analytic on a neighborhood of the
origin in C共s+1兲兩X兩. The norm 储f储w of f with weight w is defined to be 兩a兩w, where a is the
symmetric coefficient system of f. 共This definition also applies when f depends only on a
subset of the variables ␣1 , . . . , ␣s+1.兲
Remark 2.7:
共i兲
Let a be a 共not necessarily symmetric兲 coefficient system of length s + 1 and
f共␣1, . . . , ␣s+1兲 =
共ii兲
兺
共xជ 1,. . .,xជ s+1兲苸X共s+1兲
a共xជ 1, . . . ,xជ s+1兲␣1共xជ 1兲 ¯ ␣s共xជ s+1兲.
Then 储f储w 艋 兩a兩w for any weight system w. We call a a not necessarily symmetric coefficient
system for f.
In Lemma B.1 we show how to convert a norm bound on f into a supremum bound.
III. THE MAIN THEOREM
In 共1.3兲, we integrate out the last two internal fields z*, z using the measure d␮共z* , z兲 of 共1.1兲.
Recall that this measure is supported in 储z储⬁ 艋 r. The weight systems that are adapted to this
situation fulfil the Definition 3.1, below, with ␳ = 4r.
Definition 3.1: A weight system of length s + 3 “gives weight at least ␳ to the last two internal
fields” if
w共xជ 1, . . . ,xជ s ;yជ *,yជ ;xជ 兲 艌 ␳n共yជ *兲+n共yជ 兲w共xជ 1, . . . ,xជ s ;− ,− ;xជ 兲
for all 共xជ 1 , . . . , xជ s , xជ 兲 苸 X共s+1兲 and 共yជ * , yជ 兲 苸 X共2兲.
Example 3.2: Assume that d is a metric on X and ␬ j : X → 共0 , ⬁兴, j = 1 , . . . , s are weight functions. The weight system with metric d that associates the weight factor ␬ j with the field ␣ j, j
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp
053305-7
J. Math. Phys. 51, 053305 共2010兲
Complex bosonic effective actions. I
= 1 , . . . , s, the constant weight factor ␳ to the fields ␣s+1 and ␣s+2 and constant weight factor 1 to
the history field h, fulfills Definition 3.1.
We fix, for the rest of this section, a weight system w of length s + 3 that gives weight at least
4r to the last two internal fields. Furthermore, we fix the number 0 艋 s⬘ 艋 s of source fields for
Definition 2.6 of 储 · 储w.
Remark 3.3:
共i兲
If h is an analytic function for which h共0 , . . . , 0 ; z* , z ; h兲 is independent of z* and z, then
冐冕
共ii兲
h共␣1, . . . , ␣s ;z*,z;h兲d␮共z*,z兲
冐
w
艋 储h共␣1, . . . , ␣s ;z*,z;h兲储w .
Assume that the measure d␮共z* , z兲 on C is rotation invariant. For each j = 1 , 2, let
h j共␣1 , . . . , ␣s ; z* , z ; h兲 be an analytic function with h j共0 , . . . , 0 ; z* , z ; h兲 = 0. Further, assume
that the symmetric coefficient system a j共xជ 1 , . . . , xជ s ; yជ * , yជ ; xជ 兲 of h j obeys
a j共xជ 1 , . . . , xជ s ; yជ * , yជ ; xជ 兲 = 0 whenever yជ = yជ *. Then,
冕
h j共␣1, . . . , ␣s ;z*,z;h兲d␮共z*,z兲 = 0
for j = 1,2
and
冐冕
h1共␣1, . . . , ␣s ;z*,z;h兲h2共␣1, . . . , ␣s ;z*,z;h兲d␮共z*,z兲
冐
艋 储h1储w储h2储w .
w
Proof:
共i兲
共ii兲
This follows from the observation that
冏冕
冏
z共yជ *兲*z共yជ 兲d␮共z*,z兲 艋 rn共yជ *兲+n共yជ 兲
for all yជ * , yជ 苸 X共2兲.
Write
h̃共␣1, . . . , ␣s ;h兲 =
冕
=
h1共␣1, . . . , ␣s ;z*,z;h兲h2共␣1, . . . , ␣s ;z*,z;h兲d␮共z*,z兲
兺
共xជ 1,. . .,xជ s兲苸X共s兲
ã共xជ 1, . . . ,xជ s ;xជ 兲␣1共xជ 1兲 ¯ ␣s共xជ s兲h共xជ 兲
xជ 苸X共1兲
with, for each ␨ 苸 X共s兲 and xជ 苸 X共1兲,
ã共␨ ;xជ 兲 =
兺
␰,␰⬘苸X共s兲
␰ⴰ␰⬘=␨
xជ ⬙,xជ ⬘苸X共1兲
兺
共yជ ,y兲苸X共2兲
a1共␰ ;yជ *,yជ ;xជ ⬙兲a2共␰⬘ ;yជ ⬘*,yជ ⬘ ;xជ ⬘兲
冕
z共yជ * ⴰ yជ ⬘*兲*z共yជ ⴰ yជ ⬘兲d␮共z*,z兲.
*
共yជ ⬘ ,y⬘兲苸X共2兲
*
xជ ⬙ⴰxជ ⬘=xជ
We claim that only terms with supp共yជ * , yជ 兲 艚 supp共yជ ⬘* , yជ ⬘兲 ⫽ 쏗 can be nonzero. By hypothesis,
a1共␰ ; yជ * , yជ ; xជ ⬙兲 = 0 if yជ * = yជ . So we may assume that there is a y 苸 supp共yជ * , yជ 兲 with the multiplicities
of yជ * and yជ at y being different. Since 兰z共yជ * ⴰ yជ ⬘*兲*z共yជ ⴰ yជ ⬘兲d␮共z* , z兲 vanishes unless yជ * ⴰ yជ ⬘* = yជ ⴰ yជ ⬘,
the multiplicities of yជ ⬘* and yជ ⬘ at y must also be different. 关To see this, let y 苸 X and suppose that
the multiplicity, say p*, of yជ * ⴰ yជ ⬘* at y is different from the multiplicity, say p, of yជ ⴰ yជ ⬘ at y.
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp
053305-8
J. Math. Phys. 51, 053305 共2010兲
Balaban et al.
Because
d␮
is
invariant
under
z共y兲 哫 z␪共y兲 = ei␪z共y兲,
while
i共p−p 兲␪
p
p
p
p
=e
* 关z共y兲*兴 *z共y兲 , the integral 兰关z共y兲*兴 *z共y兲 d␮共z* , z兲 must vanish.兴
In particular, y 苸 supp共yជ ⬘* , yជ ⬘兲.
Consequently, by 共1.2兲,
兩ã共␨ ;xជ 兲兩 艋
兺
兺
关z␪共y兲*兴 p*z␪共y兲 p
兩a1共␰ ; ␩ ;xជ ⬙兲兩兩a2共␰⬘ ; ␩⬘ ;xជ ⬘兲兩rn共␩兲+n共␩⬘兲 .
␰,␰⬘苸X共s兲
␩,␩⬘苸X共2兲
␰ⴰ␰⬘=␨
supp共␩兲艚supp共␩⬘兲⫽쏗
xជ ⬙,xជ ⬘苸X共1兲
xជ ⬙ⴰxជ ⬘=xជ
Since w gives weight at least 2r to the last two fields,
w共␨ ;xជ 兲兩ã共␨ ;xជ 兲兩 艋
兺
兺
␰,␰⬘苸X共s兲
␩,␩⬘
w共␨ ; ␩ ⴰ ␩⬘ ;xជ 兲兩a1共␰ ; ␩ ;xជ ⬙兲兩兩a2共␰⬘ ; ␩⬘ ;xជ ⬘兲兩2−n共␩兲−n共␩⬘兲
苸X共2兲
supp共␩兲艚supp共␩⬘兲⫽쏗
␰ⴰ␰⬘=␨
xជ ⬙,xជ ⬘苸X共1兲
xជ ⬙ⴰxជ ⬘=xជ
艋
兺
兺
␰,␰⬘苸X共s兲
␩,␩⬘
w共␰ ; ␩ ;xជ ⬙兲兩a1共␨ ; ␩ ;xជ ⬙兲兩w共␰⬘ ; ␩⬘ ;xជ ⬘兲兩a2共␰⬘ ; ␩⬘ ;xជ ⬘兲兩2−n共␩兲−n共␩⬘兲
苸X共2兲
supp共␩兲艚supp共␩⬘兲⫽쏗
␰ⴰ␰⬘=␨
xជ ⬙,xជ ⬘苸X共1兲
xជ ⬙ⴰxជ ⬘=xជ
If ␰ = 共xជ 1 , . . . , xជ s兲 苸 X共s兲, we write ␰i 苸 X for 共xជ 1 ⴰ ¯ ⴰ xជ s兲i. By hypothesis, ã共␨ ; xជ 兲 = 0 if ␨ = −. So, if
there are no source fields,
储h̃储w =
兺
n1,. . .,ns+1艌0
max
max
u苸X 1艋i艋n1+¯+ns
n1+¯+ns⬎0
艋
兺
兺
w共␨ ;xជ 兲兩ã共␨ ;xជ 兲兩
␨苸Xn1⫻¯⫻Xns
xជ 苸Xns+1
␰i=u
max
max
兺
n1,. . .,ns+1艌0
u苸X 1艋i艋⌺ 共n +n⬘兲
n
n
j艋s j j ␰苸X 1⫻¯⫻X s
n1⬘,. . .,ns+1
⬘ 艌0
␰⬘苸Xn1⬘⫻¯⫻Xns⬘
m ,m艌0
*
m⬘ ,m⬘艌0
*
兺
w共␰ ; ␩ ;xជ 兲
␩苸Xm*⫻Xm
n⬘苸Xm*⬘ ⫻Xm⬘
n
n
⬘
xជ 苸X s+1, xជ 苸X s+1 supp共␩兲艚supp共␩⬘兲⫽쏗
共␰ ⴰ ␰⬘兲i=u
⌺ j艋s共n j+n⬘j 兲艌1
⫻兩a1共␰ ; ␩ ;xជ 兲兩w共␰⬘ ; ␩⬘ ;xជ ⬘兲兩a2共␰⬘ ; ␩⬘ ;xជ ⬘兲兩2−共m*+m⬘*+m+m⬘兲
Fix, temporarily, n1 , . . . , ns+1 , n1⬘ , . . . , ns+1
⬘ , m* , m , m⬘* , m⬘ 艌 0. If there are no source fields, also fix,
temporarily, 1 艋 i 艋 兺 j艋s共n j + n⬘j 兲. There are source fields if s⬘ ⬎ 0 共s⬘ was specified in Definition
⬘ 共n + n⬘兲 艌 1. In this case,
2.6兲 and 兺sj=1
j
j
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp
053305-9
J. Math. Phys. 51, 053305 共2010兲
Complex bosonic effective actions. I
兺
max
u苸X
␰苸Xn1⫻¯⫻Xns
␰⬘苸Xn1⬘⫻¯⫻Xns⬘
⬘
xជ 苸Xns+1, xជ ⬘苸Xns+1
共␰ ⴰ ␰⬘兲i=u
is replaced by
max
u j,ᐉ苸X
兺
max
u⬘j,ᐉ苸X
1艋j⬍s⬘ 1艋j⬍s
⬘
1艋ᐉ艋n j
.
␰苸Xn1⫻¯⫻Xns
␰⬘苸Xn1⬘⫻¯⫻Xns⬘
1艋ᐉ艋n⬘j
⬘
xជ 苸Xns+1, xជ ⬘苸Xns+1
共xជ j兲ᐉ=u j,ᐉ, 共xជ ⬘j 兲ᐉ⬘=u⬘j,ᐉ
⬘
1艋j艋s⬘, 1艋ᐉ艋n j, 1艋ᐉ⬘艋n⬘j
If we translate the 共xជ j兲ᐉ, 共xជ ⬘j 兲ᐉ⬘ notation into the corresponding components of ␰ , ␰⬘, we may write
both of these max/sums in the form
兺
max max
u p苸X u⬘苸X
p
p苸A
p苸A⬘
␰苸Xn1⫻¯⫻Xns
␰⬘苸Xn1⬘⫻¯⫻Xns⬘
⬘
xជ 苸Xns+1, xជ ⬘苸Xns+1
␰ p=u p, p苸A
␰⬘p=u⬘p, p苸A⬘
for some subsets A 傺 兵p 苸 N 兩 1 艋 p 艋 兺 j艋sn j其 and A⬘ 傺 兵p 苸 N 兩 1 艋 p 艋 兺 j艋sn⬘j 其 with A 艛 A⬘ nonempty.
Fix u p 苸 X for each p 苸 A and u⬘p 苸 X for each p 苸 A⬘. For notational simplicity, suppose that
p = 1 苸 A. Then,
兺
␰苸Xn1⫻¯⫻Xns
兺
w共␰ ; ␩ ;xជ 兲兩a1共␰ ; ␩ ;x兲兩w共␰⬘ ; ␩⬘ ;xជ ⬘兲兩a2共␰⬘ ; ␩⬘ ;xជ ⬘兲兩2−共m*+m*⬘ +m+m⬘兲
␩苸Xm*⫻Xm
␰⬘苸Xn1⬘⫻¯⫻Xns⬘
␩⬘苸Xm*⬘ ⫻Xm⬘
n
n
⬘
xជ 苸X s+1, xជ ⬘苸X s+1 supp共␩兲艚supp共␩⬘兲⫽쏗
␰ p=u p, p苸A
␰⬘p=u⬘p, p苸A⬘
艋
兺
␰苸Xn1⫻¯⫻Xns
兺
1艋k艋m +m
*
兺
w共␰ ; ␩ ;xជ 兲兩a1共␰ ; ␩兲;xជ 兩w共␰⬘ ; ␩⬘ ;xជ ⬘兲
␩苸Xm*⫻Xm
␰⬘苸Xn1⬘⫻¯⫻Xns⬘ 1艋ᐉ艋m +m ␩ 苸Xm⬘ ⫻Xm⬘
⬘ ⬘ ⬘
*
⬘
xជ 苸Xns+1, xជ ⬘苸Xns+1
␰ p=u p, p苸A
*
␩k=␩ᐉ⬘
␰⬘p=u⬘p, p苸A⬘
⫻兩a2共␰⬘ ; ␩⬘ ;xជ ⬘兲兩2−共m*+m⬘*+m+m⬘兲 艋
max
兺
兺
1艋k艋m +m
␰苸Xn1⫻¯⫻Xns
1艋ᐉ艋m⬘ +m⬘
␰⬘苸Xn1⬘⫻¯⫻Xns⬘ ␩ 苸Xm⬘ ⫻Xm⬘
⬘
*
*
*
⬘
xជ 苸Xns+1, xជ ⬘苸Xns+1
␰ p=u p, p苸A
w共␰ ; ␩ ;xជ 兲
␩苸Xm*⫻Xm
␩k=␩ᐉ⬘
␰⬘p=u⬘p, p苸A⬘
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp
053305-10
J. Math. Phys. 51, 053305 共2010兲
Balaban et al.
⫻a1共␰ ; ␩ ;xជ 兲兩w共␰⬘ ; ␩⬘ ;xជ ⬘兲兩a2共␰⬘ ; ␩⬘ ;xជ ⬘兲兩 艋
⫻
冢
max
*
兺
max
1艋ᐉ艋m⬘ +m⬘ y苸X
␰⬘苸Xn1⬘⫻¯⫻Xns⬘
冢
兺
w共␰ ; ␩ ;xជ 兲兩a1共␰ ; ␩ ;xជ 兲兩
␰苸Xn1⫻¯⫻Xns
␩苸Xm*⫻Xm
xជ 苸Xns+1
␰ p=u p, p苸A
w共␰⬘ ; ␩⬘ ;xជ ⬘兲兩a2共␰⬘ ; ␩⬘ ;xជ ⬘兲兩
␰⬘p=u⬘p, p苸A⬘
␩⬘苸Xm*⬘ ⫻Xm⬘
␩ᐉ⬘=y
⬘
xជ ⬘苸Xns+1
冣
冣
.
If A⬘ is empty 共i.e., there are no source fields in ␰⬘兲, the second large bracket is
冢
max
max
1艋ᐉ艋m⬘ +m⬘
y苸X
*
兺
␰⬘苸Xn1⬘⫻¯⫻Xns⬘
w共␰⬘ ; ␩⬘ ;xជ ⬘兲兩a2共␰⬘ ; ␩⬘ ;xជ ⬘兲兩
␩⬘苸Xm*⬘ ⫻Xm⬘
␩ᐉ⬘=y
⬘
xជ ⬘苸Xns+1
冣
.
If A⬘ is not empty 共i.e., there are source fields in ␰⬘兲, we bound the second large bracket by
冢
兺
␰⬘苸Xn1⬘⫻¯⫻Xns⬘
w共␰⬘ ; ␩⬘ ;xជ ⬘兲兩a2共␰⬘ ; ␩⬘ ;xជ ⬘兲兩
␰⬘p=u⬘p, p苸A⬘
␩⬘苸Xm*⬘ ⫻Xm⬘
⬘
xជ ⬘苸Xns+1
冣
.
Taking the maximum over u p’s and u⬘p’s, and possibly over i, and the remaining sums gives the
desired bound. Recall that a1共␰ ; ␩ ; xជ 兲 = 0 if ␰ = − and a2共␰⬘ ; ␩⬘ ; xជ ⬘兲 = 0 if ␰⬘ = −.
䊏
Recall that we have fixed a weight system w of length s + 3 that gives weight at least 4r to the
last two internal fields and that we have fixed the number 0 艋 s⬘ 艋 s of source fields.
1
Theorem 3.4: If f共␣1 , . . . , ␣s ; z* , z ; h兲 obeys 储f储w ⬍ 16
, then there is an analytic function
g共␣1 , . . . , ␣s ; h兲 such that
兰e f共␣1,. . .,␣s;z*,z;h兲d␮共z*,z兲
兰e f共0,. . .,0;z*,z;h兲d␮共z*,z兲
= eg共␣1,. . .,␣s;h兲
共3.1兲
and
储g储w 艋
储f储w
.
1 − 16储f储w
Proof: The proof of this theorem is virtually identical to the proof of Ref. 2 共Theorem 3.4兲. Let
a共xជ 1 , . . . , xជ s ; yជ * , yជ ; xជ 兲 be the symmetric coefficient system for f. We first introduce some shorthand
notation.
• For ␩ = 共yជ * , yជ 兲 苸 X共2兲, we write z共␩兲 = z共yជ *兲*z共yជ 兲 and
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp
053305-11
J. Math. Phys. 51, 053305 共2010兲
Complex bosonic effective actions. I
a共␩兲 =
兺
共xជ 1,. . .,xជ s,xជ 兲苸X共s+1兲
a共xជ 1, . . . ,xជ s ;yជ *,yជ ;xជ 兲␣1共xជ 1兲 ¯ ␣s共xជ s兲h共xជ 兲.
With this notation
f共␣1, . . . , ␣s ;z*,z;h兲 =
兺
␩苸X共2兲
a共␩兲z共␩兲.
By factoring e f共␣1,. . .,␣s;0,0;h兲 out of the integral in the numerator of 共3.1兲, we may assume that
a共− , −兲 = 0.
• Let X1 , . . . , Xᐉ be subsets of X. The incidence graph G共X1 , . . . , Xᐉ兲 of X1 , . . . , Xᐉ is the labeled
graph with the set of vertices 共1 , . . . , ᐉ兲 and edges between i ⫽ j whenever Xi 艚 X j ⫽ 쏗.
• For a subset of Z 傺 X, we denote by C共Z兲 the set of all ordered tuples 共␩1 , . . . , ␩n兲 for which
the incidence graph G共supp ␩1 , . . . , supp ␩n兲 is connected and for which Z
= supp ␩1 艛 ¯ 艛 supp ␩n.
Expanding the exponential
⬁
e f共␣1,. . .,␣s;z*,z;h兲 = 1 + 兺
ᐉ=1
1
兺
ᐉ! Z傺X
兺
␩1,. . .,␩ᐉ苸X共2兲
Z⫽쏗 Z=supp ␩ 艛¯艛supp ␩
1
ᐉ
a共␩1兲 ¯ a共␩ᐉ兲z共␩1兲 ¯ z共␩ᐉ兲.
As in Ref. 2 关共3.5兲兴,
冕
⬁
e
f共␣1,. . .,␣s;z*,z;h兲
d␮共z*,z兲 = 1 + 兺
n=1
1
n!
n
兺
Z1,. . .,Zn傺X
⌽共Z j兲,
兿
j=1
pairwise disjoint
where for 쏗 ⫽ Z 傺 X, the function ⌽共Z兲共␣1 , . . . , ␣s兲 is defined by
⬁
⌽共Z兲 = 兺
k=1
1
兺 a共␩1兲 ¯ a共␩k兲
k! 共␩1,. . .,␩k兲苸C共Z兲
冕
z共␩1兲 ¯ z共␩k兲d␮共z*,z兲
and ⌽共쏗兲 = 0. Again as in Ref. 2 关共3.7兲兴,
ln
冕
⬁
e f d␮ = 兺
n=1
n
1
⌽共Z j兲
兺 ␳T共Z1, . . . ,Zn兲 兿
n! Z1,. . .,Zn傺X
j=1
共3.2兲
where
␳T共Z1, . . . ,Zn兲 =
兺
共− 1兲兩g兩
g苸Cn
g傺G共Z1,. . .,Zn兲
and Cn is the set of all connected graphs on the set of vertices 共1 , . . . , n兲 that have at most one edge
joining each pair of distinct vertices and no edges joining a vertex to itself. 关In particular,
␳T共Z1 , . . . , Zn兲 = 0 if G共Z1 , . . . , Zn兲 is not connected.兴
From 共3.2兲 one determines, as in Ref. 2 关共3.10兲兴, a not necessarily symmetric coefficient
system for g. Namely,
g共␣1, . . . ␣s ;h兲 =
兺
共xជ 1,. . .,xជ s,xជ 兲苸X共s+1兲
共x1,. . .,xជ s兲⫽共−,. . .,−兲
a⬘共xជ 1, . . . ,xជ s ;xជ 兲␣1共xជ 1兲 ¯ ␣s共xជ s兲h共xជ 兲,
where for ␰ 苸 X共s兲 and xជ 苸 X共1兲,
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp
053305-12
J. Math. Phys. 51, 053305 共2010兲
Balaban et al.
⬁
a⬘共␰ ;xជ 兲 = 兺
n=1
1
n! ␰
兺
1,. . .,␰n苸X
共s兲
兺
␩1,. . .,␩n苸X共2兲
n
␳ 共supp ␩1, . . . ,supp ␩n兲 兿 ã共␰ j ; ␩ j ;uជ j兲
共3.3兲
T
j=1
␰1ⴰ¯ⴰ␰n=␰
uជ 1,. . .,uជ n苸X共1兲
uជ 1ⴰ¯ⴰuជ n=xជ
with for each ␰ 苸 X共s兲 and ␩ 苸 X共2兲,
⬁
ã共␰ ; ␩ ;xជ 兲 = 兺
k=1
1
兺
k! 共␩1,. . .,␩k兲苸C共supp ␩兲
␩1ⴰ¯ⴰ␩k=␩
兺
␰1,. . .,␰k
a共␰1 ; ␩1 ;uជ 1兲 ¯ a共␰k ; ␩k ;uជ k兲
冕
z共␩兲d␮共z*,z兲
␰1ⴰ¯ⴰ␰k=␰
uជ 1,. . .,uជ k
uជ 1ⴰ¯ⴰuជ n=xជ
共3.4兲
By Remark 2.7, part 共i兲, 储g储w 艋 兩a⬘兩w. The bounds on 兩a⬘兩w that are necessary for the proof of the
theorem are derived as in Ref. 2 关共3.12兲–共3.16兲兴.
䊏
1
Corollary 3.5: Let f共␣1 , . . . , ␣s ; z* , z ; h兲 obey 储f储w ⬍ 32
and define, for each complex number ␨
1
with 兩␨兩储f储w ⬍ 16
, the function G共␨兲 = G共␨ ; ␣1 , . . . , ␣s ; h兲 by the condition
兰e␨ f共␣1,. . .,␣s;z*,z;h兲d␮共z*,z兲
兰e␨ f共0,. . .,0;z*,z;h兲d␮共z*,z兲
= eG共␨;␣1,. . .,␣s;h兲
共3.5兲
as in Theorem 3.4. Then, G共␨兲 is a (Banach space valued) analytic function of ␨ and, for each
n 苸 N, the g共␣1 , . . . , ␣s ; h兲 = G共1兲 of Theorem 3.4 obeys
冐
g−
1 d nG
dG
共0兲 − ¯ −
共0兲
d␨
n! d␨n
冐
艋
w
冢
储f储w
1
− 储f储w
20
冣
n+1
.
We have G共0兲 = 0,
dG
共0兲 =
d␨
冕
关f共␣1, . . . , ␣s ;z*,z;h兲 − f共0, . . . ,0;z*,z;h兲兴d␮共z*,z兲
and
d 2G
共0兲 =
d 2␨ 2
冕
f共␣1, . . . , ␣s ;z*,z;h兲2d␮共z*,z兲 −
−
f共␣1, . . . , ␣s ;z*,z;h兲d␮共z*,z兲
冋冕
冕
f共0, . . . ,0;z*,z;h兲2d␮共z*,z兲
册 冋冕
2
+
f共0, . . . ,0;z*,z;h兲d␮共z*,z兲
册
2
.
If, in addition, the measure d␮共z* , z兲 on C is rotation invariant, f共0 , . . . , 0 ; z* , z ; h兲 = 0 and the
symmetric coefficient system a共xជ 1 , . . . , xជ s ; yជ * , yជ ; xជ 兲 of f obeys a共xជ 1 , . . . , xជ s ; yជ * , yជ ; xជ 兲 = 0 whenever
yជ = yជ *, then 共dG / d␨兲共0兲 = 0 and
d 2G
共0兲 =
d␨2
冕
f共␣1, . . . , ␣s ;z*,z;h兲2d␮共z*,z兲.
Proof: The analyticity of G共␨兲 is obvious since the series 共3.3兲 and 共3.4兲, with a replaced by
␨a converge in the norm 储 · 储w uniformly in ␨ for 兩␨兩储f储w bounded by any constant strictly less than
1
16 . That G共0兲 = 0 and G共1兲 = g is also obvious. So by Taylor’s formula with remainder,
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp
053305-13
J. Math. Phys. 51, 053305 共2010兲
Complex bosonic effective actions. I
g−
1 d nG
1 dn+1G
dG
共0兲 − ¯ −
共0兲
=
共u兲
d␨
n! d␨n
共n + 1兲! d␨n+1
for some 0 ⬍ u ⬍ 1. By the Cauchy integral formula,
1
1 dn+1G
n+1 共u兲 =
共n + 1兲! d␨
2␲i
冕
兩␨兩=U
G共␨兲
d␨
共␨ − u兲n+2
for any u ⬍ U ⬍ 1 / 16储f储w. Hence,
冐
g−
1 d nG
dG
共0兲 − ¯ −
共0兲
d␨
n! d␨n
冐
艋
1
1
U储f储w
储G共␨兲储w
2␲U sup
n+2 艋 U
1 − 16U储f储w 共U − 1兲n+2
2␲
兩␨兩=U 共U − 1兲
=
储f储wn+1
共U储f储w兲2
.
n+2
共U储f储w − 储f储w兲 1 − 16U储f储w
w
1
gives
Choosing U so that U储f储w = 20
冐
g−
1 d nG
dG
共0兲 − ¯ −
共0兲
d␨
n! d␨n
冐
艋
w
1
80
冉
储f储wn+1
1
− 储f储w
20
冊
n+2
艋
冢
储f储w
1
− 储f储w
20
冣
n+1
1
since 储f储w ⬍ 32
.
To get the formulas for 共dG / d␨兲共0兲 and 共d2G / d␨2兲共0兲, differentiate 共3.5兲 to give
兰f共␣1, . . . , ␣s ;z*,z;h兲e␨ f共␣1,. . .,␣s;z*,z;h兲d␮共z*,z兲
dG
共␨兲eG共␨;␣1,. . .,␣s;h兲 =
d␨
兰e␨ f共0,. . .,0;z*,z;h兲d␮共z*,z兲
−
兰f共0, . . . ,0;z*,z;h兲e␨ f共0,. . .,0;z*,z;h兲d␮共z*,z兲兰e␨ f共␣1,. . .,␣s;z*,z;h兲d␮共z*,z兲
关兰e␨ f共0,. . .,0;z*,z;h兲d␮共z*,z兲兴2
and hence
兰f共␣1, . . . , ␣s ;z*,z;h兲e␨ f共␣1,. . .,␣s;z*,z;h兲d␮共z*,z兲
dG
共␨兲 =
d␨
兰e␨ f共␣1,. . .,␣2;z*,z;h兲d␮共z*,z兲
−
兰f共0, . . . ,0,z*,z;h兲e␨ f共0,. . .,0;z*,z;h兲d␮共z*,z兲
兰e␨ f共0,. . .,0;z*,z;h兲d␮共z*,z兲
.
Setting ␨ = 0 gives the formula for 共dG / d␨兲共0兲. Differentiating again with respect to ␨ and then
setting ␨ = 0 gives the formula for 共d2G / d␨2兲共0兲. When the measure d␮共z* , z兲 is rotation invariant
and the symmetric coefficient system a共xជ 1 , . . . , xជ s ; yជ * , yជ ; xជ 兲 of f obeys a共xជ 1 , . . . , xជ s ; yជ * , yជ ; xជ 兲 = 0,
whenever yជ = yជ *, we have
冕
f共␣1, . . . , ␣s ;z*,z;h兲d␮共z*,z兲 = 0,
as in Remark 3.3, part 共ii兲, and the remaining formulas follow.
䊏
Corollary 3.6: Denote by F the Banach space of functions f共␣1 , . . . , ␣s ; z* , z ; h兲 with 储f储w
⬍ ⬁. Let f , f ⬘ 苸 F with
冕
f共␣1, . . . , ␣s ;z*,z;h兲d␮共z*,z兲 −
冕⬘
f 共␣1, . . . , ␣s ;z*,z;h兲d␮共z*,z兲 = 0
1
. Define g , g⬘ 苸 F by the conditions
and 储f储w + 储f ⬘ − f储w ⬍ 17
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp
053305-14
J. Math. Phys. 51, 053305 共2010兲
Balaban et al.
兰e f共␣1,. . .,␣s;z*,z;h兲d␮共z*,z兲
兰e f共0,. . .,0;z*,z;h兲d␮共z*,z兲
兰e f ⬘共␣1,. . .,␣s;z*,z;h兲d␮共z*,z兲
= eg共␣1,. . .,␣s;h兲
兰e f ⬘共0,. . .,0;z*,z;h兲d␮共z*,z兲
= eg⬘共␣1,. . .,␣s;h兲 ,
as in Theorem 3.4. Then,
储g⬘ − g储w 艋 4
储f ⬘ − f储w共储f储w + 储f ⬘ − f储w兲
共 171 − 储f储w − 储f ⬘ − f储w兲2
1
Proof: Define for all ␨ , ␨⬘ 苸 C with 兩␨兩 艋 U−1 ⬅ 共 17
− 储 f 储w − 储 f ⬘ − f 储w兲 / 共2储f ⬘ − f储w and 兩␨兩 艋 U⬘−1
⬅共
储 f 储w − 储 f ⬘ − f 储w兲 / 共2储f储w兲,
1
17 −
F共␨, ␨⬘ ; ␣1, . . . , ␣s ;z*,z;h兲 = ␨⬘ f共␣1, . . . , ␣s ;z*,z;h兲 + ␨共f ⬘ − f兲共␣1, . . . ␣s ;z*,z;h兲
and define G共␨兲 by
兰eF共␨,␨⬘;␣1,. . .,␣s;z*,z;h兲d␮共z*,z兲
兰e
F共␨,␨⬘;0,. . .,0;z*,z;h兲
= eG共␨,␨⬘;␣1,. . .␣s;h兲 ,
d␮共z*,z兲
as in Theorem 3.4. Then, G共␨ , ␨⬘兲 is an F-valued analytic function of ␨ , ␨⬘ since the series 共3.4兲
and 共3.3兲 with a replaced by the appropriate a共␨ , ␨⬘兲 converge in the norm 储 · 储w uniformly in ␨ , ␨⬘
1
for 兩␨⬘兩储f储w + 兩␨兩储f ⬘ − f储w bounded by any constant strictly less than 16
. Furthermore, G共0 , 1兲 = g and
G共1 , 1兲 = g⬘ so that
g⬘ − g −
冕
1
0
⳵G
共s,1兲ds =
⳵␨
冕
1
0
⳵G
共s,0兲ds +
⳵␨
冕冕
1
0
1
0
⳵ 2G
共s,s⬘兲dsds⬘ .
⳵␨⳵␨⬘
By hypothesis,
兰共f − f ⬘兲共␣1, . . . , ␣s ;z*,z;h兲es共f−f ⬘兲共␣1,. . .␣s;z*,z;h兲d␮共z*,z兲
⳵G
共s,0兲 =
⳵␨
兰es共f−f ⬘兲共␣1,. . .␣s;z*,z;h兲d␮共z*,z兲
−
兰共f − f ⬘兲共0, . . . ,0;z*,z;h兲es共f−f ⬘兲共0,. . .0;z*,z;h兲d␮共z*,z兲
兰es共f−f ⬘兲共0,. . .0;z*,z;h兲d␮共z*,z兲
vanishes when s = 0 so that
冕 冕
1
g⬘ − g =
s
ds
ds⬙
0
0
⳵ 2G
共s⬙,0兲 +
⳵␨2
冕冕
1
0
1
0
⳵ 2G
共s,s⬘兲dsds⬘
⳵␨⳵␨⬘
and
储g⬘ − g储w 艋
冐
1
⳵ 2G
sup
共s,0兲
2 s苸关0,1兴 ⳵␨2
冐
+
w
sup
s,s⬘苸关0,1兴
冐
⳵ 2G
共s,s⬘兲
⳵␨⳵␨⬘
冐
w
By the Cauchy integral formula,
1
1 ⳵ 2G
2 共s,0兲 =
2 ⳵␨
2␲i
1
⳵ 2G
共s,s⬘兲 =
⳵␨⳵␨⬘
2␲i
冕
兩␨兩=U−1
d␨
冕
兩␨兩=U−1
1
2␲i
冕
d␨
G共s + ␨,0兲
,
␨3
兩␨⬘兩=U⬘−1
d␨⬘
G共s + ␨,s⬘ + ␨⬘兲
.
␨ 2␨ ⬘2
By hypothesis, for all 0 艋 s , s⬘ 艋 1 , ␨ , ␨⬘ with 兩␨兩 艋 1 / U, 兩␨⬘兩 艋 1 / U⬘,
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp
053305-15
J. Math. Phys. 51, 053305 共2010兲
Complex bosonic effective actions. I
储F共s + ␨,s⬘ + ␨⬘兲储w = 储共s⬘ + ␨⬘兲f + 共s + ␨兲共f ⬘ − f兲储w 艋 共1 + 兩␨⬘兩兲储f储w + 共1 + 兩␨兩兲储f ⬘ − f储w 艋
1
,
17
and hence
1
储F共s + ␨,x⬘ + ␨⬘兲储w
17
= 1.
艋
储G共s + ␨,s⬘ + ␨⬘兲储w 艋
16
1 − 16储F共s + ␨,s⬘ + ␨⬘兲储w
1−
17
So
储g⬘ − g储w 艋
1
1
1
1
1 1
2␲U−1 −3 +
2␲U−1 2␲U⬘−1 −2 −2 = U2 + UU⬘ .
2␲
U
2␲
2␲
U U⬘
䊏
IV. THE HISTORY FIELD AND LINEAR TRANSFORMATIONS
In this section, we assume that we are given a metric d and weight factors ␬1 , . . . , ␬s on X. We
consider analytic functions f共␣1 , . . . , ␣s ; h兲 of the complex fields ␣1 , . . . , ␣s and the additional
history field h, which takes values in 兵0,1其. Denote by w the weight system with metric d that
associates the weight factor ␬ j to the field ␣ j, and the constant weight factor 1 to the history field
h 共see Definition 2.6兲. Also fix the number 0 艋 s⬘ 艋 s of source fields.
As pointed out in Sec. I, the purpose of the history field is to keep track of all the points that
have ever been used in the construction of a particular function or monomial. We will deal with
linear changes in the ␣-fields which may be compositions of several such changes of variables. In
each composition it may be relevant which points where involved. This is the motivation for the
following
Definition 4.1:
共i兲
An h-operator or h-linear map A on CX is a linear operator on CX whose kernel is of the
form
⬁
A共x,y兲 = 兺
兺
ᐉ=0 共x ,. . .,x 兲苸Xᐉ
1
ᐉ
共ii兲
A共x;x1, . . . ,xᐉ ;y兲h共x兲h共x1兲 ¯ h共xᐉ兲h共y兲.
The composition A ⴰ B of two h-operators A, B on CX is by definition the h-operator with
kernel
共A ⴰ B兲共x,y兲 =
兺 A共x,z兲B共z,y兲 = z苸X
兺 兺
兺
ᐉ,ᐉ⬘艌0 x1,. . .,xᐉ
y1,. . .,yᐉ⬘
z苸X
A共x;x1, . . . ,xᐉ ;z兲B共z;y1, . . . ,yᐉ ;y兲
h共x兲h共x1兲 ¯ h共xᐉ兲h共z兲h共y1兲 ¯ h共yᐉ⬘兲h共y兲.
共iii兲
Here, we used that h2 = h.
For an “ordinary” linear operator J on CX with kernel J共x , y兲, we define the associated
h-operators by
J̄共x,y兲 = h共x兲J共x,y兲h共y兲
and the associated h-exponential as
⬁
exph共J兲 = h + 兺
ᐉ=1
1 ᐉ
J̄ .
ᐉ!
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp
053305-16
共iv兲
J. Math. Phys. 51, 053305 共2010兲
Balaban et al.
If ␾ is any field on X and A an h-operator, we set
⬁
共A␾兲共x兲 =
A共x;x1, . . . ,xᐉ ;y兲h共x兲h共x1兲 ¯ h共xᐉ兲h共y兲␾共y兲.
兺 A共x,y兲␾共y兲 = ᐉ=0
兺 x ,.兺
. .,x ,y
y苸X
ᐉ
1
Definition 4.2: (Weighted L1 − L⬁ operator norms) Let ␬ , ␬⬘ : X → 共0 , ⬁兴 be weight factors and
␦ an arbitrary metric and let A be an h-linear map on CX. We define the operator norm
N␦共A; ␬, ␬⬘兲 =
冐兺
A共x;xជ ;y兲h共x兲␤l共x兲h共xជ 兲h共y兲␤r共y兲
x,y苸X
x苸X共1兲
冐
,
␻
where ␻ is the weight system with metric ␦ that associates the weight 1 / ␬ to ␤l, the weight ␬⬘ to
␤r and the weight 1 to h, and where ␤l and ␤r are thought of as internal fields. For an ordinary
operator J on CX, we set N␦共J ; ␬ , ␬⬘兲 = N␦共J̄ ; ␬ , ␬⬘兲.
Remark 4.3:
共i兲
Definition 4.2 may be equivalently formulated as follows. For each integer ᐉ 艌 0, set
1
兩A共x;xជ ;y兲兩␬⬘共y兲e␶␦共supp共x,xជ ,y兲兲 ,
␬
共x兲
ᐉ
x苸X
兺 兺
x苸X y苸X
Lᐉ共A; ␬, ␬⬘兲 = sup
ជ
1
兩A共x;xជ ;y兲兩␬⬘共y兲e␶␦共supp共x,xជ ,y兲兲 .
␬
共x兲
ᐉ
x苸X
兺 兺
y苸X x苸X
Rᐉ共A; ␬, ␬⬘兲 = sup
ជ
Then,
N␦共A; ␬, ␬⬘兲 =
共ii兲
兺
ᐉ艌0
max兵Lᐉ共A; ␬, ␬⬘兲,Rᐉ共A; ␬, ␬⬘兲其.
Clearly,
N␦共A ⴰ B; ␬, ␬⬙兲 艋 N␦共A; ␬, ␬⬘兲N␦共B; ␬, ␬⬙兲
共4.1兲
for any two h-operators A, B and any three weight factors ␬, ␬⬘, and ␬⬙.
Proposition 4.4: Let A j, 1 艋 j 艋 s, be h-operators on CX, and let f共␣1 , . . . , ␣s ; h兲 be an analytic
function on a neighborhood of the origin in C共s+1兲兩X兩. Define f̃ by
f̃共␣1, . . . , ␣s ;h兲 = f共A1␣1, . . . ,As␣s ;h兲.
Let ␬1 , . . . , ␬s, ˜␬1 , . . . , ˜␬s be weight factors. Denote by w and w̃ the weight systems with metric d
that associate to the field ␣ j the weight factors ␬ j and ˜␬ j, respectively, and the constant weight
factor 1 to the history field h 共see Definition 2.5兲.
If Nd共A j ; ␬ j , ˜␬ j兲 艋 1 for 1 艋 j 艋 s, then
储f̃储w̃ 艋 储f储w .
Proof: Let a共xជ 1 , . . . , xជ s ; xជ s+1兲 be a symmetric coefficient system for f. Define, for each n共xជ j兲
= n j 艌 0, 1 艋 j 艋 s + 1,
ã共xជ 1, . . . ,xជ s ;xជ s+1兲 =
兺
yជ 1苸Xn1
¯
兺
兺
yជ s苸Xns yជ s+1ⴰzជ 1,1ⴰ¯ⴰzជ s,ns=xជ s+1
s
a共yជ 1, . . . ,yជ s ;yជ s+1兲 兿
j=1
冋兿
nj
ᐉ=1
册
A j共y j,ᐉ ;zជ j,ᐉ ;x j,ᐉ兲 ,
where xជ j = 共x j,1 , . . . , x j,n j兲. Then, ã共xជ 1 , . . . , xជ s ; xជ s+1兲 is a coefficient system for f̃. Since
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp
053305-17
J. Math. Phys. 51, 053305 共2010兲
Complex bosonic effective actions. I
␶d共supp共xជ 1, . . . ,xជ s,yជ s+1 ⴰ zជ 1,1 ⴰ ¯ ⴰ zជ s,ns兲兲 艋 ␶d共supp共yជ 1, . . . ,yជ s,yជ s+1兲兲
+
兺
␶d共supp共y j,ᐉ ;zជ j,ᐉ ;x j,ᐉ兲兲,
1艋j艋s
1艋ᐉ艋n j
we have
˜ 共xជ 1, . . . ,xជ s ;xជ s+1兲兩ã共xជ 1, . . . ,xជ s ;xជ s+1兲兩
␻
艋
兺
兺
␻共yជ 1, . . . ,yជ s,yជ s+1兲兩a共yជ 1, . . . ,yជ s,yជ s+1兲兩
yជ j苸Xn j yជ s+1ⴰzជ 1,1ⴰ¯ⴰzជ s,ns=xជ s+1
1艋j艋s
s
⫻兿
j=1
冋兿
nj
ᐉ=1
e␶d共supp共y j,ᐉ;zជ j,ᐉ;x j,ᐉ兲兲兩A j共supp共y j,ᐉ ;zជ j,ᐉ ;x j,ᐉ兲兲兩
册
˜␬ j共x j,ᐉ兲
.
␬ j共y j,ᐉ兲
We first observe that when xជ 1 = ¯ = xជ s = −, we have
˜ 共− , . . . ,− ;xជ s+1兲兩ã共− , . . . ,− ;xជ s+1兲兩 = w共− , . . . ,− ;xជ s+1兲兩a共− , . . . ,− ;xជ s+1兲兩
␻
so that the corresponding contributions to 储f̃储w̃ and 储f储w are identical. Therefore, we may assume,
without loss of generality, that f共0 , . . . , 0 ; h兲 = 0.
If there are no source fields, we are to bound
储f̃储w̃ =
兺
n1,. . .,ns+1艌0
n1+¯+ns艌1
兺
max max max
x苸X 1艋j¯艋s 1艋ı¯艋n¯
ជ 1,. . .,xជ s+1兲苸Xn1⫻¯⫻Xns+1
j 共x
n¯j⫽0
共xជ¯j兲¯ı=x
w̃共xជ 1, . . . ,xជ s ;xជ s+1兲兩ã共xជ 1, . . . ,xជ s ;xជ s+1兲兩.
⬘ n 艌1
If there are source fields, that is, if s⬘ ⬎ 0 共s⬘ was specified in Definition 2.6兲 and 兺sj=1
j
兺
max max max
x苸X 1艋j¯艋s 1艋ı¯艋n¯
ជ 1,. . .,xជ s+1兲苸Xn1⫻¯⫻Xns+1
j 共x
n¯j⫽0
共xជ¯j兲¯ı=x
is replaced by
max
兺
1艋ᐉ艋n j
1艋j艋s⬘, 1艋ᐉ艋n j
xជ j,ᐉ苸X
共xជ 1,. . .,xជ s+1兲苸Xn1⫻¯⫻Xns+1
1艋j艋s⬘
共xជ 兲 =x̃
j ᐉ
.
j,ᐉ
Choose any n1 , . . . , ns+1 艌 0. If there are no source fields also choose a 1 艋 j̄ 艋 s with n¯j ⫽ 0 and an
1 艋 ı̄ 艋 n j. This is equivalent to choosing the singleton subset I = 兵共j̄ , ı̄兲其 of 兵共j , ᐉ兲 兩 1 艋 j 艋 s , 1 艋 ᐉ
艋 n j其. If there are source fields, set
I = 兵共j,ᐉ兲兩1 艋 j 艋 s⬘, 1 艋 ᐉ 艋 n j其.
Choose an x̃ j,ᐉ 苸 X for each 共j , ᐉ兲 苸 I.
To get 储f̃储w̃, we are to bound the sum, over the choices of n1 , . . . , ns+1, of the max, over the
choices of I and the x̃ j,ᐉ’s, of
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp
053305-18
J. Math. Phys. 51, 053305 共2010兲
Balaban et al.
兺
兺
兺
共xជ 1,. . .,xs+1兲苸Xn1⫻¯⫻Xns+1 yជ j苸Xn j, 1艋j艋s zជ j,ᐉ苸X共1兲, 1艋j艋s, 1艋ᐉ艋n j
yជ s+1ⴰzជ 1,1ⴰ¯ⴰzជ s,n =xជ s+1
共xជ j兲ᐉ=xជ j,ᐉ for 共j,ᐉ兲苸I
yជ s+1苸X共1兲
s
冋兿
册
nj
s
⫻兩a共yជ 1, . . . ,yជ s+1兲兩 兿
ᐉ=1
j=1
w共yជ 1, . . . ,yជ s+1兲
A j兩共y j,ᐉ ;zជ j,ᐉ ;x j,ᐉ兲兩 ,
共4.2兲
where, for each 1 艋 j 艋 s,
A j共y;zជ ;x兲 = e␶d共supp共y;zជ ;x兲兲兩A j共y;zជ ;x兲兩
˜␬ j共x兲
.
␬ j共y兲
Observe that
共4.2兲 =
兺
兺
兺
兺
ms+1艌0
共xជ 1,. . .,xជ s+1兲苸Xn1⫻¯⫻Xns+1
yជ j苸Xn j, 1艋j艋s zជ j,ᐉ苸Xm j,ᐉ, 1艋j艋s, 1艋ᐉ艋n j
m j,ᐉ艌0 for 1艋j艋s, 1艋ᐉ艋n j
yជ s+1ⴰzជ 1,1ⴰ¯ⴰzជ s,n =xជ s+1
共xជ j兲ᐉ=xជ j,ᐉ for 共j,ᐉ兲苸I
yជ s+1苸Xms+1
s
ms+1+⌺ j,ᐉm j,ᐉ=ns+1
s
w共yជ 1, . . . ,yជ s+1兲兩a共yជ 1, . . . ,yជ s+1兲兩 兿
j=1
冋兿
册
nj
ᐉ=1
A j兩共y j,ᐉ ;zជ j,ᐉ ;x j,ᐉ兲兩 .
In taking the sum, over the choices of n1 , . . . , ns+1, of the max, over the choices of I and the x̃ j,ᐉ’s,
of 共4.2兲, we apply “max 兺 艋 兺 max” to give
储f̃储w̃ 艋
兺
兺
n1,. . .,ns艌0
兺
max max
I
ms+1艌0
m j,ᐉ艌0 for 1艋j艋s, 1艋ᐉ艋n j
s
⫻w共yជ 1, . . . ,yជ s+1兲兩a共yជ 1, . . . ,yជ s+1兲兩 兿
j=1
兺
兺
x̃ j,ᐉ苸x 共x ,. . .,x 兲苸Xn1⫻¯⫻Xns y 苸Xn j, 1艋j艋s
zជ j,ᐉ苸Xm j,ᐉ
ជ1
ជs
ជj
共j,ᐉ兲苸I 共x 兲 =x for 共j,ᐉ兲苸I
m
ជ j ᐉ ជ j,ᐉ
yជ s+1苸X s+1 1艋j艋s, 1艋ᐉ艋n j
冋兿
册
nj
ᐉ=1
A j兩共y j,ᐉ ;zជ j,ᐉ ;x j,ᐉ兲兩 ,
where I is chosen as above.
For each 共j , ᐉ兲 苸 I and y j,ᐉ 苸 X, we have, in the notation of Remark 4.3, part 共i兲,
兺
A j共y j,ᐉ ;zជ j,ᐉ ;x j,ᐉ兲 艋 Lm j,ᐉ共A j ; ␬ j, ˜␬ j兲.
x j,ᐉ苸X
zជ j,ᐉ苸Xm j,ᐉ
Thus,
兺
兺
兺
共xជ 1,. . .,xជ s兲苸Xn1⫻¯⫻Xns yជ j苸Xn j, 1艋j艋s
zជ j,ᐉ苸Xm j,ᐉ
m
共xជ j兲ᐉ=xជ j,ᐉ for 共j,ᐉ兲苸I
yជ s+1苸X s+1 1艋j艋s, 1艋ᐉ艋n j
s
w共yជ 1, . . . ,yជ s+1兲兩a共yជ 1, . . . ,yជ s+1兲兩 兿
艋
兺
y j,ᐉ
y j,ᐉ
共j,ᐉ兲苸I
⫻
zជ j,ᐉ
冉兺
共j,ᐉ兲苸I
兺
j=1
兺
冋兿
nj
ᐉ=1
A j兩共y j,ᐉ ;zជ j,ᐉ ;x j,ᐉ兲兩
w共yជ 1, . . . ,yជ s+1兲兩a共yជ 1, . . . ,yជ s+1兲兩
yជ s+1苸Xms+1
兿
苸Xm j,ᐉ 共j,ᐉ兲苸I
A j兩共y j,ᐉ ;zជ j,ᐉ ;x j,ᐉ兲兩
兿
共j,ᐉ兲苸I
冊
册
Lm,ᐉ共A j ; ␬ j, ˜␬ j兲
共j,ᐉ兲苸I
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp
053305-19
J. Math. Phys. 51, 053305 共2010兲
Complex bosonic effective actions. I
艋
冉
兺
max
y j,ᐉ
兺
共j,ᐉ兲苸I 共j,ᐉ兲苸I
兺
兺
⫻
y j,ᐉ
共j,ᐉ兲苸I
w共yជ 1, . . . ,yជ s+1兲兩a共yជ 1, . . . ,yជ s+1兲兩
yជ s+1苸Xms+1
y j,ᐉ
兿
A j兩共y j,ᐉ ;zជ j,ᐉ ;x j,ᐉ兲兩
zជ j,ᐉ苸Xm j,ᐉ 共j,ᐉ兲苸I
兿
共j,ᐉ兲苸I
冊
Lm,ᐉ共A j ; ␬ j, ˜␬ j兲
共j,ᐉ兲苸I
艋 储wa储n1,. . .,ns,ms+1
兿
共j,ᐉ兲苸I
Rm j,ᐉ共A j ; ␬ j, ˜␬ j兲
兿
共j,ᐉ兲苸I
Lm j,ᐉ共A j ; ␬ j, ˜␬ j兲
since, for each 共j , ᐉ兲 苸 I,
兺
y j,ᐉ苸X
A j共y j,ᐉ ;zជ j,ᐉ ;x j,ᐉ兲 艋 Rm j,ᐉ共A j ; ␬ j, ˜␬ j兲.
zជ j,ᐉ苸Xm j,ᐉ
Hence,
储f̃储w̃ 艋
艋
兺
兺
max储wa储n1,. . .,nsms+1
兺
兺
储wa储n1,. . .,nsms+1 兿 max兵Rm j,ᐉ共A j ; ␬ j, ˜␬ j兲,Lm j,ᐉ共A j ; ␬ j, ˜␬ j兲其
n1,. . .,ns艌0 ms+1艌0
m j,ᐉ艌0
n1,. . .,ns艌0 ms+1艌0
m j,ᐉ艌0
I
兿
共j,ᐉ兲苸I
Rm j,ᐉ共A j ; ␬ j, ˜␬ j兲
兿
共j,ᐉ兲苸I
Lm j,ᐉ共A j ; ␬ j, ˜␬ j兲
共j,ᐉ兲
s
艋
Nd共A j ; ␬k, ˜␬ j兲n 艋 储f储w .
兺 兺 储wa储n ,. . .,n m 兿
n ,. . .,n 艌0 m 艌0
j=1
j
1
1
s
s s+1
s+1
䊏
Proposition 4.4 treats substitutions “field by field.” Corollary 4.6, below generalizes Proposition 4.4 to allow substitutions that mix fields. In preparation for the corollary, we have the
following.
Lemma 4.5: Let f共␣1 , . . . , ␣s ; h兲 be an analytic function on a neighborhood of the origin in
C共s+1兲兩X兩 with 0 艋 s⬘ ⬍ s source fields (see Definition 2.6).
共i兲
Let r 苸 N and define f̃ by
f̃共␣1, . . . , ␣s−1, ␤1, . . . , ␤r ;h兲 = f共␣1, . . . , ␣s−1, ␤1 + ¯ + ␤r ;h兲
Let ˜␬1 , . . . , ˜␬r be weight factors and assume that
r
˜␬ 共x兲
j
艋 1.
sup
兺
␬
x苸X
s共x兲
j=1
Denote by w̃ the weight system with metric d that associates the weight factor ␬i to the field
␣i, for 1 艋 i 艋 s − 1, the weight factor ˜␬ j to the field ␤ j, for 1 艋 j 艋 r, and the constant weight
factor 1 to the history field h. Then,
储f̃储w̃ 艋 储f储w .
共ii兲
Let 兵1 , . . . , s其 = I1 艛 ¯ 艛 Ir be a partition of 兵1 , . . . , s其 into disjoint nonempty subsets. If the
number of source fields s⬘ 艌 1, assume that I j = 兵j其 for all 1 艋 j 艋 s⬘. Define f̃ by
f̃共␤1, . . . , ␤r ;h兲 = 兩f共␣1, . . . , ␣s ;h兲兩
␣i=␤ j if i苸I j
1艋i艋s, 1艋j艋r
.
Let ˜␬1 , . . . , ˜␬r be weight factors and assume that ˜␬ j 艋 ␬i for all 1 艋 i 艋 s and 1 艋 j 艋 r with
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp
053305-20
J. Math. Phys. 51, 053305 共2010兲
Balaban et al.
i 苸 I j. Denote by w̃ the weight system with metric d that associates the weight factor ˜␬ j to
the field ␤ j, for 1 艋 j 艋 r, and the constant weight factor 1 to the history field h. Then,
储f̃储w̃ 艋 储f储w .
共iii兲
Let r 苸 N and let X = ⌳1 艛 ¯ 艛 ⌳r be a partition of X into disjoint subsets. Define f by
f̃共␣1, . . . , ␣s−1, ␤1, . . . , ␤r ;h兲 = f共␣1, . . . , ␣s−1,⌳1␤1 + ¯ + ⌳r␤r ;h兲.
Let ˜␬1 , . . . , ˜␬r be weight factors and assume that there is a ␯ ⬎ 1 such that
max sup
1艋j艋r x苸⌳ j
˜␬ j共x兲 1
艋 ⬍ 1.
␬s共x兲 ␯
Denote by w̃ the weight system with metric d that associates the weight factor ␬i to the field
␣i, for 1 艋 i 艋 s − 1, the weight factor ˜␬ j to the field ␤ j, for 1 艋 j 艋 r, and the constant weight
factor 1 to the history field h. Then,
储f̃储w̃ 艋 Cr,␯储f储w
with Cr,␯ =
冦
␯
冉 冊
r
e ln ␯
2r
␯
1
if ␯ 艋 er/2
if er/2 艋 ␯ ⬍ 2r
if ␯ 艌 2r .
冧
Proof:
共i兲
Let a共xជ 1 , . . . , xជ s ; zជ 兲 be a symmetric coefficient system for f. Since a is invariant under
permutation of its xs components,
f̃共␣1, . . . , ␣s−1, ␤1, . . . , ␤r ;h兲
= f共␣1, . . . , ␣s−1, ␤1 + ¯ + ␤r ;h兲
兺
=
冉 兿 冊冉兺 冊
s−1
xជ i苸x共1兲, 1艋i艋s
a共xជ 1, . . . ,xជ s ;zជ 兲
ᐉ=1
r
␣ᐉ共xជ ᐉ兲
␤ j 共xជ s兲h共zជ 兲
j=1
zជ 苸X共1兲
兺
=
兺
xជ i苸x共1兲, 1艋i艋s yជ i苸x共1兲, 1艋i艋r
xជ s=yជ 1ⴰyជ 1ⴰ¯ⴰyជ r
zជ 苸X共1兲
冉 兿 冊冉
s−1
⫻
ᐉ=1
␣ᐉ共xជ ᐉ兲
a共xជ 1, . . . ,xជ s−1,yជ 1 ⴰ ¯ ⴰ yជ r ;zជ 兲
n共yជ 1兲 + ¯ + n共yជ r兲
n共yជ 1兲, . . . ,n共yជ r兲
冊冉 兿 冊
r
␤ j共yជ j兲 h共zជ 兲,
j=1
we have that
ã共xជ 1, . . . ,xជ s−1,yជ 1, . . . ,yជ r ;zជ 兲 = a共xជ 1, . . . ,xជ s−1,yជ 1 ⴰ ¯ ⴰ yជ r ;zជ 兲
冉
n共yជ 1兲 + ¯ + n共yជ r兲
n共yជ 1兲, . . . ,n共yជ r兲
冊
is a symmetric coefficient system for f̃.
When xជ 1 = ¯ = xជ s−1 = yជ 1 = ¯ = yជ r = −, we have
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp
053305-21
J. Math. Phys. 51, 053305 共2010兲
Complex bosonic effective actions. I
w̃共− , . . . ,− ;zជ 兲兩ã共− , . . . ,− ;zជ 兲兩 = w共− , . . . ,− ;zជ 兲兩a共− , . . . ,− ;zជ 兲兩
so that the corresponding contributions to 储f̃储w̃ and 储f储w are identical. Therefore, we may
assume, without loss of generality, that f̃共0 , . . . , 0 ; h兲 = f共0 , . . . , 0 ; h兲 = 0.
Set, for each 1 艋 j 艋 r, t j = supx苸X关˜␬ j共x兲 / ␬s共x兲兴. Thus, t1 + ¯ +tr 艋 1 and ˜␬ j共x兲 艋 t j␬s共x兲 for
all 1 艋 j 艋 r and all x 苸 X. If there are no source fields, we have
储f̃储w̃ =
兺
兺
max
ni艌0, 1艋i艋s−1
x苸X
m j艌0, 1艋j艋r
1艋i艋⌺ni+⌺m j
w̃共xជ 1, . . . ,xជ s−1,yជ 1, . . . ,yជ r ;zជ 兲
zជ 苸Xᐉ
xជ i苸Xni, 1艋i艋s−1
ᐉ艌0
yជ j苸Xm j, 1艋j艋r
共xជ 1, . . . ,xជ s−1,yជ 1, . . . ,yជ r兲i=x
⫻兩ã共xជ 1, . . . ,xជ s−1,yជ 1, . . . ,yជ r ;zជ 兲兩
艋
兺
max
ni艌0, 1艋i艋s−1
x苸X
m j艌0, 1艋j艋r
1艋i艋⌺ni+⌺m j
ᐉ艌0
⫻
冉
m1 + ¯ + mr
m1, . . . ,mr
兺
艋
ni艌0, 1艋i艋s−1
冉
冊
兺
mr
1
tm
ជ 1, . . . ,xជ s−1,yជ 1 ⴰ . . . ⴰ yជ r ;zជ 兲
1 ¯ tr w共x
z苸Xᐉ
ជ
xជ i苸Xni, 1艋i艋s−1
yជ j苸Xm j, 1艋j艋r
共xជ 1, . . . ,xជ s−1,yជ 1, . . . ,yជ r兲i=x
兩a共xជ 1, . . . ,xជ s−1,yជ 1 ⴰ . . . ⴰ yជ r ;zជ 兲兩
m1 + ¯ + mr
m1, . . . ,mr
冊
mr
1
tm
1 ¯ tr
兺
max
x苸X
1艋i艋⌺ni+⌺m j
m j艌0, 1艋j艋r
ᐉ艌0
w共xជ 1, . . . ,xជ s−1,xជ s ;zជ 兲
zជ 苸Xᐉ
xជ i苸Xni, 1艋i艋s−1
xជ s苸X⌺m j
共xជ 1, . . . ,xជ s兲i=x
⫻兩a共xជ 1, . . . ,xជ s−1,xជ s ;zជ 兲兩
兺
艋
ni艌0, 1艋i艋s
兺
共t1 + ¯ + tr兲ns max
x苸X
1艋i艋⌺ni
ᐉ艌0
w共xជ 1, . . . ,xជ s ;zជ 兲兩a共xជ 1, . . . ,xជ s ;zជ 兲兩
zជ 苸Xᐉ
xជ 苸Xni, 1艋i艋s
共xជ 1, . . . ,xជ s兲i=x
艋 储f储w .
共ii兲
The case when there are source fields is similar.
It suffices to consider the case that r = s − 1, Ir = 兵s − 1 , s其, and I j = 兵j其 for all 1 艋 j 艋 s − 2. The
remaining cases follow by repeatedly reordering the field indices and applying the special
case. Let a共xជ 1 , . . . , xជ r−1 , yជ 1 , yជ 2 ; xជ r+1兲 be a symmetric coefficient system for f. As
f̃共␤1, . . . , ␤r ;h兲 = f共␤1, . . . , ␤r−1, ␤r, ␤r ;h兲 =
兺
xជ i苸x共1兲, 1艋i艋r−1
a共xជ 1, . . . ,xជ r−1,yជ 1,yជ 2 ;zជ 兲
yជ 1,yជ 2,zជ 苸X共1兲
冉兿 冊
r−1
⫻
=
ᐉ−1
␤ᐉ共xជ ᐉ兲 ␤r共yជ 1兲␤r共yជ 2兲h共zជ 兲
兺
兺
xជ i苸x共1兲, 1艋i艋r yជ 1,yជ 2苸x共1兲
xជ r=yជ 1ⴰyជ 2
zជ 苸X共1兲
冉兿 冊
r
a共xជ 1, . . . ,xជ r−1,yជ 1,yជ 2 ;zជ 兲
ᐉ−1
␤ᐉ共xជ ᐉ兲 h共zជ 兲,
we have that
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp
053305-22
J. Math. Phys. 51, 053305 共2010兲
Balaban et al.
兺
ã共xជ 1, . . . ,xជ r ;zជ 兲 =
yជ 1,yជ 2苸x共1兲
xជ r=yជ 1ⴰyជ 2
a共xជ 1, . . . ,xជ r−1,yជ 1,yជ 2 ;zជ 兲
is a, not necessarily symmetric, coefficient system for f̃. Again we may assume, without loss of
generality, that f̃共0 , . . . , 0 ; h兲 = f共0 , . . . , 0 , 0 ; h兲 = 0.
If there are no source fields, we have, by Remark 2.7, part 共i兲,
储f̃储w̃ 艋
兺
n1+¯+nr艌1
ᐉ艌0
兺
max max max
x苸X 1艋j¯艋r 1艋ı¯艋n¯
j
n¯j⫽0
w̃共xជ 1, . . . ,xជ r ;zជ 兲兩aជ 共xជ 1, . . . ,xជ r ;zជ 兲兩
xជ 苸Xᐉ
共xជ 1,. . .,xជ r兲苸Xn1⫻¯⫻Xnr
共xជ¯j兲¯j=x
艋
兺
n1+¯nr艌1
ᐉ艌0
兺
max max max
x苸X 1艋j¯艋r 1艋ı¯艋n¯
j
n¯j⫽0
兺
yជ 1,yជ 2苸X共1兲
n
j
xជ j苸X , 1艋j艋r xជ r=yជ 1ⴰyជ 2
zជ 苸Xᐉ
w共xជ 1, . . . ,xជ r−1,yជ 1,yជ 2 ;zជ 兲
共xជ¯j兲¯ı=x
⫻兩a共xជ 1, . . . ,xជ r−1,yជ 1,yជ 2 ;zជ 兲兩
=
兺
n1+¯+nr艌1
ᐉ艌0
兺
max
x苸X
兺
m1,m2艌0
zជ 苸Xᐉ
1艋i艋n1+¯+nr m +m =n
1
2 r
w共xជ 1, . . . ,xជ r−1,yជ 1,yជ 2 ;zជ 兲
yជ 1苸Xm1, yជ 2苸Xm2
xជ j苸xn j, 1艋j艋r−1
共xជ 1, . . . ,xជ r−1,yជ 1,yជ 2兲i=x
⫻兩a共xជ 1, . . . ,xជ r−1,yជ 1,yជ 2 ;zជ 兲兩
艋
兺
兺
max
x苸X
n j艌0, 1艋j艋r−1
1艋i艋n1+¯+nr−1+m1+m2
m1,m2艌0
ᐉ艌0
zជ 苸Xᐉ
w共xជ 1, . . . ,xជ r−1,yជ 1,yជ 2 ;zជ 兲
yជ 1苸Xm1, yជ 2苸Xm2
xជ j苸xn j, 1艋j艋r−1
共xជ 1, . . . ,xជ r−1,yជ 1,yជ 2兲i=x
⫻兩a共xជ 1, . . . ,xជ r−1,yជ 1,yជ 2 ;zជ 兲兩 = 储f储w .
The case when there are source fields is similar.
共iii兲
Let a共xជ 1 , . . . , xជ s ; zជ 兲 be a symmetric coefficient system for f. As in part 共i兲, we have that
冉兿 冊
r
ã共xជ 1, . . . ,xជ s−1,yជ 1, . . . ,yជ r ;zជ 兲 = a共xជ 1, . . . ,xជ s−1,yជ 1 ⴰ ¯ ⴰ yជ r ;zជ 兲
⫻
冉
n共yជ 1兲 + ¯ + n共yជ r兲
n共yជ 1兲, ¯ ,n共yជ r兲
冊
⌳ j共yជ j兲
j=1
is a symmetric coefficient system for f̃. Again, we may assume, without loss of generality,
that f̃共0 , . . . , 0 ; h兲 = f共0 , . . . , 0 ; h兲 = 0.
By hypothesis, ˜␬ j共x兲 艋 共1 / v兲␬s共x兲 for all 1 艋 j 艋 r and all x 苸 ⌳ j. If there are no source fields,
we have
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp
053305-23
储f̃储w̃ =
J. Math. Phys. 51, 053305 共2010兲
Complex bosonic effective actions. I
兺
max
ni艌0, 1艋i艋s−1
x苸X
m j艌0, 1艋j艋r
1艋p艋⌺ni+⌺m j
ᐉ艌0
兺
w̃共xជ 1, . . . ,xជ s−1,yជ 1, . . . ,yជ r ;zជ 兲
zជ 苸Xᐉ
xជ i苸Xni, 1艋i艋s−1
yជ j苸Xm j, 1艋j艋r
共xជ 1, . . . ,xជ s−1,yជ 1, . . . ,yជ r兲 p=x
⫻兩ã共xជ 1, . . . ,xជ s−1,yជ 1, . . . ,yជ r ;zជ 兲兩
艋
兺
max
ni艌0, 1艋i艋s−1
x苸X
m j艌0, 1艋j艋r
1艋p艋⌺n j+⌺m j
冉
1
v
兺
zជ 苸Xᐉ
m1+¯+mr
w共xជ 1, . . . ,xជ s−1,yជ 1 ⴰ ¯ ⴰ yជ r ;zជ 兲
xជ i苸Xni, 1艋i艋s−1
ᐉ艌0
⫻
冉冊
m
yជ j苸⌳ j j, 1艋j艋r
m1 + ¯ + mr
m1, ¯ ,mr
冊
共xជ 1, . . . ,xជ s−1,yជ 1, . . . ,yជ r兲 p=x
兩a共xជ 1, . . . ,xជ s−1,yជ 1 ⴰ ¯ ⴰ yជ r ;zជ 兲兩.
共4.3兲
We claim that
兺
max
x苸X
zជ 苸Xᐉ
1艋p艋⌺n j+⌺m j
冉
m1 + ¯ + mr
m1, ¯ ,mr
冊
w共xជ 1, . . . ,xជ s−1,yជ 1 ⴰ ¯ ⴰ yជ r ;zជ 兲兩a共xជ 1, . . . ,xជ s−1,yជ 1
xជ i苸Xni, 1艋i艋s−1
m
yជ j苸⌳ j j, 1艋j艋r
共xជ 1, . . . ,xជ s−1,yជ 1, . . . ,yជ r兲 p=x
ⴰ ¯ ⴰ yជ r ;zជ 兲兩 艋 共m1 + ¯ + mr兲
max
x苸X
1艋p艋⌺n j+⌺m j
兺
zជ 苸Xᐉ
w共xជ 1, . . . ,xជ s−1,xជ s ;zជ 兲
xជ i苸Xni, 1艋i艋s−1
xជ s苸X⌺m j
共xជ 1, . . . ,xជ s兲 p=x
⫻兩a共xជ 1, . . . ,xជ s−1,xជ s ;zជ 兲兩.
共4.4兲
To see this, fix x to be element of X and p to be the integer between 1 and 兺ni + 兺m j that give the
s−1
s−1
ni, set p̄ = p − 兺i=1
ni and let 1 艋 j̄ 艋 r obey
maximum for the left hand side. In the event that p ⬎ 兺i=1
¯
x 苸 ⌳j. Denote by n⌳共xជ 兲 the number of components of xជ that are in ⌳. Then,
兺
zជ 苸Xᐉ
兺
w共xជ 1, . . . ,xជ s−1,xជ s ;zជ 兲兩a共xជ 1, . . . ,xជ s−1,xជ s ;zជ 兲兩 艌
zជ 苸Xᐉ
xជ i苸Xni, 1艋i艋s−1
xជ i苸Xni, 1艋i艋s−1
xជ s苸X⌺m j
xជ s苸X⌺m j
共xជ 1, . . . ,xជ s兲 p=x
共xជ 1, . . . ,xជ s兲 p=x
w共xជ 1, . . . ,xជ s−1,xជ s ;zជ 兲
n⌳ 共xជ 兲=m j, 1艋j艋r
j
⫻兩a共xជ 1, . . . ,xជ s−1,xជ s ;zជ 兲兩.
To each xជ s 苸 X兺m j with n⌳ j共xជ s兲 = m j for each 1 艋 j 艋 r and with 共xជ s兲 p̄ = x 共if p̄ is defined兲 we assign
the unique permutation ␲ 苸 S兺m j with
• ␲共p̄兲 = p̄ 共if p̄ is defined兲 and
• ␲共1兲 being the index of the first component of xជ s 关other than 共xជ s兲 p̄ if p̄ is defined兴 with
共xជ s兲␲共1兲 苸 ⌳1,
• ␲共2兲 being the index of the second component of xជ s 关other than 共xជ s兲 p̄ if p̄ is defined兴 with
共xជ s兲␲共2兲 苸 ⌳1,
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp
053305-24
J. Math. Phys. 51, 053305 共2010兲
Balaban et al.
• ␲共兺m j兲 being the index of the last component of xជ s 关other than 共xជ s兲 p̄ if p̄ is defined兴 with
共xជ s兲␲共兺m j兲 苸 ⌳r.
The set ⌸ of such permutations contains exactly
冉
m1 + ¯ + mr − 1
m 1, . . .
,m¯j
− 1, . . . ,mr
冊
=
冉
共 mm+¯+m
兲 elements if p̄ is not defined and
,. . .,m
1
1
r
冊
r
冉
m1 + ¯ + mr
m1 + ¯ + mr
1
mj
艌
m1 + ¯ + mr m1, . . . ,mr
m1 + ¯ + mr m1, . . . ,mr
冊
elements if p̄ is defined. Hence, if we rename the components of xជ s 共in order兲 that are in ⌳ j to be
yជ j so that xជ s = ␲−1共yជ 1 ⴰ ¯ ⴰ yជ r兲,
兺
w共xជ 1, . . . ,xជ s−1,xជ s ;zជ 兲兩a共xជ 1, . . . ,xជ s−1,xជ s ;zជ 兲兩
zជ 苸Xᐉ
xជ i苸Xni, 1艋i艋s−1
xជ s苸X⌺m j
共xជ 1, . . . ,xជ s兲i=x
艌
兺
兺
␲苸⌸
w共xជ 1, . . . ,xជ s ;zជ 兲兩兩a共xជ 1, . . . ,xជ s ;zជ 兲兩兩xជ s=␲−1共yជ 1ⴰ¯ⴰyជ r兲
zជ 苸Xᐉ
xជ i苸Xni, 1艋i艋s−1
m
yជ j苸⌳ j j, 1艋j艋r
共xជ 1, . . . ,xជ s−1, ␲−1共yជ 1 ⴰ ¯ ⴰ yជ r兲兲 p=x
=
兺
兺
␲苸⌸
w共xជ 1, . . . ,xជ s ;zជ 兲兩兩a共xជ 1, . . . ,xជ s ;zជ 兲兩兩xជ s=yជ 1ⴰ¯ⴰyជ r
zជ 苸Xᐉ
xជ i苸Xni, 1艋i艋s−1
m
yជ j苸⌳ j j, 1艋j艋r
共xជ 1, . . . ,xជ s−1,yជ 1 ⴰ ¯ ⴰ yជ r兲 p=x
兺
= #共⌸兲
w共xជ 1, . . . ,xជ s ;zជ 兲兩兩a共xជ 1, . . . ,xជ s ;zជ 兲兩兩xជ s=yជ 1ⴰ¯ⴰyជ r
zជ 苸Xᐉ
xជ i苸Xni, 1艋i艋s−1
m
yជ j苸⌳ j j, 1艋j艋r
共xជ 1, . . . ,xជ s−1,yជ 1 ⴰ ¯ ⴰ yជ r兲 p=x
since w共xជ 1 , . . . , xជ s ; zជ 兲 and a共xជ 1 , . . . , xជ s ; zជ 兲 are invariant under permutations of the components of xជ s
and the condition 共xជ 1 , . . . , xជ s−1 , ␲−1共yជ 1 ⴰ ¯ ⴰ yជ r兲兲 p = x is equivalent to the condition
共xជ 1 , . . . , xជ s−1 , yជ 1 , . . . , yជ r兲 p = x. This yields 共4.4兲.
Substituting 共4.4兲 into 共4.3兲, we have
储f̃储w̃ 艋
兺
ni艌0, 1艋i艋s−1
冉冊
1
␯
m1+¯+mr
共m1 + ¯ + mr兲
兺
max
x苸X
1艋p艋⌺ni+⌺m j
m j艌0, 1艋j艋r
ᐉ艌0
⫻兩a共xជ 1, . . . ,xជ s ;zជ 兲兩 艋
兺
ni艌0, 1艋i艋s
共ns + 1兲r−1ns
ns艌0
1
␯
xជ i苸Xni, 1艋i艋s−1
xជ s苸X⌺m j
共xជ 1, . . . ,xជ s兲 p=x
ns
max
x苸X
1艋p艋⌺ni
ᐉ艌0
⫻兩a共xជ 1, . . . ,xជ s ;zជ 兲兩 艋 储f储w sup 共ns + 1兲r
冉冊
冉冊
1
␯
ns
w共xជ 1, . . . ,xជ s ;zជ 兲
zជ 苸Xᐉ
兺
zជ 苸Xᐉ
w共xជ 1, . . . ,xជ s ;zជ 兲
xជ i苸Xni, 1艋i艋s
共xជ 1, . . . ,xជ s兲 p=x
艋 Cr,␯储f储w ,
where
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp
053305-25
J. Math. Phys. 51, 053305 共2010兲
Complex bosonic effective actions. I
Cr,␯ =
冦
␯
冉 冊
r
e ln ␯
r
if ␯ 艋 e1/2
2r
␯
1
if e1/2 艋 ␯ ⬍ 2r
if ␯ 艌 2r .
冧
The case when there are source fields is similar, but easier since there is no condition
s−1
n i.
䊏
共xជ 1 , . . . , xជ s−1 , yជ 1 , . . . , yជ r兲 p = x with p ⬎ 兺i=1
Corollary 4.6: Let h共␥1 , . . . , ␥r ; h兲 be an analytic function on a neighborhood of the origin in
C共r+1兲兩X兩, and let ⌫ij be h-operators on CX, 1 艋 i 艋 s, 1 艋 j 艋 r. Set
冉兺
s
h̃共␣1, . . . , ␣s ;h兲 = h
冊
s
⌫i1␣i,
i=1
. . . , 兺 ⌫ri␣i ;h .
i=1
Furthermore, let ␭ j : X → 共0 , ⬁兴 for 1 艋 j 艋 r be weight factors. Let w␭ be the weight system with
metric d that associates the weight factor␭ j and the constant weight factor 1 to the history field h.
Assume that
兺
Nd共⌫ij ;␭ j, ␬i兲 艋 1
i=1,. . .,s
for each 1 艋 j 艋 r. Then,
储h̃储w 艋 储h储w␭ .
Proof: We introduce auxiliary fields 兵␣i,j其 1艋i艋s and 兵␤i,j其 1艋i艋s and define
1艋j艋r
h⬘共兵␤i,j其 1艋i艋s ;h兲 = h
1艋j艋r
冉
1艋j艋r
s
s
i=1
i=1
冊
兺 ␤i1, . . . , 兺 ␤ir ;h ,
h⬙共兵␣i,j其 1艋i艋s ;h兲 = h⬘共兵⌫ij␣i,j其 1艋i艋s ;h兲.
1艋j艋r
1艋j艋r
Then,
h̃共␣1, . . . , ␣s ;h兲 = h⬙兩共兵␣i,j其 1艋i艋s ;h兲兩
1艋j艋r
␣i,j=␣
i
1艋i艋s, 1艋j艋r
.
Set ti,j = Nd共⌫ij ; ␭ j , ␬i兲 and introduce the auxiliary weight system w⬘ with metric d that associates
the weight factor one to the history field,
• the weight factor ␬i to the field ␣i,j and
• the weight factor ti,j␭ j to the field ␤i,j.
By part 共ii兲 of Lemma 4.5,
储h̃共␣1, . . . , ␣s ;h兲储w 艋 储h⬙共兵␣i,j其 1艋i艋s ;h兲储w⬘ .
1艋j艋r
By Proposition 4.4,
储h⬙共兵␣i,j其 1艋i艋s ;h兲储w⬘ 艋 储h⬘共兵␤i,j其 1艋i艋s ;h兲储w⬘
1艋j艋r
1艋j艋r
since
Nd共⌫ij ;ti,j␭ j, ␬i兲 =
1
Nd共⌫ij ;␭ j, ␬i兲 = 1.
ti,j
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp
053305-26
J. Math. Phys. 51, 053305 共2010兲
Balaban et al.
By part 共i兲 of Lemma 4.5,
储h⬘共兵␤i,j其 1艋i艋s ;h兲储w⬘ 艋 储h共␥1, . . . , ␥r ;h兲储w␭
1艋j艋r
since, for each 1 艋 j 艋 r,
s
s
s
ti,j␭ j共x兲
= 兺 ti,j = 兺 Nd共⌫ij ;␭ j, ␬i兲 艋 1.
sup
␭
共x兲
x
j
i=1
i=1
兺
i=1
䊏
APPENDIX A: CHANGE OF VARIABLES FORMULAS IN A SIMPLE SETTING
The results of Secs. II–IV will be applied in their full generality to the construction of the
temporal ultraviolet limit of a many boson system in Ref. 4. In the second part of the paper,3 we
discuss the small field part of this construction. For it we need neither history nor source fields and
can do with constant weight factors as in 共1.4兲. For this reason, we specialize our main change of
variables formulas, Corollary 4.6 to this setting 共see Corollary A.2, below兲. We also provide
another change of variables formula, Proposition A.3, which is special for multilinear functions.
For an abstract framework, we consider analytic functions f共␣1 , . . . , ␣s兲 of the complex fields
␣1 , . . . , ␣s, none of which are history or source fields. We assume that we are given a metric d and
constant weight factors ␬1 , . . . , ␬s. Denote by w the weight system with metric d that associates the
weight factor ␬ j to the field ␣ j.
In this environment, Definition 2.6, for the norm of the function
f共␣1, . . . , ␣s兲 =
兺
共xជ 1,. . .,xជ s兲苸x共s兲
a共xជ 1, . . . ,xជ s兲␣1共xជ 1兲 ¯ ␣s共xជ s兲
with a共xជ 1 , . . . , xជ s兲 a symmetric coefficient system, simplifies to
储f储w = 兩a共− 兲兩 +
兺
n1,. . .,ns艌0
n1,. . .,ns艌1
max max max
兺
x苸X 1艋j艋s 1艋i艋n j
xជ ᐉ苸Xnᐉ
n j⫽0
1艋ᐉ艋s
兩a共xជ 1, . . . ,xជ s兲兩␬n11 ¯ ␬snse␶d共xជ 1,. . .,xជ s兲 .
共A1兲
共xជ j兲i=x
As well, for the change in variable formulas of Sec. IV, one only needs a special case of the
operator norm in Definition 4.2.
Definition A.1: Let A be a linear map on CX. We define the operator norm
再
储兩A兩储 = Nd共A; 1, 1兲 = max sup
兺 ed共x,y兲兩A共x,y兲兩,sup
兺 ee共x,y兲兩A共x,y兲兩
y苸X x苸X
x苸X y苸X
冎
.
Observe that for constant weight factors ␭ and ␬, Nd共A ; ␭ , ␬兲 = 共␬ / ␭兲储兩A兩储. Hence, Corollary 4.6 is
simplified as follows.
Corollary A.2: Let h共␥1 , . . . , ␥r兲 be an analytic function on a neighborhood of the origin in
Cr兩X兩, and let ⌫ij be linear operators on CX, 1 艋 i 艋 s, 1 艋 j 艋 r. Set
冉兺
s
h̃共␣1, . . . , ␣s兲 = h
i=1
s
⌫i1␣i,
冊
. . . , 兺 ⌫ri␣i .
i=1
Furthermore, let ␭1 , . . . , ␭r be constant weight factors and let w␭ be the weight system with metric
d that associates the weight factor ␭ j to the field ␥ j. Assume that
s
␬
i
储兩⌫ij兩储 艋 1
兺
i=1 ␭ j
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp
053305-27
J. Math. Phys. 51, 053305 共2010兲
Complex bosonic effective actions. I
for each 1 艋 j 艋 r. Then,
储h̃储w 艋 储h储w␭ .
Proposition A.3: Let h共␥1 , . . . , ␥r兲 be a multilinear form in the fields ␥1 , . . . , ␥r. Furthermore,
let
˜⌫ j,
i
⌫ij,
A j,
and
à j
共i = 1 , . . . , r; j = 1 , . . . , s兲 be linear operators on CX. Set
冉兺
s
f 1共 ␣ 1, . . . , ␣ s兲 = h
. . . , 兺 ⌫rj␣ j ,
j=1
冉兺
s
f 2共 ␣ 1, . . . , ␣ s兲 = h
j=1
冉兺
冉兺
s
A1⌫1j ␣ j,
j=1
s
−h
j=1
... ,兺
... ,兺
冊 冉
冊 冉兺
冊 冉兺
⌫rj␣ j
j=1
s
f 3共 ␣ 1, . . . , ␣ s兲 = h
j=1
s
⌫1j ␣ j,
冊
s
⌫1j ␣ j,
s
s
j=1
j=1
s
Ar⌫rj␣ j
j=1
s
Ã1⌫1j ␣ j,
−h
j=1
s
A1˜⌫1j ␣ j, . . . , 兺 Ar˜⌫rj␣ j + h
j=1
冊
− h 兺 ˜⌫1j ␣ j, . . . , 兺 ˜⌫rj␣ j ,
s
j=1
. . . , 兺 Ãr⌫rj␣ j
j=1
s
冊
冊
˜ j␣ .
Ã1˜⌫1j ␣ j, . . . , 兺 Ã⌫
r j
j=1
Furthermore, let ␭1 , . . . , ␭r be constant weight factors and let w␭ be the weight system with metric
d that associates the weight factor ␭ j to the field ␥ j. Then,
共i兲
r
储f 1储w 艋 储h储w␭ 兿
i=1
冉兺 冊
s
j=1
␬j j
储兩⌫ 兩储 .
␭i i
共ii兲
储f 2储w 艋 r储h储w␭␴␦ ␴r−1 ,
where
再
␴ = max max
i=1,. . .,r
s
兺
j=1
冎
s
␬j j
␬j ˜ j
储兩⌫i 兩储, 兺 储兩⌫
i 兩储 ,
␭i
j=1 ␭i
s
␬
j
储兩⌫ij − ˜⌫ij兩储.
兺
␭
i=1,. . .,r j=1 i
␴␦ = max
共iii兲
储f 3储w 艋 r2储h储␻␭␴␦ a␦ 共␴a兲r−1 ,
where
a = max max兵储兩Ai兩储,储兩Ãi兩储其,
i=1,. . .,r
a␦ = max 储兩Ai − Ãi兩储.
i=1,. . .,r
Proof:
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp
053305-28
共i兲
J. Math. Phys. 51, 053305 共2010兲
Balaban et al.
Define, for each 1 艋 i 艋 r, ti = 兺sj=1共␬ j / ␭i兲储兩⌫ij兩储. Set
冉兺
s
h̃共␣1, . . . , ␣s兲 = h
j=1
共ii兲
By multilinearity, f 1 = 共t1 ¯ tr兲h̃, and, by Corollary A.2, 储h̃储w 艋 储h储w␭.
Write the telescoping sum
冉兺
s
f 2共 ␣ 1, . . . , ␣ s兲 = h
共⌫1j
j=1
s
s
j=1
j=1
冊
s
j=1
冉兺
s
j=1
and apply part 共i兲 to each term.
Write the telescoping sum
冉兺
s
f 3共 ␣ 1, . . . , ␣ s兲 = h
s
A1共⌫1j
j=1
j=1
冊
s
冉兺
s
j=1
j=1
冉
s
s
j=1
j=1
兺 ˜⌫1j ␣ j, 兺 共⌫2j
s
s
j=1
j=1
˜⌫ j ␣ , 兺 ˜⌫ j ␣ , . . . , 兺 共⌫
˜ j − ˜⌫ j 兲␣
1 j
2 j
r
r j
冊 冉兺
Ã1共⌫1j
j=1
s
s
j=1
j=1
兺 A1˜⌫1j ␣ j, . . . , 兺 Ar共⌫˜ rj − ˜⌫rj兲␣ j
冊
s
冊
s
− ˜⌫1j 兲␣ j, . . . , 兺 Ar⌫rj␣ j − h
− ˜⌫1j 兲␣ j, . . . , 兺 Ãr⌫rj␣ j + ¯ + h
−h
冊 冉
− ˜⌫1j 兲␣ j, 兺 ⌫2j ␣ j, . . . , 兺 ⌫rj␣ j + h
− ˜⌫2j 兲␣ j, . . . , 兺 ⌫rj␣ j + ¯ + h
共iii兲
冊
s
1
1 j
⌫1␣ j, ¯ , 兺 ⌫rj␣ j .
t1
j=1 tr
冊
˜ j − ˜⌫ j 兲␣ .
Ã1˜⌫1j ␣ j, . . . , 兺 Ãr共⌫
r
r j
j=1
We claim that the 储 · 储w norm of each of the r lines is bounded by r储h储w␭␴␦a␦共␴a兲r−1. We
prove this for the first line. The proof for the other lines is similar. We again write a
telescoping sum
冉兺
s
h
冊 冉兺
s
A1共⌫1j
j=1
冉兺
冉兺
s
=h
− ˜⌫1j 兲␣ j, . . . , 兺 Ar⌫rj␣ j − h
j=1
j=1
s
s
Ã1共⌫1j
j=1
− ˜⌫1j 兲␣ j, . . . , 兺 Ãr⌫rj␣ j
s
s
j=1
j=1
共A1 − Ã1兲共⌫1j − ˜⌫1j 兲␣ j, 兺 A2⌫2j ␣ j, . . . , 兺 Ar⌫rj␣ j
s
+h
j=1
s
s
j=1
j=1
冊
Ã1共⌫1j − ˜⌫1j 兲␣ j, 兺 共A2 − Ã2兲⌫2j ␣ j, . . . , 兺 Ar⌫rj␣ j
+ ¯+h
冉
s
s
s
j=1
j=1
j=1
j=1
冊
冊
冊
兺 Ã1共⌫1j − ˜⌫1j 兲␣ j, 兺 Ã2⌫2j ␣ j, . . . , 兺 共Ar − Ãr兲⌫rj␣ j .
By the first bound, the 储 · 储w norm of the first term is bounded by
冉兺
s
储h储w␭
j=1
␬j
储兩共A1 − Ã1兲共⌫1j − ˜⌫1j 兲兩储
␭1
冉兺
s
艋 储h储w␭储兩A1 − Ã1兩储
j=1
冊兿 冉兺
r
s
i=2
j=1
␬j
储兩Ai⌫ij兩储
␭i
␬j j ˜ j
储兩⌫ − ⌫1兩储
␭1 1
冊兿 冉
r
i=2
冊
s
储兩Ai兩储 兺
j=1
␬2 j
储兩⌫ 兩储
␭i i
冊
艋 储h储w␭␴␦a␦共␴a兲r−1
by Remark 4.3, part 共ii兲. The norm of the second term is bounded by
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp
053305-29
J. Math. Phys. 51, 053305 共2010兲
Complex bosonic effective actions. I
冉兺
s
储h储w␭
j=1
␬j
储兩Ã1共⌫1j − ˜⌫1j 兲兩储
␭1
冊冉兺
s
j=1
␬j
储兩共A2 − Ã2兲⌫2j 兩储
␭2
冊兿 冉兺
r
s
i=3
j=1
␬j
储兩Ai⌫ij兩储
␭i
冊
艋 储h储w␭共␴␦a兲共␴a␦兲共␴a兲r−2 .
Similarly, one bounds the norms of each of the other r − 2 terms by 储h储w␭␴␦a␦共␴a兲r−1.
䊏
APPENDIX B: A SUPREMUM BOUND
Lemma B.1: Let f共␣1 , . . . , ␣s ; h兲 be a function that is defined and analytic on a neighborhood
of the origin in C共s+1兲兩X兩. Let w be the weight system with metric d that associates the weight factor
␬ j to the field ␣ j. Let S 傺 X be nonempty and assume that 兩f兩 h共x兲=0 = 0. Then,
for all x苸S
sup兵兩f共␣1, . . . , ␣s ;h兲兩 兩 兩␣ j共x兲兩 艋 ␬ j共x兲, 兩h共x兲兩 艋 1 for all x 苸 X, 1 艋 j 艋 s其 艋 Kd储f储w#S,
where Kd = supy苸X兺x苸Xe−d共x,y兲.
Proof: Denote by a共xជ 1 , . . . , xជ s ; xជ s+1兲 the symmetric coefficient system for f. Then, if 兩␣ j共x兲兩
艋 ␬ j共x兲 and 兩h共x兲兩 艋 1 for all x 苸 X and 1 艋 j 艋 s, we have
兺
兩f共␣1, . . . , ␣s ;h兲兩 艋
兺
共xជ 1,. . .,xជ s+1兲
n1,. . .,ns+1
兩a共xជ 1, . . . ,xជ s ;xជ s+1兲␣1共xជ 1兲 ¯ ␣s共xជ s兲h共xជ s+1兲兩
苸Xn1⫻¯⫻Xns+1
兺
艋
兺
共xជ 1,. . .,xជ s+1兲
n1,. . .,ns+1
兩a共xជ 1, . . . ,xជ s ;xជ s+1兲兩␬1共xជ 1兲 ¯ ␬s共xជ s兲.
苸Xn1⫻¯⫻Xns+1
It suffices to prove that, for each n1 , . . . , ns+1 艌 0,
兺
兩a共xជ 1, . . . ,xជ s ;xជ s+1兲兩␬1共xជ 1兲 ¯ ␬s共xជ s兲
共xជ 1,. . .,xជ s+1兲
苸Xn1⫻¯⫻Xns+1
艋 Kd#S储a共xជ 1, . . . ,xជ s ;xជ s+1兲w共xជ 1, . . . ,xជ s ;xជ s+1兲储n1,. . .,ns+1 .
The cases with n1 + ¯ +ns = 0 are trivial, so fix any n1 , . . . , ns+1 艌 0 with n1 + ¯ +ns ⬎ 0. For notational convenience, suppose that n1 ⬎ 0. By hypothesis a共xជ 1 , . . . , xជ s ; xជ s+1兲 = 0 unless at least one
component of xជ s+1 is in S, in which case ␶d共xជ 1 , . . . , xជ s+1兲 is greater than the distance from the first
component of xជ 1 to S. So, suppressing from the notation that 共xជ 1 , . . . , xជ s+1兲 苸 Xn1 ⫻ ¯ ⫻ Xns+1,
兺
共xជ 1,. . .,xជ s+1兲
艋
艋
兩a共xជ 1, . . . ,xជ s ;xជ s+1兲兩␬1共xជ 1兲 ¯ ␬s共xជ s兲
兺
兺
x苸X 共xជ 1,. . .,xជ s+1兲
共xជ 1兲1=x
兩a共xជ 1, . . . ,xជ s ;xជ s+1兲兩␬1共xជ 1兲 ¯ ␬s共xជ s兲
e−d共x,S兲 兺
兺
x苸X
共x ,. . .,x
ជ1
ជ s+1兲
共xជ 1兲1=x
兩a共xជ 1, . . . ,xជ s ;xជ s+1兲兩w共xជ 1, . . . ,xជ s ;xជ s+1兲
艋 储a共xជ 1, . . . ,xជ s ;xជ s+1兲w共xជ 1, . . . ,xជ s ;xជ s+1兲储n1,. . .,ns+1 兺 e−d共x,S兲 .
x苸X
The lemma now follows from
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp
053305-30
J. Math. Phys. 51, 053305 共2010兲
Balaban et al.
兺 e−d共x,y兲 艋 Kd#S.
兺 e−d共x,S兲 艋 x苸X
兺 y苸S
x苸X
䊏
1
Balaban, T., Feldman, J., Knörrer, H., and Trubowitz, E., “A functional integral representation for many boson systems.
II. Correlation functions,” Ann. Henri Poincare 9, 1275 共2008兲.
2
Balaban, T., Feldman, J., Knörren, H., and Trubowitz, E., “Power series representations for bosonic effective actions,” J.
Stat. Phys. 134, 839 共2009兲.
3
Balaban, T., Feldman, J., Knörrer, H., and Trubowitz, E., “Power series representations for complex bosonic effective
actions. II. A small field renormalization group flow,” J. Math. Phys. 51, 053306 共2010兲.
4
Balaban, T., Feldman, J., Knörrer, H., and Trubowitz, E., “The temporal ultraviolet limit for complex bosonic many-body
models,” Ann. Henri Poincare 共to be published兲.
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp