JOURNAL OF MATHEMATICAL PHYSICS 51, 053306 共2010兲
Power series representations for complex bosonic
effective actions. II. A small field renormalization group
flow
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 Zurich, Switzerland
共Received 27 October 2009; accepted 28 January 2010; published online 21 May 2010兲
In a previous paper, we developed a power series representation and estimates for
ⴱ
ⴱ
an effective action of the form ln关兰e f共␣1,. . .,␣s;z ,z兲d␮共zⴱ , z兲 / 兰e f共0,. . .,0;z ,z兲d␮共zⴱ , z兲兴.
Here, f共␣1 , . . . , ␣s ; zⴱ , z兲 is an analytic function of the complex fields
␣1共x兲 , . . . , ␣s共x兲 , zⴱ共x兲 , z共x兲 indexed by x in a finite set X and d␮共zⴱ , z兲 is a compactly supported product measure. Such effective actions occur in the small field
region for a renormalization group analysis. We illustrate the technique by a model
renormalization group flow motivated by the ultraviolet regime in many boson
systems. © 2010 American Institute of Physics. 关doi:10.1063/1.3329938兴
I. INTRODUCTION
Consider the grand canonical partition function, at temperature T and chemical potential ␮, for
a many boson system moving in a metric space X with a finite number of points and metric d.
Suppose that the Hamiltonian H is the sum of a single particle operator 共for example, the discrete
Laplacian兲 with kernel h共x , y兲 and a two body operator given by a real, symmetric, repulsive pair
potential 2v共x , y兲.
In Ref. 2 共Theorem 2.2兲, we proved the functional integral representation
Tr e−共1/kT兲共H−␮N兲 = lim
␧→0
冕
兿 关d␮˜ R 共␣␶ⴱ, ␣t兲␨␧共␣␶−␧, ␣␶兲e具␣
␶苸␧Z艚共0,1/kT兴
␧
ⴱ
ⴱ
ⴱ
␶−␧,j共␧兲␣␶典−␧具␣␶−␧␣␶v␣␶−␧␣␶典
兴
共1.1兲
for the partition function, under the convention that ␣0 = ␣1/kT and the limit ␧ → 0 is restricted to
␧’s dividing 1 / kT 关that is, 1 / ␧ 苸 共kT兲N兴. Here, N is the number operator and, for any r ⬎ 0,
˜ r共 ␣ ⴱ, ␣ 兲 =
d␮
兿
x苸X
d␣ⴱ共x兲 ∧ d␣共x兲 −␣ⴱ共x兲␣共x兲
e
␹共兩␣共x兲兩 ⬍ r兲
2␲ı
denotes the unnormalized Gaussian measure, cutoff at radius r, and ␨␧共␣ , ␤兲 is the characteristic
function of
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/053306/20/$30.00
51, 053306-1
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Balaban et al.
兵␣, ␤:X → C兩 储␣ − ␤储⬁ ⬍ p0共␧兲其.
The cutoffs R␧ and p0共␧兲 grow at an appropriate rate as ␧ → 0. 关One can think of R␧ as growing a
bit faster than 1 / 冑4 ␧ and of p0共␧兲 as a power of ln共1 / ␧兲 or a very small power of 1 / ␧.兴 Furthermore, for any ␧ ⬎ 0, the operator j共␧兲 = e−␧共h−␮兲. We write the 共R-style兲 scalar product, 具f , g典
= 兺x苸X f共x兲g共x兲 for any two fields f , g : X → C. 共Thus, the usual scalar product over C兩X兩 is 具f ⴱ , g典.兲
The representation 共1.1兲 is the first part of a program to resolve the mathematical difficulties
inherent in the time-ultraviolet limit of the formal, coherent state functional integral representation
for the partition function and correlation functions of many boson systems 关see Ref. 5, 共2.66兲 and
the discussion in the introduction to Ref. 1兴. In Ref. 4 this program is completed using techniques
of renormalization group analysis. This paper is a description of the “small field part” of that
construction.
In Ref. 4 we obtain a representation of the functional integral of 共1.1兲 which can be used to
analyze infrared problems. We do so by applying a simple version of a renormalization group
procedure, namely, “decimation.” In each decimation step we integrate out every “second” variable. In the first step, we integrate out ␣␶⬘ with ␶⬘ = ␧ , 3␧ , 5␧ , . . .. The integral with respect to these
variables factorizes into the product, over ␶ = 2␧ , 4␧ , 6␧ , . . ., of the independent integrals
冕
ⴱ
ⴱ
ⴱ
˜ R 共␣␶ⴱ−␧, ␣␶−␧兲␨␧共␣␶−2␧, ␣␶−␧兲e具␣␶−2␧,j共␧兲␣␶−␧典−␧具␣␶−2␧␣␶−␧v␣␶−2␧␣␶−␧典
d␮
␧
ⴱ
ⴱ
ⴱ
⫻e具␣␶−␧,j共␧兲␣␶典−␧具␣␶−␧␣␶,v␣␶−␧␣␶典␨␧共␣␶−␧, ␣␶兲.
That is, assuming that 1 / kT 苸 2␧N,
冕
兿 关d␮˜ R 共␣␶ⴱ, ␣␶兲␨␧共␣␶−␧, ␣␶兲e具␣
␶苸␧Z艚共0,1/kT兴
␧
=
冕
兿
␶苸2␧Z艚共0,1/kT兴
ⴱ
ⴱ
ⴱ
␶−␧,j共␧兲␣␶典−␧具␣␶−␧␣␶v␣␶−␧␣␶典
兴
˜ R 共␣␶ⴱ, ␣␶兲I1共␧; ␣␶ⴱ−2␧, ␣␶兲,
d␮
␧
where
I1共␧; ␣ⴱ, ␤兲 =
冕
˜ R 共␾ⴱ, ␾兲␨␧共␣, ␾兲e具␣
d␮
␧
ⴱ,j共␧兲␾典+具␾ⴱ,j共␧兲␤典 −␧共具␣ⴱ␾, ␣ⴱ␾典+具␾ⴱ␤, ␾ⴱ␤典兲
v
v
e
␨␧共␾, ␤兲.
共1.2兲
After n − 1 additional decimation steps we will have integrated out those ␣␶’s with
␶ 苸 共␧Z \ 共2n␧兲Z兲 艚 共0 , 1 / kT兴 leaving an integrand which is a function of the ␣␶’s with
1
兴. Thus, for kT1 苸 2n␧N, we write
␶ 苸 共2n␧兲Z 艚 关0 , kT
冕
˜ R 共␣␶ⴱ, ␣␶兲␨␧共␣␶−␧, ␣␶兲e具␣
关d␮
兿
␶苸␧Z艚共0,1/kT兴
␧
=
冕
兿
ⴱ
ⴱ
ⴱ
␶−␧,j共␧兲␣␶典−␧具␣␶−␧␣␶v␣␶−␧␣␶典
ⴱ
␶苸共2n␧兲Z艚共0,1/kT兴
˜ R 共␣␶ⴱ, ␣␶兲In共␧, ␣␶−2n␧, ␣␶兲,
d␮
␧
兴
共1.3兲
where the functions In共␧ ; ␣ⴱ , ␤兲 are recursively defined by 共1.2兲 and
In+1共␧; ␣ⴱ, ␤兲 =
冕
˜ R 共␾ⴱ, ␾兲In共␧; ␣ⴱ, ␾兲In共␧; ␾ⴱ, ␤兲.
d␮
␧
共1.4兲
The main result of Ref. 4 is the construction and description of a functional I␪共␣ⴱ , ␤兲, defined
for ␪ 苸 共0 , ⌰兴 关where ⌰ = O共1兲, independent of v兴 such that
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Complex bosonic effective actions. II
I␪共␣ⴱ, ␤兲 = lim Im共2−m␪ ; ␣ⴱ, ␤兲.
m→⬁
Then, for any p such that 1 / pkT 苸 共0 , ⌰兴,
冕 兿冋兿
p
Tr e−共1/kT兲共H−␮N兲 =
n=1
x苸X
册
d␾n共x兲ⴱ␾n共x兲 −␾ 共x兲ⴱ␾ 共x兲
ⴱ
n
e n
I共1/pkT兲共␾n−1
, ␾ n兲
2␲ı
共1.5兲
with the convention ␾0 = ␾ p. 共1.5兲 can be the starting point for an infrared analysis.
In Ref. 4 we describe the functions In共␧ ; ␣ⴱ , ␤兲 and I␪共␣ⴱ , ␤兲 as sums over “large/small field
decompositions” of X. The dominant part in the large/small field decomposition is called the “pure
small field part” and is obtained by replacing the full integrals 共1.4兲 with integrals over appropriate
neighborhoods of stationary points 共see Sec. II兲.
In this note, we discuss a toy model, which we call the “stationary phase 共SP兲 approximation,”
in which all domains of integration are restricted, simply by fiat, to neighborhoods of stationary
points. These neighborhoods will be measured by radii r共␦兲, where ␦ 哫 r共␦兲 is a positive and
monotonically decreasing function. We now give a description of the SP approximation and derive
estimates for it. To do this, introduce the notation ␧n = 2n␧ and
V␦共␧; ␣ⴱ, ␤兲 = − ␧
具关j共␶兲␣ⴱ兴关j共␦ − ␶ − ␧兲␤兴, v关j共␶兲␣ⴱ兴关j共␦ − ␶ − ␧兲␤兴典.
兺
␶苸␧Z艚关0,␦兲
共1.6兲
It will turn out that there is a function E␦共␧ ; ␣ⴱ , ␤兲 of the fields ␣ⴱ , ␤ with the property
ⴱ
兩X兩 具␣
I共SP兲
n 共␧; ␣ , ␤兲 = Z␧n共␧兲 e
ⴱ,j共␧ 兲␤典+V 共␧;␣ⴱ,␤兲+E 共␧;␣ⴱ,␤兲
n
␧n
␧n
共1.7兲
.
The function E␦共␧ ; ␣ⴱ , ␤兲 is defined for real numbers 0 ⬍ ␧ ⱕ ␦ ⱕ ⌰ such that ␦ = 2n␧ for some
integer n ⱖ 0. The normalization constant Z␦共␧兲 is defined in Appendix C. It is chosen so that
E␦共␧ ; 0 , 0兲 = 0. It is extremely close to 1. The “irrelevant” contributions E␦共␧ ; ␣ⴱ , ␤兲 to the effective
action are characterized by the recursion relation
E␧共␧; ␣ⴱ, ␤兲 = 0,
ⴱ
ⴱ
ⴱ
E2␦共␧; ␣ , ␤兲 = E␦共␧; ␣ , j共␦兲␤兲 + E␦共␧; j共␦兲␣ , ␤兲 + log
˜ r共␦兲共zⴱ,z兲e⳵A␦共␧;␣
兰d␮
˜ r共␦兲共zⴱ,z兲
兰d␮
ⴱ,␤;zⴱ,z兲
,
共1.8兲
where
⳵ A␦共␧; ␣ⴱ, ␤ ;zⴱ,z兲 = 关V␦共␧; ␣ⴱ, j共␦兲␤ + z兲 − V␦共␧; ␣ⴱ, j共␦兲␤兲兴
+ 关V␦共␧; j共␦兲␣ⴱ + zⴱ, ␤兲 − V␦共␧; j共␦兲␣ⴱ, ␤兲兴
+ 关E␦共␧; ␣ⴱ, j共␦兲␤ + z兲 − E␦共␧; ␣ⴱ, j共␦兲␤兲兴
+ 关E␦共␧; j共␦兲␣ⴱ + zⴱ, ␤兲 − E␦共␧; j共␦兲␣ⴱ, ␤兲兴.
共1.9兲
The motivation for this recursion relation comes from a SP construction and is given in Sec. II.
We estimate E␦共␧ ; ␣ⴱ , ␤兲 in terms of norms as in Ref. 3 共Definition 2.6 and, more specifically,
共A1兲兴. Assume that X is a metric space. Choose a constant m ⱖ 0 as a spatial exponential decay
rate and a positive monotonically decreasing function ␦ 哫 ␬共␦兲 to measure the radius of convergence of the expansion of E␦共␧ ; ␣ⴱ , ␤兲 in powers of the fields ␣ⴱ and ␤.
We define the norm of the power series
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053306-4
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Balaban et al.
f共␣ⴱ, ␤兲 =
兺
兺
k,ᐉⱖ0 x1,. . .,xk苸X
y1,. . .,yᐉ苸X
a共x1, . . . ,xk ;y1, . . . ,yᐉ兲␣共x1兲ⴱ ¯ ␣共xk兲ⴱ␤共y1兲 ¯ ␤共yᐉ兲
关with the coefficients a共x1 , . . . , xk ; y1 , . . . , yᐉ兲 invariant under permutations of x1 , . . . , xk and of
y1 , . . . , yᐉ兴 to be
储f共␣ⴱ, ␤兲储␦ =
兺
max max
兺
k,ᐉⱖ0 x苸X 1ⱕiⱕk+ᐉ 共xជ ,yជ 兲苸Xk⫻Xᐉ
共xជ ,yជ 兲i=x
w␦共xជ ;yជ 兲兩a共xជ ;yជ 兲兩
with the weight system
w␦共xជ ;yជ 兲 = ␬共␦兲k+ᐉem␶共xជ ,yជ 兲
for 共xជ ,yជ 兲 苸 Xk ⫻ Xᐉ ,
where ␶共xជ , yជ 兲 is the minimal length of a tree whose set of vertices contains the points of the set
兵x1 , . . . , xk , y1 , . . . , yᐉ其. In the language of Ref. 3 共Definitions 2.5 and 2.6兲, w␦ is the weight system
with metric md that associates the constant weight factor ␬共␦兲 to the fields ␣ⴱ and ␤, and the norm
储f共␣ⴱ , ␤兲储␦ is denoted 储f共␣ⴱ , ␤兲储w␦. For any operator A on CX, with kernel A共x , y兲, we define the
weighted L1 − L⬁ operator norm
再
冎
兩储A储兩 = max sup 兺 emd共x,y兲兩A共x,y兲兩,sup 兺 emd共x,y兲兩A共x,y兲兩 ,
x苸X y苸X
y苸X x苸X
共1.10兲
as in Ref. 3 共Definition A.1兲.
The quantities relevant for the estimates of E␦共␧ ; ␣ⴱ , ␤兲, in addition to the radii r共␦兲 and ␬共␦兲,
are the norm 兩储v储兩 of the interaction, a constant K j such
兩储j共␶兲储兩 ⱕ eK j␶
and
兩储j共␶兲 − 1储兩 ⱕ K j␶eK j␶
for ␶ ⱖ 0
共1.11兲
共see Corollary B.2兲, and a constant 0 ⬍ ⌰ ⱕ 1 that bounds the range for which the constructions
work. On these quantities we make the following.
Hypothesis 1.1: We assume that the monotonically decreasing functions r共t兲 and ␬共t兲 do not
decrease too quickly. Precisely,
共i兲
共ii兲
共iii兲
共iv兲
共v兲
共vi兲
1 ⱕ r共t兲 ⱕ 2r共2t兲 and 1 ⱕ ␬共t兲 ⱕ 2␬共2t兲, for all 0 ⱕ t ⱕ ⌰ / 2.
On the other hand, r共t兲 and ␬共t兲 must decrease quickly enough and r共t兲 must be sufficiently
small compared to ␬共t兲 that
etK j关␬共2t兲 / ␬共t兲兴 + 4关r共t兲 / ␬共t兲兴 ⱕ 1 for all 0 ⱕ t ⱕ ⌰ / 2 and
r共t兲关r共t兲 − r共2t兲兴 ⱖ 2 for all 0 ⱕ t ⱕ ⌰ / 2.
We also assume that there are constants KE and q ⱖ 1 such that
t兩储v储兩r共t兲␬共t兲3 ⱕ 1 / KE for all 0 ⱕ t ⱕ ⌰;
关1 / C共⌰ , KE兲兴共2 / q兲 ⱕ 共␬共t兲 / ␬共2t兲兲4 ⱕ C共⌰ , KE兲4共r共2t兲 / r共t兲兲4 for all 0 ⱕ t ⱕ ⌰ / 2, where
C共⌰ , KE兲 = e−4⌰K j关1 − 233e14K j / KE兴; and
⬁
t2兺k=0
共q / 4兲kr共t / 2k兲2␬共t / 2k兲6 converges uniformly in 0 ⱕ t ⱕ ⌰.
Example 1.2: Let v ⬎ 0.
共i兲
Suppose that
␬共t兲 =
共ii兲
1
冉冊
1
4
冑兩储v储兩 t
a␬
and
r共t兲 =
1
冉冊
1
4
冑兩储v储兩 t
ar
for some constants 0 ⬍ ar ⬍ a␬ obeying 3a␬ + ar ⬍ 1. We prove in Appendix D that there are
constants KE, ⌰, and q such that Hypothesis 1.1 is fulfilled for all nonzero v with 兩储v储兩
ⱕ v.
Suppose that
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Complex bosonic effective actions. II
1
␬共t兲 =
冑4 t兩储v储兩
冉
ln
1
t兩储v储兩
冊
冉
b
and
r共t兲 = ln
1
t兩储v储兩
冊
b
for some b ⱖ 1. Again, we prove in Appendix D that there are constants KE, ⌰, and q such
that Hypothesis 1.1 is fulfilled for all nonzero v with 兩储v储兩 ⱕ v.
We choose p0共␧兲 of the functional integral representation 共1.1兲 to be r共␧兲.
Theorem 1.3: Under Hypothesis 1.1
储E␦共␧; ␣ⴱ, ␤兲储␦ ⱕ KE␦2兩储v储兩2r共␦兲2␬共␦兲6
for all 0 ⱕ ␧ ⱕ ␦ ⱕ ⌰ for which ␦ / ␧ is a power of 2. The function E␦共␧ ; ␣ⴱ , ␤兲 has degree at least
two both in ␣ⴱ and ␤. [By this we mean that every monomial appearing in its power series
expansion contains a factor of the form ␣ⴱ共x1兲␣ⴱ共x2兲␤共x3兲␤共x4兲.]
Theorem 1.3 is proven after Proposition 3.3. The theorem is proven by induction on n, where
␦ = 2n␧. The induction step is prepared by Proposition 3.3, which is proven using the results in Ref.
3. After the proof of Theorem 1.3, we give the following proof.
Theorem 1.4: The limit
E␪共␣ⴱ, ␤兲 = lim E␪共2−m␪ ; ␣ⴱ, ␤兲
m→⬁
exists uniformly in 0 ⱕ ␪ ⱕ ⌰. It fulfills the estimate
储E␪共␣ⴱ, ␤兲储␪ ⱕ KE␪2兩储v储兩2r共␪兲2␬共␪兲6
and has degree of at least 2 in both ␣ⴱ and ␤.
To take the limit ␧ → 0 in 共1.7兲 observe that, for any fixed ␦,
lim V␦共2−n␦ ; ␣ⴱ, ␤兲 = V␦共␣ⴱ, ␤兲,
n→⬁
where
V ␦共 ␣ ⴱ, ␤ 兲 = −
冕
␦
具关j共t兲␣ⴱ兴关j共␦ − t兲␤兴, v关j共t兲␣ⴱ兴关j共␦ − t兲␤兴典dt.
0
The SP approximation to the I␪共␣ⴱ , ␤兲 constructed in Ref. 4 then is
共SP兲 −m
共2 ␪ ; ␣ⴱ, ␤兲 = Z␪兩X兩e具␣
I␪共SP兲共␣ⴱ, ␤兲 = lim Im
ⴱ,j共␪兲␤典+V 共␣ⴱ,␤兲+E 共␣ⴱ,␤兲
␪
␪
.
m→⬁
The existence of Z␪ = lim Z␪共␪ / 2m兲 is proven in Lemma C.1.
m→⬁
II. SP AND STOKES’ THEOREM
in the recursion
To motivate the recursive definition 共1.8兲 of E␦共␧ ; ␣ⴱ , ␤兲, we replace In by I共SP兲
n
relation 共1.4兲. Inserting 共1.7兲, the resulting integral
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J. Math. Phys. 51, 053306 共2010兲
Balaban et al.
共SP兲
ⴱ
ⴱ
˜ R 共␾ⴱ, ␾兲I共SP兲
d␮
n 共␧; ␣ , ␾兲In 共␧; ␾ , ␤兲
␧
= Z兩2X兩
n
= Z2兩X兩
n
冕
冕冋兿
˜ R 共␾ⴱ, ␾兲e具␣
d␮
␧
x苸X
ⴱ,j共␧ 兲␾典+具␾ⴱ,j共␧ 兲␤典 V 共␧;␣ⴱ,␾兲+V 共␧;␾ⴱ,␤兲 E 共␧;␣ⴱ,␾兲+E 共␧;␾ⴱ,␤兲
n
n
␧n
␧n
␧n
␧n
e
e
册
d␾ⴱ共x兲 ∧ d␾共x兲
ⴱ
ⴱ
␹共兩␾共x兲兩 ⬍ R␧兲 eA共␣ ,␤;␾ ,␾兲
2␲ı
共2.1兲
with Zn = Z␧n共␧兲 and
A共␣ⴱ, ␤ ; ␾ⴱ, ␾兲 = − 具␾ⴱ, ␾典 + 具␣ⴱ, j共␧n兲␾典 + 具␾ⴱ, j共␧n兲␤典 + V␧n共␧; ␣ⴱ, ␾兲 + V␧n共␧; ␾ⴱ, ␤兲
+ E␧n共␧; ␣ⴱ, ␾兲 + E␧n共␧; ␾ⴱ, ␤兲.
Here we have written A as a function of four independent complex fields ␣ⴱ, ␤, ␾ⴱ, and ␾. The
activity in 共2.1兲 is obtained by evaluating A共␣ⴱ , ␤ ; ␾ⴱ , ␾兲 with ␾ⴱ = ␾ⴱ, the complex conjugate of
␾. The reason for introducing independent complex fields ␾ⴱ and ␾ lies in the fact that the critical
point 共with respect to the variables ␾ⴱ and ␾兲 of the quadratic part
− 具␾ⴱ, ␾典 + 具j共␧n兲␣ⴱ, ␾典 + 具␾ⴱ, j共␧n兲␤典 = − 具␾ⴱ − j共␧n兲␣ⴱ, ␾ − j共␧n兲␤典 + 具␣ⴱ, j共␧n+1兲␤典
of A is “not real.” Precisely, the critical point is
ⴱ
␾crit
ⴱ = j共␧n兲␣ ,
␾crit = j共␧n兲␤
ⴱ
crit
and, in general 共␾crit
ⴱ 兲 ⫽ ␾ . To do stationary phase, we make the substitution
␾ⴱ = ␾crit
ⴱ + z ⴱ,
␾ = ␾crit + z
共2.2兲
with “fluctuation fields” zⴱ and z. With this substitution, the quadratic part of A is equal to
−具zⴱ , z典 + 具␣ⴱ , j共␧n+1兲␤典. So 共2.1兲 becomes
具␣
Z2兩X兩
n e
ⴱ,j共␧
n+1兲␤典
冋兿 冕
x苸X
M共x兲
册
dzⴱ共x兲 ∧ dz共x兲 −z 共x兲z共x兲 Ã共␣ⴱ,␤;z ,z兲
ⴱ ,
e ⴱ
e
2␲ı
共2.3兲
where
ⴱ crit
+ z兲 + E␧n共␧; ␾crit
Ã共␣ⴱ, ␤ ;zⴱ,z兲 = V␧n共␧; ␣ⴱ, ␾crit + z兲 + V␧n共␧; ␾crit
ⴱ + zⴱ, ␤兲,
ⴱ + zⴱ, ␤兲 + E␧n共␧; ␣ , ␾
ⴱ
crit
M共x兲 = 兵共zⴱ共x兲,z共x兲兲兩共␾crit
ⴱ 共x兲 + zⴱ共x兲兲 = ␾ 共x兲 + z共x兲
and
兩␾crit共x兲 + z共x兲兩 ⬍ R␧其.
共2.3兲 is the integral over a real 2兩X兩 dimensional subset in the complex 2兩X兩 dimensional space of
fields zⴱ and z.
The first step in the SP approximation is to replace, for each x 苸 X, the set M共x兲 in 共2.3兲 by the
neighborhood
ⴱ
crit
D共x兲 = 兵共zⴱ共x兲,z共x兲兲 苸 C2兩兩zⴱ共x兲兩 ⱕ r共␧n兲, 兩z共x兲兩 ⱕ r共␧n兲, 共zⴱ共x兲 + ␾crit
ⴱ 共x兲兲 = z共x兲 + ␾ 共x兲其
of the critical point. In Ref. 4 共Sec. VI兲 we show that the error introduced by this approximation
is extremely small 关even when D共x兲 is empty兴. There, we provide detailed bounds on a “large
field–small field” expansion for which the above approximation is the leading term. In Remark 2.1
关part 共a兲, below兴, we illustrate the sources of the smallness for n = 0.
ⴱ
crit
By Stokes’ theorem 关Lemma A.1 with r = r共␧n兲, ␴ = ␴ⴱ = 0, and ␳ = 共␾crit
ⴱ 兲 − ␾ 兴, one can write
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Complex bosonic effective actions. II
具␣
Z2兩X兩
n e
ⴱ,j共␧
n+1兲␤典
冋兿 冕
D共x兲
x苸X
册
dzⴱ共x兲 ∧ dz共x兲 −z 共x兲z共x兲 Ã共␣ⴱ,␤;z ,z兲
ⴱ
e ⴱ
e
2␲ı
共2.4兲
as the sum of
具␣
Z2兩X兩
n e
ⴱ,j共␧
n+1兲␤典
冋兿 冕
x苸X
具␣
and Z2兩X兩
n e
ⴱ
,j共␧n+1兲␤典
R⫽0”
⫻
册
共2.5兲
times
冋冕
兿 冋冕
兿
兺 x苸R
R傺X
兩z共x兲兩ⱕr共␧n兲
dzⴱ共x兲 ∧ dz共x兲 −兩z共x兲兩2 Ã共␣ⴱ,␤;zⴱ,z兲
e
e
2␲ı
C共x兲
x苸X\R
dzⴱ共x兲 ∧ dz共x兲 −z 共x兲z共x兲
e ⴱ
2␲i
兩z共x兲兩ⱕr共␧n兲
册
册
dz共x兲ⴱ ∧ dz共x兲 −z 共x兲z共x兲 Ã共␣ⴱ,␤;zⴱ,z兲 z 共x兲=z共x兲ⴱ
e
兩ⴱ
,
e ⴱ
2␲i
for x苸X\R
where, for each x 苸 X, C共x兲 is a two real dimensional submanifold of C2 whose boundary is the
union of “circles” ⳵D共x兲 and 兵共zⴱ共x兲 , z共x兲兲 苸 C2 兩 zⴱⴱ共x兲 = z共x兲 , 兩z共x兲兩 = r共␧n兲其. In Ref. 4 we argue that
−zⴱ共x兲z共x兲 has an extremely large negative real part whenever 共zⴱ共x兲 , z共x兲兲 苸 C共x兲. 关Also see
Remark 2.1, part 共b兲, below.兴 The second step in the SP approximation is to ignore these terms.
That is, to replace 共2.4兲 with 共2.5兲.
Thus, the SP approximation for
冕
共SP兲
ⴱ
ⴱ
˜ R 共␾ⴱ, ␾兲I共SP兲
d␮
n 共␧; ␣ , ␾兲In 共␧; ␾ , ␤兲
␧
is
具␣
共2.5兲 = Z2兩X兩
n e
ⴱ,j共␧
n+1兲␤典
冕
˜ r共␧ 兲共zⴱ,z兲eÃ共␣
d␮
n
ⴱ,␤;zⴱ,z兲
.
By construction
ⴱ
ⴱ
ⴱ
V␧n共␧; ␣ⴱ, ␾crit兲 + V␧n共␧; ␾crit
ⴱ , ␤兲 = V␧n共␧; ␣ , j共␧n兲␤兲 + V␧n共␧; j共␧n兲␣ , ␤兲 = V␧n+1共␧; ␣ , ␤兲
so that
ⴱ
ⴱ
ⴱ
V␧n共␧; ␣ⴱ, ␾crit + z兲 + V␧n共␧; ␾crit
ⴱ + z , ␤兲 = V␧n+1共␧; ␣ , ␤兲 + 关V␧n共␧; ␣ , j共␧n兲␤ + z兲
− V␧n共␧; ␣ⴱ, j共␧n兲, ␤兲兴 + 关V␧n共␧; j共␧n兲␣ⴱ + zⴱ, ␤兲
− V␧n共␧; j共␧n兲␣ⴱ, ␤兲兴.
Consequently, the SP approximation for
冕
共SP兲
ⴱ
ⴱ
˜ R 共␾ⴱ, ␾兲I共SP兲
d␮
n 共␧; ␣ , ␾兲In 共␧; ␾ , ␤兲
␧
can also be written as
具␣
Z2兩X兩
n e
ⴱ,j共␧
ⴱ
ⴱ
ⴱ
n+1兲␤典+V␧n+1共␧;␣ ,␤兲 E␧n共␧;␣ ,j共␧n兲␤兲E␧n共␧;j共␧n兲␣ ,␤兲
e
冕
˜ r共␧ 兲共zⴱ,z兲e⳵A␧n共␧;␣
d␮
n
ⴱ,␤;z ,z兲
ⴱ
.
This is compatible with 共1.7兲 and 共1.8兲 since, by the definition of Appendix C,
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Balaban et al.
Z2␧n共␧n兲 = Z␧n共␧兲2
冕
兩z兩⬍r共␧n兲
dzⴱ ∧ dz −兩z兩2
e .
2␲i
Remark 2.1:
共a兲
We now illustrate why the error introduced by the SP approximation is extremely small by
considering the case n = 0. The initial functional integral representation 共1.1兲 may be written
as
Tr e−共1/kT兲共H−␮N兲 = lim
␧→0
冕
兿
␶苸␧Z艚共0,1/kT兴
再冋 兿
x苸X
册
d␣␶ⴱ共x兲 ∧ d␣␶共x兲
␹共兩␣␶共x兲兩 ⬍ R␧兲 ␨␧共␣␶−␧, ␣␶兲
2␲ı
ⴱ
ⴱ
冎
⫻e−共1/2兲具␣␶−␧,␣␶−␧典I0共␧; ␣␶ⴱ−␧, ␣␶兲e−共1/2兲具␣␶ ,␣␶典 ,
where
I0共␧; ␣ⴱ, ␤兲 = e具␣
ⴱ,j共␧兲␤典 −␧共␣ⴱ␤, ␣ⴱ␤兲
v
e
.
Dropping one of the “time derivative small field characteristic functions” ␨␧共␣␶−␧ , ␣␶兲 would
introduce only a very small error. This is because writing ␣␶−␧ = ␣ and ␣␶ = ␤, the quadratic
ⴱ
ⴱ
part of the exponent of e−共1/2兲具␣ ,␣典I0共␧ ; ␣ⴱ , ␤兲e−共1/2兲具␤ ,␤典 obeys
Re兵− 21 具␣ⴱ, ␣典 + 具␣ⴱ, j共␧兲␤典 − 21 具␤ⴱ, ␤典其 ⬇ Re兵− 21 具␣ⴱ, ␣典 + 具␣ⴱ, ␤典 − 21 具␤ⴱ, ␤典其 = − 21 储␣ − ␤储L2 ,
2
2
2
which generates a factor on the order of e−共1/2兲p0共␧兲 = e−共1/2兲r共␧兲 when 共␣ , ␤兲 is not in support
of ␨␧共␣ , ␤兲. The quartic part −␧具␣ⴱ␤ , v␣ⴱ␤典 of the exponent is roughly −␧具␣ⴱ␣ , v␣ⴱ␣典 ⱕ 0
and so cannot generate a large factor. A similar mechanism generates small factors for the
right hand side of 共1.3兲 whenever the difference ␣␶−␧n − ␣␶ between the two arguments of In
is larger than roughly r共␧n兲.
Consequently, we apply the SP approximation to the integral
I1共␧; ␣ⴱ, ␤兲 =
冕
˜ R 共␾ⴱ, ␾兲␨␧共␣, ␾兲I0共␧; ␣ⴱ, ␾兲I0共␧; ␾ⴱ, ␤兲␨␧共共␾ⴱ兲ⴱ, ␤兲
d␮
␧
of 共1.2兲 only when the “time derivative small field condition” 储␣ − ␤储⬁ ⱕ r共2␧兲 is satisfied.
The change in variables 共2.2兲 expresses I1 as
I1共␧; ␣ⴱ, ␤兲 = e具␣
ⴱ,j共2␧兲␤典
冋兿 冕
x苸X
M共x兲
册
dzⴱ共x兲 ∧ dz共x兲 −z 共x兲z共x兲 Ã共␣ⴱ,␤;z ,z兲
ⴱ
e ⴱ
e
2␲ı
⫻␨␧共␣, j共␧兲␤ + z兲␨␧共共j共␧兲␣ⴱ + zⴱ兲ⴱ, ␤兲.
The characteristic function ␨␧共␣ , j共␧兲␤ + z兲 limits the domain of integration to z’s, obeying
储z + j共␧兲␤ − ␣储⬁ ⬍ r共␧兲.
共b兲
Since 储␣ − ␤储⬁ ⱕ r共2␧兲 ⱕ 21 r共␧兲 and 储j共␧兲␤ − ␤储⬁ ⱕ const ␧R␧ Ⰶ r共␧兲, this condition is roughly
equivalent to 储z储⬁ ⬍ r共␧兲. On the difference between these two domains of integration,
the integrand is extremely small, for reasons like those given above. Similarly, the
condition imposed by the second ␨␧ is roughly equivalent to 储zⴱ储⬁ ⬍ r共␧兲. The two conditions 储z储⬁ ⱕ r共␧兲 and 储zⴱ储⬁ ⱕ r共␧兲 are built into the domains of integration D共x兲 in 共2.4兲.
The “time derivative small field condition” 储␣ − ␤储⬁ ⱕ r共2␧兲 ⱕ 21 r共␧兲 is also used to ensure
that −zⴱ共x兲z共x兲 has an extremely large negative real part whenever 共zⴱ共x兲 , z共x兲兲 lies on C共x兲,
the side of the Stokes’ “cylinder.” This may be seen from Remark A.3 with r = r共␧兲, ␴
ⴱ
crit
= ␴ⴱ = 0, and ␳ = 共␾crit
ⴱ 兲 − ␾ = j共␧兲关␣ − ␤兴.
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J. Math. Phys. 51, 053306 共2010兲
Complex bosonic effective actions. II
III. THE INDUCTION STEP
In preparation for the proof of Proposition 3.3 below, as well as of the proofs of Theorems 1.3
and 1.4, we collect consequences of Hypotheses 1.1 in the form in which they are actually used in
these proofs.
Remark 3.1: Hypothesis 1.1 implies that for all 0 ⱕ ␦ ⱕ ⌰ / 2,
KE ⱖ 219e9K j ,
冋
共3.1a兲
冉 冊 册冉 冊
231e14␦K j
k共␦兲
+ e 2␦K j
KE
k共2␦兲
2e2␦K j
冉 冊
k共2␦兲
k共␦兲
4
r共␦兲
r共2␦兲
2
2
ⱕ 2,
共3.1b兲
+ 226e9K j␦兩储v储兩r共␦兲␬共2␦兲3 ⱕ q.
共3.1c兲
Proof: Hypothesis 1.1 关part 共v兲兴 forces C共⌰ , KE兲 ⱖ 0 and hence KE ⱖ 233e14K j, which implies
共3.1a兲. Hypothesis 1.1 关part 共v兲兴 also forces
冉 冊
k共␦兲
␬共2␦兲
2
ⱕ 2e−2⌰K j
冑
1−
冉 冊
233e14K j r共2␦兲
KE
r共␦兲
2
冋
ⱕ 2e−2␦K j 1 −
232e14K j
KE
册冉 冊
r共2␦兲
r共␦兲
2
and hence
4
冉 冊冉 冊
231e14␦K j
␬共␦兲
+ e 2␦K j
KE
␬共2␦兲
2
r共␦兲
r共2␦兲
2
ⱕ 2.
So 共3.1b兲 now follows from Hypothesis 1.1 关part 共i兲兴, which ensures that 共r共␦兲 / r共2␦兲兲2 ⱕ 4. Finally, by Hypothesis 1.1 关parts 共iv兲 and 共v兲兴,
2e2␦K j
冉 冊
␬共2␦兲
␬共␦兲
4
+ 226e9K j␦兩储v储兩r共␦兲␬共2␦兲3 ⱕ e2⌰K jqC共⌰,KE兲 +
= qe−2⌰K j − q
ⱕq−
226e9K j
KE
233e−2⌰K je14K j 226e9K j
+
KE
KE
233e12K j 226e9K j
+
KE
KE
ⱕq
since q ⱖ 1 and 0 ⬍ ⌰ ⱕ 1.
We formulate the recursion relation 共1.8兲 that defines E␧n 共␧ ; ␣ⴱ , ␤兲 more abstractly.
Definition 3.2: Let 0 ⱕ ␧ ⱕ ␦. For an action E共␣ⴱ , ␤兲, we set
ⴱ
ⴱ
ⴱ
R␦,␧关E兴共␣ , ␤兲 = E共␣ , j共␦兲␤兲 + E共j共␦兲␣ , ␤兲 + log
˜ r共␦兲共zⴱ,z兲e⳵A␦,␧共E;␣
兰d␮
䊏
ⴱ,␤;zⴱ,z兲
˜ r共␦兲共zⴱ,z兲
兰d␮
whenever the logarithm is defined. Here,
⳵ A␦,␧共E; ␣ⴱ, ␤ ;zⴱ,z兲 = 关V␦共␧; ␣ⴱ, j共␦兲␤ + z兲 − V␦共␧; ␣ⴱ, j共␦兲␤兲兴
+ 关V␦共␧; j共␦兲␣ⴱ + zⴱ, ␤兲 − V␦共␧; j共␦兲␣ⴱ, ␤兲兴
+ 关E共␣ⴱ, j共␦兲␤ + z兲 − E共␣ⴱ, j共␦兲␤兲兴
+ 关E共j共␦兲␣ⴱ + zⴱ, ␤兲 − E共j共␦兲␣ⴱ, ␤兲兴.
The recursion relation 共1.8兲 is equivalent to
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J. Math. Phys. 51, 053306 共2010兲
Balaban et al.
E␧共␧; ␣ⴱ, ␤兲 = 0,
E␧n+1共␧; ␣ⴱ, ␤兲 = R␧n,␧关E␧n共␧; ␣ⴱ, ␤兲兴.
共3.2兲
To prove Theorem 1.3, we perform induction on n to successively bound E␧n共␧ ; ·兲 for n
= 0 , . . . , log2共⌰ / ␧兲. For the induction step, we use the following.
Proposition 3.3: Assume that r共t兲 and ␬共t兲 fulfill Hypothesis 1.1. Then, for all 0 ⱕ ␧ ⱕ ␦
ⱕ ⌰ / 2, with ␦ an integer multiple of ␧, the following holds.
Let E共␣ⴱ , ␤兲 be an analytic function which has degree of at least 2 both in ␣ⴱ and ␤ and which
obeys 储E共␣ⴱ , ␤兲储␦ ⱕ 29e7␦K j␦兩储v储兩r共␦兲␬共2␦兲3. Then R␦,␧关E兴共␣ⴱ , ␤兲 is well defined, has degree at
least two both in ␣ⴱ and ␤, and satisfies the estimate
储R␦,␧关E兴储2␦ ⱕ 232e14␦K j␦2兩储v储兩2r共␦兲2␬共2␦兲6 + 2e2␦K j
冉 冊
␬共2␦兲 4
储E储␦ .
␬共␦兲
Proof: Observe that the functions V␦共␧ ; ␣ⴱ , j共␦兲␤ + z兲 − V␦共␧ ; ␣ⴱ , j共␦兲␤兲 and E共␣ⴱ , j共␦兲␤ + z兲
− E共␣ⴱ , j共␦兲␤兲 both have degree of at least 2 in ␣ⴱ, degree of at least 1 in z, and do not depend on
zⴱ. Similarly, both V␦共␧ ; j共␦兲␣ⴱ + zⴱ , ␤兲 − V␦共␧ ; j共␦兲␣ⴱ , ␤兲 and E共j共␦兲␣ⴱ + zⴱ , ␤兲 − E共j共␦兲␣ⴱ , ␤兲 have
degree of at least 2 in ␤, degree of at least 1 in zⴱ, and do not depend on z. Since the integral of
˜ r共␦兲共zⴱ , z兲 is zero unless there are the same number of z’s and zⴱ’s,
any monomial against d␮
冕
˜ r共␦兲共zⴱ,z兲 ⳵ A␦,␧共E; ␣ⴱ, ␤ ;zⴱ,z兲 = 0
d␮
共3.3兲
and
log
˜ r共␦兲共zⴱ,z兲e⳵A␦,␧共E;␣
兰d␮
ⴱ,␤;zⴱ,z兲
˜ r共␦兲共zⴱ,z兲
兰d␮
has degree of at least 2 both in ␣ⴱ and ␤. This implies that R␦,␧关E兴共␣ⴱ , ␤兲 has degree of at least 2
both in ␣ⴱ and ␤.
To estimate ⳵A␦,␧ we introduce a second auxiliary weight system wfluct. It has a metric md and
associates the constant weight factor ␬共2␦兲 to the fields ␣ⴱ and ␤ and the constant weight factor
4r共␦兲 to the fluctuation fields zⴱ and z. We abbreviate
储f共␣ⴱ , ␤ ; zⴱ , z兲储fluct = 储f共␣ⴱ , ␤ ; zⴱ , z兲储wfluct. Clearly, 储f共␣ⴱ , ␤兲储2␦ = 储f共␣ⴱ , ␤兲储fluct for functions that
are independent of the fluctuation fields.
Observe that
V␦共␧; ␣ⴱ, j共␦兲␤ + z兲 − V␦共␧; ␣ⴱ, j共␦兲␤兲 = ␧
兺
␶苸␧Z艚共0,␦兴
关具␥ⴱ␶−␧g␶, v␥ⴱ␶−␧g␶典 − 具␥ⴱ␶−␧ĝ␶, v␥ⴱ␶−␧ĝ␶典兴
with
␥ⴱ␶ = j共␶兲␣ⴱ,
g␶ = j共2␦ − ␶兲␤,
ĝ␶ = j共␦ − ␶兲共j共␦兲␤ + z兲 = j共2␦ − ␶兲␤ + j共␦ − ␶兲z.
共3.4兲
We apply Proposition A.3 关part 共ii兲兴 of Ref. 3, with d replaced by md, r = 4, s = 3, h共␥1 , . . . , ␥4兲
= 共␥1␥2 , v␥3␥4兲, ␣1 = ␣ⴱ, ␣2 = ␤, ␣3 = z, weights ␭1 = ¯ = ␭4 = 1, and
⌫11 = ⌫13 = j共␶ − ␧兲,
˜⌫1 = ˜⌫1 = j共␶ − ␧兲,
1
3
⌫22 = ⌫24 = j共2␦ − ␶兲,
˜⌫2 = ˜⌫2 = j共2␦ − ␶兲,
4
2
⌫32 = ⌫34 = 0,
˜⌫3 = ˜⌫3 = j共␦ − ␶兲
2
4
共3.5兲
with all other ⌫ij’s and ˜⌫ij’s being zero. Then
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Complex bosonic effective actions. II
再
␴ = ␬共2␦兲max 兩储j共␶ − ␧兲储兩, 兩储j共2␦ − ␶兲储兩, 兩储j共2␦ − ␶兲储兩 + 4
冉
ⱕ 1+4
冊
r共␦兲
兩储j共␦ − ␶兲储兩
␬共2␦兲
冎
r共␦兲 2K ␦
e j ␬共2␦兲,
␬共2␦兲
␴␦ = 4r共␦兲兩储j共␦ − ␶兲储兩 ⱕ 4eK j␦r共␦兲.
共3.6兲
So, for each ␶ 苸 ␧Z 艚 共0 , ␦兴,
储具␥ⴱ␶−␧g␶, v␥ⴱ␶−␧g␶典 − 具␥ⴱ␶−␧ĝ␶, v␥ⴱ␶−␧ĝ␶典储fluct ⱕ 4储h储w␭␴␦␴3 ⱕ 29e7K j␦兩储v储兩r共␦兲␬共2␦兲3 .
Here, we used that r共␦兲 ⱕ 41 ␬共␦兲 ⱕ 21 ␬共2␦兲 by Hypothesis 1.1 关parts 共i兲 and 共ii兲兴. Summing over ␶
gives
储V␦共␧; ␣ⴱ, j共␦兲␤ + z兲 − V␦共␧; ␣ⴱ, j共␦兲␤兲储fluct ⱕ 29e7K j␦␦兩储v储兩r共␦兲␬共2␦兲3 .
Similarly,
储V␦共␧; j共␦兲␣ⴱ + zⴱ, ␤兲 − V␦共␧; j共␦兲␣ⴱ + zⴱ, ␤兲储fluct ⱕ 29e7K j␦␦兩储v储兩r共␦兲␬共2␦兲3 .
Next, by Corollary A.2 of Ref. 3 for any analytic function f共␣ⴱ , ␤兲,
储f共␣ⴱ, j共␦兲␤ + z兲 − f共␣ⴱ, j共␦兲␤兲储fluct ⱕ 储f共␣ⴱ, j共␦兲␤ + z兲储fluct ⱕ 储f共␣ⴱ, ␤兲储␦
共3.7兲
since
␬共2␦兲 4r共␦兲
4共r␦兲
␬共2␦兲
+
兩储j共␦兲储兩 +
兩储1储兩 ⱕ e␦K j
ⱕ1
␬共␦兲
␬共␦兲
␬共␦兲
␬共␦兲
by Hypothesis 1.1 关part 共ii兲兴. In particular 储E共␣ⴱ , j共␦兲␤ + z兲 − E共␣ⴱ , j共␦兲␤兲储fluct ⱕ 储E储␦. Similarly
储E共j共␦兲␣ⴱ + zⴱ , ␤兲 − E共j共␦兲␣ⴱ , ␤兲储fluct ⱕ 储E储␦.
Combining the bounds of the previous two paragraphs with the assumption on 储E储␦, we get
储⳵ A␦,␧共E; ·兲储fluct ⱕ 210e7␦K j␦兩储v储兩r共␦兲␬共2␦兲3 + 2储E储␦ ⱕ 211e7␦K j␦兩储v储兩r共␦兲␬共2␦兲3 ⱕ
1
64
共3.8兲
by Hypothesis 1.1 关part 共iv兲兴 and 共3.1a兲. By 共3.3兲 and Corollary 3.5 共Ref. 3兲 with n = 1,
冐
log
˜ r共␦兲共zⴱ,z兲e⳵A␦,␧共E;␣
兰d␮
˜ r共␦兲共zⴱ,z兲
兰d␮
ⴱ,␤;z ,z兲
ⴱ
冐 冉
ⱕ
2␦
2
储⳵ A␦,␧共E; ·兲储fluct
冊
1
− 储⳵ A␦,␧共E; ·兲储fluct
20
2
ⱕ 232e14␦K j␦2兩储v储兩2r共␦兲2␬共2␦兲6 .
Combining this estimate and the estimate of Lemma 3.4, below, with f = E, we get the desired
bound on 储R␦,␧关E兴储2␦.
䊏
Lemma 3.4: Let f共␣ⴱ , ␤兲 be an analytic function that has degree of at least 2 both in ␣ⴱ and
␤. Then,
储f共␣ⴱ, j共␦兲␤兲储2␦, 储f共j共␦兲␣ⴱ, ␤兲储2␦ ⱕ e2␦K j
冉 冊
␬共2␦兲 4
储f储␦ .
␬共␦兲
Proof: Introduce the auxiliary weight system waux with, in the language of Ref. 3 关Definitions
2.5 and 2.6 and, more specifically, 共A.1兲兴, metric md that associates the constant weight factor to
the field ␣ⴱ and the constant weight factor e−␦K j␬共␦兲 to the field ␤. Since, by 共1.11兲,
关e−␦K j␬共␦兲 / ␬共␦兲兴兩储j共␦兲储兩 ⱕ 1, Corollary A.2 of Ref. 3 gives
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Balaban et al.
储f共␣ⴱ, j共␦兲␤兲储waux ⱕ 储f储␦ .
As f共␣ⴱ , j共␦兲␤兲 has degree of at least 2 both in ␣ⴱ and ␤ and e−␦K j␬共␦兲 ⱖ ␬共2␦兲, by Hypothesis 1.1
关part 共ii兲兴,
储f共␣ⴱ, j共␦兲␤兲储2␦ ⱕ
冉 冊冉
␬共2␦兲
␬共␦兲
2
冊
冉 冊
␬共2␦兲 2
␬共2␦兲 4
储f共␣ⴱ, j共␦兲␤兲储waux ⱕ e2␦K j
储f储␦ .
e ␬共␦兲
␬共␦兲
−␦K j
The estimate on 储f共j共␦兲␣ⴱ , ␤兲储2␦ is similar.
䊏
Proof of Theorem 1.3: We write ␦ = ␧n = 2n␧ and prove the statement by induction on n. In the
case n = 0, there is nothing to prove. For the induction step from n to n + 1, set ␦ = ␧n. The
hypothesis of Proposition 3.3, with E = E␦, is satisfied since,
储E␦储␦ ⱕ KE␦2兩储v储兩2r共␦兲2␬共␦兲6 ⱕ 8␦兩储v储兩r共␦兲␬共2␦兲3
共3.9兲
by the inductive hypothesis and Hypothesis 1.1 关parts 共i兲 and 共iv兲兴. Using 共3.2兲, Proposition 3.3,
and 共3.1a兲 we see that
储E␧n+1储␧n+1 ⱕ 232e14␦K j␦2兩储v储兩2r共␦兲2␬共2␦兲6 + 2e2␦K j
冋
冋
ⱕ 232e14␦K j + 2e2␦K j
=
冉 冊
␬共2␦兲 4
KE␦2兩储v储兩2r共␦兲2␬共␦兲6
␬共␦兲
冉 冊 册
冉 冊 册冉 冊
␬共␦兲 2
KE ␦2兩储v储兩2r共␦兲2␬共2␦兲6
␬共2␦兲
1 231e14␦K j
␬共␦兲
+ e 2␦K j
2
KE
␬共2␦兲
2
r共␦兲
r共2␦兲
2
KE共2␦兲2兩储v储兩2r共2␦兲2␬共2␦兲6
ⱕ KE共2␦兲2兩储v储兩2r共2␦兲2␬共2␦兲6
2
= KE␧n+1
兩储v储兩2r共␧n+1兲2␬共␧n+1兲6 .
䊏
In the proof of Theorem 1.4, we shall compare E␪共2−m−1␪ ; ␣ⴱ , ␤兲 and E␪共2−m␪ ; ␣ⴱ , ␤兲 to prove
that the sequence E␪共2−m␪ ; ·兲 is Cauchy with respect to our norm. To do so, we shall compare
E␧n共␧ / 2 ; ·兲 and E␧n共␧ ; ·兲 for each n = 0 , . . . , log2共⌰ / ␧兲. This is done by induction on n. For the
induction step, we use Proposition 3.6, below. To prepare for it, we have as follows.
Lemma 3.5: Set
W共␣ⴱ, ␤兲 = V␦共␧; ␣ⴱ, ␤兲 − V␦
冉
冊
␧
; ␣ ⴱ, ␤ .
2
Then
储W共␣ⴱ,z + j共␦兲␤兲 − W共␣ⴱ, j共␦兲␤兲储fluct ⱕ 210e9K j␧␦兩储v储兩r共␦兲␬共2␦兲3 ,
储W共zⴱ + j共␦兲␣ⴱ, ␤兲 − W共j共␦兲␣ⴱ, ␤兲储fluct ⱕ 210e9K j␧␦兩储v储兩r共␦兲␬共2␦兲3 .
Proof: We prove the first inequality. By definition
W共␣ⴱ, ␤兲 = W1共␣ⴱ, ␤兲 + W2共␣ⴱ, ␤兲
with
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Complex bosonic effective actions. II
W 1共 ␣ ⴱ, ␤ 兲 = −
␧
兺 关具␥ⴱ␶−␧␥␶, v␥ⴱ␶−␧␥␶典 − 具␥ⴱ␶−␧␥␶−␧/2, v␥ⴱ␶−␧␥␶−␧/2典兴,
2 ␶苸␧Z艚共0,␦兴
W 2共 ␣ ⴱ, ␤ 兲 = −
␧
兺 关具␥ⴱ␶−␧␥␶, v␥ⴱ␶−␧␥␶典 − 具␥ⴱ␶−␧/2␥␶, v␥ⴱ␶−␧/2␥␶典兴,
2 ␶苸␧Z艚共0,␦兴
where
␥ⴱ␶ = j共␶兲␣ⴱ,
␥␶ = j共␦ − ␶兲.
Using the notation of 共3.4兲,
W1共␣ⴱ, j共␦兲␤兲 − W1共␣ⴱ,z + j共␦兲␤兲 = −
␧
兺 关具␥ⴱ␶−␧g␶, v␥ⴱ␶−␧g␶典 − 具␥ⴱ␶−␧g␶−␧/2, v␥ⴱ␶−␧g␶−␧/2典
2 ␶苸␧Z艚共0,␦兴
− 具␥ⴱ␶−␧ĝ␶, v␥ⴱ␶−␧ĝ␶典 + 具␥ⴱ␶−␧ĝ␶−␧/2, v␥ⴱ␶−␧ĝ␶−␧/2典兴
=−
冋
␧
兺 具␥ⴱ␶−␧g␶, v␥ⴱ␶−␧g␶典
2 ␶苸␧Z艚共0,␦兴
冓 冉冊
− ␥ⴱ␶−␧ j
冉冊
␧
␧
g␶, v␥ⴱ␶−␧ j
g␶
2
2
冔
冓 冉冊
− 具␥ⴱ␶−␧ĝ␶, v␥ⴱ␶−␧ĝ␶典 + ␥ⴱ␶−␧ j
冉 冊 冔册
␧
␧
ĝ␶, v␥ⴱ␶−␧ j
ĝ␶
2
2
.
This time, we apply Proposition A.3 关part 共iii兲兴 of Ref. 3 using the ⌫ij’s and ˜⌫ij’s of 共3.5兲 and, in
addition,
A1 = Ã1 = A3 = Ã3 = 1,
A2 = A4 = 1,
Ã2 = Ã4 = j
冉冊
␧
.
2
The corollary bounds the 储 · 储fluct norm of the ␶ term by 42兩储v储兩␴␦a␦共␴a兲3 with ␴ and ␴␦ of 共3.6兲 and
再 冏 冐 冉 冊冐冏 冎
a = max 兩储1储兩,
a␦ =
j
␧
2
冏 冐 冉 冊 冐冏
j
␧
−1
2
ⱕ eK j共␧/2兲 ,
␧
ⱕ K jeK j共␧/2兲
2
by 共1.11兲. Inserting and summing over ␶, we get
储W1共␣ⴱ,z + j共␦兲␤兲 − W1共␣ⴱ, j共␦兲␤兲储fluct
ⱕ
␧
␧␦ 2
4 兩储v储兩4eK j␦r共␦兲 K jeK j共␧/2兲
2␧
2
冉冉
1+4
冊
r共␦兲 2K ␦
e j ␬共2␦兲eK j共␧/2兲
␬共2␦兲
冊
3
ⱕ 29e9K j␧␦兩储v储兩r共␦兲␬共2␦兲3 .
The same estimate holds for 储W2共␣ⴱ , z + j共␦兲␤兲 − W2共␣ⴱ , j共␦兲␤兲储fluct.
䊏
Proposition 3.6: Under the hypotheses of Proposition 3.3, assume that there is a second
analytic function Ẽ共␣ⴱ , ␤兲, which has similar properties to E and is close to E. Precisely,
we assume that Ẽ is of degree of at least 2 both in ␣ⴱ and ␤ and obeys 储Ẽ储␦
ⱕ 29e7␦K j␦兩储v储兩r共␦兲␬共2␦兲3. Then
储R␦,␧关E兴 − R␦,␧/2关Ẽ兴储2␦ ⱕ 236e18K j␧␦2兩储v储兩2r共␦兲2␬共2␦兲6 + q储E − Ẽ储␦ ,
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Balaban et al.
where q is the constant in parts (v) and (vi) of Hypothesis 1.1.
Proof: By Definition 3.2,
ⴱ
R␦,␧关E兴 − R␦,␧/2关Ẽ兴 = B共␣ , ␤兲 + log
˜ r共␦兲共zⴱ,z兲e⳵A␦,␧共E;␣
兰d␮
ⴱ,␤;zⴱ,z兲
˜ r共␦兲共zⴱ,z兲
兰d␮
− log
˜ r共␦兲共zⴱ,z兲e⳵A␦,␧/2共Ẽ;␣
兰d␮
ⴱ,␤;zⴱ,z兲
,
˜ r共␦兲共zⴱ,z兲
兰d␮
共3.10兲
where B共␣ⴱ , ␤兲 = 共E − Ẽ兲共␣ⴱ , j共␦兲␤兲 + 共E − Ẽ兲共j共␦兲␣ⴱ , ␤兲. By Lemma 3.4,
储B储2␦ ⱕ 2e2␦K j
冉 冊
␬共2␦兲 4
储E − Ẽ储␦
␬共␦兲
共3.11兲
With the notation of the previous lemma,
⳵ A␦,␧共E; ␣ⴱ, ␤ ;zⴱ,z兲 − ⳵ A␦,␧/2共Ẽ; ␣ⴱ, ␤ ;zⴱ,z兲
= 关W共␣ⴱ,z + j共␦兲␤兲 − W共␣ⴱ, j共␦兲␤兲兴 + 关W共zⴱ + j共␦兲␣ⴱ, ␤兲 − W共j共␦兲␣ⴱ, ␤兲兴 + C共␣ⴱ, ␤ ;zⴱ,z兲,
where
C共␣ⴱ, ␤ ;zⴱ,z兲 = 关共E − Ẽ兲共␣ⴱ, j共␦兲␤ + z兲 − 共E − Ẽ兲共␣ⴱ, j共␦兲␤兲兴 + 关共E − Ẽ兲共j共␦兲␣ⴱ + zⴱ, ␤兲
− 共E − Ẽ兲共j共␦兲␣ⴱ, ␤兲兴.
By 共3.7兲, 储C储fluct ⱕ 2储E − Ẽ储␦. Combining this with Lemma 3.5, we get
储⳵ A␦,␧共E; ·兲 − ⳵ A␦,␧/2共Ẽ; ·兲储fluct ⱕ 211e9K j␧␦兩储v储兩r共␦兲␬共2␦兲3 + 2储E − Ẽ储␦
ⱕ 兵211e9K j␧ + 211e7␦K j其␦兩储v储兩r共␦兲␬共2␦兲3 .
共3.12兲
Consequently, by 共3.8兲,
储⳵ A␦,␧共E; ·兲储fluct + 储⳵ A␦,␧共E; ·兲 − ⳵ A␦,␧/2共Ẽ; ·兲储fluct ⱕ 兵211e9K j␧ + 212e7␦K j其␦兩储v储兩r共␦兲␬共2␦兲3
ⱕ 213e9K j␦兩储v储兩r共␦兲␬共2␦兲3 ⱕ
1
1
−
17 32
by 共3.1a兲 and Hypothesis 1.1 关part 共iv兲兴. Therefore, the hypotheses of Ref. 3 共Corollary 3.6兲 are
satisfied and we have, using 共3.3兲 and 共3.12兲,
冐
log
˜ r共␦兲共zⴱ,z兲e⳵A␦,␧共E;␣
兰d␮
ⴱ,␤;z ,z兲
ⴱ
− log
˜ r共␦兲共zⴱ,z兲
兰d␮
˜ r共␦兲共zⴱ,z兲e⳵A␦,␧/2共Ẽ;␣
兰d␮
˜ r共␦兲共zⴱ,z兲
兰d␮
ⴱ,␤;z ,z兲
ⴱ
冐
2␦
ⱕ 2 兵储⳵ A␦,␧共E; ·兲储fluct + 储⳵ A␦,␧共E; ·兲 − ⳵ A␦,␧/2共Ẽ; ·兲储fluct其储⳵ A␦,␧共E; ·兲 − ⳵ A␦,␧/2共Ẽ; ·兲储fluct
12
ⱕ 225e9K j␦兩储v储兩r共␦兲␬共2␦兲3兵211e9K j␧␦兩储v储兩r共␦兲␬共2␦兲3 + 2储E − Ẽ储␦其.
Using this, 共3.11兲 and 共3.1c兲, we bound 共3.10兲 by
储R␦,␧关E兴 − R␦,␧/2关Ẽ兴储2␦ ⱕ 236e18K j␧␦2兩储v储兩2r共␦兲2␬共2␦兲6 + q储E − Ẽ储␦ .
Corollary 3.7: For all sufficiently small ␧ ⬎ 0 and integers 0 ⱕ n ⱕ log2共⌰ / ␧兲, we have
冐
E␧n共␧; ␣ⴱ, ␤兲 − E␧n
冉
␧ ⴱ
;␣ ,␤
2
冊冐
䊏
n
␧n
ⱕ KEqn␧2兩储v储兩2r共␧兲2␬共␧兲6 + 236e18K j␧兩储v储兩2 兺 qn−k␧2k r共␧k兲2␬共␧k兲6 .
k=1
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Complex bosonic effective actions. II
Proof: The proof is by induction on n. In the case n = 0, E␧n共␧ ; ␣ⴱ , ␤兲 = 0 and
冐 冉
E ␧n
␧ ⴱ
;␣ ,␤
2
冊冐
␧n
ⱕ KE␧2兩储v储兩2r共␧兲2␬共␧兲6
by Theorem 1.3. For the induction step from n to n + 1, apply Proposition 3.6, with ␦ = ␧n, E
= E␧n共␧兲, and Ẽ = E␧n共␧ / 2兲. This gives, using Hypothesis 1.1 关part 共i兲兴 and the induction hypothesis
on 储E␧n共␧兲 − E␧n共␧ / 2兲储␧n,
冐
E␧n+1共␧兲 − E␧n+1
冉 冊冐
␧
2
␧n+1
冐
ⱕ 236e18K j␧兩储v储兩2␧2nr共␧n兲2␬共␧n+1兲6 + q E␧n共␧兲 − E␧n
冉 冊冐
␧
2
␧n
n+1
ⱕ KEqn+1␧2兩储v储兩2r共␧兲2␬共␧兲6 + 236e18K j␧兩储v储兩2 兺 qn+1−k␧2k r共␧k兲2␬共␧k兲6 .
k=1
䊏
Proof of Theorem 1.4: By Corollary 3.7, with ␧ = 2−m␪ and n = m, we have, for sufficiently
large m,
冐冉
E␪
冊 冉
␪ ⴱ
␪
; ␣ , ␤ − E␪ m+1 ; ␣ⴱ, ␤
2m
2
冊冐
␪
ⱕ KE␪2兩储v储兩2
冉冊 冉 冊 冉 冊
兺冉 冊 冉 冊 冉 冊
q
4
␪ 2 ␪
␬ m
2m
2
m
r
6
m−1
+ 236e18K j兩储v储兩2
␪3
q ᐉ ␪ 2 ␪
r ᐉ ␬ ᐉ
m
2 ᐉ=0 4
2
2
6
and, consequently,
⬁
兺
␯=m
冐冉
E␪
冊 冉
冊冐
冋兺 冉 冊 冉 冊 冉 冊
冋兺 冉 冊 冉 冊 冉 冊
␪ ⴱ
␪
; ␣ , ␤ − E␪ ␯+1 ; ␣ⴱ, ␤
2␯
2
⬁
ⱕ const ␪ 兩储v储兩
2
ⱕ const ␪ 兩储v储兩
2
2
␯=m
⬁
2
k=m
q
4
␯
␪
␪ 2 ␪
r ␯ ␬ ␯
2
2
q k ␪
r k
4
2
2
␪
␬ k
2
⬁
冉 冊 冉 冊 冉 冊册
兺冉 冊 冉 冊 冉 冊 兺 册
␯−1
6
␪
q ᐉ ␪ 2 ␪
+ 兺 ␯兺
r ᐉ ␬ ᐉ
2
2
␯=m 2 ᐉ=0 4
6
␪
q k ␪ 2 ␪
+ m
r k ␬ k
2 k=0 4
2
2
⬁
6 ⬁
␯=0
6
1
.
2␯
Hence, by Hypothesis 1.1 关part 共vi兲兴 and the Cauchy criterion, the sequence E␪共2−m␪ ; ␣ⴱ , ␤兲 con䊏
verges uniformly in ␪. This gives Theorem 1.4.
APPENDIX A: APPENDIX ON STOKES’ THEOREM
Lemma A.1: Let r ⬎ 0 and ␴ , ␴ⴱ , ␳ 苸 CX obey 兩␳共x兲 + ␴ⴱ共x兲ⴱ − ␴共x兲兩 ⬍ 2r for all x 苸 X. Set
D␴ⴱ,␴,␳共x兲 = 兵共zⴱ共x兲,z共x兲兲 苸 C2兩兩zⴱ共x兲 − ␴ⴱ共x兲兩 ⱕ r, 兩z共x兲 − ␴共x兲兩 ⱕ r, z共x兲 − zⴱ共x兲ⴱ = ␳共x兲其,
D␴ⴱ,␴,␳ = X D␴ⴱ,␴,␳共x兲.
x苸X
Let, for each x 苸 X, C␴ⴱ,␴,␳共x兲 be any two real dimensional submanifold of C2 whose boundary is
the union of the one real dimensional submanifolds ⳵D␴ⴱ,␴,␳共x兲 and the circle 兵共zⴱ共x兲 , z共x兲兲
苸 C2 兩 zⴱⴱ共x兲 = z共x兲 , 兩z共x兲兩 = r其. [By submanifold, we really mean a submanifold with corners. The
orientation of C␴ⴱ,␴,␳共x兲 must also be chosen appropriately.] Furthermore, let f共␣1 , . . . , ␣s ; zⴱ , z兲
be a function that is holomorphic in the variables ␣1 , . . . , ␣s in a neighborhood of the origin in CsX
and in the variables 共zⴱ , z兲 苸 Xx苸XP共x兲, with, for each x 苸 X, P共x兲 being an open poly disk in C2
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that contains C␴ⴱ,␴,␳共x兲. Then,
冕
D␴ ,␴,␳
ⴱ
=
兿
x苸X
冋
兺兿
R傺X x苸R
⫻
册
dzⴱ共x兲 ∧ dz共x兲 −z 共x兲z共x兲 f共␣ ,. . .,␣ ;z ,z兲
s ⴱ
e 1
e ⴱ
2␲i
冉冕
C␴ ,␴,␳共x兲
ⴱ
兿 冉冕
兩z共x兲兩ⱕr
x苸X\R
dzⴱ共x兲 ∧ dz共x兲 −z 共x兲z共x兲
e ⴱ
2␲i
冊
冊
dz共x兲ⴱ ∧ dz共x兲 −z 共x兲z共x兲 f共␣ ,. . .,␣ ;z ,z兲 z 共x兲=z共x兲ⴱ
s ⴱ 兩 ⴱ
e ⴱ
e 1
.
2␲i
for x苸X\R
Proof: In the proof we suppress the subscripts ␴ⴱ , ␴ , ␳. For each x 苸 X there is a three real
dimensional submanifold B共x兲 傺 P共x兲 whose boundary is the union of D共x兲, C共x兲 and
DIR共x兲 = 兵共zⴱ共x兲,z共x兲兲 苸 C2兩zⴱⴱ共x兲 = z共x兲,兩z共x兲兩 ⱕ r其.
We apply Stokes’ theorem once for each point x 苸 X to the differential form
␻=
兿
x苸X
dzⴱ共x兲 ∧ dz共x兲
exp兵− 具zⴱ,z典 + f共␣1, . . . , ␣s ;zⴱ,z兲其.
2␲i
Since ␻ is a holomorphic 2兩X兩 form in C2兩X兩, d␻ = 0 and
冕
D
␻=
兺
R傺X
冕
␻,
where M R =
MR
兿 DIR共x兲 ⫻ x苸R
兿 C共x兲.
x苸R
䊏
Example A.2: In Lemma A.1, C共x兲 = C␴ⴱ,␴,␳共x兲 must be a surface whose boundary coincides
with the union of the boundaries of DIR共x兲 and D␴ⴱ,␴,␳共x兲. A possible choice of such a surface is
constructed as follows. Interpolate between DIR共x兲 and D␴ⴱ,␴,␳共x兲 by the three dimensional set
B共x兲 = 艛0ⱕtⱕ1Dt共x兲, where
Dt共x兲 = 兵共zⴱ,z兲 苸 C2兩兩zⴱ − t␴ⴱ共x兲兩 ⱕ r, 兩z − t␴共x兲兩 ⱕ r, z − zⴱⴱ = t␳共x兲其.
Then C共x兲 = 艛0⬍t⬍1⳵Dt共x兲 has the required boundary
Dσ∗ ,σ,ρ (x)
C(x)
B(x)
C(x)
DIR (x)
Remark A.3: In the above example,
Re共zⴱz兲 ⱖ 21 共r2 − 兩␳共x兲兩2兲 − r共兩␴共x兲兩 + 兩␴ⴱ共x兲兩兲
for all 共zⴱ , z兲 苸 C共x兲. Furthermore, the area of C共x兲 is bounded by 8␲r关兩␴兩 + 兩␴ⴱ兩 + 兩␳兩兴.
Proof: Let 共zⴱ , z兲 苸 C共x兲. We suppress the dependence on x. There is a 0 ⱕ t ⱕ 1 such that
max兵兩zⴱ − t␴ⴱ兩 , 兩z − t␴兩其 = r and zⴱ = zⴱ − t␳ⴱ. So
zⴱz = 兩z − t␴兩2 + 2 Re共z − t␴兲t␴ⴱ + 兩t␴兩2 − t␳ⴱz,
zⴱz = 兩zⴱ − t␴ⴱ兩2 + 2 Re共zⴱ − t␴ⴱ兲t␴ⴱⴱ + 兩t␴ⴱ兩2 + t␳zⴱ − 兩t␳兩2 .
Adding and taking the real part,
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Complex bosonic effective actions. II
2 Re共zⴱz兲 = 兩z − t␴兩2 + 兩zⴱ − t␴ⴱ兩2 + 2 Re共z − t␴兲t␴ⴱ + 2 Re共zⴱ − t␴ⴱ兲t␴ⴱⴱ + t2共兩␴兩2 + 兩␴ⴱ兩2 − 兩␳兩2兲
ⱖ r2 − 2r共兩␴兩 + 兩␴ⴱ兩兲 − 兩␳兩2 .
By construction, C共x兲 is contained in the union of the two cylinders
兵共r␨ + t␴,r␨ⴱ + t共␴ⴱ − ␳ⴱ兲兲兩兩␨兩 = 1,
t 苸 关0,1兴其,
兵共r␨ⴱ + t共␴ⴱⴱ + ␳兲,r␨ + t␴ⴱ兲兩兩␨兩 = 1,
t 苸 关0,1兴其.
The area of the first is bounded by 2␲冑2r冑兩␴兩2 + 兩␴ − ␳兩2 and the area of the second is bounded by
䊏
2␲冑2r冑兩␴ⴱ兩2 + 兩␴ⴱⴱ − ␳兩2.
APPENDIX B: PROPERTIES OF j„␶…
We discuss the decay properties of the operator j共␶兲 = e−␶共h−␮兲 using the operator norm 共1.10兲.
Lemma B.1:
(a)
For any two operators A , B : L2共X兲 → L2共X兲,
(b)
For any operator A : L 共X兲 → L 共X兲 and any complex number ␣,
兩储AB储兩 ⱕ 兩储A储兩 兩储B储兩.
2
2
兩储e␣A储兩 ⱕ e兩␣兩兩储A储兩
兩储e␣A − 1储兩 ⱕ 兩␣兩兩储A储兩e兩␣兩兩储A储兩
Proof:
共a兲
By the triangle inequality, for each x 苸 X,
兺 emd共x,y兲兩共AB兲共x,y兲兩 ⱕ y,z苸X
兺 emd共x,z兲兩A共x,z兲兩emd共z,y兲兩B共z,y兲兩
y苸X
ⱕ
emd共x,z兲兩A共x,z兲兩兩储B储兩
兺
z苸X
ⱕ 兩储A储兩 兩储B储兩.
共b兲
The other bound is similar.
By part 共a兲,
⬁
兩储e␣A储兩 ⱕ 兺
n=0
⬁
1
1
兩储␣nAn储兩 ⱕ 兺 兩␣兩n兩储A储兩n = e兩␣兩 兩储A储兩
n!
n=0 n!
and
⬁
兩储e
␣A
− 1储兩 ⱕ 兺
n=1
⬁
1
1
兩储␣nAn储兩 ⱕ 兺 兩␣兩n兩储A储兩n ⱕ 兩␣兩 兩储A储兩e兩␣兩 兩储A储兩 .
n!
n=1 n!
䊏
Corollary B.2: Let ␶ ⱖ 0,
兩储j共␶兲储兩 ⱕ e␶共兩储h储兩+␮兲,
兩储j共␶兲 − 1储兩 ⱕ ␶共兩储h储兩 + 兩␮兩兲e␶共兩储h储兩+兩␮兩兲 .
Proof: Write j共␶兲 = e␶␮e−␶h and j共␶兲 − 1 = e␶␮共e−␶h − 1兲 + e␶␮ − 1. By the previous lemma
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兩储j共␶兲储兩 = e␶␮兩储e−␶h储兩 ⱕ e␶␮e␶兩储h储兩
and
兩储j共␶兲 − 1储兩 ⱕ e␶␮兩储e−␶h − 1储兩 + 兩储e␶␮ − 1储兩 ⱕ ␶兩储h储兩e␶␮e␶兩储h储兩 + 兩e␶␮ − 1兩.
䊏
APPENDIX C: THE NORMALIZATION CONSTANT
We define the normalization constant Z␦共␧兲 by the recursion relations
Z␧共␧兲 = 1,
Z2␦共␧兲 = Z␦共␧兲2
冕
兩z兩ⱕr共␦兲
dzⴱ ∧ dz −兩z兩2
e .
2␲i
Lemma C.1: For all 0 ⬍ ␧ ⬍ ␦ with ␦ / ␧ a positive integer power of 2, we have 0 ⬍ Z␦共␧兲
⬍ 1 and
兩ln Z␦共␧兲兩 ⱕ e−r共␦兲 .
2
2
Furthermore, the limit Z␦ = limn→⬁ Z␦共␦ / 2n兲 exists and also obeys 兩ln Z␦兩 ⱕ e−r共␦兲 .
Proof: Start by fixing any ␧ ⬎ 0 and writing ␧n = 2n␧. From the inductive definition,
ln Z␧n+1共␧兲 = 2 ln Z␧n共␧兲 + ln Z␧⬘ ,
where Z␧⬘ =
n
n
冕
兩z兩ⱕr共␧n兲
dzⴱ ∧ dz −zⴱz
e
⬍1
2␲i
so that 0 ⬍ Z␧n共␧兲 ⬍ 1 for all n 苸 IN and
2−n−1兩ln Z␧n+1共␧兲兩 = 2−n兩ln Z␧n共␧兲兩 + 2−n−1兩ln Z␧⬘ 兩
n
which implies that
n−1
2−n兩ln Z␧n共␧兲兩 = 兺 2−k−1兩ln Z␧⬘ 兩.
共C1兲
k
k=0
Since
1 − Z␧⬘ =
k
冕
兩共x,y兲兩ⱖr共␧k兲
dxdy −共x2+y2兲 1
e
=
␲
␲
and 兩ln共1 − x兲兩 ⱕ 兩x兩 / 共1 − 兩x兩兲 ⱕ 2兩x兩 for all
冕 冕
⬁
2␲
dr
r共␧k兲
2
d␪re−r =
0
冕
⬁
dse−s = e−r共␧k兲
2
r共␧k兲2
兩x兩 ⱕ 21 ,
n−1
n−1
er共␧n兲 兩ln Z␧n共␧兲兩 = 兺 2n−k−1er共␧n兲 兩ln共1 − e−r共␧k兲 兲兩 ⱕ 兺 2n−ke−共r共␧k兲
2
2
2
k=0
2−r共␧ 兲2兲
n
.
k=0
By Hypothesis 1.1 关part 共iii兲兴,
n−1
n−1
n−1
p=k
p=k
p=k
r共␧k兲2 − r共␧n兲2 = 兺 共r共␧ p兲2 − r共␧ p+1兲2兲 ⱖ 兺 r共␧ p兲共r共␧ p兲 − r共␧ p+1兲兲 ⱖ 兺 2 = 2共n − k兲
so that
n−1
⬁
er共␧n兲 兩ln Z␧n共␧兲兩 ⱕ 兺 2n−ke−2共n−k兲 ⱕ 兺
2
k=0
ᐉ=1
冉冊
2
e2
ᐉ
=
2/e2
ⱕ 1.
1 – 2/e2
For the limit, we rewrite 共C1兲
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Complex bosonic effective actions. II
n−1
兩ln Z␧n共␧兲兩 = 兺 2
n
n−k−1
兩ln Z␧⬘ 兩 = 兺 2ᐉ−1兩ln Z␧⬘ 兩,
k
k=0
n−ᐉ
ᐉ=1
which implies that
冏 冉 冊冏
ln Z␦
1
␦
2n
n
= 兺 2ᐉ−1兩ln Z2⬘−ᐉ␦兩
共C2兲
ᐉ=1
and hence
冏 冉 冊冏
lim ln Z␦
n→⬁
1
␦
2n
⬁
= 兺 2ᐉ−1兩ln Z2⬘−ᐉ␦兩
ᐉ=1
⬁
= 兺 2ᐉ−1兩ln共1 − e−r共2
−ᐉ␦兲2
ᐉ=1
兲兩
⬁
ⱕe
−r共␦兲2
⬁
兺2 e
ᐉ −共r共2−ᐉ␦兲2−r共␦兲2兲
ᐉ=1
ⱕe
−r共␦兲2
兺 2ᐉe−2ᐉ
ᐉ=1
ⱕ e−r共␦兲 .
2
APPENDIX D: THE PROOF OF EXAMPLE 1.2
Example 1.2: Let v ⬎ 0.
共i兲
Suppose that
␬共t兲 =
共ii兲
冉冊
1
1
4
冑兩储v储兩 t
a␬
and
r共t兲 =
1
冉冊
1
4
冑兩储v储兩 t
ar
for some constants 0 ⬍ ar ⬍ a␬ obeying 3a␬ + ar ⬍ 1. Then there are constants KE, ⌰, and q
such that Hypothesis 1.1 is fulfilled for all nonzero v with 兩储v储兩 ⱕ v.
Suppose that
␬共t兲 =
1
冑t兩储v储兩
4
冉
ln
1
t兩储v储兩
冊
b
and
冉
r共t兲 = ln
1
t兩储v储兩
冊
b
for some b ⱖ 1. Then there are constants KE, ⌰, and q such that Hypothesis 1.1 is fulfilled
for all nonzero v with 兩储v储兩 ⱕ v.
Proof:
共i兲
We have
␬共2t兲 1
=
,
␬共t兲 2a␬
r共2t兲 1
=
,
r共t兲 2ar
t兩储v储兩r共t兲␬共t兲3 = t1−ar−3a␬,
r共t兲 a −a
= t ␬ r.
␬共t兲
So part 共i兲 of the hypothesis is trivially fulfilled if 共1 / 冑4 v兲min兵1 / ⌰ar , 1 / ⌰ak其 ⱖ 1. Part 共ii兲
of the hypothesis, namely, etK j共1 / 2a␬兲 + 4ta␬−ar ⱕ 1, is satisfied provided e⌰K j共1 / 2a␬兲
+ 4⌰a␬−ar ⱕ 1, which is the case if ⌰ is small enough. Part 共iii兲 of the hypothesis,
namely,
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J. Math. Phys. 51, 053306 共2010兲
Balaban et al.
r共t兲关r共t兲 − r共2t兲兴 =
1
冑兩储v储兩 t
1
2ar
冋 册
1−
1
ⱖ2
2 ar
is satisfied if 共1 / ⌰2ar兲关1 − 1 / 2ar兴 ⱖ 2冑v. Part 共iv兲 of the hypothesis, namely, t1−ar−3a␬
ⱕ 1 / KE, is satisfied provided that ⌰1−ar−3a␬ ⱕ 1 / KE. The uniform convergence 共for each
fixed nonzero v兲 of
⬁
t
2
兺
k=0
冉冊冉 冊 冉 冊
q k t
r k
4
2
2
t
␬ k
2
⬁
6
=t
2−2ar−6a␬
兩储v储兩
−2
兺
k=0
冉
q 2a +6a
2 r ␬
4
冊
k
is achieved whenever q ⬍ 22共1−ar−3a␬兲. Finally, to satisfy part 共v兲, we need
4
2
1
ⱕ 24a␬ ⱕ C共⌰,KE兲 4a
C共⌰,KE兲 q
2 r
or
qⱖ
共ii兲
21–4a␬
C共⌰,KE兲
and
24a␬+4ar−2 ⱕ C共⌰,KE兲.
Since a␬ + ar ⬍ a␬ + 21 a␬ + 21 ar = 21 共3a␬ + ar兲 ⬍ 21 , we have 1 – 4a␬ ⬍ 2共1 − ar − 3a␬兲 and hence
max兵1 , 21–4a␬其 ⬍ 22共1−ar−3a␬兲. Fix any q obeying max兵1 , 21–4a␬其 ⬍ q ⬍ 22共1−ar−3a␬兲. Then,
pick a C ⬍ 1 sufficiently close to 1 that q ⱖ 21–4a␬ / C and 24a␬+4ar−2 ⱕ C. Then, pick a KE
large enough that 1 − 共233e14K j / KE兲 ⬎ C. Finally, choose 0 ⬍ ⌰ ⬍ 1 that is small enough
that 共1 / 冑4 v兲min兵1 / ⌰ar , 1 / ⌰ak其 ⱖ 1, e⌰K j共1 / 2a␬兲 + 4⌰a␬−ar ⱕ 1, ⌰1−ar−3a␬ ⱕ 1 / KE, 1 / ⌰2ar关1
− 1 / 2ar兴 ⱖ 2冑v, and C共⌰ , KE兲 ⱖ C.
Set
冢
C⌰ = 1 −
ln 2
2
ln
⌰v
冣
b
.
Then, for all 0 ⱕ t ⱕ ⌰ / 2 and v with 兩储v储兩 ⱕ v,
冢
r共2t兲
= 1−
r共t兲
ln 2
1
ln
t兩储v储兩
冣
b
苸 关C⌰,1兲
so that
1
␬共2t兲
苸 4 关C⌰,1兲,
冑2
␬共t兲
冉
t兩储v储兩r共t兲␬共t兲3 = 冑t兩储v储兩 ln
The proof now continues as in part 共i兲.
4
1
t兩储v储兩
冊
4b
,
r共t兲 4
= 冑t兩储v储兩.
␬共t兲
䊏
1
Balaban, T., Feldman, J., Knörrer, H., and Trubowitz, E., “A functional integral representation for many boson systems.
I: The partition function,” Ann. Henri Poincare 9, 1229 共2008兲.
2
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.
3
Balaban, T., Feldman, J., Knörrer, H., and Trubowitz, E., “Power series representations for complex bosonic effective
actions,” J. Math. Phys. 51, 053305 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.
5
Negele, J. W. and Orland, H., Quantum Many-Particle Systems 共Addison-Wesley, Reading, MA, 1988兲.
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp