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Research ArticleDeconstructing the Gel’fand–Yaglom Method and
VacuumEnergy from a Theory Space
Nahomi Kan1 and Kiyoshi Shiraishi 2
1National Institute of Technology, Gifu College, Motosu-shi,
Gifu 501-0495, Japan2Graduate School of Sciences and Technology for
Innovation, Yamaguchi University, Yamaguchi-shi, Yamaguchi
753-8512, Japan
Correspondence should be addressed to Kiyoshi Shiraishi;
[email protected]
Received 5 March 2019; Revised 27 April 2019; Accepted 7 May
2019; Published 2 June 2019
Academic Editor: Antonio Scarfone
Copyright © 2019 Nahomi Kan and Kiyoshi Shiraishi. This is an
open access article distributed under the Creative
CommonsAttribution License, which permits unrestricted use,
distribution, and reproduction in any medium, provided the original
work isproperly cited.
The discrete Gel’fand–Yaglom theorem was studied several years
ago. In the present paper, we generalize the
discreteGel’fand–Yaglom method to obtain the determinants of mass
matrices which appear in current works in particle physics, suchas
dimensional deconstruction and clockwork theory. Using the results,
we show the expressions for vacuum energies in suchvarious
models.
1. Introduction
The Gel’fand–Yaglom method [1] for obtaining
functionaldeterminants of differential operators with boundaries
iswidely known nowadays. For nice reviews, see [2, 3].
Theapplications of theGel’fand–Yaglommethod have been inves-tigated
quite recently, to evaluate one-loop vacuum energiesin nontrivial
boundary conditions [4–7].
Among them, Altshuler examined vacuum energy inwarped
compactification [6, 7]. In recent years, it is supposedthat extra
dimensions of various types could play an impor-tant role in the
hierarchy problem, and thus the study ofphysics in nontrivial
background geometry is still advancing.
The dimensional deconstruction has appeared as a newtool for
understanding the properties of higher-dimensionalfield theories
[8–10] more than a decade ago. In such amodel of deconstruction, a
“theory space” is considered,which consists of sites and links, to
which four-dimensionalfields are individually assigned. Theory
spaces thus have thestructures of graphs [11] and can be
interpreted as the theorywith discrete extra dimensions.
Several years ago, the discrete Gel’fand–Yaglom methodfor
difference operators was reviewed and studied by Dowker[12]. We
generalize the discrete Gel’fand–Yaglom methodfor studying one-loop
vacuum energies in extended decon-structed theories and models with
discrete dimensions in
the present paper. To this end, we develop the method
ofcomputing determinants of repetitive Hermitian matriceswhich
correspond tomass matrices utilized in deconstructedtheories.
After completion of the first version of the manuscriptof the
present paper (arXiv:1711.06806), a paper which treatsthe
determinants of discrete Laplace operators appeared [13].Their
method is substantially the same as ours, because theauthor also
relies on the recurrence relation among threevariables on a lattice
(see Section 3 in the present paperand below). We recently become
aware of another similarpaper on the determinants of matrix
differential operators[14].They studied generalization of
Gel’fand–Yaglommethodto obtain the functional determinants. Their
work differsessentially from ours because they considered
differentialoperators while we treat matrices as operators. We also
pointout that they did not consider the matrices of large size
whichhave certain continuum limits.
The organization of this paper is as follows. In orderto make
the present paper self-contained, we show a shortreview of
theGel’fand–Yaglommethod for a differential oper-ator, along with
Dunne’s review [2], in Section 2. In Section 3,we give the method
to obtain determinants of tridiagonalmatrices with repeated
structure. This is a straightforwardgeneralization of description
in Ref. [12]. In Section 4, we give
HindawiAdvances in Mathematical PhysicsVolume 2019, Article ID
6579187, 15 pageshttps://doi.org/10.1155/2019/6579187
http://orcid.org/0000-0002-9761-5137https://arxiv.org/abs/1711.06806https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2019/6579187
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2 Advances in Mathematical Physics
the method to obtain determinants of periodic
tridiagonalmatrices. Determinants of extended periodic
tridiagonalmatrices are obtained in Section 5. The rest of the
presentpaper is devoted to applications to deconstructed
theoriesand discrete systems. In Section 6, free energy on a
graphis discussed by using the results of previous sections.
InSection 7, we show the method of calculation for
evaluatingone-loop vacuum energy in deconstructed models from
thedeterminants of mass matrices. In Section 8, we show afew more
examples of one-loop vacuum energies for slightlycomplicated theory
spaces. We give conclusions in the lastsection, Section 9.
2. Review of the Gel’fand–Yaglom Method [2]
Suppose that an eigenvalue equation (Δ + 𝑚2)𝜓(�푚2)�휆
(𝑥) =𝜆(�푚2)𝜓(�푚2)�휆
(𝑥) with Dirichlet-Dirichlet boundary conditions𝜓(�푚2)�휆
(0) = 𝜓(�푚2)�휆
(𝐿) = 0 in one dimension is given.Then
det [Δ + 𝑚2]det [Δ] = 𝜓(�푚2)0 (𝐿)𝜓(0)0 (𝐿) (1)
holds. Here 𝜓(�푚2)0 (𝑥) satisfies (Δ + 𝑚2)𝜓(�푚2)0 (𝑥) = 0 with
aboundary condition 𝜓(�푚2)0 (0) = 0.Example. In the region 0 ≤ 𝑥 ≤
𝐿, under the Dirichletcondition at the boundaries, we consider the
functionaldeterminant
det [−𝜕2�푥 + 𝑚2]det [−𝜕2�푥] . (2)
The solution of (−𝜕2�푥 +𝑚2)𝜓(𝑥) = 0with 𝜓(0) = 0 is sinh𝑚𝑥.Thus
according to the Gel’fand–Yaglommethod, we obtain
det [−𝜕2�푥 + 𝑚2]det [−𝜕2�푥] = sinh𝑚𝐿𝑚𝐿 = ∞∏�푛=1 𝑛2𝜋2/𝐿2 +
𝑚2𝑛2𝜋2/𝐿2 . (3)
Proof. det[Δ +𝑚2 − 𝜆] is a function of 𝜆 and has zeros at 𝜆
=𝜆(�푚2).The function𝜓�휆(𝑥)which satisfies (Δ+𝑚2−𝜆)𝜓�휆(𝑥) = 0and
𝜓�휆(0) = 0 becomes the eigenfunction when 𝜆 = 𝜆(�푚2).Then the
boundary condition at 𝑥 = 𝐿, i.e., 𝜓�휆(𝐿) = 0, issatisfied. In
otherwords,𝜓�휆 (𝐿) is a function of𝜆 and has zerosat 𝜆 = 𝜆(�푚2).
Therefore det[Δ + 𝑚2 − 𝜆] ∝ 𝜓�휆(𝐿) holds.3. Determinants of
Tridiagonal Matrices
3.1. The Discrete Gel’fand–Yaglom Method for Tridiago-nal
Matrices. Now, we show the discrete Gel’fand–Yaglommethod to obtain
determinants of finite matrices. First, weconsider the following
Hermitian tridiagonal matrix of 𝑁rows and columns:
𝐻 = (((((((
𝑐 −𝑏 0 ⋅ ⋅ ⋅ 0−𝑏∗ 𝑎 −𝑏 ⋅ ⋅ ⋅ 00 −𝑏∗ 𝑎 ⋅ ⋅ ⋅ 0... ... ... d ...𝑎
−𝑏0 0 0 ⋅ ⋅ ⋅ −𝑏∗ 𝑑)))))))
. (4)In this case, the eigenvalue equation𝐻k = 𝜆k, (5)where k�푇
= (V1, V2, . . . , V�푁), can be categorized into threeparts:
�푁∑�푗=1
𝐻1�푗V�푗 − 𝜆V1 = 0, (6)�푁∑
�푗=1
𝐻�푘�푗V�푗 − 𝜆V�푘 = 0 (2 ≤ 𝑘 ≤ 𝑁 − 1) , (7)�푁∑
�푗=1
𝐻�푁�푗V�푗 − 𝜆V�푁 = 0. (8)Here, Eq. (7) is just the recurrence
relation among three termsin V�푘 as a sequence of numbers. In the
present case, the generalsolution for the recurrence relation𝑏V�푘+1
− (𝑎 − 𝜆) V�푘 + 𝑏∗V�푘−1 = 0 (2 ≤ 𝑘 ≤ 𝑁 − 1) (9)is
V�푘 = 𝐴𝛼�푘−1 + 𝐵𝛽�푘−1, (10)where 𝐴 and 𝐵 are constants and
𝛼 = 𝑎 − 𝜆 + √(𝑎 − 𝜆)2 − 4 |𝑏|22𝑏 ,𝛽 = 𝑎 − 𝜆 − √(𝑎 − 𝜆)2 − 4
|𝑏|22𝑏 .
(11)
Note that𝛼 and𝛽 are roots of the second-order equation 𝑏𝑥2−(𝑎 −
𝜆)𝑥 + 𝑏∗ = 0 and 𝛼𝛽 = 𝑏∗/𝑏.The first row of the eigenvalue
equation, Eq. (6), deter-
mines the relation between V1 and V2; in this case, that is(𝑐 −
𝜆)V1 − 𝑏V2 = 0. If we further chooseV1 = 𝐴 + 𝐵 = 1, (12)
the coefficients 𝐴 and 𝐵 are obtained as𝐴 = 𝐴 (𝜆) = 𝑐 − 𝜆 − 𝑏𝛽𝑏
(𝛼 − 𝛽) = 𝑐 − 𝜆 − 𝑏𝛽√(𝑎 − 𝜆)2 − 4 |𝑏|2 , (13)𝐵 = 𝐵 (𝜆) = − (𝑐 − 𝜆 −
𝑏𝛼)𝑏 (𝛼 − 𝛽) = − (𝑐 − 𝜆 − 𝑏𝛼)√(𝑎 − 𝜆)2 − 4 |𝑏|2 . (14)Substituting
all of the results above into Eq. (8) in the
present case, we get
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Advances in Mathematical Physics 3
𝐴 (𝜆) [−𝑏∗ + (𝑑 − 𝜆) 𝛼] 𝛼�푁−2+ 𝐵 (𝜆) [−𝑏∗ + (𝑑 − 𝜆) 𝛽] 𝛽�푁−2 =
0. (15)Now, we set the left-hand side of Eq. (15) as 𝐷(𝜆). 𝐷(𝜆)
is zero if 𝜆 is an eigenvalue of the matrix 𝐻 in this case.
Byconstruction, 𝐷(𝜆) should be an𝑁th order polynomial of 𝜆.The
reason is the following: V2 = [(𝑐 − 𝜆)/𝑏]V1 = (𝑐 − 𝜆)/𝑏,V3 = [(𝑎 −
𝜆)/𝑏]V2 − (𝑏∗/𝑏)V1 = (𝑎 − 𝜆)(𝑐 − 𝜆)/𝑏2 − (𝑏∗/𝑏),and so on.This
observation shows V�푁 includes (−𝜆)�푁−1/𝑏�푁−1.Finally, since the
left-hand side of Eq. (8) reads −𝑏∗V�푁−1+(𝑑−𝜆)V�푁 in the present
case,𝐷(𝜆)has the term (−𝜆)�푁/𝑏�푁−1 as thehighest order term in 𝜆.
We can also directly confirm this bysetting 𝑎 = 𝑐 = 𝑑 = 0 and the
limit 𝑏 → 0 in the left-handside of Eq. (15). We then verify 𝐷(𝜆) →
(−𝜆)�푁/𝑏�푁−1.
Therefore, we conclude that 𝐷(𝜆) = 𝑏�푁−1𝐷(𝜆) =∏�푁�푝=1(𝜆�푝 − 𝜆)
is the characteristic polynomial of 𝐻, where𝜆�푝 (𝑝 = 1, 2, . . . ,
𝑁) are eigenvalues of𝐻.The determinant of𝐻 is given by 𝐷(0) =
∏�푁�푝=1𝜆�푝. In the
present case, we find
det𝐻 = 𝐷(0)= 𝑏�푁−1√𝑎2 − 4 |𝑏|2 [(𝑐 − 𝑏𝛽) (−𝑏∗ + 𝑑𝛼) 𝛼�푁−2− (𝑐 −
𝑏𝛼) (−𝑏∗ + 𝑑𝛽)𝛽�푁−2] ,
(16)
where
𝛼 = 𝑎 + √𝑎2 − 4 |𝑏|22𝑏 ,𝛽 = 𝑎 − √𝑎2 − 4 |𝑏|22𝑏 .
(17)
After a lengthy calculation, we obtain
det𝐻 = 𝐷(0) = 12�푁√𝑎2 − 4 |𝑏|2 {[2 (𝑐𝑑 − |𝑏|2)+ (𝑎 − 𝑐 − 𝑑) (𝑎 −
√𝑎2 − 4 |𝑏|2)](𝑎+ √𝑎2 − 4 |𝑏|2)�푁−1 − [2 (𝑐𝑑 − |𝑏|2)+ (𝑎 − 𝑐 − 𝑑)
(𝑎 + √𝑎2 − 4 |𝑏|2)](𝑎− √𝑎2 − 4 |𝑏|2)�푁−1}
(for a tridiagonal matrix) ,
(18)
in the present case. It is notable that the determinant
dependsonly on |𝑏| and does not depend on 𝑏∗/𝑏 in the present
case. The reason is because the eigenvalues are unchangedunder
“gauge” transformation k → 𝑃k and 𝐻 → 𝑃𝐻𝑃−1,where𝑃 = diag.(1,
𝑒�푖�휒, 𝑒�푖2�휒, . . . , 𝑒�푖(�푁−1)�휒)with an arbitrary realconstant
𝜒.
The prescription of the above method to obtain thedeterminant is
very similar to the Gel’fand–Yaglom methodfor differential
operators. Namely, solving the differentialequation corresponds to
solving the recurrence relation,putting one of the boundary
conditions corresponds to fixingthe first term of the series of
numbers, and obtaining thedeterminant at another boundary
corresponds to obtainingthe determinant as the equation of the last
row in theeigenvalue equation.Note that, becausewe are treating a
finitematrix, the idea of normalization becomes different fromthe
functional determinant treated by the Gel’fand–Yaglommethod.
The method to obtain the determinant of tridiagonalmatrices in
this section is substantially equivalent to themethod for
difference operators described by Dowker [12],except for a specific
choice for Hermitian matrix in thepresent section.
3.2. Examples. In this subsection, we show determinantsof some
simple tridiagonal matrices for example. For allthe examples below,
the eigenvalues are known and, then,one can find that the formulas1
for finite product includingtrigonometric functions are
derived.
Note that the determinant 𝐷(0) for the matrix 𝐻 = Δ +𝑚2𝐼 (where
𝐼 is the identity matrix) is equivalent to 𝐷(−𝑚2)for the matrix Δ,
and we choose explicit expressions of 𝐷(0)for𝐻 here and
hereafter.
(i) 𝑎 = 𝑐 = 𝑑 = 2 + 4 sinh2(𝑧/2) and 𝑏 = 1.2In this case,𝐻 =
Δ�퐷�퐷 + 4 sinh2(𝑧/2)𝐼�푁, where
Δ�퐷�퐷 ≡ (((((((
2 −1 0 ⋅ ⋅ ⋅ 0−1 2 −1 ⋅ ⋅ ⋅ 00 −1 2 ⋅ ⋅ ⋅ 0... ... ... d ...2
−10 0 0 ⋅ ⋅ ⋅ −1 2)))))))
, (19)and 𝐼�푁 is the𝑁×𝑁 identity matrix. Note that, for 𝑎 = 𝑐 =
𝑑,Eq. (18) becomes
𝐷(0) = 12�푁+1√𝑎2 − 4 |𝑏|2 [(𝑎 + √𝑎2 − 4 |𝑏|2)�푁+1− (𝑎 − √𝑎2 − 4
|𝑏|2)�푁+1] . (20)
We now find
det𝐻 = 𝐷(0) = sinh (𝑁 + 1) 𝑧sinh 𝑧
for 𝐻 = Δ�퐷�퐷 + 4 sinh2 𝑧2𝐼�푁. (21)
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4 Advances in Mathematical Physics
Figure 1: 𝑃8 as an example of a path graph.(ii) 𝑎 = 𝑐 = 2 + 4
sinh2(𝑧/2), 𝑑 = 1 + 4 sinh2(𝑧/2), and𝑏 = 1.3In this case,𝐻 = Δ�퐷�푁
+ 4 sinh2(𝑧/2)𝐼�푁, where
Δ�퐷�푁 ≡ (((((((
2 −1 0 ⋅ ⋅ ⋅ 0−1 2 −1 ⋅ ⋅ ⋅ 00 −1 2 ⋅ ⋅ ⋅ 0... ... ... d ...2
−10 0 0 ⋅ ⋅ ⋅ −1 −1)))))))
(22)
We find
det𝐻 = 𝐷(0) = sinh (𝑁 + 1) 𝑧 − sinh𝑁𝑧sinh 𝑧= cosh𝑁𝑧 + 2 sinh 𝑧
sinh𝑁𝑧
for 𝐻 = Δ�퐷�푁 + 4 sinh2 𝑧2𝐼�푁.(23)
(iii) 𝑎 = 2+4 sinh2 (𝑧/2), 𝑐 = 𝑑 = 1+4 sinh2 (𝑧/2), and 𝑏 =
1.4In this case,𝐻 = Δ�푁�푁 + 4 sinh2 (𝑧/2)𝐼�푁, where
Δ�푁�푁 ≡ (((((((
1 −1 0 ⋅ ⋅ ⋅ 0−1 2 −1 ⋅ ⋅ ⋅ 00 −1 2 ⋅ ⋅ ⋅ 0... ... ... d ...2
−10 0 0 ⋅ ⋅ ⋅ −1 −1)))))))
= Δ(𝑃�푁) , (24)
which is known as the graph Laplacian [15–18] for the pathgraph
with𝑁 vertices (𝑃�푁, see Figure 1).
We find
det𝐻 = 𝐷(0) = 2 tanh 𝑧2 sinh𝑁𝑧for 𝐻 = Δ�푁�푁 + 4 sinh2 𝑧2𝐼�푁
(25)
in this case. Note that, since Δ(𝑃�푁) has a zero
mode,lim�푧�㨀→0𝐷(0) = 0.
(iv) Clockwork theory [19–22].5
We consider 𝐻 = Δ �푞 + 𝑙2𝐼�푁, where Δ �푞 is the following𝑁
×𝑁matrix:
Δ �푞 ≡ ((((((((
1 −𝑞 0 ⋅ ⋅ ⋅ 0−𝑞 1 + 𝑞2 −𝑞 ⋅ ⋅ ⋅ 00 −𝑞 1 + 𝑞2 ⋅ ⋅ ⋅ 0... ... ...
d ...1 + 𝑞2 −𝑞0 0 0 ⋅ ⋅ ⋅ −𝑞 𝑞2))))))))
. (26)
We find that the determinant of𝐻 can be written as𝐷(0) = 𝑙2 ⋅
(𝛾�푁+ − 𝛾�푁− )√(1 − 𝑞2)2 + 2 (1 + 𝑞2) 𝑙2 + 𝑙4
for 𝐻 = Δ �푞 + 𝑙2𝐼�푁, (27)where𝛾±= 12 [1 + 𝑞2 + 𝑙2 ± √(1 − 𝑞2)2
+ 2 (1 + 𝑞2) 𝑙2 + 𝑙4] . (28)Of course, one can see that lim�푞�㨀→1Δ
�푞 = Δ(𝑃�푁).4. Determinants of PeriodicTridiagonal Matrices
4.1. The Discrete Gel’fand–Yaglom Method for Periodic
Tridi-agonalMatrices. In this section, we treat periodic
tridiagonalmatrices, such as
𝐻 = (((((((
𝑎 −𝑏 0 ⋅ ⋅ ⋅ −𝑏∗−𝑏∗ 𝑎 −𝑏 ⋅ ⋅ ⋅ 00 −𝑏∗ 𝑎 ⋅ ⋅ ⋅ 0... ... ... d
...𝑎 −𝑏−𝑏 0 0 ⋅ ⋅ ⋅ −𝑏∗ 𝑎)))))))
. (29)
In this case, the recurrence relation is the same as in
theprevious section. Therefore, we can write
V�푘 = 𝐴𝛼�푘−1 + 𝐵𝛽�푘−1, (30)where 𝛼 and 𝛽 are the same as the
previous ones, i.e., Eq. (11).
In the periodic case, however, the first and the last rowsof the
eigenvalue equation are also the relation among threeterms in the
sequence of numbers. In the present case, theyare reduced to𝑏 [𝐴𝛽
(1 − 𝛼�푁) + 𝐵𝛼 (1 − 𝛽�푁)] = 0, (31)𝑏 [𝐴𝛼 (𝛼�푁 − 1) + 𝐵𝛽 (𝛽�푁 − 1)]
= 0, (32)where we used the fact that 𝛼 and 𝛽 are solutions of 𝑏𝑥2
−(𝑎 − 𝜆)𝑥 + 𝑏∗ = 0 and 𝛼𝛽 = 𝑏∗/𝑏. The existence of 𝐴 and 𝐵
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satisfying the above two equations and not being 𝐴 = 𝐵 =
0requires 𝛽 (1 − 𝛼
�푁) 𝛼 (1 − 𝛽�푁)𝛼 (𝛼�푁 − 1) 𝛽 (𝛽�푁 − 1)= (𝛽2 − 𝛼2) (1 − 𝛼�푁) (𝛽�푁
− 1) = 0. (33)This equation is satisfied if 𝜆 is an eigenvalue of
the matrix𝐻.In general, we suppose 𝛼 ̸= 𝛽 and the normalization can
beknown from the limit 𝑎 = 0 and 𝑏 → 0. Then, we concludethat the
characteristic polynomial 𝐷(𝜆) = ∏�푁�푝=1(𝜆�푝 − 𝜆)(where 𝜆�푝 (𝑝 = 1,
. . . , 𝑁) are eigenvalues of𝐻) is written by𝐷 (𝜆) = 𝑏�푁 (1 − 𝛼�푁)
(𝛽�푁 − 1)= 𝑏�푁 (𝛼�푁 + 𝛽�푁) − 𝑏�푁 − 𝑏∗�푁. (34)Therefore, the
determinant of𝐻 in this case is given bydet𝐻 = 𝐷 (0)
= (𝑎 + √𝑎2 − 4 |𝑏|22 )�푁
+(𝑎 − √𝑎2 − 4 |𝑏|22 )�푁 − 𝑏�푁 − 𝑏∗�푁
(for a periodic tridiagonal matrix) .(35)
One may be aware of unnecessary arguments in abovediscussion.
From the periodic structure, 𝛼�푁 = 1 or 𝛽�푁 =1 can be concluded.
However, the discussion above can begeneralized to treat another
type ofmatrix in the next section.
4.2. Example. (i) 𝑎 = 2 + 4 sinh2 (𝑧/2) and 𝑏 = 1.6In this case,
𝐻 = Δ(𝐶�푁) + 4 sinh2 (𝑧/2)𝐼�푁, where Δ(𝐶�푁)
is the graph Laplacian of the cycle graph with𝑁 vertices
(seeFigure 2):
Δ (𝐶�푁) ≡ (((((((
2 −1 0 ⋅ ⋅ ⋅ −1−1 2 −1 ⋅ ⋅ ⋅ 00 −1 2 ⋅ ⋅ ⋅ 0... ... ... d ...2
−1−1 0 0 ⋅ ⋅ ⋅ −1 2)))))))
. (36)
We find
det𝐻 = 𝐷(0) = 4 sinh2𝑁𝑧2 = (𝑒�푁�푧/2 − 𝑒−�푁�푧/2)2for 𝐻 = Δ(𝐶�푁) +
4 sinh2 𝑧2𝐼�푁. (37)
Figure 2: 𝐶8 as an example of a cycle graph.Note that
lim�푧�㨀→0𝐷(0) = 0 because of the zero mode ofΔ(𝐶�푁).
(ii) 𝑎 = 2 + 4 sinh2 (𝑧/2) and 𝑏 = 𝑒�푖�휒.7In this case,𝐻 =
Δ(𝐶�푁, 𝜒) + 4 sinh2 (𝑧/2)𝐼�푁, whereΔ (𝐶�푁, 𝜒)
≡ ((((((((
2 −𝑒−�푖�휒 0 ⋅ ⋅ ⋅ −𝑒−�푖�휒−𝑒−�푖�휒 2 −𝑒−�푖�휒 ⋅ ⋅ ⋅ 00 −𝑒−�푖�휒 2 ⋅
⋅ ⋅ 0... ... ... d ...2 −𝑒−�푖�휒−𝑒−�푖�휒 0 0 ⋅ ⋅ ⋅ −𝑒−�푖�휒 2
))))))))
. (38)
We find
det𝐻 = 𝐷(0) = 4 sinh2 𝑁𝑧2 + 4 sin2𝑁𝜒2= 𝑒�푁(�푧+�푖�휒)/2 −
𝑒−�푁(�푧+�푖�휒)/22for 𝐻 = Δ(𝐶�푁, 𝜒) + 4 sinh2 𝑧2𝐼�푁.
(39)
5. Determinants of Extended PeriodicTridiagonal Matrices
5.1. The Discrete Gel’fand–Yaglom Method For Extended Peri-odic
Tridiagonal Matrices. In this section, we consider thefollowing (𝑁
+ 1) × (𝑁 + 1)matrix:
𝐻 = ((((((((((
𝑎 −𝑏 0 ⋅ ⋅ ⋅ −𝑏∗ −𝑑−𝑏∗ 𝑎 −𝑏 ⋅ ⋅ ⋅ 0 −𝑑0 −𝑏∗ 𝑎 ⋅ ⋅ ⋅ 0 −𝑑... ...
... d ... ...𝑎 −𝑏 −𝑑−𝑏 0 0 ⋅ ⋅ ⋅ −𝑏∗ 𝑎 −𝑑−𝑑∗ −𝑑∗ −𝑑∗ ⋅ ⋅ ⋅ −𝑑∗ −𝑑∗
𝑐
))))))))))
. (40)
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6 Advances in Mathematical Physics
The recurrence relation can be found as𝑏V�푘+1 − (𝑎 − 𝜆) V�푘 +
𝑏∗V�푘−1 = −𝑑V�푁+1(2 ≤ 𝑘 ≤ 𝑁 − 1) . (41)The general solution of this
equation is
V�푘 = 𝐴𝛼�푘−1 + 𝐵𝛽�푘−1 − 𝑑V�푁+1𝑏 + 𝑏∗ − (𝑎 − 𝜆) , (42)where 𝛼 and
𝛽 are the same as Eq. (11).
The first row of the eigenvalue equation then becomes𝑏 [𝐴𝛽 (1 −
𝛼�푁) + 𝐵𝛼 (1 − 𝛽�푁)] = 0, (43)while the𝑁th row of the eigenvalue
equation is𝑏 [𝐴𝛼 (𝛼�푁 − 1) + 𝐵𝛽 (𝛽�푁 − 1)] = 0. (44)The two
equations are exactly the same as Eqs. (31) and (32).
Now, in addition, the (𝑁 + 1)st row of the eigenvalueequation
reads−𝑑∗ (V1 + V2 + ⋅ ⋅ ⋅ + V�푁) + (𝑐 − 𝜆) V�푁+1 = 0, (45)
and, by using the general solution, this can be reduced to
− 𝑑∗ (𝐴1 − 𝛼�푁1 − 𝛼 + 𝐵1 − 𝛽�푁1 − 𝛽 − 𝑁𝑑V�푁+1𝑏 + 𝑏∗ − (𝑎 − 𝜆))+
(𝑐 − 𝜆) V�푁+1 = 0. (46)As in the previous section, we require that
a nontrivial set
of (𝐴, 𝐵, V�푁+1) exists. This leads to the following equation:𝛽
(1 − 𝛼�푁) 𝛼 (1 − 𝛽�푁) 0𝛼 (𝛼�푁 − 1) 𝛽 (𝛽�푁 − 1) 0−𝑑∗ 1 − 𝛼�푁1 − 𝛼
−𝑑∗ 1 − 𝛽�푁1 − 𝛽 𝑁 |𝑑|2𝑏 + 𝑏∗ − (𝑎 − 𝜆) + 𝑐 − 𝜆
= (𝛽2 − 𝛼2) (1 − 𝛼�푁) (𝛽�푁 − 1)⋅ [ 𝑁 |𝑑|2𝑏 + 𝑏∗ − (𝑎 − 𝜆) + 𝑐 −
𝜆] = 0.
(47)
The second left-hand side of the equation should be
pro-portional to 𝐷(𝜆), as for discussion in the previous
section.Because we have already known the normalization of 𝑏�푁(1
−𝛼�푁)(𝛽�푁 − 1), we conclude that the characteristic polynomialin
the present case is written by𝐷(𝜆) = 𝑏�푁 (1 − 𝛼�푁) (𝛽�푁 − 1)
⋅ [ 𝑁 |𝑑|2𝑏 + 𝑏∗ − (𝑎 − 𝜆) + 𝑐 − 𝜆] . (48)
Thus, the determinant of𝐻 in this section is given by𝐷(0) = (𝑐 −
𝑁 |𝑑|2𝑎 − 𝑏 − 𝑏∗)[[[(
𝑎 + √𝑎2 − 4 |𝑏|22 )�푁
+(𝑎 − √𝑎2 − 4 |𝑏|22 )�푁 − 𝑏�푁 − 𝑏∗�푁]]](for an extended periodic
tridiagonal matrix) .
(49)
5.2. Examples. (i) 𝑎 = 2𝑟+ 𝑠 + 𝑙2, 𝑐 = 𝑁𝑠+ 𝑙2, 𝑑 = 𝑠, and 𝑏 =
𝑟[23, 24].8
Suppose the (𝑁 + 1) × (𝑁 + 1)matrix [23, 24],
Δ (𝑟, 𝑠) = 𝑟((((((((((
2 −1 0 ⋅ ⋅ ⋅ −1 0−1 2 −1 ⋅ ⋅ ⋅ 0 00 −1 2 ⋅ ⋅ ⋅ 0 0... ... ... d
... ...2 −1 0−1 0 0 ⋅ ⋅ ⋅ −1 2 00 0 0 ⋅ ⋅ ⋅ 0 0 0
))))))))))
+ 𝑠((((((((((
1 0 0 ⋅ ⋅ ⋅ 0 −10 1 0 ⋅ ⋅ ⋅ 0 −10 0 1 ⋅ ⋅ ⋅ 0 −1... ... ... d
... ...0 0 0 ⋅ ⋅ ⋅ 1 −1−1 −1 −1 ⋅ ⋅ ⋅ −1 𝑁
))))))))))
,(50)
where 𝑟 and 𝑠 are constants. The determinant of𝐻 = Δ(𝑟,
𝑠)+𝑙2𝐼�푁+1 isdet𝐻 = 𝐷 (0) = 𝑙2 (1 + 𝑁𝑠𝑠 + 𝑙2 ) (𝜂�푁+ + 𝜂�푁− − 2𝑟�푁)
, (51)
where
𝜂± ≡ 12 [2𝑟 + 𝑠 + 𝑙2 ± √(2𝑟 + 𝑠 + 𝑙2)2 − 4𝑟2] . (52)(ii) 𝑎 = 3 +
𝑙2, 𝑐 = 𝑁 + 𝑙2, and 𝑏 = 𝑑 = 1.
This is the previous case with 𝑟 = 𝑠 = 1. In this case,𝐻 =
Δ(𝑊�푁+1) + 𝑙2𝐼�푁+1, where
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Advances in Mathematical Physics 7
Figure 3:𝑊9 as an example of a wheel graph.
Δ (𝑊�푁+1) ≡((((((((((
3 −1 0 ⋅ ⋅ ⋅ −1 −1−1 3 −1 ⋅ ⋅ ⋅ 0 −10 −1 3 ⋅ ⋅ ⋅ 0 −1... ... ...
d ... ...3 −1 −1−1 0 0 ⋅ ⋅ ⋅ −1 3 −1−1 −1 −1 ⋅ ⋅ ⋅ −1 −1 𝑁
))))))))))
(53)
is the graph Laplacian of the wheel graph (see Figure 3) with𝑁+
1 vertices. We do not repeat writing the expression of𝐻,which is
given by Eq. (51) with Eq. (52) when 𝑟 = 𝑠 = 1.
(iii) 𝑏 = 0.In this case, the determinant simply becomes as
det𝐻 = 𝐷(0) = 𝑎�푁𝑐 − 𝑁𝑎�푁−1 |𝑑|2 . (54)Particularly, Δ(0, 1) is
in this category and can be written as
Δ (0, 1) = ((((((((((
1 0 0 ⋅ ⋅ ⋅ 0 −10 1 0 ⋅ ⋅ ⋅ 0 −10 0 1 ⋅ ⋅ ⋅ 0 −1... ... ... d
... ...0 0 0 ⋅ ⋅ ⋅ 1 −1−1 −1 −1 ⋅ ⋅ ⋅ −1 𝑁
))))))))))= Δ(𝐾1,�푁) .
(55)
This is the graph Laplacian of a star graph 𝐾1,�푁 (Figure 4).The
eigenvalues of Δ(𝐾1,�푁) are known as𝜆0 = 0,𝜆�푝 = 1 (𝑝 = 1, 2, . . .
, 𝑁 − 1) ,𝜆�푁 = 𝑁 + 1. (56)The determinant of𝐻 = Δ(𝐾1,�푁) + 𝑙2𝐼�푁+1
is
det𝐻 = 𝐷(0) = 𝑙2 (1 + 𝑁1 + 𝑙2) (1 + 𝑙2)�푁 . (57)
Figure 4: 𝐾1,8 as an example of a star graph.6. Free Energy on a
Graph
In this section, we consider applications of the results
ondeterminants for studying discrete systems.
We first consider 𝑁 scalar degrees of freedom and definethe
action as follows:
𝑆 = 12 (𝑁2𝐿2 �푁∑�푘=1
�푁∑�푘=1
𝜙�푘Δ (𝐶�푁)�푘�푘 𝜙�푘 + 𝜇2 �푁∑�푘=1
𝜙2�푘) , (58)where Δ(𝐶�푁) is the graph Laplacian for 𝐶�푁 and𝜇 ≡
2𝑁𝐿 sinh 𝑚𝐿2𝑁, (59)where 𝐿 and 𝑚 are constants.
Then, the Gaussian free energy on 𝐶�푁 [25] is obtainedusing Eq.
(37) as
𝐹�퐶𝑁 = 12 ln [det(Δ (𝐶�푁) + 4 sinh2 𝑚𝐿2𝑁𝐼�푁)]= ln(2 sinh 𝑚𝐿2 ) +
𝑐𝑜𝑛𝑠𝑡.. (60)This is interesting because the action (58) can be
rewritten as
𝑆 = 12 [𝑁2𝐿2 �푁∑�푘=1
(𝜙�푘 − 𝜙�푘+1)2 + 𝜇2 �푁∑�푘=1
𝜙2�푘] , (61)under the “periodic” condition, 𝜙�푁+1 ≡ 𝜙1. A
continuumlimit, 𝑎0 ≡ 𝐿/𝑁 → 0, enforces (𝜙�푘+1 − 𝜙�푘)/𝑎0 →
𝜕�푥𝜙,where 𝑥 is a coordinate of one dimension with periodicity𝑥+𝐿 ∼
𝑥.Therefore, we can find that the one-loop free energyof a real
scalar field 𝜙(𝑥) with mass 𝑚 on a circle (𝑆1) withcircumference 𝐿
governed by the action
𝑆 = 12 ∫𝑑𝑥 [(𝜕�푥𝜙)2 + 𝑚2𝜙2] (62)takes the form 𝐹�푆1 = ln (2 sinh
𝑚𝐿2 ) , (63)
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8 Advances in Mathematical Physics
after some regularization [25, 26]. Note that, since
2 sinh 𝑚𝐿2 = 𝑚𝐿∏∞�푛=1 (4𝜋2𝑛2/𝐿2 + 𝑚2)∏∞�푛=1 (4𝜋2𝑛2/𝐿2) , (64)we
find that the eigenvalues of −𝜕2�푥 + 𝑚2, where −𝜕2�푥 is
theone-dimensional Laplacian on 𝑆1, are shown by4𝜋2𝑛2𝐿2 + 𝑚2 (𝑛 is
an integer) . (65)
Similarly, we can consider the other matrices. For exam-ple, the
action for complex scalar fields, defined as
𝑆 = 12 (𝑁2𝐿2 �푁∑�푘=1
�푁∑�푘=1
𝜙†�푘Δ (𝐶�푁, 𝜒)�푘�푘 𝜙�푘 + 𝜇2 �푁∑�푘=1
𝜙�푘2)= 12 [𝑁2𝐿2 �푁∑
�푘=1
𝜙�푘 − 𝑒�푖�휒𝜙�푘+12 + 𝜇2 �푁∑�푘=1
𝜙�푘2] , (66)leads to the free energy𝐹�퐶𝑁,�휒 = ln [det (Δ (𝐶�푁,
𝜒) + 4 sinh2 𝑚𝐿2𝑁𝐼�푁)]= ln(4 sinh2 𝑁𝑧2 + 4 sin2𝑁𝜒2 ) + 𝑐𝑜𝑛𝑠𝑡..
(67)Here we will avoid repeated discussion and only note that
theeigenvalue spectrum of the continuum limit of this case isgiven
by (2𝜋𝑛 − 𝜒)2𝐿2 + 𝑚2 (𝑛 is an integer) . (68)
Continuum limits exist also in other some cases.The large 𝑁
limit of the determinant of Δ�퐷�퐷 + 𝜇2𝐼�푁
(according to Eq. (21)) becomes
sinh𝑚𝐿sinh (𝑚𝐿/𝑁) → 𝑁 sinh𝑚𝐿𝑚𝐿 , (69)
which coincides with the result of the example stated inSection
2 up to the constant. We find that the continuumlimit corresponds
to the system of massive scalar field in aline 0 ≤ 𝑥 ≤ 𝐿 with
Dirichlet-Dirichlet boundary conditionsat its ends.
The large 𝑁 limit of the determinant of Δ�퐷�푁 + 𝜇2𝐼�푁(according
to Eq. (23)) becomes simply
cosh𝑚𝐿 + 2 sinh 𝑚𝐿𝑁 sinh𝑚𝐿 → cosh𝑚𝐿. (70)A comparison to a known
mathematical relation
cosh𝑚𝐿 = ∏∞�푛=0 [(𝜋2/𝐿2) (𝑛 + 1/2)2 + 𝑚2]∏∞�푛=0 [(𝜋2/𝐿2) (𝑛 +
1/2)2] (71)leads to the conclusion that the continuum limit of
spectrumis given by (𝜋2/𝐿2)(𝑛 + 1/2)2 + 𝑚2; thus the
boundarycondition of the system is Dirichlet-Neumann condition.
Finally, the determinant ofΔ(𝑃�푁)+𝜇2𝐼�푁 (according to Eq.(25))
is
2 tanh 𝑚𝐿2𝑁 sinh𝑚𝐿. (72)Since the free energy is proportional to
the logarithm of this,we drop the 𝑁 dependent term (which is log
divergent if𝑁 → ∞).The boundary condition of the continuum systemis
Neumann-Neumann condition (which can be judged fromthe existence of
a zero mode).
In the next section, we will consider the way to obtainone-loop
vacuum energy of scalar field theory with massmatrix required by
structure of a theory space with four-dimensional spacetime.
7. Vacuum Energy from a Theory Space
7.1. Formulation. One-loop vacuum energy density in quan-tum
field theory can be derived from the functional determi-nants [2].
In the present paper, we only consider scalar fieldtheories for
simplicity. As seen in the previous section, 𝑁-scalar field theory
can resemble compactification of a dimen-sion. This is the key idea
of the dimensional deconstruction[8–10].The structure of the theory
space is determined by thequadratic term of fields, i.e., the mass
matrix. Suppose that amass matrix (precisely, the (𝑚𝑎𝑠𝑠)2 matrix)
Δ/𝑎20 is given (inother words, a theory space is given).The
eigenvalues ofΔ aredenoted by 𝜆�푝, as previously. Then, using the
characteristicpolynomial 𝐷(𝜆) = ∏�푝(𝜆�푝 − 𝜆), one-loop vacuum
energydensity for real scalar fields is calculated by
𝑉 = 12 ∫ 𝑑4𝑙(2𝜋)4 ln det [Δ/𝑎20 + 𝑙2]det [Δ/𝑎20 + 𝑙2 +𝑀2]= 12𝑎40
(2𝜋2) ∫∞0 𝑙3𝑑𝑙(2𝜋)4 ln 𝐷(−𝑙2)𝐷 (−𝑙2 −𝑀2𝑎20) ,
(73)
where we used 𝑀 of the Pauli-Villars regularization, whichis
considered to be 𝑀 → ∞. The constant 𝑎0 illustrates anoverall scale
in the theory space, i.e., related to mass scale ofnew physics via
𝑎0 ∼ 𝑚−1�푛�푒�푤 �푝ℎ�푦�푠i�푐�푠.
In practice, regularization is an art of assembly of
math-ematical techniques. We adopt here the following approach.A
physical value of the vacuum energy should be
determinedindependently of the unphysical 𝑀 and the UV
divergencemust be subtracted in the expression of it. Thus, we
consider,in the denominator in log in Eq. (73), as𝐷(−𝑙2 −𝑀2𝑎20) ⇒
Asymptotic form of 𝐷(−𝑙2)
when 𝑙2 → ∞. (74)Further, if the theory contains the 𝑁 (scalar)
fields, theintegrand of the most divergent part should be
proportionalto𝑁. Thus, we extract the part of∝ (𝑙2)�푁 ⊂ 𝐷(−𝑙2) for
large𝑙2.
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Advances in Mathematical Physics 9
7.2. DimensionalDeconstruction of a Circle. Aconcrete exam-ple
is in order. We consider a theory space associated withΔ(𝐶�푁, 𝜒).
This model has widely been studied by manyauthors [8–10, 27]. We
have already obtained det[Δ(𝐶�푁, 𝜒) +4 sinh2 (𝑧/2)𝐼�푁] (∝ 𝐷(−𝑙2),
in the present case) in Eq. (39).The asymptotic behavior can be
found as
det [Δ (𝐶�푁, 𝜒) + 4 sinh2 𝑧2𝐼�푁]= 𝑒�푁(�푧+�푖�휒)/2 −
𝑒−�푁(�푧+�푖�휒)/22 →𝑒�푁(�푧+�푖�휒)/22 (= 𝑒�푁�푧) (𝑧 → ∞) .(75)
Thus, in our regularization scheme,9𝑉 (𝜒) = 1𝑎40 (2𝜋2) ∫∞0
𝑑𝑧(2𝜋)4 8 sinh3 𝑧2 cosh 𝑧2⋅ ln 𝑒�푁(�푧+�푖�휒)/2 −
𝑒−�푁(�푧+�푖�휒)/22𝑒�푁(�푧+�푖�휒)/22 ,
(76)
where we set 𝑙 = 2 sinh(𝑧/2). Now, the integration can bedone by
elementary methods as𝑉(𝜒) = 2𝜋2𝑎40 ∫∞0 𝑑𝑧(2𝜋)4 4 sinh2 𝑧2 sinh 𝑧⋅
[ln (1 − 𝑒−�푁(�푧+�푖�휒)) + ln (1 − 𝑒−�푁(�푧−�푖�휒))]
= − 116𝜋2𝑎40 ∞∑�푛=1 ∫∞0 𝑑𝑧 (𝑒�푧/2 − 𝑒−�푧/2)2 (𝑒�푧 − 𝑒−�푧)⋅ [
𝑒−�푁�푛(�푧+�푖�휒)𝑛 + 𝑒−�푁�푛(�푧−�푖�휒)𝑛 ] = − 18𝜋2𝑎40 ∞∑�푛=1cos𝑁𝑛𝜒𝑛⋅
∫∞
0𝑑𝑧 (𝑒2�푧 − 2𝑒�푧 + 2𝑒−�푧 − 𝑒−2�푧) 𝑒−�푁�푛�푧
= − 18𝜋2𝑎40 ∞∑�푛=1cos𝑁𝑛𝜒𝑛 ( 1𝑁𝑛 − 2 − 2𝑁𝑛 − 1+ 2𝑁𝑛 + 1 − 1𝑁𝑛 +
2) = − 18𝜋2𝑎40 ∞∑�푛=1cos𝑁𝑛𝜒𝑛⋅ ( 4𝑁2𝑛2 − 4 − 4𝑁2𝑛2 − 1) = − 32𝜋2𝑎40⋅
∞∑�푛=1
cos𝑁𝑛𝜒𝑛 (𝑁2𝑛2 − 1) (𝑁2𝑛2 − 4) .
(77)
This result exactly coincides with the known result [8–10,27].10
Incidentally, for large𝑁,𝑉 (𝜒) ∼ − 32𝜋2 (𝑁𝑎0)4 ∞∑�푛=1cos𝑁𝑛𝜒𝑛5 ,𝑉
(0) ∼ − 3𝜁 (5)2𝜋2 (𝑁𝑎0)4 ,
(78)
where 𝜁(𝑧) is Riemann’s zeta function. We find that thereexists
a “continuum limit,”𝑁 → ∞ as𝑁𝑎0 and𝑁𝜒 are fixed.7.3. The Clockwork
Theory. Next, we turn to consider thetheory space of the clockwork
theory [19–22] for real scalarfields. The action is
𝑆 = 12 ∫ 𝑑4𝑥 [ �푁∑�푘=1
(𝜕�휇𝜙�푘)2 + 𝑚2�푁−1∑�푘=1
(𝜙�푘 − 𝑞𝜙�푘+1)2] , (79)where 𝑚 = 𝑎−10 . Thus, the relevant
matrix determinant isgiven as Eq. (27). The subtraction of UV
divergence is subtlebecause of the complicated form of the
determinant in thiscase. We separate the vacuum energy density into
three parts,such as 𝑉 = 𝑉�푁(𝑞) + 𝑉0 + 𝑁𝑉1. Here, 𝑉�푁(𝑞) is a finite
part,
𝑉�푁 (𝑞) = 12𝑎40 (2𝜋2)∫∞0 𝑙3𝑑𝑙(2𝜋)4 ln[1 − (𝛾−𝛾+)�푁] , (80)where
𝛾± is given by Eq. (28). This will be of order of 𝑂(𝑁−4)as in the
previous case and thus will have a continuum limitin vacuum energy
density.
The change of the integration variable cosh 𝑦 = (𝑙2 + 1 +𝑞2)/2𝑞
makes the integration simple. Then, we can rewrite𝑉�푁(𝑞) as𝑉�푁 (𝑞)
= 12𝑎40 (2𝜋2) ∫∞| ln �푞| 𝑑𝑦(2𝜋)4 𝑞 sinh 𝑦⋅ (2𝑞 cosh 𝑦 − 1 − 𝑞2) ln
(1 − 𝑒−2�푁�푦) , (81)and we get the form with infinite
summations,
𝑉�푁 (𝑞) = − 𝑞216𝜋2𝑎40 ∞∑�푛=1𝑞−2�푁�푛2𝑛 [ 𝑞2(2𝑁𝑛 − 2) (2𝑁𝑛 − 1)− 2
1(2𝑁𝑛 − 1) (2𝑁𝑛 + 1) + 𝑞−2(2𝑁𝑛 + 1) (2𝑁𝑛 + 2)(𝑞 ≥ 1) ,(82)
𝑉�푁 (𝑞) = − 𝑞216𝜋2𝑎40 ∞∑�푛=1𝑞2�푁�푛2𝑛 [ 𝑞−2(2𝑁𝑛 − 2) (2𝑁𝑛 − 1)− 2
1(2𝑁𝑛 − 1) (2𝑁𝑛 + 1)+ 𝑞2(2𝑁𝑛 + 1) (2𝑁𝑛 + 2)] (𝑞 < 1) .
(83)
Note that 𝑞−2𝑉�푁(𝑞) = 𝑞2𝑉�푁(𝑞−1). The numerical resultsfor
(𝑁𝑎0)4𝑞−2(5/�푁)𝑉�푁(𝑞5/�푁) are shown in Figure 5 for 𝑁 =3, 4, . . .
, 10. These curves indicate that there is a continuumlimit𝑁 → ∞,
while𝑁𝑎0 and 𝑞1/�푁 are fixed constants. If wecan treat 𝑞 as a
dynamical variable, the effective potential of𝑞 seems to have a
minimum at 𝑞 ∼ 1 for large 𝑁, where themass matrix simply becomes
the graph Laplacian of 𝑃�푁. Notealso that 𝑉�푁(𝑞) → 0 both for 𝑞 → 0
and for 𝑞 → ∞.
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10 Advances in Mathematical Physics
0.6 0.8 1.2 1.4 1.6 1.8 2
−0.0025
−0.002
−0.0015
−0.001
−0.0005
(.;0)4K−2(5/.)6.(K
5/.)q
Figure 5:The numerical results for (𝑁𝑎0)4𝑞−2(5/�푁)𝑉�푁(𝑞5/�푁) as
func-tions of 𝑞 for𝑁 = 3, 4, . . . , 10, from lower to upper
curves.
We now estimate the separated contributions. They arewritten
as
𝑉0 = 12𝑎40 (2𝜋2) ∫∞0 𝑙3𝑑𝑙(2𝜋)4⋅ ln 𝑙2√(1 − 𝑞2)2 + 2 (1 + 𝑞2) 𝑙2
+ 𝑙4 ,(84)
and
𝑉1 = 12𝑎40 (2𝜋2)∫∞0 𝑙3𝑑𝑙(2𝜋)4 ln 𝛾+= 12𝑎40 (2𝜋2) ∫∞0 𝑙3𝑑𝑙(2𝜋)4
ln[[[
1 + 𝑞2 + 𝑙2 + √(1 − 𝑞2)2 + 2 (1 + 𝑞2) 𝑙2 + 𝑙42 ]]] .(85)
As for 𝑉0, if we use the standard formula of derivation ofthe
Coleman-Weinberg potential12 ∫ 𝑑4𝑙(2𝜋)4 ln (𝑙2
+𝑀2)�푟�푒�푔�푢�푙�푎�푟�푖�푧�푒�푑 = 𝑀464𝜋2 ln𝑀2, (86)to regularize 𝑉0,
aside from the contribution of a zero mode(as ln 𝑙2 in the
integrand), we find
𝑉0 = − 12 (64𝜋2) 𝑎40 [(1 − 𝑞)4 ln (1 − 𝑞)2+ (1 + 𝑞)4 ln (1 +
𝑞)2] . (87)It is notable that this contribution is equivalent to
subtractionof the half of vacuum energy densities due to scalar
fields withmass squared (1 − 𝑞)2/𝑎20 and (1 + 𝑞)2/𝑎20 . The UV
divergenceof this part can be regarded to be canceled by the
zero-modecontribution.
On the other hand, for the complicated form of a
genuinedivergent contribution of𝑁𝑉1 , we introduce a cut-offΛ in
theintegration over 𝑙 and find
𝑁𝑉1 = 𝑁64𝜋2𝑎40 [Λ4 (lnΛ2 − 12) + 2 (1 + 𝑞2) Λ2− (1 + 4𝑞2 + 𝑞4)
ln 2Λ21 + 𝑞2 + 1 − 𝑞2 + (1 + 𝑞2)2− 32 (1 + 𝑞2) 1 − 𝑞2] .(88)
The quartic divergence seems to be independent of thestructure
of the mass matrix and the quadratic divergence isproportional to
the trace of the mass matrix.
7.4. Latticization of a Disk. The matrix Δ(𝑟, 𝑠) is used in
[23,24] as a latticization of a disk. Using the result of Eqs. (51)
and(52), one-loop vacuum energy density of scalar field theorywith
mass matrix Δ(𝑟, 𝑠)/𝑎20 can be written formally as𝑉 = 2𝜋22𝑎40 ∫∞0
𝑙3𝑑𝑙(2𝜋)4⋅ ln [𝑙2 (1 + 𝑁𝑠𝑠 + 𝑙2 ) (𝜂�푁+ + 𝜂�푁− − 2𝑟�푁)] = 𝑉�푁 (𝑟,
𝑠)+ 𝑉0 + 𝑁𝑉1,
(89)
where
𝑉�푁 = 2𝜋22𝑎40 ∫∞0 𝑙3𝑑𝑙(2𝜋)4 ln [1 + (𝜂−𝜂+)�푁 − 2( 𝑟𝜂+ )�푁]=
2𝜋22𝑎40 ∫∞0 𝑙3𝑑𝑙(2𝜋)4 2 ln[1 − ( 𝑟𝜂+)�푁] ,(90)
with 𝜂+ ≡ 12 [2𝑟 + 𝑠 + 𝑙2 + √(2𝑟 + 𝑠 + 𝑙2)2 − 4𝑟2] , (91)𝑉0 =
2𝜋22𝑎40 ∫∞0 𝑙3𝑑𝑙(2𝜋)4 ln [𝑙2 (1 + 𝑁𝑠𝑠 + 𝑙2 )] , (92)and
𝑉1 = 2𝜋22𝑎40 ∫∞0 𝑙3𝑑𝑙(2𝜋)4 ln 𝜂+ = 2𝜋22𝑎40 ∫∞0 𝑙3𝑑𝑙(2𝜋)4⋅ ln 2𝑟
+ 𝑠 + 𝑙2 + √(2𝑟 + 𝑠 + 𝑙2)2 − 4𝑟22 .
(93)
A finite part 𝑉�푁 can be rewritten as𝑉�푁 ( 𝑠𝑟) = 4𝜋2𝑟22𝑎40 ∫∞0
𝑙3𝑑𝑙(2𝜋)4 ln[[[1
−( 22 + 𝑠/𝑟 + 𝑙2 + √(2 + 𝑠/𝑟 + 𝑙2)2 − 4)�푁]]] .
(94)
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Advances in Mathematical Physics 11
46
8
10N
0
0.5
1
1.5
2
u
−0.15−0.125−0.1
−0.075−0.05
(.;0)4L−26.(u/N)
Figure 6:Thenumerical results for (𝑁𝑎0)4𝑟−2𝑉�푁(𝑢/𝑁) as a
functionof𝑁 and 𝑢.Furthermore, introducing new variables 𝑠/𝑟 = 2
sin2 (𝑢/2)and cosh 𝑦 = 𝑙2/2 + cosh 𝑢, we find𝑉�푁 (𝑢) = 4𝜋2𝑟22𝑎40
∫∞�푢 𝑑𝑦(2𝜋)4 2 sinh 𝑦 (cosh 𝑦 − cosh 𝑢)⋅ ln [1 − 𝑒−�푁�푦] = −
𝑟28𝜋2𝑎40 ∞∑�푛=1𝑒−�푁�푛�푢𝑛⋅ [ 𝑒2�푢(𝑁𝑛 − 2) (𝑁𝑛 − 1) − 2 1(𝑁𝑛 − 1) (𝑁𝑛
+ 1)
+ 𝑒−2�푢(𝑁𝑛 + 1) (𝑁𝑛 + 2)] .(95)
The numerical result of (𝑁𝑎0)4𝑟−2𝑉�푁(𝑢/𝑁) is plotted as
afunction of 𝑁 and 𝑢 in Figure 6, where we treat 𝑁 as a con-tinuous
parameter. We find that lim�푁�㨀→∞(𝑁𝑎0)4𝑟−2𝑉�푁(0) =−3𝜁(5)/4𝜋2, while
we find no other limiting cases for general𝑟 and 𝑠; i.e., no
precise continuum limit exists in general cases.
We now turn to consider the other part of the vacuumenergy. For
𝑉0, using similar estimation as in the previoussubsection, we
obtain, up to the zero-mode contribution,
𝑉0 = 164𝜋2𝑎40 𝑠2 {(𝑁 + 1)2 ln [(𝑁 + 1)2 𝑠2] − ln 𝑠2} , (96)which
is equivalent to the contribution of a scalar field withmass
squared (𝑁+1)2𝑠2/𝑎20 minus the contribution of a scalarfield with
mass squared 𝑠2/𝑎20 .The UV divergence is canceledin these two
contributions.
The divergent part is analyzed by using the cut-off Λ andis
found to be
𝑁𝑉1 = 𝑁64𝜋2𝑎40 [Λ4 (lnΛ2 − 12) + 2 (2𝑟 + 𝑠) Λ2− (6𝑟2 + 4𝑟𝑠 + 𝑠2)
ln 2Λ22𝑟 + 𝑠 + √𝑠 (4𝑟 + 𝑠)+ (2𝑟 + 𝑠)2 − 32 (2𝑟 + 𝑠)√𝑠 (4𝑟 + 𝑠)]
.
(97)
Again, we find that the quartic divergence is independent ofthe
mass matrix and the quadratic divergence is proportionalto the
trace of the mass matrix.11
In the next section, we will exhibit one more example
ofcalculation of one-loop vacuum energy density for a
slightlycomplicated theory space.
8. Some Other Examples of Vacuum Energy
Using the additional formulas on determinants, we canfurther
obtain determinants of various matrices. In thissection, we show
some other examples below.
8.1. Adding an Edge with a Vertex to Each Vertex of a Graph.Let
Δ�푁 be an𝑁×𝑁Hermitian matrix and define a 2𝑁× 2𝑁matrix Δ 2�푁 as
follows:Δ 2�푁 ≡ (Δ�푁 + 𝐼�푁 −𝐼�푁−𝐼�푁 𝐼�푁 ) , (98)where 𝐼�푁 is an 𝑁
dimensional identity matrix. In particular,if Δ�푁 is the graph
Laplacian of a graph 𝐺, Δ 2�푁 is the graphLaplacian of the graph
generated by adding an edge with avertex to every vertex of 𝐺.
Then, the formula on determinants
det(𝐴 𝐵𝐶 𝐷) = det (𝐴 − 𝐵𝐷−1𝐶) det (𝐷) (99)tells us that
det (Δ 2�푁 − 𝜆𝐼2�푁)= (1 − 𝜆)�푁 det(Δ�푁 − 1 − (1 − 𝜆)21 − 𝜆 𝐼�푁)
. (100)Therefore, if 𝐷�푁(𝜆) ≡ ∏�푁�푝=1(𝜆�푝 − 𝜆) = det(Δ�푁 −
𝜆𝐼�푁),
where {𝜆�푝} are eigenvalues of Δ�푁, is known, 𝐷2�푁(𝜆) ≡det(Δ 2�푁
− 𝜆𝐼2�푁) is obtained as𝐷2�푁 (𝜆) = (1 − 𝜆)�푁 𝐷�푁 (1 − (1 − 𝜆)21 − 𝜆
) . (101)
For example, we will calculate vacuum energy density ofthe
scalar field theorywithmassmatrixΔ 2�푁/𝑎20 , whereΔ 2�푁 isgenerated
fromΔ�푁 = Δ(𝐶�푁); i.e., Δ 2�푁 is the graph Laplacianof the graph
shown in Figure 7.
In this case, after some manipulation, we get
𝐷2�푁 (𝜆) = 𝛼 (𝜆)�푁 [1 − (1 − 𝜆𝛼 (𝜆) )�푁]2 , (102)
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12 Advances in Mathematical Physics
Figure 7: The graph generated by adding an edge with a vertex
toevery vertex of 𝐶9.where
𝛼 (𝜆) ≡ 𝜆2 − 4𝜆 + 2 + √𝜆 (𝜆3 − 8𝜆2 + 16𝜆 − 8)2 . (103)Now, we
can obtain the vacuum energy density in this
theory by utilizing ln𝐷2�푁(−𝑙2), as in the previous section.We
separate the finite and divergent parts of vacuum energydensity
as
𝑉�푁 = 12𝑎40 (2𝜋2) ∫∞0 𝑙3𝑑𝑙(2𝜋)4 2 ln[1 − ( 1 + 𝑙2𝛼 (−𝑙2))�푁] ,
(104)and
𝑉1 = 12𝑎40 (2𝜋2)∫∞0 𝑙3𝑑𝑙(2𝜋)4 ln𝛼 (−𝑙2) . (105)The numerical
result of (𝑁𝑎0)4𝑉�푁 in the present case
is shown in Figure 8, where 𝑁 is treated as a
continuousparameter. In the limit of 𝑁 → ∞, (𝑁𝑎0)4𝑉�푁
approaches−3𝜁(5)/16𝜋2, which is quarter of the value of the large𝑁
limitof (𝑁𝑎0)4𝑉�푁 in the case of the real scalar theory based onthe
graph Laplacian Δ(𝐶�푁). The divergent part 𝑁𝑉1 can beestimated,
because 𝛼(−𝑙2) ∼ 𝑙4 for large 𝑙, as
𝑁𝑉1 = 2𝑁64𝜋2𝑎40 Λ4 (lnΛ2 − 12) + 𝑁Λ28𝜋2 + ⋅ ⋅ ⋅ . (106)The
leading term is proportional to the number of real scalarfields, as
expected. The quadratic divergence is proportionalto the trace of
the mass matrix.
8.2.The Graph Cartesian Products 𝐺 × 𝑃2. Let Δ�푁 be an𝑁×𝑁
Hermitian matrix and define a 2𝑁 × 2𝑁 matrix Δ̂ 2�푁 asfollows:
Δ̂ 2�푁 ≡ (Δ�푁 + 𝐼�푁 −𝐼�푁−𝐼�푁 Δ�푁 + 𝐼�푁) . (107)In particular,
ifΔ�푁 is the graph Laplacian of a graph𝐺, Δ̂ 2�푁 isthe graph
Laplacian of the graph Cartesian product 𝐺 × 𝑃2.12
6 8 10 12 14N
−0.05
−0.04
−0.03
−0.02(.;0)
46.
Figure 8: The numerical value of (𝑁𝑎0)4𝑉�푁 for the model inthis
subsection as a function of 𝑁. The dotted lines
indicate(1/4)(𝑁𝑎0)4𝑉�푁, where 𝑉�푁 is the vacuum energy density in
the realscalar theory whose mass matrix is Δ(𝐶�푁)/𝑎20 , and the
constant−3𝜁(5)/16𝜋2.
Then, the use of the formula on determinants
det(𝐴 𝐵𝐶 𝐷) = det (𝐴 − 𝐵𝐷−1𝐶) det (𝐷)= det (𝐴𝐷 − 𝐵𝐶) ,
(108)provided that [𝐶,𝐷] = 0, leads to
det (Δ̂ 2�푁 − 𝜆𝐼2�푁) = det ([Δ�푁 + (1 − 𝜆) 𝐼�푁]2 − 𝐼�푁)= det
(Δ�푁 − 𝜆𝐼�푁)⋅ det (Δ�푁 + (2 − 𝜆) 𝐼�푁) . (109)Therefore, if𝐷�푁(𝜆) ≡
∏�푁�푝=1(𝜆�푝 − 𝜆) = det(Δ�푁 − 𝜆𝐼�푁), where{𝜆�푝} are eigenvalues of
Δ�푁, is known, 𝐷2�푁(𝜆) ≡ det(Δ̂ 2�푁 −𝜆𝐼2�푁) is obtained as𝐷2�푁 (𝜆)
= 𝐷�푁 (𝜆)𝐷�푁 (𝜆 − 2) . (110)
The vacuum energy density of the scalar field theory withmass
matrix Δ̂ 2�푁/𝑎20 , where Δ̂ 2�푁 is generated from Δ�푁 =Δ(𝐶�푁),
i.e., Δ̂ 2�푁 is the graph Laplacian of the graph Cartesianproduct
𝐶�푁 × 𝑃2, called as the prism graph 𝑌�푁. We show thegraph 𝑌9 in
Figure 9.
We can obtain the vacuum energy density in this theoryby
utilizing ln𝐷2�푁(−𝑙2). We separate the finite and divergentparts of
vacuum energy density as𝑉�푁 = 116𝜋2𝑎40 ∫∞0 𝑙3𝑑𝑙
⋅ 2{{{{{ln[[[1 −( 22 + 𝑙2 + √(2 + 𝑙2)2 − 4)�푁]]]
+ ln[[[1 −( 24 + 𝑙2 + √(4 + l2)2 − 4)�푁]]]
}}}}} ,(111)
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Advances in Mathematical Physics 13
Figure 9: The graph Cartesian product 𝐶9 × 𝑃2, or the prism
graph𝑌9.and
𝑉1 = 116𝜋2𝑎40 ∫∞0 𝑙3𝑑𝑙 [[[ln2 + 𝑙2 + √(2 + 𝑙2)2 − 42
+ ln4 + 𝑙2 + √(4 + 𝑙2)2 − 42 ]]] .(112)
The numerical result of (𝑁𝑎0)4𝑉�푁 in the present caseis shown in
Figure 10, where 𝑁 is treated as a continuousparameter. In the
limit of 𝑁 → ∞, (𝑁𝑎0)4𝑉�푁 approachesthe vacuum energy density of
the model associated with 𝐶�푁and −3𝜁(5)/4𝜋2. The divergent part𝑁𝑉1
can be estimated as
𝑁𝑉1 = 2𝑁64𝜋2𝑎40 Λ4 (lnΛ2 − 12) + 3𝑁Λ216𝜋2 + ⋅ ⋅ ⋅ . (113)The
leading term is proportional to the number of real scalarfields, as
expected. The quadratic divergence is proportionalto the trace of
the mass matrix.
9. Conclusion
In the present paper, we showed the method of obtaining
thedeterminant of repetitive tridiagonal matrices with
concreteexamples. The concept of the method is similar to
theGel’fand–Yaglom method of obtaining functional determi-nants for
differential operators.
The repetitive matrices as mass matrices are widelyconsidered in
modern models in particle physics, in orderto attack the hierarchy
problem by adopting a theory space.We showed one-loop vacuum
energies of such models canbe evaluated by using the determinant of
the mass matricesobtained by our method stated in earlier
sections.
We have seen that there are not always genuine contin-uum limits
in large𝑁 for general theory spaces. In Section 7,we have also
found that contributions of 𝑉0 expressed inlogarithmic functions
remain in general. They can be com-pensated by addition of bosonic
or fermionic free fields withappropriate mass in some cases.13
6 8 10 12 14N
(.;0)46.
−0.08
−0.09
−0.11
−0.12
−0.13
Figure 10: The numerical value of (𝑁𝑎0)4𝑉�푁 for the model
whosetheory space is associated with 𝑌�푁 as a function of 𝑁. The
dottedlines indicate (𝑁𝑎0)4𝑉�푁, where 𝑉�푁 is the vacuum energy
densityin the real scalar theory whose mass matrix is Δ(𝐶�푁)/𝑎20 ,
and theconstant −3𝜁(5)/4𝜋2.
In future work, we wish to study one-loop energy densityin
models of deconstructed warped (theory) space [28–33].Although it
is difficult to evaluate the determinants in a closedform in such a
model, calculation based on recurrence rela-tions would be suitable
for a computer. It is also interesting toinvestigate the recent
model of deconstruction of torus withmagnetic flux [34].14
If we would like to deal with the matrices related withmore
complicated graphs or higher dimensional lattices,we confront other
difficulties.15 The graph Laplacians ofgeneric graphs cannot be
expressed by tridiagonal matrices,though, fortunately, it is known
that arbitrary square matricescan be systematically tridiagonalized
by the Householdermethod [35] (see also Refs. [36, 37]). Thus, in
principle, ourGel’fand–Yaglom-type method can be applied to the
matrixwith the general graph structure.
Finally, we add a comment on exclusion of zero modes.Zero modes
of operators which appear in quantum fieldtheory have crucial
meanings related with nonperturbativeaspects of the theory (see,
for example, the first section ofRef. [14]). In our present paper,
we considered mass termsin almost all examples and the cases with
zero modes canbe considered as the limit that the value of mass
goes tozero. Because we considered the vacuum energies and
theirdependence on the parameters in this paper, the analysis
isjust sound.Moreover, it is known that, if amatrix is expressedas
a graph Laplacian of a simple graph (as in each examplein this
paper), the matrix has a single zero mode. Therefore,further
analysis on zeromodes, if necessary, could be
fulfilledappropriately.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of
interestregarding the publication of this paper.
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14 Advances in Mathematical Physics
Endnotes
1. For example, see [38].2. In this simple case, eigenvalues are
known as𝜆�푝 = 4 sin2 𝜋𝑝2 (𝑁 + 1) + 4 sinh2 𝑧2 (𝑝 = 1, 2, . . . , 𝑁)
. (∗)3. In this case,
𝜆�푝 = 4 sin2 𝜋 (𝑝 − 1/2)2 (𝑁 + 1/2) + 4 sinh2 𝑧2 (𝑝 = 1, 2, . .
. , 𝑁) . (∗∗)4. In this case, the eigenvalues are𝜆�푝 = 4 sin2 𝜋𝑝2𝑁
+ 4 sinh2 𝑧2(𝑝 = 0, 1, 2, . . . , 𝑁 − 1) .(∗ ∗ ∗)5. In this case,𝜆0
= 𝑙2,𝜆�푝 = 𝑞2 + 1 − 2𝑞 cos𝜋𝑝𝑁 + 𝑙2(𝑝 = 1, 2, . . . , 𝑁 − 1) .(∗ ∗
∗∗)6. In this case, the eigenvalues are𝜆�푝 = 4 sin2𝜋𝑝𝑁 + 4 sinh2
𝑧2(𝑝 = 0, 1, 2, . . . , 𝑁 − 1) .(∗ ∗ ∗ ∗ ∗)
Note that degeneracy occurs.7. In this case, the eigenvalues
are
𝜆�푝 = 4 sin2 (𝜋𝑝𝑁 + 𝜒2 ) + 4 sinh2 𝑧2(𝑝 = 0, 1, 2, . . . , 𝑁 −
1) .(∗ ∗ ∗ ∗ ∗∗)8. In this case,𝜆0 = 𝑙2,𝜆�푝 = 𝑠 + 𝑟 sin2 𝜋𝑝𝑁 + 𝑙2(𝑝
= 1, 2, . . . , 𝑁 − 1) ,𝜆�푁 = (𝑁 + 1) 𝑠 + 𝑙2.
(∗ ∗ ∗ ∗ ∗ ∗ ∗)9. We now consider complex scalar fields.10. It
is notable that the regularized vacuum energy is𝑂(𝑁−4), while the
subtracted part whose integrand
including ln 𝑒�푁�푧 = 𝑁𝑧 is proportional to𝑁.
11. Note that, remembering the one zero-mode
contributionseparated from 𝑉0, the quartic divergence is found to
beproportional to𝑁 + 1.
12. The graph Cartesian product of 𝐺1 and 𝐺2 is also
oftenwritten as 𝐺1◻G2.
13. In order to apply the models to the hierarchy problem,there
must be other matter fields coupled to the fieldsin the theory
space. Therefore, it may not be a greatdifficulty in the model.
14. A partially deconstructed model with flux has beenconsidered
more than a decade ago [39].
15. Even in the case with differential operators in
higerdimensions, there is a problem of degeneracy, whichbecomes an
origin of another divergence [40].
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