Deconstructing the Gel’fand–Yaglom Method and Vacuum Energy from a Theory Space
The discrete Gel’fand–Yaglom theorem was studied several years ago. In the present paper, we generalize the discrete Gel’fand–Yaglom method to obtain the determinants of mass matrices which appear in current works in particle physics, such as dimensional deconstruction and clockwork theory. Using the results, we show the expressions for vacuum energies in such various models.
The Gel’fand–Yaglom method  for obtaining functional determinants of differential operators with boundaries is widely known nowadays. For nice reviews, see [2, 3]. The applications of the Gel’fand–Yaglom method have been investigated quite recently, to evaluate one-loop vacuum energies in nontrivial boundary conditions [4–7].
Among them, Altshuler examined vacuum energy in warped compactification [6, 7]. In recent years, it is supposed that extra dimensions of various types could play an important role in the hierarchy problem, and thus the study of physics in nontrivial background geometry is still advancing.
The dimensional deconstruction has appeared as a new tool for understanding the properties of higher-dimensional field theories [8–10] more than a decade ago. In such a model of deconstruction, a “theory space” is considered, which consists of sites and links, to which four-dimensional fields are individually assigned. Theory spaces thus have the structures of graphs  and can be interpreted as the theory with discrete extra dimensions.
Several years ago, the discrete Gel’fand–Yaglom method for difference operators was reviewed and studied by Dowker . We generalize the discrete Gel’fand–Yaglom method for studying one-loop vacuum energies in extended deconstructed theories and models with discrete dimensions in the present paper. To this end, we develop the method of computing determinants of repetitive Hermitian matrices which correspond to mass matrices utilized in deconstructed theories.
After completion of the first version of the manuscript of the present paper (arXiv:1711.06806), a paper which treats the determinants of discrete Laplace operators appeared . Their method is substantially the same as ours, because the author also relies on the recurrence relation among three variables on a lattice (see Section 3 in the present paper and below). We recently become aware of another similar paper on the determinants of matrix differential operators . They studied generalization of Gel’fand–Yaglom method to obtain the functional determinants. Their work differs essentially from ours because they considered differential operators while we treat matrices as operators. We also point out that they did not consider the matrices of large size which have certain continuum limits.
The organization of this paper is as follows. In order to make the present paper self-contained, we show a short review of the Gel’fand–Yaglom method for a differential operator, along with Dunne’s review , in Section 2. In Section 3, we give the method to obtain determinants of tridiagonal matrices with repeated structure. This is a straightforward generalization of description in Ref. . In Section 4, we give the method to obtain determinants of periodic tridiagonal matrices. Determinants of extended periodic tridiagonal matrices are obtained in Section 5. The rest of the present paper is devoted to applications to deconstructed theories and discrete systems. In Section 6, free energy on a graph is discussed by using the results of previous sections. In Section 7, we show the method of calculation for evaluating one-loop vacuum energy in deconstructed models from the determinants of mass matrices. In Section 8, we show a few more examples of one-loop vacuum energies for slightly complicated theory spaces. We give conclusions in the last section, Section 9.
2. Review of the Gel’fand–Yaglom Method 
Suppose that an eigenvalue equation with Dirichlet-Dirichlet boundary conditions in one dimension is given.
Thenholds. Here satisfies with a boundary condition .
Example. In the region , under the Dirichlet condition at the boundaries, we consider the functional determinantThe solution of with is . Thus according to the Gel’fand–Yaglom method, we obtain
Proof. is a function of and has zeros at . The function which satisfies and becomes the eigenfunction when . Then the boundary condition at , i.e., , is satisfied. In other words, is a function of and has zeros at . Therefore holds.
3. Determinants of Tridiagonal Matrices
3.1. The Discrete Gel’fand–Yaglom Method for Tridiagonal Matrices
Now, we show the discrete Gel’fand–Yaglom method to obtain determinants of finite matrices. First, we consider the following Hermitian tridiagonal matrix of rows and columns: In this case, the eigenvalue equation where , can be categorized into three parts:Here, Eq. (7) is just the recurrence relation among three terms in as a sequence of numbers. In the present case, the general solution for the recurrence relationiswhere and are constants andNote that and are roots of the second-order equation and .
The first row of the eigenvalue equation, Eq. (6), determines the relation between and ; in this case, that is . If we further choosethe coefficients and are obtained as
Substituting all of the results above into Eq. (8) in the present case, we get
Now, we set the left-hand side of Eq. (15) as . is zero if is an eigenvalue of the matrix in this case. By construction, should be an th order polynomial of . The reason is the following: , , and so on. This observation shows includes . Finally, since the left-hand side of Eq. (8) reads in the present case, has the term as the highest order term in . We can also directly confirm this by setting and the limit in the left-hand side of Eq. (15). We then verify .
Therefore, we conclude that is the characteristic polynomial of , where are eigenvalues of .
The determinant of is given by . In the present case, we find where After a lengthy calculation, we obtainin the present case. It is notable that the determinant depends only on and does not depend on in the present case. The reason is because the eigenvalues are unchanged under “gauge” transformation and , where with an arbitrary real constant .
The prescription of the above method to obtain the determinant is very similar to the Gel’fand–Yaglom method for differential operators. Namely, solving the differential equation corresponds to solving the recurrence relation, putting one of the boundary conditions corresponds to fixing the first term of the series of numbers, and obtaining the determinant at another boundary corresponds to obtaining the determinant as the equation of the last row in the eigenvalue equation. Note that, because we are treating a finite matrix, the idea of normalization becomes different from the functional determinant treated by the Gel’fand–Yaglom method.
The method to obtain the determinant of tridiagonal matrices in this section is substantially equivalent to the method for difference operators described by Dowker , except for a specific choice for Hermitian matrix in the present section.
In this subsection, we show determinants of some simple tridiagonal matrices for example. For all the examples below, the eigenvalues are known and, then, one can find that the formulas1 for finite product including trigonometric functions are derived.
Note that the determinant for the matrix (where is the identity matrix) is equivalent to for the matrix , and we choose explicit expressions of for here and hereafter.
(i) and .2
In this case, , where and is the identity matrix. Note that, for , Eq. (18) becomes We now find
(ii) , , and .3
In this case, , where We find(iii) , , and .4
We findin this case. Note that, since has a zero mode, .
We consider , where is the following matrix: We find that the determinant of can be written aswhereOf course, one can see that .
4. Determinants of Periodic Tridiagonal Matrices
4.1. The Discrete Gel’fand–Yaglom Method for Periodic Tridiagonal Matrices
In this section, we treat periodic tridiagonal matrices, such as
In this case, the recurrence relation is the same as in the previous section. Therefore, we can write where and are the same as the previous ones, i.e., Eq. (11).
In the periodic case, however, the first and the last rows of the eigenvalue equation are also the relation among three terms in the sequence of numbers. In the present case, they are reduced towhere we used the fact that and are solutions of and . The existence of and satisfying the above two equations and not being requires This equation is satisfied if is an eigenvalue of the matrix . In general, we suppose and the normalization can be known from the limit and . Then, we conclude that the characteristic polynomial (where () are eigenvalues of ) is written by Therefore, the determinant of in this case is given by
One may be aware of unnecessary arguments in above discussion. From the periodic structure, or can be concluded. However, the discussion above can be generalized to treat another type of matrix in the next section.
(i) and .6
In this case, , where is the graph Laplacian of the cycle graph with vertices (see Figure 2):
We findNote that because of the zero mode of .
(ii) and .7
In this case, , where We find
5. Determinants of Extended Periodic Tridiagonal Matrices
5.1. The Discrete Gel’fand–Yaglom Method For Extended Periodic Tridiagonal Matrices
In this section, we consider the following matrix: The recurrence relation can be found as The general solution of this equation is where and are the same as Eq. (11).
Now, in addition, the ()st row of the eigenvalue equation reads
and, by using the general solution, this can be reduced to
As in the previous section, we require that a nontrivial set of exists. This leads to the following equation: The second left-hand side of the equation should be proportional to , as for discussion in the previous section. Because we have already known the normalization of , we conclude that the characteristic polynomial in the present case is written by Thus, the determinant of in this section is given by
This is the previous case with . In this case, , where is the graph Laplacian of the wheel graph (see Figure 3) with vertices. We do not repeat writing the expression of , which is given by Eq. (51) with Eq. (52) when .
In this case, the determinant simply becomes as Particularly, is in this category and can be written as This is the graph Laplacian of a star graph (Figure 4). The eigenvalues of are known as The determinant of is
6. Free Energy on a Graph
In this section, we consider applications of the results on determinants for studying discrete systems.
We first consider scalar degrees of freedom and define the action as follows:where is the graph Laplacian for andwhere and are constants.
Then, the Gaussian free energy on  is obtained using Eq. (37) as This is interesting because the action (58) can be rewritten as under the “periodic” condition, . A continuum limit, , enforces , where is a coordinate of one dimension with periodicity . Therefore, we can find that the one-loop free energy of a real scalar field with mass on a circle () with circumference governed by the actiontakes the form after some regularization [25, 26]. Note that, since we find that the eigenvalues of , where is the one-dimensional Laplacian on , are shown by
Similarly, we can consider the other matrices. For example, the action for complex scalar fields, defined as leads to the free energy Here we will avoid repeated discussion and only note that the eigenvalue spectrum of the continuum limit of this case is given by
Continuum limits exist also in other some cases.
The large limit of the determinant of (according to Eq. (21)) becomes which coincides with the result of the example stated in Section 2 up to the constant. We find that the continuum limit corresponds to the system of massive scalar field in a line with Dirichlet-Dirichlet boundary conditions at its ends.
The large limit of the determinant of (according to Eq. (23)) becomes simply A comparison to a known mathematical relation leads to the conclusion that the continuum limit of spectrum is given by ; thus the boundary condition of the system is Dirichlet-Neumann condition.
Finally, the determinant of (according to Eq. (25)) is Since the free energy is proportional to the logarithm of this, we drop the dependent term (which is log divergent if ). The boundary condition of the continuum system is Neumann-Neumann condition (which can be judged from the existence of a zero mode).
In the next section, we will consider the way to obtain one-loop vacuum energy of scalar field theory with mass matrix required by structure of a theory space with four-dimensional spacetime.
7. Vacuum Energy from a Theory Space
One-loop vacuum energy density in quantum field theory can be derived from the functional determinants . In the present paper, we only consider scalar field theories for simplicity. As seen in the previous section, -scalar field theory can resemble compactification of a dimension. This is the key idea of the dimensional deconstruction [8–10]. The structure of the theory space is determined by the quadratic term of fields, i.e., the mass matrix. Suppose that a mass matrix (precisely, the matrix) is given (in other words, a theory space is given). The eigenvalues of are denoted by , as previously. Then, using the characteristic polynomial , one-loop vacuum energy density for real scalar fields is calculated bywhere we used of the Pauli-Villars regularization, which is considered to be . The constant illustrates an overall scale in the theory space, i.e., related to mass scale of new physics via .
In practice, regularization is an art of assembly of mathematical techniques. We adopt here the following approach. A physical value of the vacuum energy should be determined independently of the unphysical and the UV divergence must be subtracted in the expression of it. Thus, we consider, in the denominator in log in Eq. (73), as Further, if the theory contains the (scalar) fields, the integrand of the most divergent part should be proportional to . Thus, we extract the part of for large .
7.2. Dimensional Deconstruction of a Circle
A concrete example is in order. We consider a theory space associated with . This model has widely been studied by many authors [8–10, 27]. We have already obtained (, in the present case) in Eq. (39). The asymptotic behavior can be found as Thus, in our regularization scheme,9 where we set . Now, the integration can be done by elementary methods as This result exactly coincides with the known result [8–10, 27].10 Incidentally, for large , where is Riemann’s zeta function. We find that there exists a “continuum limit,” as and are fixed.
7.3. The Clockwork Theory
Next, we turn to consider the theory space of the clockwork theory [19–22] for real scalar fields. The action is where . Thus, the relevant matrix determinant is given as Eq. (27). The subtraction of UV divergence is subtle because of the complicated form of the determinant in this case. We separate the vacuum energy density into three parts, such as . Here, is a finite part, where is given by Eq. (28). This will be of order of as in the previous case and thus will have a continuum limit in vacuum energy density.
The change of the integration variable makes the integration simple. Then, we can rewrite as and we get the form with infinite summations, Note that . The numerical results for are shown in Figure 5 for . These curves indicate that there is a continuum limit , while and are fixed constants. If we can treat as a dynamical variable, the effective potential of seems to have a minimum at for large , where the mass matrix simply becomes the graph Laplacian of . Note also that both for and for .
We now estimate the separated contributions. They are written as and
As for , if we use the standard formula of derivation of the Coleman-Weinberg potential to regularize , aside from the contribution of a zero mode (as in the integrand), we find It is notable that this contribution is equivalent to subtraction of the half of vacuum energy densities due to scalar fields with mass squared and . The UV divergence of this part can be regarded to be canceled by the zero-mode contribution.
On the other hand, for the complicated form of a genuine divergent contribution of , we introduce a cut-off in the integration over and find The quartic divergence seems to be independent of the structure of the mass matrix and the quadratic divergence is proportional to the trace of the mass matrix.
7.4. Latticization of a Disk
The matrix 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 theory with mass matrix can be written formally as where with and
A finite part