Correlation function of modular Hamiltonians

We investigate varies correlation functions of modular Hamiltonians defined with respect to spatial regions in quantum field theories. These correlation functions are divergent in general. We extract finite correlators by removing divergent terms for two dimensional massless free scalar theory. We reproduce the same correlators in general two dimensional conformal field theories.


Introduction
Modular Hamiltonian of a spatial subregion is a natural and fundamental object in QFT [1]. In QFT defined with a cut-off, it is the logrithmic of the reduced dentity matrix ρ A ,Ĥ A = − log ρ A . It plays a central role in quantum information theory. Entanglement entropy, relative entropy and other important quantities are constructed from modular Hamiltonian [2][3][4][5]. It has also been used to obtain many interesting results, such as the Bousso bound [6][7][8], first law of entanglement entropy [9], the proof of varies energy conditions [10,11] and contrains of correlation functions [12]. Modular Hamiltonian is in general a highly non-local operator, though for certain symmetric situations it could be analytic. When the subregion is Rindler wedge of Minkowski spacetime and the state is in vacuum, modular Hamitonian is the boost generator which is a smeared operator of stress tensor in Rindler wedge [13]. A nice realization of this result is Unruh effect [14]. For a spherical region in a conformal field theory, an integral form could be found by conformal transformation of the Rindler wedge [15]. In two dimensional free field theory, a bilocal form of modular Hamiltonian can also be obtained for several disjoint intervals [16,17]. For a conformal field theory in a state which has a gravitational dual, a 1/G N expansion of modular Hamiltonian has also been proposed [18,19] from bulk, where the first term is an area operator of Ryu-Takayanagi [20] surface,Ĥ bulk and other higher order terms are bulk modular Hamiltonian of the corresponding bulk region. Unfortunately, modular Hamiltonian is not additive, in the sense that modular Hamitonian of two disjoint regions is in general not the summation of modular Hamiltonian in each seperate region which makes it hard to study multi-region quantities.
Rényi entropy is divergent for continuous quantum field theories 2 , however, one can still obtain interesting information by taking care of the cut-off. By definition, it is essentially a combination of correlation functions of modular Hamiltonians, the divergence is partly due to the singularity when two or more modular Hamiltonians collide in the same region. Therefore, the following correlation function Ĥ m for two finite disjoint regions. We will specify the details of the construction and compute (1.4) for massless free scalar and general conformal field theories in the following sections. The correlation function (1.4) is not directly related to Rényi entropy of two disjoint regions since (1.2). However, it may provide another way to understand the correlation between two disjoint regions.
The structure of this paper is as follows. We will introduce a generator of (1.4) in section 2 and study the generator explicity in two dimensinal massless free scalar theory in section 3. In section 4, we reproduce the same result of section 3 using technics of two dimensional conformal field theory. We will comment on the implications to operator product expansion of reduced density matrix and constraints on holograhic dual (1.1) in section 5 and 6. Conclusions and discussions are collected in the last section.

A generator of correlation functions
We consider a system with a density matrix ρ. The modular Hamiltonian corresponding to region A(or B) isĤ A (orĤ B ). A and B are assumed to be spacelike and disjoint, therefore from causality, the commutator ofĤ A andĤ B is zero. A generator of correlation function (1.4) is where we have inserted a normalization factor such that T A∪B (a, b) is zero whenever e −aĤ A e −bĤ B = e −aĤ A e −bĤ B . An alert reader may realize that there is a similar quantity in the context of Wilson loop. If we replace e −aĤ A and e −bĤ B by two seperated Wilson loops, then (2.1) is the logrithmic of the correlator of two Wilson loops. Expanding (2.1) around (a, b) = (0, 0), the coefficient before a m b n is the connected correlation function (1.4). The summation is from (m, n) = (1, 1) because T A∪B (a, 0) = T A∪B (0, b) = 0. More explicitly, we have the first few orders are 3 We don't present the correlators for m < n since the defintion is symmetric under the exchange of A and B. Generator (2.1) is not easy to evaluate in general. We will focus on two dimensional massless free scalar theory in the following section and then extend it to general conformal field theories. The advantage of two dimensional conformal system is that the modular Hamiltonian of a single region is well known.

Two dimensional massless free scalar
To get familiar with the concept T A∪B (a, b), we will compute the generator (2.1) in two dimensional massless free scalar system. The system is in vacuum with Lagrangian We use t, z to denote spacetime coordinates Region A (B) is an interval with radius R A (R B ) whose center is at z A (z B ), where the end points of the intervals are

5)
A and B are disjoint, we can assume Massless free theory is a conformal field theory, there is a unique cross ratio The cross ratio is always between 0 and 1, Another quantity which is related to cross ratio is η is between 0 and ∞, 0 < η < ∞. (3.11)

Modular hamiltonian of region A iŝ
where the stress tensor T tt is The integral is evaluated at constant t = 0 slice. Unfortunately, a direct expansion of the exponential function e −aĤ A (3.14) leads to divergent terms when two stress tensors T tt collide. However, as we will show below, the divergent terms are canceled in the generator (2.1). It would be much easier to work in momentum space for free scalar. In the following subsections, we will first review the quantization of a free scalar in general curved spacetime and then quantize a massless free scalar in an interval. After that, we will discuss the generator (2.1).

Massless free scalar field in curved spacetime
It is useful to review the general framework [22] of a free scalar field in curved spacetime since we will quantize a free scalar field in a subregion of Minkowski spacetime. We just list several key points of this framework. The spacetime we will consider is 15) where N is called lapse function and G ij is the d − 1 dimensinoal reduced metric on the hypersurface of constant time t = t 0 . The Klein-Gordon inner product of two field configurations is where Σ is the constant time hypersurface and n µ is its unit norm vector. Klein-Gordon equation in spacetime (3.15) is solved by a set of complete set of eigenmodes f i (x). Eigenmodes can be dedomposed and normalized to satisfy Therefore the scalar field φ could be decomposed in terms of eigenmodes f i by We will assume f i 's are positive frequency modes and f * i 's are negative frequency modes, then coefficients a i and a † i are annihilation and creation operators, respectively. It is easy to show Therefore, one can define vacuum |0 f as 20) The subscript f shows that the choice of positive frequency modes f i 's determines the vacuum. However, there is no unique choice of positive frequency modes in general. Suppose another complete set of positive frequency modes g I satisfy Then the scalar field φ can also be decomposed in terms of eigenmodes g I by where b I and b † I are annihilation and creation operators, respectively. The commutation relations are Therefore, one can define vacuum |0 g by b I |0 g = 0, ∀I. Vacuum |0 f and |0 g are not equivalent in general. Since f i eigenmodes are complete, g I and g * I can be decomposed in terms of f i and f * i , where α Ii and β Ii are called Bogoliubov coefficients. The inverse of (3.25) is since g I 's are also complete. Then the annihilation and creation operators a i , a † i , b I , b † I are related by Bogoliubov transformation Therefore, commutation relations (3.19) and (3.23) are equivalent to the following consistency relations of Bogoliubov coefficients α Ii and β Ii can be treated as elements of matrices α and β, respectively. Then the consistency relations (3.31) to (3.34) are and Finally, before we study massless free scalar field in an interval, we list several results of two dimensional massless free scalar in Minkowski spacetime The field can be decomposed into right moving and left moving modes where right moving and left moving modes are decouple. We will just consider right moving modes. Solving Klein-Gordon equation, a complete set of eigenmodes is where ω is frequency which is positive ω > 0. By choosing t = 0 time slices and requiring the standard Klein-Gordon bracket relations (3.17), the normalization constant N ω is The scalar field can be decomposed into linear combination of f ω and f * ω , The annihilation and creation operators satisfy the commutation relation Minkowski vacuum |0 M is annihilated by a ω ,

Massless free scalar field in region A
Now we can study a massless free scalar field in region A (3.3). The method is similar to quantize a free scalar in Rindler spacetime [23]. The interval is located at t = 0 time slice with center position z A and interval length 2R A . The coordinate transformation from Minkowski spacetime to region A is where −∞ < τ, u < ∞ are new coordinates of region A. They cover the causal development of A. To see this point, we observe that (3.46) After the coordinate transformation, the metric becomes which is conformally flat. Now we will quantize the massless free boson field in spacetime (3.47). The Klein-Gordon equation can be solved, the field is still decomposed into right moving and left moving modes. Again, we will just consider right moving modes. A complete set of positive frequency modes is where v is a positive frequency corresponding to time τ , v > 0. The normalization constant is still Therefore, field in region A could be written as The annihilation and creation operators Since field φ in region A can also be expanded in terms of Minkowski modes f ω given in previous section, the Bogoliubov transformation between f ω and g v modes is Since g v 's are not complete in Minkowski spacetime, there is no inverse of transformation (3.53). The Bogoliubov coefficients are We checked the consistency conditions (3.36) using Bogoliubov coefficients above. Note (3.35) are not satisfied since g v 's are not complete eigenmodes in Minkowski spacetime. As we have reviewed in previous subsection, annihilation and creation operators in region A are related to those in Minkowski spacetime by Bogoliubov transformation The stress tensor (3.13) in region A can also be casted into right moving and left moving part. Focusing on right moving part, we find the modular Hamiltonian (3.12) to bê The form (3.57) is very similar to the Hamiltonian of a free scalar in Minkowski spacetime. The constant term can be fixed by normalization condition Massless free scalar field in region B (3.4) is similar, the coordinate transformation from Minkowski spacetime to region B is Quantizing massless free scalar field, we useṽ to denote the frequency in region B, then the annihilation and creation operators in region B are where Bogoliubov coefficients are

Expectation value of an exponential operator
From the definition of T A∪B (a, b) and the modular Hamiltonian in momentum space (3.57) and (3.63), the relevant quantity is the expectation value of an exponential operator where we can set x I to be free real function of quantum number I at this moment. For simplicity, we will assume quantum number I to be discrete , the dimension of Hilbert space is finite, we denote the dimension to be M . The annihilation and creation operators in the original Minkowski spacetime are also labeled as discrete quantum number i whose dimension is N . The result can be easily extended to continuous limit. To introduce the final result, we will clarify some notations at first. We will define two vectors A and B as Therefore, the commutation relations (3.19) and (3.23) are where K and k are 2N × 2N and 2M × 2M matrices They are Hermitian matrices. We also note that Then the operator I x I b † I b I can be written compactlŷ The Bogoliubov transformation from a modes to b modes is where S is a 2M × 2N matrix whose elements are Bogoliubov matrices defined previously Its Hermitian conjugate is They transform K to k through consistent relations (3.36), The commutation relation ofĤ and A is where We will prove this identity in Appendix A. Using the definition of K, k, Λ and S, e −zKH can be written as where p and q are rank M diagonal matrices The matrices Φ,Θ,Φ are not relevant to our result below. The expectation value of e zĤ is We will use normal ordering and parameter differentiation method to prove this identity in Appendix B. We can simplify (3.84) further by noticing the so called matrix determinant lemma [24], where the matrix T is At the last step of (3.87), we used the Bogoliubov coefficients consistency conditions (3.36) and we can simplify (3.84) to be

Region A
From previous discussion, the expectation value of an exponential operator of the formĤ = where 1 should be understood as Dirac delta function in the continuous limit, The diagonal matrix q A (a) has the following elements Using Bogoliubov coefficients (3.54) and (3.55), we find The details of the computation is in Appendix C. So T A is a diagonal matrix. Therefore (3.92) is where δ(0) is from the diagonal element in momentum space, formally it is Dirac delta function evaluated at 0, We will regularize it by matching Z A (a) to Rényi entropy of region A. This provides a consistent check of our result. We already know the reduced density matrix ρ A is where ρ 0 can be fixed by the normalization of ρ A In momentum space, the right hand side is easy to calculate, therefore for one inverval, where is a UV cutoff. Comparing (3.105) with (3.106) and noticing that the central charge of free boson is 1, we have the following regularization rule

Region A and B
We are interested in the expectation value of the following operator From previous general discussion, we find that (3.110) The matrix M, N, C, D are Some of the matrices are already discussed in previous section, where Matrices q A and q B are Bogoliubov matrices which connect region A and region B are where the function G(x, y) is without changing the value of T A∪B (a, b). Therefore (3.132) 4 If an n × n matrix M is formed from an n × n matrix M by interchanging two rows or two columns of M , See chapter 13 of [24]. The matrix (3.130) is obtained from (3.110) by interchanging two rows (or two columns) even times, therefore the determinant is invariant.
Since M AA , M BB , N AA , N BB are diagonal matrices, we could easy obtain their inverse (3.139) Therefore, A, B, C and D are  The compact form of (3.131) can be evaluated as a series expansion where we define (3.149) Given the exact result (3.148)-(3.149), we discuss several properties of T A∪B (a, b) for two dimensional massless free scalar in the following.

Large distance expansion
When two regions A and B are far way to each other, η 1, then the matrix Therefore (3.148) can be understood as large distance expansion of T A∪B (a, b). As n increases, the contribution of T n decreases. The leading term is xy sinh πax sinh πby sinh πx sinh πy sinh π(1 + a)x sinh π(1 + b)y × F(x, x, y) which is also expected from the definition of T A∪B (a, b).
In the large distance limit, the first two terms of F(x, x, y) are independent of x and y (3.156) The first two terms are also the leading two terms of T A∪B (a, b) since T n (a, b) is at least O(η 4 ) for any n ≥ 2. Now we compute T 2 (a, b),  which removes divergent terms. We will discuss the correlation functions (3.160) in two dimensional massless free scalar theory. We first define a set of quantities

Correlation functions of modular Hamiltonians
x 2 x 2 y 2 y 2 sinh 2 πx sinh 2 πx sinh 2 πy sinh 2 πy F 2 (x, x , y, y ). A∪B for any m ≥ 1. Actually, The last step has been checked numerically for a general set of positive a and η. Then We will discuss this point later.

Two dimensional conformal field theory
We find the exact correlator T A∪B (a, b) in two dimensional massless free scalar theory by quantizing it in finite region. In this section, we will study the same correlator for more general two dimensional conformal field theories. Any two dimensional conformal field theory has a general sector which is realized by operator product expansion (OPE) of stress tensor We defined y ij = y i − y j which is the distance between points y i and y j . Any higher point functions of stress tensor could be fixed by Ward identity [25]  ). (4.5) Therefore T A∪B (a, b) should be completely fixed by conformal symmetry. The modular Hamiltonian in region A isĤ At the second step, we have used the convention [26] T tt = −2πT and changed variable y to z by z = R A (y + z A ). (4.7) To simplify computation, we already set R A = 1. We can also set R B = 1 and z B = 0, therefore the modular hamiltonian in region B iŝ It is enough to choose the branch z A > 2, therefore The leading term is The integral is ]. (4.11) For c = 1, it matches with (3.164) which is computed in a rather different way. Now we compute T " : · · · : is understood to remove divergent terms. There is a pole near y 1 = y 2 inside the integral, we regularize the integral as if there is no pole 5 . In practice, one can first do indefinite integral and then taking the limit to the integral bound. If one regularize the integral like this, the integral becomes finite ]. (4.13) It matches with (3.165) for c = 1. This is also a consistency check for the way to regularize the integral. To convince ourselves further, we compute The O(c 2 ) term has been canceled between the terms Ĥ 3 AĤ B and −3 Ĥ 2 A Ĥ AĤB . This can be checked by using the Ward identity for T (y 4 ). Using the same method to regularize the integral, we find Again, it matches with corresponding scalar result. Finally, where the polylogrithm Li n (z) is (4.17) In the large distance limit, we find which is exactly (3.168) for c = 1. One can check (4.16) is indeed (3.167) numerically. This result is also mathematically nontrivial since it is a multiple integral of product of hypergeometric functions. It is quite interesting to obtain even higher point correlators of modular Hamiltonian from position space, the result can be expressed as Li n functions similar to T A∪B . However, the computation becomes cumbersome quickly.
We will comment on the general structure of T where T (m, n; η) is independent of central charge. Technically any O(c k ), k ≥ 2 terms are from the most singular term of Ward identity, but these terms are canceled in connected correlation functions Therefore, only terms proportional to c are left which will lead to (4.19). One can also understand the property in another way. The modular Hamiltonian is O(c) for conformal field theory, therefore in the large c limit, where f is an unkown function which is O(c 0 ) at this moment. Taking the logrithmic, one should have T A∪B (a, b) ∼ O(c).
where the matrices A, B, C and D are exactly those in massless free scalar theory.

Operator product expansion of reduced density matrix
In this section, we will study the relation between operator product expansion of e −aĤ A ≡ ρ a A and the generator T A∪B (a, b) defined in this paper. We will focus on two dimensional conformal field theory. By definition, We find the correlator in previous sections. However, one can also use operator product expansion to evaluate the same quantity. Notice that ρ a A is a nonlocal operator in region A, it should be decomposed as a summation of complete orthogonal operators in region A with proper coefficients. Schematically, it is The terms in "decendants" should be fixed by conformal symmetry for each primary operators O. Formally expanding ρ a A with the powers of modular Hamiltonian, operator product expansion of stress tensor tells us that O can either be stress tensor or (quasi-)primary operators constructed from multiple stress tensors. The operator with lowest conformal weight is just the stress tensor, therefore where O means primary operators whose conformal dimension is at least 4. We have seperated a term which is related to stress tensor. We expect that the most natural way to organize stress tensor and its decendants in region A is the modular Hamiltonian. To fix the coefficient c T (a), we compute correlator T A∪B (a, b) in the large distance limit, At the second line, we used operator product expansion (5.3) for region A and B. The " · · · " terms are subleading terms in the large distance limit since they are contributed by operators with higher conformal weight. The quantity Ĥ AĤB has been discussed in previous section, we just borrow the result in large distance limit, This should match with (3.156) since it is the leading term in T A∪B (a, b), therefore There is no phase factor since c T (a) is real. Note it seems that also satisfies all the conditions. However, one can expand ρ a A for small a, therefore This fixes c T (a) to be (5.6) completely. Then This interprets the novel integral property (3.171). At the second line, we used operator product expansion of ρ a A . At the last step, we used the property that the vacuum correlation function of any two primary operators with different conformal weight is zero. To fix the coefficients before primary operator with higher conformal weight in (5.3), we should expand the exact result of T A∪B (a, b) order by order. We leave it for future work.

Comments on holographic dual
In the context of AdS 3 /CF T 2 , Newton constant of AdS 3 gravity is mapped to the central charge of a two dimensional conformal field theory living in the boundary [27], where is AdS radius and G N is Newton constant. A quantity which is proportional to central charge in conformal field theory will be said to be classical from the gravity side. The correlator T A∪B (a, b) defined in this work is proportional to central charge for general conformal field theory.This indicates the gravity dual should be a quantity evaluates on a classical configuration. The construction of the explicit quantity is beyond the scope of this paper. However, from discussion in the CF T side, there is no O(1) correction for the correlator, so the quantity will be free from any quantum corrections. This will constrain the correlation functions of operators in the bulk through (1.1). A similar bulk quantity T A∪B (a, b) can have a 1/G N ∝ c expansion, however, only terms proportional to c will be left and any other correlation functions should be canceled exactly. In higher dimensions, there will be no similar universal result of T A∪B (a, b) for general conformal field theory, however, we still expect it will constrain bulk correlation functions for any consistent quantum gravity.

Conclusion and discussion
We evaluate the exact generator T A∪B (a, b) for two dimensional massless free scalar theory. The result only depends on a set of Bogoliubov matrices. We also obtain an exact generator in two dimensional conformal field theories by noticing T A∪B (a, b) is proportional to the central charge c and matching it with massless free scalar theory. We could check the result up to m + n = 4 for general two dimensional conformal field theories. Higher point correlation functions could be found in principle though the computation will quickly become messy. As a by product, we find several exact definite integrals of multiple specific hypergeometric functions. We could check these integrals numerically but a rigorous proof is still lacking.
Our work shows that one can extract finite result from correlation functions Ĥ m AĤ n B , though the correlators themselves are divergent in general. The finite part, Ĥ m AĤ n B c are functions of cross ratio for conformal field theories. The generator T A∪B (a, b) may be meaningful by itself since its form is formally similar to a correlator of Wilson loops [28,29]. Actually, given any non-local quantities e Q A and e Q B , one can always define similar correlators. T A∪B is special as modular Hamiltonian should exist for almost any subregion QFTs. On the other hand, Wilson loops (or other non-local operators) can only be defined in limited QFTs (e.g. gauge theories). T A∪B could be extended to curved spacetimes and may characterize non-local properties of spacetime. One simple example is shock wave spacetime [30,31]. The quantization of a massless scalar field in supertranslated shock wave spacetime is studied by [32]. Assuming A is in the left region of shock wave spacetime while B is in the right region, A(B) is a flat subregion so modular HamiltonianĤ A (Ĥ B ) can be defined as in Minkowski spacetime. However, the correlator T A∪B is affected by shock wave between A and B. In this simple example, subregion observer in A(B) will detect the same phenomenon as in Minkowski spacetime while non-trivial information is stored in correlators between A and B.
Motivated by the similarity to [33], We briefly discussed the operator product expansion of e −aĤ A using the knowledge of T A∪B (a, b). A complete discussion of operator product expansion may be possible. We expect to return to this project in the near future.
In higher dimensions, T A∪B (a, b) depends on theory and shape of subregions. However, it is still possible to extract finite result in some simple examples. The technics of quantizing field theory in a subregion of Minkowski spacetime developed in this work could be extended to higher dimensions.
since normal ordering will remove any annihilation operator to the right hand side and annihilation operator will annihilate vacuum |0 M . Only e K is left since it is a number. The differential method is to taking the derivative of z of both sides of (B.2), then using the definition of normal ordering to findĤ = where means the derivative of z, S(a i ) is similarity transformation of a i Similarly, the similarity transformation of a i a j is which is a quadratic polynomial of a i and a † i . By matching the coefficients before a † i a † j , a † i a j , a i a j and identity of (B.4), we find an equation set where we have omitted the differential equations for F ij and H ij since they are not relevant in this work. G are matrix whose elements are The matrices h and g are defined as Combining with (3.76), we have So h is Hermitian and g is symmetric, Now we can take z derivative of the matrix e −zKH and using the definition of Θ, Φ, h and g, we find The differential equation of Θ is Θ = −Θh + Φg * . (B.14) The differential equation of K is then At the last step, we used the equation (B.14) and trh = trh * (B. 16) since h is Hermitian. The equation (B.15) can be integrated out explicitly with the initial condition K(0) = 0, This proves the identity (3.84).

C Bogoliubov Matrices
The Bogoliubov matrices used in this work are only quadratic in terms of α, β. Depending on the region, Bogoliubov matrices are classified into two classes. The first class is Any matrix belongs to first class is constructed from only one region (A or B). The second class is Any matrix belongs to second class is constructed from two regions (A and B). We first study matrices in first class. Since region B is similar to region A, we can focus on region A. Notice that it is enough to consider β * A β T A and α * A β † A .
At the second step, we inserted a positive imaginary part in the exponential to make the integral finite. At the third step, we changed the varaible s, s to t, t by s = tanh t, s = tanh t . (C.7) Then residue theorem has been used for the integral of t. The proof of (3.98) is similar, we will find a term which is proportional to δ(v + v ). However, since v and v are assumed to be positive, δ(v + v ) is always zero, The computation of matrices in second class is quite similar, we will just show the details for β * A β T B .