Quantum flux operators for Carrollian diffeomorphism in general dimensions

We construct Carrollian scalar field theories in general dimensions, mainly focusing on the boundaries of Minkowski and Rindler spacetime, whose quantum flux operators form a faithful representation of Carrollian diffeomorphism up to a central charge, respectively. At future/past null infinity, the fluxes are physically observable and encode rich information of the radiation. The central charge may be regularized to be finite by the spectral zeta function or heat kernel method on the unit sphere. For the theory at the Rindler horizon, the effective central charge is proportional to the area of the bifurcation surface after regularization. Moreover, the zero mode of supertranslation is identified as the modular Hamiltonian, linking Carrollian diffeomorphism to quantum information theory. Our results may hold for general null hypersurfaces and provide new insight in the study of the Carrollian field theory, asymptotic symmetry group and entanglement entropy.


Introduction
Null hypersurfaces play a central role in the causal structure of spacetime. They are important in many physical systems, such as future/past null infinity of asymptotically flat spacetime and black hole horizons. The notable feature of null hypersurfaces is that the metric is degenerate. Recently, the null geometry has been studied in the context of Carrollian manifolds [1][2][3][4][5].
The Carrollian diffeomorphism preserving the Carroll structure [6] could provide a natural separation between space and time [7]. More interestingly, it has been shown in a series of papers [6,8,9] that one may construct quantum flux operators that generate Carrollian diffeomorphism up to anomalous terms at future null infinity. The operators are defined through the flux densities which are radiated to future null infinity and they encode rich information of the time and angle dependencies about the radiation. The operators are totally defined at future null infinity and could form a representation of Carrollian diffeomorphism. Rather interestingly, the radiation fluxes are physically observables. Therefore, it indicates that all the Carrollian diffeomorphism should be regarded as large gauge transformations, extending the BMS groups [10][11][12][13][14][15][16].
However, there are still two remaining problems. In higher dimensions, there are consistent falloff conditions that prevent the supertranslations from being asymptotic symmetries [17,18]. On the other hand, the Weinberg soft theorem is believed to be valid in all dimensions [19][20][21]. There exist interesting discussions on the extended symmetry in higher dimensions [22][23][24]. Moreover, there are various null hypersurfaces (not necessarily null infinity) on which one could define "localized charges" [25][26][27]. One of the most intriguing example is the Killing horizon of black holes [28][29][30][31][32]. From the intrinsic perspective of Carrollian manifold, there are no essential differences among various null hypersurfaces [33][34][35]. Therefore, one could define field theory and find quantum operators that represent Carrollian diffeomorphism for any null hypersurfaces in general dimensions.
In this paper, we will study the quantum flux operators associated with Carrollian diffeomorphism in general dimensions. We will use two typical null hypersurfaces, the null infinity and the Rindler horizon, to present the results. More explicitly, we reduce the bulk massless scalar theory in Minkowski spacetime to future null infinity and construct the radiation fluxes. The flux operators form a faithful representation of the Carrollian diffeomorphism up to a central charge which may be regularized through the spectral zeta function on the unit sphere. The result is also consistent with the heat kernel method. We will also study the scalar field theory in the Rindler wedge and reduce it to the null boundary, a Killing horizon associated with a Lorentz boost in Minkowski spacetime. Since the Killing horizon is not at infinity, we do not impose any artificial fall-off condition except that the field is finite at the null hypersurface. The field theory can be massive since any particles can cross through the null hypersurface. Similar to the case of null infinity, the quantum flux operators form a representation of Carrollian diffeomorphism at the Killing horizon. The supertranslation flux operators form a Virasoro algebra with a divergent central charge. We use heat kernel method and Pauli-Villars method to regularize the central charge. They lead to the same conclusion that there is an effective central charge which is proportional to area of the bifurcation surface of the Killing horizons. Using the conservation law of stress tensor and Stokes' theorem, we identify the zero mode of supertranslation as the modular Hamiltonian of the Rindler wedge. The result indicates that Carrollian diffeomorphism may be related to quantum information theory. We try to calculate the expectation values of the quantum flux operators for the vacuum and excited states. The relative fluxes, which are defined as the difference of the expectation values between two quantum states, may provide new observables in Unruh effect.
The structure of the paper is as follows. In section 2, we will introduce the basic ingredients on the Carrollian manifold. In the following two sections, we will construct quantum flux operators on future null infinity and Rindler horizon and discuss their various properties. We extend the results to general null hypersurfaces in section 5 and conclude in section 6. Technical details are relegated to three appendices.

Conventions and notations
We will use the following conventions in this paper. Lorentz manifolds are denoted by M, N, etc., with signature (−, +, +, · · · , +). Tensors on Lorentz manifolds are denoted by lowercase Greek indices µ, ν, · · · . Timelike hypersurfaces are denoted by H, I, and so on. A null hypersurface is a Carrollian manifold which is denoted as H, I, etc.. Usually they can be factorized as a product such as where R is the retarded/advanced time direction and S is a Riemannian manifold. Tensors on Carrollian manifolds are labeled by lowercase Roman indices a, b, c, etc. from the first half of the alphabet. Riemannian manifolds are denoted by uppercase Roman letters. Tensors on Riemannian manifolds are indexed by uppercase Roman indices A, B, C, etc..

Preliminaries
In this section, we will review the minimal ingredients on Carrollian diffeomorphism which are relevant to later discussion. A Carrollian manifold C with topology R × S has a degenerate metric ds 2 ≡ h = h AB dθ A dθ B , A, B = 1, 2, · · · , d − 2 (2.1) as well as a null vector The metric h also defines the geometry of the d − 2 dimensional manifold S, which is assumed to be Riemannian with signature (+, +, · · · , +). The d − 2 coordinates θ A , A = 1, 2, · · · , d − 2 may also be collected as Ω = (θ 1 , θ 2 , · · · , θ d−2 ). The null vector χ generates the time direction of the Carrollian manifold. We have used the coordinate u to denote the time for this manifold. The volume form for the manifold S is denoted by dΩ = √ det hdθ 1 ∧ · · · ∧ dθ d−2 (2.3) and the unique volume ϵ for the Carrollian manifold C is given by For brevity, we will also write it as ϵ = dudΩ. where ξ f = f (u, Ω)∂ u generates general supertranslations (GSTs) and ξ Y = Y A (Ω)∂ A generates special superrotations (SSRs) within the terminology of [6]. One may also consider general superrotations (GSRs) which is generated by Y A (u, Ω)∂ A . However, this turns out to break the null structure of the Carrollian manifold and we will not focus on it in this work. The supertranslations and superrotations are borrowed from BMS transformations. Nevertheless, in general situations, they are not related to the original BMS transformations which are only defined at future/past null infinity. The vectors (2.7) form an infinite dimensional Lie algebra (2.8) whereḟ i ≡ ∂ u f i , i = 1, 2. In particular, for pure supertranslations, the corresponding Lie algebra is which is called T-Witt algebra in [36]. For pure superrotations, the Lie algebra is where ξ Y generates diffeomorphism of the manifold S. In [6,8,9], the authors find that the Carrollian diffeomorphism generated by ξ may be realized by quantum flux operators Q ξ whose algebra is exactly (2.8) up to anomalous quantum corrections We will use the conventions to distinguish supertranslations and superrotations. The operators Q ξ and the anomalous corrections K ξ 1 ,ξ 2 may depend on the field F defined on the Carrollian manifold in general For real scalar theory, the unique anomalous term is a central charge of T-Witt algebra such that the algebra becomes a Virasoro algebra with more transverse indices The identity operator in (2.14) is defined as while the Dirac delta function with argument equalling to zero is proportional to the density of states in the transverse direction, e.g. the density of states on the manifold S. Since the spectrum of the field in the transverse direction is continuous in general, the Dirac delta function δ (d−2) (0) is divergent. Note that we have already written the results in d dimensions. In the vector and gravitational theory, there are additional anomalous terms which may be interpreted as helicity flux operators whose explicit forms can be found in [8,9]. The helicity flux operators generate super-duality transformations (angle dependent duality transformations) which are essential for the field theories with nonzero spins.
To find the operators Q ξ , one may embed the Carrollian manifold into d dimensional spacetime M where the bulk theory is well defined. The stress tensor T µν of the bulk theory is conserved such that one may integrate the current over the Carrollian manifold C to obtain The vector ξ may be chosen as the generator of Carrollian diffeomorphism (2.7). If ξ is a Killing vector when extended into the bulk spacetime M, the operator Q ξ is the corresponding charge which crosses the Carrollian manifold C during the whole time. Therefore, Q ξ could be interpreted as a leaky flux from bulk to boundary.
The above results have been mainly found at null infinity in four dimensional flat spacetime.
In this paper, we will extend it to general null hypersurfaces and general dimensions.

Scalar field theory at future null infinity
In this section, we will construct various scalar field theories at future null infinity (I + ) from bulk reduction in general dimensions. The overall framework is to embed I + into Minkowski spacetime R 1,d−1 and reduce the bulk theory from R 1,d−1 to I + .

Null infinity
The future/past null infinity (I + /I − ) is a Carrollian manifold in asymptotically flat spacetime. In this work, we will discuss I + for Minkowski spacetime R 1,d−1 which can be described in Cartesian coordinates x µ = (t, x i ) where µ = 0, 1, · · · , d−1 denotes the components of spacetime coordinates and i = 1, 2, · · · , d−1 labels the components of spatial coordinates. We also use the retarded coordinate system (u, r, θ A ) and write the metric of R 1,d−1 as The future null infinity I + is a d − 1 dimensional Carrollian manifold with a degenerate metric The spherical coordinates θ A = (θ 1 , · · · , θ d−2 ) are used to describe the unit sphere S d−2 whose metric ds 2 S d−2 can be found in Appendix A. We will also use the notation Ω = (θ 1 , · · · , θ d−2 ) to denote the spherical coordinates in the context. The covariant derivative ∇ A is adapted to the metric γ AB , while the covariant derivative ∇ µ is adapted to the Minkowski metric in Cartesian frame. The integral measure on I + is abbreviated as Besides the metric (3.4), there is also a distinguished null vector which is to generate the retarded time direction.
To obtain the metric of the Carrollian manifold (3.4) from the bulk metric, one may choose a cutoff such that the induced metric on the hypersurface is We use a Weyl scaling to remove the conformal factor and take the limit while keeping the retarded time u finite such that (3.10) becomes the metric of the Carrollian manifold I + . We define the limit lim + = lim r→∞, u finite (3.12) to send the quantities on H r to I + . Similarly, taking the limit below sends the quantities on H r to I − where v is the advanced time v = t + r.

Bulk theory
We may consider a general scalar theory with the action (3.14) Only massless particles can arrive at I + , and therefore we consider massless theory with the potential We impose the fall-off condition for the bulk scalar field. The constant ∆ is the scaling dimension of the bulk field and is related to the dimension d through Now we consider the equation of motion (EOM) For the first term, one can show that which leads to For the potential term, we find Note that it does not contribute to EOM for k = 0. Moreover, for k = 1, potential also has no contribution unless ∆ = 1 (corresponding to dimension 4). If ∆ = 1, the leading EOM is λ 3 Σ 2 = 0, which results in λ 3 = 0. Otherwise, the leading EOM is trivial. Nevertheless, we write out the EOM at order O(1/r ∆+k ) In odd dimensions, ∆ is a half integer. Therefore, only the terms with n = even have contributions in the second line. In general dimensions, the leading coefficients Σ encode the dynamical propagating degree of freedom. All the subleading coefficients Σ (k) , k ≥ 2 are determined by the EOM up to initial data Σ (k) (u 0 , Ω), k ≥ 2 where u 0 is the initial time on C.

Boundary theory
The pre-symplectic form of the theory is The normal co-vector m µ of the hypersurface H r is where the null vectors n µ andn µ are defined as n µ = (1, n i ),n µ = (−1, n i ) (3.26) with n i = x i /r, i = 1, 2, · · · , d − 1 being the normal vector of S d−2 embedded into R d−1 . The symplectic form for the theory on H r is With the fall-off condition (3.16), the symplectic form becomes Ω(δΣ; δΣ; Σ) = lim + Ω Hr (δΦ; δΦ; Φ) = dudΩδΣ ∧ δΣ (3.28) at I + . This is identified as the symplectic form of the Carrollian scalar field theory at I + . We may find the following commutators for the field Σ with θ(x) being the Heaviside step function The field Σ may be expanded in a set of complete basis of I + In the expansion ( Therefore, we find the following correlators (3.35d)

Flux operators for Carrollian diffeomorphism
In this section, we will define flux operators for Carrollian diffeomorphism in general dimensions. For any conserved current j µ in the bulk, we may construct the corresponding flux radiated to I + through the limit The stress tensor for the theory (3.14) is where L is the Lagrangian of the theory. Note that we have the expansion for the field Φ which leads to where we have assumed Σ (0) = 0. The vector Y A µ is the gradient of the null vector n µ Similarly, we have The stress tensor has the following fall-off behaviour For any Killing vector ξ in R 1,d−1 , we may find a conserved current Therefore, the flux corresponding to the Killing vector ξ is 2. For a Lorentz rotation generated by ξ LT = ω µν (x µ ∂ ν − x ν ∂ µ ), the angular momentum or center of mass flux is  since they lead to the same fluxes in (3.48) after integration by parts. Following [6], we fix λ = 1 2 using the orthogonality condition Note the Poincaré group is a subgroup of Carrollian diffeomorphism for I + . We may exploit (2.19) and extend the fluxes to general Carrollian diffeomorphism. After some algebra, we find the following extended flux operators for Carrollian diffeomorphism. The time and angle dependence of the density flux operators may be transformed to the following smeared operators We add a bar in the definition to distinguish them with the one from Carrollian diffeomorphism. The operatorT f is equal to T f exactly for any smooth function f = f (u, Ω). On the other hand, the operatorM Y matches with M Y only for time independent vectors Y A = Y A (Ω). As has been noticed, for general time dependent vectors Y A = Y A (u, Ω), the corresponding operator generates non-local terms which may relate to the violation of the null structure under GSRs.
There is a rather different way to obtain the same flux operators. From Hamiltonian theory, we learn that the Hamiltonian corresponds to a general vector ξ may be obtained from the Hamilton equation δH ξ = i ξ Ω(δΣ; δΣ; Σ) (3.55) where i ξ is the interior product along the direction ξ in phase space. The variation of the boundary scalar field Σ can be obtained from bulk diffeomorphism In Minkowski spacetime R 1,d−1 , the generator of a global translation along the direction c µ is We have inserted a constant vector c µ into the expression to denote the direction of the translation, and transformed to retarded coordinates at the last step. Therefore, the boundary field Σ is transformed to We observe that the generator of the translation ξ trans. reduces to −c µ n µ ∂ u (3.59) at I + . Obviously, it is the Carrollian diffeomorphism generated by It is natural to assume that the transformation of the boundary field Σ under a GST is Note this is also the transformation of a boundary scalar field under the Carrollian diffeomorphism generated by ξ f δ f Σ(u, Ω) = ξ f (Σ) = L ξ f Σ = f (u, Ω)Σ. (3.63) In this derivation, there is no need to assume ξ being a Killing vector or an asymptotic Killing vector near I + . There would be a corresponding Hamiltonian H f related to the GST through the Hamilton equation (3.55). The infinitesimal variation of the Hamiltonian is which is integrable. The integrable flux is This is the classical version of the operator T f . Similarly, the generator of a Lorentz transformation parameterized by an antisymmetric constant tensor ω µν is where the antisymmetric tensors n µν andm µν are defined in Appendix A.4. The boundary field Σ is transformed as under Lorentz transformation. We note that the generator (3.66) reduces to Therefore, it is natural to assume the variation of Σ under SSR to be which could be reduced to (3.67) for Lorentz transformations 5 . Note the transformation law (3.71) is independent of the dimension d. However, one should notice that it is not equal to the following transformation using boundary Lie derivative. To understand this transformation, we integrate the infinitesimal transformation (3.71) and find the finite transformation Thus, the field Σ is a scalar density under superrotations with weight w = 1 2 due to the result (3.73). In contrast, Σ is a scalar under general supertranslations. In general, we may define a scalar density Λ(u, Ω) of weight w under superrotation (3.74) by the following transformation Infinitesimally, this is Interestingly, we could find the integrable flux for a SSR labeled by Y A (Ω) In quantum theory, the operators should be normal ordered and we find the same flux operator Before we close this subsection, we will shortly discuss the integrability problem for general scalar densities Λ(u, Ω). Assuming the boundary scalar theory is defined through the symplectic form Ω(δΛ; δΛ; Λ) = dudΩδΛ ∧ δΛ (3.79) and the superrotation variation of the field Λ is given by (3.76) where w is free. The corresponding Hamiltonian is The first part of above equation on the right hand side is integrable while the second part is not except for w = 1 2 .

Commutators and regularization of central charge
By computing the commutators we confirm that T f and M Y are generators for the Carrollian diffeomorphism given that Σ is a scalar density of weight 1 2 under superrotations. More explicitly, T f generates GSTs while M Y generates SSRs. They should form Lie algebra (2.8) up to central extension terms due to quantum corrections. Indeed, we find where the central charge is The Dirac delta function on S d−2 with the argument equalling to zero is divergent 6 where ℓ 1 counts the number of all possible eigenstates on S d−2 and I(0) is the area of the unit sphere. Therefore, Dirac delta function δ (d−2) (0) counts the number of states on S d−2 . Now we will try to regularize the Dirac delta function. Notice that the eigenstates on S d−2 obey the Laplace equation where Y ℓ (Ω) is the spherical harmonic function and ∆ S d−2 = ∇ A ∇ A is the Laplace operator on S d−2 . The Laplace operator is a natural elliptic operator which may be used to define the spectral zeta function 7 on S d−2 where tr ′ is to sum over all possible eigenstates except the one with ℓ m = 0. The eigenstates with zero eigenvalue is excluded otherwise the summation would be divergent for any Re(s) > 0. Using the degeneracy of the eigenstates and the eigenvalue of the spherical harmonics, we find The spectral zeta function ζ S d−2 (s) is convergent for sufficient large s and can be analytically continued to the complex plane except for a single pole. Since the degeneracy g(ℓ, d − 2) is a polynomial of ℓ of degree d − 3, we may always reduce ζ S d−2 (s) to a finite summation of the functions where a, b, c are integers. In [38,39], this function has been treated carefully and expressed as a series of Riemann zeta functions (see Appendix C). The number of eigenstates on S d−2 may be regularized to In odd dimensions, the regularization method leads to 0. In even dimensions, the results have been listed in table 1.  We may also define a heat kernel 8 associated with the Laplacian where tr is sum over all possible eigenstates, including the one with ℓ m = 0. Therefore, The summation is convergent for any Re(σ) > 0. It turns out that the heat kernel has the pole structure near σ = 0 There are only finite number of divergent terms in the expansion and we may extract the constant term by regularization. Interestingly, the constant term is always zero for odd dimensions. When d is even, the asymptotic expansion of the heat kernel can be found in the Appendix of [41]. The results matches with the ones from spectral zeta function regularization.
4 Scalar field theory on Rindler horizon 4

.1 Rindler horizon
In Minkowski spacetime R 1,d−1 , the Killing horizon associated with the generator of Lorentz boost along the direction of x d−1 is the set of points where The solution of null Killing vector condition X 2 = 0 is the union of the following two hypersurfaces whose intersection is the bifurcation surface B(X) with X = 0 and This leads to the four Killing horizons associated with X The above description is showed in figure 1.
Without loss of generality, we will study the Carrollian field theory on the Killing horizon H −− (X). This is one of the null boundary of the Rindler spacetime which is also called Rindler horizon. We will write it as H −− and omit X in the following. We note that it is also possible to define Carrollian field theories on the null hypersurface The interested reader may find more details in Appendix B.3.
Rindler spacetime is a patch of Minkowski spacetime which may be obtained from the coordinate transformation The Rindler time has an imaginary periodic 2π There is a bifurcation surface B which is the entangling surface between left and right region. whose inverse is the temperature of the thermal bath in Unruh effect [42]. The metric of Rindler spacetime is We may define a new coordinate 8) and the metric becomes With the retarded time in Rindler spacetime the metric of the Rindler spacetime may be written in retarded coordinates (u, λ, x A ⊥ ) as To obtain the metric of the Carrollian manifold H −− from bulk metric, one may choose a cutoff such that the induced metric on the hypersurface is (4.14) The Killing horizon H −− can be parameterized by d − 1 coordinates (u, x A ⊥ ) whose metric may be obtained by taking the limit ϵ → 0 for (4.14) Therefore, the Killing horizon H −− is a Carrollian manifold with a degenerate metric (4.15) as well as a null vector while in this case we have Similarly, we may define an advanced coordinate v = τ + λ, (4.18) and the metric of the Rindler spacetime in advanced coordinates (v, λ, θ A ) is By taking the limit λ → −∞ while keeping the advanced time v finite, we find the metric of the Carrollian manifold H ++ ds 2 The null vector of the Carrollian manifold H ++ is ∂ v . We will define the limits lim − and lim + to reduce the quantities from bulk hypersurface H ϵ to H −− and H ++ , respectively One can see these two limits in figure 2. An attentive reader may have the confusion that why we should reduce the theory from Rindler spacetime to H −− . It seems equivalent to reduce the theory straightforwardly from the theory in Minkowski spacetime. Indeed, one could do so, as has been shown in Appendix B.2, and what one actually gets is the theory defined on H − . One should integrate out degrees of freedom on H −+ ∪ B to find the final results.

From bulk to boundary
In this section, we will construct Carrollian scalar field theory at the Killing horizon H −− from bulk reduction in general dimensions. We still consider a general scalar theory with the action (3.14). The potential could include a mass term since massive particles can also arrive at H −− .
By embedding the theory in Minkowski spacetime, there is no need to impose any artificial boundary condition at H −− since they should be determined by the whole system. Instead, a scalar field should be finite at any position of Minkowski spacetime. Therefore, we just acquire the finiteness of Φ at H −− . Using Taylor expansion, we assume the series expansion of Φ near for the bulk scalar field. The pre-symplectic form of the theory is The normal covectorm µ of the hypersurface H ρ is dρ which implieŝ and we have normalized it to 1m ·m = 1.
The symplectic form of the theory is With the fall-off condition (4.23), the symplectic form becomes 9 at H −− . This is identified as the symplectic form of the Carrollian scalar field theory at H −− . Interestingly, this is formally the same as the one at I + though the Carrollian manifold and the boundary field Σ are not identical. We may find the following commutators for the field Σ (4.32c) 9 Similar to the constraint equations (3.22) at null infinity, we may also use the differential operator and reduce the bulk EOM to Therefore, only the leading mode Σ is free from dynamical equations. All the subleading modes Σ (k) , k ≥ 1 are determined by the equations up to initial data.
The field Σ may be expanded in a set of complete basis of In the expansion (4.33), the functions e −iωu+ik·x ⊥ and their complex conjugates form a set of complete basis of H −− . We have also checked the mode expansion (4.33) using canonical quantization method in Appendix B.3. With the canonical quantization, we may define the We have added a subscript R to denote the vacuum state of Rindler spacetime. It should be distinguished with the vacuum state of Minkowski spacetime |0⟩ defined in previous section. Therefore, we find the following correlators

Flux operators for Carrollian diffeomorphism
In this section, we will apply the formula (2.19) to the Killing horizon and calculate the flux operators for Carrollian diffeomorphism. To understand the physical meaning of the flux operators, we may study the flux operators associated with Killing vectors at first. Rindler spacetime is locally flat which has d(d + 1)/2 Killing vectors inherited from Minkowski spacetime. Therefore, we may use these generators to discuss the symmetries of Rindler spacetime. The Killing vectors are listed below which are expressed in terms of the quantities in retarded coordinates (4.37c) In the light cone coordinates we find the following generators The vectors ξ + and ξ +A blow up near H −− and they do not generate Carrollian diffeomorphism.
To be more precise, we note the field Φ transforms along the direction of the Killing vectors as Note that the variation of the field Φ along the direction of ξ + or ξ +A blows up and violates the fall-off condition (4.23). Therefore, we should exclude the generators ξ + and ξ +A . On the other hand, the variations of the field Φ along the directions ξ − , ξ A , ξ +− , ξ −A , ξ AB preserve the fall-off conditions and we find the following variations for the boundary field Σ We have inserted constant vectors b A , c A or antisymmetric tensor ω AB to balance the indices. From the explicit form of the generators, we conclude that ξ − , ξ A , ξ +− , ξ −A , ξ AB could reduce to special choice of Carrollian diffeomorphism at the boundary H −− All the transformations (4.41) could be unified by with the special choice of the f and Y A listed in (4.42). Note that the last term is always zero due to the vanishing of the term ∇ A Y A for all the choices of Y A in (4.42). Therefore, The constant w is free at this moment. Interestingly, the function f (u, x ⊥ ) is automatically time dependent from (4.42a) and (4.42d). This fact strongly indicates that one should consider time dependent supertranslations. Another noticeable fact is that the translation along Rindler time is a Lorentz boost in Minkowski spacetime. Therefore, the corresponding flux should be interpreted as center of mass flux in Minkowski spacetime. Similarly, a superrotation flux with Y A = b A should be interpreted as a momentum flux in Minkowski spacetime. Nevertheless, we can construct the leaky fluxes across H −− for the Killing vectors (4.42) by where and T µν is the stress tensor given by (3.37) and ξ is a Killing vector of Rindler spacetime. Given the fall-off conditions (4.23), we find and To compute (4.44), we need the following fall-off behaviour of stress tensor Therefore, we find the fluxes corresponding to the Killing vectors (4.39) 2ω CA x C respectively. On the other hand, the fluxes F ξ + and FcA ξ +A are finite but not related to Carrollian diffeomorphism. Now we can construct the flux operators associated with Carrollian diffeomorphism. For any GST, the flux is We may uplift to operators using normal ordering Similarly, we can find the flux associated with any SSR The corresponding flux operator is where we have chosen We should mention that one could also use Hamilton equation (3.55) to obtain the same results for Carrollian diffeomorphism. For a GST generated by the corresponding Hamiltonian is exactly the flux operator (4.52) after normal ordering. For a SSR generated by we find an integrable flux only for w = 1 2 which is again exactly the flux operator (4.54). Note the constant w is absent from the variation we already meet with this problem at future null infinity. In that case, we can fix the value w = 1 2 by considering Lorentz transformations in the bulk. However, one cannot fix it in the Killing horizon H −− as has been shown in (4.43). Nevertheless, the value w should be 1 2 due to the following reasons.

The Hamiltonian for superrotation is integrable only for
is a scalar density of weight 1. Therefore, the consistency for the commutator (4.32) has already fixed w = 1 2 .
It turns out that the commutators between T f and M Y are exactly the same as (3.82). The central charge (3.83) is also divergent. However, in this case, we find It shows that the central charge also counts the number of eigenstates in the transverse direction. To regularize it, we introduce a UV cutoff 2π/a where a has the dimension of a microscopic length scale. In lattice theory, a is the distance of adjacent lattice points. Therefore, the Dirac function becomes At the last step, we have used the fact that a d−2 is the volume of the smallest cell of the lattice, (4.60) Therefore, the central charge should be proportional to the density of states in the transverse direction, which has exactly the same physical meaning with the one at I + . We may also use the heat kernel method to define K(σ) = tr e σ∆ (4.61) with ∆ being the Laplacian operator in the transverse direction. The spectrum of the Laplacian operator is −k 2 ⊥ which is continuous. Then the heat kernel becomes As σ → 0, the heat kernel approaches the Dirac delta function δ (d−2) (0). Since the dimension of σ is length square, we may introduce a UV cutoff a with dimension length and identify σ = a 2 4π . (4.63) Combining (4.62) and (4.63), we find the same result as (4.59).

Now we choose the basis for the supertranslation function
and define the Fourier transformation of the flux operator Then we find the Virasoro algebra The Dirac delta function δ (d−2) (k ⊥ + k ′ ⊥ ) is zero for k ′ ⊥ ̸ = −k ⊥ . The divergence for the central charge only appears for k ′ ⊥ = −k ⊥ . In this case, we find where we have introduced an IR cutoff L which is the large length scale in the transverse direction. At the last step, we have switched the L d−2 to the area of the bifurcation surface B 10 . Combining with the regularization (4.59), we find the Virasoro algebra is exactly the number of states in the transverse directions. We add a bar over "c" to distinguish it with the central charge c = δ (d−2) (0) which is the density of states in the transverse directions. We will call it an effective central charge.

Relation to Modular Hamiltonian
We have defined flux operators associated with supertranslations and superrotations for the Rindler wedge. In Rindler wedge, the fundamental quantity is the modular Hamiltonian [44] H modular = 2π where T tt (t = 0, x) is the tt component of the stress tensor on the surface V which is defined by The modular Hamiltonian (4.70) is defined for general relativistic field theories and plays a central role in the context of geometric entanglement entropy. As a subregion of Minkowski spacetime, V defines the Rindler wedge whose null boundaries are exactly H −− and H ++ . There could be relations between the flux operators associated with Carrollian diffeomorphism and modular Hamiltonian. We will explore this problem in the following.
We will derive a general formula for the flux operators at first. In It will crosses V and defines a vector ξ (0) on this spacelike hypersurface From the conservation of stress tensor, we can find the following identity from Stokes' theorem where the domain of integration on the left hand side is the Rindler wedge which is denoted as "bulk". We have dropped the terms at infinity since the modes are suppressed exponentially in Rindler spacetime which is checked in Appendix B.3. Since the stress tensor is symmetric we may recap (4.74) as where we have used the definition of flux operators associated with a Carrollian diffeomorphism generated by ξ or ξ (+) . Interestingly, for the Killing vectors ζ, the flux operators should be equal to each other Q ξ = Q ξ (+) , ζ is a Killing vector. (4.78) The derivation could be extended to other bulk region. For example, when we choose the shaded region which is bounded by H −− and V in figure 3, the identity becomes where we have defined a "charge" associated with On V , the Killing vector ∂ τ becomes Therefore, we can identify the modular Hamiltonian as the flux operator T f =1 in Rindler wedge This identity strongly indicates that the flux operators T f and M Y are physically important. We may regard them as an extension of modular Hamiltonian for subsystems. The reduced density matrix for a Rindler wedge becomes As a by product, we list the one-to-one correspondence between the operators on H −− and V for the Killing vectors (4.42).

Killing vectors ζ
Flux operators (f, Y A ) Operators in the subregion

Observables in Unruh effect
The Rindler wedge is the spacetime which is observed by an accelerated observer with constant acceleration a = 1. The Unruh effect says that the accelerated observer could detect a thermal bath in the Minkowski vacuum |0⟩. From mode expansion which is reviewed in Appendix B.3, the vacuum defined in H −− is the Rindler vacuum |0⟩ R which is not |0⟩. In this section, we will compute the expectation value of the flux operators T f and M Y in the Minkowski vacuum |0⟩ and excited states to propose that they are physical observables for the accelerated observer.
The starting point is the identification of the modular Hamiltonian and the zero mode of supertranslation flux operator (4.85). In quantum information theory, considering a subsystem R which is obtained by tracing out the degrees of freedom of its complementR, we may find the reduced density matrix ρ R for the subsystem where ρ is the density matrix of the whole system R ∪R. The reduced density matrix may be rewritten as an exponential of the modular Hamiltonian which is given by (4.85) for the Rindler wedge. The entanglement entropy for the subsystem is Switching to the whole system with the relation (4.87), we conclude that the entanglement entropy is the expectation value of modular Hamiltonian. Now for the Rindler wedge, we may replace the modular Hamiltonian to the supertranslation flux operator with f = 1 Now we may extend the modular Hamiltonian to general flux operators T f and M Y , it is natural to define the following expectation values They are direct extension of entanglement entropy for Rindler wedge. These are also the fluxes across the Killing horizon observed by the accelerated observer. When the system is in the vacuum state |0⟩, the density matrix is Sometimes, we compare the fluxes for different states. Given two states characterized by density matrices ρ 1 and ρ 2 , we may define the following relative fluxes (4.93b) In the following, we will calculate the quantities (4.91) in vacuum and excited states and find the relative fluxes according to (4.93).

Minkowski vacuum
We use the mode expansion to find the operators in momentum space (4.96) The operators a R ω,k ⊥ , a R † ω,k ⊥ are related to b p , b † p by Bogoliubov coefficients The Bogoliubov coefficients are Readers can find the derivation in Appendix B.3. Therefore, we find the following expectation values in Minkowski vacuum Utilizing the integral identity In (4.94a), the integration variables ω, ω ′ are non-negative. Therefore, the contributions from are zeros and At the last step, we used the identity for Gamma function Similarly, one can obtain Since the integrand is odd, we conclude that the vacuum expectation value of M Y (ρ 0 ) vanishes. This shows that the observer will detect a thermal bath without any time and angular independent information. We notice that the expectation value T f (ρ 0 ) is divergent in Minkowski vacuum. The integral in the transverse direction can be regularized by choosing a cutoff in the UV region, similar to (4.59). To be precise, we notice f(0, 0) is a constant and the divergent integral is exactly The length dimension of x ⊥ is 1 while the retarded time u is dimensionless, following from the transformations (4.5) and (4.10). Therefore the supertranslation function f (u, x ⊥ ) is dimensionless. From the definition of f, we conclude that f * (0, 0) has the dimension of area in the transverse direction. We parameterize by a dimensionless constant γ which may be expressed as Therefore, the expectation value T f (ρ 0 ) becomes To check the consistency of the regularization, we choose f = 1 and its Fourier transform is where T and L are the IR cutoffs for the retarded time and transverse directions, respectively. Comparing with (4.106) we read Note that T is dimensionless by definition. As we have mentioned, the expectation value T f =1 (ρ 0 ) is the entanglement entropy for the Rindler wedge. It is well known that the entanglement entropy for a continuous subsystem is proportional to area of the entanglement surface [45][46][47]. Our regularization method matches with this conclusion.

Excited states in Minkowski spacetime
Considering the one particle state with definite momentum p we could find the following expectation values We can easily obtain the following relative fluxes Note that the expectation value of M Y is divergent. However, the normalization of the state |p⟩ is also divergent. We may consider the following wave packet which is the superposition of the state with definite momentum. The coefficients σ p is chosen to normalize the state |σ⟩ The density matrix becomes Using the transition matrices the relative fluxes are The relative fluxes are finite and encode the time and angle dependent information of the excited states.

Null hypersurfaces
We consider a general null hypersurface N with topology R × N whose metric is degenerate The metric h AB is also the metric of the d − 2 dimensional manifold N . We need a null vector χ = ∂ u to generate the time direction of the Carrollian manifold N . The null hypersurface may be embedded into d dimensional spacetime whose metric may be expanded near N as [48][49][50] where K, Λ A , H AB depend on the coordinates u, ρ, θ A , A = 1, 2, · · · , d − 1 and satisfy the conditions We may expand K, Λ A , H AB by Taylor series such that the near N metric is where λ A = λ B h AB . The coordinate system is usually called null Gaussian coordinates. The hypersurface N sits at ρ = 0. The components of the metric could be written out explicitly as 6a) g uρ = −1, g ρρ = 0, g ρA = 0, (5.6b) The determinant of the metric (5.2) is where det H and det h are the determinants of the metric H AB and h AB , respectively. The The inverse metric of g µν is g uu = 0, g uρ = −1, g uA = 0, (5.9a) g ρρ = −K = 2κρ + O(ρ 2 ), (5.9b) The integral measure on N is abbreviated as The normal covector of the hypersurface N is We may consider a general scalar theory with the action (3.14) whose field Φ may be expanded as Φ(u, ρ, θ) = Σ(u, θ) + ρΣ (1) (u, θ) + · · · (5.13) near N . However, we may allow a mass term since the null hypersurface N is placed at a finite position in spacetime which could be approached by massive particles similar to the Rindler spacetime. Therefore, we may expand the potential perturbatively as where λ 2 is the mass square The pre-symplectic form of the scalar theory on the null hypersurface is From the expansion of the field Φ, we find Therefore, we obtain It follows that the symplectic form is 20) which implies that the commutators between Σ andΣ are The field Σ may be expanded by the superposition of positive frequency modes and negative frequency modes with the following commutators between a ω (θ) and a † ω (θ) Indeed, one can check the commutation relations (5.21) are satisfied by virtue of the commutators (5.23). Therefore, we may interpret a ω (θ) and a † ω (θ) as annihilation and creation operators. More explicitly, we may define the vacuum state as a ω (θ)|0⟩ N = 0 (5.24) and the creation operator a † ω (θ) acts on the vacuum |0⟩ N should create a particle with frequency ω at the position θ. We add a subscript N to emphasize that the vacuum state may depend on the null hypersurface N . The correlators are Now it is straightforward to use the formula (2.19) to find the flux operators associated with Carrollian diffeomorphisms. The results are We can also obtain similar algebra (3.82) with a divergent central charge There is a set of eigenfunctions f q (θ) of the Laplace operator on the manifold N where λ q is the corresponding eigenvalue. The eigenfunctions are normalized to 1 and orthogonal to each other 11 The completeness relation for the eigenfunctions is Summing over all possible eigenstates in (5.29) and we obtain Therefore, the central charge is We may still use the heat kernel method [51] to regularize the central charge for general Riemannian manifold without boundary.

Conclusion and discussion
In this work, we have constructed the quantum flux operators for scalar field theory which form a faithful representation of Carrollian diffeomorphism for null hypersurfaces up to a non-zero central charge. We have checked the results mainly for two interesting null hypersurfaces and then extend them to general null hypersurfaces. In the first case, the flux operators associated with Carrollian diffeomorphism at future null infinity are finite in general dimensions. Therefore, our results overcome various problems in the context of higher dimensional asymptotic symmetry analysis and also consistent with graviton soft theorems. Secondly, we extend our method to the Rindler horizon and find similar flux operators which are associated with the accelerated observer. They are natural operators for the Rindler spacetime, a subsystem in the context of quantum information. We conclude that the zero mode of the supertranslation flux is actually the modular Hamiltonian of the subsystem. These results build a connection between Carrollian diffeomorphism and quantum information theory. We could define the expectation values of the supertranslation and superrotation flux operators associated with the subsystem, which extend the geometric entanglement entropy. By computing the expectation value of T f for the vacuum state explicitly, we could check the regularization method of the central charge using the area law of entanglement entropy. We also define the relative fluxes between different states which may characterize the difference between any two states, and may be new observables in Unruh effect. Finally, we also propose to regularize the central charge using spectral zeta function or heat kernel method. We could find a finite result for the theory at null infinity. For the Rindler wedge, we conclude that there is an effective central charge which is proportional to the area of the bifurcation surfacē • It should be able to extend our results to the theory with nonzero spins. In [8,9], it has been shown that more anomalous terms should be included for these theories. It would be nice to work out the details for general null hypersurfaces in general dimensions.
• Spectral geometry of Carrollian manifold. To study the regularization of the central charge, we compute the spectral zeta function associated with the Carrollian manifold where ∆ N is the Laplace operator of the Riemannian manifold N . We read out the central charge from 1 + ζ C (0). It would be nice to extract more physical information from the spectral zeta function for general Carrollian manifolds. A related question is to study the heat kernel associated with a general Carrollian manifold • Periodicity of imaginary time. For a general subsystem or a black hole, the imaginary time should be periodic and the inverse period is the temperature of the system. We still do not discuss the consequence of this condition. Taking Rindler spacetime as an example, the retarded time u is periodic in imaginary direction u ∼ u + 2πi. (6.4) To be consistent with the periodic condition, the basis function should obey the equation which implies ω = in, n = 0, ±1, ±2, · · · . (6.6) We may redefine L n,k ⊥ = iT in,k ⊥ , (6.7) then the Virasoro algebra becomes We find an infinite tower of two dimensional Virasoro algebras whose label n is discrete, which is exactly the same as two dimensional conformal field theory. It would be nice to explore this topic in the future.
• Event horizon and Hawking radiation. Our method can be extended to event horizon of black holes. It would be very interesting to study the Carrollian diffeomorphism and its associated quantum flux operators at the event horizon. A better understanding of the representation theory of the algebra may provide the key insight on Hawking radiation.
• Relation to quantum information theory. We have noticed that the quantum flux operators defined on the null hypersurface are direct generalization of the null modular Hamiltonians defined in [52]. For each spatial subregion, there is a null boundary which is exactly described by a Carrollian manifold. It would be interesting to explore the physical interpretation of the quantum flux operators in the context of quantum information theory. where The normal vector of the sphere S m is In spherical coordinates, the metric of Minkowksi spacetime is where the metric of the unit sphere S m is The components of the metric γ AB is where Y i A is the gradient of the normal vector n i The explicit metric of the sphere may be found by the following recursion formula ds 2 S 1 = dθ 2 1 , ds 2 S n = dθ 2 n + sin θ 2 n ds 2 S n−1 , n ≥ 2. (A.9) The determinant of the metric is

A.2 Spherical harmonics
The scalar spherical harmonic functions on S m are given by [53] where the vector ℓ is ℓ = (ℓ 1 , ℓ 2 , · · · , ℓ m ) (A. 12) with ℓ j , j = 1, 2, · · · , m being integers that satisfy The associated Legendre function of the first kind is defined through the hypergeometric function: 14) The coefficient c(a 1 , a 2 , a 3 ) is chosen as such that the spherical harmonic functions are normalized and orthogonal to each other The harmonic functions obey the following Laplace equation on S m Therefore, for any fixed quantum number ℓ m = L, the degeneracy of the spherical harmonic functions is [54] g(L, m) = The generating function of the degeneracy in general dimension is (A.20)

A.3 Addition theorem
As is well known, the addition theorem in four dimensions states that where γ is the intersection angle between the two directions labeled by Ω and Ω ′ . The notation C k ℓ (x) is the so-called Gegenbauer polynomial which may be defined by hypergeometric function In 4 dimensions, k takes the value 1/2 and the intersection angle γ becomes 0 when Ω ′ = Ω, thus the addition theorem reduces to The left hand side could be reduced to the familiar Legendre polynomial of order ℓ. In general d dimensions, we have a similar addition theorem, where k is related to the dimension d by and K ℓm is a constant which is not important in this work. The summation ℓ ′ is to sum over all possible quantum numbers while fixing the value of ℓ m (A.26) When Ω ′ = Ω, the addition theorem becomes The special value C (A. 29) where I(0) is the area of the unit sphere S m : Finally substituting the result above into (A.27) yields This is the density of states on S m .
As a matter of fact, this conclusion is immediately derived if we take sum for the orthogonality relation of the spherical harmonics and notice the completeness relation It follows that This method may be generalized to any compact manifold where the eigenfunctions have the similar orthogonality and completeness relations.

A.4 Conformal Killing vectors
The conformal group of S m is SO(m + 1, 1) which is isomorphic to the Lorentz group of R 1,d−1 .
The conformal algebra so(m + 1, 1) is generated by conformal Killing vectors (CKVs) of S m . The CKVs could be found by embedding S m into R 1,d−1 and making use of the Lorentz algebra of R 1,d−1 . We may define two null vectors n µ ,n µ in Cartesian coordinates n µ = (1, n i ),n µ = (−1, n i ). Hence, the Cartesian coordinates are related to the retarded coordinates by and the partial derivatives in Cartesian coordinates can be rewritten as The partial derivatives ∂ µ can also be regarded as the translation operator in R 1,d−1 . Therefore, we may also write it as We can construct the following antisymmetric tensors in which the tensors n µν andm A µν are related to Y A µν through The Lorentz algebra is generated by Using retarded coordinates and the relations (A.40)-(A.41), we find To find the CKVs of S m , we may pull back the generator L µν to S m by setting u = 0 and r = 1. Therefore, the CKVs of S m are Y A µν . More explicitly, they are denoted as Y A i and Y A ij in the context The vectors Y A ij are Killing vectors of S m and they form the algebra so(m). The vectors Y A i are strictly CKVs of S m and they obey the equation In the following, we collect several important identities about the CKVs.

Null infinity
In Minkowski spacetime R 1,d−1 , the scalar field is may be expanded in plane waves where b k and b † k are annihilation and creation operators, respectively. They obey the standard commutators The plane wave can be expanded in spherical waves by [54] where the summation ℓ is over all possible combinations labeled by ℓ. The d − 1 dimensional vectors k and x are written in spherical coordinates The function j d−1 ℓm (kr) is the d − 1 dimensional generalization of spherical Bessel function which is related to the ordinary Bessel function of the first kind by For example, when d = 4, the function j d−1 ℓm (x) becomes the ordinary spherical Bessel function. The Bessel function of the first kind J n (x) has the following asymptotic behaviour for large x Therefore, the mode expansion Φ(t, x) has the expected fall-off condition with the expansion (3.32) We could also use mode expansion to calculate the commutator between boundary scalar fields. The first step is to notice that Then one can compute the commutator straightforwardly

B.2 Null hypersurface H −
The null hypersurface H − corresponds to the surface t + x d−1 = 0 in Cartesian coordinates.
On this hypersurface, we may define the retarded time u = t − x d−1 = −2x d−1 and expand the bulk field Φ(t, x) as with v = t + x d−1 . Therefore, we can reduce the expansion (B.1) to H − by setting t + x d−1 = 0. This leads to where E = |p| 2 + m 2 = p 2 ⊥ + (p d−1 ) 2 + m 2 . (B.14) It is much more convenient to define the momentum in light cone coordinates [55] p ± = E ± p d−1 2 (B.15) such that dp + = p + E dp d−1 for fixed p ⊥ . (B.16)

B.3 Rindler horizon H −−
In this subsection, we will review the mode expansion of scalar field in Rindler wedge [55,56]. In Rindler wedge, the massless/massive scalar field can be written in the coordinate system (τ, λ, We add an upper index R to label the right Rinder wedge modes. The solution in the Rindler wedge may be expanded as where the function χ ω,k ⊥ satisfies the equation This equation may be transformed to Bessel equation of order ω using the coordinate ρ = e κ d 2 dρ 2 χ ω,k ⊥ + ρ −1 d dρ χ ω,k ⊥ − (k 2 ⊥ + m 2 )χ ω,k ⊥ + ω 2 ρ −2 χ ω,k ⊥ = 0, k ⊥ = |k ⊥ |, (B.24) whose solution is where the function K iω (k ⊥ ρ) is the Modified Bessel function of the second kind. Φ R is bounded near ρ → ∞ since K iω (k ⊥ ρ) decays in this region. More explicitly, Therefore, the stress tensor T µν decays exponentially when ρ → ∞. The normalization of the solution is chosen such that the operators c ω,k ⊥ and c † ω,k ⊥ obey the commutator The asymptotic behaviour of the function K iω (k ⊥ ρ) is The Φ is finite near H −− and we find the following mode expansion of the boundary field Σ It is easy to find that To match the expansion (B.29) with (B.13), we find the following Bogoliubov transformation a R ω,k ⊥ = dp(α * ω,k ⊥ ;p b p − β * ω,k ⊥ ;p b † p ), (B.33a) a R † ω,k ⊥ = dp(α ω,k ⊥ ;p b † p − β ω,k ⊥ ;p b p ), (B.33b) with α ω,k ⊥ ;p = 1 2π ω E e πω/2 Γ(iω)( β ω,k ⊥ ;p = − 1 2π The Bogoliubov coefficients can also be expressed as the commutators We may also expand the field Φ R near H ++ The relations between the operators a R ω,k ⊥ , a R † ω,k ⊥ on H −− and a R ω,k ⊥ , a R † ω,k ⊥ on H ++ are The relation can be transformed to where Σ(ω, k ⊥ ) and Σ + (ω, k ⊥ ) are Fourier transform of Σ(u, x ⊥ ) and Σ + (v, x ⊥ ), respectively

C Zeta-function regularizations
Usually, there are two kinds of zeta functions to proceed this regularization, namely the Riemann zeta function ζ(s) and the Hurwitz zeta function ζ H (s) which are defined in the following [39]. where B n (a) are the Bernoulli polynomials.

C.2 Regularization
When trying to calculate the spectral zeta function, one may analytically continue the expression as following f (s; a, b, c) = ∞ ℓ=1 ℓ −s+b (ℓ + a) −s+c , (C.10) where a, b, c are arbitrary real numbers. In our work, they are natural numbers. We factor out ℓ, separating the series into a finite summation and another series, and perform binomial expansion on it: f (s; a, b, c) = Note that choosing any of the scheme does not affect the final regularization result, since the series are absolutely convergent. After fixing the constants a, b, c , we can take the limit s → 0. For the first scheme, we have For the second scheme, we have ζ ∆ S 2 (s → 0) = 1 − 5 6 + 2 − 17 6 = − 2 3 .