Position and momentum operators for a moving particle in bulk

In this paper we explore how to describe a bulk moving particle in the dual conformal field theories (CFTs). One aspect of this problem is to construct the dual state of the moving particle. On the other hand one should find the corresponding operators associated with the particle. The dynamics of the particle, i.e., the geodesic equation, can be formulated as a Hamiltonian system with canonical variables. The achievements of our paper are to construct the dual CFT states and the operators corresponding to the canonical variables. The expectation values of the operators give the expected solutions of the geodesic line, and the quantum commutators reduce to the classical Poisson brackets to leading order in the bulk gravitational coupling. Our work provides a framework to understand the geodesic equation, that is gravitational attraction, in the dual CFTs.


Introduction
To understand gravity one should not only know the curved spacetime but also how matter moves in the spacetime. The AdS/CFT correspondence provides us a framework to explore both aspects of gravity in the conformal field theory (CFT) on the boundary of the asymptotically AdS spacetime [1]- [3].
In the context of AdS/CFT, previous studies mainly focus on the first aspect, that is emergence of spacetime from the non-gravitational degrees of freedom. The concepts from the quantum information theories are found to be useful. Many quantities, such as entanglement entropy, complexity, are expected to be related to the bulk geometry [4]- [8]. One could refer to the recent review [9] for more references on these studies.
It is also significant to make clear how to describe the bulk moving particle in the CFTs. To answer this question, one should construct the dual 4. The quantum commutators of the constructed operators should reduce to the classical Poisson brackets in the semiclassical limit G → 0.
In the paper we only focus on the vacuum state of AdS 3 . With the observation on the geodesic solution, we assume that the position and momentum operators can be constructed by stress energy tensor T andT , or equally the Virasoro generators. Actually, we will show below only global Virasoro generators are needed.
We expect the operators would be state-dependent [10]. In the Hamiltonian formulation the canonical variables would be associated with the metric g µν . According to the point 3 of the general rules, we would like to construct the operators associated with the canonical variables. It is nature to consider the operators should depend on the background geometry. However, they should not depend on the dual state of the particle, which is related to the initial conditions of the bulk particle.
The organization of this paper is as follows. In section.2 we will briefly discuss the so-called geometric state. One of the feature is that the correlators of stress energy tensor will satisfy the factorization property, which is important for our constructions. In section.3, the solution of the geodesic line in the global coordinate is shown. In section.4 the radial moving particle is discussed. The dual CFT state is assumed to be associated with the local bulk state with suitable regularization. We show how to construct the radial position and momentum operator in this case. In section.5 we discuss another example. The boundary locally excited state can be taken as particle starting from the AdS boundary. We also construct a new state that are expected to be dual to particle with angular momentum. The angular momentum of the particle has a dictionary with the rapidity of a boost in the CFT. The final section is conclusion and discussion. We discuss three interesting problems that are worthy to explore in the near future.

Geometric state and factorization
In the introduction we have mentioned that the background geometry is expected to be dual to a CFT state |g . We will call these kinds of CFT states geometric states. The states |Ψ that are expected to be dual to the moving particle should also be geometric states. It is expected that the particle should have backreaction on the background geometry. If the backreaction can be neglected in the semiclassical limit G → 0, it is not expected the observables in the CFT could detect the difference from the reference state |g . For example, the energy is same in both states in the limit G → 0. The construction would be meaningless. Therefore, in the following examples the mass of the particle will be taken to be O(1/G).
We construct the position and momentum operators by using stress energy tensor T andT . One of the feature of the geometric states is the factorization property. For a given geometric state |g , the expectation value of T is of order c or 1/G [11], for 2-point correlator Or we can define the scaled operator u := T /c, the expectation value of which is of order c 0 . The above condition becomes This means the operators u satisfy the factorization property in the limit c → ∞. Actually, for n-point correlator we also have the factorization property. In [12] we have shown the factorization condition is associated with the geometric state by the scaling behavior of holographic Rényi entropy. If the operators are functions of stress energy tensor, they also satisfy the factorization property. For an arbitrary operatorX as a function of T andT , if the expectation value of it is finite in the semiclassical limit c → ∞, we will call it classical operator. Two arbitrary classical operatorsX and Y , we expect the factorization lim c→∞ g|XŶ |g − g|X|g g|Ŷ |g = 0.
For given geometric states and classical operators, such as the ones that we construct in the following, one could check the above statement by direct calculation. Actually, the factorization property is general for quantum system which has a well defined classical limit [13]. The factorization property is very useful for our calculations. For example, consider X g := g|X|g , Ŷ g := g|Ŷ |g ∼ O(c 0 ). By the factorization (3) we have XŶ g ∼ O(c 0 ). Hence, we expect the commutator Now the commutator More generally, one could check The above results will be used in the following sections.

Geodesic line
We will focus on the global coordinate. The metric is where we take the radius of AdS to be 1. Consider a particle starting from (ρ 0 , φ 0 ) with velocity dφ dt | t=0 = v φ and dρ dt | t=0 = 0. The action of the particle with mass m is where the Lagrangian is The canonical momentum associated with the coordinates ρ and φ is Using these we obtain the Hamiltonian The geodesic line of the particle can be obtained by solving the Hamiltonian equations associated with the canonical variables {ρ, φ, P ρ , P φ }. The result is We can get the momentum P ρ (t) and P φ (t) by taking the solutions (12) into (10) The Hamiltonian of the particle is conserved. Thus, the energy of the particle is constant, that is given by Another constant of motion is the angular momentum P φ , which is independent with t as we can see from (13).

Radial moving particle
Firstly, let us consider the radial moving particle, that is the velocity v φ = 0. We would like to show the dual CFT state of the radial moving particle. Then we will construct the position and momentum operators corresponding to the canonical variables {ρ, P ρ }.

State dual to radial moving particle
The bulk local states have been explored in many literatures. The Hamilton-Kabat-Lifschytz-Lowe (HKLL) construction is a well known method to express the bulk local operator as CFT operators [14]- [16]. A different view on the construction is proposed in [17], for which the symmetry of AdS and CFT play an important role [18]. We will briefly review the methods and show the bulk local states with suitable regularization can be taken as the dual state of the radial moving particle. The bulk scalar operatorφ α (X µ ) satisfies the equation of motion on the background geometry g µν (X µ ), where m is the mass of scalar field. Suppose the metric g µν can be associated with a geometric state |Ψ(g µν ) . The bulk local state is defined as where X µ is the coordinate of the local operator. It is expected the bulk operatorφ α (X µ ) can be expanded by the CFT operators. Thus the bulk local state |φ α (X µ ) can be taken as states in Hilbert space of the CFT. We only focus on the vacuum state |0 . Consider the global coordinate, the state located in the origin of AdS ρ = 0, denoted by |Ψ α , can be expanded as the superposition of Ishibashi states [17] where ∆ α = h α +h α is the conformal dimension of primary operator O α , the primary state |O α := lim z→0 O α |0 . The standard AdS/CFT dictionary gives the relation m = ∆ α (∆ α − 2). The bulk local states at point (ρ, φ) can be associated with |Ψ α by a unitary transformation g(ρ, φ). The bulk local state at point (ρ, φ) is given by with the unitary operator The state |Ψ α is unnormalized since the local operatorφ α is unbounded operator. We can introduce a regulator Λ and define the state whereĤ := L 0 +L 0 , the normalization constant N(Λ) = e −Λ∆α √ 1 − e −4Λ . It is straightforward to obtain the following one-point functions of L n and Λ Ψ α |L n |Ψ α Λ = 0 for n = 0. It is also useful to evaluate the two-point functions More generally, we have As we have argued in the beginning of section.2 we are interested in the case ∆ α ∼ O(c) in the holographic CFTs with c ≫ 1. Define the operator l n := L n /c, which can be taken as the classical operator. In the regime of Λ ≫ 1, we would have the following clustering property for l n , for n = −1, 0, +1. This is a necessary condition for the geometric states as we have discussed in the introduction. Here we would like to explain the state |Ψ α Λ with ∆ α ∼ O(c) and Λ ≫ 1 to be dual to a particle with mass m at rest in the center of AdS in the global coordinate (7). We have the parameter relation ∆ α ≃ m.
The energy in the state |Ψ α Λ is given by is the Hamiltonian of the boundary CFT in the global coordinate. The constant − c 12 is the Casimir energy in the vacuum of global coordinate. This is consistent with the holographic result of a stationary particle with mass m at ρ = 0 by using the fact ∆ α ≃ m at leading order of c. Actually, it is expected an object at rest in AdS 3 is dual to primary state |O α [20]. In the limit Λ → ∞, |Ψ α Λ would approach to |O α . However, even taking Λ ∼ O(c 0 ) we find the bulk metric |Ψ α Λ still corresponds to the backreacted geometry with the stationary massive particle at ρ = 0. The bulk metric is not sensitive to the cut-off parameter Λ. For our purpose we will take the state |Ψ α Λ with Λ ≥ O(c 0 ) as the stationary massive particle at ρ = 0.
By using (17) the state of a particle located at ρ = ρ 0 , φ = φ 0 is given by where g(ρ, φ) is given by (18). In the radial moving case one could always to fix the angular coordinate φ 0 = 0. Consider its time evolution we have the state where U g (t) := e itHg is the unitary evolution operator. In the following we would like to show this state is dual to a radial moving particle in the bulk by directly constructing the associated position and momentum operators.

Position and Momentum operator
Let's calculate the expectation value of the Hamiltonian H g in the (25). It is obvious that the energy To evaluate it we need the formula By using (20) we have The first term is same with the classical particle energy (14) with v φ = 0 by taking ∆ α ≃ m. The second term is the Casimir energy in the vacuum. The expectation value of the operatorĤ = H g + c 12 gives energy of the particle. This suggestsĤ can be taken as the operator dual to the Hamiltonian (11).
One could also check the expectation value of the momentum operator P φ := L 0 −L 0 is zero by using the fact h α =h α . This is consistent with the result that P φ = 0 for v φ = 0. Now we move on to the construction of position operatorρ r and momentum operatorP ρ of the radial moving particle. To simplify the notations we will denote the expectation value The basic requirement for the CFT operatorsρ andP ρ is that where ρ(t) andP ρ (t) are given by (10) with v φ = 0. As we have discussed above the HamiltonianĤ and momentum operator P φ can be associated with energy and angular momentum of the bulk moving particle. They are constructed by the Virasoro generators L n andL n . Motivated by this we can try to buildρ r andP ρ by the same way. Actually, we only need the generators associated with global conformal symmetry, that is Firstly, let's show the following formulas that are useful for the constructions, It can be shown that where we have used (20). Our proposal of the radial momentum operator iŝ which gives the expected relation Actually, we can takeP ρ as the generator of the radial transformation g(ρ, 0) since g(ρ, 0) = e −iPρρ . We can construct the position operatorρ r from the classical Hamiltonian of the partcile. By using (10) and (11) the Hamiltonian withφ = 0 is Taking the Hamiltonian operatorĤ = L 0 +L 0 and radial momentum oper-atorP ρ (31) into the above equation, one could obtain the operatorρ r by solving the operator equation. This suggests the position operatorρ can be constructed asρ In the above expression arccosh(Â) is defined as ∞ n=1 a nÂ n where a n are Taylor coefficients of the function arccosh(x). To make arccosh(Â) to be a well defined bounded operator the series expansion should be convergent in the sense of operator algebra. In this section we only focus on the expectation value of the operators in the state |Ψ α (ρ 0 , 0, t) . For our purpose we would take the operator arccosh(Â) to be a well defined operator if the expansion ∞ n=0 a n Â n Ψα(t) is finite.

Check of our proposal
The operatorÂ are polynomials inĤ andP ρ , which are associated with the energy momentum operator T andT . Roughly, the relation isĤ,P ρ ∼ f T + fT , where f andf are some functions. The state |Ψ α (ρ 0 , 0, t) is explained as a moving bulk particle state, which obviously should be a geometric state. Therefore, using the factorization property for the operatorŝ H andP ρ , we obtain Similarly, the operatorÂ also satisfies the factorization property One could show this by direct calculations for a given n.
Taking (27) and (32) into (35), we have Using the above result and (36) we have the expected relation The expectation values of the operatorρ r andP ρ in the state |Ψ α (ρ 0 , 0, t) give the classical results (12) and (13) at the leading order of c.
It is convenient to introduce the scaled momentum operatorp ρ :=P ρ /c. They can be taken as the operators related to the particle with mass m/c.p ρ are classical operators, since their expectation values in the state |Ψ α (ρ 0 , 0, t) are finite in the limit c → ∞. We can also define more general opera-torsX(ρ r ,p ρ ). Consider two arbitrary classical operatorsX(ρ r ,p ρ ) and Y (ρ r ,p ρ ), which are functions ofρ r andp ρ . We also have the factorization property at the leading order of c, where ρ(t), p ρ (t) := P ρ (t)/c are given by (28). The proof is similar as (36). Therefore, the classical operators behave as c-number in the state |Ψ α (ρ 0 , 0, t) . The Newton constant G or 1/c plays the role as the parameterh in quantum mechanics. The commutator [X,Ŷ ] would also have a correspondence to the Poisson bracket {X, Y }. For the radial moving particle the phase space is 2dimensional, for which ρ and p ρ are canonical variables. The classical Poisson brackets of two functions X(ρ, p ρ ) and Y (ρ, p ρ ) are defined as One special case is the fundamental Poisson bracket {ρ, p ρ } = 1. Since we have constructed the position and momentum operators, their commutators can be evaluated by the Virasoro algebra. Our task is to show how to obtain the classical Poisson brackets from the quantum commutators. This is similar as the process that the quantum commutators reduce to classical Poisson brackets in the limith → 0. Let's begin with the fundamental bracket {ρ, P ρ } = 1. To evaluate the corresponding quantum commutator [ρ r ,P ρ ], we need [Â n ,P ρ ]. For n = 1 withQ For general n it is not easy to write down the results. However, if we consider the commutators in the state |Ψ α (ρ 0 , 0, t) the expression would be very simple at the leading order of c. By using the factorization property of operators, we have In the above calculation we only keep the leading order c results. In the last step we use (38) The quantum commutator reduces to classical Poisson brackets as The above result is consistent with the factorization property.

Locally excited state in CFT
According to the extrapolate dictionary of AdS/CFT, the bulk operator φ α (ρ, x) and the dual boundary CFT operator O α ( x) are related by in the global coordinate. We expect the bulk state φ α (ρ, x)|0 should reduce to the locally excited state O α ( x)|0 in CFT near the AdS boundary. To regularize this state O α ( x) one could introduce a cut-off ǫ and define where N is the normalization constant. We can take ǫ as the UV cut-off of theory with ǫ ≪ 1. The locally excited states has been studied in many literatures on the dynamics behavior of entanglement entropy, see [21]- [30] and references therein. In the following we would like to focus on such state, which is expected to be described by a point particle with the initial location near the boundary of AdS [31]. Denote the boundary coordinate as x ± = φ ± t. By a Wick-rotation we have the Euclidean coordinate w := φ − iτ andw := φ + iτ . With a conformal mapping z = e iw , the cylinder is mapped to z-plane. The state will be defined on the z-plane. The local state (48) inserted at w 0 =w 0 = 0 is given by where z 0 = e iw 0 = e ǫ ,z = e −iw 0 = e ǫ , and normalization constant N(z 0 ,z 0 ) = (z 0 z * 0 − 1) hα (z 0z * 0 − 1)h α . We can also write the above state as with the primary state |O α := lim z→0 O α |0 . For h α ∼ O(c), the energy of this state is For the static particle located at the AdS boundary ρ 0 ≫ 1, the energy is given by (14) with v φ = 0, that is Comparing with (52) we obtain the relation 1 ǫ ≃ 1 2 e ρ 0 , which provides an interpretation of log( 1 ǫ ) as the initial location of the bulk particle. This is also consistent with the UV/IR relation in the context of AdS/CFT [32].
One could check the expectation value of momentum operator P φ = L 0 −L 0 in the state (49) is vanishing. Therefore, we can interpret this state is dual to a particle moving in the radial direction in the bulk, that is v φ = 0.

State with angular momentum
It is more interesting to construct the state with non-vanishing v φ . From (13) and (11) with ρ 0 ≫ 1 we have This motivates us to construct the state with non-vanishing v φ by a boost with velocity v φ . The coordinates transform as where λ is the rapidity with v φ = tanh λ. We propose that the dual state of a moving bulk particle with initial position (ρ, φ) = (ρ 0 , 0) (ρ 0 ≫ 1) and velocity v φ is given by the locally excited state O α (w 0 e −λ ,w 0 e λ )|0 . On the z-plane the state is defined as where z λ = e iw 0 e −λ = e ǫe −λ ,z λ = e −iw 0 e λ = e ǫe λ , and normalization constant N(z λ ,z λ ) = (z λ z * λ − 1) hα (z λz * λ − 1)h α . The energy can be obtained by the replacement z 0 → z λ ,z 0 →z λ in (51). Keeping the leading order of ǫ we have Similarly, the angular momentum is given by These are consistent with the results (54). Consider the time evolution and define the time-dependent state where U g = e iHg t . It is obvious that the energy and angular momentum are independent with t.

Angular coordinate operator
We can construct the position and momentum operator as we have done for the radial moving case. We will show they can be expressed as operator functions of the global Virasoro generators.
To simplify the notation the expectation value of operatorX in the state |ψ α (t) ǫ is denoted by X ψα(t) . The following formulas are useful for our construction, In previous section we have defined two Hermitian operatorsP ρ andQ ρ which are linear combinations of the global Virasoro generators. Let's define two more independent operatorŝ By using (60) it is straightforward to evaluate the expectation values of the four Hermitian operators. The results are P ρ ψα(t) = ∆ α ǫ cosh λ sin t, The above results are consistent with the radial moving case λ = 0. Since the state (56) is a geometric state, the global Virasoro generators also satisfy the factorization property. Now we move to construct the operatorφ. The expectation φ ψα(t) is expected to give the classical solution (12). Rewrite (12) as φ = arccos 1 1 + (tanh λ tan t) 2 = arccos cosh λ cos t (cosh λ cos t) 2 + (sinh λ sin t) 2 .(63) The angular coordinate operatorφ is suggested to bê where arccosB := n b nB n , b n are Taylor coefficients of the function arccos(x). Using (62) we have the result One could check that φ ψα(t) = φ(t). For the radial moving particle we have φ Ψα(t) = 0.

Radial momentum and coordinate operator
For ρ 0 ≫ 1 the radial momentum is which is different from P ρ ψα(t) (62). We should include more terms to produce the above expected result. By using (62) we suggest the following radial momentum operator By the definition ofφ we have sinφ = 1 − cos 2φ = 1 −B 2 , which can be written as The expectation value ofP ρ is given by which is equal to (66). For the special radial moving case φ Ψα(t) = 0, we can effectively takeP ρ as the radial momentum operator. The radial position operatorρ can be constructed by using the relation The solution of the above equation for ρ actually gives an ansatz of the radial position operatorρ. We have constructed the Hamiltonian operator Ĥ , the angular momentum operatorP φ and the radial momentum operator P ρ . Taking them into the solution we obtain ρ = 1 2 arccosh(r), One could check the above expression will become (34) forP φ = 0. With some calculations we can find the expected relation

Poisson brackets
As a check of our proposals we will show how to get the Poisson brackets from the position and momentum operators. The phase space of the bulk moving particle is 4-dimensional with the canonical variables {φ, ρ, P φ , P φ }. We will focus on the fundamental Poisson brackets. Firstly, consider the commutator [φ,P φ ]. With the definitions we have the following commutation relations, With these and the definition ofB (64) we can obtain where in the second step we have used where the last step follows from (73). In the above evaluation we have used the factorization property. Therefore, the equality is established only in the leading order of c. Now we can evaluate the commutator In the last step we have used (62) and (65). Let's introduce the scaled operatorp φ :=P φ /c. The quantum commutator reduces to Poisson brackets as Using the above result we can evaluate the more general commutators such as where F (φ) is arbitrary functions ofφ. One could derive the above expression by takingp φ as i c ∂ ∂φ when evaluating the commutators. Now let's consider the commutator [p φ ,p ρ ], where we define the scaled radial momentum operatorp ρ :=P ρ /c. Using the definition ofP ρ (67) and commutation relations (73 ) and (78), we have With some calculations one can also get the following commutators, We show the details of the calculation in Appendix.A

Equation of motion
With the Poisson brackets one could easily obtain the equation of motion of the particle. For a classical system with canonical variable {q i , p i } the Hamiltonian equation can be expressed as where H is the Hamiltonian of the system. In the previous section we have constructed the position and momentum operators and shown the quantum commutators can reduce to the classical Poisson brackets in the limit c → ∞. Now we would like to find the classical equation in the same limit. Takeφ as an example. Let's define the scaled Hamiltonian operatorĥ := H/c. Using the result (101) andφ := arccosB we have where in the last step we have used (62). The final result is same as . Therefore, we find the equation of motion which can be taken as the classical limit of the Heisenberg equation. Another example is the operatorP ρ . Using (100) and (62) one could check the equation of motion The interested reader can check the other two equations associated withρ andP φ .

Conclusion and discussion
The main result of our paper is to explore the CFT dual of a bulk moving particle. As mentioned in the introduction the problem has two aspects.
Firstly, we construct the state that are expected to be dual to the moving particle. Two examples are shown. For the radial moving particle starting from arbitrary position ρ 0 we find the CFT state can be described by the regularized bulk local states that are discussed in previous paper [19]. The other one is the boundary locally excited state. This state can be explained as particle starting from the AdS boundary. In this case we find the dual state of the particle with angular momentum can be related to a boost. The rapidity of the boost is associated with the velocity in the φ direction. As far as we know the state with a boost hasn't been discussed in other papers.
The other aspect of the problem is to construct the position and momentum operators associated with the particle. We should also note that the operators in the radial moving example are special case of {ρ,P ρ ,φ,P φ }. However, we haven't successfully constructed the state dual to arbitrary moving particles in the bulk. It would be interesting to find such states and check whether the constructed operators could give the correct results.
Generally, we have no systematical method to find the operators. Therefore, we should discuss case by case at present. Of course, there are some basic constraints on the constructions.
Let us summarize some important clues. The energy and angular momentum momentum of the particle should be related to the Hamiltonian and momentum operators of the dual CFTs. The particle can be taken as excited state of the bulk. Hence, the energy and angular momentum of the particle should be equal to the difference between the excited state and the background state. In the CFTs they should be related to the expectation value of the Hamiltonian and momentum operators.
The basic requirement is that the expectation value of the constructed operators in the dual CFT states should give the classical solution at the leading order of G. Actually, this is the important guidance to the constructions. For the examples that are shown in our paper are simple, since we could find the exact classical solution. We find the operators can be constructed only by using the stress energy tensor, in fact only by the global Virasoso generators. The reason is that the dual state can be obtained by global symmetry. For the background state beyond the vacuum the symmetry is broken. We don't expect these operators should be universe, that is independent with the background geometry. But the stress energy tensor should be the building block of the position and momentum operators.
Another requirement is the quantum commutators of the constructed operators should reduce to the classical Poisson brackets in the semiclassical limit c → ∞. This can also be seen as a check of our constructions. In both examples in our paper we show the correspondence between the quantum and classical brackets.
There are many important and interesting problems that we haven't touch in this paper. In the following we will briefly discuss three such problems that are worthy to study in the future.

Other examples
We only focus on the vacuum AdS 3 in the global coordinate. It is easily to generalize to other situations, such as the Poincare coordinate, AdS Rindler. One can use the similar methods that we have used. It is an interesting excise to work out the results in different coordinate and compare with our results. In particular, the AdS Rindler will help us to understand more on the entanglement wedge construction or subregiona/subregion duality [33]- [35]. For example, the Poincare coordinate of pure AdS 3 is The state |Ψ α in this case corresponds to the bulk local excitation at point (t p , y, x) = (0, 1, 0). The bulk local state at point (t p , y, x) = (0, y, x) is given by where are the generators in the hyperbolic basis as shown in [19]. By similar method we expect one could obtain the position and momentum operators in the Poincaré coordinate. It would be interesting to see the difference and relation with the global coordinate.
More interesting case is BTZ black hole as the background geometry. One could use the thermofield double states to work out the results. In this case there is a horizon in the bulk. It is interesting to explore how to construct the corresponding operators once the particle is inside the horizon.
The generalization to higher dimension is not so straightforward, but we expect one could have a similar construction as the 3D AdS. In the vacuum case the operators are only associated with the global symmetry. In higher dimension the dual CFT also has global conformal symmetry. But in higher dimension the situation will be more complicated, thus more interesting phenomena are expected to appear.

Coordinate dependence
In the last section we discuss the generalization to other coordinate such as the Poincaré AdS. The vacuums in different coordinate are not same. The particle states are actually differnt in these coordinate.
It is obvious that our constructions of position and momentum operators depend on the coordinate. Even in the global coordinate, one could choose different coordinates. The operators should depend on the canonical variables that one choose.
For example, in the radial moving case one could choose D := arccoshρ as the position coordinate of the particle. The Lagrangian is Hence, the canonical momentum is given by The Hamiltonian of the classical particle withφ = 0 is given by Thanks to the factorization property of the geometric sate, we could guess the position operator should bê The equation (90) gives the expression of the canonical momentum operator One could check the above operators by comparing the expectation value of them with the classical geodesic solution.
The coordinate transformation is a special case of the canonical transformation of the phase space of the classical particle. One could choose any set of the canonical variables. And the corresponding position and momentum operators can be constructed by the same methods as above.

How to understand gravity in the CFTs?
Our results provide a framework to explore the explanation of gravity in the CFTs. In general relativity the dynamics of the particle is given by the geodesic equation. The geodesic equation is equal to equation of motion for the system with Lagrangian L = −m dτ , where τ is the proper time of the particle.
In our approach we assume the existence of the CFT operators that are dual to the canonical variables of the particle. Moreover, we construct the operators, the expectation values of which give the particle's position and momentum. A remarkable fact is that all these operators are constructed by the Hermitian operatorsP ρ ,Q ρ ,Ŝ φ ,T φ ,Ĥ andP φ , which are independent linear combinations of global Virasoro generators.
We also show how to obtain the classical Poisson brackets from the quantum commutators of the constructed operators. The Hamiltonian equations, i.e., the geodesic equations, can be expressed by the Poisson brackets as we have shown in section.5.5. Hence, the dynamics of the bulk particle is determined by the commutation relations algebra (73). Further, these relations can be derived from the stress energy tensor commutators, i.e., [T µν , T ρσ ]. The geodesic equation should have a correspondence to the stress energy tensor commutator, at least in our special cases. However, we have no evidence to conclude that the correspondence is also true for general background geometry. It is worth to study more general examples and make the correspondence more explicit.
Recently, Susskind proposes the size-momentum correspondence, that gives a connection between radial momentum of a bulk particle, operator size and complexity [36]. The size of the operator is expected to be proportional to the radial momentum of a bulk moving particle. One could refer to [36]- [38] for the definition of these concepts. The operator size can be evaluated in SYK models [39] [40]. For general theories, such as holographic field theory, it is expected the operator size may be associated with a Hermitian operator [41] [42]. Follow the notation of [41], the size of operator O that act on reference state |Ψ is given by whereŜ |Ψ is expected to be a semi-definite, Hermitian operator. Actually, in our approach the radial momentum of the bulk moving particle is also given by the expectation value of the operatorP ρ . It is very interesting to check whether the operator size operatorŜ |Ψ has some connection with the radial momentum operatorP ρ in our paper.

A Commutators
In this section we will show the details of the calculation of commutators(80). Consider the commutator [p ρ ,φ] ψα(t) . By definition we have It is useful to evaluate [P ρ ,B] ψα(t) . By using (73) we have Similarly, we have Using the above results we can obtain (94), the result is [P ρ ,φ] ψα(t) where in the last step we use the fact Ŝ φ cosφ −Q ρ sinφ ψα(t) = O(c −1 ). Therefore, we have the classical Poisson bracket lim c→∞ ic [p ρ ,φ] ψα(t) = 0.
We leave it as an exercise for interested reader to prove lim c→∞ ic [ρ,φ] ψα(t) = 0.
Finally, let's show the last commutator in (80).ρ is a function ofĤ,P φ and P ρ . We will need the commutator [P ρ ,Ĥ], which can be written as The expectation value of [P ρ ,Ĥ] is where we have used Now recall the definition ofr (71), we obtain the commutator [r,P ρ ] ψα(t) Thus we have