Doubled Hilbert space in double-scaled SYK

We consider matter correlators in the double-scaled SYK (DSSYK) model. It turns out that matter correlators have a simple expression in terms of the doubled Hilbert space $\mathcal{H}\otimes\mathcal{H}$, where $\mathcal{H}$ is the Fock space of $q$-deformed oscillator (also known as the chord Hilbert space). In this formalism, we find that the operator which counts the intersection of chords should be conjugated by certain ``entangler'' and ``disentangler''. We explicitly demonstrate this structure for the two- and four-point functions of matter operators in DSSYK.


Introduction
To describe a black hole in AdS, it is useful to consider the doubled (two-sided) Hilbert space of boundary CFT.In particular, the eternal black hole in AdS corresponds to the thermo-field double state [1] which is closely related to the idea of ER=EPR [2,3].Recently, the doubled Hilbert space in JT gravity and the double-scaled SYK (DSSYK) model has been extensively studied in the literature (see e.g.[4][5][6][7][8] and references therein).
In this paper, we consider matter correlators of DSSYK in the doubled Hilbert space formalism.As shown in [9], the correlators of DSSYK reduce to the counting problem of chord diagrams, which is exactly solved in terms of the q-deformed oscillator A ± .The Fock space H of the q-deformed oscillator, also known as the chord Hilbert space, can be thought of as the Hilbert space of bulk gravity theory [10].It turns out that matter correlators of DSSYK have a simple expression in the doubled Hilbert space H ⊗ H.We find that the operator which counts the intersection of chords is conjugated by the "entangler" E and the "disentangler" E −1 (see (4.4) and (4.11)).This structure is reminiscent of the tensor network of MERA [11,12].This paper is organized as follows.In section 2, we briefly review the known result of matter correlators in DSSYK.In section 3, we define a mapping of the operator X on H to the state |X⟩ in the doubled Hilbert space H ⊗ H and rewrite the matter correlators as the overlap ⟨0, 0|X⟩.In section 4, we perform this rewriting explicitly for the two-and four-point functions of matter operators.We find that the intersection-counting operator is conjugated by the entangler and the disentangler as in (4.4) and (4.11).Finally we conclude in section 5 with some discussion on the future problems.In appendix A we summarize some useful formulae used in the main text.In appendix B we explain the derivation of (4.8).In appendix C we prove the crossing symmetry of the R-matrix of U q (su(1, 1)).

Review of DSSYK
In this section we briefly review the result of DSSYK in [9].SYK model is defined by the Hamiltonian for N Majorana fermions ψ i (i = 1, • • • , N ) obeying {ψ i , ψ j } = 2δ i,j with all-to-all p-body interaction where J i 1 •••ip is a random coupling drawn from the Gaussian distribution.DSSYK is defined by the scaling limit As shown in [9], the ensemble average of the moment Tr H k reduces to a counting problem of the intersection number of chord diagrams with q = e −λ .This counting problem is solved by introducing the transfer matrix T where A ± denote the q-deformed oscillator acting on the chord number state |n⟩

.5)
Note that A ± satisfy the q-deformed commutation relations where N denotes the number operator Then the moment in (2.3) is written as The transfer matrix T becomes diagonal in the θ-basis and the overlap of ⟨n| and |θ⟩ is given by the q-Hermite polynomial H n (cos θ|q) where (q; q) n denotes the q-Pochhammer symbol (see appendix A for the definition).|θ⟩ and |n⟩ are normalized as and the measure factor µ(θ) is given by µ(θ) = (q, e ±2iθ ; q) ∞ . (2.12) As discussed in [9], we can also consider the matter operator with a Gaussian random coefficient K i 1 •••is which is drawn independently from the random coupling J i 1 •••ip in the SYK Hamiltonian.In the double scaling limit (2.2), the effect of this operator can be made finite by taking the limit s → ∞ with ∆ = s/p held fixed.Then the correlator of O ∆ 's is also written as a counting problem of the chord diagrams Note that there appear two types of chords in this computation: H-chords and O-chords coming from the Wick contraction of random couplings The O-chord is also called matter chord.
(2.15) Using the relation (2.15) is rewritten as As shown in [9], this bi-local operator commutes with T T, O ∆ e −βH O ∆ = 0. (2.18) The two-point function of matter operator O ∆ is given by Note that only the ℓ = 0 term in (2.17) contributes to the two-point function since Similarly, the uncrossed four-point function is given by (2.20) In the first equality we have used the relation (2.18) and the last equality follows from the fact that only the ℓ = 0 term in (2.17) contributes in this computation when sandwiched between ⟨0| and |0⟩.
The crossed four-point function is given by [9] ⟨0|e (2.21) Here we have suppressed the overall factor q ∆ 1 ∆ 2 coming from the intersection of the O ∆ 1chord and the O ∆ 2 -chord.
Let us take a closer look at the two-point function (2.19).Inserting the complete set 19), the two-point function becomes As discussed in [9,10], |n⟩ represents the state at a constant time-slice of the bulk geometry with n H-chords threading that slice.The factor q ∆n comes from the intersection of matter chord and n H-chords.Thus q ∆ N in (2.19) can be thought of as the operator counting the intersection of O ∆ -chord and H-chords.This operator q ∆ N plays an important role in what follows.

Doubled Hilbert space
As we reviewed in the previous section, the matter correlator of DSSYK takes the form ⟨0|X|0⟩, where X is a linear operator on the chord Hilbert space H spanned by the chord number states |n⟩ In order to study the matter correlators in DSSYK, it is useful to consider the doubled Hilbert space H ⊗ H and regard the operator X as a state |X⟩ in H ⊗ H In terms of the basis {|n⟩} n=0,1,••• , this mapping (3.2) is given by |n, m⟩⟨n|X|m⟩, where |n, m⟩ is the natural basis of In particular, the identity operator 1 corresponds to the state where E is given by (see (2.16) and (A.2)) Note that the state |1⟩ is the maximally entangled state and the operator E generates the entanglement when acting on the pure state |0, 0⟩.Similarly, the operator q ∆ N corresponds to the state with Note that we can append and/or prepend strings of operators as1 where X, Y, Z ∈ End(H) and t Z denotes the transpose of Z We should stress that we do not take the complex conjugation of Z on the right hand side of (3.9); we simply reverse the order of multiplication and take the transpose of Z in (3.9).
As an example of (3.9), let us consider the relation

.11)
Using t A ± = A ∓ and (3.9), we find 2 We can also show that (3.15) From (2.11), the state |1⟩ is written in terms of the |θ⟩-basis as and the state corresponding to the operator e −βT is given by This state |e −βT ⟩ is known as the thermo-field double state.

Matter correlators in the doubled Hilbert space formalism
In this section, we consider matter correlators of DSSYK in the doubled Hilbert space formalism.In general, the matter correlator of DSSYK takes the form ⟨0|X|0⟩ with some operator X ∈ End(H).In the doubled Hilbert space formalism, ⟨0|X|0⟩ is expressed as

Two-point function
Let us first consider the bi-local operator in (2.17), which is the basic building block of the two-point function and the uncrossed four-point function.The state |O ∆ e −βH O ∆ ⟩ corresponding to the operator in (2.17) is given by where we used the summation formula in (A.3).Using the relation 2 The state q ∆ N in (3.7) is reminiscent of the boundary state |Ba⟩ of the end of the world brane [15] |Ba⟩ = 1 (aA+; q)∞ |0⟩. (3.12) As shown in [15], the boundary state |Ba⟩ is a coherent state of the q-deformed oscillator where the parameter a is related to the tension of the brane.
(4.2) is rewritten as where E is defined in (3.6).The appearance of the operator q ∆ N ⊗ q ∆ N in (4.4) is natural since it counts the number of intersections between the H-chord and the matter chord.The important point is that this operator q ∆ N ⊗ q ∆ N should be conjugated by E This conjugation guarantees that the β → 0 limit of the state (4.4) reduces to |1⟩ in (3.5) In other words, the conjugation (4.5) is necessary for the following operator identity to hold3 Following the language of tensor networks, we call E and E −1 as "entangler" and "disentangler", respectively.Our result (4.4) shows that we have to insert the disentangler E −1 before acting the intersection-counting operator q ∆ N ⊗ q ∆ N .In the context of MERA [11,12], disentanglers are usually assumed to be unitary operators.However, our E and E −1 are not unitary.Thus, (4.5) is a similarity transformation, not a unitary transformation.See appendix B for the derivation of this expression.

Crossed four-point function
Next, let us consider the crossed four-point function (2.21) In the doubled Hilbert space formalism, this is written as where E ∆ 1 is defined in (3.8).Again, the operator q ∆ 2 N ⊗ q ∆ 2 N is conjugated by E ∆ 1 in (4.11); E ∆ 1 and (E ∆ 1 ) −1 can be thought of as the entangler and the disentangler associated with the state |q ) is schematically depicted as where the red line and the blue line correspond to the O ∆ 1 -chord and the O ∆ 2 -chord, respectively.In this picture, the bra and the ket are treated asymmetrically and some of the symmetries of G 4 are not manifest in our representation (4.11).In particular, the crossing symmetry ( 12) ↔ (34) of G 4 is not manifest in (4.11).

Conclusion and outlook
In this paper we have studied the matter correlators of DSSYK in the doubled Hilbert space formalism.In our formalism, a matter correlator of the form ⟨0|X|0⟩ is expressed as the overlap between ⟨0, 0| and the state |X⟩ ∈ H ⊗ H corresponding to the operator X, where the relation between X and |X⟩ is given by (3.3).We find that the intersection-counting operator q ∆ N ⊗ q ∆ N should be conjugated by the entangler E and the disentangler E −1 as in (4.4) (or the entangler E ∆ 1 and disentangler (E ∆ 1 ) −1 in the case of crossed four-point function (4.11)).In our representation of a matter correlator ⟨0, 0|X⟩ (4.1), the bra and the ket are treated asymmetrically and hence some of the symmetries of the correlators are not manifest.Nevertheless, the bra-ket exchange symmetry (or crossing symmetry) of the four-point function (4.16) can be shown rather non-trivially by using the Bailey transformation (C.5) of 8 W 7 .
We should stress that our formalism is different from that in [8].The authors of [8] introduced the two-sided chord Hilbert space in the presence of the matter operator, spanned by the states {|n L , n R ⟩} where n L and n R denote the number of H-chords to the left and right of the matter chord.Our |n, m⟩ in (3.4) is not equal to |n L , n R ⟩ in [8].According to the discussion in [14], our |n, m⟩ can be expanded as a linear combination of |n L , n R ⟩ in [8].It would be interesting to find a precise relation between our |n, m⟩ and |n L , n R ⟩ in [8].
The construction of the two-sided chord Hilbert space in [8] is based on a picture of cutting open the "bulk path integral".On the other hand, our formalism is based on a honest, direct rewriting of the known result of matter correlators in [9].At present we do not understand clearly how these two approaches are related.In particular, in our formalism we do not need to introduce the co-product of q-deformed oscillator A ± , which played an important role in the discussion of symmetry algebra in [8].Perhaps, (3.14) and (3.15) might be a good starting point to consider the relationship between the two approaches.We leave this as an interesting future problem.