Conformal Quantum Mechanics and Sine-Square Deformation

We revisit conformal quantum mechanics (CQM) from the perspective of sine-square deformation (SSD) and the entanglement Hamiltonian. The operators that correspond to SSD and the entanglement Hamiltonian are identified. Thus, the nature of SSD and entanglement can be discussed in a much simpler CQM setting than higher-dimensional field theories.

In [1,2], it was shown that sine-square deformation (SSD) [3] for two-dimensional (2d) conformal field theory (CFT) [4] can be understood by introducing a new quantization scheme called "dipolar quantization." 1) The basic idea was generalized in Ref. [12] to incorporate the entanglement Hamiltonian and other interesting deformations of 2d CFT. In this Letter, we examine whether the idea of dipolar quantization is applicable to the one-dimensional (1d) case, which is called conformal quantum mechanics (CQM). CQM was first studied in the seminal paper by de Alfaro, Fubini, and Furlan [13].
To put the problem in perspective, let us consider a scalar field φ(x) on general d-dimensional flat spacetime x µ (µ = 0, . . . , d−1) following the argument presented in Ref. [13]. Suppose φ(x) transforms under the scale transformation A simple invariant action for φ(x) can be obtained as 1) See Refs. [5,6] for earlier studies on SSD and Refs. [7][8][9] for more recent studies. References [10,11] study SSD in the context of string theory and conformal field theory.
where g is the dimensionless coupling constant. Because scale invariance implies conformal invariance in most cases [14], this action provides a good starting point. In addition, Eq. (2) suggests the difficulty in the Lagrangian formalism for d = 2 case, in which the energy momentum tensor is taken as the basis of the theory rather than the lagrangian.
The case of interest here is d = 1, which has the following Lagrangian: where t is the 1d "spacetime" coordinate. We also changed the notation of the "field" from φ to q(t) because we are now dealing with a quantum mechanical system. We can then show that the Lagrangian (3) possesses the following symmetry: which is a larger symmetry than scale invariance and translational invariance combined. In fact, it is 1d conformal symmetry, as we anticipated. The transformation (4) for t can be conveniently decomposed into the following three components: Translation a = d = 1 and c = 0 lead to Dilatation a = 1/d and b = c = 0 lead to t → a 2 t.
Special Conformal Transformation(SCT) a = d = 1 and b = 0 lead to The infinitesimal version of transformations (6) -(8) of the above three can be represented in terms of the differential operators as follows.
These operators form a closed algebra, which is readily isomorphic to sl(2, R) algebra or, equivalently, the subalgebra formed by the three Virasoro generators L 1 , L 0 , and L −1 : The time-translation generator P 0 should be identified with the Hamiltonian where p is the canonical momentum if one regards the commutation relation in (12) as the Poisson bracket 2) . Equation (14) was directly derived from Lagrangian (3). Using the symplectic structure, the rest of the generators may be expressed in terms of q and p: In Eq. (16) we employed symmetrization in anticipation of quantization. In Ref. [13], de Alfaro, Fubini, and Furlan introduced the new operator where a is a constant with the dimensions of time, along with two other operators. Then, R was proposed to supersede H as the time-translation operator, or the Hamiltonian. The distinction between the operator R and the original Hamiltonian H = P 0 is best clarified from the symmetry viewpoint [2,13]. First, the (quadratic) Casimir invariant for sl(2, R) algebra is Therefore, for any adjoint action of sl(2, R) algebra on the linear combination of the generators, 2) Note the difference in the factor i of the algebra (12), compared to that in Ref. [13].
the following combination remains unchanged 3) : In terms of the coefficients x (0) , x (1) , and x (−1) , the operator R is expressed as and the expression for the original Hamiltonian H (or P 0 ) is Putting these coefficients into c (2) defined in Eq. (20), we immediately find and These results imply that one cannot connect R and H by any adjoint action of sl(2, R), nor by its exponentiation, SL(2, R). In this sense, operators R and H are disconnected. Now, note the absence of constant a in expression (23), which infers that a can be changed numerically by an adjoint action of sl(2, R) or SL(2, R) action on the operator R. In fact, an infinitesimal change in a → a(1 − ǫ) evokes the commutation with D as Thus, different values of a in R are connected by the action of D. Two other actions can be applied to R, namely, P 0 and K 0 , which would produce terms corresponding to D and yield a nonzero x (0) coefficient. Hereinafter, we assume a to be unity for the sake of simplicity. We then ask if any class of operators is connected to H by the action of SL(2, R). Apparently, the answer is affirmative because the following operator 3) Note that the numerical coefficients of the quadratic form in Eqs. (18) and (20) are components of matrices that are inverse of each other.
yields c (2) = 0 as does H, which can be written in the above notation as H (a,b) is explicitly written as or, in terms of the canonical variables, where, without loss of generality, we have chosen one of the double signs that appeared in Eq. (26). The transformation between H (1,0) and H (a,b) can be interpreted in terms of classical mechanics because we have designed the system so that it accommodates conformal symmetry. In fact, the transformation can be achieved by changing the canonical coordinates as follows: where P (t) and Q(t) are the new canonical coordinates. The generating function of the above canonical transformation is Another class of generators yields negative c (2) , the simplest of which is ForR, the coefficients arē which yield c (2) = −1. 4) Now, each distinct class of c (2) can be conveniently represented by the following combination of coefficients: 4)R corresponds to −S in the notation of Ref. [13].

The corresponding generator is
Because the above generator resembles the ordinary Hamiltonian, it is clarifying to draw the graph of the potential V (q) = g 2 1 q 2 + c (2) 8 q 2 for each case , where we presume g > 0 5) . Figure 1 shows the potential for the cases where c (2) equals 1, 0, and −1, respectively. In the following, we investigate each case.  Reference [13] observed that the invariance of the Casimir invariant (18) is apparent from the expressions (14) - (16), if one imposes the commutation relation over q and p as [q, p] = iI: where I is the identity operator of the (enveloping) algebra in question 6) . Without fear of confusion, we also denote the parameter g 4 − 3 16 as C (2) . Using the notation 5) For negative g, despite the apparent unbounded potential, the corresponding Schrödinger equation for c (2) = 0 has a stable solution up to g = − 1 4 , similar to the Breitenlohner-Freedman bound [15] in higher dimensional AdS space. 6) As noted in Ref. [13], the case g = 0 yields particularly simple representations by the creation and annihilation operators [a, a † ] = 1, which are called singleton representations. Despite the ostensible lack of enough structure to accommodate the symmetry, this is an example of a spectrum generating algebra, and the symmetry algebra is represented by the transitions between the different energy states [16,17].
C (2) , one obtains which turns out to be useful for finding the eigenvalues of L 0 . Suppose a normalized eigenstate vector |E exists such that It is then straightforward to show that one can construct eigenstates with eigenvalues E ± 1 by multiplying by L ∓1 because We would like to normalize the eigenstates obtained above, so that E ± 1|E ± 1 = 1.
The normalization factor c ± can be calculated using Eq. (37), which yields This condition of positivity can be clearer if we introduce a common notation for the Casimir invariant C (2) = j(j − 1) (we assume j ≥ 0): We thus conclude that E ≥ j or E ≤ −j, and from physical considerations, we prefer positive E. Finally, as the eigenvalues of L 0 , we obtain where n = 0, 1, 2, 3, . . . 7) . Because R can be identified with L 0 , we obtain the system with a discrete spectrum using R as the Hamiltonian in stead of the original H 8) . This is fairly evident from Fig. 1 because R corresponds to the case c (2) = 1, where the range of motion is apparently limited to a finite region. Conversely, the case c (2) = 0 does not exhibit discrete energy states because the motion of the particle is not confined by the potential. Instead, it has a continuous spectrum as discussed in detail in Ref. [13]. This emergence of the continuous 7) One might consider an extension of sl(2, R) to the full Virasoro algebra on these eigenstates. See Ref. [18] for a related discussion. 8) With the identification of R as L 0 , L ±1 = (H − K 0 )/2 ± iD [13].
spectrum compelled the authors of Ref. [13] to propose R as the Hamiltonian of CQM instead of the original H, which corresponds to the case c (2) = 0. However, we prefer to propose another interpretation of H here: In light of SSD, we do not have to reject an operator just because it leads a continuous spectrum. In fact, this is the signature of SSD systems. Therefore, we propose to regard H as the SSD Hamiltonian 9) . If we accept this interpretation, the relation between radial quantization [20] and SSD in 2d conformal field theories [1,2] naturally parallels that between R and H. This interpretation offers a nice intuition on somewhat mysterious nature of the continuous spectrum of SSD: it stems from the runaway potential in the CQM case.
Next, we turn our attention to the case c (2) = −1. Since the potential for this case is unbounded below, the system is unstable, no meaningful physical interpretation is apparent. However, the sl(2, R) symmetry of the system enables the following analysis.
First, the generators H, K 0 , D, and R allows another non-trivial identification with the Virasoro subalgebra: The set of operators above satisfies the same commutation relations given in Eq. (13). Since the algebraic structure is the same, the eigenvalues for the operator L ′ 0 should be the same. However, the "Hamiltonian" in question isR, not L ′ 0 . The difference between R and L 0 is the multiplication of the the imaginary unit i. Thus we find that the spectrum ofR is 2i times that of R.
What can we make of a "Hamiltonian" with pure imaginary eigenvalues? Although imaginary eigenvalues appear unphysical, all these eigenvalues take the form 2iE n , where E n represent the eigenvalues of the "physical" Hamiltonian R, as explicitly shown in Eq. (44). If we rewrite t → β/2, the "time" translation operator can be expressed as and clearly corresponds to the thermal density matrix operator 9) See Ref. [19] for a recent discussion on quantization using H. for the original system that quantized with R; the time translation evoked byR yields the thermal density matrix operator. The relation between the density matrix operator ρ andR, is that of the so-called modular Hamiltonian (for example, see [21]) except the extra i factor. Conventionally, the modular Hamiltonian, which is an hermitian operator, is used to construct a unitary operator with additional i factor, but here the i factor is already included. Therefore, simply exponentiating theR yields an unitary operator. Since the modular Hamiltonian is also called the entanglement Hamiltonian [21][22][23][24][25][26][27][28], we infer that this caseR corresponds to the entanglement Hamiltonian. As a matter of fact, Wen, Ryu and Ludwig [12] pointed out in the study of 2d CFT that the deformation of the Hamiltonian, which corresponds tō R from the point of view of symmetry, yields the Rindler Hamiltonian, which is the extreme case of the entanglement Hamiltonian. Since they studied Euclidean field theory, the additional i-factor was absent for their case. At this point, it would be insightful to contemplate the action of sl(2, R). The sl(2, R) algebra is also the Lie algebra of the projective special linear group P SL(2, R) = SL(2, R)/{±} which is apparently a subgroup of SL(2, R). The relation between P SL(2, R) and SL(2, R) is reminiscent of the relation between SO(3) and SU (2). P SL(2, R) naturally acts on the hyperbolic plane H 2 , which is the upper half of the complex plane {z ∈ C; Imz > 0} with the Poincaré metric or the Poincaré disk with the metric The action of P SL(2, R) on H 2 gives the following automorphism: where a, b, c, d ∈ R and ad − bc = 0. The action of P SL(2, R) on the Poincaré disk that corresponds to R, H, andR respectively, is depicted in Fig. 2. Note, in particular, that the Möbius transformation is similar to the transformation above, except that it forms a complex Lie group and is isomorphic to the automorphism of the Riemann sphere Aut( C) rather than to the automorphism of the half plane.
A distinct feature that one can discern from disk, which consists the one-dimensional coordinate "t". The time for R and H respectively, flows in one direction and encircles the entire edge. On the other hand, the flow forR is divided into the two segments on the edge. There, the time flow in one segment covers only half of the entire edge. The situation parallels the case studied by Bisognano and Wichmann [29], where the modular Hamiltonian (automorphism, to be exact) maps the half of the entire space into itself. This resemblance further supports the interpretation presented above thatR corresponds to the modular Hamiltonian, or the entanglement Hamiltonian. Also, one may consider the other half "space" which cannot be reached by the time flow as the space effectively integrated out. This consideration further supports the above mentioned connection to the entanglement Hamiltonian. The observation here would also be useful in the study of SSD for the case of open strings, where the setup of the upper half plane is natural. In summary, we find the same structure in CQM as observed in 2d CFT where the choice of the Hamiltonian leads to radial quantization, the dipolar quantization or SSD, and the entanglement Hamiltonian, respectively. We identify the respective Hamiltonians in CQM using sl(2, R) symmetry. The findings here will offer a simpler setup for the study of SSD and the entanglement Hamiltonians. It would be also interesting to investigate further in the context of the conformal boot strap approach [30] or the recent discussion of the CQM correlation function [31].
H. Kawai, K. Okunishi, S. Ryu, and the participants of the iTHEMS workshop "Workshop on Sine square deformation and related topics, " for fruitful discussions and many suggestions, which greatly contributed to the present work.