Recursive Generation of the Semiclassical Expansion in Arbitrary Dimension

PHYSICAL REVIEW D 103, 085011 (2021)

Recursive generation of the semiclassical expansion in arbitrary dimension

Cihan Pazarbaşı * Physics Department, Boğaziçi University, 34342 Bebek/Istanbul, Turkey

(Received 23 January 2021; accepted 24 March 2021; published 26 April 2021)

We present a recursive procedure, which is based on the small time expansion of the propagator, in order to generate a semiclassical expansion of the quantum action for a quantum mechanical potential in arbitrary dimensions. In the method, we use the spectral information emerges from the singularities of the propagator on the complex t plane, which are handled by the iε prescription and basic complex analysis. This feature allows for generalization to higher dimensions. We illustrate the procedure by providing simple examples in nonrelativistic quantum mechanics.

DOI: 10.1103/PhysRevD.103.085011

I. INTRODUCTION quantum chaos theory,2 it also implies a limitation of the WKB theory itself. From the practical point of view, we can One of the main practical tasks of a theoretical physicist state that despite its many successful applications, WKB is constructing and solving differential or integral equa- theory remains limited to effectively one-dimensional tions. However, it is a well-known fact that exact solutions systems. to those equations are rare and perturbative methods are Another useful method to investigate the classical limit commonly used in many problems. Quantum theories are and its quantum corrections is the path integral. An no different: perturbation methods are at the center of advantage of the path integral method is its adaptability practical calculations to obtain quantum observables. to both relativistic and nonrelativistic theories in arbitrary An important perturbative approach in quantum theories dimensions, suggesting a unified picture for quantum is the semiclassical approximation,1 which assumes ℏ as its theories. However, generating perturbative corrections expansion parameter. It is also a tool to make a connection turns into a computational burden very quickly and between the quantized theory and the underlying classical computing higher order corrections becomes practically system. The relation between the quantized spectrum impossible. This limits the applicability of this beautiful and the classical equations of motion, known as Bohr- approach. Sommerfeld quantization, is older than the quantum theory At this point, it is reasonable to ask for the motivation we know today. Later, this relation was systematically behind computing higher order corrections. Why would used in nonrelativistic quantum mechanics via the WKB only the first few corrections not be enough? An important method. However, soon the limitation of Bohr-Sommerfeld reason is hidden in the resurgent structure of perturbative quantization in the presence of chaotic classical motion was expansions [1]. Resurgence theory states that in most of the realized. (See [2] for a detailed review on the subject.) cases, naive perturbation theory leads to a divergent series While this led to the development of EBK quantization and and, on its own, only provides a partial answer. A more careful investigation leads to the emergence of nonpertur- bative effects hidden in the large order behavior of the *[email protected] 1Throughout this paper, the semiclassical expansion refers to divergent expansion, and this is proposed as a possible way the perturbative series in ℏ, while in the literature the semi- to construct complete solutions. classical expansion sometimes refers to the terms of e−1=ℏ. In fact, Computation of quantum corrections based on the WKB the exponentially suppressed terms, which generally have its own approximation is a well studied subject. In fact, for one- perturbative corrections, correspond to the nonperturbative part dimensional anharmonic oscillators, it has been automa- of the semiclassical expansion, and together with the perturbative series, they form the trans-series structure of the semiclassical tized [4], based on the famous Bender-Wu approach [5]. expansion. (See [1] for a comprehensive discussion on the Similar calculations have been done on other methods subject.) based on the geometric properties of the curve represented by the Hamiltonian of the system. One of them is based on Published by the American Physical Society under the terms of the Picard-Fuchs differential equations, whose solutions the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3. 2For a nice book on the subject, see [3].

2470-0010=2021=103(8)=085011(12) 085011-1 Published by the American Physical Society CIHAN PAZARBAŞI PHYS. REV. D 103, 085011 (2021) correspond to the classical actions of the system [6]. where u represents the elements of the spectrum. Instead of Utilizing the recursion relation, the corresponding differ- dealing with DðuÞ directly, we focus on another spectral ential equation for the quantum corrections were also function, which is the quantum action,4 generated3 [7–10]. Another method is based on the recur- sive nature of the holomorphic anomaly equations of ΓðuÞ¼ln detðu − HÞ¼Tr lnðu − HÞ: ð2:2Þ topological string theory [11,12], which is related to the one-dimensional quantum mechanics in the Nekrasov- Now, the branch point of this new function carries the spectral information. One way to handle the singularity is Shatashvili limit [13]. All of these approaches utilize the −1 recursive dependence of the quantum corrections to the introducing the resolvent GðuÞ¼ðu − HÞ as 0 Z classical term, i.e., ℏ order. This is akin to topological u recursion [14,15], and it allows a systematic generation of ΓðuÞ¼ dzTrGðzÞ; ð2:3Þ quantum corrections with minimal input. u0 As we mentioned above, all of these analyses are done 5 where u0 is an arbitrary regular point of GðzÞ on the for one-dimensional nonrelativistic problems. The main complex z plane. Note that the simple poles of GðzÞ, where motivation of this paper is tackling this problem by the (discrete) spectrum appears, correspond to the branch constructing a general procedure for higher dimensional points of ΓðuÞ as demanded by construction [18]. The quantum mechanical problems that can possibly be gen- information around the branch point can be obtained by eralized to many body theories and quantum field theories. employing the iε prescription [19] [Appendix A] and For these purposes, the investigation of the time translation defining a gap for the action ΓðuÞ as operator will be at the center of this paper. The method is equivalent to a perturbative expansion via path integrals, ΔΓðuÞ¼ΓþðuÞ − Γ−ðuÞ; ð2:4Þ and this similarity supports our main objective. Before examining the propagator itself, in Sec. II,we where we defined the actions in different branches as start with its relation to spectral functions and derive an Z integral representation of the so-called WKB action. Then, uiε ΓðuÞ¼limΓðu iεÞ¼lim dzTrGðzÞ : ð2:5Þ ε→0 ε→0 in Sec. III, which is the main section of this paper, we 0iε discuss the perturbative expansion of the propagator by utilizing a small time expansion and derive the recursion It is well-known that the resolvent approach connects the relation we were looking for. Note that at first, the time classical dynamics and the quantum spectrum of H [20,21]. dependent formulation might appear to have a disadvantage However, in perturbative calculations, it may become for practical purposes despite its applicability to higher impractical. For this reason, it is more convenient to dimensional problems. However, by the construction we introduce its Fourier integral representation, Z describe in Sec. II, in our approach, the spectral information ∞ stems from the singularities of the time propagator on the GðuÞ¼i dteiðu−HÞt; ð2:6Þ complex time plane. This reduces the main computational 0 task to the integration of ordinary integrals and basic where t corresponds to a flow-time parameter conjugate to complex analysis. In Sec. IV, we will apply our method the eigenvalue u. Near the branch cut, ΓðuÞ becomes to anharmonic oscillators in arbitrary dimensions. The Z ∞ numerical results for this part are presented in dt it u iε Appendix C. Finally, in Sec. V, we finish the paper with ΓðuÞ¼−lim e ð ÞTrUðtÞ; ð2:7Þ ε→0 0 t a discussion of our analysis and an outlook to future work. where UðtÞ¼e∓iHt is the propagator,6 which governs the II. SPECTRAL PROBLEM flow generated by H. It is also possible to incorporate the In this section, we will briefly review the spectral analytical continuations into integration contours, problem of a Hermitian operator H acting on a Hilbert space. From elementary linear algebra, we know that the 4Our choice of labeling Γ as the quantum action comes from spectrum of H is given by the zeroes of the Fredholm the usage of the Fredholm determinant in QFT and many-body determinant, i.e., theories to calculate the effective actions (See, e.g., [16]). Its relation with the spectral ζ and Θ functions are also well- known [17]. D u u − H 0; : 5 ð Þ¼detð Þ¼ ð2 1Þ In another perspective, u0 can be introduced to eliminate the Γ Γ u − Γ u infinities by a redefinition of the action as fin ¼ ð Þ ð 0Þ. However, in our construction, we only encounter with the 3Note that the generation of the quantum corrections via singularities that have a physical meaning so there is no need Picard-Fuchs equation are originally done for genus-1 potentials, for any regularization procedure. 6 1 while some extension to higher genus potentials are also recently For simplicity in future calculations, the ℏ factor in the discussed [7]. exponential is canceled by scaling t → ℏt.

085011-2 RECURSIVE GENERATION OF THE SEMICLASSICAL … PHYS. REV. D 103, 085011 (2021) Z ∞iε dt Despite the simplicity of the discussion above, the ΓðuÞ¼−lim eituTrUðtÞ: ð2:8Þ ε→0 0iε t Zassenhauss formula is not a convenient way for practical calculations. Instead, we take a step back and rewrite the In this form, the spectral information arises from the propagator as a time ordered exponential, singularities on the complex t plane, which are intimately Z related to periods of classical orbits [22]. Note that the t UðtÞ¼T exp ∓ i dt0Hðp; xÞ ; ð3:5Þ integrand in (2.8) is already singular at t ¼ 0, which 0 corresponds to stationary classical motion. In Sec. IV, for quantum anharmonic oscillators, we will explain how which simplifies to an ordinary one when H is t indepen- the spectral information for a perturbative sector, which is dent. Note that (3.5) is the solution of related to the stationary classical motion, emerges from this singularity. First, we will continue our discussion with the dUðtÞ i ¼ Hðp; xÞUðtÞ: ð3:6Þ perturbative expansion of TrU and its recursive structure. dt One way to compute (3.5) is by introducing a Fourier III. EXPANSION IN D DIMENSIONS transformation between the canonical variables and – Before describing our recursive scheme for the pertur- eliminate one of them [23 25]. In the following, we will U bative expansion of TrUðtÞ, let us first investigate its present a perturbative expansion for Tr inspired by this general perturbative structure for a Hermitian operator H approach. However, instead of eliminating one of the given in the following form: variables, we will work on the phase space and integrate out x and p after computing the perturbative expansion. Hðp; xÞ¼TðpÞþVðxÞ; ð3:1Þ This approach was initiated in [26,27]; however, in these papers, the recursive structure behind the expansion of the where T and V are operator valued functions of appropri- time-ordered exponential (3.5) and its relation to the x p semiclassical expansion were not mentioned. ately chosen canonical variables and such that they T p form a 2D-dimensional phase space. From the quantum Let us start by separating the ð Þ part as mechanical point of view, H can be considered as a U t e∓itTðpÞU˜ t : generalized Hamiltonian. Moreover, from ordinary QM, ð Þ¼ ð Þð3 7Þ we know that projecting H onto x space, the operator p and rewrite (3.6) as starts acting as a derivative operator and vice versa. From this fact, one can easily deduce dU˜ ðtÞ i ¼ VU˜ ðtÞ; ð3:8Þ dt I ½p; VðxÞ ¼ −iℏ∇xVðxÞ; ½x; TðpÞ ¼ iℏ∇pTðpÞ; ð3:2Þ where we introduced the interaction picture potential,

V e∓itTðpÞV x eitTðpÞ: and the well-known commutation relation between the I ¼ ð Þ canonical variables, With these definitions, UðtÞ is expressed as xi;pj iℏδij: 3:3 Z ½ ¼ ð Þ t ∓iTðpÞt 0 U ðtÞ¼e T exp ∓ i dt VI U 0 The general structure of the perturbative expansion of Z can be examined by using the Zassenhauss formula, X ∓ i n t Yn e∓iTðpÞt ð Þ t T V t …V t : ¼ n! d i f I ð 1Þ I ð nÞg 2 n 0 0 i 1 ∓iT p t ∓itV x t T p ;V x ¼ ¼ UðtÞ¼e ð Þ e ð Þe 4 ½ ð Þ ð Þ ð3:9Þ it3 2 V x ; T p ;V x T p ; T p ;V x × e 3! ð ½ ð Þ ½ ð Þ ð Þþ½ ð Þ ½ ð Þ ð ÞÞ… ð3:4Þ The next step is projecting the operator valued functions 7 Together with (3.2), it is easy to see that the sequence of onto a 2D-dimensional phase space using exponents in (3.4) correspond to a derivative expansion. Besides this, expanding these exponentials, we can get VðxÞjxi¼VðxÞjxi; TðpÞjpi¼TðpÞjpið3:10Þ another expansion which we call coupling expansion. This simple observation shows that a perturbative analysis of and rewriting UðtÞ with an operator H as in (3.1) should be treated as a double expansion. 7See Appendix A for conventions.

085011-3 CIHAN PAZARBAŞI PHYS. REV. D 103, 085011 (2021) Z dDxdDp X ∓ i n which have no physical implications, and they are handled U t e∓iTðpÞt ð Þ Tr ð Þ¼ 2πℏ D n! by a normalization technique. In the following, we propose ð Þ n 0 Z ¼ a method that can be used for practical calculations, and t Yn only infinities we encounter will be related to the physical × dtihxjpihpjT ½VI ðt1Þ…VI ðtnÞjxi: 0 spectrum. We will extract this physical information without i¼1 : a need to normalize. ð3 11Þ We start by making use of the time-ordered exponential in (3.16). It enables us to rewrite (3.6) as This allows us to exchange the commutators with a derivative expansion. In order to do this, we insert an U˜ t X d ð Þ k ˜ identity operator for each VIðtiÞ, i ¼ ℏ WUðtÞ: ð3:17Þ dt k k¼0 hxjpihpjVI ðtiÞ Z Let us write U˜ in an ℏ expansion as well ¼hxjpieiTðpÞti dDx0hpjx0ihx0jVe∓iTðpÞti X ˜ ˜ l Z U ðtÞ¼ Ul ðtÞℏ : D 0 iT p t d x −ip·ðx0−xÞ iT p t l ¼ e ð Þ i e ℏ Vðx0Þhx0je ð Þ i : ð3:12Þ ð2πℏÞD Then, matching orders in (3.17), we get At this point, instead inserting a second identity operator ˜ Xm eiTðpÞti V x0 x0 x dUmðtÞ for , we expand ð Þ around ¼ , i ¼ WU˜ ðtÞ: ð3:18Þ dt l m−l X 1 l¼0 0 0 k Vðx Þ¼ ððx − xÞ · ∇xÞ VðxÞ: ð3:13Þ k! m 0 k¼0 At order ¼ , the solution is Z x0 t This enables us to take integral using integration by parts. ˜ 0 0 ∓iVt U0 ðtÞ¼T exp ∓ i dt W0 ðt Þ ¼ e : ð3:19Þ Then, removing the part we introduced as identity element, 0 we get ˜ −1 X For m ≥ 1, after multiplying (3.18) with ðU0 Þ , we get k hxjpihpjVIðtiÞ¼ ℏ Wk hxjpihpj; ð3:14Þ Z k 0 t ¼ ˜ ˜ 0 ˜ −1 0 0 UmðtÞ¼∓ iU0 ðtÞ dt ðU0 Þ ðt ÞRmðt Þ; ð3:20Þ 0 where where VðkÞ x W ð Þ bk p; ∇ ;t ; k ¼ k! ð p iÞ Xm ˜ RmðtÞ¼ Wl ðtÞUm−lðtÞ: b ðp; ∇p;tiÞ¼i∇p ∇pTðpÞti ∓∇pTðpÞt: ð3:15Þ l¼1

Finally, we can express TrU as a time-ordered exponen- ˜ ˜ Note that each UmðtÞ is written in terms of Ul≤mðtÞ. This tial, makes the recursive behavior of the perturbative expansion Z Z U˜ Dx t X evident. To utilize this recursive behavior, we express m in d 0 k ˜ TrU ðtÞ¼ D T exp ∓ i dt ℏ Wk ; terms of U0 and Wl only as ð2πℏÞ 0 k¼0 X ˜ ˜ ˜ ð3:16Þ UmðtÞ¼U0 ðtÞ Um;kðtÞ; ð3:21Þ k¼1 where Z where … Dp…e∓iTðpÞt: m Z Z h i ¼ d X t t1 ˜ k Um;kðtÞ¼ ð∓ iÞ dt1 dt2… 0 0 α1;…;αk¼1 α …α m ð 1þZ k¼ Þ A. Recursion relation tk−1 t W t …W t : : × d k α1 ð 1Þ αk ð kÞ ð3 22Þ Equation (3.16) is still impractical for perturbative 0 calculations. In addition to that, depending on the functions VðxÞ and TðpÞ, the volume integrals might lead to infinities In (3.22), we have used

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˜ −1 ˜ V x ðU0 Þ Wα U0 ¼ Wα ; compute it around a minimum of ð Þ. For example, in Sec. IV, we will compute the expansion for quantum ∓itV which was possible since U0 ¼ e is p independent. anharmonic oscillators around their harmonic minima. In Finally, let us define the sum of products as those cases, the x integrals will be Gaussian and with a proper analytical continuation of in complex t plane, they Xm lead to finite results. However, this would not be possible if Q W t …W t : : m;k ¼ α1 ð 1Þ αk ð kÞ ð3 23Þ we use the time-ordered exponential in (3.16) directly. Note α1;…;αk¼1 p ðα1þ…αk¼mÞ also that due to the separation in (3.7), the integrals do not need a further treatment to prevent nonphysical infinities. For each m and k, Qm;k can be written as a product of two W 1. Some remarks lower order terms. For example, let us separate α1 from the rest. Then, we get (i) Instead of the definition in (3.7), we can split the X original propagator as Qm;k ¼ Wα Qm−α ;k−1 1 1 U t e∓iVðxÞtU˜ t : : α1 ð Þ¼ ð Þ ð3 28Þ WQ W t Q ¼ 1 m−1;k−1 þ 2 ð 1Þ m−2;k−1 þ This is an equivalent definition, and the only differ- W t Q ence would be the roles of TðpÞ and VðxÞ in the þ m−ðk−1Þð 1Þ k−1;k−1 double expansion. Following the same procedure, mX−kþ1 mX−kþ1 we get the following recursion relation: ¼ Wl Qm−l;k−1 ¼ Ql;1Qm−l;k−1; ð3:24Þ l¼1 l¼1 Z mX−kþ1 t Pn;kðtÞ¼ dt1Rlðt1ÞPn−l;k−1ðtkÞ; ð3:29Þ where we set Qα;0 ¼ δα;0. This is the recursion relation we 0 l¼1 were looking for. Using this recursion relation, we can explicitly express (3.22) as where

m−k 1 Z n Z Z Xþ t X t t1 ˜ ˜ k Um;kðtÞ¼ dt1Wlðt1ÞUm−l;k−1ðt1Þð3:25Þ Pn;kðtÞ¼ ð∓ iÞ dt1 dt2… 0 0 l 1 0 α1;…;αk¼1 ¼ α …α n ð 1þZ k¼ Þ Z tk−1 mX−kþ1 VðlÞ x t t R t …R t : ð Þ l ˜ × d k α1 ð 1Þ αk ð kÞð3 30Þ ¼ dt1b ðp; ∇p;t1ÞUm−l;k−1ðt1Þ: ð3:26Þ 0 l! 0 l¼1 and ˜ ˜ Assuming Um−l;k−1 is already computed, for each Um;k,we lth TðkÞ p only need to compute one order differentiation and one R ð Þ ck x; ∇ ;t ; c x; ∇ ;t k ¼ ð x iÞ ð x iÞ tk integral. This is a crucial point to speed up practical k! calculations. Combining all of these, we write the actions in ¼ i∇x ∇xVðxÞti ∓∇xVðxÞt: ð3:31Þ an ℏ expansion, Γ X Z In this case, the actions becomes ∞iε dt ΓðuÞ¼−lim ℏm eiut Z ε→0 t X ∞iε t m 0 0iε n d iut ¼ Γ ðuÞ¼−lim ℏ e Z D m ε→0 0 iε t x X n¼0 d ∓iVðxÞt ˜ × e U ðtÞ : ð3:27Þ Z n 2πℏ D m;k dDp X ð Þ k¼1 ∓iTðpÞt × e Pn;kðtÞ ; ð2πℏÞD ˜ k¼1 Note that in Um;k, k represents the number of potentials, ð3:32Þ i.e., the order of coupling expansion, while m represents the total number of derivatives acting on these potentials. where In our arrangement, at any order k ≤ m. Higher order terms Z in the coupling expansion comes from the expansion of D ∓iV x t ˜ ∓itV … x…e ð Þ : U0 ¼ e , if the x integral could not be taken directly. h i ¼ d In addition to generating additional terms for the ˜ itV coupling expansion, U0 ¼ e in (3.27) also enables (ii) In both of (3.27) and (3.32), the order ℏ counts the us to obtain finite results for the x integration as long as we number of derivatives. But the difference is in the

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first one, it is the number of derivatives on VðxÞ, (i) In these computations, the operator, while in the latter, it is the one on TðpÞ. This b p; ∇ ;t i∇ pt ∓ pt; difference indicates that (3.27) and (3.32) are differ- ð p iÞ¼ p i ent expressions of same object. However, as long as we do not truncate one of these expansions, results serves as a generator of polynomials in p by coming from both approaches would be equal. acting on both the polynomials generated in the 2 (iii) The recursion relation (3.24) is in the same form ∓ip t lower orders and e 2 . We carried out this with the well-known WKB recursion relation [28]. procedure by using the Nest function in However, as stated before, since we consider the Mathematica contributions of branch points through the t integral, . (ii) Note that the x dependent functions are not we will take x and p integrals directly, and this affected by this procedure. Their multiplication will be our ticket to go to higher dimensions. For x completeness, we will also show the equivalence leads to the polynomials in . (iii) The ti terms in b also form polynomials. They between our method and the standard WKB in one can easily be integrated at each order. Note that dimension in Appendix B. (iv) Finally, note that the recursive behavior is an they will also contribute to the next order in the iteration. intrinsic property of the iterated integrals, which (2) Phase space integrals: At this point, each term in the stem from the time-ordered exponential, and it is expansion is written as a polynomial of p, x and t. T p V x 2 independent of the functions ð Þ and ð Þ. This ∓ip t e 2 indicates the topological nature of the derivative (i) Note that is already in Gaussian form and ∓iV x t expansions, and it is totally consistent with the e ð Þ can be made Gaussian by expanding it conjectured equivalence of topological recursion for small λ. Then, we can integrate out x and p and WKB expansion [14,15]. using the standard Gaussian integrals, Z P ∓it D z2 2n 2n D 2 k 1 D IV. AN EXAMPLE: ANHARMONIC OSCILLATORS I2n ¼ d ze k¼1 z1 …zD IN QUANTUM MECHANICS YD Z it 2 ∓ z 2nk Up to this point, we have constructed a recursive ¼ dzke 2 k zk expansion formula for the quantum action Γ and expressed k¼1 rffiffiffiffiffiffiffiffiffi each term by a number of integrals. In this section, as an n YD 1 ð2nkÞ! 2π illustrative example, we will demonstrate how the spectral ¼ ; ð4:3Þ 2it nk! it information of the D-dimensional quantum anharmonic k¼1 oscillator is obtained. Let us start with setting where we set n1 þ …nD ¼ n and in order to p2 prevent divergences in the zk integrals, the T p : t ð Þ¼ 2 ð4 1Þ analytical continuation of in appropriate directions is assumed. and (ii) For example, at the leading order in the deriva- tive expansion, we get x2 VðxÞ¼ þ λvðxÞ; ð4:2Þ Γm 0ðu; λÞ 2 ¼ Z ∞ iε D dt iut 2π 2 v x ¼ −lim e where ð Þ is a higher degree polynomial. Then, the ε→0 0 iε t it Z quantum actions in (3.27) are written as D X k d x ∓ix2t ð∓ iλtÞ k e 2 v x Z Z × D ð Þ X ∞ iε D D 2πℏ k! dt d xd p ð Þ k¼0 ΓðuÞ¼−lim ℏm eiut ε→0 D Z iut ð0Þ 0iε t ð2πℏÞ ∞iε t e X A λ m¼0 − d 2n ð Þ ; : m ¼ lim D n ð4 4Þ X 2 ε→0 t it x2 ip t 0 iε ðitℏÞ ð Þ ∓ið þλvðxÞÞt ˜ ∓ n¼0 × e 2 Um;kðtÞe 2 : k¼1 ð0Þ where A2n ðλÞ is a polynomial of λ originating We carried out the computations in three separate stages: from the Gaussian integral of the x2n term of (1) Iterated integrals: We first need to compute the vkðxÞ. iterated time integrals using the recursion relation (iii) For the higher order terms, additional contri- in (3.25). butions to both x and p polynomials come from

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the recursive procedure in stage 1. This makes (iii) Same technique can be applied to the higher the general expression more complicated, but it orders, and ΔΓðuÞ can be expressed as is still easy to handle by a computer. t 0 X∞ (iv) Observe that higher order poles at ¼ appear ΔΓ u; λ; ℏ ΔΓ u; λ ℏ2m; : in (4.4). They are critical for us since in our ð Þ¼ 2mð Þ ð4 10Þ m 0 setting they are associated with the spectrum ¼ of H. where each ΔΓ2mðu; λÞ corresponds to a series (3) From singularities to spectrum: The singularity at in u and λ. In Appendix C, we will provide t 0 ¼ is usually handled by zeta function regulari- numerical results for several anharmonic oscil- zation [29]. However, instead we will show that the lators in D ¼ 1, 2, 3 dimensions. basic contour integration techniques together with the iε prescription we mentioned in Sec. II are A. A comment on quantization conditions sufficient. (i) To explain how we handle these singularities, Setting λ ¼ 1 for convenience, (4.10) allows us to let us continue with (4.4). We start with express ΔΓ as a series in u and ℏ. However, as we reviewed introducing a cutoff Λ at the lower limit of in Sec. II, the original spectral quantity is u itself, so it is the t integral. This allows us to express (4.4) as natural to investigate a method for obtaining a perturbative u ΔΓ X series for starting from . In one dimension, this can be 0 Γ u; λ −ℏ−D Að Þ λ −u Dþnlim achieved by imposing the Bohr-Sommerfeld quantization m¼0ð Þ¼ 2n ð Þð Þ ε→0 n¼0 Λ→0 condition for (an)harmonic oscillators. In our formulation, E Λ iε it is expressed as Dþnþ1ð Þ ; : × D n ð4 5Þ ðΛ iεÞ þ 1 ΔΓ u; ℏ 2πi N : : ð Þ¼ þ 2 ð4 11Þ where we used the generalized exponential integral [30], For a simple harmonic oscillator in one dimension, we have Z ∞ e−t E z zα−1 t : : 2πiu αð Þ¼ d tα ð4 6Þ ΔΓ u; ℏ ; : z ð Þ¼ ℏ ð4 12Þ α ∈ N (ii) For , it can be expressed as and it is obvious that (4.11) and (4.12) lead to correct eigenvalues, i.e., u ¼ ℏðN þ 1Þ. For anharmonic cases, we ð−zÞα−1 2 E z E z will have an infinite series in u for each ΔΓ2m in (4.10).In αð Þ¼ α − 1 ! 1ð Þ ð Þ these cases, the perturbative expansion of u is achieved by −z Xα−2 inverting the series in (4.10) [9,31,32]. On the other hand, it e k þ ðα − k − 2Þ!ð−zÞ : appears that a meaningful generalization of the Bohr- ðα − 1Þ! k¼0 Sommerfeld quantization condition to higher dimensions ð4:7Þ remains lacking and further investigation is needed.

The second part of (4.7) is regular at z ¼ 0, V. DISCUSSION AND OUTLOOK while the first part has a branch cut due to the branch points of E1ðzÞ at z ¼ 0 and z ¼ ∞. In this paper, we investigated the recursive nature of the This branch cut leads to derivative expansion of the quantum action and showed how to implement it in practical calculations for quantum 2mπi E1ðze Þ − E1ðzÞ¼−2mπi; m ∈ Z: ð4:8Þ anharmonic oscillators in arbitrary dimensions. In quantum mechanics, the semiclassical expansion, which is repre- Note that to use (4.8) in (4.5), we interpret the sented by a derivative expansion in our language, can also analytical continuation as be obtained via WKB methods in one dimension, or path integrals in arbitrary dimensions. However, our method has Λ − iε ¼ðΛ þ iεÞe2πi: advantages over both methods since perturbative calcula- tions using path integral becomes cumbersome very Then, at the leading order, we have quickly, and the WKB method is only applicable to effectively one-dimensional problems. X ð0Þ Dþn Besides this practical advantage, the method we used 2πi A2n ðλÞu ΔΓ0ðu; λÞ¼ : ð4:9Þ separates the spectral information into two distinct parts. ℏD ðD þ nÞ! n¼0 The first part, which is identified as the recursion relation

085011-7 CIHAN PAZARBAŞI PHYS. REV. D 103, 085011 (2021) and iterated integrals, is universal. It is the same for all Boğaziçi University Research Fund under Grant quantum mechanical systems. Moreover, despite some No. 20B03D1. differences in the details, we expect the same structure to be present in many-body systems and effective quantum field theories as well. This reveals a general relation APPENDIX A: NOTATIONS between the classical action and its quantum corrections AND CONVENTIONS at different orders in a wide range of quantum theories. As e−ix·p=ℏ we mentioned in the Introduction, this relation has been hxjpi¼eix·p=ℏhpjxi¼ ðA1Þ well studied for one-dimensional quantum mechanical ð2πℏÞD models via Picard-Fuchs differential equations and the holomorphic anomaly equation [6–12]. Thus, the current hxjx0i¼δDðx − x0Þhpjp0i¼ð2πℏÞDδDðp − p0ÞðA2Þ paper can also be interpreted as an extension of those Z Z methods to higher dimensions and possibly to more dDp 1 ¼ dDxjxihxj¼ jpihpjðA3Þ complicated theories. ð2πℏÞD On the other hand, the second part, which is identified as Z Z the phase-space integrals, depends on the particular system Dp O Dx x O x d p O p chosen, and therefore, it carries information specific to that Tr ¼ d h j j i¼ 2πℏ D h j j iðA4Þ system. One important piece of information is the divergent ð Þ large order behavior of the semiclassical expansion. Although we postpone examining this to future work, our systematic construction allows an efficient computation APPENDIX B: WKB EXPANSION = of high orders and allows us to examine its hidden non- DERIVATIVE EXPANSION perturbative structure. As we have described, the perturbative spectrum gets Here, for completeness, we compute the first two non- contribution from the singular part of the small t expansion zero terms in the expansions of the one-dimensional T p p2 in (3.27). However, each order in the derivative expansion nonrelativistic quantum mechanics, i.e., ð Þ¼ 2 , for a in (3.27) contains finite contributions as well. Note that general potential VðxÞ. This will show the equivalence their order by order integration leads to a divergent series between the standard WKB approximation and the deriva- (see [33] for an example in the 1 loop effective action in tive expansion in our formalism. In addition to that, we will QFT). Although this was not important in our construction also observe how the “physical” singularities transfer from one could, before taking the t integral, obtain a function by the t integral to the x integral. summing the finite part, and it would be interesting to examine its contribution to the nonperturbative sector of the 1. Leading order spectrum through t integration. At the leading order (m ¼ 0) in one dimension (D ¼ 1), Finally, let us finish with some apparent downsides of the (3.27) simplifies to method we proposed. The first is the lack of expansions related to the nonperturbative sector. In WKB related Z Z ∞ t x p ip2t d d d ∓ iðuiε−VÞt approaches, these expansions are obtained by integrating Γ0 ðEÞ¼−ℏlim e 2 e : ðB1Þ ε→0 t 2πℏ along classically nonallowed paths, but we get the spectral 0 information from the singularity at t ¼ 0 plane, and so no p t non-perturbative term emerges in our calculations. Performing the integral, rotating the integral contour by iπ t e 2 t → However, as the resurgence theory indicates, there should and rescaling uiε−VðxÞ, we get be intimate connection between perturbative and nonper- Z Z turbative sectors. For genus one potentials, this is described ∞ −t pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∓iπ dte 2 ffiffiffiffiffiffiffiffiffi by Matone’s relation [6,34,35]. Adapting this to our Γ0 ðEÞ¼e p dx u iε − VðxÞ: ðB2Þ 0 2πt3 formalism could be useful to understand the emergence of nonperturbative terms, and it can be used to verify the t connection between perturbative and nonperturbative sec- Note that the integral is still divergent at the lower tors in more complicated theories. boundary. Remark that the branch cut information is now carried to points giving u ¼ VðxÞ in x space. Handling the divergence at t ¼ 0 by zeta regularization, we get ACKNOWLEDGMENTS ffiffiffiX I pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The author is grateful to Dieter Van den Bleeken for p ΔΓðEÞ¼ΓþðEÞ − Γ−ðEÞ¼i 2 dx u − VðxÞ; detailed discussions on the subject and comments on the i αi manuscript. He also would like to thank Can Kozçaz for various discussions. The author is supported by the ðB3Þ

085011-8 RECURSIVE GENERATION OF THE SEMICLASSICAL … PHYS. REV. D 103, 085011 (2021) Z Z α ∞ t where the each contour i is taken around the singularities 2 d itðuiεÞ ∓iVt Γm 2ðEÞ¼−limℏ e dxe at u ¼ VðxÞ, i.e., the turning points. Finally, combining ¼ ε→0 t Z 0 with the quantization condition, we get t V00 x t ð Þ b t b t × d 1 h ð 1Þ ð 1Þi 0 2 I Z Z pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 t t1 2 x u − V x 2π N ℏ; ∓ t t V0 x 2 b t b t : d ð Þ ¼ þ ðB4Þ d 1 d 2ð ð ÞÞ h ð 1Þ ð 2Þi 2 0 0

Following the same arguments as for the leading order, we which matches the Bohr-Sommerfeld formula. get I iℏ2 X V00 x V0 x 2 2. Corrections up to O ℏ2 ð Þ ð ð ÞÞ ð Þ ΔΓm 2 ¼ − pffiffiffi dx þ ¼ 2 24 u − V 3=2 32 u − V 5=2 For m ¼ 1, the action is given as i αi ð Þ ð Þ pffiffiffi I 2 X V0 x 2 −iℏ2 ð ð ÞÞ ; Z Z ¼ 26 u − V 5=2 ∞ t x i αi ð Þ d itðuiεÞ d ∓iVt Γm 1ðEÞ¼−limℏ e e ¼ ε→0 t 2πℏ Z 0 t which reproduces the well-known first quantum correction t0 W t0 ; to the WKB approximation. × d h 1ð Þi ðB5Þ 0 APPENDIX C: RESULTS where Here we present the results for several anharmonic oscillators in one, two, and three dimensions. The compu- W V0 x b p 0: tations are done by the implementation of the recursive h 1i ¼h ð Þ ð Þi ¼ formula in (3.25) to Mathematica. The results for one dimension match exactly with the ones in the literature Thus, at order ℏ, there is no contribution to the action. [10,11,32], and for two and three dimensions, to our Similarly, for m ¼ 2, we have knowledge, these results appear for the first time.

x2 2 Cubic oscillator: VðxÞ¼ 2 þ λx1x One dimension:

15u2λ2 1155u3λ4 255255u4λ6 66927861u5λ8 ΔΓ u u 0ð Þ¼ þ 4 þ 16 þ 128 þ 1024 7λ2 1365uλ4 285285u2λ6 121246125u3λ8 51869092275u4λ10 ΔΓ u 2ð Þ¼ 16 þ 64 þ 256 þ 2048 þ 16384 119119λ6 156165009uλ8 67931778915u2λ10 24568660040925u3λ12 ΔΓ u 4ð Þ¼ 2048 þ 16384 þ 65536 þ 262144

Two dimensions:

u2 ΔΓ u 2u3λ2 30u4λ4 672u5λ6 18480u6λ8 0ð Þ¼ 2 þ þ þ þ 1 uλ2 1120u3λ6 ΔΓ u − 10u2λ4 15400u4λ8 2ð Þ¼ 12 þ 3 þ þ 3 þ 11λ4 92uλ6 ΔΓ u 3240u2λ8 265408u3λ10 19347328u4λ12 4ð Þ¼ 105 þ 3 þ þ þ

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Three dimensions:

u3 35u4λ2 3003u5λ4 46189u6λ6 26558675u7λ8 ΔΓ u 0ð Þ¼ 6 þ 48 þ 320 þ 256 þ 6144 u 5u2λ2 77u3λ4 36465u4λ6 37182145u5λ8 ΔΓ u − − 2ð Þ¼ 8 96 þ 128 þ 1024 þ 24576 2369λ2 2869uλ4 265551u2λ6 30808063u3λ8 ΔΓ u − − − 4ð Þ¼ 80640 5120 28672 þ 294912

x2 2 2 Quartic oscillator: VðxÞ¼ 2 þ λðx Þ One dimension:

3u2λ 35u3λ2 1155u4λ3 45045u5λ4 969969u6λ5 ΔΓ u u − − − 0ð Þ¼ 2 þ 4 16 þ 64 128 3λ 85uλ2 2625u2λ3 165165u3λ4 10465455u4λ5 ΔΓ u − − − 2ð Þ¼ 8 þ 16 32 þ 128 512 1995λ3 400785uλ4 26249223u2λ5 1419711293u3λ6 ΔΓ u − − 4ð Þ¼ 256 þ 1024 2048 þ 4096

Two dimensions:

u2 4u3λ 1792u6λ4 ΔΓ u − 8u4λ2 − 64u5λ3 0ð Þ¼ 2 3 þ þ 3 1 2uλ 320u3λ3 4480u4λ4 ΔΓ u − − 8u2λ2 − 2ð Þ¼ 12 3 þ 3 þ 3 4λ2 2752uλ3 17536u2λ4 192512u3λ5 ΔΓ u − − 489472u4λ6 4ð Þ¼ 9 105 þ 21 9 þ

Three dimensions:

u3 5u4λ 63u5λ2 1001u6λ3 36465u7λ4 ΔΓ u − − 0ð Þ¼ 6 8 þ 16 32 þ 128 u 5u2λ 455u3λ2 8085u4λ3 435435u5λ4 ΔΓ u − − − 2ð Þ¼ 8 16 þ 96 128 þ 512 269λ 523uλ2 67479u2λ3 7501923u3λ4 136473909u4λ5 ΔΓ u − − 4ð Þ¼4480 þ 1280 2560 þ 10240 8192

x2 2 2 Quintic oscillator: VðxÞ¼ 2 þ λx1ðx Þ One dimension:

315u4λ2 692835u7λ4 9704539845u10λ6 166966608033225u13λ8 ΔΓ u u 0ð Þ¼ þ 16 þ 128 þ 4096 þ 131072 1085u2λ2 15570555u5λ4 456782651325u8λ6 6734319857340075u11λ8 ΔΓ u 2ð Þ¼ 32 þ 512 þ 16384 þ 262144 1107λ2 96201105u3λ4 4140194663605u6λ6 489884540580510075u9λ8 ΔΓ u 4ð Þ¼ 256 þ 2048 þ 32768 þ 2097152

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Two dimensions:

u2 ΔΓ u 8u5λ2 1440u8λ4 465920u11λ6 198451200u14λ8 0ð Þ¼ 2 þ þ þ þ 1 56u3λ2 17937920u9λ6 ΔΓ u − 9408u6λ4 4213780480u12λ8 2ð Þ¼ 12 þ 3 þ þ 3 þ 44uλ2 235023360u7λ6 222716628992u10λ8 ΔΓ u 20704u4λ4 4ð Þ¼ 7 þ þ 7 þ 5

Three dimensions:

u3 77u6λ2 323323u9λ4 1302340845u12λ6 7244053893505u15λ8 ΔΓ u 0ð Þ¼ 6 þ 32 þ 1024 þ 16384 þ 262144 u 805u4λ2 2263261u7λ4 34659070875u10λ6 624578793013175u13λ8 ΔΓ u − 2ð Þ¼ 8 þ 128 þ 1024 þ 32768 þ 1048576 83819u2λ2 1121359525u5λ4 1891956467895u8λ6 91473021008360675u11λ8 ΔΓ u 4ð Þ¼ 23040 þ 172032 þ 262144 þ 12582912

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