An Alternative Canonical Approach to the Ghost Problem in a Complexified Extension of the Pais-Uhlenbeck Oscillator

Our purpose in this paper is to analyze the Pais-Uhlenbeck (PU) oscillator using complex canonical transformations. We show that starting from a Lagrangian approach we obtain a transformation that makes the extended PU oscillator, with unequal frequencies, to be equivalent to two standard second ord...

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Опубліковано в: :	Symmetry, Integrability and Geometry: Methods and Applications
Дата:	2009
Автори:	Déctor, A., Morales-Técotl, H.A., Urrutia, L.F., Vergara, J.D.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут математики НАН України 2009
Онлайн доступ:	https://nasplib.isofts.kiev.ua/handle/123456789/149126
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Цитувати:	An Alternative Canonical Approach to the Ghost Problem in a Complexified Extension of the Pais-Uhlenbeck Oscillator / A. Déctor, H.A. Morales-Técotl, L.F. Urrutia, J.D. Vergara // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 43 назв. — англ.

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spelling	Déctor, A. Morales-Técotl, H.A. Urrutia, L.F. Vergara, J.D. 2019-02-19T17:33:37Z 2019-02-19T17:33:37Z 2009 An Alternative Canonical Approach to the Ghost Problem in a Complexified Extension of the Pais-Uhlenbeck Oscillator / A. Déctor, H.A. Morales-Técotl, L.F. Urrutia, J.D. Vergara // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 43 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 70H15; 70H50; 81S10 https://nasplib.isofts.kiev.ua/handle/123456789/149126 Our purpose in this paper is to analyze the Pais-Uhlenbeck (PU) oscillator using complex canonical transformations. We show that starting from a Lagrangian approach we obtain a transformation that makes the extended PU oscillator, with unequal frequencies, to be equivalent to two standard second order oscillators which have the original number of degrees of freedom. Such extension is provided by adding a total time derivative to the PU Lagrangian together with a complexification of the original variables further subjected to reality conditions in order to maintain the required number of degrees of freedom. The analysis is accomplished at both the classical and quantum levels. Remarkably, at the quantum level the negative norm states are eliminated, as well as the problems of unbounded below energy and non-unitary time evolution. We illustrate the idea of our approach by eliminating the negative norm states in a complex oscillator. Next, we extend the procedure to the Pais-Uhlenbeck oscillator. The corresponding quantum propagators are calculated using Schwinger's quantum action principle. We also discuss the equal frequency case at the classical level. This paper is a contribution to the Proceedings of the VIIth Workshop “Quantum Physics with NonHermitian Operators” (June 29 – July 11, 2008, Benasque, Spain). This work was partially supported by the following grants: CONACyT-SEP 51132F, CONACyTSEP 47211-F, CONACyT-SEP 55310, DGAPA-UNAM IN109107 and a CONACyT sabbatical grant to HAMT. AD wishes also to acknowledge support from CONACyT. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications An Alternative Canonical Approach to the Ghost Problem in a Complexified Extension of the Pais-Uhlenbeck Oscillator Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	An Alternative Canonical Approach to the Ghost Problem in a Complexified Extension of the Pais-Uhlenbeck Oscillator
spellingShingle	An Alternative Canonical Approach to the Ghost Problem in a Complexified Extension of the Pais-Uhlenbeck Oscillator Déctor, A. Morales-Técotl, H.A. Urrutia, L.F. Vergara, J.D.
title_short	An Alternative Canonical Approach to the Ghost Problem in a Complexified Extension of the Pais-Uhlenbeck Oscillator
title_full	An Alternative Canonical Approach to the Ghost Problem in a Complexified Extension of the Pais-Uhlenbeck Oscillator
title_fullStr	An Alternative Canonical Approach to the Ghost Problem in a Complexified Extension of the Pais-Uhlenbeck Oscillator
title_full_unstemmed	An Alternative Canonical Approach to the Ghost Problem in a Complexified Extension of the Pais-Uhlenbeck Oscillator
title_sort	alternative canonical approach to the ghost problem in a complexified extension of the pais-uhlenbeck oscillator
author	Déctor, A. Morales-Técotl, H.A. Urrutia, L.F. Vergara, J.D.
author_facet	Déctor, A. Morales-Técotl, H.A. Urrutia, L.F. Vergara, J.D.
publishDate	2009
language	English
container_title	Symmetry, Integrability and Geometry: Methods and Applications
publisher	Інститут математики НАН України
format	Article
description	Our purpose in this paper is to analyze the Pais-Uhlenbeck (PU) oscillator using complex canonical transformations. We show that starting from a Lagrangian approach we obtain a transformation that makes the extended PU oscillator, with unequal frequencies, to be equivalent to two standard second order oscillators which have the original number of degrees of freedom. Such extension is provided by adding a total time derivative to the PU Lagrangian together with a complexification of the original variables further subjected to reality conditions in order to maintain the required number of degrees of freedom. The analysis is accomplished at both the classical and quantum levels. Remarkably, at the quantum level the negative norm states are eliminated, as well as the problems of unbounded below energy and non-unitary time evolution. We illustrate the idea of our approach by eliminating the negative norm states in a complex oscillator. Next, we extend the procedure to the Pais-Uhlenbeck oscillator. The corresponding quantum propagators are calculated using Schwinger's quantum action principle. We also discuss the equal frequency case at the classical level.
issn	1815-0659
url	https://nasplib.isofts.kiev.ua/handle/123456789/149126
citation_txt	An Alternative Canonical Approach to the Ghost Problem in a Complexified Extension of the Pais-Uhlenbeck Oscillator / A. Déctor, H.A. Morales-Técotl, L.F. Urrutia, J.D. Vergara // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 43 назв. — англ.
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 5 (2009), 053, 22 pages An Alternative Canonical Approach to the Ghost Problem in a Complexified Extension of the Pais–Uhlenbeck Oscillator? A. DÉCTOR †, H.A. MORALES-TÉCOTL †‡, L.F. URRUTIA † and J.D. VERGARA † † Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, A. Postal 70-543, México D.F., México E-mail: dector@nucleares.unam.mx, urrutia@nucleares.unam.mx, vergara@nucleares.unam.mx ‡ Departamento de F́ısica, Universidad Autónoma Metropolitana Iztapalapa, San Rafael Atlixco 186, Col. Vicentina, CP 09340, México D.F., México E-mail: hugo@xanum.uam.mx Received November 14, 2008, in final form April 22, 2009; Published online May 05, 2009 doi:10.3842/SIGMA.2009.053 Abstract. Our purpose in this paper is to analyze the Pais–Uhlenbeck (PU) oscillator using complex canonical transformations. We show that starting from a Lagrangian approach we obtain a transformation that makes the extended PU oscillator, with unequal frequencies, to be equivalent to two standard second order oscillators which have the original number of degrees of freedom. Such extension is provided by adding a total time derivative to the PU Lagrangian together with a complexification of the original variables further subjected to reality conditions in order to maintain the required number of degrees of freedom. The analysis is accomplished at both the classical and quantum levels. Remarkably, at the quantum level the negative norm states are eliminated, as well as the problems of unbounded below energy and non-unitary time evolution. We illustrate the idea of our approach by eliminating the negative norm states in a complex oscillator. Next, we extend the procedure to the Pais–Uhlenbeck oscillator. The corresponding quantum propagators are calculated using Schwinger’s quantum action principle. We also discuss the equal frequency case at the classical level. Key words: quantum canonical transformations; higher order derivative models 2000 Mathematics Subject Classification: 70H15; 70H50; 81S10 1 Introduction Systems with higher order time derivatives (HOTD) have been studied with increasing interest because they appear in many important physical problems. In first place these systems were considered to improve the divergent ultraviolet behavior of some quantum field theories [1]. However, their energy turned out not bounded from below [2], then involving ghosts and making the theory non-unitary [3]. In spite of such drawbacks, higher order derivative theories were studied to learn on their improved renormalization properties. Even a renormalizable higher order quantum gravity theory was advanced along these lines [4] and the unitarity of a lattice form was studied in [5]. See also [6, 7, 8, 9, 10, 11] and references therein for other examples. As for dealing with ghosts, attempts can be classified according to whether the approach is perturbative [12, 13, 14] or not [15, 6, 8], but a definite answer is yet to be found. ?This paper is a contribution to the Proceedings of the VIIth Workshop “Quantum Physics with Non- Hermitian Operators” (June 29 – July 11, 2008, Benasque, Spain). The full collection is available at http://www.emis.de/journals/SIGMA/PHHQP2008.html mailto:dector@nucleares.unam.mx mailto:urrutia@nucleares.unam.mx mailto:vergara@nucleares.unam.mx mailto:hugo@xanum.uam.mx http://dx.doi.org/10.3842/SIGMA.2009.053 http://www.emis.de/journals/SIGMA/PHHQP2008.html 2 A. Déctor, H.A. Morales-Técotl, L.F. Urrutia and J.D. Vergara Since single particle quantum mechanics can be seen, in the free field or weak coupling limit, as a mini-superspace sector of quantum field theory where the spatial degrees of freedom have been frozen, it is suggestive to test new ideas on HOTD theories, by using quantum mechanics as a laboratory. The model example to do so is the Pais–Uhlenbeck (PU) oscillator [2], which consists of a one dimensional harmonic oscillator Lagrangian plus a term quadratic in acceleration. A standard quantum treatment of its degrees of freedom gives rise to a spectrum not bounded from below since it consists of the difference of two quantum harmonic oscillators spectra [2]. An interesting solution to such difficulty has been recently proposed in [16, 17, 18] (see also [19, 20, 21, 22] for a different approach) by finding a quantum transformation which gives its Hamiltonian a non-Hermitian, PT symmetric form. In this approach a technique has been developed to obtain a physical inner product and thus a Hilbert space [23]. The aim of this paper is to show that it is possible to tackle the problem of quantizing an extension of the PU oscillator within a Lagrangian and a canonical ormulation, using complex canonical transformations and non-Hermitian variables together with reality conditions. This approach is motivated by a previous proposal to build complex canonical variables for non- perturbative quantum canonical general relativity [24, 25], which achieved the nontrivial task of making polynomial the constraints of general relativity, whereas the so called reality conditions to be fulfilled by the non-Hermitian variables were proposed to determine the inner product of Hilbert space. For canonical general relativity such reality conditions imply the metric of space and its rate of change are real quantities. Some effort was devoted to exploit these complex variables (see for instance [26, 27, 28]) however the use of real variables allowed finally for important progress [29, 30]. The paper is organized as follows. In Section 2 we review the essential aspects of the general construction with special emphasis on the boundary conditions. In Section 3 we use as a toy model a complexified harmonic oscillator [25]. Although this is not a higher-order derivative problem it exhibits negative norm states. We show that using the reality conditions we can eliminate them. Furthermore, we evaluate the Green’s function of the system using the Schwinger quantum action principle and the path integral method, finding that both of these procedures lead to the same result. In Section 4 we combine the results of the previous sections in order to make a first attempt to extend the formalism to HOTD theories, successfully taking as an example the Pais–Uhlenbeck oscillator, which can be treated in a similar way as the complexified harmonic oscillator. In addition we perform the canonical analysis in an extended phase space in order to compare the cases of PU with equal and unequal frequencies. We observe that in the case of equal frequencies, the Lagrangian transformation induces a non canonical one which, at the quantum level, will lead to a non diagonalizable Hamiltonian. Finally, concluding remarks are presented in Section 5. Unless otherwise stated we use units in which ~ = 1. 2 General considerations Let us consider a system whose Lagrangian contains up to the n-th order time derivative in a quadratic form. Written in normal coordinates this Lagrangian will have the form, L(x, ẋ, . . . , x(n)) = ∑ i αi ( x(i) )2 − V (x), i = 1, . . . , n, (2.1) with x(k) = dkx dtk and the αi ∈ R being constants. We are assuming that under the transformation to normal coordinates we diagonalize the kinetic term and that the potential only depends on the coordinates. This Lagrangian yields an equation of motion of order (2n) n∑ k=0 (−1)k dk dtk ∂L ∂x(k) = 0. Higher Order Time Derivative Models 3 Assuming x and hence L to be real leads to the well known problems of the higher-order derivative models. So, as a first step in our extension we consider the analytic continuation of the variable x to the complex plane. Next we assume that the Lagrangian (2.1) plus a total time derivative will be real. This certainly does not alter the classical equations of motion, but modifies the Hamiltonian formulation by the addition of new terms in the momenta. Now, the essential points of our procedure are the following two i) we assume that there exists a nonlocal transformation from the Lagrangian (2.1) plus a total time derivative to a second order real Lagrangian with n independent configuration variables ξi. Specifically we assume the complexified description can be related to an alternative one in terms of a real Lagrangian Lξ ( ξ1, ξ̇1, . . . , ξn, ξ̇n ) through L ( x, ẋ, . . . , x(n) ) + df dt = Lξ ( ξ1, ξ̇1, . . . , ξn, ξ̇n ) , (2.2) f = f ( x, . . . , x(n) ) , (2.3) ξi = ξi ( x, . . . , x(n) ) , i = 1, . . . , n. (2.4) Here ξi, i = 1, . . . , n are real coordinates and only their first order time derivatives ξ̇i enter in Lξ in a quadratic form. In general f will be complex. ii) The second point is that the transformation (2.4) will impose reality conditions on the x(k), with k = 0, . . . , n. This implies, in particular, that terms in the potential of higher order than quadratic must be real functions of the variables, i.e., under the analytic continuation, we must consider only the real part of the interaction, for example powers like x4 should be replaced by (xx†)2. This is not necessary for the quadratic terms, since our transformation incorporates them properly. We must notice that strictly speaking, our system is different from the original one, since we have analytically continued the x variable. However, the reality conditions allow us to construct a properly well defined problem, with the same number of the original degrees of freedom together with the same classical equation for the original variable x. 2.1 Canonical transformation and boundary conditions One of the interesting properties of our nonlocal transformation (2.4) is that in the Hamiltonian– Ostrogradsky formalism [31], this transformation corresponds to a local canonical transforma- tion. To see this, we observe that in this formalism the corresponding phase space will be of dimension 2n, so the configuration space is extended to Q0 = x, Q1 = ẋ, . . . , Qn−1 = x(n−1), together with the momenta Πi = n−i−1∑ j=0 ( − d dt )j ∂L ∂q(i+j+1) , i = 0, . . . , n− 1. (2.5) Thus, the transformation (2.4) only depends on the indicated phase space variables, ξi ( x, . . . , x(n) ) = ξi(Q0, Q1, . . . , Qn−1,Πn−1), (2.6) in which only the last momentum Πn−1 appears. We also have the corresponding transformation for the momenta, Pi = Pi(Q0, Q1, . . . , Qn−1,Π0, . . . ,Πn−1), (2.7) given directly from the generating function (2.3). The transformations (2.6), (2.7) will be canoni- cal in an extended phase space because the Lagrangians L and Lξ differ only by a total derivative that depends on the coordinates and momenta. We must remark that the transformations (2.6), 4 A. Déctor, H.A. Morales-Técotl, L.F. Urrutia and J.D. Vergara (2.7) should be complex, because an analytic continuation of the original variables was required. However, these transformations define exactly the reality conditions, since we enforce that the 2n variables (ξi, Pi) will be real and, in consequence, also define the integration contour that must be used to quantize the system. To build the explicit form of the canonical transformation that we are using, we consider the equivalent of the expression (2.2) in Hamiltonian form, ΠiQ̇ i −H(Q,Π) + df(Q,Π) dt = Piξ̇ i −H(ξ, P ). (2.8) From the comparison of independent terms we find ∂f ∂Qi = Pj ∂ξj ∂Qi −Πi, ∂f ∂Πi = Pj ∂ξj ∂Πi , H = H. (2.9) These equations prescribe the 2n partial derivatives of the generating function f(Q,Π) and the Hamiltonians are equal due to the fact that the transformation is time independent. Now, to analyze the boundary conditions that are imposed by our complex canonical trans- formation, we make the variation of the left hand side of expression (2.8) and we observe that the boundary condition is associated with the total derivative d dt (( Πi + ∂f ∂Qi ) δQi + ∂f ∂Πi δΠi ) , which dictates the appropriate combinations of the variations δQi and δΠi to be fixed at the boundaries. Using the equations (2.9) we can show that the above expression is indeed d dt ( Piδξ i ) , in terms of the variation of the ξi. So, the variables that we need to fix on the boundary are those functions of Qi and Pi defined by the ξi’s. 3 Reality conditions and the complex oscillator Let us start by reviewing the example of a modified harmonic oscillator subject to a complex canonical transformation [24, 25]. This has also been studied in a form that exploits the PT symmetry in the framework of the non-Hermitian Hamiltonian approach [23, 32, 33, 34]. The strategy is based on the observation that the complex Lagrangian LC = q̇2 2 − q2 2 − iεqq̇, where ε is a real parameter, becomes the one corresponding to the harmonic oscillator after adding to it the total time derivative dfC dt , fC = iε q 2 2 . In the Hamiltonian description the canonical momentum is p = ∂LC ∂q̇ = q̇ − iεq, so p ∈ C and should be quantized as a non-Hermitian operator. 3.1 Hilbert space and reality conditions The quantum Hamiltonian becomes ĤC = p̂2 2 + q̂2 2 − 1 2 ε2q̂2 + iε 2 {p̂, q̂} , (3.1) Higher Order Time Derivative Models 5 where {p̂, q̂} = p̂q̂ + q̂p̂ so that a symmetric ordering is selected. Upon the canonical transfor- mation p̂ = P̂ − iεQ̂, q̂ = Q̂, with P̂ = dQ̂ dt , the Hamiltonian (3.1) becomes the standard one of the harmonic oscillator ĤHO = P̂ 2 2 + Q̂2 2 , which is Hermitian whenever Q̂ and P̂ are. The coordinate representation q̂ψ(q) = qψ(q), p̂ψ(q) = −idψ dq (q), leads to a non-Hermitian form of the Hamiltonian (3.1) with the usual scalar product1. The corresponding Schrödinger equation becomes ψ′′(q)− 2εqψ′(q)− ( (1− ε2)q2 + ε− 2E ) ψ(q) = 0. Hence the eigenvalue problem for (3.1) can be related to that of the harmonic oscillator by using the wave functions of the latter ϕn = Nne− q2 2 Hn(q) and defining ψn(q) := e εq2 2 ϕn, En = n+ 1 2 , n = 0, 1, 2 . . . . (3.2) A tedious but otherwise direct calculation shows that the eigenfunctions ψn do not have positive norm in the Hilbert space H0 = L2(R, dq). However, the classical role of fC as the generator of a canonical transformation motivates the introduction of a quantum transformation such that (see for example [35, 36] for a general discussion) ĤHO = e−ε q2 2 ĤCeε q2 2 . (3.3) This non-unitary canonical transformation changes the measure dq to dµ = e−εq 2 dq. Now the Hilbert space H = L2(R, dµ) ensures (3.2) have positive norm since 〈n\|m〉µ := ∫ dqe−εq 2 ψ∗nψm = ∫ dqϕ∗nϕm =: 〈n\|m〉q = δmn, where subscripts denote the appropriate measure for the correspondent Hilbert space. Moreover, the reality conditions q̂† = q̂, p̂† = p̂+ 2iεq̂ (3.4) are automatically implemented in H, where (3.1) is Hermitian [24]. It is illuminating to consider the Green’s functions corresponding, respectively, to ĤC and ĤHO. We recall they can be defined as GεC(q2, q1, E) := 〈q2\| 1 E − ĤC \|q1〉, (3.5) GHO(q2, q1, E) := 〈q2\| 1 E − ĤHO \|q1〉. (3.6) Now on account of (3.3), we can rearrange (3.5) to relate it with (3.6) as follows GεC(q2, q1, E) = 〈q2\| 1 E − Ô−1 ε ĤHOÔε \|q1〉 = 〈q2\|Ô−1 ε 1 E − ĤHO Ôε\|q1〉 = e+ε( q2 2 2 − q2 1 2 )GHO(q2, q1, E), Ôε := e−ε q̂2 2 , (3.7) where in the final line we use that q̂ acts diagonally on the basis \|q〉 and its dual. Thus the stan- dard harmonic oscillator Green’s function, GHO(q2, q1, E), is obtained from that corresponding to the complex oscillator, GεC(q2, q1, E), upon multiplying the former by eε ( q2 2 2 − q2 1 2 ) . 1Using p̂ = −i d dq + iεq would yield the usual Hamiltonian for the harmonic oscillator which, however, requires a modified inner product to be Hermitian. 6 A. Déctor, H.A. Morales-Técotl, L.F. Urrutia and J.D. Vergara 3.2 The Schwinger quantum action principle We can use Schwinger’s quantum action principle to derive some dynamical properties of the system [37]. For instance, we can calculate the propagator 〈q2, t2\|q1, t1〉. We will work in the Heisenberg picture, where we adopt the notation 〈q2, t2\|q(t2) = q2〈q2, t2\|, q(t1)\|q1, t1〉 = q1\|q1, t1〉, (3.8) 〈p∗2, t2\|p†(t2) = p∗2〈p∗2, t2\|, p(t1)\|p1, t1〉 = p1\|p1, t1〉, (3.9) To achieve our purpose, we first calculate the total variation of 〈q2, t2\|q1, t1〉, given by δ〈q2, t2\|q1, t1〉 = i〈q2, t2\|Gq2 −Gq1 −HδT \|q1, t1〉, where T = t2 − t1, and the generators Gqi are given by Gq2 = p(t2)δq2, Gq1 = p(t1)δq1. Starting from the Hamiltonian operator (3.1), we get the following Heisenberg operator equations q̇(t) = p(t) + iεq(t), ṗ(t) = −(1− ε2)q(t)− iεp(t). Fixing the final and initial conditions at t2 and t1 as q (t2) and q (t1), we obtain the solutions q(t) = A cos(t) +B sin(t), (3.10) p(t) = (B − iεA) cos(t)− (iεB +A) sin(t), (3.11) where the coefficients A and B are given by A = 1 sin (T ) (q(t1) sin (t2)− q(t2) sin (t1)) , B = 1 sin (T ) (q(t2) cos (t1)− q(t1) cos (t2)) . In this way, we can express p(t2) and p(t1) in terms of the initial and final conditions q(t1) and q(t2), getting p(t1) = 1 sin(T ) ( −(iε sin(T ) + cos(T ))q(t1) + q(t2) ) , p(t2) = 1 sin(T ) (−(iε sin(T )− cos(T ))q(t2)− q(t1)). With this relations, one can also calculate the commutator [q(t1), q(t2)] = i sin (T ) . Using all of the above, we can write the Hamiltonian as H = 1 2 sin2(T ) ( q2(t1) + q2(t2)− 2 cos(T )q(t2)q(t1) ) − i 2 cos(T ) sin(T ) . Finally, from the equations (3.8), (3.9) we get: δ〈q2, t2\|q1, t1〉 = i〈q2, t2\| {{ 1 sin(T ) (cos(T )q2 − q1)− iεq2 } δq2 Higher Order Time Derivative Models 7 + { 1 sin(T ) (cos(T )q1 − q2) + iεq1 } δq1 − { 1 2 sin2(T ) ( q21 + q22 − 2 cos(T )q2q1 ) − i 2 cos(T ) sin(T ) } δT } \|q1, t1〉 = i〈q2, t2\|δ { 1 2 sin(T ) ( (q22 + q21) cos(T )− 2q2q1 ) − iε 2 (q22 − q21)− i ln ( 1√ sin(T ) )} \|q1, t1〉. Integrating we obtain 〈q2, t2\|q1, t1〉 = 1√ 2πi sin(T ) exp { i 2 sin(T ) { (q22 + q21) cos(T )− 2q2q1 }} × exp { ε 2 (q22 − q21) } , (3.12) which is consistent with equation (3.7). Using the same procedure we now wish to obtain the quantity 〈p∗2, t2\|p1, t1〉. Again, we begin by calculating its variation δ〈p∗2, t2\|p1, t1〉 = 〈p∗2, t2\|Gp∗2 −Gp1 −HδT \|p1, t1〉, where now Gp∗2 = −q(t2)δp∗2, Gp1 = q(t1)δp1. We now need to fix the initial and final conditions at t1 and t2 as p(t1) and p†(t2). Thus, again we get (3.10) and (3.11) as solutions to the Heisenberg equations, but now with coefficients A and B given by A = 1 ((ε2 + 1) sin(T )− 2iε cos(T )) × {( cos(t2) + iε sin(t2) ) p(t1)− (cos(t1)− iε sin(t1)) p†(t2) } , B = 1 ((ε2 + 1) sin(T )− 2iε cos(T )) × {( sin(t2)− iε cos(t2) ) p(t1)− (iε cos(t1) + sin(t1)) p†(t2) } . In this way, we can express q(t1) and q(t2) in terms of the initial and final conditions p(t1) and p†(t2). The result is q(t1) = 1 ((ε2 + 1) sin(T )− 2iε cos(T )) {( cos(T ) + iε sin(T ) ) p(t1)− p†(t2) } , q(t2) = 1 ((ε2 + 1) sin(T )− 2iε cos(T )) { p(t1)− (cos(T ) + iε sin(T )) p†(t2) } . One can also calculate the commutator[ p(t1), p†(t2) ] = i (( ε2 + 1 ) sin(T )− 2iε cos(T ) ) . Using all of the above, we find the following expression for H in terms of the operators p†(t1) and p(t0) H = 1 2 ((ε2 + 1) sin(T )− 2iε cos(T ))2 8 A. Déctor, H.A. Morales-Técotl, L.F. Urrutia and J.D. Vergara × { (1− ε2)(p2(t1) + p† 2(t2))− 2 ( (ε2 + 1) cos(T ) + 2iε sin(T ) ) p†(t2)p(t1) } − i 2 ( (ε2 + 1) cos(T ) + 2iε sin(T ) ) ((ε2 + 1) sin(T )− 2iε cos(T )) . Since p†(t2) and p(t1) act according to equation (3.9) we get δ〈p∗2, t2\|p1, t1〉 = i〈p∗2, t2\| { 1 ((ε2 + 1) sin(T )− 2iε cos(T )) × { ((cos(T ) + iε sin(T ))p∗2 − p1) δp∗2 + ((cos(T ) + iε sin(T ))p1 − p∗2) δp1 } − ( 1 2 ((ε2 + 1) sin(T )− 2iε cos(T ))2 { (1− ε2)(p2 1 + p∗ 2 2 )− 2((ε2 + 1) cos(T ) + 2iε sin(T ))p∗2p1 } − i 2 ( (ε2 + 1) cos(T ) + 2iε sin(T ) ) ((ε2 + 1) sin(T ) + 2iε cos(T )) ) δT } \|p1, t1〉 = i〈p∗2, t2\|δ { 1 2 ((ε2 + 1) sin(T )− 2iε cos(T )) { (cos(T ) + iε sin(T )) (p∗ 2 2 + p2 1)− 2p∗2p1 } − i ln ( 1√ (ε2 + 1) sin(T )− 2iε cos(T ) )} \|p1, t1〉. Integration produces 〈p∗2, t2\|p1, t1〉 = C√ (ε2 + 1) sin(T )− 2iε cos(T ) × exp { i 2 ( (cos(T ) + iε sin(T ))(p∗ 2 2 + p2 1)− 2p∗2p1 ) ((ε2 + 1) sin(T )− 2iε cos(T )) } , where C is a normalization constant. We note that lim t2→t1 〈p∗2, t2\|p1, t1〉 = C √ − 1 2iε exp { − 1 4ε (p1 − p∗2) 2 } . Furthermore, in the limit ε→ 0, all eigenvalues p cease to be complex, so in the double limit we must have lim ε→0 〈p∗2, t2 → t1\|p1, t1〉 = δ (p2 − p1) . Fixing the constant C accordingly we obtain 〈p∗2, t2\|p1, t1〉 = 1√ 2πi 1√ (ε2 + 1) sin(T )− 2iε cos(T ) × exp { i 2 ( (cos(T ) + iε sin(T ))(p∗ 2 2 + p2 1)− 2p∗2p1 ) ((ε2 + 1) sin(T )− 2iε cos(T )) } . (3.13) Next we compare the standard harmonic oscillator propagator with that obtained in equa- tion (3.13). The bracket corresponding to the change of basis is readily shown to be 〈P \|p〉 = √ 1 2πε e− 1 2ε (p−P )2 , Higher Order Time Derivative Models 9 which allow us to relate the propagator expressed in terms of Hermitian variables, (P̂ , Q̂), with that written in terms of non-Hermitian ones, (p̂, q̂), as follows 〈p∗, t2\|p′, t1〉 = 1 2πε ∫ dPdP ′e− 1 2 ((p∗−P ′)2+(p′−P )2)〈P ′, t2\|P, t1〉. We have explicitly verified the above expression. The completeness relation in the non-Hermitian description results∫ d2p µ(p, p∗)\|p〉〈p∗\| = 1, µ = 1√ πε e+ 1 4ε (p−p∗)2 . 3.3 Path integral approach Consider the action for the modified harmonic oscillator in terms of the real and imaginary parts of q and p, S = ∫ t2 t1 dt ( pq̇ − ( p2 2 + 1 2 (1− ε2)q2 + iεpq )) = ∫ t2 t1 dt ( (pR + ipI) (q̇R + iq̇I) − ( (pR + ipI) 2 2 + 1 2 (1− ε2) (qR + iqI) 2 + iε(pR + ipI)(qR + iqI) )) . To compute the propagator using the path integral of this system we need to select an integration contour given by the reality conditions (3.4), that in terms of real and imaginary parts is qI = 0, pI = −εqR. These conditions are a pair of second class constraints in the framework of the Dirac’s method of quantization [38, 39], so the path integral subjected to them is exactly the Senjanovic path integral [40]. In our case we obtain 〈q2, t2 \|q1, t1 〉 = ∫ DpRDpIDqRDqIδ (qI)δ (pI + εqR) exp (iS) , since the determinant of the Poisson bracket of the constraints is one. Integrating over the imaginary parts using the delta functionals results in 〈q2, t2\|q1, t1〉 = ∫ DpRDqR exp ( i ∫ t2 t1 dt ( (pR − iεqR) q̇R − ( p2 R 2 + q2R 2 ))) , Here we see that the path integral is reduced to the usual path integral of the harmonic oscillator plus a term that contributes only to the classical action, so finally the amplitude is 〈q2, t2\|q1, t1〉 = ( 1 2πi~ sinT ) 1 2 exp ( i 2 sinT [( q21 + q22 ) cosT − 2q1q2 ] + ε 2 (q22 − q21) ) , with T = t2 − t1. In this way we recover the result (3.12). Summarizing, we have provided a complex canonical transformation taking the complex har- monic oscillator (3.1) into the ordinary harmonic oscillator so that the non-Hermitian variables fulfill reality conditions in the Hilbert space H which are consistent with the canonical trans- formation. Let us observe that PT symmetry has not been invoked in this approach. The states (3.2) possess positive definite norm in H and a unitary time evolution follows from the canonical transformation. 10 A. Déctor, H.A. Morales-Técotl, L.F. Urrutia and J.D. Vergara 4 The modified PU oscillator as a second order theory Now we proceed with the HOTD model. Let us start with the PU oscillator Lagrangian LPU = −1 2 ẍ2 + (ω2 1 + ω2 2) 2 ẋ2 − ω2 1ω 2 2 2 x2, (4.1) where we assume ω1 > ω2. LPU is connected to Lξ = 1 2 ξ̇1 2 − ω2 1 2 ξ21 + 1 2 ξ̇2 2 − ω2 2 2 ξ22 , (4.2) for real ξi, i = 1, 2, by the following relations LPU + df dt = Lξ, f = ẋẍ, (4.3) ξ1 = i(ax+ bẍ), ξ2 = cx+ bẍ. (4.4) By choosing a, b, c to be real we see that x is necessarily complex. To accomplish (4.3), (4.4) the following values are obtained, for which the same sign should be used, a ω2 2 = b = c ω2 1 = ± 1√ ω2 1 − ω2 2 . (4.5) In other words, according to (4.3), LPU in (4.1) fails to be the real Lξ in (4.2) only by a total time derivative. Note that f will give rise to the corresponding canonical transformation once it is expressed in terms of phase space variables which we now derive. According to the Ostrogradsky method applied to (4.1) we get Πx = ( ω2 1 + ω2 2 ) ẋ+ ... x , z = ẋ, Πz = −ẍ, so that {x,Πx} = 1 and {z,Πz} = 1. Clearly f = −zΠz. The quantum Hamiltonian is ĤPU = −1 2 Π̂2 z − ω2 1 + ω2 2 2 ẑ2 + ẑΠ̂x + ω2 1ω 2 2 2 x̂2. (4.6) Using the afore mentioned transformation we have x̂ = ibξ̂1 + bξ̂2, Π̂x = iaP̂1 + cP̂2, (4.7) ẑ = ibP̂1 + bP̂2, Π̂z = icξ̂1 + aξ̂2, (4.8) where P̂i, i = 1, 2, are the canonical momenta conjugated to ξ̂i obtained from (4.2). In terms of these Hermitian variables the Hamiltonian (4.6) takes the form Ĥξ = P̂ 2 1 2 + ω2 1 2 ξ̂21 + P̂ 2 2 2 + ω2 2 2 ξ̂22 . (4.9) So, starting from a Hamiltonian that is not bounded from below, extending the canonical vari- ables to the complex plane and adding a total time derivative a well defined Hamiltonian in terms of Pi and ξi is obtained. Notice that in [16] it is acknowledged there exists a similarity transformation relating the original PU oscillator with a couple of independent harmonic oscil- lators similar to our case. Nevertheless a further transformation has to be supplied in [16] to obtain the right frequencies for the oscillators. Higher Order Time Derivative Models 11 4.1 The modified PU Lagrangian in the unequal frequency case in the extended phase space Instead of performing the canonical analysis starting from equation (4.1), we will consider (4.3) which includes the additional time derivative. Namely we start from LT = 1 2 ẍ2 + 1 2 ( ω2 1 + ω2 2 ) ẋ2 − 1 2 ω2 1ω 2 2x 2 + ẋx(3). We first notice that LT = LT ( x, ẋ, ẍ, x(3) ) , so it is higher order than LPU. This also means that, according to Ostrogradsky’s formalism, there will be a six-dimensional phase space with coordinates (x, ẋ, ẍ) and momenta (p0, p1, p2) given by p0 = ( ω2 1 + ω2 2 ) ẋ+ x(3), (4.10) p1 = 0, (4.11) p2 = ẋ, (4.12) according to the general definitions given in equation (2.5). Now we notice that (4.11) and (4.12) are in fact primary constraints, which we write as φ1 = p1 = 0, (4.13) φ2 = p2 − ẋ = 0. (4.14) The time evolution of the above constraints fixes the corresponding Lagrange multipliers making them second class constraints. In this way, the reduced phase space has dimension four. We will choose to express our Hamiltonian in terms of (x, ẍ, p0, p2). The resulting canonical Hamiltonian is given by HC = p0ẋ+ p1ẍ+ p2x (3) − LT = p0p2 − 1 2 ẍ2 − 1 2 ( ω2 1 + ω2 2 ) p2 2 + 1 2 ω2 1ω 2 2x 2. Next we construct the canonical structure in the Dirac formalism, accounting for the second class constraints. To do so, we use the Dirac brackets {A,B}∗ = {A,B} − {A,φα}Cαβ {φβ, B} , α, β = 1, 2, where Cαβ is a square antisymmetric matrix such that Cµα {φα, φν} = δµν . Upon using equations (4.13) and (4.14) we obtain Cαβ = ( 0 −1 1 0 ) . The resulting Dirac brackets are {x, p0}∗ = 1, {ẍ, p2}∗ = 1, (4.15) and zero for any other combination. Next we rewrite the Lagrangian in the canonical formalism LT = ẋp0 + ẍp1 + x(3)p2 −HC and make the constraints (4.13), (4.14) strong obtaining LT = ẋp0 + x(3)p2 −HC. (4.16) 12 A. Déctor, H.A. Morales-Técotl, L.F. Urrutia and J.D. Vergara In order to make contact with the classical version of the Hamiltonian (4.6) we introduce the following canonically related new variables, leaving x unchanged, Πx = p0, z = p2, Πz = −ẍ. Using equation (4.15), we obtain as the only non-zero Dirac brackets {x,Πx}∗ = 1, {z,Πz}∗ = 1. In this way, we can consider the variables {x, z,Πx,Πz} as a set of canonical variables in the Dirac formalism. Introducing the new variables in equation (4.16) we obtain LT = ẋΠx − Π̇zz −HC, where HC corresponds to the classical counterpart of HPU given in equation (4.6). The above equation guarantees that HPU is the correct Hamiltonian in the reduced physical space and imply that the correct boundary conditions at the end points are given by fixing x and Πz in regard to the variational problem. 4.2 The Pais–Uhlenbeck transformation in matrix form The complex canonical transformation between the real variables {ξ1, ξ2, P1, P2} and the complex ones {x, z,Πx,Πz}, is given in matrix form by X = Mξ, ξ = M−1X, where X, ξ and M stand for X =  x z Πx Πz  , ξ =  ξ1 ξ2 P1 P2  , M =  ib b 0 0 0 0 ib b 0 0 ia c ic a 0 0  . Now, the matrix M satisfies MTΩM = Ω, (4.17) with Ω =  0 0 1 0 0 0 0 1 −1 0 0 0 0 −1 0 0  , where only the relation b(c− a) = 1 was needed to prove equation (4.17). This means that the matrix M belongs to the complex symplectic group Sp (2n, C), with n = 2. Since Sp (2n, C) is a 2n(2n+1) parameter group, in our case M should have 20 parameters in general. One notices that det(M) = (b(c− a))2 = 1, which verifies the unimodular character of the symplectic matrix. Higher Order Time Derivative Models 13 4.3 Hilbert space We have related the Hamiltonians (4.6) and (4.9) by means of the complex canonical transfor- mations (4.7), (4.8) and their inverse ξ1 = iax− ibΠz, ξ2 = cx− bΠz, Pξ1 = −icz + ibΠx, Pξ2 = −az + bΠx, where the coefficients a, b and c are given in (4.5). We notice that in this case the following relations are satisfied: ac = ω2 1ω 2 2b 2, b (c− a) = 1. At the quantum mechanical level, we fix the reality conditions by requiring that the operators {ξ1, ξ2, Pξ1 , Pξ2} are Hermitian. This in turn implies that: x† = b (a+ c)x− 2b2Πz, z† = −b(a+ c)z + 2b2Πx, (4.18) Π† z = 2acx− b (a+ c) Πz, Π† x = −2acz + b(a+ c)Πx. (4.19) We will quantize the system in the basis \|x,Πz〉, so we propose the following operator realizations x̂ = x, Π̂z = Πz, and Π̂x = −i ∂ ∂x , ẑ = i ∂ ∂Πz . Now, the inner product written in the basis \|x,Πz〉 can be expressed in general as 〈φ, ψ〉 = ∫ d2xd2Πz µ (x,Πz)φ∗ (x,Πz)ψ (x,Πz) , where d2x = dxRdxI , d2Πz = dΠzRdΠzI . Because of conditions (4.18) and (4.19), we should have in particular: 〈xφ, ψ〉 = 〈φ, ( b (a+ c)x− 2b2Πz ) ψ〉, (4.20) and 〈Πzφ, ψ〉 = 〈φ, (2acx− b (a+ c) Πz)ψ〉. (4.21) or ∫ d2xd2Πzµx ∗φ∗ψ = ∫ d2xd2Πzµφ ∗ {b (a+ c)x− 2b2Πz } ψ,∫ d2xd2ΠzµΠ∗ zφ ∗ψ = ∫ d2xd2Πzµφ ∗ {2acx− b (a+ c) Πz}ψ. Separating x and Πz in their real and imaginary parts inside the integral, and matching both equations, we conclude from (4.20) and (4.21) that: axR = bΠzR, (4.22) cxI = bΠzI , (4.23) which can be always accomplished if µ (x,Πz) = Cδ (axR − bΠzR) δ (cxI − bΠzI) , where C is a constant to be determined by some normalization condition. 14 A. Déctor, H.A. Morales-Técotl, L.F. Urrutia and J.D. Vergara 4.4 The Schrödinger equation Let us consider the stationary Schödinger equation HMψ (x,Πz) = E ψ (x,Πz) , or more explicitly, using the operator realizations given above −1 2 Π2 zψ + 1 2 ( ω2 1 + ω2 2 ) ∂2ψ ∂Π2 z + 1 2 ω2 1ω 2 2x 2ψ + ∂2ψ ∂x∂Πz = Eψ. Using the relation between {ξ1, ξ2} and {x,Πz}, applying the chain rule ∂ ∂x = ∑ i ∂ξi ∂x ∂ ∂ξi = ia ∂ ∂ξ1 + c ∂ ∂ξ2 , ∂ ∂Πz = ∑ i ∂ξi ∂Πz ∂ ∂ξi = −ib ∂ ∂ξ1 − b ∂ ∂ξ2 , and substituting we arrive at −1 2 ∂2ψ ∂ξ21 − 1 2 ∂2ψ ∂ξ22 + ω2 1 2 ξ21ψ + ω2 2 2 ξ22ψ = Eψ, with regular solutions: ψm,n (ξ1, ξ2) = exp { −1 2 ( ω1ξ 2 1 + ω2 2ξ 2 2 )} Hm ( √ ω1ξ1)Hn ( √ ω2ξ2) . Let us now consider the ground state, m = n = 0 ψ0,0 (ξ1, ξ2) = exp { −1 2 ( ω1ξ 2 1 + ω2 2ξ 2 2 )} , or written in terms of x, Πz ψ0,0 (x,Πz) = exp { 1 2 (( ω1a 2 − ω2c 2 ) x2 + 2b (ω2c− ω1a) Πzx+ b2 (ω1 − ω2) Π2 z )} . Separating x and Πz in their real and imaginary parts we get ψ0,0 (x,Πz) = exp { 1 2 (( ω1a 2 − ω2c 2 ) ( x2 R − x2 I ) + 2b (ω2c− ω1a) (ΠzRxR −ΠzIxI) + b2 (ω1 − ω2) ( Π2 zR −Π2 zI ))} exp {iφ (x,Πz)} , (4.24) where φ (x,Πz) is a real function of xR, xI , ΠzR and ΠzI and therefore is the imaginary part of ψ0,0 written as a phase factor. It is not easy to see from (4.24) if the ground state function vanishes at \|x\| → ±∞, \|Πz\| → ±∞. However, substituting the reality conditions (4.22), (4.23) we have ψ0,0 (z,Πz) = exp { − ω2 ( ω2 1 − ω2 2 ) 2 x2 R } exp { − ω1 ( ω2 1 − ω2 2 ) 2 x2 I } , where we verify that the ground state function indeed vanishes at infinity. Finally, in the Appendix we compute the full propagator of the PU oscillator. Higher Order Time Derivative Models 15 4.5 The modified PU Lagrangian in the equal frequency case as a third order system We now wish to follow the same procedure as in Subsection 4.1 for the restriction of the PU Lagrangian to the equal frequency case. Recapitulating, we begin from the corresponding equal frequency PU Lagrangian L̃PU plus a total time derivative L̃T = L̃PU + df̃ dt = −1 2 ẍ2 + ω2ẋ2 − 1 2 ω4x2 + df̃ dt . (4.25) The equations of motion arising from L̃PU are 0 = x(4) + 2ω2x(2) + ω4x, which are not affected by the total time derivative. At the Lagrangian level the choice f̃ = 1 2 ( ẍ− ω2x ) ẋ allows to transform (4.25) into the two oscillators system L̃PU + df̃ dt = L̃ξ = 1 2 ξ̇21 + 1 2 ξ̇22 − 1 2 Ω2 ( ξ21 + ξ22 ) , by means of the nonlocal transformation ξ1 = i (ax+ bẍ) , ξ2 = cx+ dẍ. together with the conditions ω2 = Ω2, b2 = d2, cd− ab = 1/2, c2 − a2 = ω2. For simplicity we can take d = b, which gives us a = ω2b− 1 4b , c = ω2b+ 1 4b . Then, in this case we have a free parameter in the transformation. Now, the total Lagrangian turns out to be L̃T = 1 2 ω2ẋ2 − 1 2 ω4x2 + 1 2 x(3)ẋ− 1 2 ω2xẍ, where x(3) = ... x . According to Ostrogradsky, the canonical momenta conjugated to the coordi- nates (x, ẋ, ẍ) are p̃0 = 3 2 ω2ẋ+ x(3), (4.26) p̃1 = −1 2 ( ω2x+ ẍ ) , (4.27) p̃2 = 1 2 ẋ, (4.28) respectively. Again, (4.27) and (4.28) are primary constraints, which can be put as φ1 = p̃1 + 1 2 (ω2x+ ẍ) = 0, φ2 = p̃2 − 1 2 ẋ = 0. (4.29) 16 A. Déctor, H.A. Morales-Técotl, L.F. Urrutia and J.D. Vergara As in the previous case, the time evolution of them fixes the corresponding Lagrange multipliers, thus turning (4.29) into a set of second class constraints which reduce the dimension of the phase space from six to four. We choose to express our Hamiltonian in terms of variables {x, ẍ, p̃0, p̃2}, eliminating ẋ, p̃1 from the constraints. The result is H̃C = p̃0ẋ+ p̃1ẍ+ p̃2x (3) − L̃T = 2p̃2(p̃0 + ω2p̃2)− 4ω2p̃2 2 + 1 2 ω4x2 − 1 2 ẍ2. A similar procedure as the one described in Subsection 4.1 yields the non-zero Dirac brackets {x, p̃0}∗ = 1, {ẍ, p̃0}∗ = −1 2 ω2, {ẍ, p̃2}∗ = 1 2 , in the phase space {x, ẍ, p̃0, p̃2} . In order to make contact with the original equal frequency PU problem, let us introduce the new variables x̃, z̃, Π̃x, Π̃z x̃ = x, z̃ = 2p̃2, Π̃x = p̃0 + ω2p̃2, Π̃z = −ẍ, and proceed to calculate their Dirac brackets. The only non-zero results in the phase space {x̃, z̃, Π̃x, Π̃z} are{ x̃, Π̃x }∗ = 1, { z̃, Π̃z }∗ = 1, The above indicates that we can consider the set of variables {x̃, z̃, Π̃x, Π̃z} as canonical variables in the Dirac formalism. Substituting in H̃C we get: H̃C = −1 2 Π̃2 z − ω2z̃2 + Π̃xz̃ + 1 2 ω4x̃2, (4.30) which corresponds to the original Pais–Uhlenbeck Hamiltonian (4.6) in the equal-frequency case. As pointed out in references [17, 21], (4.30) is not diagonalizable and hence cannot be related by a similarity transformation to the two oscillator system. However, the Pais–Uhlenbeck Hamiltonian in the equal frequencies case can be related to the Hamiltonian of two uncoupled oscillators H = 1 2 ( P 2 1 + P 2 2 ) + 1 2 ω2 ( ξ21 + ξ22 ) , through a transformation which is neither canonical nor of the similarity type, given by x z Πx Πz  =  ib √ 1 ω2 + b2 0 0 0 0 id d 0 0 i ( dω2 − 1 2d ) ( dω2 + 1 2d ) iω2 √ 1 ω2 + b2 bω2 0 0   ξ1 ξ2 P1 P2  , or, conversely ξ1 ξ2 P1 P2  =  ibω2 0 0 −i √ 1 ω2 + b2 ω2 √ 1 ω2 + b2 0 0 −b 0 −i ( dω2 + 1 2d ) id 0 0 − ( dω2 − 1 2d ) d 0   x z Πx Πz  . Even though we have not explored the quantization of the system in this case, we expect that the non existence of a similarity transformation between the PU oscillator and the two-oscillator system would be a signal of the non-diagonalizable property of the Hamiltonian related to the appearance of Jordan blocks, as discussed in [41, 17, 21]. Higher Order Time Derivative Models 17 5 Discussion In this work we have proposed that a complex canonical transformation applied to a class of HOTD models with Lagrangians modified by a total time derivative can solve the drawbacks of such theories, namely negative norm states or ghosts, unbounded below Hamiltonian and non-unitary time evolution, keeping unmodified the original classical equations of motion. Such a transformation requires the original canonical variables to be analytically continued and fur- ther subjected to appropriate reality conditions so that the final canonical set matches the original number of real degrees of freedom. In particular while the final canonical variables ξi are Hermitian the original ones x(i) fulfill some reality conditions dictated by the canonical transformation among such variables. The inner product in the original Hilbert space is defined in such way to reproduce the reality conditions. This idea has been applied previously to non- perturbative canonical quantum general relativity [24, 25] and the inclusion of reality conditions was mandatory to recover real gravity. Thus we have successfully provided a novel extension of such idea to the case of the PU oscillator as an example of HOTD models. An important point that we must clarify is that the analytic extension of the variables allow us to have a well defined transformation between the HOTD model and a second order theory. The interesting result is that both theories have the same degrees of freedom and are related by a complex transformation. So, our procedure to extend the original problem to the complex plane gives us the appropriate frame to find the mapping between both theories. The success of the proposed approach in the PU case motivates the study of its extension to other HOTD models, but this is beyond the scope of the present work. To illustrate the procedure we have first reviewed the case of a complexified model that reduces to an harmonic oscillator. Although not of the HOTD type, this example exhibits negative norm states which are neatly eliminated using a complex canonical transformation. In this case we determined also the corresponding propagator and contrasted it with the usual one of the harmonic oscillator. To do so we adopted two different techniques: Schwinger’s quantum action principle and the path integral approach, both of which led to the same result. Let us notice also that the path integral version of complex canonical variables for gravity has been studied [42]. Next we studied the PU oscillator (4.1) which is properly a HOTD model. After complexi- fying the original variables we constructed the complex canonical transformation (4.7), (4.8) which leads to a system of two decoupled harmonic oscillators (4.9) with just the frequencies ω1 and ω2 appearing in (4.1). The propagator of the model containing non-Hermitian variables was determined by adopting again Schwinger’s quantum action principle. The equal frequency PU oscillator has been also described in the classical case. We show that there are some differences with the unequal frequency situation. First, we have here a free parameter in the transformation, perhaps a resemblance of the Barbero–Immirzi parameter in loop quantum gravity [29, 30]. Furthermore, in the reduced phase space there does not exist a similarity transformation between the PU oscillator with equal frequency and the second order oscillators. However, we found that this system is also mapped to a couple of harmonic oscillators, using a non-similarity type transformation. Our approach to the PU oscillator is an alternative to that of [16] based on non-Hermitian but PT symmetric Hamiltonians. We find ours has the following properties: (1) It is based on non- Hermitian variables subjected to specific and well defined reality conditions arising from (4.7), (4.8) that make the higher-order Hamiltonian Hermitian in the appropriate Hilbert space with measure (A.3). (2) No PT symmetry is required. (3) It works for at least the PU oscillator. (4) If we want to consider additional interaction terms as an anharmonic contribution, our proposal still works provided that we add the prescription x4 → (xx†)2 = b4(ξ21 + ξ22) 2. This interaction term has the correct signs and the Hamiltonian is still bounded from below. (5) We must 18 A. Déctor, H.A. Morales-Técotl, L.F. Urrutia and J.D. Vergara emphasize that the complexification of the original problem together with the reality conditions are a key ingredient in our approach which constitute the prescription that define our problem. It is remarkable that this approach can be also applied to the case of a quadratic higher-order derivative free scalar field model. To see this notice that the following Lagrangian density L = −1 2 (2φ)2 − m2 1 +m2 2 2 φ2φ− m2 1m 2 2 2 φ2, is related to Lψ = −1 2 ψ12ψ1 − m2 1 2 ψ2 1 − 1 2 ψ22ψ2 − m2 2 2 ψ2 2, (5.1) by L+ ∂µf µ = Lψ, fµ = −2φ∂µφ, ψ1 = i(aφ+ b2φ), ψ2 = cφ+ b2φ, with a, b, c having the same form as in the PU model (4.5) except for the replacing ωi → mi, i = 1, 2. Clearly (5.1) is ghost-free. Our approach seems promising even in the interacting case including a term of the type (φφ∗)2 = − 1 (c−a)4 (ψ2 1 +ψ2 2) 2 [6, 9], however all these generalizations deserve further study. In closing we would like to mention some directions for further study along the lines proposed in the present work. Other physical mechanical models including at least third order derivatives might help to show the strength of the canonical approach here pursued. Also a crucial problem to come to terms with is field theory. The success in the simplest non interacting scalar field briefly described above requires full implementation at the quantum level. Finally, as it was mentioned in the introduction, higher order gravity has been one of the motivations to the present approach, which of course is a paramount challenge. Also it would be rather interesting to look for HOTD gauge field theories to test some aspects of the gravitational case. A Schwinger’s quantum action principle and the PU oscillator In this appendix we compute the propagator of the PU oscillator, a result already obtained in [43]. To do this, we apply Schwinger’s quantum action principle to calculate 〈x∗2,Π∗ z 2, t2\|x1, Πz 1, t1〉. However, in this case we need to be more careful since our variables are complex. We will work in the Heisenberg picture, where the involved operators act as: x(t1)\|x1,Πz1, t1〉 = x1\|x1,Πz1, t1〉, Πz(t1)\|x1,Πz1, t1〉 = Πz1\|x1,Πz1, t1〉, 〈x∗2,Π∗ z2, t2\|x†(t2) = 〈x∗2,Π∗ z2, t2\|x∗2, 〈x∗2,Π∗ z2, t2\|Π† z(t2) = 〈x∗2,Π′∗ z2, t2\|Π∗ z 2. As usual, we first calculate the total variation of the quantity 〈x∗2,Π∗ z2, t2\|x1,Πz1, t1〉, given by: δ〈x∗2,Π∗ z2, t\|x1,Π′′ z1, t1〉 (A.1) = i〈x∗2,Π′∗ z2, t2\| { Π† x(t2)δx ∗ 2 − z†(t2)δΠ∗ z2 −Πx(t1)δx1 + z(t1)δΠz1 −HδT } \|x1,Πz1, t1〉, where T = t2 − t1. Now, we must express {z,Πx, z †(t),Π† x(t)} in terms of {x,Πz, x †(t),Π† z(t)}. We solve the Heisenberg equations for the operators x, z, Πx and Πz. This equations are: ẋ(t) = z(t), Π̇x(t) = −ω2 1ω 2 2x(t), ż(t) = −Πz(t), Higher Order Time Derivative Models 19 Π̇z(t) = ( ω2 1 + ω2 2 ) z(t)−Πx(t). Uncoupling, we get the following equation for x(t): x(4)(t) + ( ω2 1 + ω2 2 ) ẍ(t) + ω2 1ω 2 2x(t) = 0, with general solution: x(t) = C1e iω1t + C2e −iω1t + C3e iω2t + C4e −iω2t, while for the other operators we get: z(t) = iω1 ( C1e iω1t − C2e −iω1t ) + iω2 ( C3e iω2t − C4e −iω2t ) , Πx(t) = iω1ω 2 2 ( C1e iω1t − C2e −iω1t ) + iω2 1ω2 ( C3e iω2t − C4e −iω2t ) , Πz(t) = ω2 1 ( C1e iω1t + C2e −iω1t ) + ω2 2 ( C3e iω2t − C4e −iω2t ) . We fix the constants Ci’s in terms of initial conditions at time t1: x(t1), Πz(t1) and final conditions at time t2: x†(t2) and Π† z(t2). To obtain such conditions, we use the reality properties (4.18), (4.19), valid at all times. In this way, we arrive to the following system of linear equations: x(t1) = C1e iω1t1 + C2e −iω1t1 + C3e iω2t1 + C4e −iω2t1 , Πz(t1) = ω2 1 ( C1e iω1t1 + C2e −iω1t1 ) + ω2 2 ( C3e iω2t1 + C4e −iω2t1 ) , x†(t2) = − ( C1e iω1t2 + C2e −iω1t2 ) + ( C3e iω2t2 + C4e −iω2t2 ) , Π† z(t2) = −ω2 1 ( C1e iω1t2 + C2e −iω1t2 ) + ω2 2 ( C2e iω2t2 + C4e −ω2t2 ) , Solving for the C’s and substituting, we get the following expressions for z(t1), Πx(t1), z†(t2) and Π† x(t2): z(t1) = 1 (ω2 1 − ω2 2) sin(ω1T ) sin(ω2T ) × {{−ω2 1ω2 sin(ω1T ) cos(ω2T ) + ω1ω 2 2 cos(ω1T ) sin(ω2T )}x(t1) + {ω2 sin(ω1T ) cos(ω2T )− ω1 cos(ω1T ) sin(ω2T )}Πz(t1) + {ω2 1ω2 sin(ω1T ) + ω1ω 2 2 sin(ω2T )}x†(t2) + {−ω2 sin(ω1T )− ω1 sin(ω2T )}Π† z(t2)}, Πx(t1) = 1 (ω2 1 − ω2 2) sin(ω1T ) sin(ω2T ) × {{−ω4 1ω2 sin(ω1T ) cos(ω2T ) + ω1ω 4 2 cos(ω1T ) sin(ω2T )}x(t1) + {ω2 1ω2 sin(ω1T ) cos(ω2T )− ω1ω 2 2 cos(ω1T ) sin(ω2T )}Πz(t1) + {ω4 1ω2 sin(ω1T ) + ω1ω 4 2 sin(ω2T )}x†(t2) + {−ω2 1ω2 sin(ω1t2)− ω1ω 2 2 sin(ω2T )}Π† z(t2)}, z†(t2) = 1 (ω2 1 − ω2 2) sin(ω1T ) sin(ω2T ) {{−ω2 1ω2 sin(ω1T )− ω1ω 2 2 sin(ω2T )}x(t1) + {ω2 sin(ω1T ) + ω1 sin(ω2T )}Πz(t1) + {ω2 1ω2 sin(ω1T ) cos(ω2T )− ω1ω 2 2 cos(ω1T ) sin(ω2T )}x†(t2) + {−ω2 sin(ω1T ) cos(ω2T ) + ω1 cos(ω1T ) sin(ω2T )}Π† z(t2)}, Π† z(t2) = 1 (ω2 1 − ω2 2) sin(ω1T ) sin(ω2T ) {{−ω4 1ω2 sin(ω1T )− ω1ω 4 2 sin(ω2T )}x(t1) + {ω2 1ω2 sin(ω1T ) + ω1ω 2 2 sin(ω2T )}Πz(t1) 20 A. Déctor, H.A. Morales-Técotl, L.F. Urrutia and J.D. Vergara + {ω4 1ω2 sin(ω1T ) cos(ω2T )− ω1ω 4 2 cos(ω1T ) sin(ω2T )}x†(t2) + {−ω2 1ω2 sin(ω1T ) cos(ω2T ) + ω1ω 2 2 cos(ω1T ) sin(ω2T )}Π† z(t2)}. Using these results in the Hamiltonian (4.6) and in the total variation (A.1), we get finally 〈x∗2,Π∗ z 2, t2\|x1,Πz 1, t1〉 = √ 1 sin(ω1T ) sin(ω2T ) exp { i (ω2 1 − ω2 2) sin(ω1T ) sin(ω2T ) × { {ω4 1ω2 sin(ω1T ) cos(ω2T )− ω1ω 4 2 cos(ω1T ) sin(ω2T )} { x2 1 2 + x∗ 2 2 2 } + {ω2 sin(ω1T ) cos(ω2T )− ω1 cos(ω1T ) sin(ω2T )} { Π2 z 1 2 + Π∗ 2 z 2 2 } + {ω2 1ω2 sin(ω1T ) + ω1ω 2 2 sin(ω2T )}{x∗2Πz 1 + Π∗ z 2x1} + {−ω2 1ω2 sin(ω1T ) cos(ω2T ) + ω1ω 2 2 cos(ω1T ) sin(ω2T )}{x∗2Π∗ z 2 + x1Πz 1} + {−ω4 1ω2 sin(ω1T )− ω1ω 4 2 sin(ω2T )}{x∗2x1} + {−ω2 sin(ω1T )− ω1 sin(ω2T )}{Π∗ z 2Πz 1} }} , or, for short: 〈x∗2,Π′∗ z 2, t2\|x1,Πz 1, t1〉 = Q exp { i D { F (x∗2Πz 1 + x1Π∗ z 2) +GΠz 1Π∗ z 2 + J (x∗2Π ∗ z 2 + x1Πz 1) +K ( Π∗2 z 2 + Π2 z 1 2 ) +Mx1x ∗ 2 +N ( x∗22 + x2 1 2 )}} , (A.2) with D, F , G, J , K, M , N , Q, being the following functions of T : D = (ω2 1 − ω2 2) sin(ω1T ) sin(ω2T ), F = ω2 1ω2 sin(ω1T ) + ω1ω 2 2 sin(ω2T ), G = −ω2 sin(ω1T )− ω1 sin(ω2T ), J = −ω2 1ω2 sin(ω1T ) cos(ω2T ) + ω1ω 2 2 sin(ω2T ) cos(ω1T ), K = ω2 sin(ω1T ) cos(ω2T )−ω1 sin(ω2T ) cos(ω1T ), M = −ω4 1ω2 sin(ω1T )− ω1ω 4 2 sin(ω2T ), N = ω4 1ω2 sin(ω1T ) cos(ω2T )− ω1ω 4 2 sin(ω2T ) cos(ω1T ), Q = √ 1 sin(ω1T ) sin(ω2T ) . Just as in our first example the PU propagator (A.2) can be related to the one corresponding to the Hamiltonian (4.9) by using the change of basis 〈P1, P2\|x,Πz〉 = exp [(ax− bΠz)P1 + (−icx+ ibΠz)P2] . The basis \|x,Πz〉 is complete with the measure dµPU = dxRdxIdΠzRdΠzI (2π)2 δ(bΠzR − axR)δ(bΠzI − cxI), (A.3) where xR, xI , ΠzR, ΠzI are real. Higher Order Time Derivative Models 21 Acknowledgements This work was partially supported by the following grants: CONACyT-SEP 51132F, CONACyT- SEP 47211-F, CONACyT-SEP 55310, DGAPA-UNAM IN109107 and a CONACyT sabbatical grant to HAMT. AD wishes also to acknowledge support from CONACyT. References [1] Thirring W., Regularization as a consequence of higher order equations, Phys. Rev. 77 (1950), 570. [2] Pais A., Uhlenbeck G.E., On field theories with nonlocalized action, Phys. Rev. 79 (1950), 145–165. [3] Heisenberg W., Lee model and quantisation of non linear field theories, Nuclear Phys. 4 (1957), 532–563. [4] Stelle K.S., Renormalization of higher-derivative quantum gravity, Phys. Rev. D 16 (1977), 953–969. [5] Tomboulis E.T., Unitarity in higher derivative quantum gravity, Phys. Rev. Lett. 52 (1984), 1173–1176. [6] Hawking S.W., Hertog T., Living with ghosts, Phys. Rev. D 65 (2002), 103515, 8 pages, hep-th/0107088. [7] Moeller N., Zwiebach B., Dynamics with infinitely many time derivatives and rolling tachyons, J. High Energy Phys. 2002 (2002), no. 10, 034, 39 pages, hep-th/0207107. [8] Rivelles V.O., Triviality of higher derivative theories, Phys. Lett. B 577 (2003), 137–142, hep-th/0304073. [9] Antoniadis I., Dudas E., Ghilencea D.M., Living with ghosts and their radiative corrections, Nuclear Phys. B 767(2007), 29–53, hep-th/0608094. [10] Codello A., Percacci R., Fixed points of higher-derivative gravity, Phys. Rev. Lett. 97 (2006), 221301, 4 pages, hep-th/0607128. [11] Berkovits N., New higher-derivative R4 theorems, Phys. Rev. Lett. 98 (2007), 211601, 4 pages, hep-th/0609006. [12] Jaen X., Llosa J., Molina A., A reduction of order two for infinite order Lagrangians, Phys. Rev. D 34 (1986), 2302–2311. [13] Eliezer D.A., Woodard R.P., The problem of nonlocality in string theory, Nuclear Phys. B 325 (1989), 389–469. [14] Cheng T.C., Ho P.M., Yeh M.C., Perturbative approach to higher derivative and nonlocal theories, Nuclear Phys. B 625 (2002), 151–165, hep-th/0111160. [15] Simon J.Z., Higher derivative Lagrangians, non-locality, problems and solutions, Phys. Rev. D 41 (1990), 3720–3733. [16] Bender C.M., Mannheim P.D., No-ghost theorem for the fourth-order derivative Pais–Uhlenbeck oscillator model, Phys. Rev. Lett. 100 (2008), 110402, 4 pages, arXiv:0706.0207. [17] Bender C.M., Mannheim P.D., Exactly solvable PT -symmetric Hamiltonian having no Hermitian counter- part, Phys. Rev. D 78 (2008), 025022, 20 pages, arXiv:0804.4190. [18] Bender C.M., Mannheim P.D., Giving up the ghost, J. Phys. A: Math. Theor. 41 (2008), 304018, 7 pages, arXiv:0807.2607. [19] Smilga A.V., Benign vs. malicious ghosts in higher-derivative theories, Nuclear Phys. B 706 (2005), 598–614, hep-th/0407231. [20] Smilga A.V., Ghost-free higher-derivative theory, Phys. Lett. B 632 (2006), 433–438, hep-th/0503213. [21] Smilga A.V., Comments on the dynamics of the Pais–Uhlenbeck oscillator, SIGMA 5 (2009), 017, 13 pages, arXiv:0808.0139. [22] Smilga A.V., Exceptional points in quantum and classical dynamics, J. Phys. A: Math. Theor. 42 (2009), 095301, 9 pages, arXiv:0808.0575. [23] Bender C.M., Making sense of non-Hermitian Hamiltonians, Rep. Progr. Phys. 70 (2007), 947–1018, hep-th/0703096. [24] Ashtekar A., Lectures on nonperturbative canonical gravity, Advanced Series in Astrophysics and Cosmology, Vol. 6, World Scientific, Singapore, 1991. [25] Ashtekar A., Mathematical problems of nonperturbative quantum general relativity, gr-qc/9302024. [26] Thiemann T., Reality conditions inducing transforms for quantum gauge field theory and quantum gravity, Classical Quantum Gravity 13 (1996), 1383–1403, gr-qc/9511057. http://arxiv.org/abs/hep-th/0107088 http://arxiv.org/abs/hep-th/0207107 http://arxiv.org/abs/hep-th/0304073 http://arxiv.org/abs/hep-th/0608094 http://arxiv.org/abs/hep-th/0607128 http://arxiv.org/abs/hep-th/0609006 http://arxiv.org/abs/hep-th/0111160 http://arxiv.org/abs/0706.0207 http://arxiv.org/abs/0804.4190 http://arxiv.org/abs/0807.2607 http://arxiv.org/abs/hep-th/0407231 http://arxiv.org/abs/hep-th/0503213 http://arxiv.org/abs/0808.0139 http://arxiv.org/abs/0808.0575 http://arxiv.org/abs/hep-th/0703096 http://arxiv.org/abs/gr-qc/9302024 http://arxiv.org/abs/gr-qc/9511057 22 A. Déctor, H.A. Morales-Técotl, L.F. Urrutia and J.D. Vergara [27] Ashtekar A., A generalized Wick transform for gravity, Phys. Rev. D 53 (1996), 2865–2869, gr-qc/9511083. [28] Montesinos M., Morales-Técotl H.A., Urrutia L.F., Vergara J.D., Complex canonical gravity and reality constraints, Gen. Relativity Gravitation 31 (1999), 719–723. [29] Rovelli C., Quantum gravity, Cambridge University Press, Cambridge, 2004. [30] Thiemann T., Modern canonical quantum general relativity, Cambridge University Press, Cambridge, 2007. [31] Ostrogradsky M., Mémoires sur les équations différentielles relatives aux problèmes des isopérimètres, Mem. Acad. St. Petersbourg VI 4 (1850), 385–517. [32] Swanson M.S., Transition elements for a non-Hermitian quadratic Hamiltonian, J. Math. Phys. 45 (2004), 585–601. [33] Jones H.F., On pseudo-Hermitian Hamiltonians and their Hermitian counterparts, J. Phys. A: Math. Gen. 38 (2005), 1741–1746, quant-ph/0411171. [34] Ivanov E.A., Smilga A.V., Cryptoreality of nonanticommutative Hamiltonians, J. High Energy Phys. 2007 (2007), no. 07, 036, 16 pages, hep-th/0703038. [35] Anderson A., Canonical transformations in quantum mechanics, Ann. Physics 232 (1994), 292–331, hep-th/9305054. [36] Anderson A., Quantum canonical transformations: physical equivalence of quantum theories, Phys. Lett. B 305 (1993), 67–70, hep-th/9302062. [37] Schwinger J., Quantum mechanics, symbolism of atomic measurements, Springer-Verlag, Berlin, 2001. [38] Henneaux M., Teitelboim C., Quantization of gauge systems, Princeton University Press, Princeton, 1992. [39] Dirac P.A.M., Lectures on quantum mechanics, Belfast Graduate School of Science, New York, 1964. [40] Senjanovic P., Path integral quantization of field theories with second class constraints, Ann. Physics 100 (1976), 227–261, Erratum, Ann. Physics 209 (1991), 248. [41] Mannheim P.D., Davidson A., Dirac quantization of the Pais–Uhlenbeck fourth order oscillator, Phys. Rev. A 71 (2005), 042110, 9 pages, hep-th/0408104. [42] Alexandrov S.Y., Vassilevich D.V., Path integral for the Hilbert–Palatini and Ashtekar gravity, Phys. Rev. D 58 (1998), 124029, 13 pages, gr-qc/9806001. [43] Mannheim P.D., Solution to the ghost problem in fourth order derivative theories, Found. Phys. 37 (2007), 532–571, hep-th/0608154. http://arxiv.org/abs/gr-qc/9511083 http://arxiv.org/abs/quant-ph/0411171 http://arxiv.org/abs/hep-th/0703038 http://arxiv.org/abs/hep-th/9305054 http://arxiv.org/abs/hep-th/9302062 http://arxiv.org/abs/hep-th/0408104 http://arxiv.org/abs/gr-qc/9806001 http://arxiv.org/abs/hep-th/0608154 1 Introduction 2 General considerations 2.1 Canonical transformation and boundary conditions 3 Reality conditions and the complex oscillator 3.1 Hilbert space and reality conditions 3.2 The Schwinger quantum action principle 3.3 Path integral approach 4 The modified PU oscillator as a second order theory 4.1 The modified PU Lagrangian in the unequal frequency case in the extended phase space 4.2 The Pais-Uhlenbeck transformation in matrix form 4.3 Hilbert space 4.4 The Schrödinger equation 4.5 The modified PU Lagrangian in the equal frequency case as a third order system 5 Discussion A Schwinger's quantum action principle and the PU oscillator References

An Alternative Canonical Approach to the Ghost Problem in a Complexified Extension of the Pais-Uhlenbeck Oscillator

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