Calculation of C Operator in PT -Symmetric Quantum Mechanics

Transcription

1 Proceedigs of Istitute of Mathematics of NAS of Ukraie 2004, Vol. 50, Part 2, Calculatio of C Operator i PT -Symmetric Quatum Mechaics Qighai WANG Departmet of Physics, Washigto Uiversity, St. Louis, MO 6330, USA qwag@hbar.wustl.edu If a Hamiltoia H has a ubroke PT symmetry, the it also possesses a hidde symmetry represeted by the liear operator C. The operator C commutes with both H ad PT.The ier product with respect to CPT is associated with a positive orm ad the quatum theory built o the associated Hilbert space is uitary. I this paper it is show how to costruct the operator C for the o-hermitia PT -symmetric Hamiltoia H = 2 p2 + 2 x2 + iɛx 3 usig perturbative techiques. It is also show how to costruct the operator C for H = 2 p2 + 2 x2 ɛx 4 usig operturbative methods []. Itroductio ad Backgroud It was observed i 998 [2] that with properly defied boudary coditios the Sturm Liouville differetial equatio eigevalue problem associated with the o-hermitia PT -symmetric Hamiltoia H = p 2 + x 2 (ix) ν (ν>0) () exhibits a spectrum that is real ad positive. ByPT symmetry we mea the followig: The liear parity operator P performs spatial reflectio ad thus reverses the sig of the mometum ad positio operators: PpP = p ad PxP = x. The atiliear time-reversal operator T reverses the sig of the mometum operator ad performs complex cojugatio: T pt = p, T xt = x, adt it = i. The o-hermitia Hamiltoia H i () is ot symmetric uder P or T separately, but it is ivariat uder their combied operatio; such Hamiltoias are said to possess space-time reflectio symmetry (PT symmetry). We say that the PT symmetry of a Hamiltoia H is ot spotaeously broke if the eigefuctios of H are simultaeously eigefuctios of the PT operator. I a recet letter it was show that ay Hamiltoia that possesses a ubroke PT symmetry also has a hidde symmetry [3]. This ew symmetry is represeted by the liear operator C, which commutes with both the Hamiltoia H ad the PT operator. I terms of C oe ca costruct a ier product whose associated orm is positive defiite. Observables exhibit CPT symmetry ad the dyamics is govered by uitary time evolutio. Thus, PT -symmetric Hamiltoias give rise to ew classes of fully cosistet complex quatum theories. The purpose of the preset paper is to preset a explicit calculatio of C for two otrivial Hamiltoias. First, we cosider the case of the PT -symmetric Hamiltoia H = 2 p2 + 2 x2 + iɛx 3, (2) for which we give a perturbative calculatio of the operator C correct to third order i powers of ɛ. Secod, we calculate C for the Hamiltoia H = 2 p2 + 2 x2 ɛx 4, (3)

2 Calculatio of C Operator i PT -Symmetric Quatum Mechaics 987 for which ordiary perturbative methods are ieffective ad operturbative methods must be used. The orgaizatio of this paper is straightforward. I Sectio 2 we review the formal costructio, first preseted i Ref. [3], of the C operator. I Sectio 3 we calculate C for the Hamiltoia i (2) ad i Sectio 4 we calculate C for the Hamiltoia i (3). 2 Formal derivatio of the C operator I this sectio we preset a formal discussio of PT -symmetric Hamiltoias ad we show how to costruct the C operator. I geeral, for ay PT -symmetric Hamiltoia H we must begi by solvig the Sturm Liouville differetial equatio eigevalue problem associated with H: Hφ (x) =E φ (x) ( =0,, 2, 3,...). (4) For Hamiltoias like those i () (3) the differetial equatio (4) must be imposed o a ifiite cotour i the complex-x plae. For large x the cotour lies i wedges that are placed symmetrically with respect to the imagiary-x axis. These wedges are described i Ref. [2]. The boudary coditios o the eigefuctios are that φ(x) 0 expoetially rapidly as x o the cotour. For H i (2) the cotour may be take to be the real-x axis, but for H i (3) the cotour lies i the two wedges π/3 < arg x<0ad π <arg x< 2π/3. For all, the eigefuctios φ (x) are simultaeously eigestates of the PT operator. We ca choose the phase of φ (x) such that the eigevalue of (PT ) is uity: PT φ (x) =φ (x). (5) Next, we observe that there is a ier product, called the PT ier product, with respect to which the eigefuctios φ (x) for two differet values of are orthogoal. For the two fuctios f(x) adg(x) the PT ier product (f,g) isdefiedby (f,g) dx [PT f(x)] g(x), (6) C where PT f(x) =[f( x )] ad the cotour C lies i the wedges described above. For this ier product the associated orm (f,f) is idepedet of the overall phase of f(x) ad is coserved i time. We the ormalize the eigefuctios so that (φ,φ ) = ad we discover the apparet problem with usig a o-hermitia Hamiltoia. While the eigefuctios are orthogoal, the PT orm is ot positive defiite: (φ m,φ )=( ) δ m, (m, =0,, 2, 3,...). (7) Despite the fact that this orm is ot positive defiite, the eigefuctios are complete. For real x ad y the statemet of completeess i coordiate space is ( ) φ (x)φ (y) =δ(x y). (8) This is a otrivial result that has bee verified umerically to extremely high accuracy [4]. It is importat to remark here that the argumet of the Dirac delta fuctio i (8) must be real because the delta fuctio is oly defied for real argumet. This may seem to be i coflict with the earlier remark i this sectio that the Schrödiger equatio (4) must be solved alog a cotour that lies i wedges i the complex-x plae. To resolve this apparet coflict we specify the cotour as follows. We demad that the cotour lie o the real axis util it passes the poits x ad y. Oly the may it veer off ito the complex-x plae ad eter the wedges. This choice of cotour is allowed because the wedge coditios are asymptotic coditios. The positios of the wedges are determied by the boudary coditios.

3 988 Q. Wag We costruct the liear operator C that expresses the hidde symmetry of the Hamiltoia H. The positio-space represetatio of C is C(x, y) = φ (x)φ (y). (9) The properties of the operator C are easy to verify usig (7). First, the square of C is uity: dy C(x, y)c(y, z) =δ(x z). (0) Secod, the eigefuctios φ (x) of the Hamiltoia H are also eigefuctios of C ad the correspodig eigevalues are ( ) : dy C(x, y)φ (y) =( ) φ (x). () Third, the operator C commutes with both the Hamiltoia H ad the operator PT. Note that while the operators P ad C are uequal, both P ad C are square roots of the uity operator δ(x y). Last, the operators P ad C do ot commute. Ideed, CP =(PC). The operator C does ot exist as a distict etity i covetioal Hermitia quatum mechaics. Ideed, we will see that as the parameter ɛ i (2) ad (3) teds to zero the operator C becomes idetical to P. We ca ow defie a ier product f g whose associated orm is positive: f g dx [CPT f(x)]g(x). (2) The CPT orm associated with this ier product is positive because C cotributes whe it acts o states with egative PT orm. 3 Perturbative calculatio of C i a cubic theory I this sectio we use perturbative methods to calculate the operator C(x, y) for the Hamiltoia H = 2 p2 + 2 x2 + iɛx 3. We perform the calculatios to third order i perturbatio theory. We begi by solvig the Schrödiger equatio 2 φ (x)+ 2 x2 φ (x)+iɛx 3 φ (x) =E φ (x) (3) as a series i powers of ɛ. The perturbative solutio to this equatio has the form i a φ (x) = π /4 2 /2! e [ 2 x2 H (x) ip (x)ɛ Q (x)ɛ 2 + ir (x)ɛ 3], (4) where H (x) is the th Hermite polyomial ad P (x), Q (x), ad R (x) are polyomials i x of degree +3, +6, ad + 9, respectively. These polyomials ca be expressed as liear combiatios of Hermite polyomials []. The eergy E to order ɛ 3 is E = ( ) ɛ 2 +O ( ɛ 4). (5) 8 The expressio for φ (x) mustbept -ormalized accordig to (7) so that its square itegral is ( ) : dx [φ (x)] 2 =( ) +O ( ɛ 4). (6)

4 Calculatio of C Operator i PT -Symmetric Quatum Mechaics 989 This determies the value of a i (4): a =+ 44 (2 +)( ) ɛ 2 +O ( ɛ 4). (7) We calculate the operator C(x, y) by substitutig the wave fuctios φ (x) i (4) ito (9). We the use the completeess relatio for Hermite polyomials to evaluate the sum. To third order i ɛ the result is { ( 4 3 C(x, y) = iɛ 3 x 3 +2xy ) [ 8 ɛ 2 6 x 9 x xy 3 x 4 + ( 2x 2 y 2 2 ) 2 ] x 2 [ 32 + iɛ x ( 7 8 xy 9 x x2 y 2 76 ) 5 5 x 5 ( ) x3 y xy x 3 + ( 8x 2 y ) ] +O ( ɛ 4)} δ(x + y). (8) x Hece, the coordiate-space represetatio of the operator C(x, y) is expressed as a derivative of a Dirac delta fuctio. From this expressio for C(x, y) we ca verify the followig properties: First, to order ɛ 3 the operator C(x, y) satisfies (0). That is, dy C(x, y)c(y, z) =δ(x z)+o ( ɛ 4). (9) Secod, to order ɛ 3 the operator C(x, y) satisfies (); the wave fuctios φ (x) are eigestates of C(x, y) with eigevalue ( ). That is, dy C(x, y)φ (y) =( ) φ (x)+o ( ɛ 4). (20) Third, i the limit as ɛ 0, the operator C(x, y) becomes the coordiate-space represetatio of the parity operator P(x, y) =δ(x + y). There is a somewhat simpler way to express the operator C(x, y). The derivative operator i (8) that is actig o δ(x + y) ca be expoetiated so that to order ɛ 4 (ad ot just ɛ 3 )we have C(x, y) =e iɛa iɛ3b δ(x + y)+o ( ɛ 5), (2) where the derivative operators A ad B are give by 3 A = 4 3 x 3 2x x x, B = x x 3 x 3 x +8x2 x x2 32 x. (22) 4 Noperturbative calculatio of C i a quartic theory I this sectio we explai briefly the operturbative methods that must be used to calculate the operator C(x, y) for the Hamiltoia H = 2 p2 + 2 x2 ɛx 4. We follow the approach take i Ref. [5], i which operturbative methods were used to calculate the oe-poit Gree s fuctio for this Hamiltoia.

5 990 Q. Wag 4. Failure of perturbatio theory We begi by explaiig why perturbatio theory fails to produce the operator C(x, y). Followig the approach take i Sectio 3, we expad the solutio to the Schrödiger equatio 2 φ (x)+ 2 x2 φ (x) ɛx 4 φ (x) =E φ (x) (23) as a series i powers of ɛ: φ (x) = i a π /4 2 /2! e 2 x2 [H (x)+p (x)ɛ]+o ( ɛ 2), (24) where H (x) is the th Hermite polyomial ad P (x) is a polyomial i x of degree + 4. The polyomial P (x) is a liear combiatio of Hermite polyomials []. The eergy E to order ɛ is E = ( ) ɛ +O ( ɛ 2). (25) 4 We must also PT ormalize the expressio for φ (x) accordig to (7) so that its square itegral is ( ) : dx [φ (x)] 2 =( ) +O ( ɛ 2). (26) This determies the value of a i (24). The result is very simple; to order ɛ we have a =+O ( ɛ 2). (27) Fially, we substitute φ (x) i (24) ito (9). However, we obtai the trivial result that oly the leadig term (zeroth-order i powers of ɛ) survives. More geerally, we ca show by a parity argumet that the coefficiets of all higher powers of ɛ vaish. Thus, we get the (wrog) result that C(x, y) =δ(x + y) (WRONG!). (28) We kow that this result is wrog because the operator C(x, y) is complex ad the result i (28) is real. A alterative way to see this is to ote (28) implies that C(x, y) adp(x, y) coicide; but i this PT -symmetric theory, C(x, y) ad P(x, y) are distict operators. We will see that the differece betwee C(x, y) adp(x, y) is a operturbative term of order e /(3ɛ), which is smaller tha ay iteger power of ɛ. 4.2 Noperturbative aalysis We will ow show how to perform a operturbative aalysis of the Schrödiger equatio (23). We decompose the eigefuctio φ (x) ito its perturbative part o the right side of (24) ad a operturbative part: φ (x) =φ pert (x)+φ opert (x). (29) The operturbative part of φ (x) is expoetially small compared with the perturbative part, but these two cotributios ca be easily distiguished because for real argumet x, oe is real while the other is imagiary.

6 Calculatio of C Operator i PT -Symmetric Quatum Mechaics 99 Followig the WKB aalysis i Ref. [5], we break the real-x axis ito three regios: I regio I, where x ɛ /4,wehave φ pert i (x) π /4! D ( ) x 2, φ opert ( ) (x) ib C x 2, (30) where the coefficiet of D is take from (24) ad the coefficiet ib of C will be determied by asymptotic matchig. Note that for oegative iteger idex the parabolic cylider fuctio D is expressed i terms of a Hermite polyomial H as D (x 2) = 2 /2 e 2 x2 H (x). (3) Also, for oegative iteger idex the fuctios D ad C are a pair of liearly idepedet solutios to the parabolic cylider equatio. They ca be expressed i terms of parabolic cylider fuctios as follows: D (z)! [i D (iz)+( i) D ( iz)], 2π C (z) i 2π [i D (iz) ( i) D ( iz)]. (32) I regio II, where x ɛ /2, we ca obtai the eigefuctio usig WKB theory. We write the Schrödiger equatio (23) i the form φ (x) =ω (x)φ (x) where, to leadig order i ɛ, wehaveω (x) = 2ɛx 4 + x 2 2. The, for positive x the physical-optics WKB approximatio reads [ x φ pert (x) f [ω (x)] /4 exp φ opert (x) g [ω (x)] /4 exp ds ] ω (s), x x ds ] ω (s), (33) x [ + where the costats f ad g will be determied by asymptotic matchig. The lower edpoit of itegratio, x = 2 +, is the approximate locatio of the ier turig poit. I regio III x is ear the outer turig poits at ±/ 2ɛ. For positive x we defie the variable r by x = x 2 ( 2 /3 ɛ 2/3 r ), where x 2 =/ 2ɛ. The coditio that x is ear x 2 is that r ɛ 2/3. I this regio the Schrödiger equatio becomes a Airy equatio i the variable r: φ (r) =rφ (r). The solutio i this regio reads φ pert (r) h Bi(r), φ opert (r) ih Ai(r), (34) where Ai(r) ad Bi(r) are the expoetially decayig ad growig Airy fuctios for large positive r. The fact that the same coefficiet h multiplies both Bi ad Ai is a otrivial result that is established i Ref. [5]. By asymptotically matchig the solutios i regios I ad II ad the solutios i regios II ad III we obtai the formula for the coefficiet of the operturbative part of the solutio i (30): b = i π /4 ( ) e 3ɛ. (35) 2! ɛ

7 992 Q. Wag Fially, usig the wave fuctio i regio I we ca costruct the operator C(x, y) accordig to (9): C(x, y) = =0 [ φ pert + φ opert (x)φ pert (x)φ pert (y)+φ pert (x)φ opert (y) (y)+φ opert (x)φ opert (y) ]. (36) The first sum i this equatio gives δ(x + y) to all orders i powers of ɛ as explaied above i Subsectio 4.. The last sum is egligible compared with the secod ad third sums. We thus obtai C(x, y) =δ(x + y) ie 2 ( 3ɛ 4 ) [ D (x 2)C (y 2) + C (x 2)D (y ] 2), (37) ɛ! ɛ =0 where C ad D are defied i (32). Observe that the correctio to the delta fuctio (that is, the differece betwee the P operator ad the C operator) is operturbative; it is expoetially small ad imagiary. The summatio i (37) ca be coverted to a double itegral: { 2 C(x, y) =δ(x + y)+i π 3 ɛ e 3ɛ e 2 (x2 +y 2 ) x ( 2 ) 2 2s/ɛ cos θ ix isy exp +s 2 π 0 ds dθ 0 +s 2 +(x y) }. (38) This is the leadig-order operturbative approximatio to the coordiate-space represetatio of the operator C. Ackowledgemets This work was supported by the U.S. Departmet of Eergy. [] Beder C.M., Meisiger P.N. ad Wag Q., Calculatio of the hidde symmetry operator i PT -symmetric quatum mechaics, J. Phys. A, 2003, V.36, [2] Beder C.M. ad Boettcher S., Real spectra i o-hermitia Hamiltoias havig PT symmetry, Phys. Rev. Lett., 998, V.80, [3] Beder C.M., Brody D.C. ad Joes H.F., Complex extesio of quatum mechaics, Phys. Rev. Lett., 2002, V.89, [4] Beder C.M. ad Wag Q., Commet o a recet paper by G.A. Mezicescu: Some properties of eigevalues ad eigefuctios of the cubic oscillator with imagiary couplig costat [J. Phys. A, 2000, V.33, ], J. Phys. A, 200, V.34, ; Beder C.M., Boettcher S., Meisiger P.N. ad Wag Q., Two-poit Gree s fuctio i PT -symmetric theories, Phys. Lett. A, 2002, V.302, [5] Beder C.M., Meisiger P.N. ad Yag H., Calculatio of the oe-poit Gree s fuctio for a gφ 4 quatum field theory, Phys. Rev. D, 200, V.63,