Non-Hermitian CP-Symmetric Dirac Hamiltonians with Real Energy Eigenvalues

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1 Non-Hermitian CP-Symmetric irac Hamiltonians with Real Energy Eigenvalues.. lhaidari Saudi Center for Theoretical Physics, Jeddah 438, Saudi rabia bstract: We present a large class of non-hermitian non-pt-symmetric two-component irac Hamiltoninas with real energy spectra. These Hamiltonians are invariant under the combined action of "charge" conjugation (two-component transpose and space-parity. Eamples are given from the two subclasses of these systems having localized and/or continuum states with real energies. PCS: Pm, Ge, Nk, Fd Keywords: irac equation, non-hermitian, PT-symmetry, CP-symmetry, parity, real spectrum I. INTROUCTION In an introductory course to quantum mechanics, hermiticity (precisely, selfadjointness is the property that one usually employs to show that the energy eigenvalues of the Hamiltonian are real []. Nonetheless, in a seminal work by Bender and Boettcher in 998 [], it was shown that there eist non-hermitian Hamiltonians with real energy spectra. Since then, the physics and math literature on the subject grew very rapidly and many eamples were given for such systems. For a recent review, one may consult [3] and references therein. Soon after [], it was shown that Hamiltonian operators associated with this class of problems have a common property. They are invariant under the combined action of space inversion ( and time reversal (. That is, such Hamiltonians are -symmetric. The work on this subject was not limited to the nonrelativistic theory. In fact, Non-Hermitian irac Hamiltonians with real relativistic energy spectra were also found [3-]. ll studies in this area were limited to the -symmetric class of problems. In this work, however, we show that there eists a large class of non-hermitian non--symmetric two-component irac Hamiltoninas with real energy eigenvalues. The Hamiltonian matri operators in this class are invariant under the combined action of "charge" conjugation and space inversion. We formulate the problem in the following section and then give analytic and numerical eamples in Sec. III. II. FORMULTION The irac equation in n+ space-time for a particle, of rest mass m, under the influence of an eternal potential matri (, t is obtained by variation of the invariant n + action, L ( d, whose Lagrangian reads as follows Present ddress: Keech rive, Redwood City, C 94065

2 where L icψ ψ mc ψψ + ψψ, ( γ n is the irac operator, { γ } 0 irac gamma matrices, ( c, t matrices satisfy the anti-commutation relation { ν, } are constant square matrices (the 0 is the (n+-gradient, and ψ ψ γ. The irac γ γ ν, where the n+ Mikowski space-time metric is diag( +. The resulting irac equation (in the conventional relativistic units c is ( iγ m + ψ 0. ( In + space-time, we choose the representation of the irac gamma matrices as 0 0 γ σ3 ( 0 and γ iσ 0 i ( 0, which makes the potential (, t a matri and gives the following most general irac equation in + dimension ψ+ m + S + V + W ψ+ ψ+ i t H ψ U m S V ψ ψ, (3 + + where { SV,, W, U } are four independent space-time functions. The irac Hamiltonian matri is Hermitian (i.e., H H if and only if * W U P + iq and all potential functions { SV,, PQ, } are real. In that case, the interaction Lagrangian becomes 5 ( ( ( ψ ψ S ψψ + ψγ ψ P ψγ ψ, (4 where γ ( 5 iγ 0 γ 0 0 and the potential components carry well-defined irreducible representations of the Lorentz group. The scalar, pseudo-scalar, and vector potentials are S, P, and ( V, Q, respectively. Moreover, the space-component of the vector potential, Q, could also be eliminated by a U( gauge transformation. However, from now on, we take { SV,, W, U } in Eq. (3 to be real space-time functions. Thus, the irac Hamiltonian is self-adjoint (precisely, Hermitian if and only if U W. This is because the two parts of the Lagrangian relevant to this issue are W ( ψγ ψ U ( ψγ ψ W ( ψ σ ψ U ( ψ σ ψ, (5 where γ ( 5 γ iγ, σ 0 ( 0 0 and σ ( On the other hand, let us eamine the following transformed irac Hamiltonian matri 5 5 m + S + V + U γ H γ + W m S + V. (6 5 Thus, γ transposes the Hamiltonian matri (without comple conjugation. It also acts to echange the two components of the wavefunction as: ψ 5 ψ. In fact, γ has all T the properties of the charge conjugation matri,. That is, and ( γ T γ []. However, one may not be at liberty to etend this technical analogy too far into the physics to assume conjugation of electric charges. Nonetheless, from now on we replace the designation γ 5 by. pplying the space-parity operator ( : on (6 gives

3 m + Sˆ + Vˆ ˆ + U H, (7 ˆ ˆ ˆ + W m S + V where the caret over the function means f ˆ( t, f (, t and we have used the fact that space-parity and charge conjugation commute (. Therefore, we conclude that H H if and only if: St (, St (,, V (, t V (, t, and U(, t W (, t. For eample, if all the potential functions are even in, then U W and H is trivially Hermitian with H H H. However, if W is odd, then U W and the irac Hamiltonian is not Hermitian but still satisfies the condition H H. To be specific, we consider in this work the class of problems where S V 0 and W ( t, W ( t, giving the non-hermitian irac Hamiltonian m + W (, t H + W (, t m, (8 which is -symmetric. Then, the irac equation (3 for this class of problems reads as follows in components form i tψ+ mψ+ + ψ + W (, t ψ, (9a i tψ mψ ψ+ + W (, t ψ+. (9b It is worth noting that the irac Hamiltonian (8 is not -symmetric. That is, H H, where is the time-reversal operator, which has the effect of comple conjugation, i i. It is worth mentioning that the -symmetry being discussed in the quantum mechanics literature regarding non-hermitian Hamiltonians with real spectra (including the present work should not be identified with the fundamental -symmetry discussed in quantum field theory and particle physics []. Now, for time independent potential functions, W(, we can write ψ ( t, i t e ε χ (, where ε is the system's energy. In that case, Eq. (9 gives the following relations between the two components of the wavefunction d χ ( + W ( + χ( ε m d, ε m (negative energy, (0a d χ ( W ( χ+ ( ε + m d, ε m (positive energy, (0b resulting the following second order differential equations d d + ( W U WU U + ε m χ + 0 d d, (a d d + ( W U WU + W + ε m χ 0 d d, (b where U ( W ( and the prime stands for the derivative with respect to. The question now is as follows: re there real potential functions, W(, that result in nonzero physical solutions (bound and/or continuous of Eq. ( for real energies ε? We will show (by eample in the following section that the answer is affirmative. 3

4 For positive energies, one solves Eq. (a to obtain χ + then substitute that in Eq. (0b to obtain χ. On the other hand, for negative energies, one solves Eq. (b to obtain χ then substitute that in Eq. (0a to obtain χ +. It should be emphasized that the solution of Eq. (a does NOT belong to the same energy subspace as that of Eq. (b and the complete solution space is a union of these two subspaces. In the following section, we illustrate our findings by giving various eamples (analytic and numerical of bound and continuum solutions of Eq. ( for real energies. If we define ρ ( [ ( ( ] III. EXMPLES ρ ( W y W y dy and write χ ( e φ (, then Eq. ( becomes d R R + ε m φ ( 0 d, ( where ρ ( R ( [ W ( + W ( ]. For bound state, lim e 0, whereas for scattering states ρ ( lim e is finite. The effective potential associated with φ ( is V ( ( R R, which has the structure of the superpotentials in supersymmetric quantum mechanics (SSQM [3]. Thus, besides performing the integration to find ρ(, all the tools of SSQM and shape invariance are at our disposal to find the solution of Eq. (. We divide the class of solutions of the irac equation (9 into two. The odd parity subclass associated with the potential functions W ( W ( and a larger subclass without definite parity, which is associated with W ( W (. In these cases V ( 0. Eamples in the subclass W ( W ( ρ ( ik ik and χ ( e ( e B e +, where k ε m. For ε < m, the oscillatory factor becomes growing and decaying eponentials of which we choose the latter. s a particular eample, we take W ( ( n + where and are real parameters such that > 0, and n is a non-negative integer. In such a ( case, we obtain bound states with ( ( n + ρ. nother eample is when n + W ( sinh( with > 0, then we obtain the bound states χ ( cosh( ik ik e ( e B e cos( ik ik the continuum states χ ( e ( e B e problems in this subclass. +. On the other hand if W ( sin(, then we obtain +. Table is a sample list of 4

5 B. Eamples in the subclass W ( W ( If we take W ( e such that < 0, then we obtain bounded solutions for all cosh( sinh ( sinh( + ε M φ ( 0 real energies with χ ( e φ ( and Eq. ( becomes d d, (3 where M m +. This is a Schrödinger equation for φ ( with the confining potential V ( sinh(. Thus, we epect that all real energy 8 solutions be bounded. second eample is for W ( ( e with > 0 giving ρ( [( sinh( cosh( ] and φ ( is a solution of Schrödinger equation with the effective potential V ( ( R R, where R ( ( cosh(. In Fig. and Fig., we show few plots of χ ( from this subclass, which are obtained numerically for the given set of physical parameters. Table is a sample list of problems in this subclass. CKNOWLEGEMENTS The uthor is grateful for the support granted by the Saudi Center for Theoretical Physics (SCTP. We also appreciate the help in literature search provided by S. l- Marzoug and H. Bahlouli. REFERENCES: [] See, for eample, R. Shankar, Principles of Quantum Mechanics, nd ed. (Springer-Verlag, 994; R. Liboff, Introductory Quantum Mechanics, 4 th ed. (ddison-wesley, 00;. J. Griffiths, Introduction to Quantum Mechanics, nd ed. (ddison-wesley, 004; N. Zettili, Quantum Mechanics: Concepts and pplications (Wiley & Sons, 009 [] C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 543 (998 [3] O, Yesiltas, J. Phys. 46, 0530 (03 [4] L. B. Castro, Phys. Lett. 375, 50 (0 [5] F. Cannata and. Ventura, J. Phys. 43, (00; ibid 4, (008; Phys. Lett. 37, 94 (008 [6] V. G. C. S. dos Santos,. de Souza utra, and M. B. Hott, Phys. Lett. 373, 340 (009 [7] Y. J. Ng and H. van am, Phys. Lett. B 673, 37 (009 [8] C-S Jia, Y-F iao, and J-Y Liu, Inter. J. Theor. Phys. 47, 664 (008 [9]. Sinha and P. Roy, Inter. J. Mod. Phys., 5807 (006; Mod. Phys. Lett. 0, 377 (005 [0] C-S Jia and. de Souza utra, J. Phys. 38, 877 (006 [] H. Egrifes and R. Sever, Phys. Lett. 344, 7 (005 [] See, for eample, C. Itzykson and J. B. Zuber, Quantum Field Theory (McGraw- Hill, 980; W. Greiner, Relativistic Quantum Mechanics: Wave Equations 5

6 (Springer-Verlag, 990; J.. Bjorken and S.. rell, Relativistic Quantum Mechanics (McGraw-Hill, 998; F. Gross, Relativistic Quantum Mechanics and Field Theory (Wiley & Sons, 999; F. Mandl and G. Shaw, Quantum Field Theory, nd ed. (Wiley & Sons, 00 [3] See, for eample,. Gangopadhyaya, J. V. Mallow, and C. Rasinariu, Supersymmetric Quantum Mechanics: n Introduction (World Scientific, 0; F. Cooper,. Khare, and U. P. Sukhatme, Supersymmetry in Quantum Mechanics (World Scientific, 00 TBLE CPTION: Table : Sample list of -symmetric problems in the subclass W ( W ( with χ ( ρ ( ik ik ( e e B e + and k ε m. Table : Sample list of -symmetric problems in the subclass W ( W ( with ρ ( φ χ ( e ( and ( φ are solutions of d ( R R m +. ε φ 0 d FIGURE CPTION: FIG. : Plots of χ ( for the case W ( e. The upper (lower component is shown with red (blue color. The physical parameters are m,.0m, and 0.m. The boundary conditions are chosen as χ + (0 and χ (0 0. The plots are given for (a positive energy ε +.0m, and (b negative energy ε.0m. + ( FIG. : Plots of χ( χ+ + χ for the case W (. The physical + ( parameters are m,.0m, and 0.5m. The boundary conditions are chosen as χ + (0 and χ (0 0. The plots are given for (a positive energy ε +.5m, and (b negative energy ε.5m. 6

7 Table W( ρ( Remarks ( n n + ( n + + ( > 0, localized sinh( cosh( > 0, localized sin( cos( Non-localized sinh ( ( sinh ( ( + > 0, localized ( ln( ( ln( > 0, localized Table W( ρ( R( Remarks cosh( cosh( < 0, localized e ( e [( sinh( cosh( ] ( sinh( > 0, localized ( e ( y + cosh( y y sinh( y y cosh( y < 0, y, localized + ( + ( ln + ( + ( > 0, localized 7

8 Fig. Fig. 8