PT symmetry in relativistic quantum mechanics
aa r X i v : . [ h e p - t h ] J u l P T symmetry in relativistic quantum mechanics
Carl M. Bender and Philip D. Mannheim Physics DepartmentWashington UniversitySt. Louis, MO 63130, USA electronic address: [email protected] Department of PhysicsUniversity of ConnecticutStorrs, CT 06269, USA electronic address: [email protected] (Dated: 2 July 2011)
Abstract
In nonrelativistic quantum mechanics and in relativistic quantum field theory, time t is a pa-rameter and thus the time-reversal operator T does not actually reverse the sign of t . However,in relativistic quantum mechanics the time coordinate t and the space coordinates x are treatedon an equal footing and all are operators. In this paper it is shown how to extend PT symme-try from nonrelativistic to relativistic quantum mechanics by implementing time reversal as anoperation that changes the sign of the time coordinate operator t . Some illustrative relativisticquantum-mechanical models are constructed whose associated Hamiltonians are non-Hermitian but PT symmetric, and it is shown that for each such Hamiltonian the energy eigenvalues are all real. . INTRODUCTION In nonrelativistic quantum mechanics the position x ( t ) is taken to be an operator whilethe time t is only a c -number parameter. To make quantum mechanics relativistic one musttreat time and space equivalently. There are then two possibilities: one can either demotethe spatial coordinates to parameters or promote the time coordinate to an operator. Theformer prescription is used in quantum field theory, where the field operators are treated asfunctions of the spacetime parameters x and t , but one can also construct sensible quantum-mechanical theories via the latter approach [1, 2]. In such theories a new parameter isneeded to parameterize evolution, and thus one introduces a fifth coordinate τ that is an SO (3 ,
1) Lorentz scalar. In this five-dimensional formalism the space and time coordinates x µ ( τ ) become operator functions of τ and one obtains an SO (3 , x j , p k ] = iδ j,k :[ x µ ( τ ) , p ν ( τ )] = iη µν , [ x µ ( τ ) , p ν ( τ )] = iδ µν , (1)where η µν is the SO (3 ,
1) Minkowski metric.The dynamics in this formalism is SO (3 ,
1) invariant in the four operators x µ , but isnonrelativistic in the fifth coordinate τ [because the dynamics is not SO (4 ,
1) or SO (3 , τ . However, just as the nonrelativistic quantum-mechanical operator x ( t ) can propagate forward and backward with respect to its timeparameter t , in relativistic quantum mechanics all four components of x µ ( τ ) can propagateforward and backward in τ [3]. The five-dimensional formalism of [1, 2] readily incorpo-rates forward and backward time propagation, so one can introduce antiparticles with firstquantization alone without requiring the second-quantization techniques of quantum fieldtheory.When the five-dimensional Hamiltonian operator ˆ H is Hermitian and its eigenfunctionshave the separable form ψ n ( x µ , τ ) = φ n ( x µ ) e − iE n τ and when its states obey the standardDirac completeness relation X | n ih n | = I, (2)the five-space forward propagator takes the form G ( x µf , τ ; x µi ,
0) = − iθ ( τ ) h x µf | e − i ˆ Hτ | x µi i = − iθ ( τ ) X φ n ( x µf ) φ ∗ n ( x µi ) e − iE n τ . (3)2his propagator obeys a Schr¨odinger equation that is first order in τ : (cid:16) i∂ τ + ˆ H (cid:17) G ( x µ , τ ; 0 ,
0) = δ ( τ ) δ ( x µ ) . (4)The four-dimensional propagators in the five-dimensional formalism are constructed byintegrating out the fifth coordinate. Given (3), the associated four-dimensional propagatoris then defined as G ( x µf ; x µi ) = N Z ∞−∞ dτ G ( x µf , τ ; x µi , , (5)where N is a normalization constant. Using the integral representation θ ( τ ) = − πi Z ∞−∞ dν e − iντ ν + iǫ , (6)one then obtains G ( x µf ; x µi ) = N X φ n ( x µf ) φ ∗ n ( x µi ) − E n + iǫ . (7)Finally, since the wave functions are eigenfunctions of ˆ H , G ( x µf ; x µi ) obeys − ˆ HG ( x µ ,
0) =
N δ ( x µ ) . (8)The primary objective in using the approach of [1, 2] is to choose a five-dimensional ˆ H so that G ( x µ ,
0) obeys a differential wave equation of the form D G ( x µ ,
0) = δ ( x µ ) , (9)where D is one of the familiar wave operators that appear in quantum field theory (suchwave operators are typically higher than first derivative in time). [Normalizing G ( x µ ,
0) ac-cording to (9) would fix the constant N .] Thus, in the simple case where the five-dimensionalHamiltonian has the form ˆ H = ˆ¯ p − (ˆ p ) + m , (7) is the Fourier transform of the standardfour-dimensional scalar field Feynman propagator D = ∂ µ ∂ µ . Because forward propagationin τ gives the correct iǫ prescription for the causal Feynman contour in four dimensions, D = ∂ µ ∂ µ is the usual four-dimensional Klein-Gordon operator. Using the five-dimensionalformalism, one can solve for a one-body quantum-mechanical Schr¨odinger-type propagatorin five space, and from it one can construct a many-body quantum-field-theoretic propagatorin four space. The five-space formalism also permits one to choose five-space Hamiltoniansfor which the resulting four-space propagator does not obey an equation of the form (9) witha familiar D . In this paper we construct some simple models that lead to a propagatorequation with a familiar D and some that have a more general structure.3hen Lorentz invariance was introduced in classical mechanics, it described the invari-ance properties of the line element ds = dt − d x . In addition to invariance under thecontinuous orthochronous Lorentz transformations, the line element also possesses a set ofdiscrete invariances, namely, space reflection P : x → − x , t → t , time reversal T : x → x , t → − t , and their product spacetime reflection PT : x → − x , t → − t . However, whentime reversal was introduced into quantum mechanics by Wigner, the time reflection ofthe i∂/∂t operator was achieved not by replacing t by − t but rather by taking T to bean antiunitary operator that transforms i into − i ( T : i → − i ); time t was treated as a c -number parameter that is not affected by T . In relativistic quantum field theory, timereversal is also not implemented by making the direct replacement t → − t even though theline element ds = dt − d x possesses this time-reversal invariance. As noted above, inthe five-dimensional relativistic quantum-mechanical approach used here, we treat time asan operator and thus we can implement a time-reversal operation that acts directly on thetime. We can also implement PT transformations directly on the time operator.The ability to implement a PT transformation on the time operator is appealing becauseof the implications of PT invariance for Hamiltonians that are not Hermitian. In the lastfew years it has been recognized [4–7] that a quantum-mechanical Hamiltonian that is notHermitian may still have an entirely real set of energy eigenvalues. In the cases that wereexplicitly considered in [4–7], the reality of the eigenvalues was traced to the existence of anunderlying invariance of the Hamiltonian with respect to a combined PT reflection. Thus,while Dirac Hermiticity of the Hamiltonian is sufficient for reality of eigenvalues, it is notnecessary. (Of course, Hamiltonians with entirely real eigenvalues can be both PT invariantand Dirac Hermitian.) However, recently it has been shown [8] that a Hamiltonian thatis not PT invariant cannot have an entirely real set of energy eigenvalues. This meansthat PT invariance, in contrast to Dirac Hermiticity, is necessary for the reality of energyeigenvalues [9]. (If one knows only that a Hamiltonian is not Dirac Hermitian, one can saynothing about the reality of the eigenvalues.) Thus, PT invariance of a Hamiltonian is abroader requirement than Dirac Hermiticity.In the non-Hermitian PT -invariant context we apply the five-dimensional formalism de-scribed above. To do this we recall [9] that when a Hamiltonian is PT invariant, its eigen-values are either real or they come in complex conjugate pairs. Consequently, both ˆ H andits Dirac-Hermitian conjugate ˆ H † have the same eigenspectrum, and they are related by4ome similarity transform V [10] V ˆ HV − = ˆ H † . (10)In this case, if | n i is a right eigenvector | R i of ˆ H , then h n | V rather then h n | is a left eigen-vector h L | of ˆ H . Consequently, the energy-eigenstate-completeness relation (2) is replacedby X | R ih L | = X | n ih n | V = I, (11)and (3) and (7) are replaced by G ( x µf , τ ; x µi ,
0) = − iθ ( τ ) h x µf | e − i ˆ Hτ | x µi i = − iθ ( τ ) X h x µf | n i e − iE n τ h n | V | x µi i , (12) G ( x µf ; x µi ) = N X h x µf | n ih n | V | x µi i− E n + iǫ . (13)The propagator (13) is the relevant one in the PT case, and with its PT -symmetric ˆ H , italso obeys (8).Invariance under PT reflection is a more physical requirement than Hermiticity becausethe proper orthochronous Lorentz group has a complex PT extension. Until now, this aspectof the Lorentz group has not been utilized because transformations that reverse the sign ofthe time have not been considered. In the present paper we consider such transformationsand explicitly extend PT symmetry to the relativistic quantum-mechanical domain. Inparticular we study some simple non-Hermitian but PT -symmetric SO (3 , II. A SIMPLE FIVE-DIMENSIONAL PT -SYMMETRIC HAMILTONIAN The generic five-dimensional action has the form I = R τ dτ ′ L ( τ ′ ), where τ is the endpoint of integration. We begin with a simple example that illustrates the five-dimensionalformalism. Specifically, we take a Lagrangian of the form L = m x µ ˙ x µ − mω x µ x µ − ia µ x µ − a µ a µ ) , (14)where µ = (0 , , , τ , and a µ is a real, exter-nal, τ -independent four-vector operator that commutes with x µ . As constructed, the action5s a relativistic SO (3 ,
1) scalar function of the four x µ coordinates, but it is nonrelativisticin the fifth coordinate τ . We define a canonical momentum p µ ≡ δIδ ˙ x µ = m ˙ x µ , (15)and then eliminate ˙ x µ to obtain a canonical Hamiltonian H = p µ ˙ x µ − L = 12 m p µ p µ + mω x µ x µ − ia µ x µ − a µ a µ ) . (16)The Hamiltonian (16) is not Dirac Hermitian because of the ia µ x µ term.Next, we assign P and T quantum numbers to the x µ and p µ operators, just as we dowith the nonrelativistic x and p = d x /dt ; to wit, we take the three spatial components x k to be P odd [ P x k ( τ ) P − = − x k ( τ )] and T even [ T x k ( τ ) T − = x k ( − τ )], and takethe three spatial components p k = dx k /dτ to be P odd [ P p k ( τ ) P − = − p k ( τ )] and T odd [ T p k ( τ ) T − = − p k ( − τ )]. Similarly, we take the time component x to be P even[ P x ( τ ) P − = x ( τ )] and T odd [ T x ( τ ) T − = − x ( − τ )] and take the time component p = dx /dτ to be P even [ P p ( τ ) P − = p ( τ )] and T even [ T p ( τ ) T − = p ( − τ )]. Withthese assignments the four x µ are PT odd while the four p µ are PT even. Because T alsoconverts i to − i , these assignments are consistent with the commutation algebra in (1). Wesummarize these assignments as follows: p p x x P − + − + T − + + −PT + + − − (17)Finally, we take the four-vector a µ to be PT even. For our purposes we will need a tobe P even and thus T even, and a to be P odd and thus T odd. In the five-space theHamiltonian (16) is conjugate to τ and not to x . Neither P nor T affect τ because τ is onlya parameter, so with p µ ˙ x µ = p µ p µ /m being PT even, the Hamiltonian is PT symmetric.To determine the energy eigenvalues we take the spacetime metric to be diag( η µν ) =( − , , , x µ = ( t, x, y, z ), we obtain a wave-mechanics representation of thealgebra (1) when p µ = − i∂/∂x µ ; that is, p = − i ∂∂t , p k = − i ∂∂x k . (18)6onsequently, in five-space the Schr¨odinger equation takes the form i ∂ψ ( τ, x µ ) ∂τ = " − m η µν ∂∂x µ ∂∂x ν + mω x µ − ia µ )( x µ − ia µ ) ψ ( τ, x µ ) . (19)The substitution y µ = x µ − ia µ brings (19) to the form i ∂ψ ( τ, y µ ) ∂τ = " m ∂ ∂t − ∂ ∂ y ! + mω y − t ) ψ ( τ, y µ ) , (20)and reduces the Schr¨odinger equation to a four-dimensional harmonic oscillator withMinkowski signature. Noting that " m ∂ ∂t − ∂ ∂ y ! + mω y − t ) e − mω ( y − t ) / = 2 ωe − mω ( y − t ) / , (21)we see that the t -dependent sector contributes a positive zero-point energy equal to ω/ y -dependent sector does. Because all the eigenvalues of a harmonic oscillator are real,the five-space energy eigenvalues of (19) are given by E = ( n x + n y + n z + n t + 2) ω, (22)where each of n x , n y , n z and n t ranges over the positive integers. Thus, while the Hamiltonian(16) is not Hermitian, all of its energy eigenvalues are real.For this model the five-space propagator obeys " i ∂∂τ + 12 m ∂∂x µ ∂∂x µ − mω x µ − ia µ )( x µ − ia µ ) G ( x µ , τ ; 0 ,
0) = δ ( τ ) δ ( x µ ) (23)and we show in Appendix A that G ( x µ , τ ; 0 ,
0) = θ ( τ ) 1(sin ωτ ) exp " imω cos ωτ ( x µ − ia µ )( x µ − ia µ )2 sin ωτ . (24)The propagator of the associated four-dimensional theory is then obtained via (5), and itobeys (8) with the PT -symmetric ˆ H = − ∂ µ ∂ µ / m + mω ( x µ − ia µ )( x µ − ia µ ) / PT -theory techniques described in [6], one can demonstrate the reality of theeigenvalues algebraically without actually solving the Schr¨odinger equation. To do so, onemust construct an operator e Q that possesses four key properties: (i) a similarity transfor-mation using e Q preserves the commutation relations; (ii) Q is a Hermitian operator (sothat e Q is not unitary); (iii) like V in (10), e Q effects the transformation e −Q He Q = H † ; (25)7iv) the operator ˜ H = e −Q / He Q / (26)obeys ˜ H † = ˜ H . The existence of such a Q operator implies that the energy eigenvalues of H are all real.We now construct the Q operator for our simple five-dimensional model. Note that themomentum operator will effect the transformation e − b ν p ν x µ e b ρ p ρ = x µ + ib µ , (27)and leave the commutation relations (1) untouched for any four-vector b µ that commuteswith both x µ and p µ . Given (27), we identify Q as the Hermitian operator 2 a ν p ν because e − a ν p ν He a ρ p ρ = 12 m p µ p µ + mω x µ x µ + 2 ia µ x µ − a µ a µ ) = H † . (28)Similarly, the transformation e − a ν p ν He a ρ p ρ = 12 m p µ p µ + mω x µ x µ = ˜ H (29)generates an equivalent Hamiltonian ˜ H that is manifestly Hermitian.In PT quantum mechanics one introduces an operator C that is required to obey[ C , H ] = 0 , C = I. (30)One constructs this operator by making the ansatz C = e Q P , where the operator P obeys P = I . In this form, the operator C fulfills the condition C = I provided that Q satisfies PQP = −Q . With e −Q generating e −Q He Q = H † , the operator C obeys C − H C = H if P generates P H P = H † . For the Q and H of interest here, both PQP = −Q and P H P = H † hold provided that a is P odd and a is P even. With this choice for the parity of a µ , wethen identify C = e Q P . (Previously, we had required that a µ be PT even.) Then, if both a µ and p µ are PT even, the operator Q is PT even. As constructed, C thus obeys [ C , PT ] = 0,as expected [8, 9] when all energy eigenvalues are real [11]. III. FIVE-DIMENSIONAL PAIS-UHLENBECK OSCILLATOR
In 1950 Pais and Uhlenbeck [12] explored the question of whether the Pauli-Villars reg-ulator associated with the fourth-order equation of motion( ∂ t − ∇ + M )( ∂ t − ∇ + M ) φ ( x , t ) = 0 (31)8nd propagator D ( k ) = 1( k + M )( k + M ) = 1 M − M k + M − k + M ! , (32)where k = − ( k ) + k , could be physically viable, or whether it was merely a mathematicaltechnique to regulate Feynman integrals. To this end they replaced the scalar field φ ( x , t )by a single coordinate z ( t ) and examined single momentum modes ω = k + M and ω = k + M . The resulting nonrelativistic quantum-mechanical limit of the equation ofmotion (31) and the propagator (32),( ∂ t + ω )( ∂ t + ω ) z ( t ) = 0 , G ( E ) = 1 ω − ω E − ω − E − ω ! , (33)is known as the PU oscillator.Pais and Uhlenbeck found that if the theory were quantized with a standard positive-metric Hilbert space, the energy spectrum would not be bounded below. One can evade thisnegative-energy problem by quantizing the theory in a negative-metric Hilbert space, butas the relative minus sign in (33) indicates, the disadvantage of doing so is that one obtainsstates of negative Dirac norm and evidently loses unitarity.The PU oscillator was revisited in 2008 [13, 14] and a new realization of the theory wasfound in which the Hilbert space has neither negative-energy nor negative-norm states. Inthis realization the Hamiltonian is not Dirac-Hermitian but is instead PT invariant. Thenorm is given by h L | R i = h n | V | n i , rather than by the Dirac norm h n | n i , and the complete-ness relation is given by (11) rather than by (2). In analogy with (13), the relative minussigns in (32) and (33) are generated by the presence of the V operator in the propagatorand not by quantizing with an indefinite metric. This realization took a long time (morethan half a century) to discover because the Hamiltonian of the theory appeared to be DiracHermitian even though it is not. (In Refs. [13, 14] the nonrelativistic PT realization of thePU oscillator is studied, and in Ref. [14] the relativistic scalar field theory is examined.)For the case of the nonrelativistic PU oscillator, the equation of motion (33) for thecoordinate z ( t ) can be derived by a stationary variation of the PU oscillator action I PU = γ Z dt h ¨ z − (cid:16) ω + ω (cid:17) ˙ z + ω ω z i , (34)where γ , ω and ω are positive constants. Since ˙ z serves as the conjugate of both z and ¨ z ,the action is constrained. One thus replaces ˙ z by a new variable x , and using the method9f Dirac constraints, one obtains [15, 16] the Hamiltonian H PU = p x γ + p z x + γ (cid:16) ω + ω (cid:17) x − γ ω ω z (35)with two canonical pairs that obey [ x, p x ] = i and [ z, p z ] = i .In the realization of the theory for which the energy eigenvalues are bounded below, H PU appears to be Hermitian but it is not. Specifically, one solves the Schr¨odinger equation forthe ground state of the system with energy E = ( ω + ω ) /
2. The eigenfunction is ψ ( z, x ) = exp (cid:20) γ ω + ω ) ω ω z + iγω ω zx − γ ω + ω ) x (cid:21) . (36)This eigenfunction diverges exponentially for large z , so integration by parts generates sur-face terms that cannot be discarded. Thus, one cannot represent the operator p z by − i∂ z .However, one can replace z by iz (this is equivalent to working in a Stokes wedge in thecomplex- z plane that includes the imaginary z axis but not the real one [13]), and represent p z by − i∂ iz = − ∂ z . The eigenfunction then vanishes exponentially as z becomes large. Thehighly unusual implication of the structure of (36) (and the reason it took so long to find) isthat while both conjugate pairs of coordinates are obtained from the same Lagrangian, thecommutator [ x, p x ] = i is realized by Hermitian operators, while the commutator [ z, p z ] = i is realized by anti-Hermitian operators. As a result, the p z x cross-term in (35) is not Her-mitian, and the Hamiltonian H PU is also not Hermitian.Rather than using non-Hermitian operators, we make the similarity transformation y = e πp z z/ ze − πp z z/ = − iz, q = e πp z z/ p z e − πp z z/ = ip z , (37)to construct Hermitian operators y and q that obey [ y, q ] = i . In terms of y and q theHamiltonian now takes the form H PU = p γ − iqx + γ (cid:16) ω + ω (cid:17) x + γ ω ω y , (38)where for notational simplicity we have replaced p x by p . The Hamiltonian H PU is nowmanifestly non-Hermitian.While H PU is not Hermitian, the P and T quantum-number assignments p x q y P − − + +
T − + + −PT + − + − (39)10ake H PU symmetric under PT reflection. Introducing the operator Q = αpq + βxy, α = 1 γω ω log (cid:18) ω + ω ω − ω (cid:19) , β = αγ ω ω , (40)we then find that [13, 14] the similarity-transformed PU Hamiltonian˜ H PU = e −Q / H PU e Q / = p γ + q γω + γ ω x + γ ω ω y (41)represents two uncoupled harmonic oscillators. The transformed Hamiltonian ˜ H PU in (41)is both Hermitian and manifestly positive definite. This realization of the quantum theory,which is associated with the non-Hermitian H PU , has no negative-norm or negative-energyeigenstates [17].Because the transformation with e Q / is not unitary, the propagator D ( H PU ) = h x ′ , y ′ | e − iH PU t | x, y i = h x ′ , y ′ | e Q / e − i ˜ H PU t e −Q / | x, y i (42)associated with H PU does not transform into the propagator D ( ˜ H PU ) = h x ′ , y ′ | e − i ˜ H PU t | x, y i (43)that one would ordinarily associate with a two-uncoupled-oscillator system. The state h x, y | e Q / is not the conjugate of e −Q / | x, y i , and the propagators in (42) and (43) arenot equivalent; for this realization of the PU Hamiltonian we must use (42) and not (43).The dependence on the operator V = e −Q is crucial because it generates the relative minussign in (33).We now illustrate PT invariance in relativistic quantum mechanics by applying the five-dimensional formalism to the PU oscillator. We will see that a straightforward covariantgeneralization of the PU oscillator does not lead back to (32). Consequently, in the nextsection we provide an alternate five-dimensional formalism that does.To generalize the PU oscillator to relativistic quantum mechanics we replace (34) by I = γ Z τ dτ h ¨ z µ ¨ z µ − (cid:16) M + M (cid:17) ˙ z µ ˙ z µ + M M z µ z µ i , (44)where the dot denotes differentiation with respect to τ . Because of constraints associatedwith this action, the Hamiltonian has the form H = ( p x ) µ ( p x ) µ γ + ( p z ) µ x µ + γ (cid:16) M + M (cid:17) x µ x µ − γ M M z µ z µ . (45)11ecalling the transformation in (37), we let y µ = − iz µ and q µ = i ( p z ) µ . On setting( p x ) µ = p µ we obtain two canonical pairs of operators that obey[ x µ ( τ ) , p ν ( τ )] = iη µν , [ q µ ( τ ) , y ν ( τ )] = iη µν , (46)and a Hamiltonian of the form H = p µ p µ γ − iq µ x µ + γ (cid:16) M + M (cid:17) x µ x µ + γ M M y µ y µ . (47)The assignments p p x x q q y y P − + − + + − + −T − + + − + − − + PT + + − − + + − − (48)in which x changes sign under T , then establish that H PU is PT symmetric.Next, we introduce the operator Q = αp µ q µ + βx µ y µ , α = 1 γM M log (cid:18) M + M M − M (cid:19) , β = αγ M M , (49)and find that ˜ H = e −Q / He Q / = p µ p µ γ + q µ q µ γM + γ M x µ x µ + γ M M y µ y µ . (50)Thus, the energy eigenvalues of the PT -symmetric Hamiltonian H are all real.We show in Appendix A that if we set M = M and M = 0, the five-space propagator is G ( x µ , y µ , τ ; 0 , ,
0) = θ ( τ ) e iB/A A , (51)where2 B/γ = M x µ x µ (sin M τ − M τ cos
M τ ) − M y µ y µ sin M τ + 2 iM x µ y µ (1 − cos M τ ) ,A = 2 − M τ − M τ sin
M τ. (52)The propagator of the associated four-dimensional theory may now be obtained byperforming the integral in (5), and the resulting propagator will obey (8) with ˆ H = − (1 / γ ) ∂/∂x µ ∂/∂x µ − x µ ∂/∂y µ + γM x µ x µ /
2. While of interest in itself, this propaga-tor is not of the generic Pauli-Villars form given in (32). Thus, in Sec. IV we provide analternate choice for the five-dimensional Hamiltonian that will lead to (32).12
V. ALTERNATE FORMULATION OF THE FIVE-SPACE PU OSCILLATOR
Given the structure of (31) we take the five-space ˆ H to have the operator formˆ H = − [ − (ˆ p ) + ˆ¯ p + M ][ − (ˆ p ) + ˆ¯ p + M ] . (53)For this Hamiltonian the five-dimensional energies are given by E = − [ − ( p ) + ¯ p + M ][ − ( p ) + ¯ p + M ] , (54)where the momenta in (54) are the eigenvalues of the operators in (53). Inserting theseenergies into (7), we obtain the Pauli-Villars propagator in (32), with (9) being satisfied.Equation (53) leads directly to (32), but its use here is nonstandard because it doesnot have a simple Lagrangian counterpart. In the previous examples and in the deriva-tion of the Klein-Gordon propagator, one can start with a five-dimensional action (of theform R τ dτ ˙ x µ ˙ x µ for the specific Klein-Gordon case) and by a canonical procedure derivea Hamiltonian from it. The Lagrangians in these examples are quadratic functions ofthe coordinates, so the procedure is straightforward and yields Hamiltonians that are alsoquadratic. However, the Hamiltonian (53) is not quadratic; it is quartic because the waveoperator in (31) is a fourth-order derivative operator [18]. Since the Lagrangian is given by L ( ˙ x µ ) = p µ ˙ x µ − H ( p µ p µ ) and since p µ = ∂L/∂ ˙ x µ , one can in principle construct L ( ˙ x µ ) ifone knows H ( p µ p µ ). Doing so for (53) is difficult, so we start directly with H ( p µ p µ ). Oncewe have H ( p µ p µ ), we can then use the representation in (7) without needing to know thestructure of the Lagrangian.We can recover the four-dimensional Pauli-Villars propagator, but at first it appears thatthe Hamiltonian in (53) is Hermitian. Moreover, in the second-order Klein-Gordon casewith ˆ H = − ( p ) + ¯ p + M and real E the Hamiltonian is Hermitian. However, in thefourth-order case, we note that ( p ) is given as( p ) = 12 (cid:16) E + E ± [( E − E ) − E ] / (cid:17) , (55)where E i = ¯ p + M i . Thus, now there can be real values of E for which ( p ) is complexand for which the operator (ˆ p ) , and thus ˆ H , is not Hermitian. (Note that with E beingreal, the Hamiltonian must be PT invariant.)In addition, we note that for general M and M , if we take E to be zero, the eigen-functions associated with the operator ˆ H in (53) will have the form ψ = e − iE t + i ¯ p · ¯ x and13 = e − iE t + i ¯ p · ¯ x . However, if we then set M = 0 and M = 0, there will be eigenfunc-tions of the form ψ a = e − ipt + i ¯ p · ¯ x and ψ b = n µ x µ e − ipt + i ¯ p · ¯ x , where n µ is the unit timelike vector n µ = (1 , , , ψ a and ψ b eigenfunctions, only ψ a is stationary; ψ b grows linearlyin the time coordinate, which indicates that the Hamiltonian has Jordan-block form and thatit has an incomplete set of eigenvectors. Consequently, the Hamiltonian ˆ H in (53) cannotbe diagonalized and is not Hermitian. Since ˆ H is not Hermitian when E = M = M = 0,it must also not be Hermitian for a range of values of these parameters. In Ref. [14] it wasfound that in the equal-frequency limit ω = ω of the PU oscillator, the Hamiltonian in(38) is also nondiagonalizable and non-Hermitian.The solutions to (55) thus break up into two sectors. In one sector the Hamiltonianis Hermitian and the energy eigenvalues are unbounded below (4 E < ( M − M ) ). Inthe other sector the Hamiltonian is not Hermitian and the energies are bounded below(4 E > ( M − M ) ), just as in the case of the nonrelativistic PU oscillator. If E is real,the Hamiltonian is PT invariant in both cases. In the sector where ˆ H is Hermitian the four-space propagator is given by (7). In the non-Hermitian sector the four-space propagatoris given by (13) and as before, the V operator then generates the relative minus sign inthe Pauli-Villars propagator [19]. Our five-space treatment of the Pauli-Villars propagatorbased on (53) recovers the key features of the analyses of Refs. [13, 14]. We see that onecan extend PT symmetry to the five-dimensional formalism, and while we have not directlystudied the time-reversal and PT properties of the time operator in the Pauli-Villars case,those properties follow directly from the commutation relations (1) depending on how theyare explicitly specified for (ˆ p ) . V. SUMMARY
Using a number of elementary models, we have shown in this paper that the standardtechniques of PT quantum mechanics extend and apply to relativistic quantum mechanics,where the time-reversal operator T reverses the sign of the time operator x . We concludethat relativistic PT -symmetric quantum mechanics is physically viable.The work of CMB is supported by a grant from the U.S. Department of Energy.14 ppendix A: Construction of Five-space Propagators To construct propagators that obey the five-dimensional equation (cid:16) i∂ τ + ˆ H (cid:17) G ( x µ , τ ; 0 ,
0) = δ ( τ ) δ ( x µ ), we first recall how a propagator is constructedwhen the eigenmodes of ˆ H are plane waves. For the nonrelativistic quantum-mechanicalfree particle in one space dimension there is a plane wave basis and the propagator is givenby G ( x, t ; 0 ,
0) = − iθ ( t )2 π Z dp e − ipx − ip t/m . (A1)When i∂ t acts on − iθ ( t ), we generate the δ ( t ) δ ( x ) term, while if we omit the θ ( t ) function,the rest of the propagator obeys " i ∂∂t + 12 m ∂ ∂x R ( x, t ; 0 ,
0) = 0 , (A2)where G ( x, t ; 0 ,
0) = θ ( t ) R ( x, t ; 0 , G ( x, t ; 0 ,
0) = θ ( t ) (cid:18) m πit (cid:19) / e imx / t . (A3)The term I STAT = mx / t in the exponent is the value of the classical action I =( m/ R t dt ˙ x for the stationary path ¨ x = 0 between the end points ( x = 0 , t = 0) and( x, t ).If one were to calculate this propagator as a path integral R [ dx ] e iI over a complete basisof paths between the end points, one would obtain the same e iI STAT phase, but one wouldnot know the multiplicative pre-factor. This pre-factor is determined by requiring thatthe propagator obey (A2). (If one does not have a plane-wave basis, one can evaluate thepropagator via a path integral and then use the Schr¨odinger equation to determine thepre-factor.)For the one-dimensional harmonic oscillator (where the basis is not plane waves), the pathintegral again has the form e iI STAT , where I STAT is the value of I = ( m/ R T dt [ ˙ x − ω x ] asevaluated in the stationary path ¨ x + ω x = 0 between the end points ( x = 0 , t = 0) and ( x = x f , t = T ). Noting that ˙ x − ω x = d ( x ˙ x ) /dt − x ¨ x − ω x , we obtain I STAT = mx f ˙ x f /
2. Thesolution to the equation of motion is x ( t ) = x f sin ωt/ sin ωT , ˙ x ( t ) = ωx f cos ωt/ sin ωT , sowe obtain I STAT = mωx f cos ωT / ωT . With this form for I STAT , the pre-factor evaluates15o (sin ωT ) − / and the propagator is G ( x, T ; 0 ,
0) = θ ( T ) (cid:18) ωT (cid:19) / exp imωx cos ωT ωT ! . (A4)The propagator (24) is the shifted covariant generalization of this result.The propagator associated with the PU oscillator action given in (34) has already beenreported in the literature [20], and because the action is quadratic, the R d [ z ] path integralbetween end points with fixed z and ˙ z has the form exp( iI STAT ) with the appropriate I STAT .Here, we present a simplified version of the propagator in which we set ω = ω , ω = 0. Inthis case the classical action reduces to I PU = γ Z dt (cid:16) ¨ z − ω ˙ z (cid:17) , (A5)and the stationary classical equation of motion is given by ∂ t (¨ z + ω z ) = 0 . (A6)Noting that ∂ t (cid:16) ˙ z ¨ z − z∂ t z − ω z ˙ z (cid:17) = ¨ z − ω ˙ z − z∂ t (¨ z + ω z ) , (A7)on evaluating I STAT between z = 0, ˙ z = 0 at t = 0, and z ( T ), ˙ z ( T ) at t = T , we obtain I STAT = ( γ/ (cid:16) ˙ z ( T )¨ z ( T ) − z ( T ) ∂ t z ( T ) − ω z ( T ) ˙ z ( T ) (cid:17) . (A8)Hence, introducing ωαA ( T ) = ˙ z ( T )( ωT − sin ωT ) − ωz ( T )(1 − cos ωT ) ,βA ( T ) = ˙ z ( T )(1 − cos ωT ) − z ( T ) ω sin ωT,A ( T ) = 2 − ωT − ωT sin ωT, (A9)we find that the solution to (A6) that satisfies the boundary conditions takes the form z ( t ) = − α (1 − cos ωt ) − ( β/ω ) sin ωt + βt, ˙ z ( t ) = − αω sin ωt − β cos ωt + β, ¨ z ( t ) = − αω cos ωt + βω sin ωt,∂ t z ( t ) = αω sin ωt + βω cos ωt. (A10)16n this solution I STAT obeys2 A ( T ) γ I STAT = ω ˙ z ( T )(sin ωT − ωT cos ωT ) − ω z ( T ) ˙ z ( T )(1 − cos ωT ) + ω z ( T ) sin ωT. (A11)Finally, we verify that this function is a solution to the Schr¨odinger equation associated with(35) and identify the pre-factor as A − / ( T ). The propagator is thus A − / ( T ) e iI STAT . Itscovariant generalization, obtained by using (37), is given in (51). [1] R. P. Feynman, Phys. Rev. , 440 (1950).[2] Y. Nambu, Prog. Theor. Phys. , 82 (1950).[3] For a more recent discussion of propagation and path integration in the five-dimensionalformalism see e.g. P. D. Mannheim, Phys. Lett. B , 385 (1984); Phys. Rev. D , 898(1985); and Phys. Lett. B , 191 (1986).[4] C. M. Bender and S. Boettcher, Phys. Rev. Lett. , 5243 (1998); C. M. Bender, S. Boettcher,and P. N. Meisinger, J. Math. Phys. , 2201 (1999).[5] C. M. Bender, D. C. Brody, and H. F. Jones, Phys. Rev. Lett. , 270401 (2002);Am. J. Phys. , 1095 (2003); Phys. Rev. Lett. , 251601 (2004).[6] C. M. Bender, Contemp. Phys. , 277 (2005) and Rep. Prog. Phys. , 947 (2007).[7] P. Dorey, C. Dunning and R. Tateo, J. Phys. A , L391 (2001) and , 5679 (2001);Czech. J. Phys. , 35 (2004); arXiv:hep-th/0201108; J. Phys. A: Math. Gen. , R205(2007).[8] C. M. Bender and P. D. Mannheim, Phys. Lett. A , 1616 (2010).[9] Technically, what was shown in [8] is that the secular equation that determines energy eigenval-ues is real if and only if the Hamiltonian is PT symmetric. Since the solutions to an equationthat is not real cannot all be real, PT invariance of a Hamiltonian is thus a necessary conditionfor the reality of eigenvalues. The solutions to a real equation can either be real or appear incomplex conjugate pairs. Thus, PT invariance alone does not guarantee reality. However, asnoted in Ref. [8], whether one has the real case or the complex-conjugate pair case dependson whether or not the available C operators of PT theory described in Ref. [6] commute with PT . Commutation of both the Hamiltonian and all available C operators with PT is thenboth necessary and sufficient for the reality of energy eigenvalues.
10] In P. D. Mannheim,
P T symmetry as a necessary and sufficient condition for unitary timeevolution , arXiv:0912.2635v1 [hep-th] and A. Mostafazadeh, Int. J. Geom. Meth. Mod. Phys. , 1191 (2010) the converse was shown: namely if there exists a V that effects V HV − = H † ,then H must be PT symmetric.[11] While we do not discuss it here, we note that analogous to the Hamiltonian given in (16),comparable results can be obtained for the Hamiltonian H = (1 / m )( p µ − ic µ )( p µ − ic µ ) + mω x µ x µ /
2, where c µ is a real PT -odd four-vector.[12] A. Pais and G. E. Uhlenbeck, Phys. Rev. , 145 (1950).[13] C. M. Bender and P. D. Mannheim, Phys. Rev. Lett. , 110402 (2008); J. Phys. A ,304018 (2008).[14] C. M. Bender and P. D. Mannheim, Phys. Rev. D , 025022 (2008).[15] P. D. Mannheim and A. Davidson, arXiv:hep-th/0001115.[16] P. D. Mannheim and A. Davidson, Phys. Rev. A , 042110 (2005).[17] Transforming H PU in (35) with Q = iαp x p z − iβxz to ˜ H = p x / γ + γω x / − p z / γω − γω ω z /
2, we see that if p z and z were Hermitian, the Hamiltonian would be unboundedbelow.[18] One can either represent the fourth-order derivative theory as a quadratic theory based on twosets of conjugate position and momentum pairs or as a quartic theory based on one conjugatepair alone. Both representations yield the same energy eigenvalues.[19] A relative minus sign can appear in (32) in the Hermitian Hamiltonian case where there areno states with negative norm because as noted in [14], in this case the contour for the p integration is not the standard Feynman one. Specifically, in the generic form 1 / ( A + iǫ a ) − / ( B + iǫ b ) = ( B − A ) / ( AB + iǫ ), where ǫ = Bǫ a + Aǫ b the sense of the contours depends onthe values of A and B .[20] P. D. Mannheim, Found. Phys. , 532 (2007)., 532 (2007).