Extension of the CPT Theorem to non-Hermitian Hamiltonians and Unstable States
aa r X i v : . [ qu a n t - ph ] D ec Extension of the
CP T
Theorem to non-Hermitian Hamiltonians and Unstable States
Philip D. Mannheim
Department of Physics, University of Connecticut, Storrs,CT 06269, USA. email: [email protected] (Dated: December 10, 2015)We extend the
CP T theorem to quantum field theories with non-Hermitian Hamiltonians andunstable states. Our derivation is a quite minimal one as it requires only the time-independentevolution of scalar products, invariance under complex Lorentz transformations, and a non-standardbut nonetheless perfectly legitimate interpretation of charge conjugation as an antilinear operator.The first of these requirements does not force the Hamiltonian to be Hermitian. Rather, it forces itseigenvalues to either be real or to appear in complex conjugate pairs, forces the eigenvectors of suchconjugate pairs to be conjugates of each other, and forces the Hamiltonian to admit of an antilinearsymmetry. The latter two requirements then force this antilinear symmetry to be
CP T , whileforcing the Hamiltonian to be real rather than Hermitian. Our work justifies the use of the
CP T theorem in establishing the equality of the lifetimes of unstable particles that are charge conjugatesof each other. We show that the Euclidean time path integrals of a
CP T -symmetric theory mustalways be real. In the quantum-mechanical limit the key results of the
P T symmetry program ofBender and collaborators are recovered, with the C -operator of the P T symmetry program beingidentified with the linear component of the charge conjugation operator.
I. ANTILINEAR SYMMETRY AND ENERGIES
Hermiticity of a Hamiltonian has been a cornerstoneof quantum mechanics ever since its inception. Once aHamiltonian is Hermitian it follows that all of its energyeigenvalues are real and that time evolution is unitary.Moreover, with some standard additional field-theoreticassumptions one can show that in a quantum field the-ory a Hermitian Hamiltonian is always
CP T invariant.Despite this, and due primarily to the work of Benderand collaborators [1, 2] on non-Hermitian but
P T sym-metric Hamiltonians ( P is parity, T is time reversal), ithas become apparent that it is possible to achieve bothreal eigenvalues and the time-independent evolution ofHilbert space scalar products even if a Hamiltonian isnot Hermitian, provided that it instead has an antilinearsymmetry such as P T . Consequently, while Hermiticityis sufficient to secure the reality of eigenvalues and thetime-independent evolution of scalar products it is notnecessary. In this paper we show that a similar situationholds for the
CP T theorem, with it being possible to es-tablish invariance of a Hamiltonian under an antilinear
CP T transformation even if the Hamiltonian is not Her-mitian. While
CP T symmetry is more general than
P T symmetry, whenever charge conjugation C is separatelyconserved, for non-Hermitian Hamiltonians with an un-derlying CP T symmetry one is able to recover the keyresults of the
P T symmetry program.The relation between eigenvalues and antilinear sym-metry dates back to Wigner’s work on time reversal.Specifically, if we apply some general antilinear operator A to H | ψ i = E | ψ i , we obtain AHA − A | ψ i = E ∗ A | ψ i .Then, if A commutes with H we infer that either E isreal and | ψ i = A | ψ i , or that E is complex and the eigen-vectors associated with E and E ∗ transform into eachother under A . Thus with an antilinear symmetry ener-gies are either real or appear in complex conjugate pairs, and since nothing in this analysis requires that H beHermitian, the eigenvalues could all be real even if H isnot in fact Hermitian. As a case in point consider the H = p + ix Hamiltonian studied in [1]. While notHermitian, this Hamiltonian does have a
P T symmetry(under PT p → p , x → − x , i → − i ), and it turns out(see e. g. [2]) that every one of its eigenvalues is real.Antilinear symmetry of a Hamiltonian is more far-reaching than Hermiticity (though of course Hamiltoni-ans can be both Hermitian and have an antilinear sym-metry, as many do), and as such it provides optionsfor quantum theory that are not allowed by Hermiticity,with antilinearity being able to encompass both decaysand non-diagonalizable Jordan-block Hamiltonians suchas those of relevance to fourth-order derivative theories[3]. For decays, the utility in having a complex conju-gate pair of energy eigenvalues is that when a state | A i (the state whose energy has a negative imaginary part)decays into some other state | B i (the one whose energyhas a positive imaginary part), as the population of state | A i decreases that of | B i increases in proportion. Thisinterplay between the two states is found [4] to lead tothe time-independent evolution of scalar products asso-ciated with the overlap of the two states. In contrast, intheories based on Hermitian Hamiltonians, to describea decay one by hand adds a non-Hermitian term to theHamiltonian, and again by hand chooses its sign so thatonly the decaying mode appears. One also does this forthe decays of particles that are charge conjugates of eachother, and then uses the CP T theorem to show that theirdecay rates are equal even though the standard proof ofthe
CP T theorem presupposes that the Hamiltonian isHermitian, in which case neither of the particles woulddecay at all [5]. In this paper we will address this issue byderiving the
CP T theorem without assuming Hermitic-ity. (Some alternate discussion of the
CP T theorem inthe presence of unstable states may be found in [6].)
II. ANTILINEAR SYMMETRY AND TIMEEVOLUTION
In the standard discussion of the time evolutiongenerated by a time-independent Hamiltonian, oneintroduces states | R i ( t ) i that evolve according to | R i ( t ) i = exp( − iHt ) | R ( t = 0) i , with the stan-dard Dirac scalar product h R i ( t ) | R j ( t ) i = h R i ( t =0) | exp( iH † t ) exp( − iHt ) | R j ( t = 0) i then being time in-dependent if H is Hermitian. While one can immediatelyconclude that the standard Dirac norm would not be timeindependent if H = H † , that does not preclude the ex-istence of some other scalar product that would be timeindependent. In the more general case we note that theeigenvector equation i∂ t | R ( t ) i = H | R ( t ) i only involvesthe kets and serves to identify right-eigenvectors. Sincethe bra states are not specified by an equation that onlyinvolves the kets, there is some freedom in choosing them.As discussed for instance in [4], in general one shouldintroduce left-eigenvectors of the Hamiltonian accordingto − i∂ t h L | = h L | H , and use the more general norm h L | R i , since for it one does have h L ( t ) | R ( t ) i = h L ( t =0) | exp( iHt ) exp( − iHt ) | R ( t = 0) i = h L ( t = 0) | R ( t = 0) i ,so that this particular norm is preserved in time. Whilethis norm coincides with the Dirac norm h R | R i when H is Hermitian, when H is not Hermitian one should usethe h L | R i norm instead.If | R i ( t ) i is a right-eigenvector of H with some generalenergy eigenvalue E i = E Ri + iE Ii , and h L j ( t ) | is a left-eigenvector of H with energy eigenvalue E j = E Rj + iE Ij ,then in general we can write h L j ( t ) | R i ( t ) i = h L j (0) | R i (0) i e − i ( E Ri + iE Ii ) t + i ( E Rj − iE Ij ) t . (1)If these norms are to be time independent, the only al-lowed non-zero norms are those that obey E Ri = E Rj , E Ii = − E Ij , (2)with every other norm having to obey h L j (0) | R i (0) i = 0.Thus we see that the only non-zero overlaps are preciselythose associated with eigenvalues that are purely real orare in complex conjugate pairs, with this being the mostgeneral condition under which scalar products can betime independent. And with h E Ri − iE Ii | E Ri + iE Ii i =exp( iE Ri t − E Ii t ) exp( − iE Ri t + E Ii t ) being time indepen-dent, in the complex energy sector the only non-zero over-laps are precisely between a state that decays in time andone that grows in time at the complementary rate. Aswe had noted above, this is just as needed to maintainthe time independence of the transition between them.While we had noted above that if one has an antilinearsymmetry one can establish the energy relationship givenin (2), for our purposes here we need to show that if one isgiven (2), i. e. if one is given time-independent evolutionof scalar products, then H must admit of an antilinearsymmetry. To this end we consider the eigenequation i ∂∂t | ψ ( t ) i = H | ψ ( t ) i = E | ψ ( t ) i . (3) On replacing the parameter t by − t and then multiplyingby a general antilinear operator A we obtain i ∂∂t A | ψ ( − t ) i = AHA − A | ψ ( − t ) i = E ∗ A | ψ ( − t ) i . (4)Then, because we are explicitly interested in the casewhere E ∗ is an eigenvalue of H we can set HA | ψ ( − t ) i = E ∗ A | ψ ( − t ) i , and thus obtain( AHA − − H ) A | ψ ( − t ) i = 0 . (5)Now (5) has to hold for every eigenstate of H , and ifthe set of all such eigenstates is complete, we can set[ H, A ] = 0 as an operator identity. We thus conclude thatif all left-right scalar products are time independent, then H must possess an antilinear symmetry. To determinewhat that antilinear symmetry might be, with H beinga generator of the Poincare group, we turn now to theimplications of the complex Lorentz group. III. THE COMPLEX LORENTZ GROUP
When Lorentz transformations were first introducedinto physics, they were taken to be real since one onlyconsidered transformations on real ( x, y, z, t ) coordinatesof the form x ′ µ = Λ µν x ν with real Λ µν (i. e. observermoving with a real velocity), so that the transformedcoordinates would be real also. Nonetheless, if we wereto take the velocity and Λ µν ( v ) to be complex the flatspace line element η µν x µ x ν would still be invariant.Moreover, as well as the line element, similar re-marks apply to the action I = R d xL ( x ). With L ( x )being a Lorentz scalar, this action is invariant underreal Lorentz transformations of the form exp( iw µν M µν )where the six w µν = − w νµ are real parameters and thesix M µν = − M νµ are the generators of the Lorentzgroup. Specifically, with M µν acting on the Lorentzscalar L ( x ) as x µ p ν − x ν p µ , under an infinitesimalLorentz transformation the change in the action isgiven by δI = 2 w µν R d xx µ ∂ ν L ( x ), and thus by δI =2 w µν R d x∂ ν [ x µ L ( x )]. Since the change in the action isa total divergence, the familiar invariance of the actionunder real Lorentz transformations is secured. However,we now note that that nothing in this argument dependedon w µν being real, with the change in the action still be-ing a total divergence even if w µν is complex. The action I = R d xL ( x ) is thus actually invariant under complexLorentz transformations as well and not just under realones, with complex Lorentz invariance thus being a nat-ural symmetry in physics.Further justification for the relevance of the complexLorentz group is provided by spinors, since they are con-tained not in SO (3 ,
1) itself but in its unitary and thuscomplex covering group. For spinor fields we can considera “line element” ψ Tr Cψ (see e. g. [7]) in Grassmannspace where Tr denotes transpose in the Dirac gammamatrix space and C is the Dirac gamma matrix that ef-fects C − γ µ C = − γ µ Tr . With a Dirac spinor transformingas ψ → exp( iw µν M µν ) ψ , we see that since ψ Tr Cψ doesnot involve Hermitian conjugation, it is invariant not justunder real but also complex w µν . Now as it stands thescalar quantity ¯ ψψ = ψ † γ ψ would be invariant underreal Lorentz transformations but not under complex ones.However, as will be central to our discussion below ofcharge conjugation, we note that since a Dirac spinor isreducible under the Lorentz group we can decompose itas ψ = ψ A + iψ B , where ψ A and ψ B are self-conjugateMajorana spinors that in the Majorana representation ofthe Dirac gamma matrices (see e. g. [8]) obey ψ † A = ψ A and ψ † B = ψ B . With this decomposition we understanda complex Lorentz transformation to be implemented onthe separate ψ A and ψ B , with ¯ ψψ then being invariantunder complex Lorentz transformations too. Thus in thefollowing we shall consider the implications of complexLorentz invariance.Complex Lorentz invariance is of significance to both P T and
CP T transformations, and both will be neededfor the
CP T theorem, since under
CP T the argument ofa field changes from x µ to − x µ , just as required by the P T part of the
CP T transformation. For
P T transfor-mations first, we note that on applying the specific se-quence of Lorentz boosts: first x ′ = x cosh ξ + t sinh ξ , t ′ = t cosh ξ + x sinh ξ , then y ′ = y cosh ξ + t sinh ξ , t ′ = t cosh ξ + y sinh ξ , and finally z ′ = z cosh ξ + t sinh ξ , t ′ = t cosh ξ + z sinh ξ , each with a complex boost an-gle ξ = iπ , we generate ( x, y, z, t ) → ( − x, − y, − z, − t ).On defining πτ = Λ ( iπ )Λ ( iπ )Λ ( iπ ), πτ effects πτ : x µ → − x µ . However, though this transformationdoes indeed reverse the signs of all four of the coordinatesjust as a P T transformation does, πτ itself is not the P T transformation of interest to physics since time reversalhas to be an antilinear operator rather than a linear one.Nonetheless, we can always represent an antilinear op-erator as a linear operator times complex conjugation.On introducing an operator K T that conjugates com-plex numbers, up to intrinsic system-dependent phaseswe can then set P T = πτ K T , i. e. we can represent P T as a complex Lorentz boost times complex conjugation,to thus give a
P T transformation an association with thecomplex Lorentz group [9].With C , P , and T respectively acting on spinors as 1, γ , and γ γ γ in the Majorana basis of the Dirac gammamatrices, for spinors CP T effects
CP T ψ ( x )[ CP T ] − = − iγ ψ † ( − x ). Then with M i = i [ γ , γ i ] /
4, Λ i ( iπ ) =exp( − iπγ γ i /
2) = − iγ γ i , quite remarkably we findthat in the Dirac gamma matrix space we recognize thepreviously introduced complex Lorentz transformationΛ ( iπ )Λ ( iπ )Λ ( iπ ) = iγ γ γ γ = γ as being noneother than the linear part of a CP T transformation inspinor space, with
CP T thus having a natural connectionto the complex Lorentz group.As well as identify the linear part of a
CP T trans-formation we also need to consider its conjugation as-pects, and initially it would appear that C would differfrom T since T involves complex conjugation of complexnumbers while C involves charge conjugation of quan- tum fields. However, the two types of conjugation canactually be related, since charge conjugation converts afield into its Hermitian conjugate, and Hermitian conju-gation does conjugate factors of i . If for instance weconsider a charged scalar field φ ( x ), then under C ittransforms as φ ( x ) → Cφ ( x ) C − = φ † ( x ). However,suppose we break φ ( x ) into two Hermitian componentsaccording to φ ( x ) = φ ( x ) + iφ ( x ). Now since C effects φ ( x ) → φ † ( x ), we can achieve this in two distinct ways.We can have C act linearly on φ ( x ) and φ ( x ) accord-ing to φ ( x ) → φ ( x ), φ ( x ) → − φ ( x ) while having noeffect on the factor of i , or we can have C act antilinearlyon i according to i → CiC − = − i while having no effecton the Hermitian φ ( x ) and φ ( x ). For our purposes herethe latter interpretation is not only the more useful as ithelps us keep track of factors of i in quantities such as φ ± ( x ) = φ ( x ) ± iφ ( x ), as we will see below, it will provecrucial to our derivation of the CP T theorem. Moreover,we note that with an antilinear interpretation for C wedo not even need φ ( x ) and φ ( x ) to actually be Hermi-tian fields at all. They could instead, for instance, bedefined as being self-conjugate under C or self-conjugateunder CP T .In addition to the complex conjugation effected by K C , C could also effect a linear transformation κ as well, andso we can write CP T as κπτ K , where K = K C K T com-plex conjugates everything it acts on, c-numbers and q-numbers alike [10]. With this analysis also holding forMajorana spinors (cf. ψ = ψ A + iψ B ), and with Ma-jorana spinors being able to serve as the fundamentalrepresentation of the Lorentz group (a Majorana spinorcan be written as a Weyl spinor plus its charge conjugate[8]), we can represent CP T as the generic κπτ K whenacting on any representation of the Lorentz group.While C as defined here effects Cφ ± ( x ) C − = φ ∓ ( x ), C does not complex conjugate the individual φ i ( x ) them-selves. However, T still can, and in fact must, sincethe [ x, p ] = i commutator for instance is preserved un-der T according to x → x , p → − p , i → − i . To seehow T explicitly achieves this, we set x = ( a + a † ) / / , p = i ( a † − a ) / / , [ a, a † ] = 1. Thus we need T to effect a → a , a † → a † , i → − i , and this is achieved by theantilinear K T . In the Fock space labelled by | Ω i , a † | Ω i ,..., where a | Ω i = 0, a and a † can both be represented byinfinite-dimensional matrices that are purely real. With x being real and symmetric and p being pure imaginaryand anti-symmetric in this Fock space, in this Fock spaceonly the i in the operator p is affected by T . With thesame analysis also holding for commutators of the genericform [ φ (¯ x, t = 0) , π (¯ x ′ , t = 0)] = iδ (¯ x − ¯ x ′ ), we see thatdue to our treating C as antilinear, for every function F that is built out of canonical quantum fields, it followsthat KF K − = F ∗ (i. e. K C K T conjugates all factors of i ). It is this specific feature that will enable us to derivethe CP T theorem.
IV. DERIVATION OF THE
CP T
THEOREM
As noted for instance in [11], under
CP T every irre-ducible representation of the Lorentz group transformsas
CP T φ ( x )[ CP T ] − = η ( φ ) φ † ( − x ) with a φ -dependentphase η ( φ ) that depends on the spin of each φ and obeys η ( φ ) = 1; with spin zero fields (both scalar and pseu-doscalar) expressly having η ( φ ) = 1 [12]. Since the mostgeneral Lorentz invariant Lagrangian must be built outof sums of appropriately contracted spin zero products offields with arbitrary numerical coefficients, and since it isonly spin zero fields that can multiply any given net spinzero product an arbitrary number of times and still yieldnet spin zero, all net spin zero products of fields musthave a net η ( φ ) equal to one [13]. Generically, such prod-ucts could involve φφ or φ † φ type contractions. However,requiring the Lagrangian and thus the Hamiltonian to beHermitian then forces the contractions to be Hermitian(only φ † φ ) while forcing the coefficients to all be real,with the Hamiltonian then being CP T invariant [11].In order to extend the
CP T theorem to non-HermitianHamiltonians, we note first that even in the non-Hermitian case Lorentz invariance still requires everyterm in the Lagrangian to have a net η ( φ ) equal to one.With CP T effecting
CP T φ ± ( x )[ CP T ] − = η ( φ ) φ ∓ ( − x )we will need some non-Hermitian-based reason in orderto be able to exclude any φ ± ( x ) φ ± ( x ) type contractions.To this end we note that with the linear part of a CP T transformation having been identified as the particularcomplex Lorentz transformation Λ ( iπ )Λ ( iπ )Λ ( iπ ),under this transformation every net spin zero term in aLorentz invariant action I = R d x L ( x ) will transformso that I → R d x L ( − x ) = R d x L ( x ) = I , with theaction thus being left invariant. Then, under the full CP T transformation, and precisely because of our hav-ing taken C to be antilinear, every term in the actionwill transform so that I → R d xK L ( x ) K − . Thus undera CP T transformation, the full Hamiltonian will trans-form as H → KHK − . Since at this point we havenow arrived at (4) with A being identified with K , wesee that the requirement of time-independent evolutionof scalar products given in (5) then follows, with H obey-ing H = KHK − . The CP T invariance of H is thus se-cured, with there thus being only φ ± ( x ) φ ∓ ( x ) type termsand no φ ± ( x ) φ ± ( x ) type ones allowed in H , and with allcoefficients again being real [14]. With our definition of K we see that H obeys H = H ∗ , while not being re-quired to obey H = H † . The CP T theorem is thusextended to non-Hermitian Hamiltonians that generatetime-independent evolution of scalar products [15].
V. IMPLICATIONS
When a time-independent Hamiltonian is real (aswould be the case if H = KHK − = H ∗ ), for Euclideantimes τ = it the time evolution operator exp( − iHt ) =exp( − Hτ ) is real. Consequently, the associated Eu- clidean time path integrals and Green’s functions are realtoo. Even though the Euclidean time path integral is realthat does not mean that all energy eigenvalues are nec-essarily real, since if they appear in complex conjugatepairs and have complex conjugate wave functions, theEuclidean time path integral would still be real. In factthis is the most general way in which the Euclidean pathintegral could be real if the Hamiltonian is not Hermitian,and is just as required of antilinear CP T symmetry.We had earlier referred to the
P T symmetric H = p + ix . Since it is the quantum-mechanical limit of arelativistic theory with appropriate Hamiltonian density H = Π + i Φ , it is CP T invariant. With Φ being un-charged, this H is separately κK C invariant, and thus it isindeed P T symmetric. Now we can realize the [ x, p ] = i commutator by x = i ( b − b † ) / / , p = ( b † + b ) / / where [ b, b † ] = 1. (This realization is unitarily equivalentto x = ( a + a † ) / / , p = i ( a † − a ) / / .) In the Fockspace where b | Ω i = 0, x is pure imaginary and antisym-metric, p is real and symmetric, and thus even though H is not Hermitian, in this particular occupation numberspace H = p + ix can be represented by an infinite-dimensional matrix all of whose elements are real. Hence,despite its appearance H = p + ix obeys H = H ∗ [16].As an example of a Hamiltonian that is CP T invariantwhile having complex conjugate energy pairs, considerthe fourth-order Pais-Uhlenbeck two-oscillator ( p z , z and p x , x ) model studied in [3, 17]. Its Hamiltonian is givenby H PU = p x / γ + p z x + γ (cid:0) ω + ω (cid:1) x / − γω ω z / ω and ω are real (this Hamiltonian isthe quantum-mechanical limit of a covariant fourth-orderneutral scalar field theory [3]). H PU turns out not tobe Hermitian but to instead be P T symmetric [3, 17],with all energy eigenvalues nonetheless being given bythe real E ( n , n ) = ( n + 1 / ω + ( n + 1 / ω . Inaddition, H PU is CP T symmetric since C plays no role([ κK C , H P U ] = 0), while thus descending from a neutralscalar field theory that is also
CP T invariant. If we nowset ω = α + iβ , ω = α − iβ with real α and β , we see that( ω + ω ) / α − β and ω ω = ( α + β ) both re-main real. In consequence H PU remains CP T invariant,but now the energies come in complex conjugate pairs asper E ( n , n ) = ( n +1 / α + iβ )+( n +1 / α − iβ ). Itis also of interest to note that when ω = ω = α with α real, the Hamiltonian becomes of non-diagonalizable, andthus of manifestly non-Hermitian, Jordan-block form [3],with its CP T symmetry not being impaired. Thus for ω and ω both real and unequal, both real and equal, or be-ing complex conjugates of each other, in all cases one hasa non-Hermitian but CP T -invariant Hamiltonian thatdescends from a quantum field theory whose Hamiltonianwhile not Hermitian is nonetheless
CP T symmetric.The
P T studies of Bender and collaborators are mainlyquantum-mechanical ones in which the field-theoreticcharge conjugation operator plays no role. In these stud-ies it has been found [2] that as well as be
P T symmetric,the Hamiltonian is also symmetric under a specific dis-crete linear operator also called C , which obeys [ C, H ] =0 and C = 1. In addition, this C obeys [ C, P T ] = 0when all energies are real, and obeys [
C, P T ] = 0 whenenergies are in complex pairs [18]. The CP T symmetryof any given relativistic theory ensures the
P T symme-try of any C -invariant quantum-mechanical theory thatdescends from it, while guaranteeing that it must possessa linear operator, viz. our previously introduced κ , thatobeys [ κ, H ] = 0 and κ = 1, and so we can now identify κ (or a similarity transform of it) with the C operator of P T theory [19]. Our work thus puts the
P T symmetrystudies of theories with non-Hermitian Hamiltonians ona quite secure quantum-field-theoretic foundation.
VI. APPLICATIONS
Once one extends the
CP T theorem to non-HermitianHamiltonians, the most interesting applications of ourideas are to situations that can never be encompassed byHermitian Hamiltonians. The currently most exploredsuch area is in applications of
P T symmetry in the com-plex conjugate energy pair situation, where there areboth growing and decaying modes. In the
P T literaturesuch modes are referred to as gain and loss, with manyexperimental examples having been identified [20, 21].Moreover, typically in these cases, as one adjusts parame-ters one can transition to the region where all eigenvaluesare real. At the point of the transition, known as an ex-ceptional point in the
P T literature, the Hamiltonian be-comes of a non-diagonalizable, and thus manifestly non-Hermitian, Jordan-block form, and experimental effectsdue to exceptional points have also been discussed in theliterature.For relativistic quantum theory our results can be ap-plied to particle decays such as those encountered inthe K meson system. Specifically, the time-independenttransition matrix elements that we obtain precisely pro-vide for probability conserving transitions between de-caying states and the growing states into which they de-cay, with the CP T theorem that we have derived here then requiring that the transition rates for the decaysof particles and their antiparticles be equal. To be morespecific, we note that as well as provide an explicitly solv-able model that is non-Hermitian but
CP T invariant,the two-oscillator Pais -Uhlenbeck model can also serveas a prototype for discussing decays. In the region where ω = α + iβ , ω = α − iβ , the Hamiltonian is given by the CP T -symmetric H PU = p x / γ + p z x + γ (cid:0) α − β (cid:1) x − γ ( α + β ) z /
2. Not only does this model contain bothdecaying ( ω = α − iβ ) and growing modes ( ω = α + iβ ),as per (1) and (2) it leads to time-independent transitionsbetween them, and thus describes the decay of one modeinto the other. If we now make the momentum and posi-tion operators be complex, which we can do in a chargeconjugation invariant manner, the field-theoretic general-ization of the model will then contain both particles andantiparticles, with the CP T invariance of the Hamilto-nian then requiring that the decay rates for particles andtheir antiparticles be equal.Another case that cannot be described by a Hermi-tian Hamiltonian is encountered in the currently viablefourth-order derivative conformal gravity theory, a con-formal invariant, general coordinate invariant theory ofgravity that has been advanced as candidate alternatetheory of gravity [22]. The conformal gravity actionis given by I W = − α g R d x ( − g ) / C λµνκ C λµνκ where C λµνκ is the Weyl conformal tensor, and its Hamilto-nian is a relativistic generalization of the equal-frequencyPais-Uhlenbeck Hamiltonian [23]. Consequently, the con-formal gravity Hamiltonian is of a non-Hermitian, non-diagonalizable, Jordan-Block form [3, 23], to thus serveas an explicit field-theoretic example of a Hamiltonianthat is not Hermitian but is CP T symmetric.
Acknowledgments
The author wishes to thank Dr. Carl Bender for somevery helpful discussions. [1] C. M. Bender and S. Boettcher, Phys. Rev. Lett. ,5243 (1998).[2] C. M. Bender, Rep. Prog. Phys. , 947 (2007).[3] C. M. Bender and P. D. Mannheim, Phys. Rev. D ,025022 (2008).[4] P. D. Mannheim, Phil. Trans. R. Soc. A , 20120060(2013).[5] Given a Hermitian weak interaction Hamiltonian H W ,through standard use of the CP T theorem one can showthe equality of matrix elements such as h K | H W | π + π − i and h ¯ K | H W | π + π − i . With the Hamiltonian being Hermi-tian, neither kaon should decay. And so to force them todo so one chooses a boundary condition (decaying modeonly and no growing mode) that is not CP T invariant.[6] M. Selover and E. C. G. Sudarshan,
Derivation of theTCP Theorem using Action Principles , arXiv:1308.5110 [hep-th], August 2013.[7] P. D. Mannheim, Phys. Rev. D , 898 (1985).[8] P. D. Mannheim, Int. J. Theor. Phys. , 643 (1984).[9] We note that no such complex association exists for ei-ther P ( ~x → − ~x ) or T ( t → − t ) separately, since, witheach Lorentz transformation involving an even numberof spacetime coordinates, there is no transformation thatcould reverse the sign of an odd number of them.[10] In the early days of the CP T theorem, an antilinear in-terpretation of charge conjugation had been consideredby authors such as Schwinger, Bell, and DeWitt (see [6]).[11] S. Weinberg,
The Quantum Theory of Fields: Volume I ,Cambridge University Press, Cambridge, U. K., 1995.[12] Since [ φ (¯ x, t = 0) , π (¯ x ′ , t = 0)] = iδ (¯ x − ¯ x ′ ), the conju-gate π ( x ) always has the opposite CP T to φ ( x ).[13] With a fermion in the fundamental representation of the Lorentz group transforming as CP T ψ ( x )[ CP T ] − = − iγ ψ † ( − x ), one can anticipate that every irreduciblerepresentation of the Lorentz group will obey an analo-gous relation since γ commutes with the Lorentz gen-erators i [ γ µ , γ ν ] /
4. With the
CP T phases of ¯ ψψ , ¯ ψiγ ψ ,¯ ψγ µ ψ , ¯ ψγ µ γ ψ , and ¯ ψi [ γ µ , γ ν ] ψ respectively being +1,+1, − −
1, and +1, for integer spin representations ofthe Lorentz group
CP T phases alternate with spin, sothat spin zero combinations (combinations that must al-ways contain even numbers of fermions) will all have a
CP T phase equal to +1. No analogous statement can bemade for any other transformation involving C , P , or T .[14] We note that even with antilinear operators, commuta-tion relations are preserved under similarity transforms.If we set H ′ = SHS − , K ′ = SKS − = S [ S − ] ∗ K = KS ∗ S − , K ′ = I , then [ K ′ , H ′ ] = S [ K, H ] S − , withthe operator K thus transforming along with H . In con-sequence, the commutation relation [ CP T, H ] = 0 is pre-served under a similarity transform.[15] The axiomatic field theory proof of the
CP T theorem(R. F. Streater and A. S. Wightman,
PCT, Spin andStatistics, and all that , W. A. Benjamin, New York 1964)also involves the complex Lorentz group as the coordi-nates in the Wightman functions h Ω | φ ( x ) . . . φ ( x n ) | Ω i are continued to complex values. However, in this ap- proach one also has to assume that H is Hermitian.[16] Starting from an initial CP T (and
P T ) symmetric andthus real Hamiltonian density H = − Π + Φ with a CP T even neutral pseudoscalar field Φ and its
CP T oddconjugate Π, the substitution Φ → − i Φ ≡ − ix , Π → i Π ≡ ip yelds the CP T (and
P T ) odd x , and the CP T (and
P T ) even p , and the still CP T (and
P T ) symmetricand still real H → Π + i Φ ≡ p + ix .[17] C. M. Bender and P. D. Mannheim, Phys. Rev. Lett. ,1616 (2010).[19] Our point here is that we can use joint CP T and C invari-ance to obtain P T invariance even if H is non-Hermitian.[20] Special issue on quantum physics with non-Hermitian op-erators , C. Bender, A. Fring, U. G¨unther, and H. Jones(Guest Editors), J. Phys. A: Math. Theor. , 444001 -444036 (2012).[21] Theme issue on PT quantum mechanics , C. M. Ben-der, M. DeKieviet, and S. P. Klevansky (Guest Edi-tors), Phil. Trans. R. Soc. A , issue 1989 (2013).[22] P. D. Mannheim, Found. Phys. , 388 (2012).[23] P. D. Mannheim, Gen. Rel. Gravit.43