aa r X i v : . [ qu a n t - ph ] J a n Crypto-Hermitian Approach to the Klein-GordonEquation
Iveta Semor´adov´a
Nuclear Physics Institute, Czech Academy of Science, ˇReˇz nearPrague, Czech Republic Faculty of Nuclear Science and Physical Engineering, CzechTechnical University in Prague, Czech Republic
E-mail: semorive@fjfi.cvut.cz
Abstract
We explore the Klein-Gordon equation in the framework of crypto-Hermitian quantum mechanics. Solutions to common problems with prob-ability interpretation and indefinite inner product of the Klein-Gordonequation are proposed.
The urge to unite special theory of relativity with quantum theory emergedshortly after their discovery. The first relativistic wave equation was introducedin 1926 simultaneously by Klein [1], Gordon [2], Kudar [3], Fock [4][5] and deDonder and Van Dungen [6]. Schr¨odinger himself formulated it earlier in hisnotes together with the Schr¨odinger equation [7]. However, with the introduc-tion of the Klein-Gordon equation arose several problems. For given momentumequation allows solutions with both positive and negative energy, it has an ex-tra degree of freedom due to presence of both first and second derivatives andmainly its density function is indefinite and therefore cannot be consistentlyinterpreted as probability density. Also, the predictions based on this equa-tion seemed to disagree with experiments (cf., e.g., the historical remark in [8]).Therefore, few years later all the attention shifted to the Dirac equation.More than ninety years old problem of proper probability interpretation ofthe Klein-Gordon equation was first solved in 1934 by Pauli and Weisskopf [9] byreinterpreting the Klein-Gordon equation in the context of quantum field theory.Quantum mechanical approach to the Klein-Gordon equation was forgotten untilAli Mostafazadeh brought it back in 2003 [10]. In his work, he made use ofpseudo or quasi-Hermitian approach to quantum mechanics.Mathematical ideas of quasi-Hermitian theory originate from works of Dieudonn´e[11] and Dyson [12], though it wasn’t until 1992 when the theory was consis-tently explained and applied in nuclear physics by Scholtz, Geyer and Hahne113]. This groundbreaking work initiated fast growth of interest popularized in1998 by Bender and Boettcher [14]. Nowadays the application of the theoryis moving away from quantum mechanics to other branches of physics, such asoptics.We would like to return to the problem of proper interpretation of the Klein-Gordon equation in the framework of Quantum mechanics only. Several publi-cations concerning this subject appeared [15, 16, 17, 18] or [19, 20, 21]. But eventhese studies did not provide an ultimate answer to all of the open questions.Some of them will be addressed in what follows.
The Klein-Gordon equation for free particle can be written in common form( (cid:3) + m c ~ ) ψ ( t, x ) = 0 , (1)where (cid:3) = c ∂ t − ∆ = ∂ µ ∂ µ is the d’Alembert operator. From now on we willuse the natural units c = ~ = 1, furthermore we can denote K = − ∆ + m andrewrite (1) as ( i∂ t ) ψ ( t, x ) = Kψ ( t, x ) . (2)The fact that the Klein-Gordon equation is differential equation of second orderin time gives it an extra degree of freedom. Feshbach and Villars [22] sug-gested solution to this problem by introducing two-component wave functionand therefore making the extra degree of freedom more visible. Following theirideas together with even earlier ideas of Foldy [23], we can replace the Klein-Gordon equation with two differential equation of first order in time. Inspiredby convention introduced in [19] we putΨ (1) = i∂ t ψ , Ψ (2) = ψ . (3)Now, equation (2) can be decomposed into a pair of partial differential equations i∂ t Ψ (1) = K Ψ (2) , (4) i∂ t Ψ (2) = Ψ (1) , (5)which, written in the matrix form, become i∂ t (cid:18) Ψ (1) Ψ (2) (cid:19) = (cid:18) KI (cid:19) (cid:18) Ψ (1) Ψ (2) (cid:19) . (6)Hamiltonian of the quantum system takes form H = (cid:18) K (cid:19) , (7)2nd enters the Schr¨odinger equation i∂ t Ψ( t, x ) = H Ψ( t, x ) , Ψ = (cid:18) Ψ (1) Ψ (2) (cid:19) . (8)Two-component vectors Ψ( t ) belong to H = L ( R ) ⊕ L ( R ) (9)and the Hamiltonian H may be viewed as acting in H .The so called Schr¨odinger form of the Klein-Gordon equation (8) is equiv-alent to the original Klein-Gordon equation (1). It is in more familiar form,although, new challenge arises with the manifest non-Hermiticity of Hamilto-nian (7). New form of the Klein-Gordon equation (8) has many benefits. One of themis simplification of calculation of its eigenvalues to mere solving the eigenvalueproblem for operator
K Kψ n = ǫ n ψ n . (10)The relationship between eigenvalues ǫ n of the operator K and eigenvalues E n of the non-Hermitian operator H of the Schr¨odinger form of the Klein-Gordonequation (cid:18) KI (cid:19) (cid:18) Ψ (1) Ψ (2) (cid:19) = E (cid:18) Ψ (1) Ψ (2) (cid:19) (11)can be easily seen. Equation (11) is formed from two algebraic equations K Ψ (2) = E Ψ (1) , Ψ (1) = E Ψ (2) . (12)After insertion of the second one to the first one we obtain K Ψ (2) n = E n Ψ (2) n , (13)which compared with equation (10) gives us following relation between eigen-values ǫ n = E n . (14)We can see, that eigenvalues E n remain real under assumption of ǫ n > H Ψ ( ± ) n = E ( ± ) n Ψ ( ± ) n , Ψ ( ± ) n = (cid:18) ±√ ǫ n ψ n ψ n (cid:19) (15)is also easy to see. 3 .2 Free Klein-Gordon equation In case of free Klein-Gordon equation operator K = − ∆ + m (16)acting on H = L ( R ) is positive and Hermitian. It has continuous and de-generate spectrum. As suggested in [10], we identify the space R with thevolume of a cube of side l , as l tends to infinity. Than we can treat the contin-uous spectrum of K as the limit of the discrete spectrum corresponding to theapproximation. The eigenvalues are given by ǫ k = k + m (17)and corresponding eigenvectors ψ k = Ψ (2) k are ψ k ( x ) = h x | k i = (2 π ) − / e i k . x , (18)where k ∈ R and k . k = k . We can see that ψ k / ∈ L ( R ). They aregeneralized eigenvectors, i.e. vectors which eventually becomes 0 if ( K − λI ) isapplied to it enough times successively, describing scattering states [10].Vectors ψ k satisfy orthonormality and completeness conditions h k | k ′ i = δ ( k − k ′ ) , Z d k | k ih k | ) = 1 (19)and operator K can be expressed by its spectral resolution as K = Z d k ( k + m ) | k ih k | . (20)From the relations (14) and (15) we see that eigenvalues and eigenvectors of H are given by E ( ± ) k = ± q k + m , Ψ ( ± ) k = ± q k + m ! ψ k . (21)The eigenvectors Φ ( ± ) k of adjoint operator H † areΦ ( ± ) k = ± q k + m ! ψ k , (22)which form together with Ψ ( ± ) k complete biorthogonal system h Φ ( ν ) k ′ | Ψ ( ν ′ ) k i = δ ( k − k ′ ) δ νν ′ E ( ν ) k , (23)where ν, ν ′ = ±
1. 4
Crypto-Hermitian approach
Apparent non-Hermiticity of Hamiltonian (7) can be dealt with by means of thecrypto-Hermitian theory (sometimes also called quasi-Hermitian [24] or PT -symmetric [25]).Hamiltonian is non-Hermitian H = H † only in the false Hilbert space H ( F ) =( V, h·|·i ). The underlying vector space of states is fixed, given by the physicalsystem. However, we have a freedom in the choice of inner product. If werepresent our Hamiltonian in different secondary Hilbert space H ( S ) = ( V, hh·|·i ),with newly defined inner product hh·|·i = h ϕ | Θ | ψ i , (24)it may become Hermitian. So called metric operator Θ must be positive definite,everywhere-defined, Hermitian and bounded with bounded inverse. Operatorsfor which such inner product exist will be called crypto-Hermitian (c.f. [26]).They satisfy the so called Dieudon´ee equation H † Θ = Θ H (25)and they are similar to Hermitian operators h = Ω H Ω − , (26)where Θ = Ω † Ω is invertible and h = h † .In such scenario, the problem of negative probability interpretation of theKlein-Gordon equation can be reinterpreted as the problem of the wrong choiceof metric operator Θ. If we would be able to find more appropriate choice ofrepresentation space H ( S ) , this problem would disappear. One of the possible ways how to construct metric operator Θ for given crypto-Hermitian Hamiltonian H is by summing the spectral resolution series. It re-quires the solution of eigenvalue problem for H † . In what follows, we try toconstruct the metric operator for free Klein-Gordon equationΘ = Z d k (cid:16) α (+) | Φ (+) k ih Φ (+) k | + α ( − ) | Φ ( − ) k ih Φ ( − ) k | (cid:17) , (27)where we insert eigenvectors Φ ( ± ) k as computed in (22)Θ = Z d k (cid:18) α β √ k + m β √ k + m α ( k + m ) (cid:19) | k ih k | , (28)where α = α (+) + α ( − ) , β = α (+) − α ( − ) . By means of equation (20) we obtainfamily of metric operators Θ = (cid:18) α βK / βK / αK (cid:19) , (29)5here K / = Z d k p k + m | k ih k | . (30)With the knowledge of the metric operator (29), we can construct positivedefinite inner product defining Hilbert space H ( S ) hh Ψ | Φ i = α ( h ψ | K | ϕ i + h ˙ ψ | ˙ ϕ i )+ iβ ( h ψ | K / | ˙ ϕ i − h ˙ ψ | K / | ϕ i ) , (31)where ˙ ϕ, ˙ ψ denote corresponding time derivatives (In fact, this equation is justan explicit version of equation (24)). Unfortunately, the metric operator (29) is unbounded and therefore doesn’tsatisfy all the requested properties we put upon metric operator. As was em-phasized in [27], boundedness of metric operator Θ is very important property,it guarantees that convergence of Cauchy sequences is not affected by introduc-tion of new inner product (24). The possibility of the use of unbounded metricsis treated e.g. in the last chapter of [28].To overcome the problems with unboundedness of the metric operator (29),we choose to shift our attention to a discrete model. In the discrete approxima-tion the metric operator stays bounded. We make use of equidistant, Runge-Kutta grid-point coordinates x k = kh , k = 0 , ± , ± . . . , (32)Laplacian can be expressed as − ψ ( x k +1 ) − ψ ( x k ) + ψ ( x k − ) h , (33)The explicit occurrence of the parameter h will be important for the study ofthe continuum limit in which the value of h would decrease to zero. Otherwisewe may set h = 1 in suitable units. Following further ideas from [29], Laplaceoperator ∆ can be discretized into matrix form∆ ( n ) = − − − − − − (34)Matrix (34) is Hermitian and therefore diagonalizable, i.e. similar to diagonalmatrix. Hence for our purposes it is enough to compute with n × n real diagonal6atrix K = a · · · a · · · · · · a n . (35)Let A , B , C be real matrices n × n , where A = A T , B = B T . Than we canwrite the Dieudonn´e equation (25) by means of block matrices (cid:18) IK (cid:19) (cid:18) A C T C B (cid:19) = (cid:18) A C T C B (cid:19) (cid:18) KI (cid:19) . (36)We obtain following conditions C = C T , KC = C T K, B = KA = AK . (37)Real symmetric matrix which commutes with diagonal matrix must be di-agonal. Thus the form of our metric operator is as followsΘ = α · · · β · · · · · · α n · · · β n β · · · a α · · · · · · β n · · · a n α n . (38)It depends on 2 n parameters α . . . α n , β . . . β n . Requirement of positive-definitness of the metric put following conditions on our parameters α i > , a i α i > β i , i = 1 , , . . . , n . (39)We can construct corresponding inner product hh ψ | ϕ i = n X i =1 α i ψ ∗ i ϕ i + n X i =1 β i ( ψ ∗ i ϕ n + i + ψ ∗ n + i ϕ i )+ n X i =1 a i α i ψ ∗ n + i ϕ n + i , (40)where ψ = ( ψ , ψ , . . . ψ n ) T , ϕ = ( ϕ , ϕ . . . , ϕ n ) T are complex vectors.7 Conclusions
In our work, we familiarized the reader with the crypto-Hermitian approach tothe Klein-Gordon equation. We computed metric operator in both continuousand discrete cases. Corresponding positive definite inner product for free Klein-Gordon equation was also computed. That is considered a crucial step in properprobability interpretation of the Klein-Gordon equation.The next step of this process would be construction of appropriate metricoperator for the Klein-Gordon equation with nonzero potential V as was donefor special cases in [19, 21, 16, 17]. It is also possible to broaden the formalismby adding manifest non-Hermiticity in operator K = K † , as was shown in [20].Related complicated problems with locality, definition of physical observ-ables and attempts to construct conserved four-current can be thoroughly stud-ied in further references [16, 17]. The problems become much simpler if wenarrow our attention to real Klein-Gordon fields only. It was shown that insuch a case, inner product is uniquely defined [16, 30]. Acknowledgements:
The work of Iveta Semor´adov´a was supported by theCTU grant Nr. SGS16/239/OHK4/3T/14.
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