Acta Polytechnica


doi:10.14311/AP.2017.57.0462
Acta Polytechnica 57(6):462–466, 2017 © Czech Technical University in Prague, 2017

available online at http://ojs.cvut.cz/ojs/index.php/ap

CRYPTO-HERMITIAN APPROACH
TO THE KLEIN–GORDON EQUATION

Iveta Semorádováa,b

a Nuclear Physics Institute, Czech Academy of Science, Řež near Prague, Czech Republic
b Faculty of Nuclear Science and Physical Engeneering, Czech Technical University in Prague, Czech Republic
correspondence: semorive@fjfi.cvut.cz

Abstract. We explore the Klein-Gordon equation in the framework of crypto-Hermitian quantum
mechanics. Solutions to common problems with probability interpretation and indefinite inner product
of the Klein-Gordon equation are proposed.

Keywords: Klein-Gordon equation; probability interpretation; metric operator; crypto-Hermitian
operator; quasi-Hermitian operator.

1. Introduction
The urge to unite special theory of relativity with
quantum theory emerged shortly after their discovery.
The first relativistic wave equation was introduced in
1926 simultaneously by Klein [1], Gordon [2], Kudar
[3], Fock [4][5] and de Donder and Van Dungen [6].
Schrödinger himself formulated it earlier in his notes
together with the Schrödinger equation [7]. However,
with the introduction of the Klein-Gordon equation
arose several problems. For given momentum equation
allows solutions with both positive and negative en-
ergy, it has an extra degree of freedom due to presence
of both first and second derivatives and mainly its
density function is indefinite and therefore cannot be
consistently interpreted as probability density. Also,
the predictions based on this equation seemed to dis-
agree 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 prob-
ability interpretation of the Klein-Gordon equation
was first solved in 1934 by Pauli and Weisskopf [9]
by reinterpreting the Klein-Gordon equation in the
context of quantum field theory. Quantum mechanical
approach to the Klein-Gordon equation was forgotten
until Ali Mostafazadeh brought it back in 2003 [10].
In his work, he made use of pseudo or quasi-Hermitian
approach to quantum mechanics.

Mathematical ideas of quasi-Hermitian theory orig-
inate from works of Dieudonné [11] and Dyson [12],
though it wasn’t until 1992 when the theory was con-
sistently explained and applied in nuclear physics by
Scholtz, Geyer and Hahne [13]. This groundbreaking
work initiated fast growth of interest popularized in
1998 by Bender and Boettcher [14]. Nowadays the
application of the theory is moving away from quan-
tum mechanics to other branches of physics, such as
optics.
We would like to return to the problem of proper

interpretation of the Klein-Gordon equation in the

framework of Quantum mechanics only. Several pub-
lications concerning this subject appeared [15–18] or
[19–21]. But even these studies did not provide an
ultimate answer to all of the open questions. Some of
them will be addressed in what follows.

2. Klein-Gordon equation in
Schrödinger form

The Klein-Gordon equation for free particle can be
written in common form

(� +
m2c2

~2
)ψ(t,x) = 0 , (1)

where � = 1
c2
∂2t − ∆ = ∂µ∂µ is the d’Alembert op-

erator. From now on we will use the natural units
c = ~ = 1, furthermore we can denote K = −∆ + m2
and rewrite (1) as

(i∂t)2ψ(t,x) = Kψ(t,x) . (2)

The fact that the Klein-Gordon equation is differ-
ential equation of second order in time gives it an
extra degree of freedom. Feshbach and Villars [22]
suggested solution to this problem by introducing two-
component wave function and therefore making the
extra degree of freedom more visible. Following their
ideas together with even earlier ideas of Foldy [23],
we can replace the Klein-Gordon equation with two
differential equation of first order in time. Inspired by
convention introduced in [19] we put

Ψ(1) = i∂tψ, Ψ(2) = ψ. (3)

Now, equation (2) can be decomposed into a pair of
partial differential equations

i∂tΨ(1) = KΨ(2), (4)
i∂tΨ(2) = Ψ(1), (5)

which, written in the matrix form, become

i∂t

(
Ψ(1)
Ψ(2)

)
=
(

0 K
I 0

)(
Ψ(1)
Ψ(2)

)
. (6)

462

http://dx.doi.org/10.14311/AP.2017.57.0462
http://ojs.cvut.cz/ojs/index.php/ap


vol. 57 no. 6/2017 Crypto-Hermitian Approach to the Klein–Gordon Equation

Hamiltonian of the quantum system takes form

H =
(

0 K
1 0

)
, (7)

and enters the Schrödinger equation

i∂tΨ(t,x) = HΨ(t,x), Ψ =
(

Ψ(1)
Ψ(2)

)
. (8)

Two-component vectors Ψ(t) belong to

H = L2(R3) ⊕L2(R3) (9)

and the Hamiltonian H may be viewed as acting in
H.

The so called Schrödinger form of the Klein-Gordon
equation (8) is equivalent to the original Klein-Gordon
equation (1). It is in more familiar form, although,
new challenge arises with the manifest non-Hermiticity
of Hamiltonian (7).

2.1. Eigenvalues
New form of the Klein-Gordon equation (8) has many
benefits. One of them is simplification of calculation of
its eigenvalues to mere solving the eigenvalue problem
for operator K

Kψn = �nψn. (10)

The relationship between eigenvalues �n of the opera-
tor K and eigenvalues En of the non-Hermitian oper-
ator H of the Schrödinger form of the Klein-Gordon
equation (

0 K
I 0

)(
Ψ(1)
Ψ(2)

)
= E

(
Ψ(1)
Ψ(2)

)
(11)

can be easily seen. Equation (11) is formed from two
algebraic equations

KΨ(2) = EΨ(1), Ψ(1) = EΨ(2). (12)

After insertion of the second one to the first one we
obtain

KΨ(2)n = E
2
nΨ

(2)
n , (13)

which compared with equation (10) gives us following
relation between eigenvalues

�n = E2n. (14)

We can see, that eigenvalues En remain real under
assumption of �n > 0.
Relationship between corresponding eigenvectors

HΨ(±)n = E
(±)
n Ψ

(±)
n , Ψ

(±)
n =

(
±
√
�nψn
ψn

)
(15)

is also easy to see.

2.2. Free Klein–Gordon equation
In case of free Klein–Gordon equation operator

K = −∆ + m2 (16)

acting on H = L2(R3) is positive and Hermitian. It
has continuous and degenerate spectrum. As sug-
gested in [10], we identify the space R3 with the vol-
ume of a cube of side l, as l tends to infinity. Than
we can treat the continuous spectrum of K as the
limit of the discrete spectrum corresponding to the
approximation. The eigenvalues are given by

�~k = k
2 + m2 (17)

and corresponding eigenvectors ψ~k = Ψ
(2)
~k

are

ψ~k(~x) = 〈~x|~k〉 = (2π)
−3/2ei

~k.~x, (18)

where ~k ∈ R3 and ~k.~k = k2. We can see that ψ~k /∈
L2(R3). They are generalized eigenvectors, i.e. vectors
which eventually becomes 0 if (K−λI) is applied to it
enough times successively, describing scattering states
[10].

Vectors ψ~k satisfy orthonormality and completeness
conditions

〈~k|~k′〉 = δ(~k −~k′),
∫
d3k|~k〉〈~k|) = 1 (19)

and operator K can be expressed by its spectral reso-
lution as

K =
∫
d3k(k2 + m2)|~k〉〈~k|. (20)

From the relations (14) and (15) we see that eigenval-
ues and eigenvectors of H are given by

E
(±)
~k

= ±
√
~k2 + m2, Ψ(±)

~k
=
(
±
√
~k2 + m2

1

)
ψ~k .

(21)
The eigenvectors Φ(±)

~k
of adjoint operator H† are

Φ(±)
~k

=
(

1
±
√
~k2 + m2

)
ψ~k, (22)

which form together with Ψ(±)
~k

complete biorthogonal
system

〈Φ(ν)
~k′
|Ψ(ν

′)
~k
〉 = δ(~k −~k′)δνν′ 2E

(ν)
~k
, (23)

where ν,ν′ = ±1.

3. Crypto-Hermitian approach
Apparent non-Hermiticity of Hamiltonian (7) can be
dealt with by means of the crypto-Hermitian theory
(sometimes also called quasi-Hermitian [24] or PT -
symmetric [25]).

463



Iveta Semorádová Acta Polytechnica

Hamiltonian is non-Hermitian H 6= H† only in the
false Hilbert space H(F) = (V,〈·|·〉). The underlying
vector space of states is fixed, given by the physical
system. However, we have a freedom in the choice
of inner product. If we represent our Hamiltonian in
different secondary Hilbert space H(S) = (V,〈〈·|·〉),
with newly defined inner product

〈〈·|·〉 = 〈ϕ|Θ|ψ〉, (24)

it may become Hermitian. So called metric opera-
tor Θ must be positive definite, everywhere-defined,
Hermitian and bounded with bounded inverse. Oper-
ators for which such inner product exist will be called
crypto-Hermitian (c.f. [26]). They satisfy the so called
Dieudonée equation

H†Θ = ΘH (25)

and they are similar to Hermitian operators

h = ΩHΩ−1, (26)

where Θ = Ω†Ω is invertible and h = h†.
In such scenario, the problem of negative probability

interpretation of the Klein-Gordon equation can be
reinterpreted as the problem of the wrong choice of
metric operator Θ. If we would be able to find more
appropriate choice of representation space H(S), this
problem would disappear.

3.1. Computation of the metric
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
requires the solution of eigenvalue problem for H†. In
what follows, we try to construct the metric operator
for free Klein-Gordon equation

Θ =
∫
d3k

(
α(+)|Φ(+)

~k
〉〈Φ(+)

~k
| + α(−)|Φ(−)

~k
〉〈Φ(−)

~k
|
)
,

(27)
where we insert eigenvectors Φ(±)

~k
as computed in (22)

Θ =
∫
d3k

(
α β

√
k2 + m2

β
√
k2 + m2 α(k2 + m2)

)
|~k〉〈~k|,

(28)
where α = α(+) + α(−), β = α(+) −α(−). By means
of equation (20) we obtain family of metric operators

Θ =
(

α βK1/2

βK1/2 αK

)
, (29)

where
K1/2 =

∫
d3k
√
k2 + m2|~k〉〈~k|. (30)

With the knowledge of the metric operator (29), we
can construct positive definite inner product defining
Hilbert space H(S)

〈〈Ψ|Φ〉 = α(〈ψ|K|ϕ〉 + 〈ψ̇|ϕ̇〉)
+ iβ(〈ψ|K1/2|ϕ̇〉−〈ψ̇|K1/2|ϕ〉) ,

(31)

where ϕ̇, ψ̇ denote corresponding time derivatives (In
fact, this equation is just an explicit version of equa-
tion (24)).

3.2. The discrete case
Unfortunately, the metric operator (29) is unbounded
and therefore doesn’t satisfy all the requested proper-
ties we put upon metric operator. As was emphasized
in [27], boundedness of metric operator Θ is very im-
portant property, it guarantees that convergence of
Cauchy sequences is not affected by introduction of
new inner product (24). The possibility of the use of
unbounded metrics is 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 atten-
tion to a discrete model. In the discrete approximation
the metric operator stays bounded. We make use of
equidistant, Runge-Kutta grid-point coordinates

xk = kh , k = 0,±1,±2 . . . , (32)

Laplacian can be expressed as

−
ψ(xk+1) − 2ψ(xk) + ψ(xk−1)

h2
, (33)

The explicit occurrence of the parameter h will be
important for the study of the continuum limit in
which the value of h would decrease to zero. Otherwise
we may set h = 1 in suitable units. Following further
ideas from [29], Laplace operator ∆ can be discretized
into matrix form

∆(n) =




2 −1
−1 2 −1

−1 2
...

... ... −1
−1 2


 (34)

Matrix (34) is Hermitian and therefore diagonalizable,
i.e. similar to diagonal matrix. Hence for our purposes
it is enough to compute with n×n real diagonal matrix

K =



a1 0 · · · 0
0 a2 · · · 0
...

...
...

...
0 0 · · · an


 . (35)

Let A, B, C be real matrices n×n, where A = AT ,
B = BT . Than we can write the Dieudonné equation
(25) by means of block matrices(

0 I
K 0

)(
A CT

C B

)
=
(
A CT

C B

)(
0 K
I 0

)
. (36)

We obtain following conditions

C = CT , KC = CTK, B = KA = AK. (37)

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vol. 57 no. 6/2017 Crypto-Hermitian Approach to the Klein–Gordon Equation

Real symmetric matrix which commutes with diag-
onal matrix must be diagonal. Thus the form of our
metric operator is as follows

Θ =




α1 · · · 0 β1 · · · 0
...

...
...

...
...

...
0 · · · αn 0 · · · βn
β1 · · · 0 a1α1 · · · 0
...

...
...

...
...

...
0 · · · βn 0 · · · anαn




. (38)

It depends on 2n parameters α1 . . .αn, β1 . . .βn.
Requirement of positive-definitness of the metric put
following conditions on our parameters

αi > 0 , aiα2i > β
2
i , i = 1, 2, . . . ,n . (39)

We can construct corresponding inner product

〈〈ψ|ϕ〉 =
n∑
i=1

αiψ
∗
i ϕi

+
n∑
i=1

βi(ψ∗i ϕn+i + ψ
∗
n+iϕi)

+
n∑
i=1

aiαiψ
∗
n+iϕn+i ,

(40)

where ψ = (ψ1,ψ2, . . .ψ2n)T , ϕ = (ϕ1,ϕ2 . . . ,ϕ2n)T
are complex vectors.

4. Conclusions
In our work, we familiarized the reader with the crypto-
Hermitian approach to the Klein-Gordon equation.
We computed metric operator in both continuous
and discrete cases. Corresponding positive definite
inner product for free Klein-Gordon equation was
also computed. That is considered a crucial step in
proper probability interpretation of the Klein-Gordon
equation.

The next step of this process would be construction
of appropriate metric operator for the Klein-Gordon
equation with nonzero potential V as was done for
special cases in [16, 17, 19, 21]. It is also possible
to broaden the formalism by adding manifest non-
Hermiticity in operator K 6= K†, as was shown in
[20].

Related complicated problems with locality, defini-
tion of physical observables and attempts to construct
conserved four-current can be thoroughly studied in
further references [16, 17]. The problems become
much simpler if we narrow our attention to real Klein-
Gordon fields only. It was shown that in such a case,
inner product is uniquely defined [16, 30].

Acknowledgements
The work of Iveta Semorádová was supported by the CTU
grant SGS16/239/OHK4/3T/14.

References
[1] O. Klein. Quantentheorie und fünfdimensionale
relativitätstheorie. Zeitschrift für Physik
37(12):895–906, 1926. doi:10.1007/BF01397481.

[2] W. Gordon. Der comptoneffekt nach der
Schrödingerschen theorie. Zeitschrift für Physik
40(1-2):117–133, 1926.
doi:https://doi.org/10.1007/BF01390840.

[3] J. Kudar. Zur vierdimensionalen formulierung der
undulatorischen mechanik. Annalen der Physik
386(22):632–636, 1926. doi:10.1002/andp.19263862208.

[4] V. Fock. Über die invariante form der wellen-und der
bewegungsgleichungen für einen geladenen massenpunkt.
Zeitschrift für Physik 39(2-3):226–232, 1926.
doi:10.1007/BF01321989.

[5] V. Fock. Zur Schrödingerschen wellenmechanik.
Zeitschrift für Physik A Hadrons and Nuclei
38(3):242–250, 1926.

[6] T. De Donder, H. van den Dungen. La quantification
déduite de la gravifique Einsteinienne. Comptes rendus
183:22–24, 1926.

[7] E. Schrödinger. Quantisierung als eigenwertproblem.
Annalen der physik 385(13):437–490, 1926.
doi:10.1002/andp.19263851302.

[8] F. Constantinescu, E. Magyari. Problems in Quantum
Mechanics. Elsevier, 2013. doi:10.1119/1.1986548.

[9] W. Pauli, V. Weisskopf. Über die quantisierung der
skalaren relativistischen wellengleichung. Helv Phys
Acta 7:709–731, 1934.

[10] A. Mostafazadeh. Hilbert space structures on the
solution space of Klein-Gordon type evolution equations.
Class Quantum Grav 20:155–171, 2003.
doi:10.1088/0264-9381/20/1/312.

[11] J. Dieudonné. Quasi-hermitian operators. Proc
Internat Sympos Linear Spaces (Jerusalem, 1960),
Pergamon, Oxford 115122, 1961.

[12] F. J. Dyson. General theory of spin-wave
interactions. Phys Rev 102(5):1217, 1956.
doi:10.1103/PhysRev.102.1217.

[13] F. Scholtz, H. Geyer, F. Hahne. Quasi-hermitian
operators in quantum mechanics and the variational
principle. Ann Phys 213:71–101, 1992.
doi:10.1016/0003-4916(92)90284-S.

[14] C. M. Bender, S. Boettcher. Real spectra in
non-hermitian hamiltonians having PT -symmetry. Phys
Rev Lett 80(24):5243, 1998.
doi:10.1103/PhysRevLett.80.5243.

[15] A. Mostafazadeh. Quantum mechanics of
Klein-Gordon-type fields and quantum cosmology. Ann
Phys (New York) 309:1–48, 2004.
doi:10.1016/j.aop.2003.08.010.

[16] A. Mostafazadeh, F. Zamani. Quantum mechanics of
Klein–Gordon fields I: Hilbert space, localized states,
and chiral symmetry. Ann Phys 321(9):2183–2209,
2006. doi:10.1016/j.aop.2006.02.007.

[17] A. Mostafazadeh, F. Zamani. Quantum mechanics of
Klein–Gordon fields II: Relativistic coherent states. Ann
Phys 321(9):2210–2241, 2006.
doi:10.1016/j.aop.2006.02.008.

465

http://dx.doi.org/10.1007/BF01397481
http://dx.doi.org/https://doi.org/10.1007/BF01390840
http://dx.doi.org/10.1002/andp.19263862208
http://dx.doi.org/10.1007/BF01321989
http://dx.doi.org/10.1002/andp.19263851302
http://dx.doi.org/10.1119/1.1986548
http://dx.doi.org/10.1088/0264-9381/20/1/312
http://dx.doi.org/10.1103/PhysRev.102.1217
http://dx.doi.org/10.1016/0003-4916(92)90284-S
http://dx.doi.org/10.1103/PhysRevLett.80.5243
http://dx.doi.org/10.1016/j.aop.2003.08.010
http://dx.doi.org/10.1016/j.aop.2006.02.007
http://dx.doi.org/10.1016/j.aop.2006.02.008


Iveta Semorádová Acta Polytechnica

[18] A. Mostafazadeh. A physical realization of the
generalized PT -, C-, and CPT -symmetries and the
position operator for Klein-Gordon fields. International
Journal of Modern Physics A 21(12):2553–2572, 2006.
doi:10.1142/S0217751X06028813.

[19] M. Znojil. Relativistic supersymmetric quantum
mechanics based on Klein-Gordon equation. J Phys A:
Math Gen 37:9557–9571, 2004.
doi:10.1088/0305-4470/37/40/016.

[20] M. Znojil. Solvable relativistic quantum dots with
vibrational spectra. Czech J Phys 55:1187–1192., 2005.
doi:10.1007/s10582-005-0127-6.

[21] M. Znojil, H. Bíla, V. Jakubský. Pseudo-hermitian
approach to energy-dependent Klein-Gordon models.
Czech J Phys 54(10):1143–1148, 2004.
doi:10.1023/B:CJOP.0000044017.33267.58.

[22] H. Feshbach, F. Villars. Elementary relativistic wave
mechanics of spin 0 and spin 1/2 particles. Rev Mod
Phys 30(1):24, 1958. doi:10.1103/RevModPhys.30.24.

[23] L. L. Foldy. Synthesis of covariant particle equations.
Phys Rev 102(2):568, 1956.
doi:10.1103/PhysRev.102.568.

[24] A. Mostafazadeh. Pseudo-hermitian representation of
quantum mechanics. Int J Geom Meth Mod Phys
7:1191–1306, 2010. doi:10.1142/S0219887810004816.

[25] C. M. Bender. Making sense of non-hermitian
hamiltonians. Reports on Progress in Physics 70(6):947,
2007. doi:10.1088/0034-4885/70/6/R03.

[26] M. Znojil. Three-hilbert-space formulation of
quantum mechanics. Symmetry, Integrability and
Geometry: Methods and Applications 5(001):19, 2009.

[27] R. Kretschmer, L. Szymanowski. Quasi-hermiticity in
infinite-dimensional hilbert spaces. Phys Lett A
325(2):112–117, 2004.

[28] F. Bagarello, J.-P. Gazeau, F. H. Szafraniec,
M. Znojil. Non-selfadjoint operators in quantum physics:
Mathematical aspects. John Wiley & Sons, 2015.

[29] M. Znojil. N-site-lattice analogues of V (x) = ix3.
Ann Phys 327(3):893–913, 2012.
doi:10.1016/j.aop.2011.12.009.

[30] F. Kleefeld. On some meaningful inner product for
real Klein-Gordon fields with positive semi-definite
norm. Czech J Phys 56(9):999–1006, 2006.
doi:10.1007/s10582-006-0395-9.

466

http://dx.doi.org/10.1142/S0217751X06028813
http://dx.doi.org/10.1088/0305-4470/37/40/016
http://dx.doi.org/10.1007/s10582-005-0127-6
http://dx.doi.org/10.1023/B:CJOP.0000044017.33267.58
http://dx.doi.org/10.1103/RevModPhys.30.24
http://dx.doi.org/10.1103/PhysRev.102.568
http://dx.doi.org/10.1142/S0219887810004816
http://dx.doi.org/10.1088/0034-4885/70/6/R03
http://dx.doi.org/10.1016/j.aop.2011.12.009
http://dx.doi.org/10.1007/s10582-006-0395-9

	Acta Polytechnica 57(6):462–466, 2017
	1 Introduction
	2 Klein-Gordon equation in Schrödinger form
	2.1 Eigenvalues
	2.2 Free Klein–Gordon equation

	3 Crypto-Hermitian approach
	3.1 Computation of the metric
	3.2 The discrete case

	4 Conclusions
	Acknowledgements
	References