Crypto-Hermitian Approach to the Klein-Gordon Equation

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.

Full text

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
aNuclear Physics Institute, Czech Academy of Science, Řež near Prague, Czech Republic
bFaculty 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∂2
t−
∆ =
∂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)=0K
I0Ψ(1)
Ψ(2).(6)
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Hamiltonian of the quantum system takes form
H=0K
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
0K
I0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=E2
nΨ(2)
n,(13)
which compared with equation
(10)
gives us following
relation between eigenvalues
n=E2
n.(14)
We can see, that eigenvalues
En
remain real under
assumption of n>0.
Relationship between corresponding eigenvectors
HΨ(±)
n=E(±)
nΨ(±)
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=k2+m2(17)
and corresponding eigenvectors ψ~
k= Ψ(2)
~
kare
ψ~
k(~x) = h~x|~
ki= (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 0if (
K−λI
)is applied to it
enough times successively, describing scattering states
[10].
Vectors
ψ~
k
satisfy orthonormality and completeness
conditions
h~
k|~
k0i=δ(~
k−~
k0),Zd3k|~
kih~
k|)=1 (19)
and operator
K
can be expressed by its spectral reso-
lution as
K=Zd3k(k2+m2)|~
kih~
k|.(20)
From the relations
(14)
and
(15)
we see that eigenval-
ues and eigenvectors of Hare given by
E(±)
~
k=±q~
k2+m2,Ψ(±)
~
k= ±p~
k2+m2
1!ψ~
k.
(21)
The eigenvectors Φ(±)
~
kof adjoint operator H†are
Φ(±)
~
k= 1
±p~
k2+m2!ψ~
k,(22)
which form together with Ψ
(±)
~
k
complete biorthogonal
system
hΦ(ν)
~
k0|Ψ(ν0)
~
ki=δ(~
k−~
k0)δνν 02E(ν)
~
k,(23)
where ν, ν0=±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
H6
=
H†
only in the
false Hilbert space
H(F)
= (
V, h·|·i
). 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, hh·|·i
),
with newly defined inner product
hh·|·i =hϕ|Θ|ψi,(24)
it may become Hermitian. So called metric opera-
tor Θmust 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†Ωis 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 Θfor 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 = Zd3kα(+)|Φ(+)
~
kihΦ(+)
~
k|+α(−)|Φ(−)
~
kihΦ(−)
~
k|,
(27)
where we insert eigenvectors Φ
(±)
~
k
as computed in
(22)
Θ = Zd3kα β√k2+m2
β√k2+m2α(k2+m2)|~
kih~
k|,
(28)
where
α
=
α(+)
+
α(−)
,
β
=
α(+) −α(−)
. By means
of equation
(20)
we obtain family of metric operators
Θ = α βK1/2
βK1/2αK ,(29)
where
K1/2=Zd3kpk2+m2|~
kih~
k|.(30)
With the knowledge of the metric operator (29), we
can construct positive definite inner product defining
Hilbert space H(S)
hhΨ|Φi=α(hψ|K|ϕi+h˙
ψ|˙ϕi)
+iβ(hψ|K1/2|˙ϕi−h˙
ψ|K1/2|ϕi),(31)
where
˙ϕ, ˙
ψ
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 Θis 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=




a10·· · 0
0a2·· · 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
0I
K0A CT
C B =A CT
C B 0K
I0.(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·· · 0β1·· · 0
.
.
.....
.
..
.
.....
.
.
0·· · αn0·· · βn
β1·· · 0a1α1·· · 0
.
.
.....
.
..
.
.....
.
.
0·· · βn0·· · anαn










.(38)
It depends on 2
n
parameters
α1. . . αn
,
β1. . . βn
.
Requirement of positive-definitness of the metric put
following conditions on our parameters
αi>0, aiα2
i> β2
i, i = 1,2, . . . , n . (39)
We can construct corresponding inner product
hhψ|ϕi=
n
X
i=1
αiψ∗
iϕi
+
n
X
i=1
βi(ψ∗
iϕn+i+ψ∗
n+iϕi)
+
n
X
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
K6
=
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.
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