Page 1
1
Dirac and Klein-Gordon Equations in Curved Space
A. D. Alhaidari a and A. Jellal a,b,c
a Saudi Center for Theoretical Physics, Jeddah, Saudi Arabia
b Physics Department, College of Sciences, King Faisal University, Alahssa 31982, Saudi Arabia
c Theoretical Physics Group, Faculty of Sciences, Chouaib Doukkali University, 24000 El Jadida, Morocco
Abstract: We introduce matrix operator algebra involving a universal curvature
constant. Using elements of the algebra, we write the Dirac equation without the need
for spin connections or vierbeins. Iterating the equation and using the algebra leads to
the Klein-Gordon equation in curved space in its canonical from (without first order
derivatives). We obtain exact solutions of the Dirac and Klein-Gordon equations for a
static diagonal metric.
PACS: 04.60.-m, 04.62.+v, 04.20.Cv, 03.65.Pm, 04.20.Jb
Keywords: Curved space, Dirac equation, Klein-Gordon equation, matrix operator
algebra, static diagonal metric
I. INTRODUCTION
Understanding the connection between quantum theory (mechanics and fields)
and gravity continues to be one of the main undertakings in physics that proved to be
highly nontrivial and very demanding. Formulation of quantum gravity is still far from
being successful or even satisfactory. A consistent unification of quantum theory and
gravity must first address the state of a single elementary particle in a gravitational
background. Consequently, sustained efforts have been applied to find a systematic and
appropriate formulation of the relativistic equation of motion for the lowest spin
particles (spin-0 and spin-1
2) in a curved space-time. That is, the extension of the Klein-
Gordon and Dirac equations from flat space to a curved space. One of the interesting
problems in this connection is the extent to which spin has an effect on the quantum
gravitational phenomena. For example, it has been shown in [1] that the spectrum of
spin-0 and spin-1/2 particles in a constant gravitational field differ by an amount of
mg c
, where m is the rest mass of the particle and g is the acceleration of gravity.
Although weak, this is a significant difference that shows the influence of spin in
gravitational interaction. Moreover, unambiguous observation of the influence of
gravity on the behavior of fermions is one of the major motivations to study the Dirac
equation in curved space. An example is the quantum effects on neutrons in a classical
gravitational field [2-6]. In recent years, investigation of Dirac particles in virtual
gravitational fields have been at the center of interest in condensed matter physics in the
context of studies of the amazing properties of graphene [7-10]. It was shown that it is
possible to simulate some of these properties by coupling the Dirac fermions to an
"artificial" gravitational field; specifically, to consider the physics of massless Dirac
particles in a 2+1 curved space-time. These results show rare and direct connection
between gravity and quantum mechanics and constitute another strong motivation to the
study of the Dirac equation in curved space.
Page 2
2
The greatest difficulties in these studies arise from the covariant generalization of
the Dirac equation [11,12] and its uniqueness. Due to the complexity of the Dirac
equation (a system of partial differential equations) the number of exact solutions even
in the special theory of relativity remained very limited. There are two types of
difficulties that occur in the solution of the Dirac equation in special relativity. The first
is due to the physical nature of the problem; in particular, the geometry of the external
field. The second is purely mathematical and is related to the choice of coordinates. On
the other hand, a complete theory of separation of variables for the Dirac equation in a
curved space-time has yet to be developed. Nonetheless, it is common knowledge that
separation of variables in the Dirac equation is easier for the massless case and in the
context of the Kerr geometry [13-15]. The connection between separation of variables
and matrix first-order differential operators commuting with the Dirac Hamiltonian has
traditionally been the main focus in such developments. However, in [16] the separation
problem was solved provided that the Dirac squared equation (or the Klein-Gordon
equation) is reduced to two independent differential equations of second order (i.e., it
admits diagonalization).
The equation of relativistic quantum mechanics was formulated in the early part
of last century by Paul Dirac [17]. It describes the state of electrons in a way consistent
with quantum mechanics and special relativity. The physics and mathematics of the
Dirac equation is very rich, illuminating and provides a theoretical framework for
different physical phenomena that are not present in the nonrelativistic regime such as
the Klein paradox, super-criticality [17-19] and the anomalous quantum Hall effect in
graphene [20,21]. In its classical representation, the Dirac equation is simply the square
root of the relativistic statement
p m c
, where p is the space-time linear momentum
, where
n space-time by
summed over). In flat space, the metric is constant and, thus, the matrices
independent of space and time. Thus, quantization is straightforward. However, in a
curved space, where the metric is not constant, these matrices are space-time dependent.
Thus, quantization of the classical term
( )
involve ordering ambiguity. However, it is known that symmetric quantization of the
classical phase space function product f(x)g(p) is not ambiguous if and only if f(x) or
g(p) is linear; which is the case here. Consequently, quantization of the Dirac equation
in curved space does not pose a problem. On the other hand, covariant generalization
does. Compatibility of symmetric quantization with covariance gives a special
representation of the spin connections
that enter in the covariant derivative of the
spinor wavefunction in terms of the matrices
More precisely, one obtains
1
n
x
the Dirac equation to curved space was independently developed long ago by Weyl [22]
and by Fock [23], which is known in the literature as Dirac-Fock-Weyl (DFW)
equation. Recently, two alternative versions of the Dirac equation in a curved space-
time were proposed in [24]. These obey the equivalence principle in a direct and explicit
sense, whereas the DFW equation obeys the same only in an extended sense.
222
vector. It is written as
p mc
0
n
is a set of square matrices that are
,2g
1 (repeated indices are
related to the metric of the
1
are
x p
becomes a non-trivial issue that may
and their first order derivatives.
. The covariant generalization of
1
( )tr
Page 3
3
The present work, which is complementary to those cited above, may constitute a
measurable contribution in the pursuit of a systematic and more appropriate formulation
of the Dirac and Klein-Gordon equations in a curved space. Specifically, we introduce a
matrix operator algebra involving the Dirac gamma matrices with a universal length
scale constant as a measure of the curvature of space. As a result, we find that spin
connections or vierbeins are no longer required for writing down the Dirac equation. It
also allows us to derive the Klein-Gordon equation in its canonical form without first
order derivatives. The option of not using vierbeins has appeared in the earlier literature
[25-29] though often without any proof that spin connections exist. We conclude this
work with an example where we choose a static diagonal metric in 1+1 space-time and
obtain exact solutions for free spin-0 and spin-1
gravitational background. In the following section, we start by defining the matrix
algebra and point out its correspondence with the classical Poisson bracket algebra and
the quantum mechanical algebra.
II. MATRIX OPERATOR ALGEBRA: DIRAC & KLEIN-GORDON
EQUATIONS
The covariant generalization of the Dirac equation in a curved space-time of
dimension 1
n for free spin 1
2 particle in the units
im
.
Here, however, we intend to formulate the Dirac equation in curved space by employing
a proper matrix algebra without the need for spin connections. For that, we propose the
following alternative equation
im
,
where is a space-time dependent matrix which is going to play a central role in the
matrix algebra that will be introduced below. is a universal real constant of inverse
length dimension that gives a measure of the curvature of space (e.g., the inverse of the
"effective radius of curvature" of space). The transformation properties of is the same
as that of
and results from the covariance of Eq. (2) under the general coordinate
transformation and local spinor transformations. Iteration of Eq. (2) (i.e., squaring the
equation) gives
,
Gi
where G
and B
,
,2
then Eq. (3) becomes
gm
,
which is the Klein-Gordon equation in a curved space in its simple canonical form (no
first order derivatives). It is thus required that
coupling (spin coupling) is present in the Klein-Gordon equation. Furthermore, we
extend the 2
n dimensional algebra (4), which is defined for a given positive integer n
2 relativistic particles in this
1
c
gives
(1)
(2)
222
iBm
, (3)
. If we define ˆD
and require that
ˆ
iD
, (4a)
2
ˆ
iD
, (4b)
222
(5)
2
be diagonal so that no component
Page 4
4
by the matrix and the metric
space of matrices of our problem. That is, any matrix , which is an element of this
algebra must satisfy
,
This algebra has correspondence with ordinary nonrelativistic quantum mechanics
where, for example,
,
t
iH
and
goes to anti-commutator,
, and functions are replaced by matrices. However, in
non-relativistic quantum mechanics we also have
m
, which maintains the correspondence only in the massless case (m = 0).
Another correspondence could also be drawn with the Poisson bracket algebra of
classical mechanics, where the Poisson bracket corresponds to the anti-commutator
[31]. In Ref. [32], the spin connection and the Riemann-Christoffel connection
related as
[,]0
,
where
ggg
matrix algebra (4) and the Dirac equation (1) gives an alternative representation of the
spin connections as
2g
Thus, in the Dirac equation (1), the matrix nature of the spin connection is in the linear
combination of the
not in
itself.
Therefore, Eq. (2) with the 2
1
2
,
g
, to any square matrix that belongs to the
ˆ
iD
. (6)
,
ip
[30]. Thus, the commutator
1
t
iH
whereas here
ˆ
iD
are
(7)
1
2g
. Compatibility of this relation with the
1
, which is a set of functions rather than matrices.
n matrices
0
,
n
satisfying the matrix
algebra (4) is taken here as the Dirac equation in a curved space-time whose metric is
g and curvature parameter . The corresponding Klein-Gordon equation for spin-0
particle is Eq. (5). One of the advantages of the algebra (4) is the absence of first order
derivatives in the resulting Klein-Gordon equation. The presence of these derivatives in
the conventional formalism is a source of technical difficulty for obtaining the solution
of the equation. Consequently, disposing of the spin connections and vierbeins in favor
of the matrix operator algebra (6) resulted in the Dirac equation (2) in a curved space
that lead naturally to the simple canonical form of the Klein-Gordon equation (5). Note
that in the flat space limit (
0
) we recover the traditional Dirac and Klein-Gordon
equations of special relativity provided that
0
limg
, where
is the flat space-
time metric.
For a given space-time metric g and a square matrix (with
remains a challenging task in representation theory to account for the complete space of
square matrices
satisfying the algebra (6) where the
elements of the same algebra and satisfy
give a simple example of our formulation in 1+1 space-time with a static diagonal
metric.
2
diagonal), it
1
n matrices
1. In the following section, we
are
,2g
Page 5
5
III. AN EXAMPLE: 1+1 SPACE-TIME WITH STATIC DIAGONAL METRIC
For
1
n , let us consider the quantum mechanical behavior of a relativistic free
particle influenced only by the gravitational field in 1+1 space-time with a static metric.
The requirement that
is diagonal dictates that must either be diagonal or its two
2
diagonal entries are the negative of each other. That is, either
0
b
0
a
or
a b
c
a
.
We choose the representation
( )
0
0
( )
a x
a x
with
0
x . The algebra (4) results in the
following
0
1
0
0
1
,
1
0 1
1 0
i
, (8)
where
the derivative with respect to x. The static metric becomes
2
1
2
Writing
( )
a xe
simplifies the solution of the nonlinear equation
b(x) as a linear function in x. Thus, we can write
is a length parameter for the curved 1+1 space-time measured in units of the universal
constant
. Thus, the line element in this space is
the flat space limit is obtained when
0
, R such that
( , )( )
x tex
in Eq. (5), we obtain the following Klein-Gordon equation
1
( )0
22
dy
where y x R
. In terms of the new variable
this equation becomes
2
2
dz
dz
The solution is obtained directly as
( )
z
2
2
a a
and the function a(x) must satisfy
2
aaa
. A prime stands for
1
0
0
g
. (9)
( )
b x
2
aa
Re
a
, where R
giving
x R
ae
and 2
x R
1
22222
2(
)
x R
ds dt R e
R
dx
. Writing
and
12
iEt
222
2
22
y
dEm
ey
, (10)
y
ze
, where
2222
( ) 2
Em
,
22
1
( )
z
0
dd
zzz
. (11)
11
22
( )
z
( )
z A JB N
, where A and B are
normalization constants.
respectively. For E
( )
z
, the variable z is real and we obtain the continuum scattering
J
and ( )
z
N
are the Bessel and Neumann functions,
m
solution. On the other hand, no bound states solutions exist for E
becomes pure imaginary. This behavior is expected since the particle is free in the
gravitational background.
On the other hand, the two-component Dirac equation (2) for the same
representation of the algebra reads as follows
dy
Giving the following set of equations
dy
mE
m
where z
1 2
0
1 2
y
y
d
dy
m E
d
m E
e
e
. (12)
1 2
y
d
e
, (13a)
Page 6
6
222
2
22
1
4
0
4
y
d
dy
Em
e
. (13b)
Note the difference between the second order differential equation (13b) and the
corresponding Klein-Gordon equation (10). Moreover, the two equations (13a) and
(13b) with the top/bottom signs are valid only for positive/negative energy since in
(13a) we should impose the constraint that E
positive/negative energy is
11
22
z
where
ze
and ( ) 4
Em
obtained by substituting (14) into (13a) with the top/bottom sign for positive/negative
energy giving
( )cossin
z
m E
zAzBz
Again, boundary conditions dictate that we obtain continuum scattering solutions for
Em
but no bound states for Em
. In Fig. 1, we compare the solution of the
Dirac equation (the upper component, Eq. 14) with that of the Klein-Gordon equation
and the flat space sinusoidal solution.
IV. CONCLUSION
We introduced an algebra based on a matrix whose square is diagonal and
involving the Dirac gamma matrices defined on curved space. This algebra is rich in
sense that it allowed us to formulate the Dirac equation without the need for spin
connections or vierbeins. Iterating the Dirac equation and using the algebra lead to the
Klein-Gordon in its canonical form without first order derivatives. As an illustration of
our findings, we studied one relativistic particle in 1+1 space-time with static diagonal
metric. By choosing an appropriate representation of the elements of our algebra, we
obtained exact solutions for both equations. The present work will shortly be extend and
followed by a study of relativistic particles in 2+1 curved space in the presence of an
electromagnetic field. An interesting application of such a study is in connection with
the work on massless Dirac fermions in graphene sheets in a magnetic field as well as in
external potential structures.
ACKNOWLEDGEMENT
The generous support provided by the Saudi Center for Theoretical Physics (SCTP) is
highly appreciated by the Authors. AJ acknowledges partial support by King Faisal
University.
REFERENCES:
[1] M. Khorrami, M. Alimohammadi and A. Shariati, Ann. Phys. 304, 91 (2003).
[2] A. W. Colella, R. Overhauser and S. A. Werner, Phys. Rev. Lett. 34, 1472 (1975).
[3] S. A. Werner, J. A. Staudenmann and R. Colella, Phys. Rev. Lett. 42, 1103
(1979).
m
. The solution of Eq. (13b) for
2
( )
z
( )( )
z
sincos
A J z B NAzBz
, (14)
y
22222
2 . The other component of the spinor is
2
2
. (15)
Page 7
7
[4] F. Riehle, Th. Kisters, A. Witte, J. Helmcke and Ch. J. Bordé, Phys. Rev. Lett. 67,
177 (1991).
M. Kasevich and S. Chu, Phys. Rev. Lett. 67, 181 (1991).
V. V. Nesvizhevsky et al., Nature 415, 297 (2002).
K. S. Novoselov et al., Nature 438, 197 (2005).
M. I. Katsnelson, K. S. Novoselov and A. K. Geim, Nat. Phys. 2, 620 (2006).
F. Zhou and G. Semenoff, Phys. Rev. Lett. 97, 180411 (2006).
[10] A. H. C. Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov and A. K. Geim, Rev.
Mod. Phys. 81, 109 (2009).
[11] A. Lichnerowicz, Bull. Sot. Math. France 92, 11 (1964).
[12] Y. Choquet-Bruhat and C. Dewitt-Morette, Analysis, Manifolds and Physics I
(North-Holland, Amsterdam, 1989).
[13] S. A. Teukolsky, Astrophys. J. 185, 635 (1973).
[14] W. G. Unruh, Phys. Rev. Lett. 31, 1265 (1973).
[15] S. Chandrasekhar, Proc. R. Soc. London, Ser. A 349, 571 (1976).
[16] V. G. Bagrov and V. V. Obukhov, J. Math. Phys. 33, 2279 (1992).
[17] P. A. M. Dirac, Proc. Roy. Soc. A 117, 610 (1928); W. Greiner, Relativistic
Quantum Mechanics: Wave Equations (Springer, Berlin, 1994).
[18] O. Klein, Z. Phys. 53, 157 (1929); N. Dombey, P. Kennedy and A. Calogeracos
Phys. Rev. Lett. 85, 1787 (2000).
[19] N. Dombey and A. Calogeracos, Phys. Rep. 315, 41 (1999).
[20] Y. Zheng and T. Ando, Phys. Rev. B 65, 245420 (2002).
[21] V. P. Gusynin and S. G. Sharapov, Phys. Rev. Lett. 95, 146801 (2005).
[22] H. Weyl, Z. Phys. 56, 330 (1929).
[23] V. A. Fock, Z. Phys. 57, 261 (1929).
[24] M. Arminjon, Found. Phys. 38, 1020 (2008).
[25] E. Schrödinger, Sitzungsber. Preuss. Akad. Wiss., Phys.-Math. Kl. XI, 105
(1932).
[26] F. J. Belinfante, Physica (Amsterdam) 7, 305 (1940).
[27] H. S. Green, Nucl. Phys. 7, 373 (1958).
[28] H. Pagels, Ann. Phys. (N.Y.) 31, 64 (1965).
[29] J. S. Dowker and Y. P. Dowker, Proc. Phys. Soc. London 87, 65 (1966).
[30] See, for example, A. Messiah, Quantum Mechanics, Vol. I (Wiley, New York,
1966); E. Merzbacher, Quantum Mechanics (Wiley, New York, 1970).
[31] See, for example, H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley,
Reading, 1980); E. A. Desloge, Classical Mechanics, Vol. 2 (Wiley, New York,
1982).
[32] H. A. Weldon, Phys. Rev. 63, 104010 (2001).
[5]
[6]
[7]
[8]
[9]
Page 8
8
FIGURE CAPTION:
FIG. 1: The solid curve is the upper component of the Dirac wavefunction (14) for an
arbitrary normalization. The dashed curve is the Klein-Gordon wavefunction, whereas
the dashed-dotted curve is the free wavefunction in flat space, sin(
coordinate x is measured in units of
and we took
2)
xE . The
10
1
2 1.5
and
1
R
.
Fig. 1
Download full-text