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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.

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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

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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

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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

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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)

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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.

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. 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)

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[5]

[6]

[7]

[8]

[9]

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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

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