Lagrangian Formalism for the New Dirac Equation

Lagrangian Formalism for the New Dirac Equation

OUSMANE MANGA Adamou

Department of Physics, Faculty of Sciences

Abdou Moumouni University of Niamey, P.O. Box 10662, Niger [email protected]

SAMSONENKO Nicolai Vladimirovich

Department of Theoretical Physics, Russian Friendship University 3, Ordjonokidze,117923 Moscow, Russia

MOUSSA Aboubacar

Department of Mathematics and Computer Sciences, Faculty of Sciences Abdou Moumouni University of Niamey P.O. Box 10662, Niger

[email protected]

Abstract

The expressions for the energy-impulse tensor and the spin operators are obtained for the particle described by the new Dirac equation in the framework of the Lagrange formalism.

Mathematics Subject Classification: 83C57, 47B15, 47B25

1. Introduction

The new relativistic wave equation proposed by Dirac in 1971 (see [3]) is not symmetric in term of positive and negative values of energy. This equation describes a spinless particle with positive energy, internal structure and non-zero rest mass. The equation has the following form:

r

0 r

m q 0

x α x β ψ

⎧ ∂ ₊ ∂ ₊ ⎫ ₌

⎨_{∂ ∂} ⎬

⎩ ⎭ (1)

where α_r ( r=1 , 2 , 3 ) are real 4 × 4 matrices and β is an antisymmetric matrix given by

0 0 1 0

0 0 0 1

1 0 0 0

0 1 0 0

β

⎛ ⎞

⎜ ⎟

⎜ ⎟

=

⎜− ⎟

⎜ ₋ ⎟

⎝ ⎠

(2)

Note that β2 = −1. The quantity q is a colon vector. The symbol q ~ will denote a line vector ( q ,q ,q ,q )_{1 2 3 4} where q ,q_{1 3} = p₁ and q ,q_{2 4} = p₂ are the dynamic variables of two harmonic oscillators describing the internal structure of the particle. The quantities q ( a_a =1 ,2 ,3 ,4 ) satisfy following commuting law

[

]

The wave function ψ is one-component and depends on x , x_{0 r} and two commuting quantities q_a (for example q₁ and q₂). The matrices α_r( r=1 ,2 ,3 )

and β satisfy the Clifford-Dirac algebra relations:

r s s r rs

r r

2

0

α α α α δ

α β βα

+ =

Introducing the notations

x

μ μ

∂ ∂ ≡

∂ and α0 =I the unity matrix, the equation (1) takes the form:

(

)

The multiplying equation (5) by matrix β at the right we get:

(

)

2. Lagrangian

The new Dirac equation, like the old one (see [4]) can be considered as an equation of some field. This field is described by functions q_aψ and ψq_b. With a corresponding Lagrange density function we can obtain the field equation (1) using variational method.

From the Lagrange density function L_D (see [1]) given by:

(

)

D

1 1

L q q q q q m q

4 2

μ μ

μ μ

ψ α β ψ ψ α β ψ ψ ψ

= − ∂ − ∂ + (7)

we can obtain the Lagrange equation in term of the field function ψq~ as following:

(

)

L L

0 xμ μψq ψq

⎛ _∂ ⎞ _∂

∂ ⎜ ⎟ − =

⎜ ⎟

∂ _⎝∂ ∂ _⎠ ∂ , (8)

where D

~

L 1 1

q mq

q 4 2

μ μ

α β ψ ψ

ψ

∂

= − ∂ +

∂ ,

(

)

~

L 1

q

x q 4

μ μ μ

μ

α β ψ

ψ

⎛ _∂ ⎞

∂ ⎜ ⎟ = ∂

⎜ ⎟

∂ _⎝∂ ∂ _⎠ .

So we obtain the equation (6).

3. Basic physical quantities

coordinates vanishing the variation of the action correspond s dynamic invariants, i.e. time-conserved combinations of field functions and their derivatives (see [2]).

Consider the following infinitesimal transformation of the field functions and coordinates:

a a a a

, , , ,

a a a a

x ,

xμ μ xμ xμ

μ μ μ μ

δ

ϕ ϕ ϕ δϕ

ϕ ϕ ′ ϕ δϕ

′

⎯⎯→ = +

′

⎯⎯→ = +

⎯⎯→ = +

(9)

where a a a a

q , a 1,2 ,3 ,4 ;

q , a 5,6 ,7 ,8 .

ϕ ψ

ϕ ψ

= =

= =

The variations δx_μ and δϕ_a are expressed in terms of the linearly independent infinitesimal transformation parameters δωn using the formulas:

( n ) n 1 n s

a a( n ) n 1 n s

x_μ Xμ

δ δω

δϕ Ψ δω

≤ ≤

≤ ≤

= =

∑

∑

Note that δϕ_a,μ is not a derivate of δϕ_a, i.e. operators ∂ ∂/ x and δ do not commute. The fact is that δϕ_a is a variation of the field function as by changing its shape and by the argument.

Denote the variations of the form of the field functions as:

(

)

, ,

a a a a a a( n ) a ( n ) n n

x X

τ τ τ

τ

δϕ =ϕ ϕ′ − =δϕ ϕ δ− =

∑

The operations δ and ∂ ∂/ x do commute. We now define the variation of the action:

D D

I L dx L dx

δ =

_∫

_∫

where

,

D D a a D D

L′ =L (ϕ ϕ′, μ′)=L +δL . The total variation δL_D is equal to:

,

D D D

D a , a D

a a

L L dL

L L x

dx

μ

μ μ

μ

δ δϕ δϕ δ δ

ϕ ϕ

∂ ∂

= + = +

∂ ∂ .

Here δL_D is the variation of L_D due to the variations of the forms ϕ_a and ϕ_a,μ: ,

D D

D a , a

a a

L L

L _μ μ

δ δϕ δϕ

ϕ ϕ

∂ ∂

= +

In result we obtain: D

D D D D D

x

dL d

I ( L L x )( 1 )dx L dx ( L ( L x ))dx

dx x dx

μ μ μ μ μ μ δ δ = +δ + δ +∂ − = δ + δ ∂

∫

∫

∫

D D D

D , a , a , a

a a a

L L L

L

x_μ μ μ x_μ x_μ μ

δ δϕ δϕ δϕ ϕ ϕ ϕ ⎛ ∂ ⎞ ∂ ⎛ ∂ ⎞ ∂ ∂ ∂ = _{⎜ ⎟} + = _{⎜ ⎟} ∂ _⎝∂ _⎠ ∂ ∂ ∂ _⎝∂ _⎠

Substituting this expression for δL_D in (13) we obtain:

(

)

(

)

a( n ) a ( n ) n D ( n ) n ,

n n

a

, D

a( n ) a ( n ) D ( n ) n ,

n a

L

I L x dx

x

L d

X L X dx

dx

L d

X L X dx

dx μ μ μ τ τ μ μ μ τ τ μ μ μ δ δϕ δ ϕ Ψ ϕ δω δω ϕ Ψ ϕ δω ϕ ⎛ ∂ ⎞ ∂ = _⎜ + _⎟ ∂ _⎝∂ _⎠ ⎡∂ ⎤ = _⎢ − + _⎥ ∂ ⎣ ⎦ ⎡∂ ⎤ = _⎢ − + _⎥ ∂ ⎣ ⎦

∫

∑

∑

∫

∑∫

Since δI =0, then by the linear independence of transformation parameters δω_n and the arbitrariness of the integration domain, we have:

(

)

D

a( n ) a ( n ) D ( n ) ,

a L d

X L X 0

dx τ τ μ μ μ Ψ ϕ ϕ ⎡∂ _{− +} ⎤₌ ⎢_∂ ⎥

⎣ ⎦ (14)

We introduce the notations

(

)

D

( n ) , a( n ) a ( n ) D ( n ) a

L

X L X

μ τ τ μ μ Θ Ψ ϕ ϕ ∂ = − + ∂ (15)

Then (14) takes the form: ( n ) d

0 dx

μ μ

Θ = (16)

So

( n ) d dx 0 dx μ μ Θ =

∫

1 2

( n ) ( n )

d _μ μ d _μ μ 0

σ σ

σ Θ − σ Θ =

∫

∫

Here dσ_μ is the projection of the element of surface σ in a 3-plane perpendicular to the axis x_μ. This equation shows that the surface integrals

n ( n )

C ( ) dx_μ μ

σ

σ =

∫

do not depend on the surface σ. a) Energy-impulse tensor

Consider an infinitesimal space-time translation x_μ′ = x_μ+δx_μ. Choosing δx_μ as the transformation parameters, we have:

x_μ X_νμ _ν X_νμ x_ν

ν ν

δ =

∑

∑

Since the field functions are not converted, then Ψ_aν =0.

With in mind, and in this particular case, from (15) we get a second order tensor:

(

)

(

)

(

)

D D

( n ) a b D

a b

~ ~

D

L L

T q q L

q q

1

q q q q L

4

μ ν ν

μν μ μ νμ

ν ν

μ μ νμ

Θ ψ ψ δ

ψ ψ

ψ α β ψ ψ α β ψ δ

∂ ∂

⎯⎯→ = − ∂ − ∂ +

∂ ∂ ∂ ∂

= ∂ − ∂ +

As for qψ and ~

q

ψ satisfying the field equations we have L_D≡0, then

(

)

1

T q q q q

4

ν ν

μν = ψ α β ψμ ∂ − ∂ψ α β ψμ (19)

Hence

~ ~

00

1 q q

T q q

4 t t

i

2 t t

ψ ψ

ψ β β ψ

ψ ψ

ψ ψ

⎛ ∂ ∂ ⎞

= _⎜ − _⎟

∂ ∂

⎝ ⎠

∂ ∂

⎛ ⎞

= _⎜ − _⎟

∂ ∂

⎝ ⎠

(20)

In the same way we obtain the expressions for the remaining components of the tensor:

n0

n n i

T , n 1 , 2 , 3

2 x x

ψ ψ

ψ ψ

⎛ ∂ ∂ ⎞

= _⎜ − _⎟ =

∂ ∂

⎝ ⎠ (21)

The conserved quantity in this case is: 3

0

Pμ =

∫

{

}

2 2 2 2

1 2 1 2 1 1 2 2 1 2 0 3

1

( x ,q ,q ) k exp q q ip ( q q ) 2ip q q /( p p ) exp ip x

2

μ

μ μ

ψ = ⎧_⎨− _⎣⎡ + + − − ⎤_⎦ + _⎬⎫× −

⎩ ⎭

(22) and the normalisation condition

3

d x 1

ψψ =

∫

we obtain the expression for the energy

3 3

0 00 0 0

P =

∫

∫

and 3-momentum

3 3

r r0 r r

P =

∫

∫

x a x ,

x x

a , ν

μ μ μ μν

μ μ μ ν

μν ′

⎯⎯→ = + ′ ⎯⎯→

∂ ∂ = ∂ + ∂ (23)

where a_μν = −a_νμ. Thus, in this case deal with a 6-parameter transformations group. Then

(

)

x X a x a x a

x a x a

x x a

μ ρσ μν σν

μ ρσ ν ν σμ

ρ σ

σν σν

ν σμ ν σμ σ ν σ ν

σν ν σμ σ νμ σ ν

δ δ

δ δ

δ δ

<

< >

<

= = = =

= + =

= −

∑

∑

∑

∑

Consequently, we have:

X_ρσμ =x_{σ ρμ}δ −x_{ρ σμ}δ (24)

We now find an expression for Ψ_aρσ. Since ϕ_a′ =ϕ δϕ_a+ _a, then the requirement of relativistically invariant field equations in coordinate transformation (23), the field functions are converted as following [3]:

(

)

qψ′ = −1 βN qψ , where N 1a 4

ρσ ρ σ

α βα

= ,

hence

dc eb

a ab b ad ce b

1

( N ) a

4

ρσ

ρ σ

δϕ = − β ϕ = −

∑

We also have

a a a , ( a 1 , 2 , 3 , 4 )

ρσ ρσ

Comparing these expressions for δϕ_a, we get: dc eb

a ad ce b

1 4

ρσ ρ σ

Ψ = − β α β α ϕ (25)

Given the transformation law for functions ϕ%_b, we get a similar expression for bρσ

Ψ :

mn k

b m n kb

1 4

ρσ ρ σ

Ψ = ϕ α β α βl

l

% (26)

In this case, the tensor Θ_{( n )}μ transforms into the tensor M_ρσμ :

(

)

(

)

D

( n ) a a

a D

b b D

b

L

M q X

q

L

q X L X

q μ μ τ τ ρσ _μ ρσ ρσ τ τ μ ρσ ρσ ρσ μ Θ Ψ ψ ψ Ψ ψ ψ ∂ _{⎡ ⎤} ⎯⎯→ = _⎣ − ∂ _⎦+ ∂ ∂ ∂ _{⎡ ⎤} + _⎣ − ∂ _⎦+ ∂ ∂ (27) where

(

)

(

)

a a a

q X q x x x x q

τ τ τ ρ σ ρσ σ ρτ ρ στ σ ρ ψ ψ δ δ ψ ∂ = ∂ − = ∂ − ∂ . So

(

)(

)

(

)(

)

(

)

₍

₎

₍

₎

D a b

a b

~ ~ 2

L L

M x x q x x q

q q

L L

L x x

q q

1 1

x T x T q q q q

16 16

x T x T S

μ σ ρ σ ρ ρσ _μ ρ σ _μ ρ σ σ ρμ ρ σμ _μ ρσ _μ ρσ σ ρμ ρ σμ ρ σ μ μ ρ σ μ σ ρμ ρ σμ ρσ ψ ψ ψ ψ δ δ Ψ Ψ ψ ψ ψ α βα βα β ψ ψ α β α βα ψ ∂ ∂ = ∂ − ∂ + ∂ − ∂ + ∂ ∂ ∂ ∂ ∂ ∂ + − + = ∂ ∂ ∂ ∂ = − + + = = − + (28)

Here x T_{σ ρμ}−x T_{ρ σμ} is the orbital angular momentum of the particle. The tensorS_ρσμ corresponds to the spin angular momentum of the particle.

Consider the spatial part of the spin angular momentum: 0 1 ~

S q q

8

ρσ = − ψ α βα ψρ σ

We can write the antisymmetric form in term of indices ρ and σ:

(

)

0 1 ~

S q q

16

ρσ = − ψ α βαρ σ −α βασ ρ ψ

(

)

~ 3

1

S q q d x

16

ρσ = −

∫

From this we can define the spin operators as follows:

(

)

~ ~

1

ˆS q q

16

1 i

q q g

8 4

ρσ ρ σ σ ρ

ρ σ ρσ

α βα α βα

α βα

= − − =

= − +

(29)

4. Conclusion

Thus the Lagrange formalism allows us to obtain expressions for all physical quantities, and these expressions are identical to those formulas obtained by Dirac without using the variational method. The resulting formulas give the opportunity to generalize the considered case of classical field for a more interesting, from a physical point of view, case of quantized field, i.e. carry out the procedure of second quantization. However, it should be emphasized that in this approach remains an unsolved problem of including the interaction of the field with known physical fields (see, for example [4]).

References

[1] N.N. Bogolubov, D.V. Shirkov, Introduction to the theory of quantized fields (in Russian), Hauka, Moscow, 1984.

[2] J.E. Castillo H. and A.H. Salas, A Covariant Relativistic Formalism for the New Dirac Equation, Adv. Studies Theor. Phys., Vol. 5, no. 8, (2011), 399-404.

[4] P.A.M. Dirac, The Quantum Theory of the Electron, Proc. R. Soc. A.117 (1928), 610-624.