Dirac equation

Dirac equation

The

Dirac equation

is an equation derived by Paul Dirac in 1927 that describes relativistic spin

1

/

2

particles (fermions). It is given by:

(

γ

μ

⁢

∂

μ

-

i

⁢

m

)

⁢

ψ

=

0

The

Einstein summation convention

is used.

0.1

Derivation

Mathematically, it is interesting as one of the first uses of the spinor

calculus

in mathematical physics. Dirac began with the relativistic equation of total energy:

E

=

p

2

⁢

c

2

+

m

2

⁢

c

4

As Schrödinger had done before him, Dirac then replaced

p

with its quantum mechanical

operator

,

p

^

⇒

i

⁢

ℏ

⁢

∇

. Since he was looking for a Lorentz-invariant equation, he replaced

∇

with the D’Alembertian or

wave operator

□

=

∇

2

-

1

c

2

⁢

∂

2

∂

⁡

t

2

Note that some authors use

□

2

for the D’Alembertian. Dirac was now faced with the problem of how to take the

square root

of an expression containing a

differential operator

. He proceeded to factorise the d’Alembertian as follows:

∇

2

-

1

c

2

⁢

∂

2

∂

⁡

t

2

=

(

a

0

⁢

∂

∂

⁡

x

+

a

1

⁢

∂

∂

⁡

y

+

a

2

⁢

∂

∂

⁡

z

+

a

3

⁢

i

c

⁢

∂

∂

⁡

t

)

2

Multiplying this out, we find that:

(

a

0

)

2

=

(

a

1

)

2

=

(

a

2

)

2

=

(

a

3

)

2

=

1

And

a

0

⁢

a

1

+

a

1

⁢

a

0

=

a

0

⁢

a

2

+

a

2

⁢

a

0

=

a

0

⁢

a

3

+

a

3

⁢

a

0

=

a

1

⁢

a

2

+

a

2

⁢

a

1

=

a

1

⁢

a

3

+

a

3

⁢

a

1

=

a

2

⁢

a

3

+

a

3

⁢

a

2

=

0

Clearly these relations cannot be satisfied by scalars, so Dirac sought a set of four matrices which satisfy these relations. These are now known as the

Dirac matrices

, and are given as follows:

γ

0

=

-

i

⁢

a

0

=

(

1

0

0

0

0

1

0

0

0

0

-

1

0

0

0

0

-

1

)

,

γ

1

=

-

i

⁢

a

1

=

(

0

0

0

1

0

0

1

0

0

-

1

0

0

-

1

0

0

0

)

γ

2

=

-

i

⁢

a

2

=

(

0

0

0

-

i

0

0

i

0

0

i

0

0

-

i

0

0

0

)

,

γ

3

=

a

3

=

(

0

0

1

0

0

0

0

-

1

-

1

0

0

0

0

1

0

0

)

These matrices are also known as the generators of the special unitary group of order 4, i.e. the group of

4

×

4

matrices with unit

determinant

.
Using these matrices, and switching to natural units (

ℏ

=

c

=

1

) we can now obtain the Dirac equation:

(

γ

μ

⁢

∂

μ

-

i

⁢

m

)

⁢

ψ

=

0

0.2

Feynman slash notation

Richard Feynman developed the following convenient notation for terms involving Dirac matrices:

γ

μ

⁢

q

μ

:=

q

Using this notation, the Dirac equation is simply

(

∂

-

i

⁢

m

)

⁢

ψ

=

0

0.3

Relationship with Pauli matrices

The Dirac matrices can be written more concisely as matrices of

Pauli matrices

, as follows:

γ

0

=

(

σ

0

0

0

-

σ

0

)

γ

1

=

(

0

σ

1

-

σ

1

0

)

γ

2

=

(

0

σ

2

-

σ

2

0

)

γ

3

=

(

0

σ

3

-

σ

3

0

)

Title

Dirac equation

Canonical name

DiracEquation

Date of creation

2013-03-22 17:54:46

Last modified on

2013-03-22 17:54:46

Owner

Raphanus (20453)

Last modified by

Raphanus (20453)

Numerical id

21

Author

Raphanus (20453)

Entry type

Definition

Classification

msc 35Q40

Classification

msc 81Q05

Related topic

Spinor

Related topic

KleinGordonEquation

Related topic

SchrodingersWaveEquation

Related topic

PauliMatrices

Defines

Feynman slash notation

Defines

Dirac matrices

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