The equations of motion of a charged particle can be expressed in the following general form (∂∏
κ
/∂
q
λ
- ∂∏
λ
/∂
q
κ
)
dq
λ
/
dτ
= 0, (1) where ∏
κ
=
p
κ
+
e
A
κ
,
p
κ
=
m
0
g
κ
λ
dq
λ
/
dτ
, and A is the potential of the electromagnetic field in which the particle is moving,
e
is the charge on the particle,
m
0
is its proper mass and
τ
is the proper time. Where summation is implied it is extended over values of the indices from
λ
= 1 to
λ
= 4. Equations (1) do not represent geodesics in the space-time continuum except in the special case where ∏
κ
reduces to the "mechanical momentum"
p
κ
. A very slight formal modification suffices, however, to convert them into equations representing a geodesic in a 5-dimensional continuum. To accomplish this we extend the metrical tensor
g
κ
λ
by introducing new components defined by
g
5
κ
=
g
κ
5
= A
κ
, as well as a component
g
55
, the significance of which will be left for subsequent investigation. The momentum ∏ can now be put in the form ∏
κ
=
m
0
g
κ
λ
dq
λ
/
dτ
,(λ = 1, 2. 3, 4, 5). Since
g
κ
5
= A
κ
, we must have
m
0
dq
5
/
dτ
=
e
. (2)
Quantum Charged Particle in a Flat Box under Static Electromagnetic Field with Landau’s Gauge and Special Case with Symmetric Gauge
