L GWS ¼
X
f
À
Ψ f iγ
μ
∂ μ À m f
À
Á
ψ f À eQ f ψ f γ
μ
ψ f A μ
Á þ
þ
g
ffiffi ffi
2
p
X
i
a
À1
L γ
μ b
i
L W
þ
μ þ b
i
L γ
μ a
i
L W
À
μ
þ
g
2C ω
X
f
Ψ f γ
μ I
3
f À 2S
2
ω Q f À I
3
f γ 5
Ψ f Z μ þ
À
1
4
∂ μ A ν À ∂ ν A μ À ie W
À
μ W
þ
ν À W
þ
μ W
À
ν
2 À
1
2
j ∂ μ W
þ
ν À ∂ ν W
þ
μ þ
Ài:e: W
þ
μ þ A ν À W
þ
ν þ A μ
þ ig
0 c ω W
þ
μ Z ν À W
þ
μ W
À
ν
2 þ
À
1
4
∂ μ Z ν À ∂ ν Z μ þ ig
0 c ω W
À
μ W
þ
ν À W
þ
μ W
À
ν
2 þ
À
1
2
M
2
η η
2
À
gM
2
η
8M W
η
3
À
g
0 2 M
2
η
32M W
η
4
þ M W W
þ
μ þ
g
2
η W
þ
μ
2 þ
þ
1
2
∂ μ η þ iM Z Z μ þ
ig
2C ω
η Z μ
2
À
X
f
g m F
2 M W
Ψ f Ψ f η
ð2:1Þ
which includes the Dirac Lagrangian amongst other things.
We pointed out that all these have been on the basis of the usual point spacetime
which is what may be called commutative. But in recent years several authors
including in particular the present author has worked with a noncommutative
spacetime which originates back to Snyder in the late forties itself. (This was an
attempt to overcome the divergences.)
We first observed that it was Dirac (1958) who pointed out two intriguing features
of his equation: (1) The Compton wavelength and (2) Zitterbewegung.
For the former, his intuition was that we can never make measurements at space
or time points. We need to observe over an interval to get a meaningful definition of
momentum for example. This interval was the Compton region (Sidharth and Das
2017). Next, his solution was rapidly oscillatory, what is called Zitterbewegung.
This oscillatory behaviour disappears on averaging over spacetime intervals over the
Compton region. Once this is done while meaningful physics appears, we are left
with not points but minimum intervals.
This leads to a noncommutative geometry. One model for this is that of Snyder
(1947). Applied at the Compton wavelength this leads to the so-called Snyder–
Sidharth dispersion relation, the geometry being given by Sidharth (2008)
x i , x j
Â
à ¼ β ij :l
2
ð2:2Þ
As described in detail in Sidharth (2010), this leads to a modification in the Dirac
and also the Klein–Gordon equation. This is because Eq. (2.2) in particular leads to
the following energy momentum relation (cf. Sidharth 2008)
18
B. G. Sidharth
X
f
À
Ψ f iγ
μ
∂ μ À m f
À
Á
ψ f À eQ f ψ f γ
μ
ψ f A μ
Á þ
þ
g
ffiffi ffi
2
p
X
i
a
À1
L γ
μ b
i
L W
þ
μ þ b
i
L γ
μ a
i
L W
À
μ
þ
g
2C ω
X
f
Ψ f γ
μ I
3
f À 2S
2
ω Q f À I
3
f γ 5
Ψ f Z μ þ
À
1
4
∂ μ A ν À ∂ ν A μ À ie W
À
μ W
þ
ν À W
þ
μ W
À
ν
2 À
1
2
j ∂ μ W
þ
ν À ∂ ν W
þ
μ þ
Ài:e: W
þ
μ þ A ν À W
þ
ν þ A μ
þ ig
0 c ω W
þ
μ Z ν À W
þ
μ W
À
ν
2 þ
À
1
4
∂ μ Z ν À ∂ ν Z μ þ ig
0 c ω W
À
μ W
þ
ν À W
þ
μ W
À
ν
2 þ
À
1
2
M
2
η η
2
À
gM
2
η
8M W
η
3
À
g
0 2 M
2
η
32M W
η
4
þ M W W
þ
μ þ
g
2
η W
þ
μ
2 þ
þ
1
2
∂ μ η þ iM Z Z μ þ
ig
2C ω
η Z μ
2
À
X
f
g m F
2 M W
Ψ f Ψ f η
ð2:1Þ
which includes the Dirac Lagrangian amongst other things.
We pointed out that all these have been on the basis of the usual point spacetime
which is what may be called commutative. But in recent years several authors
including in particular the present author has worked with a noncommutative
spacetime which originates back to Snyder in the late forties itself. (This was an
attempt to overcome the divergences.)
We first observed that it was Dirac (1958) who pointed out two intriguing features
of his equation: (1) The Compton wavelength and (2) Zitterbewegung.
For the former, his intuition was that we can never make measurements at space
or time points. We need to observe over an interval to get a meaningful definition of
momentum for example. This interval was the Compton region (Sidharth and Das
2017). Next, his solution was rapidly oscillatory, what is called Zitterbewegung.
This oscillatory behaviour disappears on averaging over spacetime intervals over the
Compton region. Once this is done while meaningful physics appears, we are left
with not points but minimum intervals.
This leads to a noncommutative geometry. One model for this is that of Snyder
(1947). Applied at the Compton wavelength this leads to the so-called Snyder–
Sidharth dispersion relation, the geometry being given by Sidharth (2008)
x i , x j
Â
à ¼ β ij :l
2
ð2:2Þ
As described in detail in Sidharth (2010), this leads to a modification in the Dirac
and also the Klein–Gordon equation. This is because Eq. (2.2) in particular leads to
the following energy momentum relation (cf. Sidharth 2008)
18
B. G. Sidharth