We can estimate the distance an electron might jump as it undergoesthis process. First the time for
which the virtual pair exists can be estimated from the uncertaintyprinciple. Energy conservation is
violated by 2mc^2 at least so ∆t= 2 mch ̄ 2 (which is approximately the reciprocal of the Zitterbewegung
frequency). The distance the electron appears to jump then is ofthe order ofc∆t= 2 mc ̄hc 2 = 0. 002
Angstroms. This is the aproximate size of the fast back and forth motion of Zitterbewegung.
36.22The QED LaGrangian and Gauge Invariance
The LaGrangian for electrons, photons, and the interaction between the two is the LaGrangian of
Quantum ElectroDynamics.
L=− ̄hcψ ̄(
γμ∂
∂xμ+
mc
̄h)
ψ−1
4
FμνFμν−ieψγ ̄μAμψQED is our first complete example of an interacting Quantum Field Theory. It taught us a great
deal about the laws of physics.
The primary difference between Quantum Mechanics and Quantum Field Theory is that particles
can be created and destroyed. The probability to find an electron or a photon integrated over space
does not have to be one. It can change with time. We have written the fields of the photon and the
electron in terms of creation and annihilation operators.
Aμ =1
√
V
∑
kα√
̄hc^2
2 ωǫ(μα)(
ak,α(t)ei
~k·~x
+a†k,α(t)e−i
~k·~x)ψ(~x,t) =∑
~p∑^2
s=1√
mc^2
EV(
b(~ps)u(~ps)ei(~p·~x−Et)/ ̄h+d(~ps)†v(~ps)e−i(~p·~x−Et)/ ̄h)
ψ†(~x,t) =∑
~p∑^2
s=1√
mc^2
EV(
b(~ps)†u(~ps)†e−i(~p·~x−Et)/ ̄h+d(~ps)v(~ps)†ei(~p·~x−Et)/ ̄h)
Note that in the interaction term−ieψγ ̄μAμψphotons can be created or destroyed singly but that
electrons must be created and destroyed along with a positron.
Phase (or Gauge) symmetry can be studied very simply from this LaGrangian. We have shown that
the phase transformation
ψ→eiλ(x)ψAμ→Aμ−̄hc
e∂λ(x)
∂xμleaves the Schr ̈odinger equation invariant. This can be most directly studied using the LaGrangian.
We can deduce from the above transformation that
ψ ̄→e−iλ(x)ψ ̄Fμν=∂Aν
∂xμ−
∂Aμ
∂xν→Fμν−̄hc
e(
∂
∂xμ∂λ(x)
∂xν−
∂
∂xν∂λ(x)
∂xμ)
=Fμν