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In the following, we shall employ
,
,
to represent the Cartesian coordinates
,
,
, respectively, and
to represent
, where
is the velocity of light in vacuum.
The timedependent wavefunction then takes the form
. Adopting
standard relativistic notation, we write the four
's as
, for
[93]. A spacetime
vector with four components that transform under Lorentz transformation in an analogous manner to the four spacetime coordinates,
, is termed a 4vector [93], and its components are written like
(i.e., with an upper
Greek suffix). We can lower the suffix according to the rules


(11.1) 


(11.2) 


(11.3) 


(11.4) 
Here, the
are called the contravariant components of the 4vector
, whereas the
are termed the covariant components. Two 4vectors
and
have the Lorentzinvariant scalar product

(11.5) 
a summation being implied over a repeated letter suffix [93]. The metric tenor,
, is defined


(11.6) 


(11.7) 


(11.8) 


(11.9) 
with all other components zero [93].
Thus,

(11.10) 
Likewise,

(11.11) 
where
,
, with all other components zero.
Finally,
if
, and
otherwise.
In the Schrödinger representation (see Section 2.4), the momentum of a particle, whose Cartesian components are written
,
,
, or
,
,
,
is represented by the operators

(11.12) 
for
. Now, the four operators
form the covariant components of a
4vector whose contravariant components are written
. (See Exercise 1.) So, to make
expression (11.12) consistent with relativistic theory, we must first write it with its
suffixes balanced,

(11.13) 
and then extend it to the complete 4vector equation,

(11.14) 
where
.
According to standard relativistic theory, the new operator
, which forms a 4vector when
combined with the momenta
, is interpreted as the energy of the particle divided by
[93].
Next: Dirac Equation
Up: Relativistic Electron Theory
Previous: Introduction
Richard Fitzpatrick
20160122