Preliminary Analysis

In the following, we shall employ $x^{\,1}$ , $x^{\,2}$ , $x^{\,3}$ to represent the Cartesian coordinates $x$ , $y$ , $z$ , respectively, and $x^{\,0}$ to represent $c\,t$ , where $c$ is the velocity of light in vacuum. The time-dependent wavefunction then takes the form $\psi(x^{\,0},\,x^{\,1},\,x^{\,2},\,x^{\,3})$ . Adopting standard relativistic notation, we write the four $x$ 's as $x^{\,\mu}$ , for $\mu= 0,\,1,\,2,\,3$ [93]. A space-time vector with four components that transform under Lorentz transformation in an analogous manner to the four space-time coordinates, $x^{\,\mu}$ , is termed a 4-vector [93], and its components are written like $a^{\,\mu}$ (i.e., with an upper Greek suffix). We can lower the suffix according to the rules

$$a_{0} =+ a^{\,0},$$ (11.1)

$$a_{1} =-a^{\,1},$$ (11.2)

$$a_{2} = -a^{\,2},$$ (11.3)

$$a_{3} = -a^{\,3}.$$ (11.4)

Here, the $a^{\,\mu}$ are called the contravariant components of the 4-vector $a$ , whereas the $a_{\mu}$ are termed the covariant components. Two 4-vectors $a^{\,\mu}$ and $b^{\,\mu}$ have the Lorentz-invariant scalar product

$$a^{\,0}\,b^{\,0}-a^{\,1}\,b^{\,1}-a^{\,2}\,a^{\,2}-a^{\,3}\,b^{\,3} = a^{\,\mu}\,b_{\mu}= a_{\mu}\,b^{\,\mu},$$ (11.5)

a summation being implied over a repeated letter suffix [93]. The metric tenor, $g_{\,\mu\,\nu}$ , is defined

$$g_{\,00} =+1,$$ (11.6)

$$g_{\,11} =-1,$$ (11.7)

$$g_{\,22} =-1,$$ (11.8)

$$g_{\,33} =-1,$$ (11.9)

with all other components zero [93]. Thus,

$$a_{\mu} = g_{\,\mu\,\nu}\,a^{\,\nu}.$$ (11.10)

Likewise,

$$a^{\,\mu} = g^{\,\mu\,\nu}\,a_{\,\nu},$$ (11.11)

where $g^{\,00}=1$ , $g^{\,11}=g^{\,22}=g^{\,33}=-1$ , with all other components zero. Finally, $g_{\,\nu}^{\,~\mu}=g^{\,\mu}_{\,~\nu} =1$ if $\mu=\nu$ , and $g_{\,\nu}^{\,~\mu}=g^{\,\mu}_{\,~\nu}=0$ otherwise.

In the Schrödinger representation (see Section 2.4), the momentum of a particle, whose Cartesian components are written $p_x$ , $p_y$ , $p_z$ , or $p^{\,1}$ , $p^{\,2}$ , $p^{\,3}$ , is represented by the operators

$$p^{\,i} = -{\rm i}\,\hbar\,\frac{\partial}{\partial x^{\,i}},$$ (11.12)

for $i=1,\,2,\,3$ . Now, the four operators $\partial/\partial x^{\,\mu}$ form the covariant components of a 4-vector whose contravariant components are written $\partial /\partial x_{\mu}$ . (See Exercise 1.) So, to make expression (11.12) consistent with relativistic theory, we must first write it with its suffixes balanced,

$$p^{\,i} = {\rm i}\,\hbar\,\frac{\partial}{\partial x_{\,i}},$$ (11.13)

and then extend it to the complete 4-vector equation,

$$p^{\,\mu} = {\rm i}\,\hbar\,\partial^{\,\mu},$$ (11.14)

where $\partial^{\,\mu}\equiv \partial/\partial x_\mu$ . According to standard relativistic theory, the new operator $p^{\,0}={\rm i}\,\hbar\,\partial/\partial x_{\,0}={\rm i}\,(\hbar/c)\,\partial/\partial t$ , which forms a 4-vector when combined with the momenta $p^{\,i}$ , is interpreted as the energy of the particle divided by $c$ [93].