Dear sheelbe999， your problem is very interesting, and I will do some mathematical formulation below.\\
Denote the rest mass of electron and positron as $m$, the critical (minimum) energy of the photon as $E_0$, and we turn to the Lorentz invariance $E^2-c^2p^2$. Throughout the reaction, the generated electron and positron couple are at rest with respect to the center of their momenta, hence
$$
E_0^2-c^2p_0^2=(2mc^2)^2 （Eq1) ,
$$
where $p_0$ is the momentum of the photon, and $p_0=E_0/c$.
Therefore Eq1 leads to:
$$
0=4m^2c^4.
$$
This is nothing more than ridiculous! So the process is impossible at all. This contradiction originating from $E^2-c^2p^2$ means that, conservation of energy and momentum don't hold at the same time. As a matter of fact, as is shown below, it is conservation of momentum that is violated.\\
Suppose the process below is permitted:
$$
\gamma \to e^-+e^+.
$$
We set the centroid of $e^-$ and $e^+$ at translation to be the reference frame, in which the momenta of the system $\{e^-,e^+\}$ is
vanishing. However, on the contrary, there's no reference in which a photon is at rest! Hence, this process fails to respect conservation of momentum and is therefore forbidden.\\
Comments:\\
1. An extra particle whose rest mass in non-vanishing must be introduced to help a photon decay into electron-positron couple. Your tutor is correct.\\
2. The argument just above, \textsl{there's no reference in which a photon is at rest}, on the other hand, means a photon, or a flash of light, couldnot act as reference frame, not to say an inertial one, although light travels at constant rapidity in arbitrary directions. This is due to quantum effect.\\
To set a photon as reference frame, its states and parameters of motion should be determined. The $x$ component of momentum $p_x$ reads $p_x=\frac{\hbar}{c} v_x$, where the constant $c$ is speed of light. Hence, $\Delta p_x=\frac{\hbar}{c}\Delta v_x$. The velocity of the photon reference is constant $v=c$, so does its $x$ exponent
$v_x$, hence $\Delta p_x=0$. However, according to uncertainty principle $\Delta p_x\cdot\Delta x\geq \hbar/2$, we have $\Delta x=\infty$. For $y$ and $z$ components, this phenomena hold, too. That is to say, when a photon acts as reference, whose momentum is determined, we canot locate the reference any more! Hence, a photon couldnot act as reference, nor could a flash of light where the
numerous photons share a similarly regular motion.
