Is quantum field theory a subset of quantum mechanics?

Quantum Mechanics (QM) is being used in two different ways and they're both true statements. The first way, which is saying that QM is an underpinning of Quantum Field Theory (QFT) is using the mathematical framework of QM as the scope of QM: states, operators, Hilbert spaces, etc.  This point of view is true.  The second way is using QM as the quantum mechanics of point particles and using point particle Hamiltonians (or Lagrangians) to describe nature.  This is the majority of the work that you do in undergraduate QM classes -- you start off with [math]H= p^2/2m + V(x)[/math] and go from there.  QFT is the relativistic completion of all these different theories.  When you combine Special Relativity with Quantum Mechanics, you are immediately led to a field theory because you can create particles in a relativistic theory -- e.g. emitting a photon, which implies you can create an electron positron pair.  So this point of view is also true.

It is really the same thing to a large extent. In fact you can think of "the field" as a sort of many-body QM system. Take a field with a simple Lagrangian [math]\mathcal L=\int_{-\infty}^\infty dx\left\{\frac{1}{2}( \partial_t\phi(x,t))^2-\frac{1}{2}(\partial_x \phi(x,t))^2 \right\}[/math] (I'm working in natural units where dimension of time and space are the same. To convert you need a velocity factor in front of the second term above.) Now this can really be viewed as a chain of Harmonic Oscillators (HO), by writing the above Lagrangian as [math]\mathcal L=\lim_{a\rightarrow 0}\sum_{i}\left\{ \frac{a}{2}\dot\phi_i^2+\frac{1}{2a}\left({\phi_{i+1}-\phi_i}\right)^2\right\}[/math] If you would to treat the above lagrangian before taking the limit [math]a\rightarrow 0[/math], then it would be just a system of many HOs. Quantizing this system using usual quantum mechanics is the same as quantizing the quantum field theory, except that in QM you usually have descrete number of degrees of freedom (i.e. particles) but in QFT each point in space is a degree of freedom. Note that it is because of this limit that QFT suffers from infinities and things such as renormalization are needed. The infinities that occur there are precisely the infinities when you take the limit [math]a\rightarrow 0[/math], which confused physicists for a long time, and apparently even had many of them thinking that QFT is utter nonsense, requiring unphysical infinities to be subtracted, and then claiming that there is some sense in the finite part. Then when it turned out that it agrees with experiment to an amazing precision, people started thinking that this renormalization is a physical thing, requiring it for physical theories. Today we turned the whole circle, and nobody who understand QFT still thinks that the theories are valid at arbitrarily small scale. In other words that the limit of [math]a\rightarrow 0[/math] has any sense (although, unfortunatelly, still many textbooks do not emphesize this point). Nevertheless we understand why we can make sense of the theory by subtracting the infinite part. This revolutionary view of renormalization was discovered by Wilson, who discovered that any theory, even a non-renormalizable one, will look renormalizable at small energies (a technical way of saying this is that all non-renormalizable couplings are irrelevant in the low energy region). But there is no way to reconstruct the high-energy theory from the low energy definition. Perhaps the universe is fundamentally such a mattress of harmonic oscillators, but that the distance between them is so tiny that we don't see them. Perhaps it is String Theory. The point is we don't know, since we don't have an particle collider which can probe such high energies.