Re: When does a quantum field decide to be a wavefunction

Your question requires two answers, one about relativity and one about 
quantum field theory.

1.  At what velocity can a relativistic equation be treated in a non-
relativistic approximation.?  The key word here is approximation.  The 
appropriate v/c depends on how good an approximation you want.  There is 
no single value of v/c that fits all cases.  If not too much accuracy is 
needed v/c~1/10 (or even somewhat larger) is often enough.  For more 
accuracy v/c must be smaller.  Like any approximation, it depends on the 
context.  One word of advice.  Don't rush into the approximation.  The 
relativistic equation is often easier to use.

2.  How is Quantum Field Theory (QFT) related to Quantum Mechanics (QM)?
QM here refers to what could be called "First Quantized" and QFT is called
"Second Quantized".  The relation between QFT and QM has very little to do 
with relativity, so v/c is not important.  QM is the original quantum 
mechanics in which classical variables such as p or x became operators on 
a new quantity, the quantum wave function.  In the early 1930's Jordan and 
Pauli developed a theory in which the quantum wave function was considered 
an operator which acted on a new function which was the real wave 
function.  This process was called "Second Quantization" because it 
quantized what had already been quantized.  This new theory led to  QFT.  
The three main first quantized wave equations, Schrodinger, Klein-Gordon, 
Dirac can all be second quantized. The main new feature of QFT is that the 
old wave function, now considered an operator, can  create and destroy 
particles.   Now, to answer the question of when QFT can be treated as old 
fashioned QM:  When energies are not large enough to create particles 
[That is where relativity (E=mc^2) comes in.] then the single particle 
wave equations (Schrodinger, KG,  Dirac) can be used. Classical E&M also 
has a wave equation for the potentials A and phi.  If A and phi are 
considered to be operators that can create and destroy photons, then the 
QFT that arises is called Quantum Electrodynamics (QED).