# Relativistic Quantum Mechanics vs Quantum Field Theory

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Tio Barnabe
What's the difference between relativistic quantum mechanics and quantum field theory?

In principle, my guess is that to do the former, one needs to express the Hamiltonian in a relativistic, Lorentz invariant, form, because it seems to be the only frame-related term in the wave equation.

(Is that correct?)

Would quantum field theory be everything we do after this procedure?

Demystifier
Gold Member
One (slightly oversimplified) answer is that in relativistic QM the number of particles is fixed, while in QFT it does not need to be so. (Now @vanhees71 will tell you that relativistic QM is wrong, but that is not an answer to your question.)

Another answer concerns the classical objects that the quantum theory attempts to quantize. In relativistic QM the corresponding classical objects are relativistic particles. In QFT the corresponding classical objects are fields.

Tio Barnabe
in relativistic QM the number of particles is fixed, while in QFT it does not need to be so
because relativistic energy allows for creation and anihilation of particles, correct?

Demystifier
Gold Member
because relativistic energy allows for creation and anihilation of particles, correct?
Not exactly. Even if there is no enough energy to create new particles, in QFT you can have a state with an uncertain number of particles.

Demystifier
Gold Member
In principle, my guess is that to do the former, one needs to express the Hamiltonian in a relativistic, Lorentz invariant, form, because it seems to be the only frame-related term in the wave equation.
That's wrong, the Hamiltonian is never Lorentz invariant, neither in relativistic QM nor in QFT.

The most obvious relativistic QM theory is S-matrix theory which does indeed allow for particle creation and annihilation and is quite independent of field theory. The creation and annihilation of particles follows in relativistic QM from the same principles that allow chemical reactions. Although you can, if you wish, derive S-matrix properties from field theory, it isn't necessary. S-matrix theory, per se, stands by itself and was the basis of the concept of "particle democracy" and the ground from which string theory was developed.

Demystifier