Relativistic Quantum Mechanics vs Quantum Field Theory

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Discussion Overview

The discussion centers on the differences between relativistic quantum mechanics (QM) and quantum field theory (QFT). Participants explore theoretical distinctions, implications for particle behavior, and the foundational principles underlying each framework.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant suggests that relativistic QM requires expressing the Hamiltonian in a Lorentz invariant form, questioning if QFT encompasses everything that follows this procedure.
  • Another participant notes that in relativistic QM, the number of particles is fixed, whereas in QFT, it is not, indicating a fundamental difference in how particle states are treated.
  • There is a discussion about the implications of relativistic energy allowing for particle creation and annihilation, with some participants clarifying that QFT can accommodate states with an uncertain number of particles even without sufficient energy for creation.
  • One participant asserts that the Hamiltonian is never Lorentz invariant in either relativistic QM or QFT, challenging the earlier assumption about its form.
  • A later reply introduces S-matrix theory as a relativistic QM framework that allows for particle creation and annihilation independently of field theory, suggesting that it is based on principles similar to those governing chemical reactions.

Areas of Agreement / Disagreement

Participants express differing views on the nature of the Hamiltonian and the treatment of particles in relativistic QM versus QFT. There is no consensus on the definitions or implications of these frameworks, indicating ongoing debate.

Contextual Notes

Some claims rely on specific interpretations of relativistic QM and QFT, and the discussion includes unresolved questions about the definitions and implications of particle states and the Hamiltonian's properties.

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?
 
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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.
 
Demystifier said:
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?
 
Tio Barnabe said:
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.
 
Tio Barnabe said:
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.
 
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