Recovering QM from QFT: David Tong Notes

In summary, David Tong's lecture notes on QFT discuss the recovery of QM from QFT and the use of integrals to define momentum and position operators. However, the intermediate steps of calculations are not shown and may be difficult to determine. In the Wilsonian spirit, the Feynman path integral is used to argue that Galilean symmetry is emergent at low energies and low speeds. The use of non-relativistic physics is a mix of theory and guesswork confirmed by experiment. Similar to the derivation of chiral perturbation theory, the relationship between non-relativistic QM and relativistic QM can be found in Weinbger's QFT volumes. However, relativistic QM is not a coherent theory
  • #1
victorvmotti
155
5
Reading through David Tong lecture notes on QFT.On pages 43-44, he recovers QM from QFT. See below link:

[QFT notes by Tong][1] [1]: http://www.damtp.cam.ac.uk/user/tong/qft/qft.pdfFirst the momentum and position operators are defined in terms of "integrals" and after considering states that are again defined in terms of integrals we see that the ket states are indeed eigen states and the eigen values are therefore position and momentum 3-vectors.

What is not clear to me is the intermediate steps of calculations not shown in the lecture notes, in particular, the computation of integrals involving operators as their integrand, to obtain the desired results.
 
  • #3
The full way is very hard, and maybe may not exist.

In the Wilsonian spirit, one would use the Feynman path integral to argue that Galilean symmetry is emergent at low energies and low speeds. Then one would write down all terms consistent with Galilean symmetry. If one could really do the maths, one would be able to determine which of those terms are important and which are not. I don't know if one can do the maths, or even if one can, whether anyone has done it. So the present use of non-relativistic physics must be treated as a mix of theory (Galilean symmetry emergent at low energies) and guesswork confirmed by experiment (determination of which terms are important).

An analogous case is the derivation of the chiral perturbation theory lagrangian. There are very interesting comments, including the non-relativistic case, in "Foundations of Chiral Perturbation Theory" in http://www.scholarpedia.org/article/Chiral_perturbation_theory.


Another interesting place to look is the chapter on bound states in external fields in the first of Weinbger's 3 volumes on QFT. Then one has to also know the relationship between non-relativistic QM which can be derived from relativistic QM which can be derived from relativistic QFT. Relativistic QM, the intermediate theory, is not a coherent theory, but for some strange reason, non-relativistic QM is a coherent theory.
 
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1. What is the relationship between QM and QFT?

Quantum mechanics (QM) and quantum field theory (QFT) are both theories that describe the behavior of particles at the subatomic level. QFT is an extension of QM that takes into account the interactions between particles and their surrounding fields. QFT can be seen as a more complete and fundamental theory, with QM being a special case of QFT in certain situations.

2. How does QFT recover QM?

In the context of David Tong's notes, QFT is seen as a more fundamental theory that can be used to describe systems at all energy scales. In the limit of low energies, QFT reduces to QM and can recover all of its predictions. This is because QM can be seen as a non-relativistic approximation of QFT, where the effects of relativity and interactions between particles are negligible.

3. What is the role of fields in QFT?

In QFT, particles are seen as excitations of fields that permeate all of space. These fields are described by mathematical objects that have both a value and a direction at each point in space and time. The interactions between particles are then described as the exchange of these field excitations. This allows for a more unified understanding of particles and their behavior.

4. What are the main differences between QFT and classical field theory?

The main differences between QFT and classical field theory lie in their underlying principles and assumptions. QFT is based on quantum mechanics and takes into account the discrete nature of energy and the uncertainty principle. Classical field theory, on the other hand, is based on classical mechanics and does not incorporate these quantum effects. Additionally, QFT incorporates special relativity, while classical field theory does not.

5. How does QFT explain the behavior of particles?

In QFT, particles are seen as excitations of fields. These fields have specific properties and interactions, which determine the behavior of particles. The equations of QFT, such as the Dirac equation and the Klein-Gordon equation, describe how these fields evolve over time and how particles interact with each other. By solving these equations, we can predict the behavior of particles and their interactions at the subatomic level.

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