Lagrangian Mechanics, Continuous Particle Paths and QFT

[Islam Hassan]

TL;DR Summary

How are Lagrangian mechanics --which posit a continuous particle path-- integrated within the framework of QFT if continuity of a particle's movement is not established/addressed in QM.

In the following thread:

https://www.physicsforums.com/threads/particle-movement-in-quantum-mechanics.1054807/

the discussion established/confirmed that the matter of a particle’s continuity of movement in Quantum Mechanics QM is not a scientific question in QM, refer to PeroK post # 6:

“Orthodox QM goes further than that. Asking what a particle does between measurements makes no sense as a scientific question. For example, only measurements of position make sense. Asking where a particle was when you didn't measure its position is not a scientific question.”

How is the foregoing reconciled with the application of Lagrangian mechanics in Quantum Field Theory which ––if I understand correctly–– posits continuous particle movement?


IH

 

[PeroK]

How is the foregoing reconciled with the application of Lagrangian mechanics in Quantum Field Theory which ––if I understand correctly–– posits continuous particle movement?

Lagrangian mechanics is a generalised approach, which is not tied to continuous classical paths. In any case, in field theory, the Lagrangian applies to the fields themselves. Here's an example where the Lagrangian applies to the Schroedinger field (effectively wave functions):

https://en.wikipedia.org/wiki/Schrödinger_field

 

[Islam Hassan]

Maybe I misunderstood regarding the requirement for continuous paths in Lagrangian mechanics...will check.

In the field-theoretic application of Lagrangian mechanics, I suppose the fields are taken as quantised and not continuous then?


IH

 

[PeroK]

In the field-theoretic application of Lagrangian mechanics, I suppose the fields are taken as quantised and not continuous then?

A field in physics is, by assumption, generally continuous - including quantum fields. A quantum field is a complicated object, as described here:

https://www.damtp.cam.ac.uk/user/tong/whatisqft.html

Let's take a simple example of the quantum harmonic oscillator. It's the energy that is quantized. A measurement of energy on a quantum harmonic oscillator must obtain an integer number of the lowest energy. The system itself still has a continuous (and infinitely differentiable) wave function.

 

[PeterDonis]

Lagrangian mechanics --which posit a continuous particle path

Only in the classical version, not in the quantum version. In the quantum version, the paths in the path integral formulation are not "particle paths"; they are one way of computing probability amplitudes.

 

[Demystifier]

Quantum Field Theory which ––if I understand correctly–– posits continuous particle movement?

QFT does not posit continuous particle movement. I have no idea why do you think it does.

 

[haushofer]

How is the foregoing reconciled with the application of Lagrangian mechanics in Quantum Field Theory which ––if I understand correctly–– posits continuous particle movement?

IH

In quantum (field) theory you basically apply Huygens' principle,

https://en.wikipedia.org/wiki/Huygens–Fresnel_principle

I.e. the propagation of the wave function can be derived from treating every point on a wave front as a new wave front. Every configuration is weighed with a certain amount with a phase factor, and this phase factor is determined by the action.

But quantum (field) theory doesn't tell you anything about paths. It tells you what you can expect if you perform a measurement (unless you adhere to other interpretations). So "continuous particle movement" in ordinary quantum mechanics is as well-defined as the colour of the wind.