Month: January 2025

For particles moving independently and freely subject only to specified forces, the motion of each particle obeys Newton’s law F=ma which can be written in terms of the Cartesian coordinates in the form x”=(1/m)dV/dx where V is the potential energy (and the form is the same for all components).
For particles constrained to be parts of a rigid body we might prefer to use more natural variables like the angular orientation of the body, but then the equations become more complicated (eg with centripetal and coriolis “forces” coming in, so the equation for the r component is not just
r”=(1/m)dV/dr ). And it is often not easy to work out the translation of Newton’s eq into the new coordinates.
The Lagrangian approach rewrites the equations of motion in an equivalent form which has the same structure in all coordinate systems – which turns out to be given for every component q by d/dt(dL/dq’)-dL/dq=0 with L=T-V where V is again the potential energy and T the kinetic energy (which in Cartesian coords would be (m(x’)^2)/2 ). This provides a systematic way of getting the appropriate equations rather than having to work them out in terms of the desired coordinates by a complicated conversion process from the Newtonian form.
It turns out that the Lagrangian equations are also equivalent to a variational principle (as mentioned by other responders) – namely that the actual trajectory be a stationary point (eg min or max) of the path integral of the Lagrangian function L.

Source: (1001) Alan Cooper’s answer to In layman’s terms, what is a Lagrangian? – Quora

Feynman diagrams are labels for a way of breaking a complex integral into parts, each of which can be evaluated by applying fairly simple rules. As such they make it much easier to evaluate expressions which originally looked very difficult, and so make it possible for theoretical predictions to be made (and checked) much more quickly (and by people with lower technical skill levels). This led to a rush of progress in the theory and application of Quantum Electrodynamics, and paved the way for much more rapid progress in applying and testing other Quantum Field Theories (such as those used to model the strong and weak interactions of elementary particles).

Source: (1001) Alan Cooper’s answer to Why did Feynman diagrams revolutionize particle physics? – Quora

It is still an open question as to whether or not the theory that is commonly used for calculations in high energy physics can be interpreted as an actual field theory in the sense of being able to produce well defined predictions for experiments at arbitrarily high energy-momentum scales – or equivalently arbitrarily short distance-time scales (which is why such theories are sometimes called strictly local field theories).

In fact no interacting strictly local quantum field theory has yet been constructed in a four dimensional spacetime, and some theories which do make sense in two and three dimensions have been shown not to do so in four. But the Yang-Mills type theory that underlies the Standard Model has not been either excluded or shown to be incompatible with the conditions of such a theory.

Current approaches to actual calculations combine perturbation theory for versions of the theory with cutoffs with a non-perturbative “renormalization group” analysis of how the predictions vary with changes of cutoff and coupling parameters. These may be sufficient for most practical purposes but answers which portray that as somehow winning a competition against the question of what mathematical framework is appropriate for working at all scales are in my opinion both misleading and pathetically misguided.

Source: (1001) Alan Cooper’s answer to What is the current status of a purely algebraic form of non-perturbative interacting quantum field theory in 4-dimensional spacetime? – Quora

Alan Feldman has a lot of good Quora answers.

Source: (1001) Search