Feynman diagrams and sum over paths

[Husserliana97]

Hi !

In a Feynman diagram, can we consider that the propagator specifying the transition amplitude of a particle (let's say, of a "real" electron, or of a "virtual" photon) between two points or two vertices, is in fact itself the sum of a multiplicity of probability amplitudes, each one corresponding to the contribution, in this sum, of a certain trajectory or world line?

Without going so far as to say that the electron effectively follows "all the world lines at once" -- a phrase which would be at best an interpretation, at worst a gross error of understanding -- it seems to me legitimate to assert that the particle does not follow a well-determined trajectory ; and that, as such, the symbol of a very straight and rigid line for fermions in diagrams could well mislead the beginner in quantum physics (which I am, by the way).
In this respect, the idea of a sum over all paths seems to me to be quite respectful of Heisenberg's principle ("no definite trajectory", which can also be translated as: let us consider all possible trajectories in the initial and final conditions, without privileging any of them); and to draw a well-determined line is not more than a convenience of representation.
But on balance, it would perhaps be better to think about each of these straight lines as one of the "space-time tubes", by means of which we represent the integral of the paths...

What do you think ? Thanks in advance !

 

[vanhees71]

Feynman diagrams are just very clever notations for the formulae to be evaluated for getting the S-matrix of scattering processes (i.e., finally cross sections) in perturbation theory. Internal lines stand for propagators.

The path-integral formulation of relativistic QFT does not "sum over all paths" but over "all field configurations", and thinking in terms of trajectories of particles is even more misguided than in non-relativistic QM.

 

[Husserliana97]

Thanks for your answer!
I understand that the image of a "trajectory" between two points, even multiplied to infinity, is misleading, and even more so in QFT.
However, some people assume it (at least it seems to me!), and well beyond the popularization.
For example (that, I heard form Lubos Motl) : in many articles and contributions, Witten refers to another “sum over histories” which is not about configurations of fields ; it is a sum over configurations of particles. Arguably, if you move from momentum space to position space, you may also imagine that a field has the “states” composed of many particles at points, and they wiggle through spacetime, over all possible trajectories, and have a one-particle-like action for each particle !

 

[topsquark]

Thanks for your answer!
I understand that the image of a "trajectory" between two points, even multiplied to infinity, is misleading, and even more so in QFT.
However, some people assume it (at least it seems to me!), and well beyond the popularization.
For example (that, I heard form Lubos Motl) : in many articles and contributions, Witten refers to another “sum over histories” which is not about configurations of fields ; it is a sum over configurations of particles. Arguably, if you move from momentum space to position space, you may also imagine that a field has the “states” composed of many particles at points, and they wiggle through spacetime, over all possible trajectories, and have a one-particle-like action for each particle !

I agree that there is much room for confusion for the general populace (even for some Physicists learning the material) but are you suggesting that we don't use them because of this? It's a convenient way to represent the amplitude. And as far as anything being taught that does reflect reality, hey, the Bohr model has so many inaccuracies that it really should be scrapped. But it does do a reasonable job of representing the Rutherford model and introduces at least some of the basic concepts and the Mathematics are fairly simple to work with. And even though the Mathematics of the Feynman diagrams is fairly advanced they pictorally do a reasonable job of representing the interaction.

-Dan

 

[vanhees71]

When it comes to quantum mechanics or even relativistic QFT you should read the works of people following the no-nonsense approach to QT, i.e., who talk about physics and the necessary mathematical formalism in a sufficiently precise way without diving into obscure "philosophical" gibberish to confuse the serious student. Most standard-QFT textbooks are of this kind. My favorites are

for introduction to the topic:

B. G. Chen et al (editors): Quantum Field Theory of Sidney Coleman, World Scientific (2019)

M. D. Schwartz, Quantum field theory and the Standard
Model, Cambridge University Press, Cambridge, New York
(2014).

M. Peskin and D. V. Schroeder, An Introduction to Quantum
Field Theory, Addison-Wesley Publ. Comp., Reading,
Massachusetts (1995).

If you prefer a path-integral-only approach

D. Bailin and A. Love, Introduction to Gauge Field Theory,
Adam Hilger, Bristol and Boston (1986).

To get the more subtle details with good use of group and representation theory:

S. Weinberg, The Quantum Theory of Fields, vol. 1,
Cambridge University Press (1995).

S. Weinberg, The Quantum Theory of Fields, vol. 2,
Cambridge University Press (1996).

A. Duncan, The conceptual framework of quantum field
theory, Oxford University Press, Oxford (2012).

 

[Husserliana97]

Thank you for your answers and reading suggestions!
I'll probably go for Bailin and Love, as S. Weinberg's books are falling off my radar (they're still far too dense and detailed for me)