Working with Lagrangian we often encounter derivatives of particles fields, for example let's consider the first term of the LO chiral Lagrangian $$ \mathcal{L}_{B\phi}^{LO}=\text{Tr}[\overline{B}(i\gamma^\mu \nabla_{\mu} - M_B)B]$$ where $B$ is the baryons matrix, and $\nabla_\mu = \partial_{\mu}B + [\Gamma_{\mu}, B]$ is the covariant derivative. Here I do not understand the meaning of $\partial_\mu B$. What does it mean (physically and non) to compute the derivative of a matrix whose elements are particles?
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1$\begingroup$ a matrix whose elements are particles Its elements aren’t particles. Its elements are mathematical functions describing particles, and these functions are differentiable. $\endgroup$– GhosterMay 2 at 1:02
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$\begingroup$ yeah, fields as I said at the beginning, right? $\endgroup$– SchieleMay 2 at 1:04
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3$\begingroup$ Yes, so why are you confused that you can compute the derivative of a field? $\endgroup$– GhosterMay 2 at 3:31
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$\begingroup$ You're right, perhaps I thought there was something more profound about it. I should have thought more by myself. $\endgroup$– SchieleMay 3 at 18:06
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2 Answers
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The confusion revealed by the OP is something that runs deep in the conceptual understanding of quantum field theory (QFT). Although the term "particle" is often used to refer to the different fields in a theory formulated in terms of QFT, the term does not mean a little solid object. The formulation of theories in terms of QFT represents all the different particles as fields. In other words, they are smooth differentiable functions. The fields interact with each other in localized points which are integrated over all spacetime. So the particle aspect, i.e., the little solid point object, never appears as such in QFT. It still allows us to think of the fields as some probability distribution for such particle, but it doesn't enforce it. In this way QFT avoids the pitfall of our ignorance about this matter.
When experimental observations are made to be compared to predictions from QFT, we measure distinct events. The statistics of these events are compared with predictions of the theories. These events are normally interpreted as manifestations of particles in the traditional understanding. However, even then it is possible to form an understanding of such events that do not require the concepts of particles as little solid objects.
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Derivatives impart dynamics to a field (or any other quantity in physics). Without derivatives on a field, you get an equation of motion which is algebraic in that field. The solution to this gives a constant number throughout spacetime. This is neither interesting nor, more importantly, what our universe corresponds to, where particles are constantly being created, destroyed, propagating through spacetime, and imparting energy and momentum to other particles. When you have derivatives in the Lagrangian, you get a differential equation, whose solution is dependent on space and time (that is, it is propagating).
