Then, as I understand, they are not "scalar fields", but rather just numbers.Demystifier said:
If you have a global coordinate system/frame then you can think of them as scalar fields. But in a different frame there will be a different set of four fields. It is a bit sloppy. Which book is this?Hill said:
"QFT and the SM" by Schwartz.martinbn said:
Yes, you can say it so.Hill said:
More precisely, a scalar field is a mapping of numbers to points in spacetime. The number assigned to a given point doesn't change when you change coordinates.Hill said:
Thank you. I get this. My point is, that such arrangement contradicts the earlier definition,PeterDonis said:
No, it doesn't. Read this again, carefully:Hill said:
PeterDonis said:
Lorentz transformation changes coordinates and, by the definition, should not affect the scalar fields.PeterDonis said:
And it doesn't.Hill said:
Yes. Which means the coordinate transformation has not affected any scalar fields, as it shouldn't.Hill said:
Because "scalar fields" is the term that physicists have used for a long time in this context. Yes, it means basically the same thing as "scalar function on spacetime".Hill said:
No. Any scalar function on spacetime, i.e., any mapping of numbers to points in spacetime, has to be Lorentz invariant, by construction: there is simply nothing for a coordinate transformation to change.Hill said:
A vector field assigns a vector to every point in space and time, whereas a scalar field assigns a single value (a scalar) to every point in space and time. In the context of quantum field theory (QFT), vector fields are often associated with gauge bosons, like the photon, while scalar fields are often associated with particles like the Higgs boson.
In the Standard Model of particle physics, scalar fields and vector fields interact in various ways. For instance, the Higgs field (a scalar field) interacts with gauge bosons (vector fields) to give them mass through the Higgs mechanism. These interactions are described by the Lagrangian of the Standard Model, which includes terms representing these interactions.
Vector fields are necessary in the Standard Model because they represent the force carriers for the fundamental interactions. For example, the photon is the vector field for the electromagnetic force, the W and Z bosons are the vector fields for the weak force, and the gluons are the vector fields for the strong force. Without these vector fields, the Standard Model would not be able to describe these fundamental forces.
In the Standard Model, scalar fields do not exist independently of vector fields because the interactions between them are essential for the theory's consistency and for explaining observed phenomena. For example, the Higgs field (a scalar field) interacts with the W and Z bosons (vector fields) to give them mass. Without these interactions, the Standard Model would fail to accurately describe particle masses and interactions.
In quantum field theory, both vector fields and scalar fields are quantized by promoting the classical fields to operators that create and annihilate particles. For scalar fields, this involves creating scalar particles, while for vector fields, it involves creating vector bosons. The quantization process ensures that the fields obey the principles of quantum mechanics, and the resulting particles interact according to the rules defined by the theory's Lagrangian.