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Can one measure the Higgs field? Or is the higgs field not measurable?

I know that the higgs boson is a indication for the presence of the higgs field. But it is the only way of the presence of the higgs field?

When the higgs field is not measurable, what is the reason?

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Your question depends on what you mean by "measure". No field in science has ever been measured directly. What we do measure is the effects due to the field. For example, we don't directly measure the gravitational field.

Instead, we predict that the gravitational field will create a force on objects and thus change their velocity. We can measure this change in velocity by measuring the object's change in position with respect to time. All measurements basically boil down to something along those lines.

However, most physicists do consider the fields to exist because even though we cannot measure them directly, we can measure their effects.

As for the Higgs field and the Higgs boson... The Higgs boson arises out of the Higgs field under certain conditions. Both are linked and you cannot have one without the other just as you can't have the force of gravity without a gravitational field.

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