Restrictions Placed on a Scalar Field by the Vacuum

[QuantumSkippy]

Hi Everyone!

I have been told that even for an entirely LOCAL scalar field φ with Lagrangian density say of the form,

L = ∂/∂x^μ∂/∂x_μφ ± φ⁴ + Aφ³ + Bφ². +. Cφ + D,

that it is really bad, bad, bad because the coefficient (C) of φ is not zero!

That is, ∂/∂φ(L) ≠ 0 when φ = 0 is very bad because the vacuum will not be stable as a result of the fact that the minimum value of φ is not zero. They are even saying that the particles of this field cannot be real physical particles!

I can certainly see this for the Higgs Field which is supposed to be everywhere in the Universe. It is a big deal in that case.

My point is, how does this matter much for AN ENTIRELY LOCAL FIELD, as an entirely local field comes into and out of existence as and when? Why does it matter if it's minimum is not zero, if it only exists for 10^-15 seconds, or whatever, coming into existence whenever and going out of existence whenever?

Please let me know what is the go here! Please supply references wherever you can; I can then chase them up and I will then LEARN!

How does it matter for an entirely local field?

Thanks in advance or all your help. Please impress me with your knowledge!

 

[AndyW]

Thank you for your question about the stability of a local scalar field. It is true that a non-zero coefficient (C) of φ in the Lagrangian density can lead to instability in the vacuum state. This means that the minimum value of φ is not zero, and the field is not in its lowest energy state. In this case, the field can spontaneously fluctuate and create particles, which can have a negative impact on the stability of the vacuum state.

This issue is not limited to the Higgs field, but can also apply to other local scalar fields. However, the impact of this instability can vary depending on the specific field and its interactions with other particles.

In general, the stability of a local field is an important consideration in theoretical physics, as it can affect the predictions and behavior of the field. Therefore, it is important to carefully study and understand the properties of a field, including its minimum value and stability, in order to make accurate predictions and interpretations.

I would recommend looking into the concept of spontaneous symmetry breaking, which is closely related to the stability of local fields. You can also refer to the works of renowned physicists such as Peter Higgs and Robert Brout, who have extensively studied this topic.

I hope this helps to answer your question. Please let me know if you have any further inquiries. Thank you for your interest in this topic and your enthusiasm to learn more.