Here is some background for my question, which is as I currently understand it (meaning, it may be wrong):

In a self-interacting scalar field, the formula for potential energy can be given as (μ^2 * (Φ^2)/2) + (λ * (Φ^4)/4).

In a field with an imaginary value for μ, μ^2 becomes negative, leading to the "Mexican hat potential":

in which there are two potential rest points for the "ball", v and -v. However, adjacent points in space must contain the same rest value, otherwise the kinetic energy between them (being partly a derivative of charge Φ with respect to the space dimensions) would have a very high value, and hence the overall energy density (Lagrangian) would be high, and therefore not at rest. So all points pick v, rather than arbitrarily v or -v, leading to a single vacuum expectation value. This is the case with the Higgs field, for instance.

However not all fields have just one charge dimension. Vector fields (such as the electromagnetic field, having a 3 dimensions of charge for the magnetic component and 1 for the electric component) can have multiple. Here is a diagram of what a Mexican hat potential would be shaped like if there were two dimensions to Φ:

Here, we can see that, rather than two noncontinuous points v and -v, there is a *continuous* circle of possible rest charges for minimal potential energy at a given point in space. Therefore, *I think*, in such a field it would be possible for various points to have different rest values, because the values could still be *continuous* (it wouldn't jump anywhere from v right to -v, but could proceed in either direction around the field space circle, as one travelled through actual space). Basically, what I'm saying is, if you represented each angle of the rest point circle (the dip in the hat) with a hue:

then the rest values in some 2D-sliced plane of space might look something like this (this was the best picture I could find, please ignore that there are some dark colors in it):

as long as the colors are continuous everywhere with no jumps. Is that right? The kinetic energy would then be nonzero (due to some gradual difference in Φ in any direction of physical space), but still low.

Or, is not that possible, and would all the points in space still pick one single rest value from the 2-dimensional circle of possible rest values?

Anyway, if it is possible, then here is my actual question:

The Higgs field has a single rest value (the Higgs vev), as in the first scalar field example, and yet, Φ for Higgs is actually a complex number. A complex number really has two dimensions to it, *regardless* of whether one thinks of the complex plane as just an abstraction or not. There are still two values, whether they're the real and imaginary components (e.g. a + bi), or the radius and angle θ in polar coordinates, etc. So a field with a complex charge Φ can still be represented by a *3D hat* as pictured above (two dimensions for Φ, and the vertical one for potential energy), no matter whether one takes the two dimensions of Φ to represent the real and imaginary components, or the radius and angle, etc. Since it can be therefore represented by a potential graph (hat) with a *continuous* circular spectrum of potential rest values (Φ coordinates in which the potential energy is at its minimum), why can't this "scalar" field have different vev values at different points, just like a vector field could, so long as the Φ rest value at adjacent points is always adjacent, so that Φ is always contiuous with a low derivative, as in the color example above? That would lead to a non-constant Higgs vev through space.

Sorry if my understanding is totally wrong.

Thanks!

In a self-interacting scalar field, the formula for potential energy can be given as (μ^2 * (Φ^2)/2) + (λ * (Φ^4)/4).

In a field with an imaginary value for μ, μ^2 becomes negative, leading to the "Mexican hat potential":

in which there are two potential rest points for the "ball", v and -v. However, adjacent points in space must contain the same rest value, otherwise the kinetic energy between them (being partly a derivative of charge Φ with respect to the space dimensions) would have a very high value, and hence the overall energy density (Lagrangian) would be high, and therefore not at rest. So all points pick v, rather than arbitrarily v or -v, leading to a single vacuum expectation value. This is the case with the Higgs field, for instance.

However not all fields have just one charge dimension. Vector fields (such as the electromagnetic field, having a 3 dimensions of charge for the magnetic component and 1 for the electric component) can have multiple. Here is a diagram of what a Mexican hat potential would be shaped like if there were two dimensions to Φ:

Here, we can see that, rather than two noncontinuous points v and -v, there is a *continuous* circle of possible rest charges for minimal potential energy at a given point in space. Therefore, *I think*, in such a field it would be possible for various points to have different rest values, because the values could still be *continuous* (it wouldn't jump anywhere from v right to -v, but could proceed in either direction around the field space circle, as one travelled through actual space). Basically, what I'm saying is, if you represented each angle of the rest point circle (the dip in the hat) with a hue:

then the rest values in some 2D-sliced plane of space might look something like this (this was the best picture I could find, please ignore that there are some dark colors in it):

as long as the colors are continuous everywhere with no jumps. Is that right? The kinetic energy would then be nonzero (due to some gradual difference in Φ in any direction of physical space), but still low.

Or, is not that possible, and would all the points in space still pick one single rest value from the 2-dimensional circle of possible rest values?

Anyway, if it is possible, then here is my actual question:

The Higgs field has a single rest value (the Higgs vev), as in the first scalar field example, and yet, Φ for Higgs is actually a complex number. A complex number really has two dimensions to it, *regardless* of whether one thinks of the complex plane as just an abstraction or not. There are still two values, whether they're the real and imaginary components (e.g. a + bi), or the radius and angle θ in polar coordinates, etc. So a field with a complex charge Φ can still be represented by a *3D hat* as pictured above (two dimensions for Φ, and the vertical one for potential energy), no matter whether one takes the two dimensions of Φ to represent the real and imaginary components, or the radius and angle, etc. Since it can be therefore represented by a potential graph (hat) with a *continuous* circular spectrum of potential rest values (Φ coordinates in which the potential energy is at its minimum), why can't this "scalar" field have different vev values at different points, just like a vector field could, so long as the Φ rest value at adjacent points is always adjacent, so that Φ is always contiuous with a low derivative, as in the color example above? That would lead to a non-constant Higgs vev through space.

Sorry if my understanding is totally wrong.

Thanks!

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