UPDATE July 6, 2022: Our reasoning about the oscillation frequency of the "hump" is incorrect. We did not take into account that simultaneity is different for a moving observer.
In several sources it is proved that solutions of the massive Klein-Gordon equation are Lorentz invariant. That means that a solution in the laboratory frame is a solution in a moving frame, too, if we map spacetime points with the Lorentz transformation.
To determine the energy of a solution, we have to integrate over a spatial volume. Length contraction makes the volume smaller in a moving frame, which may mean smaller energy. How can we handle this? A moving particle should have more energy than a static particle.
In the case of an electric field which is normal to the movement of the frame, we below were able to reconcile things. It required transforming the electric field to the moving frame. In the case of a scalar field, there is no such transformation.
The lecture above by C. Foudas (2007) solves this mystery. The probability density of the massive Klein-Gordon field is not the square of the wave function but a more complex formula involving derivatives! That allows the energy density to be correct in the moving frame. C. Foudas notes on page 6 that the integral of the probability density ρ(x) over a spatial volume V has to be constant.
We still have to check if there is a problem with the Mexican hat form of the Higgs potential. The massive Klein-Gordon equation assumes a simple harmonic potential. Is there a problem with the complicated Mexican hat potential?
----
Let us have a volume where the 4-potential A of the electroweak field is non-zero in the Coulomb gauge. It is kind of a "hump" in the potential.
Let us use the framework of the Kien Nguyen paper (2009).
Above, v is the vacuum expectation value of the Higgs field. The charge density of the Higgs field is e. The real variables h and Χ (khi) vary a little around zero.
In the scalar field Higgs model, v and e determine the mass term
1/2 e² v² = 1/2 m²
of the field A.
How should we Lorentz boost this interaction to a fast-moving frame?
The Lorentz transformation of the electromagnetic field
Let us look at the electric field when we do a Lorentz boost.
In the formulae, the boosted frame moves to the direction of the x axis. The speed of light c is assumed to be 1. E is the electric field.
z
^
| | | |
| | | | electric field lines
| | | |
|
---------------------------> x
-----------> v moving frame
Let us have a static electric field in the direction of the z axis.
Let the Lorentz factor
γ = 1 / sqrt(1 - v² / c²)
be 1.01 for the moving frame. The energy in the moving frame includes the kinetic energy, which is 1% of the energy in the static frame.
The volume of the electric field contracts 1%. Its strength grows by 1%. The energy density is
~ E²
The energy of the field grows by 1% in the boosted frame. That is the correct value.
An electric field moving to the direction of the field lines
E field strength
+| |-
+| |-
+| |-
-----------> moving frame
Let us then imagine a capacitor where the electric field points to the x direction. We Lorentz transform the field to a frame which moves to the x direction.
The electric field E is the same in the moving frame and the laboratory frame. The volume between the capacitor plates is length-contracted in the moving frame. We have a problem now: the total energy of the field between the plates seems to be less in the moving frame!
Question. Is it so that the formula
~ E²
does not calculate the field energy correctly in the moving frame?
The solution to the Question may be that the extra energy is in the field which escapes from the edges of the capacitor plates. Suppose that the capacitor is moving to the x direction. Then the magnetic field, and consequently, the Poynting vector is almost exactly zero between the plates.
But the Poynting vector is not zero for the field which escapes from the edges. The Poynting vector there can explain the energy flow from the left to the right in the field. This might make up for the missing energy.
The famous 4/3 problem may complicate the analysis.
In our blog we have suggested that the energy of a static electric field is zero, and all the mass-energy is concentrated at the charges. That would solve the 4/3 problem. That would also mean that the energy density formula ~ E² is not right in all cases.
The Lorentz transformation of arbitrary matter fields
Suppose that we have 1 kg of iron. It is described by a bunch of fields: electromagnetic, quark, gluon, and so on.
The Lorentz transformation of all these fields is necessarily very complicated.
If we want to calculate the total energy of the whole system in a moving frame, we measure its mass when it is static, and multiply it by the Lorentz factor γ of the moving frame.
Could we do the same with the Higgs field?
If someone disturbs the Higgs field and pumps energy and momentum into it, we do have a preferred coordinate frame where the excitation of the Higgs field is approximately static.
We could use that frame to do calculations and treat the Higgs field as "almost" scalar. The Mexican hat "potential" would be applicable in the preferred frame.
A detailed analysis of a "hump" in the Higgs field, and a moving frame
We again use Kien Nguyen's (2009) framework. Above is the lagrangian for a small radial displacement h from the minimum potential of the Higgs field.
The lagrangian is for the massive Klein-Gordon wave equation.
Let us use the letter ψ instead of h.
What does a "hump" look like in the Higgs field approximated with the h displacement variable?
Let us assume that the hump spans a large spatial volume. Then spatial second derivatives, the term 𝝯² above, are small.
The equation is like for a harmonic oscillator: the acceleration of the displacement ψ is proportional to -ψ.
The energy density, or the "potential" of the field, is proportional to ψ².
ψ = 1
______
_____/ \____ 0
V
-----------> moving frame
Let us assume that in the laboratory frame at the time t = 0, the value of ψ is the maximum = 1 in a certain volume V. The value of ψ will then oscillate like a harmonic oscillator.
Let us have another frame which moves very fast relative to the laboratory frame along the x axis. An observer in this frame will see ψ to oscillate slowly between 1 and -1. The maximum 1 is not attained simultaneously at every point in V.
Furthermore the volume V appears smaller for the moving observer.
Suppose that the frequency of the oscillation of ψ in the laboratory frame is very slow, and the volume V is very elongated along the x axis.
The moving observer will see the "hump" in much the same way as the laboratory observer, except that the notion of simultaneity is different for the moving observer.
The edges of the volume V move rapidly in the eyes of the moving observer.
Now we notice a contradiction: the oscillation of the middle portion appears faster in the laboratory frame than in the moving frame!
The wave equation of ψ is the same in both frames because ψ is a scalar field. The oscillation of the middle portion should have the same frequency in both frames.
This means that the Lorentz covariance is breached by the system.
Length contraction makes V to appear smaller in the moving frame, and the energy of the hump smaller. That is another breach.
Claim. A scalar field which is described by the massive Klein-Gordon equation is not Lorentz covariant.
In our analysis we did not refer to the properties of the Higgs field at all, except that it is described by the massive Klein-Gordon equation.
A massive scalar field does work well in non-relativistic contexts.
A scalar field is defined as a field whose value stays the same if we change the frame. Now we see that the concept does not make sense for a massive field if we allow relativistic speeds.
The Higgs model may work quite well in non-relativistic cases
A scalar field is easy to work with. Suppose that we can divide the system into independent parts, each of which is non-relativistic under its own coordinates. Then we can use the current Higgs scalar field model in each part.
In the interaction between the parts we have to take into account possible large kinetic energies, length contraction, time dilation and so on. In this solution we do not try to define a Lorentz covariant general Higgs field.
Conclusions
Special relativity works with 4-vectors. Based on this fact we may suspect that a scalar field cannot be Lorentz covariant.
One of the consequences is that the Higgs field cannot be scalar. The spin of the Higgs boson must be 1, not 0.
Why the LHC only observed spin 0 Higgs particles? Spin value 0 photons are created by a charge which moves back and forth. Spin 1 photons are produced in a circular movement. It might be that the processes in the LHC are back-and-forth, not circular.
Defining a Lorentz covariant massive field may be complicated. We need to study literature to find out how it can be done.
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