No Higgs Boson?

[little_monkey]

Like who ever heard of mass-less bacon anyhow?

This Higgs thing doesn't make any sense.

sm

OREALLY???

TELL ME WHAT'S WRONG WITH THE THEORY?

HERE'S A GENERAL RUNDOWN OF THE THEORY

(1) in QM: x → operator
But to satisfy Relativity, in which time is on an equal footing with space, in QFT: x → parameter, and Φ(x) → operator. Now Φ(x), a function of x, is called the “field”.

(2) L = T – V. The Lagrangian plays an important role. From Noether’s theorem, we know that if the Lagragian is invariant under a symmetry, this symmetry points to a conservation law.

Corresponding to L there is a Hamiltonian, H = T + V. The Hamiltonian is known to measure the energy of a system.

(3)In classical mechanics, let v = dx/dt, then L = (1/2) mv² – V(x). The corresponding Hamiltonian is, H = (1/2) mv² + V(x). Quantizing this, (ℏ =1),we get the Schroedinger equation:

i∂Ψ(x)/∂t =( -(1/2m)∆ ² + V(x)) Ψ(x).

(4) In Relativity, the energy equation is:

E²= p²c² + m²c⁴.

Quantizing this, (c =1) yields the K-G equation:

(1/2) (∂_μΦ)(∂^μΦ) + (1/2) mΦ² = 0.

From this, the Lagrangian can be deduced as: L = (1/2) (∂μΦ) ² – (1/2) mΦ².


(5) In QFT, the general Lagrangian is:

L = (1/2) (∂μΦ) ² – V(Φ).

(6) Comparing (5) and (4), if V(Φ) contains any terms with Φ ², its coefficient is taken to be the mass of the field quanta (particles).

Gauge theory:

From electromagnetism, it was known that Maxwell’s equations were gauge invariant. In QM, gauge invariance of the Lagragian involves three important steps:

(7) the wave function is transformed as Φ → e^iqXΦ
(8) the operator ∂_μ → ∂_μ + iqA^μ
(9) the electromagnetic field A^μ → A^μ - ∂^μX

(10) In QED, in equation (5), V(Φ) → - ¼ F_μν F^μν, where F^μν = ∂^μA^ν - ∂^νA^μ

If you apply, 7,8,9,10 to equation (5), you get the invariance of the Lagrangian under gauge transformation, in which the photon mediates the electromagnetic force. Note that the photon has no mass.

In the weak force, the bosons involved have mass, and one had to figure out how to include a mass term, keeping the Lagrangian gauge invariant.

There is where number (6) comes into play under the notion of SPONTANEOUS SYMMETRY BREAKING.



Higgs Mechanism:

Basically, I will only look at U(1) symmetry. Electroweak interactions need a U(1) x SU(2) symmetry, but SU(2) requires 2 by 2 matrices, and the software on this forum is inadequate to deal with matrices. But you can get the flavor just by doing U(1) symmetry and how mass is introduced in the Lagrangian of equation (5).

I will rewrite this equation as:

(11) L = ∂_μΦ^† ∂^μΦ - ¼ F_μν F^μν – V(Φ^†Φ).

(12) where V(Φ^†Φ) = (m²)/(2φ²) [Φ^†Φ - φ² ] ²

Three important things to note:

(13) The field Φ is now a complex number, denoted by (Φ₁, Φ₂) or Φ = Φ₁ + iΦ₂ ( i being the imaginary number, square root of – 1), and Φ^† = Φ₁ – iΦ₂.

(14) the minimum field energy is obtained when Φ^†Φ = φ².

(15) The number of possible vacuum states is infinite. We break this symmetry by requiring that Φ is real, we take the vacuum state to be (φ,0), and expand:

Φ = φ + (1/2^½)h

Substituting 7,8,9, 12, and 15 into 11, we get

(17) L = {(∂_μ - iqA_μ)( φ + (½ ^½)h)}{( ∂^μ + iqA^μ)( φ + (½ ^½)h} - ¼ F_μν F^μν - (m²)/(2φ²) [2^½φh + ½h² ]²

After calculating the Lagrangian, we separate it into two parts:

(18) L = L_free + L_int

where

(19) L_free = ½∂_μh∂^μh - m²h² - ¼ F_μν F^μν + q²φ²A_μA^μ

All the remaining terms are lumped into L_int, which offer no interest.

So, we can see that by breaking the symmetry, we end up with two massive particles. In equation 19, the second term refers to a scalar particle with mass equal to 2^½m, associated with h (the higgs field) and the fourth term, a vector boson with mass 2^½qφ, associated with A^μ( the electromagnetic field).

NOTE: in the Weinberg electroweak theory, with SU(2), equation 19 would have three extra terms for the vector boson instead of a single term, each one was identified with the W⁺, W ^-, and Z bosons. This prediction, which was confirmed subsequently in the following years, earned Weinberg, Salam and Glashow the Nobel prize.

[Leonhard]

Well beyond flattering yourself with a lot mathematical physics, I highly doubt you've made the Higgs mechanism easier to understand for those who aren't schooled in the Standard Model. But for the note, you're right and Shadowmaster is wrong. The Higgs mechanism does make sense, its just not obvious.

[little_monkey]

Well beyond flattering yourself with a lot mathematical physics, I highly doubt you've made the Higgs mechanism easier to understand for those who aren't schooled in the Standard Model. But for the note, you're right and Shadowmaster is wrong. The Higgs mechanism does make sense, its just not obvious.

SM is supposed to have a PhD in physics. So my post to him is quite appropriately at his level.

Secondly, the Higgs mechanism was necessary to make the Lagrangian gauge invariant under a local symmetry, and the Hamiltonian renormalizable. There are other higless theories on the market, notably Technicolor, but none of them are renormalizable.

Also from the point of view that the three bosons for the electroweak interaction were predicted by the theory and then observed, it would be inconsistent that the higgs boson, present also in the same equation, shouldn't exist. Should the higgs boson be ruled out - a possibility that could take place by the end of this year- then some new physics would need to be developped to account for this negative result, one of which could be along the line of quark confinement. However the difficulty here is that the strong force is just that, strong enough to prevent the quarks to be free. In the electroweak interaction, the force is too weak to make the higgs boson confined to some bound state. Some new principle would need to be invoked.

EDIT: there's no flattery here. This stuff has been around for more than 50 years. It is standard stuff in any QFT textbook.

[Adrift]

EDIT: there's no flattery here. This stuff has been around for more than 50 years. It is standard stuff in any QFT textbook.

Never mind the fact that you posted the exact same formula with the same wording on atheistforums.org a few days ago where you admitted to showing off, I'm pretty sure shadowmaster was probably joking around since his post was in reply to mass-less bacon.

[little_monkey]

Never mind the fact that you posted the exact same formula with the same wording on atheistforums.org a few days ago where you admitted to showing off, I'm pretty sure shadowmaster was probably joking around since his post was in reply to mass-less bacon.

I post on many forums. I also have a website. Is that against the law?

Secondly, why do you need to come into SM's defense? Can't he answer for himself? My understanding is that he is supposed to be more qualified than you in this area, so let the man speak for himself.

[Leonhard]

SM is supposed to have a PhD in physics.

Supposed?

[phank]

Supposed?

I think he said once that he has a physics degree. Others have pointed out that his (invariably unsupported) assertions seem inconsistent with this claim.

[shadowmaster]

OREALLY???

TELL ME WHAT'S WRONG WITH THE THEORY?

HERE'S A GENERAL RUNDOWN OF THE THEORY

(1) in QM: x → operator
But to satisfy Relativity, in which time is on an equal footing with space, in QFT: x → parameter, and Φ(x) → operator. Now Φ(x), a function of x, is called the “field”.

(2) L = T – V. The Lagrangian plays an important role. From Noether’s theorem, we know that if the Lagragian is invariant under a symmetry, this symmetry points to a conservation law.

Corresponding to L there is a Hamiltonian, H = T + V. The Hamiltonian is known to measure the energy of a system.

(3)In classical mechanics, let v = dx/dt, then L = (1/2) mv² – V(x). The corresponding Hamiltonian is, H = (1/2) mv² + V(x). Quantizing this, (ℏ =1),we get the Schroedinger equation:

i∂Ψ(x)/∂t =( -(1/2m)∆ ² + V(x)) Ψ(x).

(4) In Relativity, the energy equation is:

E²= p²c² + m²c⁴.

Quantizing this, (c =1) yields the K-G equation:

(1/2) (∂_μΦ)(∂^μΦ) + (1/2) mΦ² = 0.

From this, the Lagrangian can be deduced as: L = (1/2) (∂μΦ) ² – (1/2) mΦ².


(5) In QFT, the general Lagrangian is:

L = (1/2) (∂μΦ) ² – V(Φ).

(6) Comparing (5) and (4), if V(Φ) contains any terms with Φ ², its coefficient is taken to be the mass of the field quanta (particles).

Gauge theory:

From electromagnetism, it was known that Maxwell’s equations were gauge invariant. In QM, gauge invariance of the Lagragian involves three important steps:

(7) the wave function is transformed as Φ → e^iqXΦ
(8) the operator ∂_μ → ∂_μ + iqA^μ
(9) the electromagnetic field A^μ → A^μ - ∂^μX

(10) In QED, in equation (5), V(Φ) → - ¼ F_μν F^μν, where F^μν = ∂^μA^ν - ∂^νA^μ

If you apply, 7,8,9,10 to equation (5), you get the invariance of the Lagrangian under gauge transformation, in which the photon mediates the electromagnetic force. Note that the photon has no mass.

In the weak force, the bosons involved have mass, and one had to figure out how to include a mass term, keeping the Lagrangian gauge invariant.

There is where number (6) comes into play under the notion of SPONTANEOUS SYMMETRY BREAKING.



Higgs Mechanism:

Basically, I will only look at U(1) symmetry. Electroweak interactions need a U(1) x SU(2) symmetry, but SU(2) requires 2 by 2 matrices, and the software on this forum is inadequate to deal with matrices. But you can get the flavor just by doing U(1) symmetry and how mass is introduced in the Lagrangian of equation (5).

I will rewrite this equation as:

(11) L = ∂_μΦ^† ∂^μΦ - ¼ F_μν F^μν – V(Φ^†Φ).

(12) where V(Φ^†Φ) = (m²)/(2φ²) [Φ^†Φ - φ² ] ²

Three important things to note:

(13) The field Φ is now a complex number, denoted by (Φ₁, Φ₂) or Φ = Φ₁ + iΦ₂ ( i being the imaginary number, square root of – 1), and Φ^† = Φ₁ – iΦ₂.

(14) the minimum field energy is obtained when Φ^†Φ = φ².

(15) The number of possible vacuum states is infinite. We break this symmetry by requiring that Φ is real, we take the vacuum state to be (φ,0), and expand:

Φ = φ + (1/2^½)h

Substituting 7,8,9, 12, and 15 into 11, we get

(17) L = {(∂_μ - iqA_μ)( φ + (½ ^½)h)}{( ∂^μ + iqA^μ)( φ + (½ ^½)h} - ¼ F_μν F^μν - (m²)/(2φ²) [2^½φh + ½h² ]²

After calculating the Lagrangian, we separate it into two parts:

(18) L = L_free + L_int

where

(19) L_free = ½∂_μh∂^μh - m²h² - ¼ F_μν F^μν + q²φ²A_μA^μ

All the remaining terms are lumped into L_int, which offer no interest.

So, we can see that by breaking the symmetry, we end up with two massive particles. In equation 19, the second term refers to a scalar particle with mass equal to 2^½m, associated with h (the higgs field) and the fourth term, a vector boson with mass 2^½qφ, associated with A^μ( the electromagnetic field).

NOTE: in the Weinberg electroweak theory, with SU(2), equation 19 would have three extra terms for the vector boson instead of a single term, each one was identified with the W⁺, W ^-, and Z bosons. This prediction, which was confirmed subsequently in the following years, earned Weinberg, Salam and Glashow the Nobel prize.

It was a joke goofball.
Did you think that "mass-less bacon" was a serious comment?

But your knowledge is impressive.

SM

[little_monkey]

It was a joke goofball.
Did you think that "mass-less bacon" was a serious comment?

Less we forget that you also wrote: "This Higgs thing doesn't make any sense."

But your knowledge is impressive.

SM

Flattery won't get you anywhere. How about expanding your knowledge about gauge theory, spontaneous symmetry breaking or the higgs mechanism? Or are you clueless as always when it comes to physics?

[shadowmaster]

I post on many forums. I also have a website. Is that against the law?

Secondly, why do you need to come into SM's defense? Can't he answer for himself? My understanding is that he is supposed to be more qualified than you in this area, so let the man speak for himself.

Xru is right. Theologyweb atheists do lack a sense of humor.

Sigh.

I am surprised that I have to do this with such well versed scientists.
I have gone to local store and picked up some bacon and new scale.
Attached is a photo showing that the bacon definitely has weight -- and therefore mass.
If it were mass-less then the scale would show that.
Plus any fool knows that mass-less bacon would not taste as good bacon does.

QED


SM

[Sparko]

If the possible vacuum states are infinite, then why can't the Theta function be conditional on L= ∂μΦ / (1/2) (∂μΦ) 2

(sorry I can't get the superscript to work)

[little_monkey]

If the possible vacuum states are infinite, then why can't the Theta function be conditional on L= ∂μΦ / (1/2) (∂μΦ) 2

(sorry I can't get the superscript to work)

Well, Sparko in spite of our differences and insults we have exchanged in the past, you've made some effort to ask a decent question, unlike SM who continues to post nonsense.

If you go back to equation 12 and 13, the Φ function can take an infinite number of values in the complex plane, hence an infinite number of vacuum states. Such a device would have no value, that is, how would anyone measure something with infinite possibilities. The way out, which Peter Higgs came with in the early 1960's, (forgot off-hand the exact date, maybe 1964 ???), is that this symmetry -- think of a circle, which the Φ function would actually be in the complex plane -- would be broken by choosing a definite direction. Think: you have a ball on top of a hill. It is at rest, but its stability is precarious. A single push in one direction makes it roll down -- this is the meaning behind spontaneous symmetry breaking.

Now once that is all done, we get that the Lagrangian in equation 5 to be invariant. The origin of this idea is quite simply, in spite of the mathematical machinery it requires to accomplish it. Suppose I write my equations taking the earth as my point of origin. Suppose someone else uses another point of reference, like the sun, and he writes his laws of nature. We have two sets of laws: one from the earth, the other from the sun. We believe that these two sets of laws should be equivalent: this is rephrased as a transformation of coordinates, and that should leave the equations of physics invariant. Now, Emmy Noether added to this concept that if the transformation involves some kind of symmetry, associated to this invariance is a conservation law. All the conservations laws - energy, momentum, angular momentum, spin, etc. - can be shown to be the result of a symmetry. So it was imperative that the equations that would describe the weak force be invariant. This had already been accomplished in QED, but QED does not have a mass term. The higgs mechanism fixed that in a very neat way.

So to answer your question: the vacuum state is not dependent on φ, it is a choice we make when we choose Φ to be real, not complex. This gives the condition in equation 14.

Hope this helps.

[Sparko]

Well, Sparko in spite of our differences and insults we have exchanged in the past, you've made some effort to ask a decent question, unlike SM who continues to post nonsense.

If you go back to equation 12 and 13, the Φ function can take an infinite number of values in the complex plane, hence an infinite number of vacuum states. Such a device would have no value, that is, how would anyone measure something with infinite possibilities. The way out, which Peter Higgs came with in the early 1960's, (forgot off-hand the exact date, maybe 1964 ???), is that this symmetry -- think of a circle, which the Φ function would actually be in the complex plane -- would be broken by choosing a definite direction. Think: you have a ball on top of a hill. It is at rest, but its stability is precarious. A single push in one direction makes it roll down -- this is the meaning behind spontaneous symmetry breaking.

Now once that is all done, we get that the Lagrangian in equation 5 to be invariant. The origin of this idea is quite simply, in spite of the mathematical machinery it requires to accomplish it. Suppose I write my equations taking the earth as my point of origin. Suppose someone else uses another point of reference, like the sun, and he writes his laws of nature. We have two sets of laws: one from the earth, the other from the sun. We believe that these two sets of laws should be equivalent: this is rephrased as a transformation of coordinates, and that should leave the equations of physics invariant. Now, Emmy Noether added to this concept that if the transformation involves some kind of symmetry, associated to this invariance is a conservation law. All the conservations laws - energy, momentum, angular momentum, spin, etc. - can be shown to be the result of a symmetry. So it was imperative that the equations that would describe the weak force be invariant. This had already been accomplished in QED, but QED does not have a mass term. The higgs mechanism fixed that in a very neat way.

So to answer your question: the vacuum state is not dependent on φ, it is a choice we make when we choose Φ to be real, not complex. This gives the condition in equation 14.

Hope this helps.

actually I just made up a bunch of gobbletygook, randomly cutting and pasting parts of your post and that doesn't mean a durn thing (I watch too much star trek I guess) but since you pretended it actually made sense, I guess that means your entire post was a bunch of gobbletygook too that was intended to fool people into thinking you actually understood what you were saying.

gotcha.

[little_monkey]

actually I just made up a bunch of gobbletygook, randomly cutting and pasting parts of your post and that doesn't mean a durn thing (I watch too much star trek I guess) but since you pretended it actually made sense, I guess that means your entire post was a bunch of gobbletygook too that was intended to fool people into thinking you actually understood what you were saying.

gotcha.

Naw, Sparko you didn't fool anyone. You just gave me an opportunity to put SM to shame.

Ciao, have a nice life!

[Sparko]

Naw, Sparko you didn't fool anyone. You just gave me an opportunity to put SM to shame.

Ciao, have a nice life!

right. ...as I did you.