Understanding Mass and Vectors in Quantum Mechanics: The Role of the Higgs Field

[BruceW]

Isn't it Stephen Hawking said information can't be lost even inside the event horizon?

haha, yeah not lost, but probably not worth the effort of collecting the information together again :)

 

[kye]

yeah, the free Dirac field just has mass as a parameter, without needing the Higgs field. So in this way, the electron just has its own mass. When we use a Higgs field, we say the electron has zero mass, but acquires mass by the Higgs mechanism. So in both cases, we can say the electron effectively ends up having the same mass (which is determined experimentally).

If you only have the free Dirac field, then that is not a gauge theory. So there is not any symmetry to be lost anyway. Once we introduce photons, then we have QED which is a gauge theory. And it is invariant under gauge transformations. So the symmetry is not lost, even though we are using mass as a parameter (and without any Higgs). The reason we need the Higgs mechanism is when we try to make an electroweak theory, if we let the electron have nonzero mass, then the symmetry is lost, in the sense that our theory is no longer gauge invariant. Therefore, they say the electron has zero mass, but acquires mass through the Higgs mechanism, so that the electroweak theory is gauge invariant. So if you don't care about the weak interaction, you can just use QED, where the electron does have mass and there is no Higgs particle. This is fine for a lot of cases, for example Compton scattering, you don't really care about the weak interaction, since you just want to find out what happens when an electron and photon collide

Thanks for this illuminating piece. I'll memorize this word for word. :)

I'm not sure what you mean by "the General relativity formula that makes it as given without giving the interaction details". I think the standard thing to do is if you have several fields, then you work out how they interact with each other, and 'plug in' the resulting energy density and pressure into the Friedmann equations, then solve as if it was a general relativity equation which involved normal energy and pressure. I don't know a lot about this stuff, but it seems pretty interesting.

Some kind of duality? Remember General Relativity is pure geometry, and you can't talk of interaction fields because if you do, you are not talking about GR where gravity is literally geometry curvature and gravity fields don't exist... but maybe they can be tied up using some sort of duality where you can transform field and geometry, is this what you are saying?

 

[BruceW]

Thanks for this illuminating piece. I'll memorize this word for word. :)

hehe, I might have said already, but you should definitely try to get your hands on the book 'quantum field theory' by Mandl and Shaw. There might be a lot better books out there, I don't know, I haven't searched very hard. But this one is a really good basic introduction for physics fans like me, who want to get a basic idea of some of the physics involved, without going into too much detail. It does have some good physics (it has equations as well as word explanations). And in the equations, it will omit the longer proofs and extra detail that I just wouldn't find as interesting.

Some kind of duality? Remember General Relativity is pure geometry, and you can't talk of interaction fields because if you do, you are not talking about GR where gravity is literally geometry curvature and gravity fields don't exist... but maybe they can be tied up using some sort of duality where you can transform field and geometry, is this what you are saying?

In stuff I've seen, it looks like they keep General Relativity as pure geometry, but the stress-energy tensor results from the various fields. In Einstein's field equations:
R_{\mu \nu} - \frac{1}{2} g_{\mu \nu} R = \frac{8\pi G}{c^4} T_{\mu \nu}
all the stuff on the left-hand-side is related to the curvature e.t.c. like normal. And the stuff inside $T_{\mu \nu}$ contains all the stuff from the energy density and pressure due to the fields. And they usually mention that this only holds true in regions where the energy density is not too great and the curvature is not too great, otherwise we would need a theory of quantum gravity to describe what is going on in that region. The book 'particle physics and inflationary cosmology' by Andrei Linde is really good. But I know even less about this kind of stuff, so yeah, I can't really say much for certain. P.S. he put his book on the internet for free. You can find it at arxiv if you're interested. I definitely recommend reading a bit about quantum field theory first though.

 

[kye]

hehe, I might have said already, but you should definitely try to get your hands on the book 'quantum field theory' by Mandl and Shaw. There might be a lot better books out there, I don't know, I haven't searched very hard. But this one is a really good basic introduction for physics fans like me, who want to get a basic idea of some of the physics involved, without going into too much detail. It does have some good physics (it has equations as well as word explanations). And in the equations, it will omit the longer proofs and extra detail that I just wouldn't find as interesting.


In stuff I've seen, it looks like they keep General Relativity as pure geometry, but the stress-energy tensor results from the various fields. In Einstein's field equations:
R_{\mu \nu} - \frac{1}{2} g_{\mu \nu} R = \frac{8\pi G}{c^4} T_{\mu \nu}
all the stuff on the left-hand-side is related to the curvature e.t.c. like normal. And the stuff inside $T_{\mu \nu}$ contains all the stuff from the energy density and pressure due to the fields. And they usually mention that this only holds true in regions where the energy density is not too great and the curvature is not too great, otherwise we would need a theory of quantum gravity to describe what is going on in that region. The book 'particle physics and inflationary cosmology' by Andrei Linde is really good. But I know even less about this kind of stuff, so yeah, I can't really say much for certain. P.S. he put his book on the internet for free. You can find it at arxiv if you're interested. I definitely recommend reading a bit about quantum field theory first though.

Real forces are vector, the GR geometry is described by the metric, a rank-2 tensor. So GR is pure geometry where there are really no forces. Some people get confused with this. Do you think GR is some kind of duality to the forces or do you treat the tensor as like wave function just to represent measurements?

 

[BruceW]

$T_{\mu \nu}$ is the stress-energy tensor, so this is where forces enter the equation (well actually energy, momentum and stress, but they are related to forces). In GR, there is no gravitational force, but there are other forces.

edit: uh, I guess forces don't directly affect the metric. But they do affect it indirectly. For example, if there was two separate dust clouds, both positively charged, then they would have an electromagnetic force that tries to push them away from each other, and if this causes them to move apart, then energy, stress and momentum are affected, so $T_{\mu \nu}$ is affected, and so the metric is affected.

 

[kye]

$T_{\mu \nu}$ is the stress-energy tensor, so this is where forces enter the equation (well actually energy, momentum and stress, but they are related to forces). In GR, there is no gravitational force, but there are other forces.

edit: uh, I guess forces don't directly affect the metric. But they do affect it indirectly. For example, if there was two separate dust clouds, both positively charged, then they would have an electromagnetic force that tries to push them away from each other, and if this causes them to move apart, then energy, stress and momentum are affected, so $T_{\mu \nu}$ is affected, and so the metric is affected.

Is it right to think that $T_{\mu \nu}$ relationship to energy, stress and momentum can be taken as analogy to the wave function relationship to mass/energy meaning they represent the object and not really the object... and in future theories energy, stress and momentum and relationship to gravity can be due to others and not exactly $T_{\mu \nu}$? Just want bird eye view of it before taking 10 years to understand the math of GR (it took Einstein 10 years to understand the math too)

 

[BruceW]

energy, stress and momentum is $T_{\mu \nu}$. So it is not analogous to the wavefunction, which just contains the information about energy, stress and momentum.

 

[kye]

energy, stress and momentum is $T_{\mu \nu}$. So it is not analogous to the wavefunction, which just contains the information about energy, stress and momentum.

When Einstein was contemplating about it and how to enclose energy, stress and momentum as geometry, how do you think went through his mind? Could this be the reason he spent 40 years wanted to make the United Theory a theory of geometry because he wants to geometrize everything? In a few words, how can you geometrize energy,stress and momentum when these are not geometrical units... in wave function, they turn energy into Hamiltonian and says the wave has the energy, what is equivalent of this in GR (something like the first thought that hit Einstein when he realized this is possible)?

 

[BruceW]

I don't know much about the history of it. that's an interesting question, how did Einstein come up with the Einstein field equations? I know that in special relativity, Einstein used the idea that electromagnetism is essentially 'correct' and there is no preferred frame of rest, so whether a magnet is moving into a coil or the coil is moving over a magnet, both situations are physically the same. And from this, you can derive most of special relativity in a pretty natural way. And so for general relativity, I'd guess he did a similar thing. (But also using the equivalence principle). And then derived the Einstein field equations from making reasonable generalisations from special relativity. This is just my guess though. You could probably find some of Einstein's original work, if you want the real answer.

Also, I think he was trying to make a unified theory of general relativity and classical electromagnetism (so that they would both turn out to be different facets of the same theory). And I think that at the time, Einstein thought that classical electromagnetism was the only force, so in Einstein's mind, $T_{\mu \nu}$ would have just been due to simply the movement of mass or electromagnetic stresses and momentum. So yeah, I guess he was trying to make a unified theory where the two concepts of 'energy, stress and momentum' and 'spacetime curvature' could be described as two sides of the same coin. I don't think he had much success unfortunately.

I'd say the wavefunction and the Hamiltonian operator in quantum physics are totally different to GR. I don't think Einstein was looking for something like this for his unified theory. He was just thinking of classical (non-quantum) things.

 

[kye]

I don't know much about the history of it. that's an interesting question, how did Einstein come up with the Einstein field equations? I know that in special relativity, Einstein used the idea that electromagnetism is essentially 'correct' and there is no preferred frame of rest, so whether a magnet is moving into a coil or the coil is moving over a magnet, both situations are physically the same. And from this, you can derive most of special relativity in a pretty natural way. And so for general relativity, I'd guess he did a similar thing. (But also using the equivalence principle). And then derived the Einstein field equations from making reasonable generalisations from special relativity. This is just my guess though. You could probably find some of Einstein's original work, if you want the real answer.

Also, I think he was trying to make a unified theory of general relativity and classical electromagnetism (so that they would both turn out to be different facets of the same theory). And I think that at the time, Einstein thought that classical electromagnetism was the only force, so in Einstein's mind, $T_{\mu \nu}$ would have just been due to simply the movement of mass or electromagnetic stresses and momentum. So yeah, I guess he was trying to make a unified theory where the two concepts of 'energy, stress and momentum' and 'spacetime curvature' could be described as two sides of the same coin. I don't think he had much success unfortunately.

I'd say the wavefunction and the Hamiltonian operator in quantum physics are totally different to GR. I don't think Einstein was looking for something like this for his unified theory. He was just thinking of classical (non-quantum) things.

Say. Do you totally understand the math or equations of GR? According to an expert in GR. The field in GR and in conventional is not the same. There is no field in GR or at least it is in the geometrical sense only. Quoting a guy in sci.physics archive:

"When we say "GR is a field theory", at base we are using the GEOMETRICAL
meaning of the word "field": a function on the manifold. Yes,
historically physicists invented this usage for this word (in math there
are several completely different meanings of this word). But that was
really "vector field" (c.f. Faraday et al).

In GR, most of the tensor quantities of interest are really tensor
fields on the manifold. This is what permits us to write field
equations, which are differential equations relating those tensor fields
to each other."

So the tensor is really some kind of dual to energy, stress momentum in the geometrical analysis. Do you know what he is saying? Perhaps the analogy is like

vector in abstract Hilbert space
tensor in abstract spacetime manifold

?

 

[BruceW]

I think that all that guy is saying is there are a lot of tensor fields in GR. They don't all correspond to something like an electromagnetic field, or stress-energy. For example, the Ricci curvature tensor $R_{\mu \nu}$ tells us about the curvature of spacetime. And on the other hand, another tensor field is $T_{\mu \nu}$ This is the energy, stress and momentum at some point in our manifold. So you see some tensor fields correspond to stuff you would usually think of as physical and some tensor fields correspond to stuff you would usually think is related to curvature.

So, I think what that guy is saying, is that if you see the word 'field', then don't immediately assume it is related to an electromagnetic field, or a quantum field, because the word 'field' has many meanings in different contexts. Also, the vector in abstract Hilbert space only contains the information about energy, e.t.c. but the tensor $T_{\mu \nu}$ actually is the stress-energy. But $T_{\mu \nu}$ is not the only tensor, so you can't say that any general tensor is the stress-energy.

edit: p.s. $T_{\mu \nu}$ is technically a tensor field. But sometimes I will just call it 'tensor' (even though that's not technically correct), because it is usually obvious whether someone is speaking about a tensor or a tensor field. So for this reason, people will sometimes shorten 'tensor field' to 'tensor', but you should keep in mind that they are actually talking about a tensor field.

 

[kye]

I think that all that guy is saying is there are a lot of tensor fields in GR. They don't all correspond to something like an electromagnetic field, or stress-energy. For example, the Ricci curvature tensor $R_{\mu \nu}$ tells us about the curvature of spacetime. And on the other hand, another tensor field is $T_{\mu \nu}$ This is the energy, stress and momentum at some point in our manifold. So you see some tensor fields correspond to stuff you would usually think of as physical and some tensor fields correspond to stuff you would usually think is related to curvature.

But according to him (Tom Roberts), $T_{\mu \nu}$ is not directly the energy, stress and momentum of the manifold but what it represents. He described it after searching for this at sci.physics

"The stress-energy tensor, also called the energy-momentum tensor, is a
REPRESENTATION of physical objects in the model. Remember that physical
theories like GR are trying to model the world we inhabit (or an
idealized and simpler world). The key points are:
A) it is a tensor, so it is independent of coordinates (Nature
clearly uses no coordinates, so any representation of a physical
quantity must likewise be independent of them).
B) for a pointlike object, in its rest frame, the components of
this tensor reduce to a single value, the rest energy mc^2 of
the object, so it is an appropriate representation of energy.
C) Einstein's remarkable insight was that one could relate this
tensor mathematically to a particular curvature tensor, which
describes the curvature of the manifold.

The basic idea is that of a differential equation: at each point the
local stress-energy determines the local curvature tensor, but both must
be continuous and satisfy the consistency equation everywhere. By
solving the field equation one determines the metric of the manifold
everywhere, which gives its geometry. So the distribution of
stress-energy (energy and momentum) determines the geometry of the manifold.


The stress-energy tensor (aka the energy-momentum tensor) is a
REPRESENTATION of the objects in the world being modeled. So, for
instance, it is zero in vacuum regions of the world, and is proportional
to the density of matter in regions where mass is present."


What da you think? So the stress-energy tensor is not stress-energy on the manifold but what it represents in the manifold, meaning this is some kinda of duality. (?)

So, I think what that guy is saying, is that if you see the word 'field', then don't immediately assume it is related to an electromagnetic field, or a quantum field, because the word 'field' has many meanings in different contexts. Also, the vector in abstract Hilbert space only contains the information about energy, e.t.c. but the tensor $T_{\mu \nu}$ actually is the stress-energy. But $T_{\mu \nu}$ is not the only tensor, so you can't say that any general tensor is the stress-energy.

edit: p.s. $T_{\mu \nu}$ is technically a tensor field. But sometimes I will just call it 'tensor' (even though that's not technically correct), because it is usually obvious whether someone is speaking about a tensor or a tensor field. So for this reason, people will sometimes shorten 'tensor field' to 'tensor', but you should keep in mind that they are actually talking about a tensor field.

 

[BruceW]

$T_{\mu \nu}$ is directly the stress, energy and momentum.

 

[kye]

$T_{\mu \nu}$ is directly the stress, energy and momentum.

Is it possible there are different interpretations of General Relativity just as there are different interpretations of Quantum Mechanics?

In QM Orthodox, the wave function represent the system statistically in a collection
In QM Bohmian, the wave function guide the particles literally

Likewise,

In GR Orthodox, $T_{\mu \nu}$ is directly the stress, energy and momentum
In GR Representation, $T_{\mu \nu}$ represent the stress, energy and momentum geometrically

Is this a valid distinction? Note Tom Roberts is a Ph.D physicist and work at a research university at which he is an Associate Research Professor (in High Energy Physics).

 

[BruceW]

to me, it looks like he is also saying that $T_{\mu \nu}$ is directly the stress, energy and momentum.

 

[kye]

to me, it looks like he is also saying that $T_{\mu \nu}$ is directly the stress, energy and momentum.

Ok. Let's say it is. Can you give other equations like GR with the same relationship.. like perhaps the Hamiltonian of the quantum system is directly the energy and the wave function encompass it although we can't measure the entire wave function at once, but only the classical eigenvalues? So in GR, we can't see the entire spacetime curvature at once.. but only parts of it broken in space and time? If you won't agree with this QM analogue. Please give other examples. Thanks.

 

[BruceW]

yeah, I don't really see the analogy there. what kind of analogy are you looking for? In GR, we have the concept of stress-energy being proportional to something which tells us about curvature of spacetime. So you are looking for an analogy to this? I think it's John Wheeler who said: "Matter tells space how to curve, and space tells matter how to move." So... I'm trying to think of an analogy to this... Maybe electrodynamics, in the sense that the charges tell the EM field how to act, and the EM field tells the charges how to move. So it is slightly analogous to the Einstein field equations. Not a perfect analogy, but there is never a perfect analogy.

 

[kye]

$T_{\mu \nu}$ is directly the stress, energy and momentum.

I searched for archive here and saw this statement by a pf advisor (Bcromwell)

"Theory first. GR says that gravitational fields are described by curvature of spacetime, and that this curvature is caused by the stress-energy tensor. The stress-energy tensor is a 4x4 matrix whose 16 entries measure the density of mass-energy, the pressure, the flux of mass-energy, and the shear stress."

So $T_{\mu \nu}$ is not directly the stress, energy and momentum. Cromwell that it only measure them. Do you agree? In F=ma... m is directly the mass and "a" directly the acceleration, and "f" directly the force.. But not in GR where the stress-energy tensor just measure the energy and not directly the energy.

 

[BruceW]

no, it actually is the energy. It does not measure the energy. it is the energy. just like in F=ma, m is directly the mass, e.t.c. $T_{\mu \nu}$ is directly the energy (and those other things).

 

[kye]

no, it actually is the energy. It does not measure the energy. it is the energy. just like in F=ma, m is directly the mass, e.t.c. $T_{\mu \nu}$ is directly the energy (and those other things).

Hope Bcrowell and others can comment on this. Bcrowell said the stress-energy tensor just measure the energy, stress, momentum. Meaning, it just converts them into tensor format to serve as parameters to the amount of curvature of spacetime. But BruceW said the energy-stress tensor is directly the energy, stress and momentum. If so. Why not just make spacetime curvature = energy, stress and momentum. The way I understand it is that tensor converts the value of the energy to something compatible with spacetime curvature format. Please others comments on this. Thank you.

 

[remocat2222]

Isn't it Stephen Hawking said information can't be lost even inside the event horizon? شركة تنظيف خزانات بالرياض

You mean the free Dirac field also use mass as parameter (from the kinetic-mass), without needing the Higgs? Won't this make it weigh less? But even for the Dirac equation, won't the symmetry would be lost if you introduce mass by force...



It's said that "mass tells spacetime how to curve... spacetime tells mass how to move".. so how do fields interact with geometry, what mathematical language do you use for the interface besides the General Relativity formula that makes it as given without giving the interaction details?

Then it doesn't make sense to contemplate how electrons don't have positions (before measurements) in the atoms yet have fixed mass? This is also ad hoc and the complete analysis can only be done in the quantum field theory interactions between the higgs field and leptics field via the higgs mechanism?

 

[kye]

no, it actually is the energy. It does not measure the energy. it is the energy. just like in F=ma, m is directly the mass, e.t.c. $T_{\mu \nu}$ is directly the energy (and those other things).

Bruce, just want to understand your context... do you consider spacetime as fundamental as force? Do you consider General Relativity to be describing real gravitational field?

 

[BruceW]

in GR, there is no gravitational field like there is in Newtonian gravity. Instead, stress-energy causes spacetime to curve (and the spacetime curvature tells matter how to move). There are forces between matter, and there is spacetime. In GR, there is no gravitational force, and this idea of spacetime curvature gives us Einstein's version of gravitation. So some object will both experience forces from other objects and the curvature of spacetime. So the motion of the object will be determined by the combination of both: forces from other objects and due to the curvature of spacetime. For example, a small charged object which is acted on by an electromagnetic force, and exists in a generally curved spacetime will have an equation of motion like this:
{d^2 x^\mu \over ds^2} =- \Gamma^\mu {}_{\alpha \beta}{d x^\alpha \over ds}{d x^\beta \over ds}\ +{q \over m} {F^{\mu \beta}} {d x^\alpha \over ds}{g_{\alpha \beta}}
So, the first term on the right hand side is due to the curvature of spacetime, and the second term is due to the electromagnetic force on the small object. Anyway, I hope this answered your question, I was not 100% certain what you meant about if I consider spacetime as fundamental as force.

 

[kye]

in GR, there is no gravitational field like there is in Newtonian gravity. Instead, stress-energy causes spacetime to curve (and the spacetime curvature tells matter how to move). There are forces between matter, and there is spacetime. In GR, there is no gravitational force, and this idea of spacetime curvature gives us Einstein's version of gravitation. So some object will both experience forces from other objects and the curvature of spacetime. So the motion of the object will be determined by the combination of both: forces from other objects and due to the curvature of spacetime. For example, a small charged object which is acted on by an electromagnetic force, and exists in a generally curved spacetime will have an equation of motion like this:
{d^2 x^\mu \over ds^2} =- \Gamma^\mu {}_{\alpha \beta}{d x^\alpha \over ds}{d x^\beta \over ds}\ +{q \over m} {F^{\mu \beta}} {d x^\alpha \over ds}{g_{\alpha \beta}}
So, the first term on the right hand side is due to the curvature of spacetime, and the second term is due to the electromagnetic force on the small object. Anyway, I hope this answered your question, I was not 100% certain what you meant about if I consider spacetime as fundamental as force.

Thanks for the clarifications. I got your general point. But do you believe that space and time should truly be united as spacetime and this is not just for sake of modeling but is truly is fundamental like mass, acceleration where it is definite? By fundamental I mean not breakable into more constituent like temperature being statistical movement of the atoms?

Or let me rephrase my words so you'd understand my point. General Relativity is just a theory. Is it possible gravity can still be caused by a real field (such as weak, strong field) and General Relativity is due to the symmetry of the theory.. or more specifically being a gauge symmetry, the curvature is just due to the Lie group math and in the future there can be a new math where curvature is not necessary (akin to our QED being perturbative now and in the future a non-perturbative QED possibly existing?)

Or in short, do you believe like the others that we can only access reality by the math and physics is about the math and one shouldn't care what is truly behind the math because it is no longer physics.. like we don't need to know what is really the wave function (like Bohmian or MWI) as what is important is just a way to model systems via ensemble?

 

[BruceW]

right now, I believe that GR is 'truly fundamental'. I don't think there is a more fundamental principle which can explain GR. Like you say, statistical mechanics is a more fundamental way to explain things like temperature. And I think that GR is as fundamental as it gets, there is no underlying theory which is more fundamental than GR.

Having said that, if they ever create a quantum theory of gravity, then yes that would be more fundamental, because it would unify quantum principles with gravity. But since they are no-where near making such a theory, at the moment, GR is as fundamental as it gets.

 

[kye]

So you agree with the following analogy?

wave function deals with real Hamiltonian energy (in addition to others like spin)
spacetime curvature deals with real energy, stress and momentum.

wave function and spacetime are just models of reality and we don't even know if they are really physical or literal.

Do you agree with the analogy? Inside Planck scale, these two theories break down so it's good to contemplate on QM and GR at the same time to make our mind get used to roads to a final elegant theory of quantum gravity.

 

[BruceW]

the energy eigenvalues of the wavefunction are the possible results of a measurement of energy. That's why quantum concepts are different to the non-quantum concepts. If the wavefunction happens to be an energy eigenstate, then you can say that the system actually has energy of some definite value. But otherwise, the system does not have a definite energy.

 

[Dale]

The original question has been resolved and this thread is now degenerating into philosophy.

This is the QM forum, not the existentialism forum. Some discussion about QM interpretations is permitted, but this current discussion is not even an interpretations discussion and is therefore not in keeping with the forum.