What would happen if the speed of light were different?

[Buzz Bloom]

if the fine structure constant changes, it is not "due to" a change in c, e, or h. Which of c, e, and h change if the fine structure constant changes is a matter of choice of units, not physics. The physics is all in the fine structure constant.

Hi Peter:

I would very much like to understand the meaning of this quote. I do not want to introduce philosophy, so I will just mention briefly what I perceive to be the problem with my mental ability to understand this quote. It seems to have logical implications that contradict my philosophical view of reality.

No, it just means the value of c in SI units cannot change, because it's not measured any more, it's defined to be a specific number.

I was assuming that any alternate system of defining units would relate to SI units by constants. For example, in cgs units, 1 centimeter = 1/100 of a meter, and 1 gram = 1/1000 of a kilogram. Also in British Imperial units, 1 foot = 0.3048 meters, and 1 pound = 0.45359237 kilograms. If this is a generally correct assumption, then using any unit from a system of units with this kind of definition relative to SI units would leave the value of c unchanged.

How would you measure this?

I assume a suitable device which measures the time difference in picoseconds (ps) between two signals, where these ps units are carefully controlled to be in terms of the cesium definition of a second. I assume a straight underground tunnel which contains as close to a vacuum as practically possible. The length of the tunnel is assumed to be unchanging over time. There is an apparatus at one end which emits a laser pulse when a button is pushed by an experimenter outside of the tunnel. The button push also starts the counting of picoseconds by a suitable digital counter. The apparatus also includes a radar pulse detector which stops the counter when it detects a pulse. At the other end of the tunnel is a mirror that reflects the radar pulse from the emitter back to the detector. The use of this apparatus is repeated from time to time to measure the time (in picoseconds) it takes for a pulse to move from the emitter to the mirror to the detector. There may also be auxiliary devices to detect possible disturbances that would effect the time measurement, for example, a device for detecting seismological disturbances due to earthquakes.

But you just said the fine structure constant was unchanged. That's inconsistent with this sentence.

The quote I posted in #29 was an answer by Dale to a similar but different question (in which α did not change) posted in a different thread which I cited. I quoted it because @Dale's answer suggested to me that he was likely to be able to answer the different question (where α did change) which I asked in that context.

Regards,
Buzz

 

[PeterDonis]

I was assuming that any alternate system of defining units would relate to SI units by constants.

You assume incorrectly. SI units define a meter as the distance light travels in 9192631770/299792458 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom. Other systems of units do not define their unit of distance this way, so their relationship to SI units might not be constant. For example, the original definition of the meter was, IIRC, 1/40,000,000 of the polar circumference of the Earth. But the Earth's polar circumference changes over time by small amounts for various reasons; as it does, the relationship of a meter defined this way to the current SI meter changes. The same would be true for other distance units defined in terms of some physical object, or by some other method that did not involve the propagation of light.

If this is a generally correct assumption, then using any unit from a system of units with this kind of definition relative to SI units would leave the value of c unchanged.

No, it wouldn't, because 299,792,458 is a different number from 1. So 299,792,458 meters per second is not the same, numerically, as 1 light-second per second. They both describe the same speed, but they're not numerically the same; they are different choices of units. Just as, for example, 100 kilometers per hour and 62.137 miles per hour both describe the same speed, but they're not numerically the same; they are different choices of units.

In other words, "the value of c" means "the number we use to describe the speed of light". So 299,792,458 is not the same value of c as 1. I don't see why that is difficult to grasp; it seems to me that you are making this much more difficult than it needs to be by overthinking and confusing yourself. The value of c is not the same as the general method by which c is defined.

The length of the tunnel is assumed to be unchanging over time

Why would you assume that? It seems obvious that the length of the tunnel could change for all sorts of reasons. (Maybe the temperature was slightly different; maybe there was a small tectonic shift; etc., etc.) If this is the only measuring device you have, you have no way of telling whether changes in your readings are due to an actual change in a physical constant, or just to changes in the tunnel.

There may also be auxiliary devices to detect possible disturbances that would effect the time measurement, for example, a device for detecting seismological disturbances due to earthquakes.

And how do you know you are detecting all possible disturbances? You can't.

This is why I said that physical constants like $\alpha$ tell you about the relationships between different measurements made in different ways. You can't tell if $\alpha$ changed just from a single kind of measurement, because you can't distinguish changes in the physical constant from changes in the measuring apparatus.

 

[PeterDonis]

The quote I posted in #29 was an answer by Dale to a similar but different question (in which α did not change) posted in a different thread which I cited. I quoted it because @Dale's answer suggested to me that he was likely to be able to answer the different question (where α did change) which I asked in that context.

I have no idea what this is supposed to mean. You said in one sentence that $\alpha$ was assumed to be unchanged, and in the very next sentence, with no signal of any change in context, that $\alpha$ was assumed to change. That doesn't make sense.

I wouldn't bother belaboring this if it were just an isolated instance, but it's not. It's the same pattern you have shown throughout this discussion: you make statements which are confused and don't make sense, which should make you stop and think about the possibility that it's because your thinking about this whole issue is confused and doesn't make sense. Which means, as I've said, that you really, really need to take a big step back and forget everything you think you know about this subject and listen really carefully to what we are telling you, in particular those two italicized statements I made. If those statements appear to contradict your "philosophical view of reality", then perhaps you should consider discarding your philosophical view of reality.

 

[Buzz Bloom]

If those statements appear to contradict your "philosophical view of reality", then perhaps you should consider discarding your philosophical view of reality.

Hi Peter:

I would be happy to do what you suggest if I could. The problem is doing so would leave me with no coherent view of reality since I am unable to grasp the reasons for the first quoted statement from my post #31 for which I have failed to grasp it meaning. In particular I do not grasp the meaning of "physics" in this context. What comes to mind from this is that α represents something that has meaning in the field of physics (i.e., physical reality) while c does not. I do not think this is what you intend that quite to mean.

The examples you have given in response to my posts suggest another possible interpretation to this quoted statement. It would help if I might understand the boundaries of the following premises.

1) No measurement involving units (e.g. c) can be made with "sufficient" accuracy because the measuring apparatus and/or the environment in which the measurements are made may always introduce unsatisfactory errors/uncertainties.
2) It is possible for measurements of unitless values (e.g. α) to avoid this limitation because many (most, all?) of the sources of error/uncertainties are related to units.​

Regards,
Buzz

 

[PeterDonis]

In particular I do not grasp the meaning of "physics" in this context.

When we say the physics is all in the fine structure constant, we mean what I said in my first italicized statement:

The only way for experimental results to change is if the fine structure constant changes.

So if you are thinking of experimental results changing with the fine structure constant remaining unchanged, you are doing it wrong and need to stop.

The examples you have given in response to my posts suggest another possible interpretation to this quoted statement.

Uncertainties in measurement are a different issue and you should ignore them for this discussion. Trying to include them will only confuse you further.

 

[PeterDonis]

if you are thinking of experimental results changing with the fine structure constant remaining unchanged, you are doing it wrong and need to stop

Uncertainties in measurement are a different issue and you should ignore them for this discussion

Please note that the first statement is made with the second statement in mind. We are assuming that experimental results are not changing due to mundane factors like the temperature of the apparatus changed or a tectonic shift changed the length of a tunnel, or anything like that. As I said, all those things are a different issue and you should ignore them for this discussion.

 

[PeterDonis]

We are assuming that experimental results are not changing due to mundane factors like the temperature of the apparatus changed or a tectonic shift changed the length of a tunnel, or anything like that. As I said, all those things are a different issue and you should ignore them for this discussion.

And how do you know you are detecting all possible disturbances? You can't.

This is why I said that physical constants like $\alpha$ tell you about the relationships between different measurements made in different ways. You can't tell if $\alpha$ changed just from a single kind of measurement, because you can't distinguish changes in the physical constant from changes in the measuring apparatus.

And to further clarify, while you can assume, to simplify your thinking about the issue, that experimental results are not changing due to mundane factors, that doesn't mean you can ignore mundane factors when thinking about what counts as "experimental results changing" in the first place.

For example, consider the "light bouncing off a mirror inside a tunnel" experiment. Basically the "result" of this experiment is the number of periods of radiation corresponding to the transition between two hyperfine levels of the cesium atom that it takes for the light to return after bouncing off the mirror. But, as I said, if this number changes with repeated measurements over time, that by itself does not tell you that $\alpha$ changed; it's also possible that the tunnel could have changed due to some mundane factor. When thinking about how a change in $\alpha$ would affect this number, you can (and should) ignore the possibility of some other mundane factor affecting the number; but when thinking about how you would know if $\alpha$ changed in the real world, you can't ignore that possibility, because that possibility is what tells you that, to really know if $\alpha$ changed, you have to combine numbers obtained from multiple experiments, that involve different phenomena, so that any change in $\alpha$ will change the relationship between the numbers obtained from those multiple experiments (whereas changes in mundane factors would not).

 

[Buzz Bloom]

Uncertainties in measurement are a different issue and you should ignore them for this discussion. Trying to include them will only confuse you further.

Hi Peter:

The reason I raised the "uncertainties" question was because of the response you gave regarding my effort to explain how to detect a change in the speed of liight in responses to your question about that.

Why would you assume that? It seems obvious that the length of the tunnel could change for all sorts of reasons. (Maybe the temperature was slightly different; maybe there was a small tectonic shift; etc., etc.) If this is the only measuring device you have, you have no way of telling whether changes in your readings are due to an actual change in a physical constant, or just to changes in the tunnel.

This is why I said that physical constants like α\alpha tell you about the relationships between different measurements made in different ways. You can't tell if α\alpha changed just from a single kind of measurement, because you can't distinguish changes in the physical constant from changes in the measuring apparatus.

So if you are thinking of experimental results changing with the fine structure constant remaining unchanged, you are doing it wrong and need to stop.

This is not what I assume. I assume for the purpose of discussing the OP's question that that the speed of light changes, and also the fine structure constant changes correspondingly based on

α = 2π e2 / h c,​

that is, c and α vary reciprocally assuming no changes in e and h.

And to further clarify, while you can assume, to simplify your thinking about the issue, that experimental results are not changing due to mundane factors, that doesn't mean you can ignore mundane factors when thinking about what counts as "experimental results changing" in the first place.

I understand that the influence of mundane factors will affect experimental results. I may be mistaken about the vocabulary, but I believe these factors are generally referred to as "experimental errors". This point reminds me of an incident my wife experienced when she was an undergraduate at MIT. She was thinking of majoring in physics, but the incident while performing a measurement of the speed of light in the physics lab changed her mind, and she instead majored in math. The incident was that after one effort of measurement she obtained a result that was not reasonably close enough to the known value, so she looked for a problem with the set uo and made some changes, and then repeated the measurement. After several iterations of this process, she obtained a result that was "satisfactory". Then she stopped looking for problems in the set up. She realized that if she had not known the "right" answer in advance, she would not know when to stop experimenting.

Suppose I made multiple measurements as I described for the speed of light, and over a period of time I found small changes which averaged close to zero, threby suggesting a small distribution range of "experimental errors" with the set up. After a while, on one occasion, an anomalous value was measured. Subsequently, all measurements were distributed by small amounts close to this anomalous value, with a distribution similar to the original "experimental errors" measured values were all close to this new value with a significant change only once, afterwards, the this new value was measured every subsequent time

 

[Buzz Bloom]

Uncertainties in measurement are a different issue and you should ignore them for this discussion. Trying to include them will only confuse you further.

Hi Peter:

The reason I raised the "uncertainties" question was because of the response you gave regarding my effort to explain how to detect a change in the speed of liight in responses to your question about that.

Why would you assume that? It seems obvious that the length of the tunnel could change for all sorts of reasons. (Maybe the temperature was slightly different; maybe there was a small tectonic shift; etc., etc.) If this is the only measuring device you have, you have no way of telling whether changes in your readings are due to an actual change in a physical constant, or just to changes in the tunnel.

This is why I said that physical constants like α\alpha tell you about the relationships between different measurements made in different ways. You can't tell if α\alpha changed just from a single kind of measurement, because you can't distinguish changes in the physical constant from changes in the measuring apparatus.

The following is not what I assume.

So if you are thinking of experimental results changing with the fine structure constant remaining unchanged, you are doing it wrong and need to stop.

I assume for the purpose of discussing the OP's question that that the speed of light changes, and also the fine structure constant changes correspondingly based on

α = 2π e² / h c,​

that is, c and α vary reciprocally assuming no changes in e and h.

And to further clarify, while you can assume, to simplify your thinking about the issue, that experimental results are not changing due to mundane factors, that doesn't mean you can ignore mundane factors when thinking about what counts as "experimental results changing" in the first place.

I understand that the influence of mundane factors will affect experimental results. I may be mistaken about the vocabulary, but I believe these factors are generally referred to as "experimental errors". This point reminds me of an incident my wife experienced when she was an undergraduate at MIT. She was thinking of majoring in physics, but the incident while performing a measurement of the speed of light in the physics lab changed her mind, and she instead majored in math. The incident was that after one effort of measurement she obtained a result that was not reasonably close enough to the known value, so she looked for a problem with the set uo and made some changes, and then repeated the measurement. After several iterations of this process, she obtained a result that was "satisfactory". Then she stopped looking for problems in the set up. She realized that if she had not known the "right" answer in advance, she would not know when to stop experimenting.

Suppose I made multiple measurements as I described for the speed of light, and over a period of time I found small changes which were distributed with an average close to the known value, and with a small standard deviation. variance. It seems reasonable that this would suggest that the range of values shows small "experimental errors" with the set up. After a while, on one occasion, an anomalous value was measured with a value significantly different from the previous mean (and known correct) value. Subsequently, all measurements were distributed by small amounts close to this anomalous value, with a standard deviation similar to the original "experimental errors" previously found before the anomaly. One would of course do a search for an explanation involving some failure in the measuring apparatus, but assume nothing of this kind was found.

Parallel with this activity, suppose measurements were "directly" made of the fine-structure constant with apparatus not depending in any way on the speed of light. Suppose that at the same time (more or less) the the speed of light anomaly was found, a similar anomaly is found with the measurement of α. Furthermore. the product of the new mean values for α and c was approximately (with a small error) equal to the former known value for

2π e² / h.
​

What could one conclude from this? Would a reasonable interpretation be that the value of α had changed, and that this change was due specifically to a corresponding change in the speed of light, or vice versa.

Regards,
Buzz

 

[PeterDonis]

I assume for the purpose of discussing the OP's question that that the speed of light changes, and also the fine structure constant changes

And these two assumptions embody, not only a change in physics, but a change in the choice of units. What we are trying to get you to do is separate these two choices. You cannot say that $\alpha$ changes and $c$ changes, but $e$ and $h$ do not, without making a particular choice of units, whether you realize it or not. But you can say that $\alpha$ changes without making any choice of units at all. See further comments below.

After several iterations of this process, she obtained a result that was "satisfactory". Then she stopped looking for problems in the set up. She realized that if she had not known the "right" answer in advance, she would not know when to stop experimenting.

This happened in at least one significant case historically. Millikan's original oil drop experiment gave a result for $e$ that was different (though not by a lot) from the current value. Other experimenters that tried to replicate his results ended up looking for issues to fix until they got a result that was close enough to his, and then stopping. It took decades for physicists to realize that they should have continued looking for issues to fix because Millikan's value was not quite right.

Suppose I made multiple measurements as I described for the speed of light, and over a period of time I found small changes which averaged close to zero, threby suggesting a small distribution range of "experimental errors" with the set up. After a while, on one occasion, an anomalous value was measured. tSubsequently, all measurements were distributed by small amounts close to this anomalous value, with a distribution similar to the original "expeimental errorhe measured values were all close to this new value with a significant change only once, afterwards, the this new value was measured every subsequent time

Then you would have some interesting data which you would need to investigate further. The way to investigate further would be look at other measurements of $\alpha$ using different phenomena that were made over the same time period, to see if they changed, and if so, how. Ultimately you would look for changes in the relationships between the different measurements that could not be explained by mundane factors.

However, there is another aspect of your proposed measurement method that I haven't yet mentioned. When you say that your scheme measures "the speed of light", that is not correct in terms of SI units. As I said before, what it actually measures--the raw number that comes out of the process, before any interpretation is applied to it--is the number of periods of the cesium hyperfine transition that it takes for the light to return after bouncing off the mirror. By the definition of SI units, this is not a measurement of the speed of light, because the speed of light is not measured, it is defined to be the number 299,792,458. So what you are actually measuring, in SI units, is the length of the tunnel in meters. So if the number that comes out of your measurement changes, according to SI units, your hypothesis should not be that the speed of light changed, but that the length of the tunnel changed.

Now suppose that you also have a very long rod that is attached at both ends of the tunnel when you start taking measurements. You attach a very sensitive strain gauge to the rod, so that you can measure very precisely the stress on it; at the start of measurements, the strain gauge reads zero, indicating that, at that instant of time, the rod's unstressed length is exactly equal to the length of the tunnel.

Now consider some possibilities when you find, later on, that the output of your light-mirror setup has changed. If the rod registers a change that corresponds to the change in light-mirror output (for example, the number of periods output by the light-mirror setup increases, and the rod registers tension, indicating that it is being stretched), then you would conclude that the change is due to a change in the tunnel (in the case just described, that the tunnel length is increased). But if the rod strain gauge output does not correspond to the change in light-mirror output (for example, zero stress, or compression, but the number of periods output by the light-mirror setup increases), then you would conclude that $\alpha$ might have changed.

But in the latter case, could you conclude, more specifically, that $c$ had changed, while $e$ and $h$ did not? No, you can't. The internal stresses in the rod depend on $\alpha$, and they involve interactions between electrons and nuclei in the rod, and quantum energy levels in the rod's atoms, and there is no way, in general, to disentangle all that and say, of the change in $\alpha$, this much was due to a change in $e$, this much was due to a change in $h$, this much was due to a change in $c$. They are all tangled together, and how you separate them is a choice of units. As I've already said, in SI units $c$ is defined to be constant, so by definition it can't change; so in those units any change in the relationship between the rod strain gauge output and the light-mirror setup output must be due to a change in $e$ or $h$ or both. But if, for example, we took the rod as our standard of length--our unit of length is defined to be whatever the length of the rod is--then any change in the ratio of rod length (as determined by strain gauge output) to light-mirror number of periods would be by definition a change in $c$ (assuming we keep the SI second as our unit of time). And that change in choice of units would have to also change how much $e$ or $h$ changed given the change in rod strain gauge output and light-mirror setup output. But in all of this, the change in $\alpha$ would be the same--a given change in the relationship between rod strain gauge output and light-mirror setup output means a given change in $\alpha$, regardless of our choice of units. That is what it means to say that the physics is all in $\alpha$: the change in $\alpha$ is what tells you how much the stuff you actually observed changed.

 

[Buzz Bloom]

And these two assumptions embody, not only a change in physics, but a change in the choice of units.

Hi Peter:

I was careful to say the speed of light changed, not that c changed. Using SI units, c cannot change, but why does this imply that the speed of light has not changed. I tried to explain an experimental sequence at the end of my post #39 (last 3 paragraphs) to clarify the meaning of these assumptions. The explanation assumes repeated experiments. In the context of these paragraphs, what could have changed other than the speed of light along with α.

Regards,
Buzz

 

[PeterDonis]

I was careful to say the speed of light changed, not that c changed.

Um, these are the same thing. There is no such thing as "the speed of light" independent of your choice of units. If you think there is, go back and read my italicized statements again. And again.

In the context of these paragraphs, what could have changed other than the speed of light along with α.

Sigh. You are still not thinking clearly.

If you have not read and considered in detail the extended example I gave in post #40, go do that now. I'll wait.

Ok, ready? Here we go. Consider the case where you have the following combination of measurements: at the start of taking measurements, the light clock-mirror setup gave an output of ( 9192631770 / 299792458 ) * 100. I.e., the tunnel is 100 SI meters long. And the rod strain gauge reads zero.

After some time, the light clock-mirror setup output changes to ( 9192631770 / 299792458 ) * 100.001 (i.e., the tunnel is now 100.001 SI meters long, i.e., its SI length has increased by 1 millimeter). But the rod strain gauge still reads zero.

Now two physicists are arguing about what these results mean.

Physicist #1: Since $c$ is unchanged, the increase in the light clock-mirror output means the tunnel's length must have increased. Since the rod strain gauge still reads zero, that means $e$ and $h$ must have changed to make the physical properties of the rod different, such that its unstressed length is now 1 millimeter longer than it was before.

Physicist #2: Since the rod strain gauge still reads zero, the physical properties of the rod must be the same. That means $e$ and $h$ must be unchanged, and the rod's length, and hence the tunnel's length, must be unchanged. So the increase in light clock-mirror output means $c$ must have decreased.

Now, first of all, these two physicists are obviously using different units; #1 is using SI units, while #2 is using units in which the rod is the standard of length (or at least in which unstressed objects are assumed to have constant length no matter what else happens).

Second, note that you can substitute "speed of light" for $c$ in the above and it changes nothing. "Speed of light" and $c$ are the same thing.

Third: you are probably bursting to ask me, so, ok, which of these physicists is right? (And that might be the fundamental question you have been groping towards in this discussion.) The correct answer to that question is "mu". The question assumes that there is a "right" answer to the issue they are arguing about; but in fact, both of them are just confused. There is no "right" answer to the issue they are arguing about, because there is no "right" answer to the question of which system of units you should choose. Physically, what has happened in this hypothetical scenario is that $\alpha$ has changed; but, as I said before, how you divide up that change into changes in $c$, $e$, and $h$ depends on your choice of units. Physicists #1 and #2 are not (though they think they are) making different claims about "what really happened". They are just choosing different units. What "really happened" is the same either way: the light clock-mirror output number increased while the rod strain gauge stayed at zero. Describing this as a change in $\alpha$ relates that "real" change to the physically meaningful dimensionless constant in our models.

 

[Buzz Bloom]

Hi Peter:

I send you a Conversation message to thank you for your help.

I am now curious as to whether this phenomenon also applies to other unitless constants.
One example:
α_G = (m_e/m_P)
https://en.wikipedia.org/wiki/Gravitational_coupling_constant

Regards,
Buzz

 

[Dale]

This means that by definition the value of c (using any units) cannot change.

Almost. It means that the value of c cannot change in the current SI units. I could make a different system of units whose unit of length in meters is a function of time and c would change in those units.

If it should (hypothetically) happen that the number of seconds (or picoseconds) it takes light to travel (in vacuum) over an actual specific physical distance is measured to have changed, c would not change. What would then change is the actually physical distance corresponding to a meter.

Yes.

Assuming α changes due to the change in c, but no other "constants" change (e and h), would this change in α change the time measured by the cesium clock which would make the measured time for light to travel the physical distance to be unchanged?

If the fine structure constant is changed then the measurements will be physically different. A pulse of light from a Cesium clock would take a different number of oscillations to cross the same number of atoms in a rod.

I was assuming that any alternate system of defining units would relate to SI units by constants.

This is incorrect, even with commonly used unit systems. Take, for example, SI units and CGS units. The SI unit of charge, the coulomb, is not a constant multiple of the CGS unit, the statcoulomb. The SI system treats the coulomb as a base unit with its own dimensionality while CGS treats the statcoulomb as a derived unit based only on the gram, centimeter, and second.

I assume a suitable device which measures ...

The raw measurements obtained from this device depend only on the value of the fine structure constant

What comes to mind from this is that α represents something that has meaning in the field of physics (i.e., physical reality) while c does not.

What I would mean by it is that the outcome of a physical measurement depends on the fine structure constant, not the speed of light. That includes measurements purportedly designed to measure the speed of light.

 

[PeterDonis]

I am now curious as to whether this phenomenon also applies to other unitless constants.

The same reasoning I described would apply to any unitless constant.

α_G = (m_e/m_P)

Note that, while this ratio is indeed dimensionless, the Planck mass itself is not; and if you try to construct a field theory for gravitation, the coupling constant you end up with in that field theory, the analogue of $\alpha$ for electromagnetism, is not dimensionless; it is in fact the inverse Planck mass squared (in "natural" QFT units, in which $c = \hbar = 1$). Many physicists have speculated that this indicates that what we know of as gravitation is not really a fundamental interaction; that it "emerges" from some other underlying theory, in which there is a coupling constant that is dimensionless.

This same sort of issue appears in the Standard Model of particle physics, where (in the low energy version that we use to analyze experiments we can actually do) there has to be at least one constant with units of mass. This is usually taken to be the Higgs mass, and the other masses are given as dimensionless numbers times the Higgs mass. The currently accepted explanation for this is that the Higgs mass is emergent from spontaneous electroweak symmetry breaking; at high enough energies (as in the very early universe, before electroweak symmetry breaking occurred), all of the Standard Model fields are massless.

 

[nikkkom]

The currently accepted explanation for this is that the Higgs mass is emergent from spontaneous electroweak symmetry breaking; at high enough energies (as in the very early universe, before electroweak symmetry breaking occurred), all of the Standard Model fields are massless.

This does not explain why there is one dimensionful parameter in SM.
However you slice it, there is one explicitly dimensionful constant in SM Lagrangian: the constant coefficient at Higgs quadratic term. Its value is ~88.45 GeV squared (= Higgs mass squared / 2)

 

[PeterDonis]

there is one explicitly dimensionful constant in SM Lagrangian

Yes, there is, but it's not correct to just say it's the Higgs quadratic term.

In the low energy Lagrangian (the one that describes the SM at energies well below the electroweak symmetry breaking energy), yes, there is a Higgs quadratic term with coefficient $m_H^2$. Here $m_H$ is based on the vacuum expectation value of the Higgs field.

In the high energy SM Lagrangian, however, the Higgs boson does not have a mass; electroweak symmetry is not broken and the vacuum expectation value of the Higgs field is zero. In this Lagrangian, there are no mass terms anywhere; all the fields, including the Higgs (which has four degrees of freedom in this Lagrangian, not just one), only have kinetic terms, not quadratic mass terms.

What the SM cannot explain is why the Higgs field well below the electroweak symmetry breaking energy takes the particular vacuum expectation value it does. That is really the dimensionful parameter that the theory cannot derive from something else but has to put in by hand.

 

[Dale]

In the high energy SM Lagrangian, however, the Higgs boson does not have a mass

In that regime all of the coupling constants are dimensionless?

 

[PeterDonis]

In that regime all of the coupling constants are dimensionless?

As I understand it, yes, that's the case.

 

[nikkkom]

In the high energy SM Lagrangian, however, the Higgs boson does not have a mass; electroweak symmetry is not broken and the vacuum expectation value of the Higgs field is zero. In this Lagrangian, there are no mass terms anywhere.

I never saw it. Where can I see the high energy SM Lagrangian?

IIRC in general Lagrangian should contain all terms compatible with postulated symmetries of the theory. A quadratic term of Higgs field is allowed. Why is it absent in high energy Lagrangian?

 

[PeterDonis]

Where can I see the high energy SM Lagrangian?

See, for example, here:

https://en.wikipedia.org/wiki/Electroweak_interaction#Lagrangian

A quadratic term of Higgs field is allowed.

As I understand it, it isn't, because before EW symmetry breaking, the Higgs field is not a single scalar field. It's a complex SU(2) doublet, and a direct quadratic mass term would break SU(2) gauge invariance.

 

[nikkkom]

See, for example, here:

https://en.wikipedia.org/wiki/Electroweak_interaction#Lagrangian

As I understand it, it isn't, because before EW symmetry breaking, the Higgs field is not a single scalar field. It's a complex SU(2) doublet, and a direct quadratic mass term would break SU(2) gauge invariance.

In that article, L_h actually has a term quadratic in Higgs field h, with $\lambda v^2$ dimensionful coefficient. THAT is exactly the coefficient I was talking about, (88.45 GeV)².

 

[PeterDonis]

In that article, Lh actually has a term quadratic in Higgs field h, with λv2\lambda v^2 dimensionful coefficient.

No, $\lambda v^2$ is not a Higgs quadratic mass term. $v$ is not the Higgs scalar, it's a constant in the potential for the Higgs field. The value of $\lambda$ is not the Higgs mass squared; the "after electroweak symmetry breaking" shows the Higgs mass term with $m_H$.

 

[nikkkom]

No, $\lambda v^2$ is not a Higgs quadratic mass term. $v$ is not the Higgs scalar, it's a constant in the potential for the Higgs field.

Indeed, $\lambda v^2$ is not the full Higgs quadratic mass term. It's a part of it - its constant coefficient.
Please do follow the link in question and see how L_h is defined there:

https://en.wikipedia.org/wiki/Electroweak_interaction#Lagrangian

Expand the squared parentheses, and you'll get a quartic term, a quadratic term, and a constant one. Quadratic one ends up with dimensionful coefficient.

 

[PeterDonis]

Quadratic one ends up with dimensionful coefficient.

Ah, I see. Yes, this is correct. This dimensionful coefficient ends up determining the vacuum expectation value of the Higgs field after spontaneous symmetry breaking. But the value of this coefficient is not the same as the mass $m_H$ that we observe for the Higgs boson now; the field $h$ in the high energy Lagrangian is not the same as the field $H$ in the low energy Lagrangian.

 

[nikkkom]

This dimensionful coefficient ends up determining the vacuum expectation value of the Higgs field after spontaneous symmetry breaking. But the value of this coefficient is not the same as the mass $m_H$ that we observe for the Higgs boson now

Sure. They are related via $\mu^2 = m_H^2/2 = \lambda v^2$. My point is that SM has one *dimensionful* parameter (while other 18+ parameters are all dimensionless).

 

[PeterDonis]

My point is that SM has one *dimensionful* parameter

Yes, I agree. How it appears in the Lagrangian differs depending on whether you are in the high energy or low energy regime. I'm not sure I would call it a "coupling constant" in either regime, since $v$ is a term in the potential and $m_H$ is a mass, which appears in a quadratic term that involves only one field, not multiple fields. But there is a dimensionful parameter, yes. By the reasoning I gave earlier in the thread, this would be an indication that the SM is not fundamental, but emergent from a lower level theory.

 

[nikkkom]

That's different from this post:

The currently accepted explanation for this is that the Higgs mass is emergent from spontaneous electroweak symmetry breaking; at high enough energies (as in the very early universe, before electroweak symmetry breaking occurred), all of the Standard Model fields are massless.

Here, you said that Higgs mass is emergent *in SM*. It is not. Higgs mass is a direct consequence of $\mu^2$ coefficient.

I agree that SM may be supplanted by an extended theory without dimensionful parameters. Koide rule and other mass relations are signs of it.

 

[Buzz Bloom]

Higgs mass is a direct consequence of μ² coefficient.

Hi nikkkom:

I would much appreciate seeing a generally accessible reference which explains the above quote.

The closest discussion I can find on the Internet regarding μ² is

https://en.wikipedia.org/wiki/Coefficient_of_variation .​

However, this seems to be completely unrelated to the Higgs mass.

Regards,
Buzz

 

[PeterDonis]

Here, you said that Higgs mass is emergent *in SM*. It is not.

I think this is a matter of terminology. The dimensionful coefficient that appears in the SM Lagrangian at high energy is not called the "Higgs mass" and is not described as a "mass" of the Higgs field; descriptions of the SM at those energies typically say, as I said, that all of the fields are massless there. But you are correct that there is still a dimensionful coefficient at those energies, with units of mass, that does appear in a quadratic term, so the usual terminology is sloppy.