Does the plank length have any special significance in string theory?
I just looked up Planck length in the index of Brian Greene's book, The Elegant Universe, and find that it has twenty-five cited occurances, more than any other term found at a quick glance through the index. I guess the answer is yes, of course it does. It has eighteen entries in Greene's more recent book, The Fabric of The Cosmos.
Perhaps it is becoming less important.
As I recall, the string is thought to be on the order of the Planck length.
Why do you ask?
The planck length does come into the stringy pricture, but strings are not at the planck length. If they were, you wouldn't have the continuous motion that traces the nice smooth world sheets upon which nearly all the stringy math depends. The actual dimension of the strings is not AFAIK an output of the theory, but some people have opined that the planck length is to the string size approximately as the string size is to the proton diameter. Both those ratios are somewhere around 1015. As Feynman said in another context, there's plenty of room at the bottom.
The plank length is formed by combining the three constants c, h and G referring to the speed of light, plank's constant and Newton's gravitational constant respectively, which are the three constants that come out of SR, QM and GR. I recalled reading about them and figured out the equation for fun. After looking into it some I found that it is in fact very significant. It appears in loo quantum gravity and string theory.
the NIST.gov website (Nat'l Inst. Standards and Tech) that lists the current best estimates of physics constants
gives values for four Planck quantities (actually more if you count the various versions of Planck's hbar constant but I mean besides that)
these are formed as metrictensor indicates except that
it has become common for physicists to use hbar rather than h
so the Planck quantities are often defined using hbar, not h.
(I cant actually remember ever seeing them defined using h, which would make the length etc differ by factors of 2pi and the sqrt of 2pi, but it probably happens)
defining the Planck temperature requires a fourth constant, k.
the NIST website gives formulas for the various Planck quantities,
showing how they arise from four fundamental physical constants
hbar, c, G, and k.
this is a good source for physics constants in general
that is an interesting conjecture
(I personally am skeptical that strings exist---not sure string theory will ever be able to explain or predict anything---but assuming stringy math does eventually become useful for depicting nature, one can ask how long strings are imagined to be...)
so they speculate that the string size is on the order of
ONE QUADRILLION PLANCK LENGTHS???!!!
I didnt realize strings were supposed to be so big!
1015 planck length units seems enormous to me.
I have an idea of how to resolve it so I am not so boggled.
the ratio between proton mass and planck mass is actually 1019
(more exactly 1.3 x 1019 but I am only doing orders of magnitude)
so if one took the geometric mean between proton energy and planck energy the two ratios would be 109.
the two ratios would each be a billion, not a quadrillion.
so these people who opine that string length is geometric mean between proton Compton and Planck length, they could be saying
"we think string length is a billion Planck lengths, and then proton (Compton) size is a billion times string length"
I can picture that.
Ive read various estimates of string scale ranging from
Electroweak scale (TeV = 1012 eV) to Planck scale (1028 eV)
if string scale were what they conjecture about being geometric mean between proton and Planck then it would be 1018 or 1019 eV
dont have much grasp so that's about all I can say
On a somewhat related issue I think that the conceptual approach of LQG is interesting. As I understand it LQG gives no priority to any field, in particular the gravitational field. In QFT fields are defined "on" or "in" spacetime but in GR the gravitational field is not defined on any other field. Spacetime the gravitational field are the same entity. The reason I think this developed is due to our everyday experience that massive objects move through "space". The essential conceptual problem for me is if GR says that spacetime is the gravitational field then why does the EM field need to be defined upon another field but GR does not? This asymmetry either needs to be explained or done away with. LQG, as I understand it, does away with this asymmetry by defining all fields and particles in relation to each other, not in relation to an fixed backgroud. In his draft of a Quantum Gravity book Carlo Rovelli clarified many of these point very beautifully. In section 1.1.3 he gives a very nice explanation of what I tried to capture above. Here is a link to his book.
You actually have a very good grasp; I was pulling numbers out of my memory ( if not some lower site) and I accept your calculation of a billion as the ratio. Does a billion planck lengths satisfy you better, Marcus?
that section 1.1.3, with its parable of the animals is a fine explanation!
a pleasure to have it recalled.
Also some other history in Rovelli's book, about the invention of absolute space and time by Newton. I dont remember what section of the book. Rovelli quotes Newton's own writings about this and how Newton thought long and hard about it before he proposed the idea of an absolute space.
I believe this history of ideas useful to contemplate so i will give my version of it, in case anyone reading this isnt familiar with it. I got most of this from Chapter 2 in Rovelli.
Absolute space is a rigid empty frame which is presumed to exist on its own and lets other things be defined on it. Quite possibly it is not really there. Newton eventually imagined it absolute space to be the mind of God.
He needed it to get his work done. At first he had his doubts about postulating it but eventually became quite convinced, as did a lot of other people.
Newton put forth this (then novel but now commonplace) idea around 1680 and for several centuries physics made great progress by defining things on this frame of absolute space.
the very different notion of a "field" was invented by Faraday around 1830 or 1840.
This time the image is of a field of grass or a head of hair.
The ground, or scalp, is absolute space
and the field is the hair, or grass, growing on it.
Initially a "field" was a function defined at every point of absolute space which tells a vector or arrow at every point-------as if you take a piece of graph paper (absolute space) and in every box you draw an arrow.
the length and direction of the arrows is the field.
Einstein discovered how to define a field without using absolute space. he was able to describe the gravitational field without first setting up a rigid empty frame in which to define it. this was a new departure and came around 1915. He used a shapeless, non-rigid, thing called a manifold---and on that he defined the METRIC which gives shape to the manifold and IS the gravitational field. Then he shows that the manifold wasnt essential and that all that is there is the metric. The manifold was just an arbitrarily chosen convenience and so he throws it away and keeps the gravitational field (defined as an equivalence class of metrics which can be defined on different manifolds but give the same results and are the same)
So in a very strong way Einstein gave priority to one special field, the gravitational field. In Rovelli's fable, it is the whale.
Other fields which historically were once defined as fields of arrows written on Absolute Space, now are to be defined in relation to the gravitational field. Things are located in the context of the field. Rotation is defined with respect to the field.
this is radically new. things used to be located in absolute space or in the spacetime of Einstein's earlier (1905) contribution Special Relativity.
Rotation used to be defined relative to newtons absolute space or relative to the 4D space of Special Relativity.
But as of 1915 we have something completely different. All that stuff is to be defined not relative to an absolute space or a special relativity setup,
but instead is to be defined relative to a FIELD
(something originally analogous to the hair on a scalp, but now there is no scalp)
some how the idea of a framework has been eradicated, or seriously diminished.
Faraday's original fields were the magnetic field lines which you see in iron filings on a sheet of paper when you put a magnet under.
static electricity also makes fields, which you see when someones hair stands up because of static charge
the field is not the scalp or the piece of paper, it is the arrows.
(and then later with Gen Rel it is the shape described by the metric)
That's pretty much the way I see it too, marcus. Without gravity, spacetime can no longer be described - i.e., becomes dimensionless.
Great post, very helpful and clear. I have to say that most posts here are overly technical and my opionion is that this is because the people making the posts don't have much of a physical understanding.
You post made me realize that maybe the postulate regarding the invariance of light to all inertial observers is not so strange if we look at things from a new point of view. If space and time form "spacetime" which turns out to be the gravitational field then why should it be absolute. Fields can vary and in this case the relation between the electromagnetic and gravitational fields simply inply that their relationship results in the second postualte.
After reading and thinking about this stuff it seems so obvious. I can't help wonder if this is how Einstein might have felt when he realized he could use the equivalence principle.
Yes, a billion is good at least as the square root of 1018
and approximate square root of 1019
this number 1019 is etched on my brain as the ratio of planck scale to proton scale
at my level of comprehension it is a stark unexplainable fact about the universe
that number could be said to be almost more important than this or that comtemporary attempts at constructing theories
Frank Wilczek had a series of 3 articles in Physics Today that were explicitly about that number 1019
You probably saw them and we may have discussed them.
the series was called "Scaling Mount Planck"
It asked why is the ratio of planck to proton so large? Or why is the inverse ratio so small? Why is 10-19 so small.
I dont believe he mentioned string theory but he did mention QCD. In the end, I could not understand his reasoning. He seemed to offer some explanation from QCD about why.
In the variant Planck units I am trying out in that other thread, the ratio is 2.6E18
2.6 x 1018
so it is really quite close to a billion billion
if those people want to imagine "string scale" as smack in the middle between proton and planck it could be quite convenient (if it turned out they were right)
I found this quote from a Ph.D at http:physlink.com by google on Planck mass.
"Since the Planck length has this special property of being the length scale where we can't ignore quantum gravity effects, it is typically taken to be the size of a fundamental string, in string theory."
It isn't where I "learned" this bit of misinformation, but it shows that the misconception may be widespread, not merely my own whimsy, a comfort to me at least.
Thanks for the learned discourse!
Oh yes! I "learned" that too, some years back from a CalTech professor's website called "Level5 Knowledge-base". I believed it for some years and repeated it to others.
It may, after all be true. I do not suppose that string theory will turn out to be useful. I suspect that it has proven unsatisfactory and will gradually be relegated to pure math. But if (to my surprise) some stringy theory turns out to have physical content then it still could be that the string scale and the planck scale are the same!
It also could be that they differ by a factor of a billion.
The trouble is people really dont know.
So these CalTech professors and PhDs who are so eager to explain things can easily fail to qualify what they are saying and end up by confusing us!
Oh well, unleash the dogs. I think string theory is a top down approach that does not produce a viable return on investment. It is elegant, fascinating and allows for a near infinite number of solutions. Finding one that actually works, however, appears to be harder than finding the proverbial needle in the haystack. I prefer the bottom up approach - let observation drive the effort.
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