Wednesday, September 28, 2011

On the universal length appearing in the theory of elementary particles - in 1938

Special relativity and quantum mechanics are characterized by two universal constants, the speed of light, c, and Planck's constant, ℏ. Yet, from these constants one cannot construct a constant of dimension length (or mass respectively as a length can be converted to a mass by use of ℏ and c). In 1899, Max Planck pointed out that adding Newton's constant G to the universal constants c and ℏ allows one to construct units of mass, length and time. Today these are known as Planck-time, Planck-length and Planck-mass respectively. As we have seen in this earlier post, they mark the scale at which quantum gravitational effects are expected to become important. But back in Planck's days their relevance was in their universality, since they are constructed entirely from fundamental constants.

In the early 20th century, with the advent of quantum field theory, it was widely believed that a fundamental length was necessary to cure troublesome divergences. The most commonly used regularization was a cut-off or some other dimensionful quantity to render integrals finite. It seemed natural to think of this pragmantic cut-off as having fundamental significance, though the problems it caused with Lorentz-invariance. In 1938, Heisenberg wrote "Über die in der Theorie der Elemtarteilchen auftretende universelle Länge" (On the universal length appearing in the theory of elementary particles), in which he argued that this fundamental length, which he denoted r0, should appear somewhere not too far beyond the classical electron radius (of the order some fm).

This idea seems curious today, and has to be put into perspective. Heisenberg was very worried about the non-renormalizability of Fermi's theory of β-decay. He had previously shown that applying Fermi's theory to the high center of mass energies of some hundred GeV lead to an "explosion," by which he referred to events of very high multiplicity. Heisenberg argued this would explain the observed cosmic ray showers, whose large number of secondary particles we know today are created by cascades (a possibility that was discussed at the time of Heisenberg's writing already, but not agreed upon). We also know today that what Heisenberg actually discovered is that Fermi's theory breaks down at such high energies, and the four-fermion coupling has to be replaced by the exchange of a gauge boson in the electroweak interaction. But in the 1930s neither the strong nor the electroweak force was known. Heisenberg then connected the problem of regularization with the breakdown of the perturbation expansion of Fermi's theory, and argued that the presence of the alleged explosions would prohibit the resolution of finer structures:

"Wenn die Explosionen tatsächlich existieren und die für die Konstante r0 eigentlich charakeristischen Prozesse darstellen, so vermitteln sie vielleicht ein erstes, noch unklares Verständnis der unanschaulichen Züge, die mit der Konstanten r0 verbunden sind. Diese sollten sich ja wohl zunächst darin äußern, daß die Messung einer den Wert r0 unterschreitenden Genauigkeit zu Schwierigkeiten führt... [D]ie Explosionen [würden] dafür sorgen..., daß Ortsmessungen mit einer r0 unterschreitenden Genauigkeit unmöglich sind."

("If the explosions actually exist and represent the processes characteristic for the constant r0, then they maybe convey a first, still unclear, understanding of the obscure properties connected with the constant r0. These should, one may expect, express themselves in difficulties of measurements with a precision better than r0... The explosions would have the effect... that measurements of positions are not possible to a precision better than r0.")

In hindsight we know that Heisenberg was, correctly, arguing that the theory of elementary particles known in the 1930s was incomplete. The strong interaction was missing and Fermi's theory indeed non-renormalizable, but not fundamental. Today we also know that the standard model of particle physics is perturbatively renormalizable and know techniques to deal with divergent integrals that do not necessitate cut-offs, such as dimensional regularization. But lacking that knowledge, it is understandable that Heisenberg argued gravity had no role to play for the appearance of a fundamental length:

"Der Umstand, daß [die Plancklänge] wesentlich kleiner ist als r0, gibt uns das Recht, von den durch die Gravitation bedingen unanschaulichen Zügen der Naturbeschreibung zunächst abzusehen, da sie - wenigstens in der Atomphysik - völlig untergehen in den viel gröberen unanschaulichen Zügen, die von der universellen Konstanten r0 herrühren. Es dürfte aus diesen Gründen wohl kaum möglich sein, die elektrischen und die Gravitationserscheinungen in die übrige Physik einzuordnen, bevor die mit der Länge r0 zusammenhängenden Probleme gelöst sind."

("The fact that [the Planck length] is much smaller than r0 gives us the right to leave aside the obscure properties of the description of nature due to gravity, since they - at least in atomic physics - are totally negligible relative to the much coarser obscure properties that go back to the universal constant r0. For this reason, it seems hardly possible to integrate electric and gravitational phenomena into the rest of physics until the problems connected to the length r0 are solved.")

Today, one of the big outstanding questions in theoretical physics is how to resolve the apparent disagreements between the quantum field theories of the standard model and general relativity. It is not that we cannot quantize gravity, but that the attempt to do so leads to a non-renormalizable and thus fundamentally nonsensical theory. The reason is that the coupling constant of gravity, Newton's constant, is dimensionful. This leads to the necessity to introduce an infinite number of counter-terms, eventually rendering the theory incapable of prediction.

But the same is true for Fermi's theory that Heisenberg was so worried about that he argued for a finite resolution where the theory breaks down - and mistakenly so since he was merely pushing an effective theory beyond its limits. So we have to ask then if we are we making the same mistake as Heisenberg, in that we falsely interpret the failure of general relativity to extend beyond the Planck scale as the occurence of a fundamentally finite resolution of structures, rather than just the limit beyond which we have to look for a new theory that will allow us to resolve smaller distances still?

If it was only the extension of classical gravity, laid out in many thought experiments (see eg. Garay 1994), that made us believe the Planck length is of fundamental importance, then the above historical lesson should caution us we might be on the wrong track. Yet, the situation today is different from that which Heisenberg faced. Rather than pushing a quantum theory beyond its limits, we are pushing a classical theory and conclude that its short-distance behavior is troublesome, which we hope to resolve with quantizing the theory. And several attempts at a UV-completion of gravity (string theory, loop quantum gravity, asymptotically safe gravity) suggest that the role of the Planck length as a minimal length carries over into the quantum regime as a dimensionful regulator, though in very different ways. This feeds our hopes that we might be working on unraveling another layer of natures secrets and that this time it might actually be the fundamental one.


Aside: This text is part of the introduction to an article I am working on. Is the English translation of the German extracts from Heisenberg's paper understandable? It sounds funny to me, but then Heisenberg's German is also funny for 21st century ears. Feedback would be appreciated!

29 comments:

Steven Colyer said...

Hi bee,

You have a gift for exposition, for explaining things clearly and in logical order, on par with Peter Woit's, which is why this and yours and Stefan's blogs are my personal favorites in the physics weblogging community, so thanks.

Arun said...

The Heisenberg stuff is clear enough.

If you want to change it, e.g.,

"The fact that [the Planck length] is much smaller than r0 gives us the right to leave aside the obscure properties of the description of nature due to gravity"

gives us the right -> makes it valid, makes it acceptable

Uncle Al said...

Literal translation versus style is slippery. Your English has no faults given 100-year old text.

"problems it caused with Lorentz-invariance" (only validated by massless photons). Galactic rotation - MOND, Milgrom acceleration - suggests trace anisotropic vacuum toward mass, then non-zero minimum angular momentum. Physics postulats vacuum isotropy toward mass, then inserts symmetry breakings. Noether’s theorems ignore absolute discontinuous symmetries. Parity!

A massed sector vacuum left foot discerns opposite shoes. If they melt into identical socks, transition energies diverge. Chemically and macroscopically identical, opposite geometric parity atomic mass distributions, paired enantiomorphic space groups P3(1) | P3(2); P3(1)12 | P3(2)12; P3(1)21 | P3(2)21, originate outside physics. Euclid fails on a 2-sphere.

http://www.mazepath.com/uncleal/shoes2.png
http://www.mazepath.com/uncleal/benzil.png

Paired differential scanning calorimeters measure 95°C ΔH(fusion) of fresh P3(1)21 versus P3(2)21 benzil single crystals. Run every 30 minutes for 24 hours. Baseline non-zero ΔΔH(fusion) with a 24-hour sinusoidal EP violation superposed results. Null is powdered racemic benzil both sides.

Somebody should look.

Neil Bates said...

Thanks, Bee. AFAICT, the biggest embarrassment to the attempt to use the "universal" scales (which also produce a Planck mass - oddly enough about that of a flea, which Penrose tries to make use of in objective collapse theories.) Well, the length scale and mass give an "energy density of space" (on order of one PM per cubic Planck length) that was enormous, and exceeded the current dark energy lambda by something like 10^120.

Note the "classical electron radius", the only other significant physical length albeit based on a specific particle and not combinations of constants per se. The CEL is very much bigger than the PL.

Another odd "embarrassment" is why the fundamental charge e is not the "logically appealing" simple value of sqrt(hbar*c) or maybe just h subbed. Instead it has sqrt(1/137) of that value, relating to the fine structure constant 1/137. (I sometimes flub the squares or inverses, double check that.)

BTW I'm so glad blogger uses a more readable word test than the horrendous Captcha.

Mud said...

It's like gravity has no short term memory, it only has long term memory! I think, I don't know, what was I saying, oh yeah, hold on I have to tie my shoes..........

Jochen said...

I think the translations are spot on, and it's a truly clear and precise expository piece.

Out of curiosity, what do you think about Dvali et al.'s proposal to use the physical interpretation of the Planck units -- the formation of black holes -- to provide a UV completion for Einstein gravity?

Bee said...

Uncle, Arun,

Thanks!

Bee said...

Hi Jochen,

Thanks for the feedback. I've only briefly looked at Dvali's paper (it's on my pile with things to read but not currently near the top), so can't say anything sensible about it at that point, sorry. Best,

B.

Phillip Helbig said...

The English is fine (I am fluent in both languages). Let's see your Swedish version!

Bee said...

Tack!

Jag talar inte Svenska så bra...

Phil Warnell said...

Hi Bee,

Although what you’re saying is perfectly linguistically correct below find my own suggestions as bolded. I do find it somewhat interesting that when it comes to Heisenberg we are again left with uncertainty; only in this case not with anything physical yet rather of his meaning:-)

"If the explosions actually exist and represent the processes characteristic for the constant r0, then they may convey a first, still unclear, understanding of the obscure properties connected with the constant r0. These should, as one might expect, express themselves in difficulties of measurements with a precision better than r0... The explosions would have the effect... that measurements of positions are not possible to a precision better than r0.")

("The fact that [the Planck length] is significantly smaller than r0 gives us license to ignore) the obscure properties of the description of nature due to gravity, since they - at least in atomic physics - are totally negligible relative to the much coarser obscure properties that go back to the universal constant r0. For this reason, it seems hardly possible to integrate electric and gravitational phenomena into the rest of physics until the problems connected to the length r0 are solved.")

Best,

Phil

Bee said...

Hi Phil,

Thanks so much. You are right of course that "wesentlich" should be "significant." As to the "may" however, he does indeed write "so vermitteln sie vielleicht" (so they maybe convey) rather than "so könnten sie vermitteln" (so they may convey). Best,

B.

Phil Warnell said...

Hi Bee,

I’m glad I was able to help and might have done better if I was able to read German; well perhaps in my next time around in the universe if Penrose is correct :-)

Bet,

Phil

Bee said...

Hi Phil,

And as to the "might": thing is that Heisenberg uses the word "wohl" which is as I've learned a Modalpartikel (look, a new particle!), and quite difficult to translate. On the above website they suggest "certainly" which is actually not a bad suggestion. Either way, "might" does not express Heisenberg's meaning at all. Best,

B.

Phil Warnell said...
This comment has been removed by the author.
Phillip Helbig said...

"Wohl" is interesting. It can either mean "for sure, certainly" or "perhaps", depending on the context. (No wonder Heisenberg used it. :-)) Similar to "grundsätzlich", which can mean "with absolutely no exceptions" or "as a rule, but there are exceptions".

Phil Warnell said...
This comment has been removed by the author.
Phil Warnell said...

Hi Bee,

Thanks for this further clarification as it might lend some insight as to why QM at times is so difficult to comprehend. This is reminiscent of when Anton Ziegler, during his PI public lecture, stated that the word “entanglement” improperly expressed the situation, as the German word is more closely captured as being a handshake; however he neglected to make clear as to what has the particles stick to their agreement :-) have

Best,

Phil

Rastus Odinga Odinga said...

Hi,in English there exists the expression "might well", which has a subtle meaning, something like, "I´m just about certain about this but I don´t want to sound over-confident." Despite containing the word "might", it doesn´t really have any connotations that you aren´t sure. eg "The OPERA results might well be due to systematic errors." Perhaps wohl = might well?

Phillip Helbig said...

"Perhaps wohl = might well?"

In some cases, yes.

Bee said...

Not here though. What Heisenberg means is basically "How could it be any different?" Depending on whether you read the question as rhetoric, it means certainty or exactly the opposite.

Plato said...

Hi Bee,

Putting Planck in your search feature is also very helpful. I always appreciate the historical developments and where this has lead us too till today.

The correlative views of cascading particle expressions is well founded relation that has been carefully nurtured in the development of the LHC.

So you have this experimental backdrop with which to share two views of the world as we know it, "hidden or otherwise" you have to look carefully what all this science is saying.. or not saying, and admit that our thinking is directed with regard to the phenomenological approach we are dealing with. This is very responsible.

So you look for locations that are compelling as to describing the nature of the universe and as to what it is doing? So if geometrically the nature of the universe is doing this....what ever that means in the understanding of the Friedmann equations then what proof have you as to such motivations leading our perspective about it's direction geometrically expressed. The universe speeding up?

It is just such a imaginative mind that such an expression can flash across this universe with such speed to have examples of such expressions in our detectors as to the distance measured "between the event in the cosmos and our back board measure here on earth?"

What appear in the detectors as such flashes are constraints in the mediums of expression, yet such expressions are not held to consider such flashes as significant considerations?

I know our scientists know better and that some are better prepared.

Best,

Plato said...

In very high energy collisions, excitations may be shot out into the bulk space. They leave the MU, much like when rogue waves on the ocean hit each other and water splashes far above the surface of the ocean. The splashed water can travel much faster than any of the waves ever could inside the ocean.Neutrinos CAN Go Faster Than Light Without Violating Relativity

How does one resolve the realization with them self with regard to such distances?:)

Best,

Plato said...

Sorry Bee...one last quote,


Right-handed neutrinos, with the intrinsic spin oriented in the direction of motion, have yet to be observed, but if they do exist then they could make neutrino superfluids possible. Joe Kapusta of the University of Minnesota has shown that such an exotic medium could arise because the right-handed particles could exchange Higgs bosons with the well known left-handed neutrinos and pair up to make bosons, which could then form a superfluid.The right spin for a neutrino superfluid

Best,

Uncle Al said...

What is physics' fundamental distance? Planck distance effects are inferred not measured. Leptons are point particles. Hadrons are required. Chirality is an emergent phenomenon – length! Observed chiral vacuum background toward mass validates vacuum structure. Observation trumps postulates and derivations.

A holographic universe has information proportional to surface area; volume is illusory. Tessellate a high-symmetry (low entropy) 2-D surface with mathematically perfectly chiral (pseudo)scalar fields: Fullerenes C44,C52,C92,C100 (point group T, not Th or Td); fullerenes C140,C260 (point group I, not the full icosahedral group Ih); etc. Point group I obtains 12 unique rotations by 72°, 12 rotations by 144°, 20 rotations by 120°, and 15 rotations by 180°. Excluded are an inversion point, 12 unique rotoreflections by 108°, 12 rotoreflections by 36°, 20 rotoreflections by 60°, and 15 reflection planes. Geometric parity is the proper test of vacuum structure.

Somebody should look.

Uncle Al said...

What is physics' fundamental distance? Planck distance effects are inferred not measured. Leptons are point particles. Hadrons are required. Chirality is an emergent phenomenon – length! Observed chiral vacuum background toward mass validates vacuum structure. Observation trumps postulates and derivations.

A holographic universe has information proportional to surface area; volume is illusory. Tessellate a high-symmetry (low entropy) 2-D surface with mathematically perfectly chiral (pseudo)scalar fields: Fullerenes C44,C52,C92,C100 (point group T, not Th or Td); fullerenes C140,C260 (point group I, not the full icosahedral group Ih); etc. Point group I obtains 12 unique rotations by 72°, 12 rotations by 144°, 20 rotations by 120°, and 15 rotations by 180°. Excluded are an inversion point, 12 unique rotoreflections by 108°, 12 rotoreflections by 36°, 20 rotoreflections by 60°, and 15 reflection planes. Geometric parity is the proper test of vacuum structure existence.

Somebody should look.

Thomas Larsson said...

"Jag talar inte Svenska så bra..."

Quite ok, except that we don't capitalize neither nouns, months, weekdays, nor languages.

One of my favorite ideas is the appearence of a non-universal scale - the observer's mass. Zero in comparison to GR, infinite in comparison to QFT.

Steven Colyer said...

Hi Uncle Al,

Regarding your first comment on this thread.... correct me if I'm wrong but hasn't MOND been discredited? I think it was in late '09 in planetary science when MOND predicted something in the gas giants' orbits that was falsified. Correct me if I'm wrong someone, thanks in advance.

Uncle Al said...

MOND is one parameter galaxy dynamics implying non-conservation of angular momentum. Noetherian failure is bench top testable. Dark matter is fantastical curve fitting upon which hundreds of $(US)millions give no resolution. Theory endlessly debates both (MOND, arxiv:1108.5588, 1108.4021, 1107.2934, 1107.2109, 1106.4108, 1106.2966, etc.).

Euclid fails on a torus because his Fifth Postulate is incomplete. Nothing within theory tests postulates, for then they would not be postulated. One must test outside derivation.

Test for a vacuum background chiral only toward mass. This sources all physics' parity anomalies within 90 days (parity Eotvos experiment) or one day (parity calorimetry experiment). The worst it can do is succeed, ending the epicycles.