Cover photo
Cliff Harvey
Works at Functor AB
Attended Worcester Polytechnic Institute
Lives in Stockholm, Sweden
28,457 followers|1,788,975 views


Cliff Harvey

Shared publicly  - 
The geometry of music revealed!  The red lines connect notes that are a major third apart.  The green lines connect notes that are a minor third apart. The blue lines connect notes that are a perfect fifth apart.

Each triangle is a chord with three notes, called a triad.  These are the most basic chords in Western music.   There are two kinds:

A major triad sounds happy.  The major triads are the triangles whose edges go red-green-blue as you go around clockwise.

A minor triad sounds sad.  The minor triads are the triangles whose edges go green-red-blue as you go around clockwise.

This pattern is called a tone net, and this one was created by David W. Bulger.  There's a lot more to say about it, and you can read more in this Wikipedia article:

and this great post by Richard Green:

The symmetry group of this tone net is important in music theory, and if you read these you'll know why!
94 comments on original post
Matt McIrvin's profile photo
Interesting to see this post making the rounds again--we had a really entertaining discussion in the comments of John's original post!
Add a comment...

Cliff Harvey

Videos: lectures, interviews, panels  - 
A pretty good collection of various lectures on physics including strings, QFT, and several other things.

Via +Giotis Mth.

(I guess I can't just share to communities directly anymore? What gives?)
Black holes, quantum entanglement, and worlds with 11 dimensions - get to know the amazing physics that governs our universe.
Stam Nicolis's profile photoAndreas Geisler's profile photo
The blurb says : the amazing physics that governs our universe
That should be the amazing physics that models our universe :D
Add a comment...

Cliff Harvey

Shared publicly  - 
The trouble with QED

If you're trying to understand charged particles and radiation in a way that takes special relativity and quantum mechanics into account, you need QED.

That stands for quantum electrodynamics. Feynman, Schwinger and Tomonaga invented this theory - with lots of help - around 1948. In QED we often compute answers to physics problems as power series in the fine structure constant

α ≈ 1/137.036

This number says how strong the electric force is. For example, if you have an electron orbiting a proton, on average it's moving about 1/137.036 times the speed of light.

We can compute lots of things using QED. A great example is the magnetic field produced by an electron. The electron is a charged spinning particle, so it has a magnetic field in addition to its electric field. How strong is this magnetic field?

With a truly heroic computation, physicists have used QED to compute this quantity up to order α⁵. This required computing and adding up over 13,000 integrals. If we also take other Standard Model effects into account we get agreement with experiment to roughly one part in a trillion!

This is often called the most accurate prediction of science. However, if we continue adding up terms in this power series, there is no guarantee that the answer converges. Indeed, in 1952 Freeman Dyson gave a heuristic argument that makes physicists expect that the series diverges, along with most other power series in QED!

I explain that argument in this blog article. I'm especially happy because I think I've made it a bit more precise. But it's not a proof: just an argument that something very strange must happen if the answer converges.

Currently, the consensus among physicists is that ultimately QED is inconsistent. I explain why. But again, there's no proof. We need some mathematicians to help settle these questions!
34 comments on original post
Add a comment...

Cliff Harvey

Shared publicly  - 
The signal has probably reached you that the long search for gravitational waves has finally paid off!

LIGO observed a dramatic signal on September 14, consistent with black holes of 29 and 36 solar masses merging, with 3 solar masses of the combined energy converted into gravitational waves. They traveled for something like 1.3 billion years before being detected on Earth.

One of the facts that always amazes me about these processes and the waves they produce is that they actually lie in the audible frequency range. So playing this signal as sound is an especially meaningful way to experience it, as this video they released shows. It first plays the signal in its true form a couple times, then plays it shifted up in tone to be more audible. You can also really clearly see the signals as detected at the two LIGO sites (along with their Fourier transforms).

All the details of the discovery, including uncertainties in the numbers I mentioned can be found in the paper:


I don't have anything novel to say about this, other than spacetime and general relativity are amazing, science is awesome, and congrats to everybody who made this happen!
Add a comment...

Cliff Harvey

Shared publicly  - 
I hope this holds up. It would be awesome to know that humanity has succeeded in directly measuring vibrations in spacetime, and, longer term, to learn what's within earshot at LIGO's new level of sensitivity (since September).
This is as big as the detection of the Higgs, if it turns out that is what they are seeing. 

I got into physics so I could do general relativity, which lead me to this: and thence to being the category theorist I am today :-) But gravitational waves are incredibly cool, and if I'm not wrong, the direct detection of which is just about the last major un-verified prediction of GR.
5 comments on original post
Amber Peall's profile photoCliff Harvey's profile photoPeter Schmidt's profile photo
+Cliff Harvey you are the second person today to tell me something like that, sigh. 😉 I was writing software for the COBE interferometer at the time. 
Add a comment...

Cliff Harvey

String Physics Discussion  - 
M&m and little strings

It is amazing how efficiently String theorists build on previous work of their colleagues constructing step by step in a consistent way this beautiful framework of ideas and tools we call String theory.

There is an interesting troika of papers [1], [2], [3] following the pioneered work of Vafa et al [4], [5], [6] on M-strings (intersections of open M2 and M5 branes, refer to figure 2) which relate the M-strings of 6d SCFTs, the monopole Strings of 5d Supersymmetric Yang-Mills theories (coined m-strings) and the 6d little strings.

The basic result is that there is a direct correspondence between these type of Strings in the sense that monopole and little strings are basically bound states of M-strings!

In [1] it is proposed that the BPS degeneracies of bound-state of M-strings provide the elliptic genus of the moduli space of m-strings in the Nekrasov-Shatashvili limit.

This is based on the fact that BPS states of the 5d gauge theory are related via S-duality to magnetic monopole m-strings of the same theory but also via the appropriate compactification scheme to the M-strings in 6d.

The partition function of the 5d Supersymmetric Yang-Mills theory corresponds to an index that counts the degeneracies of BPS bound-states of W-bosons with instanton particles. Then the S-dual m-string picture can be shown to be the elliptic genus of certain moduli spaces.
The authors of [1] manage to match the partition function of the M-strings with this elliptic genus and thus establishing the correspondence.

To calculate the M-strings partition function they rely on the original work of Vafa et al [4], [5], [6] and use the powerful topological vertex formalism to calculate the topological string partition function of certain elliptically fibered Calabi-Yau 3-folds.

The key point here is that if we manage to geometrically engineer the 5d theory by compactifying M-theory on this CY the topological string partition function will give the partition function of the gauge theory. The gauge theory partition function on the other hand appears in the world-volume of certain (p,q) 5 branes of IIB which are associated to the corresponding M-theory branes configurations where the M-strings appear. From the IIB we can uplift to M-theory in 11 dimensions and thus find the appropriate compactification on the corresponding CY (it is well known that IIB (p,q) brane web diagrams are associated to CY toric diagrams in the context of toric geometry-branes correspondence).

Thus via the 5d theory the topological String partition function on the CY captures the BPS degeneracies of M-strings. This was suggested in the quite famous troika of the "parent" papers [4], [5], [6] where the partition function of the 5d theory was derived via three different methods, the topological string partition function, localization techniques in the 5d field theory and the (2,0) elliptic genus of the M-Strings (refer to figure 1).

Similar arguments are made in [3].

Now how the little strings come into the picture in [2]?

This can be achieved by compactifying the dimension along which the M5 branes are separated. Then even in case the M5 branes coincide (and thus M-strings become tensionless) there can always be M2 branes suspended between the first and last M5 and thus tensile strings; these are the little strings (little string theories are six-dimensional non-local quantum theories with non-gravitational string excitations which at energies far below the string scale flow to the SCFTs counterparts).

Another way for getting LSTs is in the world-volume of NS5 branes in IIA and IIB. The two LSTs (IIa and IIb respectively) are related by T-duality much like the parent string theories. This T-duality is reflected in the M5-M2 branes configuration setup by exchanging suitable compactification circles and in the topological string partition function computation by a fiber-base duality of the corresponding elliptically fibered CY.

There is a illuminating analogy that the authors make with Dp branes and open strings. Here is the relevant excerpt:

They compare multiple M5-branes on a transverse circle with multiple Dp-branes on a transverse circle and they interpret the M-strings (i.e. open M2-branes) as noncritical counterparts of open fundamental strings, while a little string ground state is the noncritical counterpart of a closed fundamental string. In the same way as multiple open fundamental strings on the Dp-branes can form a closed string and move freely in ambient ten-dimensional bulk spacetime, multiple open M2-branes ending on M5-branes can form a closed M2-brane and move freely in eleven dimensional spacetime but what makes the little strings very different from fundamental strings is that, in the decoupling limit (i.e. g-->0) the little strings are confined inside the five-brane worldvolume, i.e. the six dimensional spacetime the little string theories live in.

Now, by checking the compact partition functions (corresponding to little strings) to their non-compact counterparts (corresponding to M-strings) studied in [1], it is found that the counting function of compact BPS configurations can fully be constructed as a linear superposition of the non-compact ones. This implies that little strings can be viewed as bound-states of M-strings.

The overall result is stunning. As the authors state the fact that the little strings can be viewed as bound-states of M-strings means that "for the purpose of BPS counting of IIb little strings at least, one only needs to know BPS excitations of the (2,0) superconformal field theory, which is just the low-energy limit of the IIb little string theory".

As far as I know this is the first time that such relation between little strings and M-strings is proposed. In my mind this will have far reaching implications for the study of the elusive 6d SCFT where M-strings become tensionless.

Much more and other beautiful insights inside.

Figures were taken from [6] plus a bonus, a rare picture of uncle Albert giving a lecture on M5/M2 branes, M-strings and 6d SCFT.

[1] M String, Monopole String and Modular Forms

[2] Instanton-Monopole Correspondence from M-Branes on S1 and Little String Theory

[3] From strings in 6d to strings in 5d

[4] M-strings

[5] On orbifolds of M-Strings

[6] M-strings, Elliptic Genera and N=4 String Amplitudes
View original post
Edward NSZY's profile photo
Any TENSOR matrix of 4d draw the space time you want. Each terms is a string with itself clock = dimension = pole = string = fold axis, so you has 16 terms or strings, this do the framework into the multiplex topologic math could draw the increible number of 4^4 shapes of differents space time with the same matrix, just change your relativity speed clocking system to measure the fold. But like you have 16 terms in 16 positions you could have 16^16 space time in balance to draw the intradimensional leap, so on now all must know that it must exist the inverse matrix of energy TENSOR field, guessing that appear the other 16^16 if we operate mixing every clocking will get the full 32^32 strings = 2^5 x 2^5 = 4^10, how ever must exist the first reference to begin the secuence it would be 2^20 +1, but like any fold is a infinite we must renormalizate it with impliciti NEPER base, which reach a ladder to up amid infinities.
The TENSOR Matrix of manifold or variety or energy field put in their main diagonal position the reference polar direction or the principal, building the rest with their vectorial product doing it antisimetric if stay up or down, you only has to get the trace with the terms you prefer to paint the string into the infinite variety to proyect it in RIEMMAN - FRIEDMAN continous geometric shapes to begin the construction of SPACE structure to allow the moving of TIME passing to get absolute more power theorie of 6d in 4d making up a 24 dimension.
But if you join the first 4x4 to itself inverse in NEPER imaginary form, knowing that each clocks defines a imaginary base to compute itself maths, you could open the matrix to a 8x8 clocking system then blow your mind up to 64^64 dimension or stings in relativity moving to solve it with a good logic is needing use the CHESS PLAY, with this play each figure is one full fold the table is the GENERAL RELATIVITY SPACE TIME so each one has itself logic physic law to solve their movement problems.
Space is mass and light is time so are inverse meanings that paint a logic structure in algebra.
Now but what is time? Time is the direction of space, because without time is impossible to space exist but time can be without time.
To change relativity time clocking only is need direction in space not speed in moving, so you could change yourself clock without moving, just with a magnetic polar shift should be easier convert from light or from time any mass or space, if you want to move throw time you must convert in light, exist some polar interval in every mass that will do the same effect. With polar phase is easy transform a mass gravity into inertial and back.
;;))) Thanks for share!!

Add a comment...
In his circles
1,091 people
Have him in circles
28,457 people
Bernardo Valente's profile photo
Debra Sasser's profile photo
john franz's profile photo
Pamela Hanaman's profile photo
Debbie Michaels's profile photo
Adolfo Hernandes's profile photo
Joshua Scruggs's profile photo
‫احمد ممدوح‬‎'s profile photo
whatsanme's profile photo

Cliff Harvey

Shared publicly  - 
A Capella Science crushes it once again, not only musically, but in every other way.

Via +Jenny Winder+Jennifer Ouellette
Add a comment...

Cliff Harvey

Shared publicly  - 
Learn about Category Theory from a master, French mathematician Pierre Schapira, writing for the new Inference magazine: "Here I will consider uniqueness, or, rather, the concept of identity, and how it functions. I will therefore address the status of equality in mathematics and its variants, namely isomorphism, equivalence, and so on. This issue has until recently been totally ignored, but is of such importance that it may potentially lead us to question set theory itself." Fun and illuminating piece. The French version is available as well:
Set theory, category theory, and topology. Pierre Schapira explores the concept of identity within category theory, and what it means for the properties to be satisfied only up to homotopy.
3 comments on original post
Claudia Martinez's profile photo
Me gusts
Add a comment...

Cliff Harvey

Shared publicly  - 
Einstein's stance on Gravitational Waves (For +Urs Schreiber, extracted from Kennefick's Traveling at the Speed of Thought.)

Citation marks refer to the book above.

1912 Max Abraham worked on his own version of relativistic theory of gravity, coming to the conclusion that gravitational radiation cannot be a component since there does not exist dipole radiation.

1915 Publication of Einstein's paper on GR

1915-1916 Discover of Schwarzschild solution

February 19, 1916 In communication to Schwarzschild, Einstein indicated that he did not believe in gravitational radiation, for a reason which essentially boils down to the impossibility of dipole gravitational radiation, which he deduced from the non-existence of negative mass. [K p39]

Mid 1916 Publication of "Approximate Integration for the Field Equations", in which Einstein discussed linearization of his namesake equations, and showed that the linearized equations can be solved with retarded potentials, thereby showing existence of gravitational waves.

1918 Publication of new paper which corrected an error of the 1916 paper, after a letter from Nordstrom. (The formula in 1916 paper seems to indicate the existence of monopole radiation. The 1918 paper correctly found that the lowest order terms are quadrupole.) In particular he recognized that certain apparent monopole and dipole radiation that does not appear to carry energy can be gauged away with coordinate transforms. [K p65]

mid 1936 In a letter to Max Born, Einstein claims that gravitational waves in fact do not exist in the full theory. He claims that the full nonlinear theory puts additional constraints such one cannot upgrade the solutions to the linearized system to a solution of the full system. He arrived at this conclusion after studying plane-wave solutions with Nathan Rosen, and found that the solutions must be singular. [K p79]

(Their paper was sent to John Tate for Physical Review. Tate sent it to HP Robertson to referee, and some comments were returned. Einstein threw a hissy fit and withdrew the submission on July 30, 1936.)

Early 1937 Publication of the aforementioned paper, now however with the conclusion altered to support the existence of gravitational waves, in a different journal. This is the Einstein-Rosen wave paper.

It turns out that HP Robertson spotted a mistake in the original submission, that the singularity they observed is really just a coordinate singularity and can be removed if one assumes one is working on a cylindrical, instead of rectangular, coordinate system. [K p81 - 88]


As far as I am aware that is the summary of Einstein's involvement in the theory. There are papers published in the twenties concerning exact gravitational wave solutions (PP wave solutions seem to have been first written down by Brinkmann in '25), but Einstein was apparently oblivious to those papers during his investigation with Rosen.

For obvious reasons Einstein was not involved in the "controversies" about gravitational waves between the 50s and the later 70s.

After 1937, Einstein's focus on GR (besides the grand unified theory) seems to be on studying stationary solutions (where among other things, in the linearized regime he proved the rigidity part of the positive mass theorem) and on studying particle motions (the series of papers by Einstein, Infeld, and Hoffman, in some combination). 
1 comment on original post
Add a comment...

Cliff Harvey

Shared publicly  - 
Signal Processing with GW150914 Open Data
Besides securing a prominent position in scientific history, LIGO looks to be taking a lead role in the movement for open science. They've put out a tutorial in the form of an iPython notebook where they show you how to process the raw data from the gravitational wave signal, compare it to a numerical relativity prediction, convert it to audio, and so on:

This is a phenomenal thing to do. Communicating your science to the public shouldn't be limited to just explaining it in absolute basic language (though thats surely important too), but I also think there should be some effort put towards helping people with intermediate levels of knowledge and a desire to learn be able to do so. And of course, it is also extremely desirable to be able to see and verify the steps taken to analyze the data, so that it's easier to reproduce, understand, and ultimately be made that much more accountable.

I'm not sure to what extent this notebook exactly matches what they did to prepare their paper – it would certainly be especially great if that were the case. But either way this looks like a great contribution by LIGO for learning both about gravitational wave science and about the concepts involved in their data analysis.

It would be very cool to see something like this from +ATLAS Experiment or +CMS Experiment!
Add a comment...

Cliff Harvey

Shared publicly  - 
The relation between the icosahedral group and E8 in classical discussions remains a bit mysterious. N=2 super Yang-Mills theory clarifies it.

Various seemingly unrelated structures in mathematics fall into an "ADE classification" [1]. Notably finite subgroups of SU(2) and compact simple Lie groups do. The way this works usually is that one tries to classify these structures somehow, and ends up finding that the classification is goverened by the combinatorics of Dynkin diagrams.

While that does explain a bit, it seems the statement that both the icosahedral group and the Lie group E8 are related to the same Dynkin diagram somehow is still more a question than an answer. Why is that so?

The first key insight is due to Kronheimer in 1989 [3]. He showed that the (resolutions of) the orbifold quotients C^2/Gamma for finite subgroups Gamma of SU(2) are precisely the generic form of the gauge orbits of the direct product of U(ni)-s acting in the evident way on the direct sum of Hom(C^ni, C^nj)-s, where i and j range over the vertices of the Dynkin diagram, and (i,j) over its edges.

This becomes more illuminating when interpreted in terms of gauge theory: in a "quiver gauge theory" the gauge group is a product of U(ni) factors associated with vertices of a quiver, and the particles which are charged under this gauge group arrange, as a linear representation, into a direct sum of Hom(C^ni, C^nj)-s, for each edge of the quiver. 

Pick one such particle, and follow it around as the gauge group transforms it. The space swept out is its gauge orbit, and Kronheimer says that if the quiver is a Dynkin diagram, then this gauge orbit looks like C^2/Gamma.

On the other extreme, gauge theories are of interest whose gauge group is not a big direct product, but is a "simple" Lie group, in the technical sense [4], such as SU(N) or E8. The mechanism that relates the two classes of examples is spontaneous symmetry breaking ("Higgsing"): the ground state energy of the field theory may happen to be achieved by putting the fields at any one point in a higher dimensional space of field configurations, acted on by the gauge group, and fixing any one such point "spontaneously" singles out the corresponding stabilizer subgroup [5]. 

Now here is the final ingredient: it is N=2 super Yang-Mills gauge theories [6] ("Seiberg-Witten theory") which have a potential that is such that its vacua break a simple gauge group such as SU(N) down to a Dynkin diagram quiver gauge theory. One place where this is reviewed, physics style, is section 2.3.4 in the thesis [7].

More precisely, these theories have two different kinds of vacua, those on the "Coulomb branch" and those on the "Higgs branch" depending on whether the scalars of the "vector multiplets" (the gauge field sector) or of the "hypermultiplet" (the matter field sector) vanish. The statement above is for the Higgs branch, but the Coulomb branch is supposed to behave "dually".

So that then finally is the relation, in the ADE classification, between the simple Lie groups and the finite subgroups of SU(2): start with an N=2 super Yang Mills theory with gauge group a simpe Lie group. Let it spontaneously find its vacuum and consider the orbit space of the remaining spontaneously broken symmetry group. That is (a resolution of) the orbifold quotient of C^2 by a discrete subgroup of SU(2).

7 comments on original post
Add a comment...

Cliff Harvey

Shared publicly  - 
This is may be one of the more interesting and important developments in theoretical quantum gravity research lately. Stephen Hawking and Andrew Strominger both judge this to be a major step in understanding the black hole information paradox, though not yet decisive. I'm inclined to agree, to a point, but major questions remain.

The paper: Soft Hair on Black Holes

Quick Recap: The black hole info paradox is the fact that naively it seems like black holes are characterized by only a few numbers (mass, charge, angular momentum) while other considerations make it pretty clear they actually need to carry a huge amount of information (corresponding to everything it's gobbled up). To loose the information preserving property of physical laws would be huge.

The new insight here has to do with extra information that might be stored in zero-energy – so-called soft – photons and gravitons. Andrew Strominger describes his and Hawking's perspective on this in good depth in this interview. Pretty informative reading.

Like so many other things in physics, symmetry is at the heart of this. The basis of the new work are some previously-neglected symmetries of the laws of nature; of quantum field theory. And not just any symmetry but an infinite-dimensional symmetry in which "the point at infinity" plays a key role.

It seems pretty solid that these new properties of quantum field theory are basically right at the mathematical level, and it also seems likely that the importance for the black hole information paradox might be approximately what the authors argue (incomplete as the argument is).

However not everyone is fully on board with the exact interpretation presented by the authors. Here are a couple of responses from the usual theorist bloggers who cover these things.

Im particularly partial to Lubos's quips that no solution to the black hole information paradox should really privilege points on the black hole horizon as special, since there isn't anything fundamentally special about them. Especially since the information in question is associated with soft particles, and zero-energy particles are supposed to be maximally-delocalized in spacetime according to the uncertainty principle. Also there seems to be some schizophrenic quality to these particles since they're supposed to save information that would otherwise be lost, but at the same time they also must be "not physically realizable" in some sense.

Like everyone else, my understanding of this is quite incomplete, but those are some of the questions that seem natural to ask after reading this new paper and interview.

Its always great to see more progress on black hole information, and great to see still more evidence that even insanely well-tested standard theories like general relativity and electromagnetism can continue to give up significant insights after all these decades. Quantum field theory continues to be a beautiful and mysterious beast.

I hope everybody had a great start to 2016, and plan to get more blogging in this year. Hopefully there will be many more great developments to discuss...

Via +Jenny Winder, +Omar Loisel .
The Harvard physicist explains the collaboration's long-awaited research on the black-hole information paradox
Cliff Harvey's profile photoValdis Klētnieks's profile photoJohn Baez's profile photo
As a mathematician, I take products of spaces at all hours of day or night, so that's how I normally talk... but normally I try to restrain my use of jargon in polite company!
Add a comment...
Cliff's Collections
In his circles
1,091 people
Have him in circles
28,457 people
Bernardo Valente's profile photo
Debra Sasser's profile photo
john franz's profile photo
Pamela Hanaman's profile photo
Debbie Michaels's profile photo
Adolfo Hernandes's profile photo
Joshua Scruggs's profile photo
‫احمد ممدوح‬‎'s profile photo
whatsanme's profile photo
  • Worcester Polytechnic Institute
Basic Information
Co-Founder of Functor AB. I love high energy physics, computer science and giant robots
Above all, Im passionate about physics. More generally Im interested in furthering understanding of this world at all levels, from particle physics to economics to galactic superclusters and everything in between. I have an artistic side I once cultivated aggressively but has somewhat fallen by the wayside. My inner artist/musician is howling to get let out soon.

As a physics major at WPI, I did a couple major research projects on quantum information theory, specifically on proofs of Bell's theorem involving 4 and 5 qubits. Those projects heavily shaped the way I view physics in general, and its deep foundational issues in particular. Despite the difficulty of the subject, I am convinced there is much more confusion than is necessary, as sloppy reasoning and explanations persist.

At the moment Im mostly spending my time learning quantum field theory and string theory. I am consistently amazed by the unity of physical and mathematical logic, and as I've studied the structure of this logic I've inexorably gravitated to the stringy school of thought. Despite the challenges, it seems to me an essentially indispensable set of puzzle pieces that allow the whole structure to make sense. It appears so deeply enmeshed that extrication just does not seem very likely. But, as a scientist, I of course try to challenge any presumptions of mine aggressively.

Ill forever be a theorist at heart, but I also want to find a rewarding way to use my skills for something more down to earth. I think becoming a better computer scientists may be one of the better ways for this to happen.

And I believe that those of us awake to the challenges facing the human race – especially intellectuals, anyone who analyzes things systematically – has a certain obligation to actively stand for the truth and what is right. As a US citizen Im especially focused on the issue of campaign finance as fundamental to all other political problems in the US. 

"There are a thousand hacking at the branches of evil to one who is striking at the root."

Profile shortcuts:
  • Functor AB
Map of the places this user has livedMap of the places this user has livedMap of the places this user has lived
Stockholm, Sweden
Milford, Connecticut - Worcester, Massachusetts