Mathematical picture language program
31 lainon 21 hrs 7
http://www.pnas.org/content/early/2017/12/18/1710707114.full
Arthur M. Jaffea,1,2 and Zhengwei Liua,1,2 aHarvard University, Cambridge, MA 02138 Contributed by Arthur M. Jaffe, August 9, 2017 (sent for review June 18, 2017; reviewed by Jacob D. Biamonte, John Ewing, and Alina Vdovina)
We give an overview of our philosophy of pictures in mathematics. We emphasize a bidirectional process between picture language and mathematical concepts: abstraction and simulation. This motivates a program to understand different subjects, using virtual and real mathematical concepts simulated by pictures.
Pictures appear throughout mathematical history, and we recount some of this story. We explain insights we gained through using mathematical pictures. Our picture language program has the goal to unify ideas from different subjects. We focus here on the 3D quon language (1). This language is a topological quantum field theory (TQFT) in 3D space, with lower-dimensional defects. A quon is a 2D defect on the boundary of a 3D manifold.
We believe quon and other languages will provide a framework for increased mathematical understanding, and we expect that looking further into the role of the mathematics of pictures will be productive. Hence, in 4: Questions, we pose a number of problems as the basis for a picture language research program.
Pictures have been central for visualizing insights and for motivating proofs in many mathematical areas, especially in geometry, topology, algebra, and combinatorics. They extend from ancient work in the schools of Euclid and Pythagoras to modern ideas in particle physics, category theory, and TQFT. See the interesting recent account in Silver (2). Nevertheless, we mention two aspects of mathematical pictures that we feel merit special study.
First is the importance we ascribe to mathematical analysis of pictures. We explain how we have begun to formulate a theory of mathematical analysis on pictures, in addition to the study of their topology and geometry. For example, the analytic aspect of pictures in TQFT is a less-developed area than its topological and algebraic aspects. Yet it has great potential for future advances.
Second is the notion of proof through pictures. In focusing on general mathematical properties of pictures, we wish to distinguish this quality from using pictures in a particular concrete mathematical theory. In other words, we aim to distinguish the notion of the properties of a picture language L, on the one hand, from its use through a simulation S to model a particular mathematical reality R. We thank one referee for pointing out that the distinction between L and R parallels the distinction in linguistics between syntax and semantics.
We propose that it is interesting to prove a result about the language L, and thereby, through simulation, ensure results in R. One can use a single picture language L to simulate several different mathematical areas. In fact, a theorem in L can ensure different theorems in different mathematical subjects R1, R2, etc., as a consequence of different simulations S1, S2, etc. This leads to the discussion of the simulation clock in What Next? Different configurations of the hands of the clock reveal the interrelation between picture proofs for seemingly unrelated mathematical results.
We also discuss the important distinction between two types of concepts in R that we simulate by a given S. These may be ‘‘real’’ concepts, or they may be ‘‘virtual.’’ This distinction is not absolute but depends on what language and simulation one considers. We give some examples, both in mathematics and in physics, in Real and Virtual.
We hope these remarks about pictures can enable progress in understanding both mathematics and physics. Perhaps this can even help other subjects, such as the neurosciences, where “One picture is worth one thousand symbols.”
We propose two basic components to understanding that we call L and R. Here L denotes abstract concepts or language, while R stands for the concrete subjects or reality, which we desire to understand. We could also think of them as left and right. Simulation S represents a map from L to R.
Our universe provides a great reservoir for ideas about the real world. We can consider this as R. We understand these ideas through abstraction, including theories of mathematics, physics, chemistry, and biology. To deal with these real concepts, one often requires virtual concepts that have no meaning in the real world. These virtual concepts may not have an immediate real meaning, yet they may provide key insight to understanding the real structures.
Abstraction is a method to study complicated ideas and goes from R to L. For example, chemistry as L may provide a logical language to abstract certain laws in biology as R, with a simulation S: L →→ R. One can continue this chain where we regard chemistry as a new R and physics as its abstraction as a new L with a new simulation S. Abstraction can be repeated yet again with mathematics as L and physics as R. At each step, one learns the axioms in L from the real concepts in R. In order to do computation, we often need to introduce virtual concepts in L to understand the concepts in R.
We are especially interested in the case that L is a picture language. Then one can represent the concepts in L by pictures that play the role of logical words, with axioms as their grammar. The axioms should be compatible with pictorial intuition. One can develop a picture language as an independent theory like Euclidean geometry.
A good simulation for a particular mathematical subject R should satisfy two conditions. Mathematical concepts in R should be simulated by simple pictures in L, and mathematical identities in R should arise from performing elementary operations on the pictures in L. Abstraction and simulation could be considered as inverse processes of each other.
The idea of whether a concept is real or virtual plays an important role. This is not an absolute concept but depends on the simulation. For example, over the real-number field, the square root xx of a positive number xx is real, while the square root of a negative number xx is virtual. By enlarging the field to complex numbers, one understands this virtual concept in a fruitful way. One gets used to the power of complex numbers, and, in a new simulation, one can regard them as “real.”
In a picture language, it may be that a picture can represent a real concept in one theory and a virtual one in another. As an example, consider Feynman diagrams, the pictures that one uses to describe particle interactions. A contribution to the scattering of two physical electrons is described by the first electron emitting a photon (quantum) that is absorbed by the second electron. However, conservation of energy and momentum preclude the exchanged photon from being real: It must be virtual. Fig. 1 shows the original diagram in ref. 3.
Analytic continuation (Wick rotation) to imaginary time puts the real and virtual particles on equal footing. In that situation, the above example only involves virtual particles. In this way, one will obtain many virtual concepts, which could become real based on a proper simulation.
When Atiyah gave a mathematical definition of TQFT, he wrote “it may well be that such topological understanding is a necessary prerequisite to building the analytical apparatus of the quantum theory” (7). Today, we want to move forward to reach a full quantum field theory. Our long-term goal is to construct quantum field theory using pictures. However, that may be too hard to achieve as the initial step.
We emphasize two properties of QFT that shed light on pictures. The first theme is “symmetry.” To understand continuum symmetry, it is useful to understand discrete symmetry that can approximate the continuum.
The second theme, “positivity,” is especially interesting, as positivity provides the basis for analysis. The striking fact is that the analysis of pictures is a mathematical theory with great potential, but is still in its infancy. On one hand, we expect to abstract concepts from analysis to pictures. This will enrich the theory of picture language in one additional dimension. On the other hand, we want to simulate analysis using picture language and provide pictorial tools.
How do we connect pictures with analysis? Usually, pictures without boundaries are scalars. How can we formulate a measurement pictorially? A main lesson from quantum field theory is the importance of reflection positivity. An elementary pictorial interpretation of reflection positivity is that gluing a picture to its mirror image is positive.
In constructive quantum field theory, extensive analysis is performed through estimating Feynman diagram pictures in terms of subdiagrams; this may use a pictorial Schwarz inequality, or some more sophisticated operator norm. These estimates are central to the proof of most results in the subject. In the framework of our present discussion, such analysis takes place on pictures in R. We call this using ‘‘pictures in analysis.’’
However, we are really interested in whether, and to what extent, one can do analysis on pictures. This means that we need to do computations in L, without reference to R. The discussion of rotation, Fourier analysis, multiplication, and convolution indicate that some progress can be made.
Do we obtain interesting analysis based on this minimal requirement? In fact, the surprise is that we already have interesting results inspired by Fourier theory. As mentioned, the 2D pictorial operation on square-like pictures is compatible with Fourier analysis.
Compactifying the plane to a sphere defines a measurement based on reflection positivity. Pictorial consistency on the sphere implies that the Fourier transform is a unitary. Many other results in Fourier analysis carry over to pictures.
Freeman Dyson described two types of mathematicians: birds and frogs (17). They do mathematics in different ways. Birds soar between different fields with unifying ideas, like Yuri Manin in his book Mathematics as Metaphor (18). Frogs sort out details to achieve great depth of understanding. In fact, it is good to attempt to encompass both metaphors! One does find both of these ingredients in our story of picture language. While we began in quantum information, we ended up traveling through much of the colorful landscape of mathematics.
In our context of language, the bird flies back and forth to discover new Rs, Ls, and Ss. The frog uses an S to understand some important problem. The shape of pictures provides key hints and insights for finding these connections. A popular article by Peter Reuell described picture language as Lego-like mathematics (19).
The authors began our collaboration 2 years ago in late July 2015 with much discussion at Harvard. Our first goal was to understand reflection positivity for parafermions (20, 21) in a pictorial way (15). If we call this mathematical problem R1, then it led us to define the picture language L1, which we call planar para algebra. This is a generalization of planar algebra, but with strings replaced by charged strings, and topological isotopy replaced by para isotopy. The map from L1 to R1 we call the simulation S1.
In the simulation S1, we found an elementary explanation that a 90° rotation of pictures represented Fourier transformation, and takes multiplication to convolution. We used this fact to give a geometric, picture proof of reflection positivity (15).GraphicWe also found elementary picture representations for d×dd×d unitary Pauli matrices X,Y,ZX,Y,Z with eigenvalues qjqj, where q=e2πi/dq=e2πi/d, and j=0,1,…,d−1∈ℤdj=0,1,…,d−1∈ℤd.
Alex Wozniakowski pointed out that our work seemed related to quantum information. This led to fruitful collaboration among the three of us, in which we used the language L1 to simulate quantum communication in the mathematical framework that we name R2. Using this simulation S2, one picture deformed by isotopy into different shapes simulates different concepts in quantum information. In this way, we reproduced the standard teleportation protocol of Bennett et al. (22) by a topological design in L1 (23). With these concepts in place, we could also design other protocols, including multipartite, teleportation protocols (24).
The important point for our understanding of R2 was to follow pictorial intuition. This led to finding a natural candidate for a resource state that we called |Max⟩|Max⟩. Our picture Max in L1 for the entangled state |Max⟩|Max⟩ is simple and natural, as well as suggesting entanglement in a pictorial fashion. Our two-string picture for |Max⟩n|Max⟩n (with nn qudits) isGraphic
The algebraic formula for the simulation S2Maxn in R2 is|Max⟩n=1d(n−1)/2∑|k→|=0|k→⟩n,|Max⟩n=1d(n−1)/2∑|k→|=0|k→⟩n,[3]where we call |k→|=k1+⋯+kn|k→|=k1+⋯+kn the total charge in ℤdℤd. It involves dn−1dn−1 terms for a resource state with nn qudits, so, algebraically, the simulation of Maxn is complicated (23).
We then left Massachusetts to spend 4 months visiting the Research Institute for Mathematics (FIM) at the Eidgenössische Technische Hochschule (ETH) in Zurich, Switzerland.
We finally had time to begin writing up these results (15) and later results (23, 24). We then learned that Greenberger et al. (25) had long before introduced another multiqubit resource state in quantum information. This |GHZ⟩|GHZ⟩ state appears to be algebraically simpler, as it is a sum of only dd terms, independent of the number nn of qubits,|GHZ⟩n=1d1/2∑k∈Zd|k,k,…,k⟩n.|GHZ⟩n=1d1/2∑k∈Zd|k,k,…,k⟩n.[4]Both |Max⟩|Max⟩ and |GHZ⟩|GHZ⟩ generalize the Bell states, and they have some very similar properties. This led us eventually to observe that |GHZ⟩|GHZ⟩ and |Max⟩|Max⟩ are related by a change of basis—in fact by Fourier transformation on ℤdℤd,|Max⟩n=F⊗n|GHZ⟩n.|Max⟩n=F⊗n|GHZ⟩n.[5]To understand this further, we were intrigued by our generalization of “Kitaev’s map” from Majoranas to the Pauli spin matrices X,Y,ZX,Y,Z. We had discovered a natural generalization to represent a single qudit as a neutral parafermion/antiparafermion pair (15). Neutrality provides an elegant way to reduce the d2d2-dimensional space of states for two virtual parafermions to the correct dd-dimensional space for one real qudit. It involved introducing a “four-string” planar language L2 to describe a single qudit, and a corresponding simulation S3L2 = R3. The model R3 contains dd real and d2−dd2−d virtual one-qudit states. The representations of the qudit Pauli X,Y,ZX,Y,Z matrices are neutral, so they act on the neutral (real) subspace of dd dimensions. Another hint that neutrality holds the key is that the SFT equals the discrete Fourier transform in the neutral subspace of L2.
However, the language L2 and simulation S3 pose a problem for describing more than one qudit. Braiding charged strings from different qudits destroys neutrality of the individual excitations. This would allow transitions from the real nn-qudit space of dimension dndn into the virtual nn-qudit space of dimension d2nd2n. Thus the obstruction to describing multiqudit states boiled down to the question: How can one ensure that real multiqudit states evolve into real multiqudit states? When we met Daniel Loss in Basel, we found that he was also considering this question. We did not find the answer immediately.
After Zurich, we had the opportunity to spend 6 weeks visiting two institutes in Bonn. We discovered the answer to the puzzle described in ETH Zurich during June 2016, perhaps receiving inspiration from working in Fritz Hirzebruch’s former office at the Max Planck Institute for Mathematics.
During that time, we encountered two more hints about L2. The first is that the Frobenius algebra for the mm-interval Jones–Wassermann subfactor in CFT (26) isγ=⊕X→dim(X→)X→.γ=⊕X→dim(X→)X→.[6]Here X→=X1⊗⋯⊗XmX→=X1⊗⋯⊗Xm, a tensor of simple objects in a modular tensor category (MTC) (8), and dim(X→)dim(X→) is the multiplicity of 1 in X→X→. This formula coincides with |Max⟩|Max⟩ for the group ZdZd, but they have completely different meanings. Secondly, we learned the relations for bi-Frobenius algebras in the manuscript for ref. 27, which one of the authors had shared with Alex Wozniakowski.
The construction of Jones–Wassermann subfactors for an MTC requires an extension of pictures from 2D to 3D space. Here the coincidence between |Max⟩|Max⟩ and γγ is explained through an m−nm−n duality (28). Going from 2D to 3D also gives a natural explanation of the bi-Frobenius algebras relation.
The last piece in solving the puzzle came by adding three manifolds to these 3D pictures; this was inspired by TQFT. Finally, we unified all these ideas by formulating the 3D quon language L3, and a simulation S4 to quantum information. Moreover, we extended our approach from ZdZd symmetry to MTCs based on work about the Jones–Wassermann subfactor.
We designed L3 using ideas from different Rs: quantum information, subfactor theory, TQFT, and CFT. Therefore, we expected to simulate those Rs using L3 as well as others.
In the quon language, the bi-Frobenius algebra relation has a topological interpretation. Both |Max⟩|Max⟩ and |GHZ⟩|GHZ⟩ are represented by single pictures Max and GHZ, where one picture is a 90° rotation of the other. Algebraically, one resource state is the Fourier transform of the other. Moreover, the complicated resource state |Max⟩|Max⟩ can be computed by using its relation to the elementary resource state |GHZ⟩|GHZ⟩. For three qudits, the quon state pictures are just rotations of one another, as illustrated inGraphic[7]
Furthermore, the quon language expresses elegantly in pictures the duality between orthonormal bases for the Pauli matrices XX and ZZ, so one can begin to imagine pictures describing quantum coordinates.
Our first discovery by S4 was the topological nature of the Feynman, or quantum controlled NOT (CNOT), gate. Fig. 2, drawn by Lusa Zheglova, captures our representation of the quantum teleportation protocol. It illustrates classical communication in the foreground. In the background, it shows the quon representation of a Bell state, a CNOT gate, the Fourier transform, a measurement, and Pauli matrices (corresponding to Kitaev’s map) used in the recovery map. Together, they give the widely used quantum teleportation protocol of Bennett et al. (22), and illustrate its elegant 3D topological interpretation. The details can be found in ref. 1.
This quon language L3 provided an opportunity to collect ideas and to think about their meaning. It became apparent that the quon language L3 was important on its own, as it could have applications in other areas of mathematics and physics besides quantum information. Furthermore, the algebraic identity between |GHZ⟩|GHZ⟩ and |Max⟩|Max⟩ given by the rotation of the diagrams could provide insight in other subjects through use of other simulations.
What is the meaning of this interesting algebraic identity? It actually is the Verlinde formula. Mathematically, one can generalize the pictorial construction to define GHZ and Max on higher-genus surfaces. Then the pictorial Fourier duality between GHZ and Max leads to the generalized Verlinde formula (29) for any MTC on the genus gg surface,Maxn,g=𝔉S⊗nGHZn,g,⟹dim(k→,g)=∑k(∏i=1nSki,k)Sk,02−n−2g.Maxn,g=𝔉S⊗nGHZn,g,⟹dim(k→,g)=∑k(∏i=1nSki,k)Sk,02−n−2g.Here, on the first line, nn is the number of quons, 𝔉S𝔉S is the SFT, and, on the second line, k→=(k1,k2,…,kn)k→=(k1,k2,…,kn) are punctures (or marked points) on the genus gg surface, SS is the modular transformation, and dim is the dimension of the associated (moduli) space.
This pictorial Fourier duality also coincides with the duality of graphs on the sphere. The dual graph of the tetrahedron is also a tetrahedron. Applying the quon language to this graphic duality, we obtain a general algebraic identity Eq. 8 for 6j6j symbol self-duality. With X¯X¯ denoting the dual object to XX in an MTC,|(X6X3X5X2X4X1)|2=∑Y→(∏k=16SXkYk)|(Y1Y2Y3Y4Y5Y6)|2.|(X6X3X5X2X4X1)|2=∑Y→(∏k=16SXkYk)|(Y1Y2Y3Y4Y5Y6)|2.[8]In the special case of quantum SU(2)SU(2), this was discovered by Barrett (30), based on an interesting identity of Roberts (31). The general formulation and proof of Eq. 8 is in section 6 of ref. 32.
For each graph on a surface, the graphic duality gives a new algebraic identity for MTCs in quon language. Most of these identities have virtual meanings. Each graph can also be considered as a linear functional generalizing integration. The pictures generalize the symbol ∫∫ and capture additional pictorial relations, such as graph duality mentioned above. It would be interesting to understand these new identities and integrations in some new R.
We are grateful to Lusa Zheglova for the use of her drawing. We thank the Operator Algebras: Subfactors and their Applications (OAS) program at the Isaac Newton Mathematical Institute for hospitality. This research was supported in part by Grants TRT0080 and TRT0159 from the Templeton Religion Trust.
Footnotes
Author contributions: A.M.J. and Z.L. designed research, performed research, and wrote the paper.
Reviewers: J.D.B., Skolkovo Institute of Science and Technology; J.E., Mathematics for America; and A.V., University of Newcastle.
The authors declare no conflict of interest.
↵∗The argument depends on the inscribed angle theorem: Three points B, D, C on a circle determine the angle BDC = θθ, and the angle BOC = ψψ, where O is the center of the circle. Then, always, 2θ=ψ2θ=ψ. The proof is as follows: In the special case that BD passes through O, symmetry shows that the triangle COD is isosceles. As the sum of angles in a triangle is ππ, both 2θ2θ and ψψ complement the same angle, so 2θ=ψ2θ=ψ. The general case then follows by drawing the diameter through BO, and considering the sum or difference of two special cases.
This is an open access article distributed under the PNAS license.
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