Friday, March 19, 2021

Supremacy of physics-like proofs

They lead to the greatest generalizations, deepest understanding

Theoretical physics is close enough to pure mathematics and the subjects greatly influence one another (although this intercourse was hyped as politically incorrect by several waves of misguided ideologues in mathematics such as the Bourbaki group). Many interesting findings in mathematics are inspired by physics and physics-like systems; physics needs the ideas and tools of mathematics to understand things rigorously.

One aspect of this relationship is the existence of mathematical proofs that are either originally found by physicists; or that are made simpler and intuitive by physicists. There are many examples but let us pick the identity that Carl Jacobi, a classical liberal who died young, incredibly proved in 1829.

Let us define a function of \(w\in\CC\)\[ f_R(w) = 8 \prod_{m=1}^\infty \zav{ \frac{1+w^m}{1-w^m} }^8. \] We may call it the "Ramond partition sum", just for fun. It is an infinite product over positive integers \(m\). The factors in the product are eighth powers of a ratio. The numerator \(1+w^m\) and the denominator only differ by the relative sign. And then there is the overall factor of eight.

Now, let us define a similar function, the "Neveu-Schwarz partition sum", which has a prefactor \(1/2 \sqrt w\) instead of eight, the rest is a difference between two similar infinite products, and the numerators are slightly shifted relatively to one in \(f_R\): they have \(1\pm w^{m-1/2}\) with the extra shift by one-half. \[ f_{NS}(w) = \frac{1}{2\sqrt w} \zzav{ \prod_{m=1}^\infty \zav{ \frac{1\!+\!w^{m-1/2}}{1-w^m} }^{\!8}\! - \prod_{m=1}^\infty \zav{ \frac{1\!-\!w^{m-1/2}}{1-w^m} }^{\!\!8} }. \] Surprisingly enough,\[ f_{R} = f_{NS} \] and this identity was called "a rather obscure formula" (aequatio identica satis abstrusa). The expressions are complicated – infinite products of eighth powers etc. – so it surely looks terrifying. But maybe the identity can be proven by some totally straightforward manipulation? No, it cannot. Shockingly, it only works when the exponent is eight; when the shifts are one-half; there exists no straightforward way to generalize the identity.

It's incredible that in 1829, Jacobi considered expressions like that at all – and he even managed to prove the identity. He had to create a mental framework in his head that was a part of the usual "thinking of a perturbative string theorist". What is the most characteristically physics-inspired proof of the identity?

I have already leaked the spoilers. The functions are "partition sums". These objects are sums of the form\[ \sum_{k} \exp(-\beta H_k) = \sum_k w^{H_k}, \quad k\equiv \exp(-\beta) \] where the sum goes over "states" labeled by \(k\) and the summand is the probability that the physical system at the absolute temperature \(T = 1/\beta\) chooses the state \(k\). Partition sums (which may also be called "logarithms of free energy", up to some decoration) appear as the denominators in various "expectation values" in thermodynamics. The numerators are similar but in their summand, the exponential is multiplied by another property that depends on the state \(k\) such as \(H_k\) itself (the energy of the state).

The states \(k\) may be identified with basis vectors of a Hilbert space and in the "simplest" cases including this complicated formula, they are labeled by a collection of (infinitely) many integers obeying some conditions and inequalities (some "discrete data") while the energy \(H_k\) is some rather linear combination of all these integers. In this case, it is something like\[ H_k = \sum_m m N_m.\quad k\equiv (N_1,N_2,N_3,\dots) \] Partition sums therefore routinely look like functions that are infinite products over all integers \(m\). And the factors for a given \(m\) often end up looking as similar ratios of functions that resemble \(1-w^m\) above.

We would like to get more details about this identity. Why is it correct, physically speaking? Because both \(f_R\) and \(f_{NS}\) (they are equal!) are partition sums for a string in superstring theory. The dynamics of the string is an infinite-dimensional harmonic oscillator. These infinitely many dimensions are labeled by this very integer \(m\) and this integer may be identified as a "number of waves along a string" or the index in the Fourier expansion. (The simplest string that produces the identity directly happens to be an open string but morally equivalent identities exist for closed strings and some modified strings.)

So the state of the string is encoded in the number of excitations \(N_m\) for every integer \(m\in\ZZ\). Sometimes \(N_m\) is any non-negative integer (in the case of bosonic oscillators); sometimes \(N_m\) is either zero or one (fermionic oscillators, the Pauli exclusion principle demands the social distancing and two excitations in one place is already too many). The different directions of the infinite-dimensional oscillator labeled by \(m\) are independent from each other which means that the total Hilbert space is a tensor product of factor Hilbert spaces over \(m\); and the total Hamiltonian for a basis vector \(k\) is simply a sum over \(m\), \(\sum_k H_{k\to m}\) where \(k\to m\) means that the energy only depends on \(N_m\), a subset of the information contained in \(k \equiv (N_1,N_2,N_3,\dots)\).

Because the energy \(H_k\) of a state of the string is a sum over \(m\), the partition sum (which sums the exponentials of the energy, with some temperature-related coefficient in the exponent) ends up being the product over \(m\). The previous sentence is an omnipresent idea that is used in statistical physics (the microscopic explanation of thermodynamics) many times a day (so it was understood well before strings became important).

In the partition sums, we know where the integer \(m\) comes from: it is the Fourier index labeling the number of waves on the string in the Fourier decomposition of functions like \(x^\mu(\sigma)\). We know why there is an infinite product over \(m\): it's because the Hilbert space is a tensor product and the energy is a sum over \(m\). And we may also understand why the factors from a given \(m\) look like \((1+w^m)/(1-w^m)\). Those are smallest possible yet similar partition sums for harmonic oscillators.

In the fermionic harmonic oscillator case, you only have two states, \(N_m=0\) and \(N_m=1\), which may be considered occupied and unoccupied. The partition sum for a fermionic harmonic oscillator therefore reduces to \(1+\exp(-\beta H_{N_m=1})\), the sum of one (the exponential of zero) and the exponential of the energy of the excited state. For the bosonic harmonic oscillator, the occupation number may be \(N_m = 0,1,2,\dots\) i.e. an arbitrarily high non-negative integer. The partition sum ends up being a geometric series and that is why you have \(1/(1-w^m)\) there as well.

So both partition sums, \(f_R\) and \(f_{NS}\), are rather similar and correspond to some partition sums of infinite-dimensional harmonic oscillators with directions for each positive integer \(m\). For each number of waves along the string, \(m\), you will have eight bosonic oscillators and eight fermionic oscillators (because things like \(1\pm w^m\) are "eight times" in the numerator and "eight times" in the denominator).

At this point, I should already say that "eight bosons and eight fermions" (only physical, "transverse" excitations are counted) is exactly what you get in the \(D=10\) superstring (which has to propagate in a ten-dimensional spacetime for string theory to be consistent). Out of the ten space dimensions \(X^\mu(\sigma)\) on the two-dimensional string world sheet labeled by \(\sigma,\tau\), one spacetime dimension may be eliminated by identifying it with \(\tau\) in some way; and another condition is automatically imposed on us once we do it which eliminates another dimension. Roughly speaking, we might say that two of the ten dimensions are identified with \(\sigma,\tau\) and that is why eight dimensions \(X^i(\sigma)\) are left on the string. And they also have the eight corresponding fermionic fields \(\psi^i(\sigma)\). (Here we use the fermions which are spacetime vectors, it is the NSR formalism; the same string is equally conveniently described by the Green-Schwarz or GS formalism where the fermions \(\theta^a(\sigma)\) are spacetime \(SO(8)\) spinors.)

Finally, we may switch to finer details such as the difference between \(f_R\) and \(f_{NS}\). The former has a coefficient eight in front of everything because it is the degeneracy of the ground state i.e. from the fermionic degrees of freedom with \(m=0\) (zero waves along the string). The latter has the numerators shifted by \(1/2\) because we Fourier expand not periodic fermions but antiperiodic ones. So the resulting partition sums for the fermions end up being \(1\pm w^{m-1/2}\), with the exponents that are half-integer instead of integer. Finally, the Neveu-Schwarz partition sum also has the square root because the energy \(H_k\) must be shifted by an overall constant; and this partition sum is a difference of two terms because the physical spectrum consists of one-half of the states only; the other one has to be eliminated by a GSO projection. The sum over the physical spectrum may be achieved by inserting a projection operator \((1+(-1)^F)/2\) and the effect of the \((-1)^F\) is to change the sign in the numerator (the sign coming with each fermionic excitation) only. The functions \(f_R,f_{NS}\) differ by the (anti)periodicity along the \(\sigma\)-direction while the two terms in \(f_{NS}\) differ by the (anti)periodicity in the periodic world sheet time \(\tau\)-direction (the switch to the other periodicity is equivalent to the insertion of \((-1)^F\)).

The physical interpretation of Jacobi's functions is a bit complicated – and maybe more complicated than big parts of the "uninspired", mathematical proofs. But when you do it correctly, everything agrees precisely. The two partition sums \(f_R\) and \(f_{NS}\) are partition sums of the superstring in two different sectors, "overall fermionic" (R) and "overall bosonic" (NS) sectors, and the identity says that at each level, the string has an equal number of fermionic excitations and bosonic excitations (these numbers at each level may be directly extracted as coefficients in front of \(w^E\) in a power law expansion of the partition sums \(f\)). And these identities (between the number of bosonic and fermionic states) directly follow from the spacetime supersymmetry.

A funny, subtle issue is that the spacetime supersymmetry is not quite trivial in this NSR description of the superstring and that is why the identity expressing the "seemingly trivial" consequence of supersymmetry, the equal number of bosons and fermions, looks so hard. However, we may prove that the NSR superstring is supersymmetic, e.g. by showing that it is equivalent to the GS superstring where the supersymmetry (and the equal numbers) is trivial (because we may get bosonic states from fermionic ones by a simple action of \(\psi^a_0\) which doesn't change the energy at all).

How do we prove that the NSR superstring (with the vector-like fermions \(\psi^\mu(\sigma)\); and obfuscated spacetime supersymmetry) is equivalent to the GS superstring (with spinor-like \(\theta^a(\sigma)\); and with a manifest spacetime supersymmetry)? We may bosonize the 8 fermions into 4 equivalent bosons; rotate them through a rotation that encodes the "triality" in the 4-dimensional Cartan subalgebra of \(SO(8)\), and back fermionize the bosons into 8 different fermions. The bosonization and fermionization is a rather potent, universal enough trick but unlike the original "products of traces" rules at the top, this is rather characteristic for strings because it only works in the straightforward way when the "world volume" is a "world sheet" (it has 2 dimensions, like the stringy world sheet).

Schematicallly, the fermionization and bosonization works as follows: the field operators for fermions \(\theta_\pm(\sigma)\) are written as some (normal-ordered) exponentials \(\exp(\pm\phi(\sigma)/2)\) of the world sheet bosons; insert the imaginary unit \(i\) to the exponent if you want to get complex fermions that are Hermitian-conjugates of each other. Or inversely, the (derivatives) of the bosons may be written as \(\partial_\sigma\phi(\sigma) = \theta_+(\sigma)\theta_-(\sigma) \); it is a current for a \(U(1)\) symmetry and you need to integrate over \(\sigma\) to get \(\phi\) itself. Again, it looks extremely strange and awkward to a beginner (at least it did look strange to me) why such a seemingly important equivalence requires some quantum fields to be exponentiated and why the exponentiation becomes a quadratic function in the opposite direction! But for quantum fields, it simply does work like that. The fermionic field is a branch cut created in the quantum field of the boson; the boson is a kink in the fermionic field, and you may make these things rather rigorous. At the end, similar "CFTs" have to be equivalent because the set of simple enough CFTs is limited and if we consider CFTs with \(U(1)\) currents (which must be expressed in these sigma-derivative-of-phi or quadratic-fermionic ways), the objects are even more constrained. Aside from this local relationship (on the world sheet), you should be careful about the allowed (periodic or antiperiodic) boundary conditions for the bosons and fermions, and that is how you get the two terms in \(f_{NS}\); and how you get all the relative signs in the numerators, too.

The details may get intense and boring (and another level of complexity and boredom is required to make sure that the "physically correctly sounding" statements are translated to mathematically reliable, quantitative identities: lots of simple prefactors must be added, among other things) but we shouldn't get lost here. The point is that the proof of Jacobi's rather obscure identity depends on the representation of both functions as partition sums in the \(D=10\) superstring; on the identification of this string as an infinite-dimensional oscillator; on the infinite-product formula for the partition sum of such composite systems; on the geometric series for the bosonic oscillator partition sum; and on the bosonization or fermionization in two-dimensional conformal field theory. Some parts of the proof may be done a bit differently, of course.

Surprisingly, there are several other, completely different proofs of Jacobi's identity and most of them – so far? – don't look like physically meaningful operations (like "operations done with strings or string theory"). In some cases, the proofs could actually be "physical" even though people don't appreciate it yet. What do I mean by "physical"? I mean "understandable to the thinking of a physicist, using ideas that will be very important in many branches of physics". There may even be a deeper physical unification of all the distinct mathematical strategies to prove this identity (and maybe all identities in related branches of mathematics!) that sort of reflect the "consolidated wisdom of all string-like constructions in the Universe".

Whether it is the case or not, we know for sure that the superstringy proof of the identity has lots of advantages. We feel that "we know what we are doing". Because all the objects and steps have some physical interpretation – properties of quantum fields, their excitations, and their energies that we may "touch" (or that belong to the same universality class of things that we may really "touch"), and that leads to lots of extra possibilities. Because we may touch them, it is not hard to see that we may also twist, choke, consume, recycle, beat, fudge, and do many other things (verbs) with them.

One consequence is that the single identity relating two functions of \(w\in\RR\) or \(w\in\CC\) may be generalized to much grander propositions. Let me remind you that the complex identity encoded the "equal number of bosons and fermions on each level of the spectrum of a free superstring". This "equal number" follows from the spacetime supersymmetry but spacetime supersymmetry has many deeper, more extensive implications. It implies that all the scattering and correlation functions are related by some supersymmetry-based identities. That is true for infinitely many Green's functions on the world sheet that allow us to do "many things on the single string's world sheet simultaneously", but even for spacetime Green's functions that "also allow the strings to be split and merged". There are "infinity to some power" generalizations of Jacobi's identity, generalizations with one or infinitely many or "infinity to infinity" additional parameters and variables.

In some naive counting, you could say that Jacobi's identity is just an "infinitely small part of the wisdom" that we may get from the spacetime supersymmmetry in string theory, when expressed in different variables (and even "all of consequences of SUSY" may be argued to be an infinitesimal part of the wisdom of string theory). However, if we attach some pragmatic measures proportional to our learning time and our feelings, the fraction must be greater than "one over infinity" because once we look at the system from the right perspective, all the generalizations become rather obvious because it is "common sense" to make things on the string interact with each other, it's normal for strings to split and join, and it's normal to do all the usual things that physicists are imagining to do with particles and fields.

It is possible that whenever some proofs of mathematical theorems and identities of related kinds don't look physical, it either means that these theorems and identities aren't really deep; or we don't understand them properly yet. Einstein said that if you can't explain something to your grandmother, you don't understand it (did he really say it?). I must add that it is true for grandkids of a grandma-theoretical physicist. If you are a mathematician and you can't explain a proof to your grandma who is a theoretical physicist, then you don't really understand this piece of mathematics.

This assertion might be said to be a belief; or even an expression of the theoretical physicists' arrogance. But I view it differently; for me, this assertion is basically a tautology, a something that directly follows from my definition of the understanding. Physics is the enterprise of dealing with real objects and processes and their mathematical representations in all conceivable ways that are mathematically or physically possible (allowed by God or Nature), and especially the search for these laws (defining limits where you can move; truly unavoidable restrictions on your freedom as long as you stay in the physical world) themselves. And you need to get the power to play with the objects and processes in all allowed ways if you want to be in charge. Otherwise you are just an assistant fulfilling some very limited tasks in some very limited context(s) that someone has prescribed for you.

So that's why I think it's important to try to find physics-like proofs of mathematical theorems; and to try to find some hidden new physical ideas in mathematical proofs that look like non-physics-based ones. Even sociologically, I think that a greater fraction of the "mathematicians" who are dealing with certain topics in mathematics should be trained within the culture of theoretical physics – exactly because the culture of pure mathematicians may tend to downgrade the people into the intellectual working class if not slaves who were thrown into socially distanced, Covid-style cages.

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