1 Boson-Fermion Correspondence and JTP

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1 1 Boson-Fermion Correspondence and JTP Recall that the Jacobi Triple Product (JTP) is the following mathematical identity n= q n2 /2 z n = (1 q n )(1 + zq n 1/2 )(1 + z 1 q n 1/2 ). n=1 Over the semester, we have seen that this identity can be proved using different methods (for example, via the q-binomial theorem) and has many applications (a special case is Euler s pentagonal number theorem). This demonstrates the immense amount of symmetry encoded in this identity. Today, we will show JTP to be a consequence of the correspondence between bosons and fermions. This idea developed by Richard Borcherds in Bosons and Fermions Let us begin by defining the two classes of objects in this model. A fermion is a particle which satisfies the Pauli exclusion principle, the rule that two objects cannot co-exist in the same state at the same location at the same time. They also have half spins (, 1/2, 1/2, 3/2, ), a consequence of the Pauli exclusion principle for reasons not fully understood. Quarks and leptons are fermions. You are likely more familiar with composite quarks, like protons and neutrons. Electrons are a kind of lepton. A boson is a particle which does not follow the Pauli exclusion principle and, therefore, has integer-valued spins ( 1, 0, 1, ). Two famous examples are photons and Higgs bosons. Bosons are often referred to as force carriers. In particle physics, forces between particles arise from an exchange of bundles of energy of a particular kind of field. For example, a photon carries energy of an electromagnetic field. One way to organize these terms in your mind is with the diagram on the following page. In the standard model, everything is broken down into bosons and fermions. Fermions are further broken down into leptons and quarks. The examples are those on the bottom. Math 597B: Q-Series and Partitions 1 of 7

2 Everything Bosons Fermions Leptons Quarks Higgs Bosons Photons Electrons Composite Quarks Neutrons Protons Physicists like to summarize all the different kinds of particles in the following table: Math 597B: Q-Series and Partitions 2 of 7

3 While fermions and bosons are distinct objects, there exists a correspondence between them that is describe by JTP. This description arises from a representation called the Dirac Sea. As we will see, fermion systems are represented as particular configurations, or states, while bosons are represented by a charge-preserving transformation. 1.2 Dirac Sea Let each value i Z + 1/2 be an energy level. Each energy level is either occupied (1) or not (0). If it is occupied, we call that energy level a particle. Otherwise, it is a hole. The energy level of each particle is exactly i, the value of that level. The configuration where exactly all negative energy levels are occupied is called the vacuum state. This is the lowest possible energy state and we define it to be zero. We represent it as follows: A (fermionic) state, S, is called admissible if it is different from the vacuum state on finitely many levels. Associated with each admissible state is Q(S): The charge of S, which is the number of occupied levels relative to the vacuum state, meaning we add the number of positive particles and subtract the number of negative holes. H(S): The total energy of the state relative to the vacuum state, meaning we add the positive energy levels and subtract the negative energy levels (rendering them positive values). Exercise 1.1. Consider the following state. Is it admissible? If not, explain. If so, find Q(S) and H(S). Assume all states lower than 9/2 are 1 and all states lower than 9/2 are 0. Math 597B: Q-Series and Partitions 3 of 7

4 Let us now consider the partition function of these fermionic systems as represented by this model. Z(q, z) = z Q(S) q H(S) S admissible Fermionic Evaluation of Z(q, z) In the fermionic version, each level is either occupied or empty due to the Pauli exclusion principle. For i Z + 1/2, Z(q, s) = (1 + z +1 q e i ) + z i>0 i<0(1 1 q e i ) (1) = (1 + zq n 1/2 )(1 + z 1 q n 1/2 ) = ( z q; q) ( z 1 q; q) (2) n=1 The first product of (1) describes the particles in positive levels while the second product describes the holes present in the negative levels. In equation (2), we plug in the energy levels and simplify. Let s consider some of the terms in this series to convince ourselves this is correct. We get Z(q, s) = 1 + zq 1/2 + z 1 q 1/2 + q +. The first term corresponds to the vacuum state, for which there is only one configuration. The second term corresponds to the state in which there is a positive charge and an energy level of 1/2. There is only one possible state, namely Similarly, the third term corresponds to the state in which there is a negative charge and and an energy level of 1/2. This state is only depicted in one way Exercise 1.2. Draw the state corresponding to the term q (i.e. zero charge and energy level 1) Math 597B: Q-Series and Partitions 4 of 7

5 Bosonic Evaluation of Z(q, z) To adapt bosons to this model, we will describe them as transformations of constant charge from vac(n) to S n, where n represents the charge. The following is a construction of an admissible state. For n Z, we describe the vacuum state of n as the configuration that achieves charge n with the lowest energy level. That is, vac(n) := all levels < n are occupied. The description of the transformation from vac(n) to state S n is as follows: Start with the state vac(n). Move the top occupied level (i = n 1/2) to n 1/2+λ 1, the top occupied level in the desired state S n. Move the next occupied level (i = n 3/2) to n 3/2 + λ 2. And so on. Put another way, an admissible state is one where λ 1 λ 2 > 0 and each λ i Z. We now present the following exercise as an example. Exercise 1.3. From vac(1), construct the state in exercise 0.1. What are the values of λ 1, λ 2,? Here is vac(1) and the state from exercise 0.1: Math 597B: Q-Series and Partitions 5 of 7

6 All admissible boson states of charge n are described by these finite dimensional vectors λ Sn = (λ 1, λ 2, ). Notice that λ Sn is one partition of λ Sn := i λ i. The energy for state S n with fixed charge n is H(S n ) = n2 2 + λ i The first term in this identity corresponds to the total energy of the vacuum state. The second term is the additional energy from the particular configuration. We can now evaluate the partition function under the bosonic formulation (that of transformations). Z(q, z) = n Z z n S n admissible q n2 /2+ λ i = z n n Z λ Sn = ( ) z n q n2 /2 p(m)q m n Z m 0 1 = z n q n2 /2 (q; q) n Z q H(Sn) (3) Note that from equation (5) to (6), we are simply using the identity of the partition function p(n)q n = n=1 1 (1 q n ). We have now defined the partition function of admissible states by their description and by a transformation. If we equate the bosonic (6) and fermionic (2) evaluations, we get 1 z n q n2 /2 = ( z q; q) ( z 1 q; q), (q; q) n Z which is precisely the Jacobi Triple Product identity. 2 Remarks At the crux of this model is the notion of spin. But why do we represent the spins of bosons as elements in Z and the spins of fermions as elements in Z + 1/2? We know that this fact relates to the Pauli exclusion principle, but that doesn t shed much light on this question. Math 597B: Q-Series and Partitions 6 of 7 (4) (5) (6)

7 Let s revisit the Pauli exclusion principle. It was described earlier as a rule where two objects cannot co-exist in the same state at the same location at the same time. This phrasing is not rigorous nor mathematical. An equivalent definition is that a system of particles is antisymmetric with respect to the exchange of particles. This gives us a description from which we can build a mathematical structure; however, why are they equivalent? For simplicity, let s consider two particles, x and y, which have only two different spins (so you can imagine that they are electrons). Let 1 and 0 represent the two states spin up and spin down, respectively. Then the system of two particles is described by ψ = a a a a If you have heard of the wave function, ψ is one way to represent it (there are many different ways!). The information for ψ is encoded in the coefficients in this expression. Let A(x, y) represent this (second-order) tensor for given configurations of x and y. The Pauli exclusion principle is the statement that A(x, x) = 0, meaning ψ = 0. This is not a solution with physical meaning, so we say it is not possible. For the second formulation, we d like to show that this statement implies A(x, y) = A(y, x). This turns out to be a straightforward exercise from functional analysis. That is, we consider 0 = A(x + y, x + y) = A(x, x) + A(x, y) + A(y, x) + A(y, y) = A(x, y) + A(y, x) To justify why A can expand as it does, the reader need only remember that A is a tensor. It should also be clear that antisymmetry implies A(x, x) = 0. Therefore, these are equivalent notions. Na lively, exchange properties of particles should have little to do with spin properties. However, the spin-statistics theorem (taken from Wikipedia) makes this connection. Theorem 2.1. The wave function of a system of identical half-integer spin particles changes sign when two particles are swapped (antisymmetry). For integer spin particles, the sign does not change (symmetry). Unfortunately, rigorous proofs of this statement require the reader to be familiar with mathematical machinery not required for this course. The core observation for the proof is that if the operators used to describe these systems satisfy antisymmetric relations (equivalent to anticommutivity), then they must have spectrums in Z + 1/2. Math 597B: Q-Series and Partitions 7 of 7