Bosons, Fermions and Clifford Algebras

A minus sign can make a huge difference. Einstein discovered that the difference between space and time is all due to a minus sign.

Another amazing fact is that the difference between ‘matter particles’ (or more precisely fermions, like electrons, quarks, etc.) and ‘force particles’ (bosons, like photons, gluons, etc.) is mainly due to the fact that when you switch two fermions their quantum state gets multiplied by -1, while when you switch two bosons it get multiplied by 1.

This was discovered by Pauli, who realized that there must be some reason why the electrons in atoms go into ‘shells’ – why all the electrons in a big atom like iron don’t all fall into the same lowest-energy state. The reason is that if two electrons were in the same state, switching them would do nothing but also multiply that state by -1: a contradiction. This rule, that fermions can’t be in the same state, is called the Pauli exclusion principle.

Bose and Einstein realized that on the contrary, bosons actually like to be in the same lowest energy state at low temperatures! This is called Bose-Einstein condensation. Similarly, a laser beam has many photons in the same state.

Later people realized that if we replace vector spaces (like the Hilbert space of quantum states of some system) by ‘super vector spaces’, where every vector is a sum of a bosonic and fermionic part, we can impose a rule saying that switching two fermionic vectors should always introduce an extra minus sign.

It turns out that this rule is not arbitrary—it’s mathematically very natural and it’s lurking around all over in mathematics, even in contexts that superficially have nothing to do with bosons or fermions!

A nice example of how ‘super’ thinking can clarify things is in the study of Clifford algebras and their representations.

In the Clifford algebra Cliffₙ we start with the real numbers and then throw in n anticommuting square roots of -1. For example:

Cliff₀ = ℝ, Cliff₁ = ℂ, Cliff₂ = ℍ (the quaternions), etc.

With the quaternions, once you throw in i and j with i² = j² = -1 and make them anticommute (ij = -ji) you get k = ij for free.

Each Clifford algebra has ‘representations’: roughly, real vector spaces where elements of the Clifford algebra act as linear operators. The most famous representations of Cliffₙ are the ‘pinor’ representations, which describe spin-1/2 particles in n-dimensional space along with how reflections act on these particles. You get all the other representations by taking direct sums of pinor representations.

For example, in 2d space Cliff₂ = ℍ has a representation on itself, and this is the only pinor representation in 2d space. All other representations of Cliff₂ are direct sums of this one – so its category of representations is the category of quaternionic vector spaces!

This chart shows the categories of representations of the Clifford algebras up to dimension 7. After that they repeat.

The symbol ≃ means that two algebras have equivalent categories of representations. For example, Cliff₆ is the category of 8×8 real matrices! So it’s not isomorphic to ℝ, but you can show they have equivalent categories of representations.

One last thing: Cliff₃ is isomorphic to ℍ⊕ℍ, meaning an element is a pair of quaternions. So a representation is a pair of quaternionic vector spaces – or equivalently, a quaternionic vector space that’s been ‘split’ as the direct sum of two pieces. That’s what I mean by ‘split’ here.

Do you see the surprising pattern in the above chart? It has bilateral symmetry across the diagonal line from Cliff₃ to Cliff₇!

Why such a weird diagonal line? It’s because we’re doing things a bit wrong! The Clifford algebras aren’t just algebras: they are ‘superalgebras’, meaning that every element is a sum of two parts, which we can jazzily call the bosonic and the fermionic part, and multiplication obeys these rules:

bosonic × bosonic = bosonic
bosonic × fermionic = fermionic
fermionic × bosonic = fermonic
fermionic × fermionic = bosonic

These rules are motivated by pure math and also what happens in nature when you combine bosons and fermions into bigger particles.

How do we make the Clifford algebras into superalgebras? We just decree that the square roots of -1 we throw in are fermionic. In Cliff₂ this means that i and j are fermionic and k is bosonic. That may seem weird, but that’s because we’re getting the quaternions from studying 2-dimensional space, which is also a bit weird. (In 3d space it turns out that the quaternions are the bosonic part of Cliff₃, and this is closer to Hamilton’s original thoughts.)

Believe it or not, working with superalgebras and their super-representations takes our chart and rotates it a bit, so the weird diagonal line becomes a vertical line!

It’s super time!

A ‘super vector space’ is a vector space V that’s the direct sum of two subspaces:

V = V₀ ⊕ V₁

We call vectors in V₀ bosonic or ‘even’ and vectors in V₁ fermionic or ‘odd’. And a superalgebra can have a super-representation on a super vector space. It’s just like an ordinary representation except that when we let guys in our superalgebra act on guys in our superalgebra, we require these by now familiar rules:

bosonic × bosonic = bosonic
bosonic × fermionic = fermionic
fermionic × bosonic = fermonic
fermionic × fermionic = bosonic

The chart above shows the categories of super-representations of the Clifford algebras. Amazingly, it’s just like the previous chart rotated an eighth of a turn clockwise! Now the axis of symmetry is the vertical line!

This doesn’t explain yet why the symmetry exists in the first place, but it’s a step in the right direction.

As a quick sanity check, think about super-representations of Cliff₀. The chart claims these are ‘split real vector spaces’. Why is that true? Cliff₀ is just ℝ, with every element bosonic. So a super-representation of Cliff₀ is just a super-vector space where you can multiply by real numbers. But this is just a super-vector space. But a super-vector space is just the same as a split real vector space: it’s a vector space V split as a direct sum of two parts:

V = V₀ ⊕ V₁

which we call fermionic and bosonic. So yes, the category of super-representations of Cliff₀ is just the category of split real vector spaces!

So at least that case checks out.

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