Torsors and affine spaces, or: I keep forgetting where I started

The topic of this post is torsors, which occur naturally throughout mathematics and physics whenever we have natural notions of relative — but not absolute — sizes, positions, temperatures and so forth. This post owes a lot to this 2009 post by John Baez, and so I’ll shamelessly steal borrow some (but not all) examples from him:

• Even after choosing a unit of voltage (e.g. the SI unit, the volt), it makes no sense to say that the voltage at some point p in a circuit is, say, 7V. It does, however, make sense to say that the voltage at p, relative to that at another point q, is 7V. Relative to that chosen reference value at q, voltages are real numbers — but we are free to change the reference point, and without a reference point, voltages are not real numbers, but they do live in a real torsor.
• A good geographical example is longitude: we habitually describe longitude on Earth as longitude using units of degrees and relative to the Greenwich meridian. However, the choice of the Greenwich meridian is basically arbitrary, and if we were to change to the Cairo, Paris, or Washington meridian instead, it would not change the difference in longitude between any two points on Earth. Longitudes are not elements of the circle group S1 (angles); it is longitude differences that are angles in S1, whereas longitudes live in an S1-torsor.
• Both the previous two examples indicate that, whatever a “torsor” is, it’s like a well-behaved algebraic structure (like the real line ℝ or circle group S1) in which the usual reference point, the origin (0 in ℝ and 1 in S1) has been “forgotten”. The usual setting of plane geometry going all the way back to ancient Greece is like this, too: there is no preferred origin for plane Euclidean geometry: you are free to work relative to one corner of your graph paper, or relative to some point in the ground in your Athenian sand-pit.

So… what’s going on here?

Null vectors and spinors

My recent reading on the topic of spin and angular momentum in quantum mechanics led me to the concept of a spinor. It turns out that spinors are fearsomely nasty objects to wrap one’s head around in full generality, requiring Clifford algebras and other ingredients, although the three-dimensional case is quite accessible and is described below. The (slightly unhelpful) heuristic is that spinors behave like vectors except that they change sign under rotation through an angle of 2π — a somewhat confusing property that will be made clearer later.

First, some general notions: let V be a vector space over a field K, equipped with a bilinear map b: V × VK and hence a quadratic form q: VK given by q(v) ≔ b(v, v) for all v ∈ V. A vector vV is called a null vector or isotropic if q(v) = 0. Recall that the standard Euclidean bilinear form on ℝn is

$\displaystyle b \bigl( (x_{1}, \dots, x_{n}) , (x'_{1}, \dots, x'_{n}) \bigr) \equiv x \cdot x' := \sum_{j = 1}^{n} x_{j} x'_{j} .$

This bilinear form has no non-trivial null vectors (i.e. the only null vector is the zero vector), but two close relatives of the Euclidean bilinear form do have interesting null vectors.

[Somehow, WordPress deleted large chunks of this post. Apologies! If the text below differs from what you saw earlier, then blame the post-deletion restoration effort.]

These are some notes, mostly for my own benefit, on annihilation, creation, and ladder operators in quantum mechanics, with a few remarks towards the end on angular momentum, spin and Clebsch–Gordan coefficients.

First, the abstract definition: if T, LV → V are linear operators on a vector space V over a field K, then L is said to be a ladder operator for T if there is a scalar cK such that the commutator of T and L satisfies

$\displaystyle [T, L] := TL - LT = cL.$

The operator L is called a raising operator for T if c is real and positive, and a lowering operator for T if c is real and negative.

The motivation behind this definition is that if (λv) ∈ K × V is an eigenpair for T (i.e. Tv = λv), then a quick calculation reveals that (λ + cLv) is an eigenpair for T:

$\displaystyle T(Lv) = (TL)v = (LT - [T,L])v = LTv + cLv = (\lambda + c) (L v).$

Ladder operators come up in quantum mechanics because many of the elementary operations on quantum systems act as ladder operators and increase or decrease the eigenvalues of other operators. Those eigenvalues often encode important information about the system, and the increments and decrements provided by the ladder operators often come in discrete, rather than continuous, values. Annihilation and creation operators are a prime example of this phenomenon.

Preprint: Stratified graphene-noble metal systems for low-loss plasmonics applications

Lauren Rast, Vinod Tewary and I have just posted to the arXiv a preprint of our joint paper “Stratified graphene-noble metal systems for low-loss plasmonics applications”, which has been accepted for publication in Physical Review B later this year.

Graphene — which is basically just a sheet of carbon graphite just a few atomic layers thick — is often hailed in the press as a “wonder material”, usually because of its material properties such as its exceptionally high strength-to-weight ratio. Another area of interest, however, is the study of the electronic and optical properties of graphene.

Graphene has a very high carrier mobility, meaning that if you excite it by shooting some light at it, then the electrons in the graphene sheet will wiggle about in a very pleasant manner (called a plasmon) and thereby transmit a large chunk of the energy contained in the incident light as electrical current through the graphene sheet. This is exactly what you want a solar cell to do if you want to generate electricity from sunlight — and a graphene-based “thin film” solar cell would be much thinner and lighter than today’s solar cells.

But there’s a tiny problem: graphene doesn’t do this very well at visible frequencies of light. Silver, on the other hand, does respond very sympathetically to visible light, but is much more lossy. But, a-ha! What is one were to make a graphene and silver sandwich? Could one have the best of both worlds, i.e. absorption in the visible range like silver, and low electronic energy loss like graphene?

Our paper is a theoretical and numerical exploration of such sandwich structures, and we show that such composites do go a long way towards this best of both worlds.

Time Evolution of Quantum States

My previous posts on quantum mechanics, and specifically uncertainty principles, were essentially about the quantum state ψ of the system at a fixed time. This post concerns the time evolution of quantum systems.

The key evolution equation here is a Hilbert-space-valued ordinary differential equation that consists of two key ingredients: the differential operator i ℏ ∂t (the energy operator) and a Hamiltonian operator H that describes the energetics of the system. (Sorry, “H” has changed from denoting a Hilbert space to denoting the Hamiltonian operator. So many concepts, so few letters…) The time-dependent Schrödinger equation with Hamiltonian H is

i ℏ ∂tψ = Hψ.

Often, the Hamiltonian is itself a differential operator: a good example is the Schrödinger equation for a single non-relativistic particle of mass m moving in a scalar potential V: ℝn → ℝ:

i ℏ ∂tψ = − (ℏ2 ⁄ 2m) Δψ + V ψ

The Hamiltonian in this case is the familiar “kinetic energy + potential energy” one. V is fairly obviously the potential energy term. The kinetic energy term is the usual “½ × mass × velocity2” but in an interesting form: it is 1⁄2m times the dot product of the momentum operator P := −iℏ∇ with itself, hence the “− (ℏ2 ⁄ 2m) Δ”, where Δ denotes the spatial Laplacian.

Simply put, the time-dependent Schrödinger equation is an absolute pain to solve in all but the simplest settings. Life gets slightly easier if we search for so-called “stationary states”, which, despite the name, are not actually solutions ψ that are constant in time, but rather are eigenstates of the Hamiltonian operator H. (In a sense to be made precise in a moment, these stationary states are constant from the point of view of any observation operator, even though they are themselves non-constant.)

A stationary state is a solution ψ to the time-independent Schrödinger equation, i.e. to the eigenvalue problem

Hψ = Eψ.

Here the eigenvalue E ∈ ℝ is the energy of the quantum state ψ. If H is compact and self-adjoint, then the usual remarks about there being at most countable many eigenvalues apply. In any case, the state ψ with least energy E is called the ground state of the system, and E is called ground state energy or zero-point energy of the system; the other eigenstates are called excited states.

Note that stationary states are not actually constant in time: substituting the definition into the time-dependent Schrödinger equation reveals that a stationary state ψ satisfies the (complex) ordinary differential equation

i ℏ ∂tψ = Eψ,

to which the solution, given ψ at some initial time t0, is

ψ(t) = eiE(tt0)⁄ℏ ψ(t0).

So, stationary states actually evolve by “rotation”, with “angular velocity” E⁄ℏ; but note that the probability density |ψ(t)|2 is independent of time t. Indeed, if A is any linear observation operator, then

Aψ(t) = 〈Aψ(t), ψ(t)〉 = eiE(tt0)⁄ℏ e+iE(tt0)⁄ℏAψ(t0), ψ(t0)〉 = 〈Aψ(t0).

The Spectral Theorem and the Fredholm Alternative

My recent reading on uncertainty principles in quantum mechanics, where a lot of things are expressed in terms of operators on Hilbert spaces, has required me to refresh my memory with the basic facts about the spectral theory of self-adjoint and compact operators on Hilbert and Banach spaces.

Let H and K be Hilbert spaces and T: HK a linear operator. The adjoint of T is the linear operator T: KH defined by

Tx, y〉 = 〈x, Ty

for all xH and yK. An operator T: HH is said to be self-adjoint if T = T, i.e.

Tx, y〉 = 〈x, Ty

for all x, yH. If we fix bases of H and K, then the matrix of T is the conjugate transpose of that of T. Self-adjoint operators have the following very nice eigenvalue/eigenvector property:

Theorem. If T: HH is a self-adjoint operator on a Hilbert space H, then all the eigenvalues of T are real and the eigenvectors corresponding to distinct eigenvalues are orthogonal in H.

Another property that plays very nicely with self-adjointness is compactness:

Let X and Y be Banach spaces. A linear operator T: XY is said to be a compact operator if, for every bounded set BX, the image T(B) ⊆ Y is a compact subset of Y. Equivalently, every norm-bounded sequence (xn)n∈ℕ in X is such that the image sequence (Txn)n∈ℕ in Y has a convergent subsequence.

Clearly, compact operators are always bounded. Operators that are both compact and self-adjoint are especially nice:

Theorem. If T is a compact self-adjoint operator on a Hilbert space, then either ‖T‖ or −‖T‖ is an eigenvalue of T. Furthermore, the set of all eigenvalues of T (the point spectrum of T) is either finite or consists of a countable sequence tending to 0.

[Incidentally, while I had no problem remembering the “furthermore” part about the point spectrum, I had completely forgotten the simpler fact that, up to sign, the operator norm of T is always one of its eigenvalues.]

The next result, the Spectral Theorem, is essentially the “diagonalization” result for compact, self-adjoint operators, analogous to results from finite-dimensional linear algebra on the diagonalization of (conjugate-)symmetric square matrices.

Spectral Theorem. If T is a compact self-adjoint operator on a Hilbert space H, then there exists a finite or infinite orthonormal sequence (xn) of eigenvectors of T with corresponding real eigenvalues (λn) such that, for all xH,

Tx = ∑n λnx, xnxn.

Note that the orthonormal sequence of the Spectral Theorem can be extended by the usual basis extension and orthogonalization procedures to an orthonormal basis of H.

The other key fact about eigenvalue problems for compact (not necessarily self-adjoint) operators is the Fredholm Alternative, which in finite-dimensional settings follows from the Rank-Nullity Theorem. The Fredholm Alternative asserts a dichotomy: either λ is an eigenvalue of T, or else it lies in the domain of the resolvent of T. More precisely,

Fredholm Alternative. Let T: XX be a compact operator on a Banach space X, and let λ ∈ ℂ be non-zero. Then precisely one of the following holds true:

• (eigenvalue) there is a non-zero solution xX to the eigenvalue equation Tx = λx;
• (resolvent) the operator Tλ has bounded inverse (Tλ)−1.

Uncertainty Principles II: Entropy Bounds

In my previous post on uncertainty principles, the lower bounds were on the standard deviations of self-adjoint linear operators on a Hilbert space H. The most general such inequality was the Schrödinger inequality

σA2 σB2 ≥ | (〈{A, B}〉 − 2〈A〉〈B〉) ⁄ 2 |2 + | 〈[A, B]〉 ⁄ 2i |2,

and the classic special case was the (Kennard form of) the Heisenberg Uncertainty Principle, in which A and B are the position and momentum operators Q and P respectively:

σP σQ ≥ ℏ⁄2.

One problem with the Robertson and Schrödinger bounds, though, is that the lower bound depends upon the state ψ; this deficiency is obscured in the Kennard inequality because the commutator of the position and momentum operators is a constant, namely −iℏ. It would be nice to have uncertainty principles for more general settings in which the lower bound does not depend on the quantum state. Also, we don’t have to restrict ourselves to standard deviation as the only measure of non-concentration of a (measurement of a) distribution.