Separability and quantum mechanics

Tuesday, Apr 21st
Fernando Barbero, CSIC, Madrid 
Title: Separability and quantum mechanics 
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by Juan Margalef-Bentabol, UC3M-CSIC, Madrid

Classical vs Quantum: Two views of the world

In classical mechanics it is relatively straightforward to get information from a system. For instance, if we have a bunch of particles moving around, we can ask ourselves: where is its center of mass? What is the average speed of the particles? What is the distance between two of them? In order to ask and answer such questions in a precise mathematical way, we need to know all the positions and velocities of the system at every moment; in the usual jargon, we need to know the dynamics over the state space (also called configuration space for positions and velocities, or phase space when we consider positions and momenta). For example, the appropriate way to ask for the center of mass, is given by the function that for a specific state of the system, gives the weighted mean of the positions of all the particles. Also, the total momentum of the system is given by the function consisting of the sum of the momenta of the individual particles. Such functions are called observables of the theory, therefore an observable is defined as a function that takes all the positions and momenta, and returns a real number. Among all the observables there are some ones that can be considered as fundamental. A familiar example is provided by the generalized position and momenta denoted as $q^i$ and $p_i$.

In a quantum setting answering, and even asking, such questions is however much trickier. It can be properly justified that the needed classical ingredients have to be significantly changed:

1. The state space is now much more complicated, instead of positions and velocities/momenta we need a (usually infinite dimensional) complex vector space $\mathcal{H}$ with an inner product that is complete. Such vector space is called a Hilbert space and the vectors of $\mathcal{H}$ are called states (up to a complex multiplication).
2. The observables are functions $A$ from $\mathcal{H}$ to itself that "behave well" with respect to the inner product (these are called self-adjoint operators). Notice in particular that the outputs of the quantum observables are complex vectors and not numbers anymore!
3. In a physical experiment we do obtain real numbers, so somehow we need to retrieve them from the observable $A$ associated with the experiment. The way to do this is by looking at the spectrum of $A$, which consists of a set of real numbers called eigenvalues associated with some vectors called eigenvectors (actually the number that we obtain is a probability amplitude whose absolute value squared is the probability of obtaining as an output a specific eigenvector).

The questions that arise naturally are: how do we choose the Hilbert space? how do we introduce fundamental observables analogous to the ones of classical mechanics? In order to answer these questions we need to take a small detour and talk a little bit about the algebra of observables.

Algebra of Observables

Given two classical observables, we can construct another one by means of different methods. Some important ones are:

• By adding them (they are real functions) $h_1=f+g$
• By multiplying them $h_2=f\cdot{}g$
• By a more sophisticated procedure called the Poisson bracket $h_3=\{f,g\}$

The last one turns out to be fundamental in classical mechanics and plays an important role within the Hamiltonian form of the dynamics of the system. A basic fact is that the set of observables endowed with the Poisson bracket forms a Lie algebra (a vector space with a rule to obtain an element out of two other ones satisfying some natural properties). The fundamental observables behave really well with respect to the Poisson bracket, namely they satisfy simple commutation relations $\{q^i,p_j\}=\delta^i_j$ i.e. if we consider the $i$-$th$ position observable and "Poisson-multiply" it by the $j$-$th$ momentum observable, we obtain the constant function $1$ if $i=j$, or the constant function $0$ if $i\neq j$.

One of the best approaches to construct a quantum theory associated with a classical one, is to reproduce at the quantum level some features of its classical formulation. One way to do this is to define a Lie algebra for the quantum observables such that some of such observables mimic the behavior of the Poisson bracket of some classical fundamental observables. This procedure (modulo some technicalities) is known as finding a representation of this algebra. In order to do this, one has to choose:

1. A Hilbert space $\mathcal{H}$.
2. Some fundamental observables that reproduce the canonical commutation relations when we consider the commutator of operators.

In standard Quantum Mechanics the fundamental observables are positions and momenta. It may seem that there is a great ambiguity in this procedure, however there is a central theorem due to Stone and von Neumann that states that, under some reasonable hypothesis, all the representations are essentially the same.

Separability

One of the hypotheses of the Stone-von Neumann theorem is that the Hilbert space $\mathcal{H}$ must be separable. This means that it is possible to find a countable set of orthonormal vectors in $\mathcal{H}$ (called Hilbert basis) such that any state -vector- of $\mathcal{H}$ can be written as an appropriate countable sum of them. A separable Hilbert space, despite being infinite dimensional, is not "too big", in the sense that there are Hilbert spaces with uncountable bases that are genuinely larger. The separability assumption seems natural for standard quantum mechanics, but in the case of quantum field theory -with infinitely many degrees of freedom- one might expect to need much larger Hilbert spaces i.e. non separable ones. Somewhat surprisingly, most of the quantum field theories can be handled with our beloved and "simple" separable Hilbert spaces with the remarkable exception of LQG (and its derivative LQC) where non separability plays a significant role. Henceforth it seems interesting to understand what happens when one considers non separable Hilbert spaces [3] in the realm of the quantum world. A natural and obvious way to acquire the necessary intuition is by first considering quantum mechanics on a non-separable Hilbert space.

The Polymeric Harmonic Oscillator

The authors of [2,3] discuss two inequivalent (among the infinitely many) representations of the algebra of fundamental observables which share a non familiar feature, namely, in one of them (called the position representation) the position observable is well defined but the momentum observable does not even exist; in the momentum representation the roles of positions and momenta are exchanged. Notice that in this setting, some familiar features of quantum mechanics are lost for good. For instance, the position-momentum Heisenberg uncertainty formula makes no sense at all as both position and momentum observables need to be defined.

To improve the understanding of such systems and gain some insight for the application to LQG and LQC, the authors of [1] (re)study the $1$-dimensional Harmonic Oscillator (PHO) in a non separable Hilbert space (known in this context as a polymeric Hilbert space). As the space is non separable, any Hilbert basis should be uncountable. This leads to some unexpected behaviors that can be used to obtain exotic representations of the algebra of fundamental observables.

The motivation to study the PHO is kind of the same as always: the HO, in addition to being an excellent toy model, is a good approximation to any 1-dimensional mechanical system close to its equilibrium points. Furthermore, free quantum field theories can be thought of as ensembles of infinitely many independent HO's. There are however many ways to generalize the HO to a non separable Hilbert space and also many equivalent ways to realize a concrete representation, for instance by using Hilbert spaces based on:

• the Bohr compactification of the real line.
• the Besicovitch almost periodic functions.
• constructions that generalize the usual separable Hilbert spaces based on sequences ($\ell^2(\mathbb{R})$ spaces).

The eigenvalue equations in these different spaces take different forms: in some of them they are difference equations, whereas in others they have the form of the standard Schrödinger equation with a periodic potential. It is important to notice nonetheless that writing Hamiltonian observables in this framework turn out to be really difficult, as only one of the position or momentum observables can be strictly represented. This means that for the other one it is necessary to rely on some kind of approximation (that can be obtained by introducing an arbitrary scale) and choosing a periodic potential with minima corresponding to the one of the quadratic operator. The huge uncertainty in this procedure has been highlighted by Corichi, Zapata, Vukašinac and collaborators. The standard choice leads to an equation known as the Mathieu equation but other simple choices have been explored, as the one shown in the figure.

Energy eigenvalues (bands) of a polymerized harmonic oscillator. The horizontal axis shows the position (or the momentum depending on the chosen representation), the vertical axis is the energy and the red line represents the particular periodic extension of the potential used to approximate the usual quadratic potential of the HO. The other lines plotted in this graph correspond to auxiliary functions that can be used to locate the edges of the bands that define the point spectrum in the present example.

As we have already mentioned, the orthonormal bases in non separable Hilbert spaces are uncountable. A consequence of this is the fact that the orthonormal basis provided by the eigenstates of the Hamiltonian must be uncountable, i.e. the Hamiltonian must have an uncountable infinity worth of eigenvalues (counted with multiplicity). A somewhat unexpected result that can be proved by invoking classical theorems on functional analysis in non-separable Hilbert spaces is the fact that these eigenvalues are gathered in bands. It is important to point out here that only the lowest-lying part of the spectrum is expected to mimic reasonably well the one corresponding to the standard HO, however it is important to keep also in mind the huge difference that persists: even the narrowest bands contain a continuum of eigenvalues.

Some physical consequences

The fact that the spectrum of the polymerized harmonic oscillator displays this band structure is relevant for some applications of polymerized quantum mechanics. Two main issues were mentioned in the talk. On one hand the statistical mechanics of polymerized systems must be handled with due care. Owing to the features of the spectrum, the counting of energy eigenstates necessary to compute the entropy in the microcanonical ensemble is ill defined. A similar problem crops up when computing the partition function of the canonical ensemble. These problems can probably be circumvented by using an appropriate regularization and also by relying on some superselection rules that eliminate all but a countable subset of energy eigenstates of the system.

A setting where something similar can be done is in the polymer quantization of the scalar field (already considered by Husain, Pawłowski and collaborators). As this system can be thought of as an infinite ensemble of harmonic oscillators, the specific features of their (polymer) quantization will play a significant role. A way to avoid some difficulties here also relies on the elimination of unwanted energy eigenvalues by imposing superselection rules as long as they can be physically justified.

Bibliography

[1] J.F. Barbero G., J. Prieto and E.J.S. Villaseñor, Band structure in the polymer quantization of the harmonic oscillator, Class. Quantum Grav. 30 (2013) 165011.
[2] W. Chojnacki, Spectral analysis of Schrodinger operators in non-separable Hilbert spaces, Rend. Circ. Mat. Palermo (2), Suppl. 17 (1987) 135–51.
[3] H. Halvorson, Complementarity of representations in quantum mechanics, Stud. Hist. Phil. Mod. Phys. 35 (2004) 45-56.