Berry Phase and Density Matrix in Quantum Mechanics

Since Tachikawa-san from UTokyo IPMU @yujitach posted the following article, I would like to provide some comments.
Postscript (2023/11/28): I have since had a long discussion with Tachikawa-san, and after gaining more clarity, I have revised and added to the text.

In my textbook, I emphasize that the probability distribution of physical quantities observable in experiments is the foundation of quantum mechanics, and that state vectors, wave functions, and density matrices are merely one way of representing them.

This is in the same sense that Maxwell's equations of electromagnetism can be expressed using quaternions or exterior differential forms; vectors, matrices, and operators are just one way of representing quantum mechanics. Quantum states are first defined from a collection of probability distributions of physical quantities. In the textbook, the flow then proceeds to represent quantum states as density matrices using quantum state tomography. Then, as a special example of such a density matrix, we take up quantum states whose spectral representation can be written with a single state vector, and define that as a pure state. A wave function is merely a single component representation of a state vector in a pure state of a point particle. Therefore, the content is that not only state vectors and wave functions, but also density matrices are 'convenient descriptive tools' introduced after defining quantum states via probability theory.

From the perspective of mathematical physics, Tachikawa-san introduces the labels 'density matrix first' school and 'state vector first' school in his article, but I myself belong to neither, and rather to the 'probability distribution of observable physical quantities first' school.

In general, not limited to quantum mechanics, probability distributions undergo collapse as a result of information being updated through observation, so it is natural that state vectors and wave functions also collapse upon observation. As a result, we can see that the 'measurement problem,' which was discussed extensively in the last century without producing any fruitful results, did not exist in the first place.

Tachikawa-san writes his article focusing on the difference between the formulation using density matrices and state vectors, rather than the formulation based on probability distributions. In particular, he examines an interesting physical phenomenon related to the topology of mathematics called the Berry phase.

The Berry phase is a phase factor applied to the vector of an energy eigenstate that appears in the process of deforming the Hamiltonian of a quantum system very slowly and finally returning it to the original Hamiltonian. As is also in Tachikawa-san's notes, in the Born-Oppenheimer approximation in quantum many-body systems, it can also be explained as a phase factor given to the dynamics of fast-moving degrees of freedom by slowly moving degrees of freedom.

Let's look at this more concretely using the example Tachikawa-san used in the latter half of his notes. This is an example where the slowly moving degrees of freedom become a two-level spin particle on the space of a 3D Klein bottle. Let's write its wave function as

And suppose the Schrödinger equation that emerges from the Born-Oppenheimer approximation takes the form accompanied by a Berry connection field.

Since the space is a Klein bottle, the values of each spatial coordinate are subject to

periodicity and

an identification incorporating a topological twist.

Furthermore, since we are considering a two-level spin, we will use the following Pauli matrices hereafter:

The quantum mechanical expectation value of each of these components is calculated by

In this setup, Tachikawa-san's boundary conditions for each expectation value are the following three:

In Tachikawa-san's notes, to satisfy these boundary conditions for the expectation values, the following three boundary conditions are imposed on the wave function:

Tachikawa-san proved in his notes that these three boundary conditions cannot be satisfied simultaneously due to the following contradiction. First, by using the periodicity condition in the first direction and then the twist condition in the third direction, the wave function is calculated as follows.

On the other hand, if the twist deformation in the third direction is performed first and then the periodicity condition in the first direction is used, the result is a different phase factor depending on the position coordinate in the second direction, even though it is the same wave function, which leads to a contradiction.

For this reason, it can be seen that there is no wave function or corresponding state vector that satisfies the periodic boundary conditions of equations (4), (5), and (6).

I think this result by Tachikawa-san is very interesting as mathematical physics, but this result can also be derived from the density matrix. This is because the same contradiction can be shown in the same way with a similar analysis.

First, the important point is that in the Born-Oppenheimer approximation, the slowly moving coordinate degrees of freedom are also treated equally as physical quantities. Since spatial degrees of freedom and spin degrees of freedom are closely intertwined through topology, it is not possible to extract and treat only the spin degrees of freedom. Therefore, as an argument for the density matrix ρ, spatial degrees of freedom must also be incorporated as follows.

If we use the periodicity condition in the first direction for this ρ first, and then use the twist condition in the third direction, a phase factor that depends on the position coordinate in the second spatial direction appears.

If we perform the twist deformation in the third direction with the same ρ and then use the periodicity condition in the first direction, we get the following.

Since the phase factor depending on the position coordinate in the second spatial direction differs between the two equations even though it should be the same density matrix, the contradiction was proven just like with the wave function. In other words, it does not matter whether you are in the "density matrix first" camp or the "state vector first" camp.

Note that one should be careful that the definition of observables changes depending on whether this theory of particles on a Klein bottle is regarded as an approximation for some degrees of freedom embedded in a condensed matter system, like the theory of Berry phase, or as an SU(2) non-commutative gauge theory in pure mathematics. In a condensed matter system, the Pauli matrices of a two-level spin are regarded as observable physical quantities. In this position, since each expectation value of the spin component is also a physical quantity, it must be single-valued on the Klein bottle. Therefore, the two-valued conditions of equations (1) and (2) set by Tachikawa-san for the spin components will never be realized in physical experiments. Even in a Klein bottle space, the expectation value of a physical quantity is properly determined as a single value, so a contradiction arises.

(Note: Please distinguish this discussion from cases where the topology of a torus is considered in the identification of the Brillouin zone in wave number space that appears in condensed matter crystal systems. The physics of crystal systems is not the physics of quantum mechanics in a real torus space, but merely a mathematical identification of a large wave number Brillouin zone and a small wave number Brillouin zone. Therefore, it is possible to consider multi-valued functions and physical quantities on the torus of wave number space. However, in actual physical experiments, different wave numbers can be distinguished directly by measurement, so it is not really assuming a single torus space. It is just a situation where different tori are connected in countless ways, like a Riemann surface with cuts.)

However, if we consider it not as a theory of condensed matter systems but as a purely mathematical SU(2) gauge theory, there is no problem with equations (1), (2), and (3). Since each Pauli matrix changes under its gauge transformation, it is not a gauge invariant. Since observable physical quantities in gauge theory should be gauge invariant, the physical quantities of the Pauli matrices are not observed. Even if the gauge-dependent Pauli matrices change in a two-valued manner in the Klein bottle space as in equations (1) and (2), there is no contradiction because they are not observed.

However, there remains something to be careful about. Since each Pauli matrix is not observable in gauge theory, the expectation value of that Pauli matrix does not appear in the tomography of the quantum state in that gauge theory. The quantum state of the theory is determined by the probability distribution and expectation values of other gauge-invariant observables. Sufficient attention must be paid to this point.

In summary, in the formulation based on "probability distribution" handled in my textbook, both the density matrix and the state vector as a special case appear as its notation, but the difference between the two is only apparent. Physical phenomena of a quantum system in a pure state can be described without omission by either. However, for mixed states handled by the density matrix, the notes in the article below are necessary. In quantum mechanics, properties that did not exist in classical statistical mechanics also appear in mixed states.