Saturday, November 23, 2013

Holonomy in quantum mechanics

Bohm-Aharonov effect

Let’s start with the definition of holonomy: if you walk in  closed loop and the object you carry changes when you complete the loop then you have experienced a holonomy.

Now this sounds plain crazy so a simple example can illustrate it. Suppose you are a hunter living on the Equator and you go on a quest to explore the Earth. You walk a quarter of the Earth circumference on the Equator going east, you travel north all the way to the North Pole and then you go straight down to the starting point of your journey. During your journey you carry with you your spear always making sure it is pointing in the same direction. For definiteness sake, let’s say that originally your spear was pointing towards the North Pole. When you walk on the Equator and then towards the North Pole your spear is pointing north. However, on the last part of the journey, your spear will be pointing west. So upon your arrival the spear has a different orientation even though you always carried it parallel with itself. This is the result of Earth’s curvature.

Now something very similar happens in Einstein’s general relativity: the presence of mass curves the space-time and although you travel on geodesics in a straight line, nearby curves are not parallel. We feel this lack of parallelism as gravity.

Fine, we understand this, but what does holonomy have to do with quantum mechanics? Suppose I have a box with a quantum device which when I press a button can flash either a red light or a blue light. Suppose that every time I press the button only the red light will flash. Now I go on a similar quest on a closed space loop and when I press the button only the blue light is flashing. This would happen every time I would circle a zone of magnetic field although I cannot detect any magnetic field anywhere on my path.

Now this is downright freaky: there are no forces whatsoever along my path and still there is a measurable effect. Welcome to the wonderful Bohm Aharonov effect.

Mathematicians usually refers to geometric phases and call this effect a topological one. But surprisingly it has a nice mathematical explanation in terms of boundary conditions and domains in standard quantum mechanics. When learning quantum mechanics, pesky boundary conditions and domains tend to be ignored as pedantic crossing of t’s and dotting the i’s. This is the typical cavalier physicist’s attitude towards mathematical rigor. Don’t believe me? Ask any physicist to tell you the difference between Hermitean and self-adjoint. If one in one hundred knows the difference you are lucky.

The point is that self-adjointness demands the domains of the operator and its hermitean counterpart (complex conjugate and transposed) to be identical!!!
A hermitean operator may have even an infinity of different self-adjoint extensions. And the eigenvalues (the observed values) are all distinct.

Here is an example of how sloppiness can get you into trouble:

Now for the Bohm-Aharonov effect, it is the boundary condition which selects the eigenvalues. And this boundary condition comes from the magnetic flux carried by the solenoid.

An excellent review of those topics can be found in Asher Peres’ classic: “Quantum Theory: Concepts and Methods” and I hope I wetted your appetite to read this outstanding book.

Now, although true, changes without forces bother a lot a people. Last year a proposal was made to restore the role of forces in the Bohm-Aharonov effect: The paper has a clever idea: the apparatus and not the particle feels the force, but it is fundamentally flawed because any forces will give out the “which way information” which will destroy the interference pattern. Holonomy is a fundamental property of Nature and it is not explainable away.


  1. Florin,

    I am pondering whether partial isometries might play a role in what I wrote about over a week ago. I am pondering the idea that maybe the orthogonal complements of the domain and range of the operator U may differ as N_+(U) = dim dom(U)^┴ and N_-(U) = dim range(U)^┴ according to the size of the quantum error correction code (QECC) space. If the QECC space is small, or with nonvacuum states N_e << N for the horizon states then N_+(U) – N_-(U) = N_e << N and the space is nearly unitary. The problem is that this space grows, or better put the number of occupied states therein grows. I have this idea I am kicking around that the quantum state space is such that the operators are not entirely unitary, or unital, but meromorphic. There is a pole or singularity involved with the departure from unitarity. An integration around the pole for an amplitude results in

    A(k) = (n-1)!/( 2πi)∮f(γe, p)/(p – k)^n.

    The temperature of the system is then evaluated by such an integral or amplitude and this temperature is periodic on the set of Riemann sheets. In this way I think the growth in the quantum error correction code space can be renormalized. This renormalization prevents the growth in the error space.

    I am not an expert on this area. Yet this seems plausible. This matter gets close to the whole matter of C* algebras of adjoint operators. I also put this on your page about the Berry phase or holonomy. The Eguchi-Hanson metric is a holonomy with SU(3) group structure and is STU dual to QCD. This matter with the singularity is related in some way to that.

    Cheers LC

  2. Lawrence,

    I am not an expert in quantum information and error codes either. A good introductory reference book is "Geometry of Quantum States" by Ingemar Bengtsson and Karol Zyczkowski.

    However, the moment you talk about temperature the whole things changes to field theory which is a different beast described not by C* algebras but by C* Hilbert modules and different notions of convergence enter there. Type 2 and 3 von Neumann algebras play a role there and the math is much harder than in the case of regular QM.

  3. Hi,

    A part of this is I think that maybe QFT of gravity is simply not a closed system. The difference between n^+ and n^- that defines the partial isometry (perfect unitarity if n^+ - n^- = 0) is a measure of the quantum information that is entangled in some unknown manner. This entanglement is with the Einstein-Rosen bridge, or similarly with the measure ~ e^{-|x-x'|} for the intertwining of entanglement and metric.

    I ponder whether the singularity or pole that induces the departure from unitarity to meromorphic function has a pole that is some measure of the information content in this difference n^+ - n^- = dim(H_c) for H_c the dimension of the quantum error correction code space. Showing this is some sort of invariant is the challenge.

    Cheers LC