Harmonic Oscillators Can Be Converted into Free Particles: The Concept of Duality

In theoretical physics, symmetry is an important concept. If a system has translational symmetry in time coordinates, Noether's theorem yields an important conservation law for energy, and if it has translational symmetry in spatial coordinates, a conservation law for momentum is derived. On the other hand, there is also a concept called "duality," which is different from symmetry. By replacing the degrees of freedom of a physical system with other degrees of freedom, or by rewriting the spacetime coordinate system into a curved coordinate system as is done in general relativity, the dynamics of one physical system can sometimes become the dynamics of another. In this case, it is said that there is a duality between the two physical systems. Symmetry corresponds to the case where the two systems connected by this duality happen to be the same single system. As a non-trivial example of this duality, the AdS/CFT correspondence is known, which states that quantum gravity theory with a negative cosmological constant is equivalent to conformal field theory of quantum matter in a flat spacetime with one less spacetime dimension.

In this article, I would like to introduce the "Takagi transformation," which is the simplest example of such duality: the duality between a harmonic oscillator and a free particle. (The reference is the paper [1] below.)

First, we write the action of a free particle of unit mass moving in one-dimensional space as follows.

Here, the position coordinate of the particle is Q, and the time coordinate is τ. Next, letting ω be a positive constant, we rewrite the time coordinate from τ to another time coordinate t as follows.

At the same time, let us rewrite the position coordinate of the particle from Q to another coordinate q as follows.

If we ignore the boundary terms of the action that do not affect the dynamics, by performing these two transformations, S transforms into the action of a harmonic oscillator with angular frequency ω as follows.

Here, as can be seen from Equation (1), the motion of the half-period of the harmonic oscillator's oscillation is stretched in time and mapped from the infinite past to the infinite future, and the q(t) of the harmonic oscillator, which was changing within a finite spatial region, is mapped by Equation (2) to Q(τ), which undergoes uniform motion from negative infinity to positive infinity (or vice versa) on the real line. Therefore, in each of these time domains, it can be said that there is indeed a duality between the free particle and the harmonic oscillator.

Above, we confirmed the duality between a harmonic oscillator with angular frequency ω and a free particle as ω=0, but once we notice this relationship, the duality relationship between a harmonic oscillator with angular frequency ω and a harmonic oscillator with another angular frequency Ω can also be derived automatically. Once the harmonic oscillator system with angular frequency ω is moved to a free particle system by the transformations of Equation (1) and Equation (2), we can then apply the inverse transformations of Equation (1) and Equation (2) with ω replaced by Ω to the free particle system to move it to the harmonic oscillator system with angular frequency Ω.

Specifically, we transform time t and position coordinate q into time t' and position coordinate q' using the following equations.

Then, indeed, after the transformation, the action S becomes as follows, and the duality can be confirmed.

There is such a duality relationship even between harmonic oscillators with different angular frequencies.

Once a duality is found, it becomes possible to export the symmetry in one physical system as a dynamical symmetry of the other physical system. In the following, we will consider exporting the symmetry of the free particle system to the harmonic oscillator system.

Although I have explained this using the Lagrangian formalism so far, it is, of course, exactly the same in the Hamiltonian formalism. Let us look at it in the Hamiltonian formalism. The canonical conjugate momentum of a free particle is given as follows.

And the Hamiltonian of a free particle system with unit mass is given as follows.

The equations of motion in this Hamiltonian formalism are the following canonical equations.

From these canonical equations, we obtain the following equation of motion for a free particle, which is the same as the Euler equation in the Lagrangian formalism.

The general solution to this equation is

Here, the constants appearing on the right-hand side are the initial position and initial momentum at τ=0, respectively.

Since this is a free particle system, the existence of symmetry under the following Galilean transformation, where v is a constant, for example, is known.

If we actually perform this Galilean transformation on the general solution in Equation (6), the result is as follows.

Since the Galilean transformation is a symmetry of this free particle system, the time function in Equation (7) naturally satisfies the same equation of motion as Equation (5) even after the transformation.

In the symmetry argument above, a single transformation connects different time coordinates or position coordinates, but if we change our perspective, we can see that this is just changing the initial conditions in the general solution as follows.

This perspective can be applied not only to Galilean transformations but also to general transformations. Let us transform the initial conditions in the general solution of Equation (6) generally as follows.

The general solution of Equation (6) after this transformation is

but since only the initial conditions have been changed, this naturally satisfies the exact same equation of motion as follows.

If you want to make the transformation of initial conditions in Equation (8) a canonical transformation,

then you just need to impose the following condition on the transformation function of Equation (8).

Here, let us recall that the following transformation, which performs time reversal to shift the position and momentum at time τ to the position and momentum at time τ=0, was a canonical transformation.

Then, we perform the transformation of Equation (8) at time τ=0.

After that transformation, the following transformation, which evolves the system from time τ=0 to time τ, is also a canonical transformation.

From the composition of these three transformations,

a general transformation that maps a free particle solution to another solution can be constructed. Here, if Equation (9) is restricted to a canonical transformation, the composed transformation of Equation (10) is also a canonical transformation. Furthermore, from the discussion above, the Hamiltonian H does not change its form under this canonical transformation of Equation (10). Therefore, it can be seen that Equation (10) provides the most general symmetry in canonical form.

The Galilean symmetry of the free particle system and the general symmetry of Equation (10) can also be exported as dynamical symmetries of the harmonic oscillator system with angular frequency ω through the transformations of Equations (1) and (2). One simply needs to map the harmonic oscillator system to a free particle system, perform a Galilean transformation there, and then return to the original harmonic oscillator system. The symmetry transformation obtained in the harmonic oscillator system in this way also merely changes the initial conditions in the original general solution on the harmonic oscillator side. Moreover, in any Hamiltonian system described by canonical equations, the group of canonical transformations based on all canonical transformations at time τ=0 is realized as a dynamical symmetry. While it is common in applications to focus analysis only on rotation groups or conformal groups whose properties are well known, these transformation groups are not special as dynamical symmetries; all canonical transformations can be regarded as equivalent dynamical symmetry transformations.

Although we treated duality in classical mechanics this time, it is known that the duality between free particles and harmonic oscillators holds in quantum mechanics as well. (Refer to paper [1] for this.) In Chapter 10 of my book, "Introduction to Modern Quantum Mechanics," the quantum version of the duality between a two-dimensional harmonic oscillator and a charged particle in a uniform magnetic field is presented.

Also, for the sake of simplicity, I used the example of a free particle and a harmonic oscillator, but if duality exists, it is always possible, at least in classical mechanics, to construct the symmetry of another physical system B, which can be reached via duality, from the symmetry of a physical system A. First, we evolve time backward from time τ to time τ=0 using the Hamiltonian of A. Next, at time τ=0, we apply the general canonical transformation of Equation (9) to the initial values of position and momentum. After that, we start from those new initial values and evolve time forward to time τ using the Hamiltonian of B. The transformation obtained by the composition of these three canonical transformations can be regarded as a duality transformation connecting A and B.

Combining duality and symmetry in this way to capture the dynamics of more complex physical systems is still frequently done today in various fields of theoretical physics.

[1] Shin Takagi, Progress of Theoretical Physics, Vol 85, No 4, 723 (1991).