Is the "Hamiltonian" special in physics?

In analytical mechanics, a physical quantity called the Hamiltonian appears when dealing with classical objects. It is usually denoted by H, taken from the first letter of the English word. This Hamiltonian appears in the canonical equations that describe the motion of an object.

When given specific initial conditions to predict future behavior, solving these canonical equations is important. Therefore, the Hamiltonian H, which governs dynamics, might at first glance seem like a physical quantity with a special status. While H is certainly a very useful quantity, if we ask again, "Is H special as a physical quantity?", the answer is actually no.

The Hamiltonian H depends on the physical variables p and q. However, if H, as a function of p and q, does not depend on time t, then the value of H (energy) E does not depend on time for the solutions of the canonical equations. This is the so-called law of conservation of energy.

The law of conservation of energy is an important concept that appears frequently in the study of physics. However, energy is not the only physical quantity that is conserved. For example, if the canonical equations are preserved even when the symmetry is rotated, angular momentum is conserved. Also, if the canonical equations are preserved when position coordinates are shifted by a constant amount (spatial translation), momentum is conserved. Thus, the law of conservation of energy is not the only special one; there are conservation laws corresponding to other physical quantities as well.

In analytical mechanics, it is shown that if there is a symmetry under which the canonical equations are invariant under a certain general transformation, then a conservation law for the physical quantity corresponding to that symmetry holds. Mathematically, it can be seen that a specific physical quantity called the generator of this transformation does not change over time. The law of conservation of energy is just one example of such a conservation law.

For example, a transformation that shifts the reference point of time is called a time translation. If the canonical equations are invariant under this time translation, the system is said to have time translation symmetry. And in a system with this time translation symmetry, the Hamiltonian H, which is the generator of the time translation, is conserved.

What should be emphasized here is the following point: even when considering one specific system, the Hamiltonian, which is the generator of the time translation, is not uniquely determined. This is because different time coordinates can be defined by changing how time flows according to the initial conditions of the physical variables (p, q). As a result, different Hamiltonians corresponding to each time coordinate are defined. If we let H be the Hamiltonian in the original time coordinate, the Hamiltonian corresponding to the new time coordinate can be expressed as F(H) using an arbitrary monotonically increasing function F(x). For details, please refer to the following article.

The fact that many Hamiltonians exist as generators is actually not a property unique to the Hamiltonian. For a physical quantity A, which is a generator that produces other transformations, it is similarly possible to define many different generators F(A). Let us look at this below.

First, consider a general physical quantity A that has no time dependence when viewed as a function of (p, q).

For any initial value (p(0), q(0)), let us perform the following transformation that depends on a parameter b.

Here, b is a real number and is assumed to be continuously variable. This transformation is defined to satisfy the following equation for A.

In this case, b is also called the canonically conjugate quantity to A in analytical mechanics. And when equation (3) is satisfied, it can be seen that A does not depend on the value of b.

Here, b does not necessarily have to be time, but equation (4) can be regarded as a conservation law of A with respect to changes in b.

Now, let us create a physical quantity F(A) using a monotonically increasing function F(x). Since A is conserved with respect to changes in b, the following relationship also holds.

Here, we introduce a new parameter β at each point (p,q) in phase space using the following equation.

The function N(a) that appeared on the right side of the above equation is the lapse function when b is treated as a time coordinate.

This N(A) is also invariant with respect to changes in b. Following the same calculation performed on the Hamiltonian H in the note article above, the following relationship is obtained.

In other words, it is shown that the transformation that shifts the new parameter β is a canonical transformation with F(A) as the generator. I will verify this below.

First, equation (5) is transformed into

Furthermore, using the conservation law of the lapse function N with respect to changes in b, and using the value A(p(0),q(0)) = a, which is determined solely by the values of (p(0),q(0)) before the transformation, it can also be written as

And after dividing both sides by N(a),

using the relationship between b and β, the canonical transformation by A in the original equation (3) is indeed obtained.

Therefore, not only for the Hamiltonian H but also for any physical quantity A, by choosing different values of the parameter b for each point in phase space, we can consider the generator F(A) and its transformation. In this sense as well, the Hamiltonian H cannot be said to be a special physical quantity.

In the coming era, the realization of macroscopic quantum control systems such as quantum computers is expected. Along with this, it will become possible to freely design Hamiltonians. As a result, it will become possible to construct time evolution in which an arbitrarily chosen physical quantity is conserved over short discrete time intervals.

If one learns quantum mechanics from canonical quantization as in the last century, it might seem that there is a deep meaning in the fact that the Hamiltonian H is written as the sum of kinetic energy K and potential energy U. However, in the modern understanding of quantum mechanics, it is important to recognize that the form of the Hamiltonian H is not fixed, but is something that is "freely designed".