The Classical Lagrangian Basis of Quantum Mechanics
Some explorations of the classical mechanics Lagrangian basis for quantum theory, as preparation for the next post.
OK, I have managed to keep the mathematics to a relative minimum up until this point. But that has to change now. There is no way to convey the sheer beauty of what is going on with Lagrangian theory without the mathematics. So strap in…
The Lagrangian approach is based on (Hamilton’s) Principle of Least Action (strictly, Stationary Action), widely considered one of the most important principles in physics (Feynman 1963–1965; Lanczos 1970; Coopersmith 2017). And note that point: it is the action that is the basis of this most important principle. Planck’s constant $h$, which is the defining feature of quantum physics, has the units of action, which also tells us that one of the most fundamental aspects of the universe is this thing called action. We therefore need to take that primary fact seriously from the outset.
Hamilton’s Principal Function, $S$, from Hamilton-Jacobi (HJ) theory is also the action, a function defining a space-filling congruence of (non-intersecting) trajectories in configuration space, usually denoted $Q$. Every trajectory in HJ theory traces itself back through various transformations of coordinates and functions from the initial minimised-action trajectories that arise from the Lagrangian formalism. As a result, every point in the configuration space $Q$ of HJ theory defines a single, allowable configuration of the entire $N$-particle dynamics occurring in 3D space that are being represented in $Q$, which representation is attained by the various coordinate transformations that get us from 3D to $Q$ (as was discussed in the previous post). Therefore, the $S$ of the HJ formalism contains a vast amount of information about the extent and scope of the physically-allowed particles dynamics. I have noted before how powerful this formalism is, but that power comes at the cost of increased abstractness and remoteness from the underlying reality which is being represented. And as also noted before, the single-particle case can be conceived of as occurring in 3D space, but the moment we go multiple-particle, it’s off to configuration space $Q$ we are forced to go.
In the Lagrangian formalism, the action is found from an integral equation, specifically an integral of the Lagrangian function $L$, with $L = T – V$ (kinetic minus potential energy), which is a function of the $i=1\dots n$ generalised coordinates $q_i(t)$, their time derivatives $\dot{q}_i(t)$, and time $t$ by way of
$$
S[q] = \int_{t_i}^{t_f} L\left( q_i(t), \dot{q}_i(t), t \right)\; dt\;.
$$
The square brackets indicate that $S$ is a functional; that is, it is a function that takes a function as input. Here the endpoints on the integral are the initial time $t_i$ and final time $t_f$. The generalised coordinates are defined by suitable transformations that map the ordinary 3D space coordinates of our direct experience into the more abstract generalised ones. When we have these transformations, the mapping is very easy to reverse (mathematicians usually like to call things that are this easy trivial). The scalar functional $S$ is (so to speak) ‘inert’ when it comes to the actual dynamics, which only emerge when the action is varied, by way of a variational procedure. It is (so to speak) the ‘seed’ that underpins the dynamics that are implicit in the formalism, which are waiting to be extracted.
The actual physical (classical) path $q^{\text{cl}}(t)$ is the one that makes the action ‘stationary’, that is, when the variation of $S$ vanishes, $\delta S=0$. That is,
$$
\delta S = \delta \int_{t_i}^{t_f} L\left(q, \dot{q}, t \right)\, dt\; = 0.
$$
The standard approach in Mechanics is to vary the action by a small perturbation $\delta q$ thus:
$$
q(t) \to q(t) + \delta q(t)\;.
$$
Then, the standard result is that $\delta S$ becomes
$$
\int_{t_i}^{t_f} \left( \frac{d}{dt} \left(\frac{\partial L}{\partial\dot{q}}\right) – \frac{\partial L}{\partial q}\right) \delta q\, dt + \left[ \frac{\partial L}{\partial\dot{q}}\delta q \right]_{t_i}^{t_f} = 0,
$$
where the term in square brackets is the boundary term. Note that the boundary term contains the expression $(\partial L/\partial\dot{q})\delta q$ evaluated at the endpoints. The $\partial L/\partial\dot{q}$ term is the definition of the so-named generalised momentum $p$ which is conjugate to the generalised coordinates $q$. Therefore, the boundary term is of the form
$$
p(t_f)\delta q(t_f) – p(t_i)\delta q(t_i) \; .
$$
(This will become important in the next post.) In Mechanics, the usual assumption is that the variations $\delta q$ at the endpoints vanish identically, i.e., $\delta q(t_i) = \delta q(t_f) = 0$. Therefore, the square bracket also vanishes identically which means only the integral remains. In order for it to equal zero, the expression inside it must equal zero; that is,
$$
\frac{d}{dt} \left( \frac{\partial L}{\partial\dot{q}}\right) – \frac{\partial L}{\partial q} = 0.
$$
These are the Euler-Lagrange Equations, which are (local) second-order differential equations in time and are the equivalent of the Newtonian equations of motion, expressed in generalised coordinates. But notice: the action is a global (i.e., non-local) quantity defined at two different times (and places), and the classical path $q^{\text{cl}}(t)$ which minimises this action must obey the local Euler-Lagrange Equations at each point along that path. In this remarkable way, a global condition manifests as local constraints on the motions of the particles. Feynman relates, in Chap. 19 of the second volume of his famed Lectures on Physics, that he has always found this fascinating, ever since his high school physics teacher told him about it (Feynman 1963–1965, p.19–1). And this is indeed a remarkably fascinating thing—nature chooses the path that minimises the (abstract concept of) action.
The point to note here is that this is the classical limit, where the boundary term is silenced (made zero) because the variations at the boundary are assumed to be zero. In the next post, however, we will let it speak to see what it says…
It is also possible to take ‘higher’ degrees of variation than just the ‘first’ one, $\delta S$. The sign of the second variation, $\delta^2 S$, is used to determine the nature of the stationary value defined by $\delta S=0$, such as if it is a local minimum (positive definite $\gt0$), a local maximum (negative definite $\lt0$), or saddle point (indefinite).
Typically, the analysis tends to stop there, and it is rare for it to proceed to higher variations. For the most part, Mechanics is concerned pretty much only with the major consequence of the first variation, $\delta S$, namely the extraction of the Euler-Lagrange Equations when this variation is made to vanish. In functional analysis, though, the second variation is called the Hessian (usually represented by a matrix), containing the second-order partial derivatives of $S$, which encodes information about the local ‘curvature’ of the trajectory. If the function is an expression of the action for a particular dynamical system, say in Hamiltonian mechanics or HJ theory, then the Hessian encodes information about the stability of the classical trajectories. Related to the Hessian is the Van Vleck Determinant, usually denoted $\Delta_{VV}$ or $D_{VV}$, which contains information about, among other things, the spreading or converging of those trajectories (Visser 1993). In places where the Van Vleck Determinant vanishes, there is what is called a caustic. An example is where light is focused to a point by a lens in geometrical optics; the idealised rays converge and intersect, which a mathematical congruence really should not do. The appearance of caustics in the Hessian (one or more zero eigenvalues) or Van Vleck Determinant (where it vanishes) is an example of the wonderfully-named catastrophe theory, about which I’ll say more when we explore the nature of caustics in the post after the next one (i.e., in Part IX).
From this starting point in Lagrangian theory, the rest of Mechanics can be formulated by way of different types of transformations, as noted in Part I, leading all the way up to HJ theory, and its master function $S$, the “orchestrating field” (Norsen’s term) of HJ-based Mechanics.
It is possible to expand the variational derivative operator $\delta$ as a Taylor series, thus:
$$
S[q+\delta q] = \sum_{n=0}^\infty \frac{1}{n!}\delta^n S[q] = S[q] + \delta S[q] + \frac{1}{2!}\delta^2 S[q] + \frac{1}{3!}\delta^3 S[q] + \dots \;.
$$
The $\delta^n S$ are the Fréchet derivatives of the functional $S[q]$, and, since the $e^\delta$ operator takes the explicit expanded form
$$
e^\delta = \sum_{n=0}^\infty \frac{1}{n!}\delta^n = \mathbb{1} + \delta + \frac{1}{2!}\delta^2 + \frac{1}{3!}\delta^3 + \dots\, ,
$$
where $\mathbb{1}$ is the identity operator, this can be written in the much more compact form $e^\delta S$, so that the expansion can be rendered
$$
S[q+\delta q] = e^\delta S[q] \;.
$$
This bears a strong resemblance to the shift operator which can be found in the theory of Lie Groups, specifically the exponential map from the Lie algebra of variations to the Lie group of finite transformations. But I should probably refrain from calling it that because I am sure there are so many caveats to the correct formal use of the term that I’d best avoid that potential headache. So, what I will say is that since it behaves a lot like the shift operator, I will just choose to call it that by a convenient abuse of the language (with suitable disclaimers etc etc, and apologies to all of my past mathematics lecturers who thought that physicists tend to be a bit too sloppy with respect to proper rigour. My bad; let’s move on).
Note that the $n=0$ term in the summation defining the exponential map of the variational derivative $\delta$ acting on the action functional $S[q]$ just returns $S$ again, since $\frac{1}{0!}\delta^0 S\equiv \mathbb{1}S = S$. If we remove the ‘inert’ ‘seed’ term $S$ from the “shift” operator (note the quotes), we can extract the ‘dynamics’ that lives in the infinite series of variations. Therefore, let us define the operator $\mathscr{D}$ which ‘generates’ the Dynamics (hence the $\mathscr{D}$) from the ‘inert’ ‘seed’, $S$:
$$
\mathscr{D} \equiv e^\delta – \mathbb{1} = \sum_{n=1}^\infty \frac{1}{n!}\delta^n = \delta + \frac{1}{2!}\delta^2 + \frac{1}{3!}\delta^3 + \dots\, ,
$$
so that the full series could be written as
$$
S[q+\delta q] = e^\delta S[q] = S[q] + (e^\delta – \mathbb{1})S[q] = S[q] + \mathscr{D}S[q]\;.
$$
The operator $\mathscr{D}$ is (so to speak) the ‘engine’ of the physics, since it extracts all of the dynamics that are encoded into the action. The expression $S+\mathscr{D}S$ could perhaps be thought of informally as ‘seed plus the dynamical response’. So, in this sense, $\mathscr{D}$ is the Dynamical Response Operator, and $\mathscr{D}S$ is the Dynamical Response Functional.
With this setup, (classical) Mechanics emerges as the $n=1$ step with definite sharp boundaries and vanishing variations at those boundaries. The rest of the terms in the ‘Dynamical Response’ functional $\mathscr{D}S$ are either discarded once the Euler-Lagrange Equations are extracted, or treated as negligible perturbations, except perhaps for the $n=2$ case which yields the ‘curvature’ of the action functional and, in the case of ray optics, for example, reveals the existence of caustics.
Now I pose the question: what happens if we don’t assume the variations vanish at the endpoints of the integral, combined with wondering what happens if the endpoints are not sharp and point-like a classical particle? What would happen, for example, if the endpoints were extended, say, the way that quantons are assumed to have continuous extension (or pantopy)? What physics might emerge from asking such a simple question and making such a simple change to the elegant mathematics of classical Lagrangian theory? Hmm…
In the next post, we will adopt what I will call the Extended Boundary Condition, and begin to explore the further reaches of the Dynamical Response Operator $\mathscr{D}$. Let’s find out where that leads to …
Next: Part VIII – The Transition |M⟩ → |Q⟩