A Brief Introduction
Recently, I have been fascinated by understanding mechanics and field theory from the perspective of Lie symmetry, and I have obtained some interesting results which I would like to share with you here. I found that with only an infinitesimal generator describing symmetry and some most basic assumptions, one can almost completely derive the entire relativistic mechanics, and even the entire (classical) field theory. This is truly incredible. I can now basically appreciate the meaning of the title of the book Fearful Symmetry written by Master Anthony Zee. The power of symmetry is so great that we truly have to revere it. When conceiving the title for this article, I also thought about using “Fearful Symmetry” as the title, but it felt a bit like plagiarism and a cliché. Later, I remembered a manga called A Manga Ventures into the World; so I changed “Manga” to “Symmetry” and “World” to “Physics,” which seems to express my feelings toward “symmetry” quite well.
Symmetry is the property of remaining invariant under certain transformations. For example, special relativity requires all physical laws to remain invariant in all inertial frames. This is equivalent to requiring the equations describing physical laws to remain invariant under coordinate transformations of uniform motion. Combined with the requirement of the constancy of the speed of light, we can derive the Lorentz transformation, thereby completely describing the symmetry within special relativity. However, it is not always possible to derive a complete transformation as one does with the Lorentz transformation. Fortunately, Lie symmetry does not require a complete description of the symmetry; it only needs an “infinitesimal transformation” (meaning we can ignore higher-order terms), which correspondingly produces an “infinitesimal generator.” Using this infinitesimal generator is sufficient to fully construct the physics we need. This wonder of the “infinitesimal” determining the “broad” and the “local” determining the “global” still feels incredible to me. (Regarding the basic concepts of Lie symmetry and infinitesimal generators, you may first read: Lie Symmetry Methods for Solving Differential Equations).
The Generator of Special Relativity
Einstein stated that all physical laws remain invariant in all inertial frames. This actually implies a one-parameter transformation group (of course, this group also exists in Newtonian mechanics). Since velocity is arbitrary (we haven’t discussed the constancy of the speed of light yet), this is a one-parameter transformation group with velocity as the parameter. Let us consider our inertial frame (x, t), and suppose there exists an inertial frame (x', t') moving away from us at an infinitesimal velocity \epsilon. The Galilean transformation tells us: x' = x + \epsilon t We know that in special relativity, this is not correct. However, assuming we do not know what the correct transformation is, we only know that the above transformation needs correction. Nevertheless, the above transformation has passed a large number of “tests,” so we believe that it is correct at least to the “first order.”
Special relativity also requires the constancy of the speed of light, which is a correction to t' = t. This is equivalent to requiring: \frac{dx'}{dt'} = c \iff \frac{dx}{dt} = c By analogy with x' = x + \epsilon t, we can only assume: t' = t + \epsilon kx where k is a constant. That is: \begin{aligned} c = \frac{dx'}{dt'} &= \frac{dx + \epsilon dt}{dt + \epsilon kdx} = \frac{dx/dt + \epsilon}{1 + \epsilon k dx/dt} \\ &= \frac{c + \epsilon}{1 + \epsilon kc} \approx (c + \epsilon)(1 - \epsilon kc) \\ &\approx c + \epsilon(1 - kc^2) \end{aligned} This indicates that under the first-order approximation, we require 1 - kc^2 = 0, hence k = 1/c^2, which means: t' = t + \epsilon \frac{x}{c^2}
Thus, we have found the infinitesimal generator corresponding to the symmetry required by special relativity: X = \frac{x}{c^2}\frac{\partial}{\partial t} + t\frac{\partial}{\partial x}
Readers might think: \frac{1}{c^2} is also an infinitesimal quantity and should be discarded! But this is incorrect, because \frac{1}{c^2} is a “small quantity” rather than an “infinitesimal quantity.” We say \epsilon is an infinitesimal quantity because it represents the relative velocity of the reference frame, which can indeed be arbitrarily small. Once the unit of length is chosen, \frac{1}{c^2} has a fixed value; it cannot be arbitrarily small. Therefore, this term cannot be ignored. In other words, we feel \frac{1}{c^2} is small only because the chosen units are inappropriate; if we chose “light-seconds” as the unit, then c=1, and at this point \frac{1}{c^2}=1, which is clearly “not small.”
First and Second Order Prolongations
Mechanical laws usually contain variables such as velocity and acceleration. Therefore, the generator above, which only contains displacement and time, is not sufficient. We need to prolong it, at least to the second derivative of displacement with respect to time.
We have (denoting \frac{dx}{dt} = \dot{x}): \begin{aligned} \frac{dx'}{dt'} &= \frac{dx + \epsilon dt}{dt + \epsilon dx/c^2} = \frac{\dot{x} + \epsilon}{1 + \epsilon \dot{x}/c^2} \\ &\approx (\dot{x} + \epsilon)(1 - \epsilon \dot{x}/c^2) \\ &\approx \dot{x} + \epsilon(1 - \dot{x}^2/c^2) \end{aligned} This shows that the first-order prolongation is: X^{(1)} = \frac{x}{c^2}\frac{\partial}{\partial t} + t\frac{\partial}{\partial x} + \left(1 - \frac{\dot{x}^2}{c^2}\right)\frac{\partial}{\partial \dot{x}}
Similarly: \begin{aligned} \frac{d^2 x'}{dt'^2} &= \frac{d[\dot{x} + \epsilon(1 - \dot{x}^2/c^2)]}{dt + \epsilon dx/c^2} = \frac{\ddot{x} - 2\epsilon \dot{x}\ddot{x}/c^2}{1 + \epsilon \dot{x}/c^2} \\ &\approx (\ddot{x} - 2\epsilon \dot{x}\ddot{x}/c^2)(1 - \epsilon \dot{x}/c^2) \\ &\approx \ddot{x} - 3\epsilon \dot{x}\ddot{x}/c^2 \end{aligned} This shows that the second-order prolongation is: X^{(2)} = \frac{x}{c^2}\frac{\partial}{\partial t} + t\frac{\partial}{\partial x} + \left(1 - \frac{\dot{x}^2}{c^2}\right)\frac{\partial}{\partial \dot{x}} - \frac{3\dot{x}\ddot{x}}{c^2}\frac{\partial}{\partial \ddot{x}}
In the following, we will use these operators to construct relativistic mechanics. Of course, this will be placed in the next article; for now, let’s leave it as a cliffhanger.
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