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          "markdown": "An invariant of this sort, that fixes some notion of placing reliable rulers on the space, is a **metric**. For normal rotations and translations, these relationships are intuitive, but a symmetry may preserve a less intuitive metric. If we define invariant length not as \\(x^2+y^2\\) but as \\(t^2-\\mathbf{x}^2\\), we have a \"hyperbolic rotation.\" Separation between points stretches out to infinity as the rotation approaches its asymptotes, leaving the invariant interval unchanged even though the separation appears to grow using our everyday notion of distance. Such a hyperbolic metric, as we will see, encodes the asymptotic limit of light speed in the relativistic geometry **spacetime**."
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          "markdown": "On the other hand, the space of functions on a number line does form a linear representation. Instead of translating the point $x$, we translate a function by shifting its argument:"
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          "markdown": "```math\n(T_a f)(x)=f(x-a).\n```"
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          "markdown": "| $(x,y)$ basis: components mix | $(u,v)$ basis: components do not mix |\n|---|---|\n| $\\displaystyle \\begin{aligned}\\mathbf r&=\\frac12\\begin{pmatrix}s+s^{-1}&s-s^{-1}\\\\s-s^{-1}&s+s^{-1}\\end{pmatrix}\\mathbf r_{\\mathrm{in}}\\\\&=\\frac12\\left[(s+s^{-1})x+(s-s^{-1})y\\right]\\hat{\\mathbf x}\\\\&\\quad+\\frac12\\left[(s-s^{-1})x+(s+s^{-1})y\\right]\\hat{\\mathbf y}\\end{aligned}$ | $\\displaystyle \\begin{aligned}\\mathbf r&=\\begin{pmatrix}s&0\\\\0&s^{-1}\\end{pmatrix}\\mathbf r_{\\mathrm{in}}\\\\&=su\\hat{\\mathbf u}+s^{-1}v\\hat{\\mathbf v}\\end{aligned}$ |"
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          "start_line": 899,
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          "markdown": "We can also discover and define phase directly from our symmetry group's commutation relations, which gives us a useful algebraic packaging of the group structure. Let's ask the question:"
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      "caption": "Seeing stationarity emerge in closely spaced paths",
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          "markdown": "We can repeat the same procedure with many more paths. As the path deviates more from a straight, minimum length path, it has a greater first-order change in phase. (This is the common result from calculus that near a function's minimum, there is no change to the value of the function in the first order of the argument.) When the candidate paths are far from the stationary value, their phases vary greatly, effectively cancelling out their contributions to the total sum. On the other hand, the phases of the paths near the stationary path align and dominate the sum. The green line in the tip-to-tail pane of the animation shows the sum of each of these contributions, giving the amplitude at $B$. The resulting intensity on the projection screen is the square of this magnitude."
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          "markdown": "We can extend this procedure to its limit and include infinitely many screens with infinitely many slits, and recover unobstructed propagation. In the following animation, we start with a plane wave and recover that same wave. This construction, which provides a bridge to Feynman’s path integral formulation of quantum mechanics, has its roots in Huygens’ wavelets from the late 1600s, extended by Fresnel to include interference in the early 1800s."
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          "markdown": "Let us now ask one more question. What happens to our tip-to-tail diagram of the path contributions when we vary the wavelength relative to the slit? As the wavelength becomes small, even a slight change in path length can produce a large phase change:"
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          "markdown": "```math\n\\phi=\\frac{2\\pi L}{\\lambda}=2\\pi n+\\theta,\n\\qquad 0\\leq\\theta<2\\pi\n```"
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          "markdown": "Action, the quantity extremized by a physically valid path, can be constructed from the structure of **spacetime**, as articulated in the theory of special relativity, which will be the topic of our next chapter. Crudely speaking, because the quantity to be extremized must be agreed upon by all observers, it is natural that it should be an invariant of symmetry actions on spacetime. This leads to the result that the action is, in free motion, for massive bodies, proportional to an invariant built from translations — the time elapsed along a path as measured in a body's rest frame — times a dual invariant built from time and space translation generators. The former quantity is called **proper time** while the latter is the body's **mass**."
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}