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because it tells you there's a second solution
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and you cannot throw them out."
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And it turns out the second solution,
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with negative energy, was when the theory is telling you,
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hey, there are particles and there are anti-particles,
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and the negative energy when properly interpreted will describe anti-particles.
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So the equations are very smart.
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The way the physics works is
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you will find some laws of motion in mathematical form,
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you put in the initial conditions of whatever,
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you solve the equations, and the answer that comes,
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you have no choice.