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A complex scalar field that describes a quantum mechanical system. The square of the modulus of the wave function gives the probability of the system to be found in a particular state. DO NOT USE THIS TAG for classical waves.
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Antisymmetry and coordinate invariance
How can we prove that $\phi(y_1,y_2) = -\phi(y_2,y_1)$?
You cannot. Consider
$$f_1(y_1,y_2) = \frac{1}{2}(y_1 + y_2),\qquad f_2(y_1,y_2) = \frac{1}{2}(y_1 - y_2).$$
Then, let us take $x_2=0$. In gen …