# What does it mean to add the wave function of axion particles together?

I have been recently following works studying axion field. In many works studying axions, the authors assume the axion field is a bunch of plane wave, for example in this paper, equation 7 (though this paper is about vector particle, but it's similar for scalar particle like axion).

Typically, the authors will say the total axion field is something like the following $$a_{total}(t,\vec{x})=\sum_{i}a_i=\sum_{i}A_i\sin(\omega_i t-\vec{k}_i\cdot\vec{x}+\phi_i)$$ Where $$A_{i,0}$$ is the amplitude of the plane wave while $$\phi_i$$ is some random phase. $$\omega_i=\sqrt{k_i^2+m_a^2}$$

Typically, the authors claim $$a_i$$ to be the wave function of the $$i^{th}$$ axion particle (as claimed in the above cited paper), i.e. $$A_i\sin(\omega_i t-\vec{k}_i\cdot\vec{x}+\phi_i)$$. But it confuses me what does it mean to add together the wave functions of a bunch of independent particles. According to quantum mechanics, if these are the wave functions of different particles, shouldn't the total wave function be the direct product of them (not addition)? If on the other hand, $$A_i\sin(\omega_i t-\vec{k}_i\cdot\vec{x}+\phi_i)$$ is not the wave function of a single axion particle, what each plane wave physically represent?

Edit for the bounty: We know that there is coherent state $$| C\rangle$$ for the field, that gives a plane wave field expectation value (For construction of the state in field theory, see this paper), i.e. $$\langle C_i|a(x,t)|C_i\rangle=A_i\sin(\omega_i t-\vec{k}_i\cdot\vec{x}+\phi_i)$$. Then does this mean the field is in a mixed state described described by the density matrix $$\rho=\sum_{i}|C_i\rangle \langle C_i|$$?

• If you take an inner product of any of the sine fucntions $sin ( \omega_{i} t - k_{i} x +\phi_{i})$ you will notice that it is zero whenver $i\neq j$. The latter menas that $\{sin ( \omega_{i} t - k_{i} x +\phi_{i})\}_{i}$ forms a linnearly independent basis. The direct sum (Not the direct product) is the space $a_{total}(t, x)$ lives in. $span \{ sin ( \omega_{i} t - k_{i} x +\phi_{i})\}_{i} = \bigoplus _{i}sin ( \omega_{i} t - k_{i} x +\phi_{i})$ Jun 18 at 14:28
• @Hldngpk I am actually more interested in what each sin function physically represents Jun 18 at 14:39
• Short answer: those aren’t the wave functions of individual particles, they are the expectation values of Fourier modes of the quantum field. Jun 20 at 14:39
• @knzhou Is the field represented by a pure state or a mixed state? Jun 20 at 14:41
• Certainly mixed, just like almost everything else in nature, because it has interacted with other systems that you're not keeping track of. Jun 20 at 15:55

The axion field mentioned is a bunch of fluctuations and superposition of all the waves, imagine it as a Fourier series where each sin has its own contributions. Each plane is one of the contributions that adds to the final sum. For Then does this mean the field is in a mixed state described described by the density matrix . Answer: Yes, it a mixed state but it's not described by the matrix shown. For above reasons.