Let's consider interference between two photons. If we "rotate" one of the wavefunctions with a phase factor, this will affect the interference, and the result will depend on the phase factor. So, the interference between $|1\rangle$ and $|2\rangle$ is different than the interference between $|1\rangle$ and $e^{i\vartheta}|2\rangle$. This can be checked experimentally by using a phase shifter.

Let $\frac 1 {\sqrt 2}\left(|1\rangle|2\rangle + |2\rangle|1\rangle\right)$ be two photons. The same state can be represented in the Fock space as $\frac 1 {\sqrt 2}\left(|1'\rangle|2'\rangle + |2'\rangle|1'\rangle\right)$, where $|1'\rangle=e^{i\varphi}|1\rangle$, and $|2'\rangle=e^{-i\varphi}|2\rangle$, and $\varphi$ is an arbitrary overall phase.

Let's write the interference:

$$|1'\rangle+|2'\rangle = e^{i\varphi}|1\rangle + e^{-i\varphi}|2\rangle = e^{i\varphi}\left(|1\rangle + e^{-2i\varphi}|2\rangle\right).$$

This can't be equal to $|1\rangle+|2\rangle$, not even up to a phase factor.

Apparently, the interference between photons depends on the way we decompose the state from the Fock space: in terms of $|1\rangle,|2\rangle$ we obtain different interference than in terms of $|1'\rangle,|2'\rangle$. So, either the Hilbert space does not contain all the needed information, or I am missing something (which is by far more likely). Could you please explain me where I am wrong, and what is the correct description of interference using the Fock space?

*P.S. Let me emphasize that I am fully aware that the Hilbert (and Fock in particular) space formalism has been tested successfully by so many experiments, and that most likely I am missing something. I just don't know what.*


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**Update.**

Apparently Dirac said that a photon can only interfere with itself. This is consistent with the tensor product of two photon spaces representation. But it is known that there is [interference between distinct sources][1], [between two photons][2], [between two photons again][3], and between [two electrons][4]. This is related to [this SE question][5], and [this one][6].


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**Update 2.**

Suppose there is an algorithm that receives as input a two-photon state, and calculates the interference pattern. Since

$$\frac 1 {\sqrt 2}\left(|1\rangle|2\rangle + |2\rangle|1\rangle\right)=\frac 1 {\sqrt 2}\left(|1'\rangle|2'\rangle + |2'\rangle|1'\rangle\right),$$

it seems that the interference should not depend on the particular choice of $|1\rangle$ and $|2\rangle$ to represent the two-photon state. So the interference should remain unaffected if we shift the phase of one of the two photons, as described in the original question. I cannot see how this can be avoided, even if we don't add the two wavefunctions, since the algorithm should return the same interference pattern, when the same two-photon state is inputed. Does the interference pattern depend on the representation of the two-photon state?


  [1]: http://ajp.aapt.org/resource/1/ajpias/v68/i3/p245_s1?isAuthorized=no
  [2]: http://www.haverford.edu/physics/love/teaching/Physics302PJL2009Recitation/06%20Uncertainty%20Principle%20-%20Pfleegor%20Mandel/PfleegorMandelp1084_1.pdf
  [3]: http://en.wikipedia.org/wiki/Hong%E2%80%93Ou%E2%80%93Mandel_effect
  [4]: http://www.weizmann.ac.il/condmat/heiblum/papers/nature05955.pdf
  [5]: http://physics.stackexchange.com/questions/630/is-it-possible-to-observe-interference-from-2-independent-optical-lasers
  [6]: http://physics.stackexchange.com/questions/6234/does-a-photon-interfere-only-with-itself