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Dirac said that a photon can only interfere with itself. This is consistent with the tensor product of two photon spaces representation. On the other hand, it is known that there is interference between distinct sources, between two photons, between two photons again, and between two electrons. This is related to this SE question, and this one.

Let $\frac 1 {\sqrt 2}\left(|1\rangle|2\rangle + |2\rangle|1\rangle\right)$ be a two-photon state.

Let's consider interference between the two photons. If one "rotates" one of the wavefunctions with a phase factor, one expects that this will affect the interference, and the result will depend on the phase factor. So, the interference between $|1\rangle$ and $|2\rangle$ should be different than the interference between $|1\rangle$ and $e^{i\varphi}|2\rangle$, and that between $e^{i\varphi}|1\rangle$ and $|2\rangle$ (where $\varphi$ is an arbitrary overall phase). This can be checked experimentally by using a phase shifter.

Suppose that there is an algorithm to calculate the interference between the two photons, and that this algorithm receives as input the two-photon state. The state $\frac 1 {\sqrt 2}\left(|1\rangle|2\rangle + |2\rangle|1\rangle\right)$ is equivalent to $\frac 1 {\sqrt 2}\left(|1\rangle|2'\rangle + |2'\rangle|1\rangle\right)$, where $|2'\rangle=e^{i\varphi}|2\rangle$. If the algorithm which gives us the interference depends only on the two-photon state, then it apparently gives the same result, even when we phase-shift the second photon.

Apparently, the interference between photons depends on the way we decompose the two-photon state in two photons: in terms of $|1\rangle,|2\rangle$ we obtain different interference than in terms of $|1\rangle,e^{i\varphi}|2\rangle$, respectively $e^{i\varphi}|1\rangle,|2\rangle$. 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 two-photon state?

P.S. Let me emphasize that I am fully aware that QM has been tested successfully by so many experiments, and that most likely I am missing something. I just don't know what.

Dirac said that a photon can only interfere with itself. This is consistent with the tensor product of two photon spaces representation. On the other hand, it is known that there is interference between distinct sources, between two photons, between two photons again, and between two electrons. This is related to this SE question, and this one.

Let $\frac 1 {\sqrt 2}\left(|1\rangle|2\rangle + |2\rangle|1\rangle\right)$ be a two-photon state.

Let's consider interference between the two photons. If one "rotates" one of the wavefunctions with a phase factor, one expects that this will affect the interference, and the result will depend on the phase factor. So, the interference between $|1\rangle$ and $|2\rangle$ should be different than the interference between $|1\rangle$ and $e^{i\varphi}|2\rangle$, and that between $e^{i\varphi}|1\rangle$ and $|2\rangle$ (where $\varphi$ is an arbitrary overall phase). This can be checked experimentally by using a phase shifter.

Suppose that there is an algorithm to calculate the interference between the two photons, and that this algorithm receives as input the two-photon state. The state $\frac 1 {\sqrt 2}\left(|1\rangle|2\rangle + |2\rangle|1\rangle\right)$ is equivalent to $\frac 1 {\sqrt 2}\left(|1\rangle|2'\rangle + |2'\rangle|1\rangle\right)$, where $|2'\rangle=e^{i\varphi}|2\rangle$. If the algorithm which gives us the interference depends only on the two-photon state, then it apparently gives the same result, even when we phase-shift the second photon.

Apparently, the interference between photons depends on the way we decompose the two-photon state in two photons: in terms of $|1\rangle,|2\rangle$ we obtain different interference than in terms of $|1\rangle,e^{i\varphi}|2\rangle$, respectively $e^{i\varphi}|1\rangle,|2\rangle$. 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 two-photon state?

P.S. Let me emphasize that I am fully aware that QM has been tested successfully by so many experiments, and that most likely I am missing something. I just don't know what.

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.

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, between two photons, between two photons again, and between two electrons. This is related to this SE question, and this one.

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?

Dirac said that a photon can only interfere with itself. This is consistent with the tensor product of two photon spaces representation. On the other hand, it is known that there is interference between distinct sources, between two photons, between two photons again, and between two electrons. This is related to this SE question, and this one.

Let $\frac 1 {\sqrt 2}\left(|1\rangle|2\rangle + |2\rangle|1\rangle\right)$ be a two-photon state.

Let's consider interference between the two photons. If one "rotates" one of the wavefunctions with a phase factor, one expects that this will affect the interference, and the result will depend on the phase factor. So, the interference between $|1\rangle$ and $|2\rangle$ should be different than the interference between $|1\rangle$ and $e^{i\varphi}|2\rangle$, and that between $e^{i\varphi}|1\rangle$ and $|2\rangle$ (where $\varphi$ is an arbitrary overall phase). This can be checked experimentally by using a phase shifter.

Suppose that there is an algorithm to calculate the interference between the two photons, and that this algorithm receives as input the two-photon state. The state $\frac 1 {\sqrt 2}\left(|1\rangle|2\rangle + |2\rangle|1\rangle\right)$ is equivalent to $\frac 1 {\sqrt 2}\left(|1\rangle|2'\rangle + |2'\rangle|1\rangle\right)$, where $|2'\rangle=e^{i\varphi}|2\rangle$. If the algorithm which gives us the interference depends only on the two-photon state, then it apparently gives the same result, even when we phase-shift the second photon.

Apparently, the interference between photons depends on the way we decompose the two-photon state in two photons: in terms of $|1\rangle,|2\rangle$ we obtain different interference than in terms of $|1\rangle,e^{i\varphi}|2\rangle$, respectively $e^{i\varphi}|1\rangle,|2\rangle$. 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 two-photon state?

P.S. Let me emphasize that I am fully aware that QM has been tested successfully by so many experiments, and that most likely I am missing something. I just don't know what.

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.

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, between two photons, and between two electrons. This is related to this SE question, and this one.

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?

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.

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, between two photons, between two photons again, and between two electrons. This is related to this SE question, and this one.

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?