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As I understand, there are 2 types of photons in our (3+1) space with photon helicity $\pm 1$. How many photons exist in another spaces like (2 + 1) or (1 + 1)? Can we apply the same for gravitons?

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    $\begingroup$ It might be simpler if you ask just about photons. A lot of people know the answer to that question. One could ask how many polarizations gravitational waves have in other dimensions, and somewhat less people would know that, but probably still someone on this site could answer. But given that gravitons are not well-accepted to exist, adding that second question makes the answer a whole lot more complicated and significantly reduces the number of people who would be willing to answer. There might be solid answers in string theory or perturbative quantum gravity though. $\endgroup$
    – AXensen
    Commented Jan 14 at 20:23
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    $\begingroup$ ok, I'll be satisfied just by photons $\endgroup$ Commented Jan 14 at 20:28
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    $\begingroup$ I'm not sure if spin hence helicity is even meaningful in n<3 dimensional space. $\endgroup$
    – g s
    Commented Jan 14 at 21:41
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    $\begingroup$ I don't think photons exist at all in dimensions other than 3+1. In 2+1 even the distinction between bosons and fermions seems to be lost (and instead there are "anyons"). The meaning of symmetric and anti-symmetric products is completely different in n>3, yet again, so I don't even know how one would form something similar to classical electrodynamics in which formulas with differential operators like rotB could appear. The cross product doesn't even exist in any other than dimensions 3 and 7, I believe... $\endgroup$ Commented Jan 17 at 5:00

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In the space (2+1) we introduce the tensor \begin{equation} F^{\mu \nu}=\left(\begin{array}{ccc} 0 & -E^x / c & -E^y / c \\ E^x / c & 0 & -B \\ E^y / c & B & 0 \end{array}\right) \end{equation} As a consequence, the magnetic field is no longer a vector, but a scalar. Maxwell's equations are then written in the form \begin{equation} \begin{aligned} & \partial_\nu F^{\mu \nu}=\mu_0 j^\mu, \\ & \partial_\nu \tilde{F}^\nu=0, \end{aligned} \end{equation} where $j^\mu$ is surface current. By differentiating the first equation we can obtain an analogue of the continuity equation $$ \partial_\mu j^\mu=0, $$ Maxwell's complete equations would look like: \begin{equation} \begin{aligned} & \boldsymbol{\nabla} \cdot \boldsymbol{E}=\frac{\sigma}{\epsilon_0} \\ & \boldsymbol{\nabla}_{\perp} \cdot \boldsymbol{E}=\frac{\partial B}{\partial t}, \\ & \boldsymbol{\nabla}_{\perp} B=\mu_0 \boldsymbol{j}+\frac{1}{c^2} \frac{\partial \boldsymbol{E}}{\partial t}, \end{aligned} \end{equation} Where \begin{equation} \nabla_{\perp}=\left(\partial_y,-\partial_x\right) \end{equation} From this, in a similar way as for our space, wave equations are obtained \begin{equation} \nabla^2 \boldsymbol{E}-\frac{1}{c^2} \partial_{t t}^2 \boldsymbol{E}=\frac{1}{\epsilon_0} \nabla \sigma+\mu_o \partial_t \boldsymbol{j}, \end{equation} \begin{equation} \nabla^2 B-\frac{1}{c^2} \partial_{t t}^2 B=\mu_0 \nabla_{\perp} \cdot j \end{equation} If there are no charges, then the equations are independent of each other and you can get a whole set of photons with different phase shifts of the two waves. In space (1+1), the EM field tensor will look like: \begin{equation} F^{\mu \nu}=\left(\begin{array}{cc} 0 & -E \\ E & 0 \end{array}\right) \end{equation} and Maxwell's equations will be \begin{equation} \frac{\partial E}{\partial x}=0 \end{equation} The wave equation will be \begin{equation} -\frac{\partial^2 E}{\partial t^2}+\frac{\partial^2 E}{\partial x^2}=0 \end{equation} Given the constant field, photons will not exist in (1+1) space.

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