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The canonical commutation relations in gravity are sometimes written $$ [\gamma_{ij}(x),\pi^{kl}(y)]=\frac{i}{2}(\delta_i^k\delta_j^l+\delta_i^l\delta_j^k)\delta(x-y)\tag{0} $$ where $\gamma_{ij}$ is the 3-metric. This means \begin{align} [\gamma_{11}(x),\pi^{11}(y)]&=i\delta(x-y) \tag{1} \\ [\gamma_{12}(x),\pi^{12}(y)]&=\frac{i}{2}\delta(x-y) \tag{2} \end{align} for example.

Eq. (1) looks standard so I would conclude that $\gamma_{11}$ is canonically conjugate (c.c.) to $\pi^{11}$.

Eq. (2), however, has the factor of $\frac{1}{2}$ which doesn't look standard at all. It would seem to suggest that $\gamma_{12}$ and $\pi^{12}$ are not canonically conjugate. Any comments?

EDIT: I think the simple conclusion is that $\gamma_{ij}$ is c.c. to $2\pi^{ij}$ when $i\neq j$ and c.c. to $\pi^{ij}$ when $i=j$.

The canonical commutation relations in gravity are sometimes written $$ [\gamma_{ij}(x),\pi^{kl}(y)]=\frac{i\hbar}{2}(\delta_i^k\delta_j^l+\delta_i^l\delta_j^k)\delta^3(x-y),\tag{0} $$ where $\gamma_{ij}$ is the 3-metric. This means \begin{align} [\gamma_{11}(x),\pi^{11}(y)]&=i\hbar\delta^3(x-y) \tag{1} \\ [\gamma_{12}(x),\pi^{12}(y)]&=\frac{i\hbar}{2}\delta^3(x-y) \tag{2} \end{align} for example.

Eq. (1) looks standard so I would conclude that $\gamma_{11}$ is canonically conjugate (c.c.) to $\pi^{11}$.

Eq. (2), however, has the factor of $\frac{1}{2}$ which doesn't look standard at all. It would seem to suggest that $\gamma_{12}$ and $\pi^{12}$ are not canonically conjugate. Any comments?

EDIT: I think the simple conclusion is that $\gamma_{ij}$ is c.c. to $2\pi^{ij}$ when $i\neq j$ and c.c. to $\pi^{ij}$ when $i=j$.

The canonical commutation relations in gravity are sometimes written $$ [\gamma_{ij}(x),\pi^{kl}(y)]=\frac{i}{2}(\delta_i^k\delta_j^l+\delta_i^l\delta_j^k)\delta(x-y)\tag{0} $$ where $\gamma_{ij}$ is the 3-metric. This means \begin{align} [\gamma_{11}(x),\pi^{11}(y)]&=i\delta(x-y) \tag{1} \\ [\gamma_{12}(x),\pi^{12}(y)]&=\frac{i}{2}\delta(x-y) \tag{2} \end{align} for example.

Eq. (1) looks standard so I would conclude that $\gamma_{11}$ is canonically conjugate to $\pi^{11}$.

Eq. (2), however, has the factor of $\frac{1}{2}$ which doesn't look standard at all. It would seem to suggest that $\gamma_{12}$ and $\pi^{12}$ are not canonically conjugate. Any comments?

The canonical commutation relations in gravity are sometimes written $$ [\gamma_{ij}(x),\pi^{kl}(y)]=\frac{i}{2}(\delta_i^k\delta_j^l+\delta_i^l\delta_j^k)\delta(x-y)\tag{0} $$ where $\gamma_{ij}$ is the 3-metric. This means \begin{align} [\gamma_{11}(x),\pi^{11}(y)]&=i\delta(x-y) \tag{1} \\ [\gamma_{12}(x),\pi^{12}(y)]&=\frac{i}{2}\delta(x-y) \tag{2} \end{align} for example.

Eq. (1) looks standard so I would conclude that $\gamma_{11}$ is canonically conjugate (c.c.) to $\pi^{11}$.

Eq. (2), however, has the factor of $\frac{1}{2}$ which doesn't look standard at all. It would seem to suggest that $\gamma_{12}$ and $\pi^{12}$ are not canonically conjugate. Any comments?

EDIT: I think the simple conclusion is that $\gamma_{ij}$ is c.c. to $2\pi^{ij}$ when $i\neq j$ and c.c. to $\pi^{ij}$ when $i=j$.

The canonical commutation relations in gravity are sometimes written $$ [\gamma_{ij}(x),\pi^{kl}(y)]=\frac{i}{2}(\delta_i^k\delta_j^l+\delta_i^l\delta_j^k)\delta(x-y) $$ where $\gamma_{ij}$ is the 3-metric. This means \begin{align} [\gamma_{11}(x),\pi^{11}(y)]&=i\delta(x-y) \tag{1} \\ [\gamma_{12}(x),\pi^{12}(y)]&=\frac{i}{2}\delta(x-y) \tag{2} \end{align} for example.

Eq. (1) looks standard so I would conclude that $\gamma_{11}$ is canonically conjugate to $\pi^{11}$.

Eq. (2), however, has the factor of $\frac{1}{2}$ which doesn't look standard at all. It would seem to suggest that $\gamma_{12}$ and $\pi^{12}$ are not canonically conjugate. Any comments?

The canonical commutation relations in gravity are sometimes written $$ [\gamma_{ij}(x),\pi^{kl}(y)]=\frac{i}{2}(\delta_i^k\delta_j^l+\delta_i^l\delta_j^k)\delta(x-y)\tag{0} $$ where $\gamma_{ij}$ is the 3-metric. This means \begin{align} [\gamma_{11}(x),\pi^{11}(y)]&=i\delta(x-y) \tag{1} \\ [\gamma_{12}(x),\pi^{12}(y)]&=\frac{i}{2}\delta(x-y) \tag{2} \end{align} for example.

Eq. (1) looks standard so I would conclude that $\gamma_{11}$ is canonically conjugate to $\pi^{11}$.

Eq. (2), however, has the factor of $\frac{1}{2}$ which doesn't look standard at all. It would seem to suggest that $\gamma_{12}$ and $\pi^{12}$ are not canonically conjugate. Any comments?