# Energy-momentum $T_{00}$ component for a scalar field in 2-dimensional spacetime

I'm reading the book "Advances in the Casimir Effect" by M. Bordag. In section 2.4 it says that the $$00$$-component of the energy-momentum tensor is given by (eq. 2.59): $$T_{00}(t,x)=\frac{\hbar c}{2}\left(\frac{1}{c^2} [\partial_t \varphi(t,x)]^2 + [\partial_x \varphi(t,x)]^2 + \frac{m^2c^2}{\hbar^2}\varphi^2(t,x)\right)$$

I've tried to get this starting from the general form of the tensor:

$$T^{\mu\nu} = \frac{\hbar^2}{m}(g^{\mu\alpha}g^{\nu\beta}+g^{\mu\beta}g^{\nu\alpha}-g^{\mu\nu}g^{\alpha\beta})\partial_\alpha\bar{\phi}\partial_\beta\phi-g^{\mu\nu}mc^2\bar{\phi}\phi$$

Where in our case $$\phi = \varphi(t,x) = \sum_n \left[\varphi_n^+ a_n + \varphi_n^- a_n^{\dagger}\right]$$ is the scalar field that satisfies the Klein-Gordon equation, $$a_n$$ and $$a_n^\dagger$$ are the anihilation and creation operators, and the metric tensor is given by:

$$g^{\mu \nu}=\begin{pmatrix} -\frac{1}{c^2} & 0 \\ 0 & 1 \end{pmatrix}$$

This way I get: $$T_{00}(t,x)=\frac{\hbar^2 }{m c^4}\left([\partial_t \varphi(t,x)]^2 + c^2[\partial_x \varphi(t,x)]^2\right) + m\varphi^2(t,x)$$

And that's not what the book says. Any idea of what I'm doing wrong?

In units $$c=\hbar=1$$, it's clear the expression you found that the one presented in the book differ by an overall factor of $$m$$ (and a 2). In principle there's nothing wrong with this as both tensors will still be conserved and this difference can be considered a choice in normalization of the fields and/or of the stress tensor's definition.
However I will point out that it's conventional for the term $$m^2\psi^2$$ to have units of "energy," and for $$T_{00}$$ to have units of energy (the 1/2 in the book's expression is also conventional). So, the expression from the book would be the more canonical form for the stress tensor.